Author name code: goldreich ADS astronomy entries on 2022-09-14 author:Goldreich, Peter ------------------------------------------------------------------------ Title: DAVs: Red Edge and Outbursts Authors: Luan, Jing; Goldreich, Peter Bibcode: 2018ApJ...863...82L Altcode: 2017arXiv171106367L As established by photometric surveys, white dwarfs with hydrogen atmospheres and surface gravity, log(g) ≈ 8.0 pulsate as they cool across the temperature range of 12,500 K ≳ T eff ≳ 10,800 K. Known as DAVs or ZZ Ceti stars, their oscillations are attributed to gravity modes excited by convective driving. Overstability requires convective driving to exceed radiative damping. Previous works have demonstrated that ω ≳ max(τ c -1, L ℓ,b ) is a necessary and sufficient condition for overstability. Here τ c and L ℓ,b are the effective thermal timescale and Lamb frequency at the base of the surface convection zone. Below the observational red edge, L ℓ,b ≫ τ c -1, so overstable modes all have ωτ c ≫ 1. Consequently, their photometric amplitudes are reduced by that large factor rendering them difficult to detect. Although proposed previously, the condition ω ≳ L ℓ,b has not been clearly interpreted. We show that modes with ω < L ℓ,b suffer enhanced radiative damping that exceeds convective driving rendering them damped. A quasi-adiabatic analysis is adequate to account for this enhancement. Although this approximation is only marginally valid at the red edge, it becomes increasingly accurate toward both higher and lower {T}eff}. Recently, Kepler discovered a number of cool DAVs that exhibit sporadic flux outbursts. Typical outbursts last several hours, are separated by days, and release ∼1033-1034 erg. We attribute outbursts to limit cycles arising from sufficiently resonant 3-mode couplings between overstable parent modes and pairs of radiatively damped daughter modes. Limit cycles account for the durations and energies of outbursts and their prevalence near the red edge of the DAV instability strip. Title: Dense Regions in Supersonic Isothermal Turbulence Authors: Robertson, Brant; Goldreich, Peter Bibcode: 2018ApJ...854...88R Altcode: 2018arXiv180105440R The properties of supersonic isothermal turbulence influence a variety of astrophysical phenomena, including the structure and evolution of star-forming clouds. This work presents a simple model for the structure of dense regions in turbulence in which the density distribution behind isothermal shocks originates from rough hydrostatic balance between the pressure gradient behind the shock and its deceleration from ram pressure applied by the background fluid. Using simulations of supersonic isothermal turbulence and idealized waves moving through a background medium, we show that the structural properties of dense, shocked regions broadly agree with our analytical model. Our work provides a new conceptual picture for describing the dense regions, which complements theoretical efforts to understand the bulk statistical properties of turbulence and attempts to model the more complex features of star-forming clouds like magnetic fields, self-gravity, or radiative properties. Title: Enceladus: three-act play and current state Authors: Luan, J.; Goldreich, P. Bibcode: 2017AGUFM.P51F..04L Altcode: Eccentricity (e) growth as Enceladus migrates deeper into mean motion resonance with Dione results in increased tidal heating. As the bottom of the ice shell melts, the rate of tidal heating jumps and runaway melting ensues. At the end of run-away melting, the shell's thickness has fallen below the value at which the frequency of free libration equals the orbital mean motion and e has damped to well below its current value. Subsequently, both the shell thickness and e partake in a limit cycle. As e damps toward its minimum value, the shell's thickness asymptotically approaches its resonant value from below. After minimum e, the shell thickens quickly and e grows even faster. This cycle is likely to have been repeated multiple times in the past. Currently, e is much smaller than its equilibrium value corresponding to the shell thickness. Physical libration resonance resolves this mystery, it ensures that the low-e and medium-thickness state is present for most of the time between consecutive limit cycles. It is a robust scenario that avoids fine tuning or extreme parameter choice, and naturally produces episodic stages of high heating, consistent with softening of topographical features on Enceladus. Title: Classification of Satellite Resonances in the Solar System Authors: Luan, Jing; Goldreich, Peter Bibcode: 2017AJ....153...17L Altcode: Several pairs of solar system satellites occupy mean motion resonances (MMRs). We divide these into two groups according to their proximity to exact resonance. Proximity is measured by the existence of a separatrix in phase space. MMRs between Io-Europa, Europa-Ganymede, and Enceladus-Dione are too distant from exact resonance for a separatrix to appear. A separatrix is present only in the phase spaces of the Mimas-Tethys and Titan-Hyperion MMRs, and their resonant arguments are the only ones to exhibit substantial librations. Could there be a causal connection between the libration amplitude and the presence of a separatrix? Our suspicions were aroused by Goldreich & Schlichting, who demonstrate that sufficiently deep in a MMR, eccentricity damping could destabilize librations. However, our investigation reveals that libration amplitudes in both the Mimas-Tethys and Titan-Hyperion MMRs are fossils. Although the Mimas-Tethys MMR is overstable, its libration amplitude grows on the tidal damping timescale of Mimas’s inclination, which is considerably longer than a Hubble time. On the other hand, the Titan-Hyperion MMR is stable, but tidal damping of Hyperion’s eccentricity is too weak to have affected the amplitude of its libration. Title: Understanding Enceladus Authors: Luan, Jing; Goldreich, Peter Bibcode: 2016DPS....4822504L Altcode: We offer answers to the following questions?1, How did the global ocean form?2, Why is thermal activity concentrated at the south pole?3, What maintains the current small orbital eccentricity?4, How is the thickness of the ice shell changing?5, Why are the tiger stripes so hot?6, What sets the area of the south polar terrain? Title: Thermal Conductivity of Rubble Piles Authors: Luan, Jing; Goldreich, Peter Bibcode: 2015ApJ...814...36L Altcode: 2015arXiv151005295L Rubble piles are a common feature of solar system bodies. They are composed of monolithic elements of ice or rock bound by gravity. Voids occupy a significant fraction of the volume of a rubble pile. They can exist up to pressure P≈ {ɛ }Yμ , where {ɛ }Y is the monolithic material's yield strain and μ its rigidity. At low P, contacts between neighboring elements are confined to a small fraction of their surface areas. As a result, the effective thermal conductivity of a rubble pile, {k}{con}≈ k{(P/({ɛ }Yμ ))}1/2, can be orders of magnitude smaller than the thermal conductivity of its monolithic elements, k. In a fluid-free environment, only radiation can transfer energy across voids. It contributes an additional component, {k}{rad}=16{\ell }σ {T}3/3, to the total effective conductivity, {k}{eff}={k}{con}+{k}{rad}. Here ℓ, the inverse of the opacity per unit volume, is of the order of the size of the elements, and voids. An important distinction between {k}{con} and {k}{rad} is that the former is independent of the size of the elements, whereas the latter is proportional to it. Our expression for {k}{eff} provides a good fit to the depth dependence of thermal conductivity in the top 140 cm of the lunar regolith. It also offers a good starting point for detailed modeling of thermal inertias for asteroids and satellites. Measurement of the response of surface temperature to variable insolation is a valuable diagnostic of a regolith. There is an opportunity for careful experiments under controlled laboratory conditions to test models of thermal conductivity such as the one we outline. Title: Secular Evolution of the Pulsar Triple System J0337+1715 Authors: Luan, Jing; Goldreich, Peter Bibcode: 2014ApJ...790...82L Altcode: 2014arXiv1405.2374L The pulsar triple system J0337+1715 is remarkably regular and highly hierarchical. Secular interactions transfer angular momentum between inner and outer orbits unless their apsidal lines are parallel or anti-parallel. These choices correspond to orthogonal eigenmodes p and are characterized by e p, 1/e p, 2 ~ a 1/a 2 and e a, 1/e a, 2 ~ (a 1/a 2)-3/2(m 2/m 1). Mode p dominates the current state so e 1/e 2 remains close to e p, 1/e p, 2. A small contribution by Mode a causes e 1 and e 2 to oscillate with a period of ~103 yr which should be apparent in a few years. These will reveal the effects of general relativity, and possibly the distortion of the inner white dwarf (WD). Phinney proposes that the epicyclic energy of a WD-pulsar binary reaches equipartition with the kinetic energy of a single convective eddy when the WD's progenitor fills its Roche lobe. We extend Phinney's theory to apply to modes rather than individual orbits. Thus we predict that Mode p and Mode a achieved equipartition with eddies in the giant envelopes of the progenitors of the outer and inner WD, respectively. The most effective eddies are those with lifetimes closest to the orbit period. These were far more energetic in the progenitor of the outer WD. This explains why Mode p overwhelms Mode a, and also why the inner binary's orbit is far more eccentric than orbits of other WD-pulsar binaries with similar orbit periods. Mode a's small but finite amplitude places a lower bound of Q ~ 106 on the tidal quality parameter of the inner WD. Title: Physical Constraints on Fast Radio Bursts Authors: Luan, Jing; Goldreich, Peter Bibcode: 2014ApJ...785L..26L Altcode: 2014arXiv1401.1795L Fast radio bursts (FRBs) are isolated, ms radio pulses with dispersion measure (DM) of order 103 pc cm-3. Galactic candidates for the DM of high latitude bursts detected at GHz frequencies are easily dismissed. DM from bursts emitted in stellar coronas are limited by free-free absorption and those from H II regions are bounded by the nondetection of associated free-free emission at radio wavelengths. Thus, if astronomical, FRBs are probably extragalactic. FRB 110220 has a scattering tail of ~5.6 ± 0.1 ms. If the electron density fluctuations arise from a turbulent cascade, the scattering is unlikely to be due to propagation through the diffuse intergalactic plasma. A more plausible explanation is that this burst sits in the central region of its host galaxy. Pulse durations of order ms constrain the sizes of FRB sources implying high brightness temperatures that indicates coherent emission. Electric fields near FRBs at cosmological distances would be so strong that they could accelerate free electrons from rest to relativistic energies in a single wave period. Title: Overstable Librations can Account for the Paucity of Mean Motion Resonances among Exoplanet Pairs Authors: Goldreich, Peter; Schlichting, Hilke E. Bibcode: 2014AJ....147...32G Altcode: 2013arXiv1308.4688G We assess the multi-planet systems discovered by the Kepler satellite in terms of current ideas about orbital migration and eccentricity damping due to planet-disk interactions. Our primary focus is on first order mean motion resonances, which we investigate analytically to lowest order in eccentricity. Only a few percent of planet pairs are in close proximity to a resonance. However, predicted migration rates (parameterized by \tau _n=n/{|\dot{n}|}) imply that during convergent migration most planets would have been captured into first order resonances. Eccentricity damping (parameterized by \tau _e=e/{|\dot{e}|}) offers a plausible resolution. Estimates suggest τ e /τ n ~ (h/a)2 ~ 10-2, where h/a is the ratio of disk thickness to radius. Together, eccentricity damping and orbital migration give rise to an equilibrium eccentricity, e eq ~ (τ e /τ n )1/2. Capture is permanent provided e eq <~ μ1/3, where μ denotes the planet to star mass ratio. But for e eq >~ μ1/3, capture is only temporary because librations around equilibrium are overstable and lead to passage through resonance on timescale τ e . Most Kepler planet pairs have e eq > μ1/3. Since τ n Gt τ e is the timescale for migration between neighboring resonances, only a modest percentage of pairs end up trapped in resonances after the disk disappears. Thus the paucity of resonances among Kepler pairs should not be taken as evidence for in situ planet formation or the disruptive effects of disk turbulence. Planet pairs close to a mean motion resonance typically exhibit period ratios 1%-2% larger than those for exact resonance. The direction of this shift undoubtedly reflects the same asymmetry that requires convergent migration for resonance capture. Permanent resonance capture at these separations from exact resonance would demand μ(τ n /τ e )1/2 >~ 0.01, a value that estimates of μ from transit data and (τ e /τ n )1/2 from theory are insufficient to match. Plausible alternatives involve eccentricity damping during or after disk dispersal. The overstability referred to above has applications beyond those considered in this investigation. It was discovered numerically by Meyer & Wisdom in their study of the tidal evolution of Saturn's satellites. Title: A Simple Model for Orbital Resonances with Dissipation and its Application to Multiple-Planet Systems Authors: Schlichting, Hilke; Goldreich, P. Bibcode: 2013DPS....4511303S Altcode: Data from multi-planet systems discovered by Kepler show that most planets currently do not reside in or close to mean motion resonances. There is however a significant excess of planet pairs with period ratios close to but 1-2% larger than those of exact resonance. We present a simple model for orbital resonance including dissipation and apply it to exoplanet systems. Our results demonstrate that there are three possible outcomes for a pair of planets encountering a resonance. These three different outcomes are due to eccentricity damping that can drive an instability of the amplitude of libration about exact resonance. Damping promotes instability because the effective potential is maximal at exact resonance. The instability manifests itself in systems for which eccentricity damping occurs more rapidly than semi-major axis migration. Our results indicate that only about 10% of planet pairs should currently reside in or near mean motion resonances. Thus the small fraction of near commensurate orbits in multi-planet systems discovered by Kepler is fully consistent with disk migration of low mass planets. Furthermore, for planet pairs massive enough to be permanently captured into resonance, our model can account for the deficit of planet pairs at exact resonance and for the corresponding excess of pairs with slightly greater separation. Title: Adiabatic Heating of Contracting Turbulent Fluids Authors: Robertson, Brant; Goldreich, Peter Bibcode: 2012ApJ...750L..31R Altcode: 2012arXiv1203.4815R Turbulence influences the behavior of many astrophysical systems, frequently by providing non-thermal pressure support through random bulk motions. Although turbulence is commonly studied in systems with constant volume and mean density, turbulent astrophysical gases often expand or contract under the influence of pressure or gravity. Here, we examine the behavior of turbulence in contracting volumes using idealized models of compressed gases. Employing numerical simulations and an analytical model, we identify a simple mechanism by which the turbulent motions of contracting gases "adiabatically heat," experiencing an increase in their random bulk velocities until the largest eddies in the gas circulate over a Hubble time of the contraction. Adiabatic heating provides a mechanism for sustaining turbulence in gases where no large-scale driving exists. We describe this mechanism in detail and discuss some potential applications to turbulence in astrophysical settings. Title: Why do we see the man in the Moon? Authors: Aharonson, Oded; Goldreich, Peter; Sari, Re'em Bibcode: 2012Icar..219..241A Altcode: Numerical simulations and analysis show that the Moon locks into resonance with a statistical preference of facing either the current near-side or far-side toward Earth. The near-side is largely covered by dense, topographically low, dark mare basalts, the pattern of which to some, resembles the image of a man's face. Although the Moon is locked in this configuration at present, the opposite one, with the current far-side facing Earth, is of lower potential energy and hence might be naively expected. Instead, we find that the probability of selecting each configuration depends upon the ratio of the asymmetry of the potential energy maxima, dominated by the octupole moment of the Moon, to the energy dissipated per tidal cycle within the Moon. If this ratio is small, the two configurations are equally likely. Otherwise, interesting dynamical behavior ensues. In the Moon's present orbit, with the best-estimated geophysical parameters and dissipation parameter Q = 35, trapping into the current higher-energy configuration is preferred. With Q = 100 in analogy with the solid Earth, the current configuration is nearly certain. The ratio of energies and corresponding probabilities were different in the past. Relative crater counts on the leading and trailing faces indicate an impact may have unlocked the Moon before it settled into the present configuration. Our analysis constrains the geophysical parameters at the time of the last such event. Title: Why do We See the Man in the Moon? Authors: Aharonson, O.; Goldreich, P.; Sari, R. Bibcode: 2012LPI....43.2532A Altcode: Numerical simulations and analysis show that the Moon locks into resonance with a statistical preference of facing either the current nearside or farside toward Earth. Title: Why do We See the Man in the Moon? Authors: Aharonson, O.; Goldreich, P.; Sari, R. Bibcode: 2010AGUFM.P51C1464A Altcode: The Moon's current spin-orbit resonance results in it continuously pointing the same side towards the Earth. This is the side largely covered by dense, dark mare basalts, the pattern of which, to some, resembles the shape of a man's face. For dynamical stability, the Moon must spin about its axis of maximum moment of inertia, and point its axis of minimum inertia in direction of the Earth, leaving two options of facing the current near-side or far-side towards us. While both configurations are local potential energy minima, we show that under certain conditions, breaking and locking into resonance from a state of rapid rotation preferentially selects one configuration over the other. We analytically and numerically simulate the locking of the Moon into resonance and consider the dependence on the relevant geophysical parameters at the time of last resonance locking. The probability of each configuration depends upon the gravitational potential energy asymmetry dominated by the octupole moment of the Moon, and by the energy dissipated per tidal cycle dominated by the internal properties of the Moon. If these energies are widely separated, the two configurations are equally likely. If these energies are comparable, interesting dynamical behavior results. For example, we find that in the current orbit with the current best-estimated parameters, the Moon is more likely to fall into the higher of the two energy configurations facing the near-side towards the Earth with a preference of ~3:1. Title: Elastic ice shells of synchronous moons: Implications for cracks on Europa and non-synchronous rotation of Titan Authors: Goldreich, Peter M.; Mitchell, Jonathan L. Bibcode: 2010Icar..209..631G Altcode: 2009arXiv0910.0032G A number of synchronous moons are thought to harbor water oceans beneath their outer ice shells. A subsurface ocean frictionally decouples the shell from the interior. This has led to proposals that a weak tidal or atmospheric torque might cause the shell to rotate differentially with respect to the synchronously rotating interior. Applications along these lines have been made to Europa and Titan. However, the shell is coupled to the ocean by an elastic torque. As a result of centrifugal and tidal forces, the ocean would assume an ellipsoidal shape with its long axis aligned toward the parent planet. Any displacement of the shell away from its equilibrium position would induce strains thereby increasing its elastic energy and giving rise to an elastic restoring torque. In the investigation reported on here, the elastic torque is compared with the tidal torque acting on Europa and the atmospheric torque acting on Titan. Regarding Europa, it is shown that the tidal torque is far too weak to produce stresses that could fracture the ice shell, thus refuting an idea that has been widely advocated. Instead, it is suggested that the cracks arise from time-dependent stresses due to non-hydrostatic gravity anomalies from tidally driven, episodic convection in the satellite's interior. Two years of Cassini RADAR observations of Titan's surface have been interpreted as implying an angular displacement of ∼0.24° relative to synchronous rotation. Compatibility of the amplitude and phase of the observed non-synchronous rotation with estimates of the atmospheric torque requires that Titan's shell be decoupled from its interior. We find that the elastic torque balances the seasonal atmospheric torque at an angular displacement ≲0.05°, effectively coupling the shell to the interior. Moreover, if Titan's surface were spinning faster than synchronous, the tidal torque tending to restore synchronous rotation would almost certainly be larger than the atmospheric torque. There must either be a problem with the interpretation of the radar observations, or with our basic understanding of Titan's atmosphere and/or interior. Title: Elastic Ice Shells of Synchronous Moons Authors: Mitchell, Jonathan L.; Goldreich, P. M. Bibcode: 2009DPS....41.6605M Altcode: A growing number of water-rich, synchronous moons are thought to contain subsurface water oceans below an icy shell, most notably Europa and Titan. Under tidal and rotational stresses, these shells deform and store elastic energy unless the shell yields or viscously deforms on a timescale that is fast compared to the forcing period. Otherwise, the deformed shell experiences a back-torque tending to keep it aligned with the satellite-planet line. The shell will rotate an angle given by the balance between the elastic torque and the spinup torque. For Europa, it has been widely advocated that stresses arising from non-synchronous rotation of the shell are responsible for the large cracks on its surface. The angle giving balance between the tidal and elastic torques is exceedingly small -- of order 10^-6 radians -- and associated stresses are only of the order 1 Pa, well below that needed to crack the shell; this poses a challenge to the prevailing ideas of surface cracking on Europa. Time-dependent gravity anomalies from mantle convection in Europa is a more likely mechanism for producing large surface cracks. On Titan, it has been reported that the surface spin is faster than synchronous, which has been interpreted as resulting from a seasonally varying atmospheric torque on the surface shell. The maximum atmospheric torque could rotate an elastic shell relative to the underlying hydrostatic shape of order 10^-3 radians, implying the shell is effectively coupled to the interior. Therefore, the atmospheric torque forces a libration of the entire figure that is 180 degrees out-of-phase with the forcing and with a smaller amplitude than has been reported. Title: Tidal Evolution of Rubble Piles Authors: Goldreich, Peter; Sari, Re'em Bibcode: 2009ApJ...691...54G Altcode: 2007arXiv0712.0446G Many small bodies in the solar system are believed to be rubble piles, a collection of smaller elements separated by voids. We propose a model for the structure of a self-gravitating rubble pile. Static friction prevents its elements from sliding relative to each other. Stresses are concentrated around points of contact between individual elements. The effective dimensionless rigidity, \tilde{μ}_rubble, is related to that of a monolithic body of similar composition and size, \tilde{μ} by \tilde{μ}_rubble ∼ \tilde{μ}^{1/2} ɛ_Y^{-1/2}, where epsilon Y ~ 10-2 is the yield strain. This represents a reduction in effective rigidity below the maximum radius, R max ~ [μepsilon Y /(Gρ2)]1/2 ~ 103 km, at which a rubble pile can exist. Our model for the rigidity of rubble piles is compatible with laboratory experiments on the speed of shear waves in sand. Densities derived for binary asteroids imply that they are rubble piles. Thus their tidal evolution proceeds faster than it would if they were monoliths. Binary orbit evolution is also driven by torques resulting from the asymmetrical scattering and reradiation of sunlight (YORP and BYORP effects). The tidal torque probably overcomes the radiative (YORP) torque and synchronizes the spins of secondaries in near-Earth binary asteroids and it definitely does so for secondaries of main-belt binary asteroids. Synchronization is a requirement for the radiative (BYORP) torque to act on the binary orbit. This torque clearly dominates the tidal torque for all near-Earth binary asteroids and for some binaries in the main belt. For other main-belt binaries, the tidal torque appears to be at least comparable in strength to the BYORP torque. An exciting possibility is that in these systems the angular momentum added to the orbit by the tidal torque might be removed by the radiative torque. Title: MHD turbulence Authors: Goldreich, Peter Bibcode: 2009SchpJ...4.2350G Altcode: No abstract at ADS Title: Constraints on deep-seated zonal winds inside Jupiter and Saturn Authors: Liu, Junjun; Goldreich, Peter M.; Stevenson, David J. Bibcode: 2008Icar..196..653L Altcode: 2007arXiv0711.3922L The atmospheres of Jupiter and Saturn exhibit strong and stable zonal winds. How deep the winds penetrate unabated into each planet is unknown. Our investigation favors shallow winds. It consists of two parts. The first part makes use of an Ohmic constraint; Ohmic dissipation associated with the planet's magnetic field cannot exceed the planet's net luminosity. Application to Jupiter (J) and Saturn (S) shows that the observed zonal winds cannot penetrate below a depth at which the electrical conductivity is about six orders of magnitude smaller than its value at the molecular-metallic transition. Measured values of the electrical conductivity of molecular hydrogen yield radii of maximum penetration of 0.96R and 0.86R, with uncertainties of a few percent of R. At these radii, the magnetic Reynolds number based on the zonal wind velocity and the scale height of the magnetic diffusivity is of order unity. These limits are insensitive to difficulties in modeling turbulent convection. They permit complete penetration along cylinders of the equatorial jets observed in the atmospheres of Jupiter and Saturn. The second part investigates how deep the observed zonal winds actually do penetrate. As it applies heuristic models of turbulent convection, its conclusions must be regarded as tentative. Truncation of the winds in the planet's convective envelope would involve breaking the Taylor-Proudman constraint on cylindrical flow. This would require a suitable nonpotential acceleration which none of the obvious candidates appears able to provide. Accelerations arising from entropy gradients, magnetic stresses, and Reynolds stresses appear to be much too weak. These considerations suggest that strong zonal winds are confined to shallow, stably stratified layers, with equatorial jets being the possible exception. Title: How much oxygen is too much? Constraining Saturn's ring atmosphere Authors: Farmer, Alison J.; Goldreich, Peter Bibcode: 2007Icar..188..108F Altcode: The discovery of a molecular oxygen atmosphere around Saturn's rings has important implications for the electrodynamics of the ring system. Its existence was inferred from the Cassini in situ detection of molecular oxygen ions above the rings during Saturn Orbit Insertion in 2004. Molecular oxygen is difficult to observe remotely, and theoretical estimates have yielded only a lower limit ( N≳10 cm) to the O 2 column density. Comparison with observations has previously concerned matching ion densities at spacecraft altitudes far larger than the scale height of the neutral atmosphere. This is further complicated by charged particle propagation effects in Saturn's offset magnetic field. In this study we adopt a complementary approach, by focusing on bulk atmospheric properties and using additional aspects of the Cassini observations to place an upper limit on the column density. We develop a simple analytic model of the molecular atmosphere and its photo-ionization and dissociation products, with N a free parameter. Heating of the neutrals by viscous stirring, cooling by collisions with the rings, and torquing by collisions with pickup ions are all included in the model. We limit the neutral scale height to h≲3000 km using the INMS neutral density nondetection over the A ring. A first upper limit to the neutral column is derived by using our model to reassess O 2 production and loss rates. Two further limits are then obtained from Cassini observations: corotation of the observed ions with the planet implies that the height-integrated conductivity of the ring atmosphere is less than that of Saturn's ionosphere; and the nondetection of fluorescent atomic oxygen over the rings constrains the molecular column from which it is produced via photo-dissociation. These latter limits are independent of production and loss rates and are only weakly dependent on temperature. From the three independent methods described, we obtain similar limits: N≲2×10 cm. The mean free path for collisions between neutrals thus cannot be very much smaller than the scale height. Title: Spontaneous axisymmetry breaking of the external magnetic field at Saturn Authors: Goldreich, Peter; Farmer, Alison J. Bibcode: 2007JGRA..112.5225G Altcode: 2007JGRA..11205225G; 2006astro.ph..8296G Saturn's magnetic field is remarkably axisymmetric. Early evidence for nonaxisymmetry came from the periodicity of Saturn's kilometric radio bursts (SKR). Subsequently, percent-level variations of the SKR period were found to occur on timescales of years. A recent breakthrough has been the direct detection of a nonaxisymmetric component of the field that rotates with a period close to that of the SKR and whose magnitude varies only weakly with distance from Saturn. The latter implies that it must be supported by currents external to the planet. We explore the hypothesis that centrifugally driven convection spontaneously breaks the axisymmetry of the external magnetic field at Saturn. The density of the outflowing plasma close to its source is assumed to contain a substantial part that varies as cos? and rotates uniformly. We demonstrate that the plasma stream must narrow with distance from the planet, while the field-aligned currents joining stream to ionosphere increase rapidly. These currents produce a nonaxisymmetric component of magnetic field whose magnitude varies inversely with radial distance in the planet's equatorial plane. For a rate of plasma outflow 104 ≲ ? ≲ 105g s-1, this component's strength is compatible with that observed. Additionally, we postulate that the SKR is associated with the narrow range of longitudes over which large currents flow along magnetic field lines connecting the tip of the outflow to the auroral ionosphere. Title: Imbalanced Strong MHD Turbulence Authors: Lithwick, Y.; Goldreich, P.; Sridhar, S. Bibcode: 2007ApJ...655..269L Altcode: 2006astro.ph..7243L We present a phenomenological model of imbalanced MHD turbulence in an incompressible magnetofluid. The steady state cascades, of waves traveling in opposite directions along the mean magnetic field, carry unequal energy fluxes to small length scales, where they decay as a result of viscous and resistive dissipation. The inertial range scalings are well understood when both cascades are weak. We study the case in which both cascades are, in a sense, strong. The inertial range of this imbalanced cascade has the following properties: (1) The ratio of the rms Elsässer amplitudes is independent of scale and is equal to the ratio of the corresponding energy fluxes. (2) In common with the balanced strong cascade, the energy spectra of both Elsässer waves are of the anisotropic Kolmogorov form, with their parallel correlation lengths equal to each other on all scales, and proportional to the two-thirds power of the transverse correlation length. (3) The equality of cascade time and wave period (critical balance) that characterizes the strong balanced cascade does not apply to the Elsässer field with the larger amplitude. Instead, the more general criterion that always applies to both Elsässer fields is that the cascade time is equal to the correlation time of the straining imposed by oppositely directed waves. (4) In the limit of equal energy fluxes, the turbulence corresponds to the balanced strong cascade. Our results are particularly relevant for turbulence in the solar wind. Spacecraft measurements have established that in the inertial range of solar wind turbulence, waves traveling away from the Sun have higher amplitudes than those traveling toward it. Result 1 allows us to infer the turbulent flux ratios from the amplitude ratios, thus providing insight into the origin of the turbulence. Title: Spontaneous Axisymmetry Breaking of Saturn's External Magnetic Field Authors: Farmer, A. J.; Goldreich, P. Bibcode: 2006AGUFMSM22B..08F Altcode: Saturn's magnetic field is remarkably axisymmetric: its dipole axis is inclined by less than 0.2° with respect to its rotation axis. Recent Cassini observations have revealed small non-axisymmetric field components which rotate slightly more slowly than any atmospheric feature. We show that rotationally driven convection of magnetospheric plasma is responsible for the axisymmetry-breaking. Field-aligned currents transfer angular momentum from the planet to a tongue of outflowing plasma. This transfer slows the rate of rotation of the ionosphere relative to that of the underlying atmosphere. The currents are the source for the non- axisymmetric components of the field. The common rotation rates of these components and Saturn's kilometric radio (SKR) bursts is that of the plasma near the orbit of Enceladus, and by extension the rotation rate in the ionosphere to which this plasma is coupled. That rate tells us nothing about the rotation rate of Saturn's deep interior. Of that we remain ignorant. Magnetic perturbations with magnitudes similar to those observed by Cassini are produced for \dot M ~ 104 g/s, a value similar to estimates for the rate of production of plasma from Saturn's E-ring. Enhancement of the SKR occurs in a narrow range of longitudes where the tip of the outgoing plasma stream connects to the auroral ionosphere, via field lines that are bowed outwards by currents that supply the plasma's centripetal acceleration. Title: Ohmic Dissipation Constraint on Deep-seated Zonal Winds in Jupiter and Saturn Authors: Liu, Junjun; Goldreich, P. M.; Stevenson, D. J. Bibcode: 2006DPS....38.0205L Altcode: 2006BAAS...38..483L The atmospheres of Jupiter and Saturn exhibit strong and stable zonal winds. Busse (1976) suggested that they might be the surface expression of deep flows on cylinders. This hypothesis experiences difficulty when account is taken of the electrical conductivity of molecular hydrogen as measured in shockwave experiments. The deep zonal flow of an electrically conducting fluid would produce a toroidal magnetic field, an associated poloidal electrical current, and Ohmic dissipation. In steady state, the total Ohmic dissipation cannot exceed the planet's net luminosity. If we assume that the observed zonal flow penetrates along cylinders until it is truncated to (near) zero at some spherical radius, the upper bound on Ohmic dissipation constrains this radius to be no smaller than 0.96 Jupiter radius and 0.86 Saturn radius. The truncation of the cylindrical flow in the convective envelope requires an appropriate force to break the Taylor-Proudman constraint. We have been unable to identify any plausible candidate. The Lorentz force is much too weak. Order of magnitude considerations suggest that both divergence of the Reynolds stress and the buoyancy force are inadequate. Therefore, we claim that the assumed deep-seated flows do not exist. Equatorial jets could maintain constant velocities on cylinders through the planet provided their half-widths were no greater than 21° for Jupiter and 31° for Saturn. These boundaries are similar to those of the equatorial jets observed in the planets’ atmospheres. We speculate that the Reynolds stress associated with turbulent convection promotes differential rotation throughout the interiors of the giant planets. Along cylinders that pass through the maximum penetration depth, the Maxwell stress balances the Reynolds stress resulting in small differential rotation except in the stably stratified atmosphere. Equatorial jets are unencumbered by the Maxwell stress. They pass through the planets and maintain velocities limited by parasitic instabilities. Title: Magnetospheric Eclipses in the Double Pulsar System J0737-3039 Authors: Rafikov, R. R.; Goldreich, P. Bibcode: 2006IAUJD...2E..32R Altcode: Recently discovered double pulsar system J0737-3039 consisting of millisecond and normal pulsars in a 2.4 hour orbit provides us with unprecedented tests of general relativity and magnetospheric effects. One of the most interesting phenomena observed in this system is the eclipse of the millisecond pulsar in the radio at its conjunction with the normal pulsar. I will describe a theory which explains this observation as a result of synchrotron absorption of the millisecond pulsar radiobeam in the magnetosphere of the normal pulsar. Absorption is "induced" in a sense that the intense radiobeam of millisecond pulsar itself strongly modifies the properties of the plasma in the closed part of the normal pulsar magnetosphere: absorption of high-brightness temperature radio emission heats up particles already present there and also allows additional pair plasma to be trapped in this region by magnetic bottling effect. This theory self-consistently predicts the size of the eclipsing region which agrees very well with the observed duration of eclipse. Recent observations of the variability of transmission during the eclipse modulated at the rotation period of the normal pulsar have been interpreted as resulting from the absorption by the rigidly rotating dipolar-shaped magnetosphere which is in perfect agreement with our theory. Title: Spherical Accretion Authors: Sari, Re'em; Goldreich, Peter Bibcode: 2006ApJ...642L..65S Altcode: 2006astro.ph..3293S We compare different examples of spherical accretion onto a gravitating mass. Limiting cases include the accretion of a collisionally dominated fluid and the accretion of collisionless particles. We derive expressions for the accretion rate and density profile for semicollisional accretion, which bridges the gap between these limiting cases. Particle crossing of the Hill sphere during the formation of the outer planets is likely to have taken place in the semicollisional regime. Title: Folded Fields as the Source of Extreme Radio-Wave Scattering in the Galactic Center Authors: Goldreich, Peter; Sridhar, S. Bibcode: 2006ApJ...640L.159G Altcode: 2006astro.ph..2532G A strong case has been made that radio waves from sources within about half a degree of the Galactic center undergo extreme diffractive scattering. However, problems arise when standard (``Kolmogorov'') models of electron density fluctuations are employed to interpret the observations of scattering in conjunction with those of free-free radio emission. Specifically, the outer scale of a Kolmogorov spectrum of electron density fluctuations is constrained to be so small that it is difficult to identify an appropriate astronomical setting. Moreover, an unacceptably high turbulent heating rate results if the outer scale of the velocity field coincides with that of the density fluctuations. We propose an alternative model based on folded magnetic field structures that have been reported in numerical simulations of small-scale dynamos. Nearly isothermal density variations across thin current sheets suffice to account for the scattering. There is no problem of excess turbulent heating, because the outer scale for the velocity fluctuations is much larger than the widths of the current sheets. We speculate that interstellar magnetic fields could possess geometries that reflect their origins: fields maintained by the Galactic dynamo could have large correlation lengths, whereas those stirred by local energetic events might exhibit folded structures. Title: Understanding the behavior of Prometheus and Pandora Authors: Farmer, Alison J.; Goldreich, Peter Bibcode: 2006Icar..180..403F Altcode: 2005astro.ph.11055F We revisit the dynamics of Prometheus and Pandora, two small moons flanking Saturn's F ring. Departures of their orbits from freely precessing ellipses result from mutual interactions via their 121:118 mean motion resonance. Motions are chaotic because the resonance is split into four overlapping components. Orbital longitudes were observed to drift away from predictions based on Voyager ephemerides. A sudden jump in mean motions took place close to the time at which the orbits' apses were antialigned in 2000. Numerical integrations reproduce both the longitude drifts and the jumps. The latter have been attributed to the greater strength of interactions near apse antialignment (every 6.2 yr), and it has been assumed that this drift-jump behavior will continue indefinitely. We re-examine the dynamics of the Prometheus-Pandora system by analogy with that of a nearly adiabatic, parametric pendulum. In terms of this analogy, the current value of the action of the satellite system is close to its maximum in the chaotic zone. Consequently, at present, the two separatrix crossings per precessional cycle occur close to apse antialignment. In this state libration only occurs when the potential's amplitude is nearly maximal, and the "jumps" in mean motion arise during the short intervals of libration that separate long stretches of circulation. Because chaotic systems explore the entire region of phase space available to them, we expect that at other times the Prometheus-Pandora system would be found in states of medium or low action. In a low action state it would spend most of the time in libration, and separatrix crossings would occur near apse alignment. We predict that transitions between these different states can happen in as little as a decade. Therefore, it is incorrect to assume that sudden changes in the orbits only happen near apse antialignment. Title: Spoke formation under moving plasma clouds Authors: Farmer, Alison J.; Goldreich, Peter Bibcode: 2005Icar..179..535F Altcode: 2005astro.ph..9306F Goertz and Morfill [Goertz, C.K., Morfill, G., 1988. Icarus 53, 219-229] propose that spokes on Saturn's rings form under radially moving plasma clouds produced by meteoroid impacts. We demonstrate that the speed at which a plasma cloud can move relative to the ring material is bounded from above by the difference between the Keplerian and corotation velocities. The radial orientation of new spokes requires radial speeds that are at least an order of magnitude larger than this upper limit, thus the model advanced by Goertz and Morfill fails to make radial spokes. Title: Magnetospheric Eclipses in the Double-Pulsar System PSR J0737-3039 Authors: Rafikov, Roman R.; Goldreich, Peter Bibcode: 2005ApJ...631..488R Altcode: 2004astro.ph.12355R We argue that eclipses of radio emission from the millisecond pulsar A in the double-pulsar system PSR J0737-3039 are due to synchrotron absorption by plasma in the closed field line region of the magnetosphere of its normal pulsar companion B. On the basis of a plausible geometric model, pulsar A's radio beam only illuminates pulsar B's magnetosphere for about 10 minutes surrounding the time of eclipse. During this time it heats particles at r>~109 cm to relativistic energies and enables extra plasma, beyond that needed to maintain the corotation electric field, to be trapped by magnetic mirroring. An enhancement of the plasma density by a factor of ~102 is required to match the duration and optical depth of the observed eclipses. The extra plasma might be supplied by a source near B through Bγ pair creation by energetic photons produced in B's outer gap. Relativistic pairs cool by synchrotron radiation close to where they are born. Reexcitation of their gyrational motions by cyclotron absorption of A's radio beam can result in their becoming trapped between conjugate mirror points in B's magnetosphere. Because the trapping efficiency decreases with increasing optical depth, the plasma density enhancement saturates even under steady state illumination. The result is an eclipse with finite, frequency-dependent optical depth. After illumination by A's radio beam ceases, the trapped particles cool and are lost. The entire cycle repeats every orbital period. We speculate that the asymmetries between eclipse ingress and egress result in part from the magnetosphere's evolution toward a steady state when illuminated by A's radio beam. We predict that A's linear polarization varies with both eclipse phase and B's rotational phase. Title: Spokes in Saturn's Rings Authors: Farmer, A. J.; Goldreich, P. Bibcode: 2004AAS...205.4307F Altcode: 2004BAAS...36.1412F The ``spokes'' in Saturn's rings have eluded explanation since their discovery by the Voyager spacecraft. These ghostly radial dust lanes spring into existence in under five minutes, then fade away in hours as the Keplerian rotation of the rings distorts them. I shall present a new model for spoke formation which looks set to reproduce the observations. In particular we can account for the occurrence of spokes around the radius of corotation of the planet with its rings, a point which has evaded previous studies. New data from the Cassini mission should provide even more stringent tests of our theory. Title: Chaos in the F Ring Authors: Rappaport, N. J.; Goldreich, P. Bibcode: 2004DPS....36.0704R Altcode: 2004BAAS...36.1078R We have been applying the methods developed in our studies of the chaotic motion of Prometheus and Pandora to study the chaotic motion of particles in the F ring. This work is still in progress at the time this abstract is written. We hope to present results that will be consistent with some of the characteristic features of the F ring. Title: Final Stages of Planet Formation Authors: Goldreich, Peter; Lithwick, Yoram; Sari, Re'em Bibcode: 2004ApJ...614..497G Altcode: 2004astro.ph..4240G We address three questions regarding solar system planets: What determined their number? Why are their orbits nearly circular and coplanar? How long did they take to form?

Runaway accretion in a disk of small bodies resulted in a tiny fraction of the bodies growing much larger than all the others. These big bodies dominated the viscous stirring of all bodies. Dynamical friction by small bodies cooled the random velocities of the big ones. Random velocities of small bodies were cooled by mutual collisions and/or gas drag. Runaway accretion terminated when the orbital separations of the big bodies became as wide as their feeding zones. This was followed by oligarchic growth during which the big bodies maintained similar masses and uniformly spaced semimajor axes. As the oligarchs grew, their number density decreased, but their surface mass density increased. We depart from standard treatments of planet formation by assuming that as the big bodies got bigger, the small ones got smaller as the result of undergoing a collisional fragmentation cascade. It follows that oligarchy was a brief stage in solar system evolution.

When the oligarchs' surface mass density matched that of the small bodies, dynamical friction was no longer able to balance viscous stirring, so their velocity dispersion increased to the extent that their orbits crossed. This marked the end of oligarchy. What happened next differed in the inner and outer parts of the planetary system. In the inner part, where the ratios of the escape velocities from the surfaces of the planets to the escape velocities from their orbits are smaller than unity, big bodies collided and coalesced after their random velocities became comparable to their escape velocities. In the outer part, where these ratios are larger than unity, the random velocities of some of the big bodies continued to rise until they were ejected. In both parts, the number density of the big bodies eventually decreased to the extent that gravitational interactions among them no longer produced large-scale chaos. After that their orbital eccentricities and inclinations were damped by dynamical friction from the remaining small bodies.

The last and longest stage in planet formation was the cleanup of small bodies. Our understanding of this stage is fraught with uncertainty. The surviving protoplanets cleared wide gaps around their orbits that inhibited their ability to accrete small bodies. Nevertheless, in the inner planet system, all of the material in the small bodies ended up inside planets. Small bodies in the outer planet system probably could not have been accreted in the age of the solar system. A second generation of planetesimals may have formed in the disk of small bodies, by either collisional coagulation or gravitational instability. In the outer planet system, bodies of kilometer size or larger would have had their random velocities excited until their orbits crossed those of neighboring protoplanets. Ultimately they would have either escaped from the Sun or become residents of the Oort Cloud. An important distinction is that growth of the inner planets continued through cleanup, whereas assembly of the outer planets was essentially complete by the end of oligarchy. These conclusions imply that the surface density of the protoplanetary disk was that of the minimum solar mass nebula in the inner planet region but a few times larger in the outer planet region. The timescale through cleanup was set by the accretion rate at the geometrical cross section in the inner planet region and by the ejection rate at the gravitationally enhanced cross section in the outer planet region. It was a few hundred million years in the former and a few billion years in the latter. However, since Uranus and Neptune acquired most of their mass by the end of oligarchy, they may have formed before Earth!

A few implications of the above scenario are worth noting. Impacts among protoplanets of comparable size were common in the inner planet system but not in the outer. Ejections from the outer planet system included several bodies with masses in excess of Earth after oligarchy and an adequate number of kilometer-size bodies to populate the Oort comet cloud during cleanup. Except at the very end of cleanup, collisions prevented Uranus and Neptune from ejecting kilometer-size objects. Only Jupiter and, to a much lesser extent, Saturn were capable of populating the Oort Cloud with comets of kilometer size. Title: Planet Formation by Coagulation: A Focus on Uranus and Neptune Authors: Goldreich, Peter; Lithwick, Yoram; Sari, Re'em Bibcode: 2004ARA&A..42..549G Altcode: 2004astro.ph..5215G Planets form in the circumstellar disks of young stars. We review the basic physical processes by which solid bodies accrete each other and alter each others' random velocities, and we provide order-of-magnitude derivations for the rates of these processes. We discuss and exercise the two-groups approximation, a simple yet powerful technique for solving the evolution equations for protoplanet growth. We describe orderly, runaway, neutral, and oligarchic growth. We also delineate the conditions under which each occurs. We refute a popular misconception by showing that the outer planets formed quickly by accreting small bodies. Then we address the final stages of planet formation. Oligarchy ends when the surface density of the oligarchs becomes comparable to that of the small bodies. Dynamical friction is no longer able to balance viscous stirring and the oligarchs' random velocities increase. In the inner-planet system, oligarchs collide and coalesce. In the outer-planet system, some of the oligarchs are ejected. In both the inner- and outer-planet systems, this stage ends once the number of big bodies has been reduced to the point that their mutual interactions no longer produce large-scale chaos. Subsequently, dynamical friction by the residual small bodies circularizes and flattens their orbits. The final stage of planet formation involves the clean up of the residual small bodies. Clean up has been poorly explored. Title: Planet-Disk Symbiosis Authors: Sari, Re'em; Goldreich, Peter Bibcode: 2004ApJ...606L..77S Altcode: 2003astro.ph..7107S Planets form in disks around young stars. Interactions with these disks cause them to migrate and thus affect their final orbital periods. We suggest that the connection between planets and disks may be deeper and involve a symbiotic evolution. By contributing to the outward transport of angular momentum, planets promote disk accretion. Here we demonstrate that planets sufficiently massive to open gaps could be the primary agents driving disk accretion. Those having masses below the gap opening threshold drift inward more rapidly than the disk material and can only play a minor role in its accretion. An even more intimate symbiosis involving gap opening planets may result if they acquire most of their mass prior to gap formation. Given a small initial eccentricity, just a fraction of a percent, the orbital eccentricity of a massive planet may grow rapidly once a mass in excess of the planet's mass has been repelled to form a gap around the planet's orbit. Then, as the planet's radial excursions approach the gap's width, subsequent eccentricity growth slows so that the planet's orbit continues to be confined within the gap. Title: Wave Damping by Magnetohydrodynamic Turbulence and Its Effect on Cosmic-Ray Propagation in the Interstellar Medium Authors: Farmer, Alison J.; Goldreich, Peter Bibcode: 2004ApJ...604..671F Altcode: 2003astro.ph.11400F Cosmic rays scatter off magnetic irregularities (Alfvén waves) with which they are resonant, that is, waves of wavelength comparable to their gyroradii. These waves may be generated either by the cosmic rays themselves, if they stream faster than the Alfvén speed, or by sources of MHD turbulence. Waves excited by streaming cosmic rays are ideally shaped for scattering, whereas the scattering efficiency of MHD turbulence is severely diminished by its anisotropy. We show that MHD turbulence has an indirect effect on cosmic-ray propagation by acting as a damping mechanism for cosmic-ray-generated waves. The hot (``coronal'') phase of the interstellar medium is the best candidate location for cosmic-ray confinement by scattering from self-generated waves. We relate the streaming velocity of cosmic rays to the rate of turbulent dissipation in this medium for the case in which turbulent damping is the dominant damping mechanism. We conclude that cosmic rays with up to 102 GeV could not stream much faster than the Alfvén speed but 106 GeV cosmic rays would stream unimpeded by self-generated waves, unless the coronal gas were remarkably turbulence-free. Title: Sackler Lecture: Final Stages of Planet Formation Authors: Goldreich, Peter Bibcode: 2004cfa..collE..15G Altcode: My lecture will address three major questions regarding solar system planets. What determined their number? Why are their orbits nearly circular and coplanar? How long did they take to form? Answers to these will be given in terms of: stability against large scale chaos; dynamical friction by small bodies; and the accretion rate at the geometrical cross section in the inner planet region and the ejection rate at the gravitationally enhanced cross section in the outer. Title: Gauge freedom in the N-body problem of celestial mechanics Authors: Efroimsky, M.; Goldreich, P. Bibcode: 2004A&A...415.1187E Altcode: 2003astro.ph..7130E The goal of this paper is to demonstrate how the internal symmetry of the N-body celestial-mechanics problem can be exploited in orbit calculation.

We start with summarising research reported in (Efroimsky \cite{Efr02}, \cite{Efr03}; Newman & Efroimsky \cite{New03}; Efroimsky & Goldreich \cite{Efro03}) and develop its application to planetary equations in non-inertial frames. This class of problems is treated by the variation-of-constants method. As explained in the previous publications, whenever a standard system of six planetary equations (in the Lagrange, Delaunay, or other form) is employed for N objects, the trajectory resides on a 9(N-1)-dimensional submanifold of the 12(N-1)-dimensional space spanned by the orbital elements and their time derivatives. The freedom in choosing this submanifold reveals an internal symmetry inherent in the description of the trajectory by orbital elements. This freedom is analogous to the gauge invariance of electrodynamics. In traditional derivations of the planetary equations this freedom is removed by hand through the introduction of the Lagrange constraint, either explicitly (in the variation-of-constants method) or implicitly (in the Hamilton-Jacobi approach). This constraint imposes the condition (called ``osculation condition'') that both the instantaneous position and velocity be fit by a Keplerian ellipse (or hyperbola), i.e., that the instantaneous Keplerian ellipse (or hyperbola) be tangential to the trajectory. Imposition of any supplementary constraint different from that of Lagrange (but compatible with the equations of motion) would alter the mathematical form of the planetary equations without affecting the physical trajectory.

However, for coordinate-dependent perturbations, any gauge different from that of Lagrange makes the Delaunay system non-canonical. Still, it turns out that in a more general case of disturbances dependent also upon velocities, there exists a ``generalised Lagrange gauge'', i.e., a constraint under which the Delaunay system is canonical (and the orbital elements are osculating in the phase space). This gauge reduces to the regular Lagrange gauge for perturbations that are velocity-independent.

Finally, we provide a practical example illustrating how the gauge formalism considerably simplifies the calculation of satellite motion about an oblate precessing planet. Title: Origin of chaos in the Prometheus-Pandora system Authors: Goldreich, Peter; Rappaport, Nicole Bibcode: 2003Icar..166..320G Altcode: 2003astro.ph..7259G We demonstrate that the chaotic orbits of Prometheus and Pandora are due to interactions associated with the 121:118 mean motion resonance. Differential precession splits this resonance into a quartet of components equally spaced in frequency. Libration widths of the individual components exceed the splitting, resulting in resonance overlap which causes the chaos. Mean motions of Prometheus and Pandora wander chaotically in zones of width 1.8 and 3.1 deg yr -1, respectively. A model with 1.5 degrees of freedom captures the essential features of the chaotic dynamics. We use it to show that the Lyapunov exponent of 0.3 yr -1 arises because the critical argument of the dominant member of the resonant quartet makes approximately two separatrix crossings every 6.2 year precessional cycle. Title: Gauge symmetry of the N-body problem in the Hamilton-Jacobi approach Authors: Efroimsky, Michael; Goldreich, Peter Bibcode: 2003JMP....44.5958E Altcode: 2003JMPS...44.5958E; 2003JMP...44..5958E; 2003astro.ph..5344E In most books the Delaunay and Lagrange equations for the orbital elements are derived by the Hamilton-Jacobi method: one begins with the two-body Hamilton equations in spherical coordinates, performs a canonical transformation to the orbital elements, and obtains the Delaunay system. A standard trick is then used to generalize the approach to the N-body case. We reexamine this step and demonstrate that it contains an implicit condition which restricts the dynamics to a 9(N-1)-dimensional submanifold of the 12(N-1)-dimensional space spanned by the elements and their time derivatives. The tacit condition is equivalent to the constraint that Lagrange imposed ``by hand'' to remove the excessive freedom, when he was deriving his system of equations by variation of parameters. It is the condition of the orbital elements being osculating, i.e., of the instantaneous ellipse (or hyperbola) being always tangential to the physical velocity. Imposure of any supplementary condition different from the Lagrange constraint (but compatible with the equations of motion) is legitimate and will not alter the physical trajectory or velocity (though will alter the mathematical form of the planetary equations). This freedom of nomination of the supplementary constraint reveals a gauge-type internal symmetry instilled into the equations of celestial mechanics. Existence of this internal symmetry has consequences for the stability of numerical integrators. Another important aspect of this freedom is that any gauge different from that of Lagrange makes the Delaunay system noncanonical. In a more general setting, when the disturbance depends not only upon positions but also upon velocities, there is a ``generalized Lagrange gauge'' wherein the Delaunay system is symplectic. This special gauge renders orbital elements that are osculating in the phase space. It coincides with the regular Lagrange gauge when the perturbation is velocity independent. Title: Gauge-invariant disturbing function in precessing frames of reference. Authors: Efroimsky, M.; Goldreich, P. Bibcode: 2003DDA....34.0903E Altcode: 2003BAAS...35.1042E In most books the Lagrange and Delaunay systems of equations for the orbital elements are derived in the Hamilton-Jacobi approach: one begins with two-body Hamilton equations in spherical or Cartesian coordinates; then carries out a canonical transformation to the orbital elements and, thus, arrives to the Delaunay or Lagrange system. A standard trick then enables one to generalize the approach to the N-body case. We carefully re-examine this step and demonstrate that it contains an implicit condition which restricts the orbit to a certain 9(N-1)-dimensional submanifold of the 12(N-1)-dimensional space spanned by the orbital elements and their time derivatives. This tacit assumption is equivalent to the so-called Lagrange constraint, one that Lagrange imposed ``by hand'' in order to remove the excessive freedom, when he was deriving his system of equations by the method of variation of parameters.

The physical meaning of this implicit condition, tacitly present also in the Hamilton-Jacobi treatment of the N-body problem, is transparent: it is the condition of the orbital elements being osculating (i.e., of the velocity being expressed through the orbital elements in the same manner as in the two-body case). The imposure of any supplementary condition, which is different from the Lagrange constraint (but is compatible with the equations of motion), is legitimate. However, it will alter the form of the Lagrange and Delaunay equations (Efroimsky 2002, Newman & Efroimsky 2003) and will have consequences for numerical integrators (Efroimsky 2002, Murison & Efroimsky 2003).

Another important alteration of the Lagrange and Delaunay systems will be in order when the disturbing function depends not only upon the coordinates but also upon the velocities, i.e., when the orbital elements are defined in a non-inertial coordinate system (Goldreich 1965).

In the current presentation we consider interplay between these two issues: the freedom of gauge fixing and the freedom of reference-system choice. We apply our results to description of a satellite motion about a precessing planet. Title: The origin of chaos in the dynamics of the Prometheus-Pandora System Authors: Rappaport, N. J.; Goldreich, P. Bibcode: 2003DPS....35.4603R Altcode: 2003BAAS...35.1010R In 2002, Rappaport and Goldreich demonstrated that the motions of Prometheus and Pandora are chaotic. The chaos arises from the gravitational interactions between the satellites. Sudden changes in mean motions occur every 6.2 years when the periapsis of Pandora's orbit is aligned and on the same side of Saturn with the apoapsis of Prometheus' orbit, and the gravitational interactions between the two bodies are greatly enhanced.

We followed up this work by a search for the origin of the chaos. We found that the chaos is due to overlapping of the four resonances associated with the same 121:118 ratio of mean motions and the four possible combinations of periapsis longitudes of Prometheus and Pandora. Mean motion of Prometheus and Pandora wander chaotically in zones of width 1.8 and 3.1 degrees per year, respectively.

Furthermore, we demonstrated that the whole system can be represented by a single degree of freedom model that captures the full chaotic dynamics.

References: (1) Rappaport, N.J., and P. Goldreich 2002: The Chaotic Dynamics of Prometheus and Pandora, BAAS 34, 883. (2) Goldreich, P., and N. Rappaport 2003: Chaotic motions of Prometheus and Pandora, Icarus 162, 391-399.

This research was supported by NASA Planetary Geology and Geophysics grant 344-30-53-02 and by NSF grant AST-0098301. Title: Formation of Kuiper Belt Binaries Authors: Sari, R.; Goldreich, P.; Lithwick, Y. Bibcode: 2003DPS....35.4909S Altcode: 2003BAAS...35.1017S At least several percent of large Kuiper belt objects are members of wide binaries. We argue that binaries formed during runaway accretion. Collisionless gravitational interactions provide two channels for binary formation. The initial step is always the formation of a transient binary when two large bodies penetrate each other's Hill spheres. Then, either dynamical friction due to small bodies or the scattering of a third large body can carry away the energy needed for permanent binding. We predict that most objects of size comparable to those currently observed in the Kuiper belt are members of multiple systems: outside of a critical separation of 3'', the binary probability is about 0.3%; inside that separation, the probability increases inversely with separation. This prediction is compatible with results from current surveys. Title: Chaotic motions of prometheus and pandora Authors: Goldreich, Peter; Rappaport, Nicole Bibcode: 2003Icar..162..391G Altcode: Recent HST images of the saturnian satellites Prometheus and Pandora show that their longitudes deviate from predictions of ephemerides based on Voyager images. Currently Prometheus is lagging and Pandora leading these predictions by somewhat more than 20°. We show that these discrepancies are fully accounted for by gravitational interactions between the two satellites. These peak every 24.8 days at conjunctions and excite chaotic perturbations. The Lyapunov exponent for the Prometheus-Pandora system is of order 0.3 year -1 for satellite masses based on a nominal density of 0.63 g cm -3. Interactions are strongest when the orbits come closest together. This happens at intervals of 6.2 years when their apses are antialigned. In this context, we note the sudden changes of opposite signs in the mean motions of Prometheus and Pandora at the end of 2000 occurred around the time their apsidal lines were antialigned. Title: Eccentricity Evolution for Planets in Gaseous Disks Authors: Goldreich, Peter; Sari, Re'em Bibcode: 2003ApJ...585.1024G Altcode: 2002astro.ph..2462G At least several percent of solar-type stars possess giant planets. Surprisingly, most move on orbits of substantial eccentricity. We investigate the hypothesis that interactions between a giant planet and the disk from which it forms promote eccentricity growth. These interactions are concentrated at discrete Lindblad and corotation resonances. Interactions at principal Lindblad resonances cause the planet's orbit to migrate and open a gap in the disk if the planet is sufficiently massive. Those at first-order Lindblad and corotation resonances change the planet's orbital eccentricity. Eccentricity is excited by interactions at external Lindblad resonances that are located on the opposite side of corotation from the planet, and damped by co-orbital Lindblad resonances that overlap the planet's orbit. If the planet clears a gap in the disk, the rate of eccentricity damping by co-orbital Lindblad resonances is reduced. Density gradients associated with the gap activate eccentricity damping by corotation resonances at a rate that initially marginally exceeds that of eccentricity excitation by external Lindblad resonances. But the corotation torque may be reduced as the result of the trapping of fluid in libration around potential maxima. This nonlinear saturation can tip the balance in favor of eccentricity excitation. A minimal initial eccentricity of the order of 1% is required to overcome viscous diffusion, which acts to unsaturate corotation resonances by reestablishing the large-scale density gradient. Thus, eccentricity growth is a finite-amplitude instability. Formally, the apsidal resonance, which is a special kind of co-orbital Lindblad resonance that exists in pressure-dominated disks, appears to damp eccentricity faster than external Lindblad resonances can excite it. However, the wavelength of the apsidal wave in a pressure-dominated disk is so long that it does not propagate. A self-gravity-dominated disk does not have an apsidal resonance. Nevertheless, apsidal waves are excited at gap edges. Although these propagate, their long wavelengths suggest that they are likely to be reflected at disk edges to form standing waves. Viscous damping of standing waves results in eccentricity damping, but at level far below that which traveling waves would produce. Although the level of eccentricity damping due to apsidal waves is reduced to a modest level in both pressure- and self-gravity-dominated disks, whether it drops well below that of Lindblad resonances depends sensitively on the disk's thickness and planet's mass. However, our analysis shows that with reasonable parameters, planet-disk interactions can promote eccentricity growth. Title: Imbalanced Weak Magnetohydrodynamic Turbulence Authors: Lithwick, Yoram; Goldreich, Peter Bibcode: 2003ApJ...582.1220L Altcode: 2002astro.ph..8046L Weak MHD turbulence consists of waves that propagate along magnetic field lines, in both directions. When two oppositely directed waves collide, they distort each other, without changing their respective energies. Each wave suffers many collisions before cascading; by contrast, in strong MHD turbulence, waves cascade on the same timescale at which they collide. ``Imbalance'' means that more energy is going in one direction than the other. In general, MHD turbulence is imbalanced. Yet imbalanced MHD cascades are not understood. For example, turbulence in the solar wind is observed to be imbalanced, so solar wind turbulence will not be understood until a theory of the imbalanced cascade is developed. We solve weak MHD turbulence that is imbalanced. Of crucial importance is that the energies going in both directions are forced to equalize at the dissipation scale. This ``pinning'' of the energy spectra was discovered by Grappin and coworkers. It affects the entire inertial range. Weak MHD turbulence is particularly interesting because perturbation theory is applicable. Hence, it can be described with a simple kinetic equation. Galtier and coworkers derived this kinetic equation. We present a simpler, more physical derivation, based on the picture of colliding wavepackets. In the process, we clarify the role of the zero-frequency mode. We also explain why Goldreich & Sridhar claimed that perturbation theory is inapplicable, and why this claim is wrong. (Our ``weak'' is equivalent to Goldreich & Sridhar's ``intermediate.'') We perform numerical simulations of the kinetic equation to verify our claims. We construct simplified model equations that illustrate the main effects. Finally, we show that a large magnetic Prandtl number does not have a significant effect, and that hyperviscosity leads to a pronounced bottleneck effect. Title: Formation of Kuiper-belt binaries by dynamical friction and three-body encounters Authors: Goldreich, Peter; Lithwick, Yoram; Sari, Re'em Bibcode: 2002Natur.420..643G Altcode: 2002astro.ph..8490G The Kuiper belt is a disk of icy bodies that orbit the Sun beyond Neptune; the largest known members are Pluto and its companion Charon. A few per cent of Kuiper-belt bodies have recently been found to be binaries with wide separations and mass ratios of the order of unity. Collisions were too infrequent to account for the observed number of binaries, implying that these binaries formed through collisionless interactions mediated by gravity. These interactions are likely to have been most effective during the period of runaway accretion, early in the Solar System's history. Here we show that a transient binary forms when two large bodies penetrate one another's Hill sphere (the region where their mutual forces are larger than the tidal force of the Sun). The loss of energy needed to stabilize the binary orbit can then occur either through dynamical friction from surrounding small bodies, or through the gravitational scattering of a third large body. Our estimates slightly favour the former mechanism. We predict that five per cent of Kuiper-belt objects are binaries with apparent separations greater than 0.2arcsec, and that most are in tighter binaries or systems of higher multiplicity. Title: The Chaotic Dynamics of Prometheus and Pandora Authors: Rappaport, N. J.; Goldreich, P. Bibcode: 2002DPS....34.2401R Altcode: 2002BAAS...34..883R The longitudes of the F-ring shepherd satellites Prometheus and Pandora measured in recent HST images are different from predictions based on Voyager images. The discrepancies are of order 20 degrees. In a paper submitted to Icarus (Goldreich, P., and N. Rappaport 2002: Chaotic Motions of F-Ring Shepherds), we showed that these discrepancies are fully accounted for by the gravitational interactions between these satellites. These peak every 24.8 days at conjunctions and excite chaotic perturbations. Interactions are strongest when the orbits come close together. This happens at intervals of 6.2 years when the apses are anti-aligned. We found that the sudden changes of opposite changes in the mean motions of Prometheus and Pandora at the end of 2000 occurred shortly after their apsidal line was anti-aligned. The Lyapunov exponent for the Prometheus-Pandora system is of order 0.35 year-1 for satellite masses based on a nominal density of 1.3 g/cm3. In this presentation we will characterize the chaos as a function of the parameters and discuss new results pertaining to the understanding of the mechanism giving rise to it. Rappaport acknowldeges funding by NASA Planetary Geology and Geophysics Program, Grant 344-30-53-02. Goldreich acknowledges NSF Grant AST-0098301 and NASA Grant NAG5-12037. Title: Chaotic Motions of F-Ring Shepherds Authors: Goldreich, Peter; Rappaport, Nicole Bibcode: 2002astro.ph..5330G Altcode: Recent HST images of the Saturnian satellites Prometheus and Pandora show that their longitudes deviate from predictions of ephemerides based on Voyager images. Currently Prometheus is lagging and Pandora leading these predictions by somewhat more than 20 degrees. We show that these discrepancies are fully accounted for by gravitational interactions between the two satellites. These peak every 24.8 days at conjunctions and excite chaotic perturbations. The Lyapunov exponent for the Prometheus-Pandora system is of order 0.35 inverse years for satellite masses based on a nominal density of 1.3 gm/cm^3. Interactions are strongest when the orbits come closest together. This happens at intervals of 6.2 years when their apses are anti-aligned. In this context we note the sudden changes of opposite signs in the mean motions of Prometheus and Pandora at the end of 2000 occured shortly after their apsidal lines were anti-aligned. Title: Fundamental Models Of MHD Turbulence Authors: Goldreich, Peter Bibcode: 2002APS..APR.J2001G Altcode: The inertial range of incompressible MHD turbulence is most conveniently described in terms of counter propagating waves. Shear Alfvén waves control the cascade dynamics. Slow waves play a passive role and adopt the spectrum set by the shear Alfvén waves. Cascades composed entirely of shear Alfvén waves do not generate a significant measure of slow waves. MHD turbulence is anisotropic with energy cascading more rapidly along k_⊥ than along k_allel. Anisotropy increases with k_⊥ such that the excited modes are confined inside a cone bounded by k_allel∝ k_⊥^2/3. The opening angle of the cone, θ(k_⊥)∝ k_⊥-1/3, defines the scale dependent anisotropy. MHD turbulence is generically strong in the sense that the waves which comprise it are critically damped. Nevertheless, deep inside the inertial range, turbulent fluctuations are small. Their energy density is less than that of the background field by a factor θ^2(k_⊥)<< 1. MHD cascades are best understood geometrically. Wave packets suffer distortions as they move along magnetic field lines perturbed by counter propagating wave packets. Field lines perturbed by unidirectional waves map planes perpendicular to the local field into each other. Shear Alfvén waves are responsible for the mapping's shear and slow waves for its dilatation. The former exceeds the latter by θ-1(k_⊥)>> 1 which accounts for dominance of the shear Alfvén waves in controlling the cascade dynamics. Title: Tidal Evolution of the Planetary System around HD 83443 Authors: Wu, Yanqin; Goldreich, Peter Bibcode: 2002ApJ...564.1024W Altcode: 2001astro.ph..8499W Two planets with an orbital period ratio of approximately 10:1 have been discovered around the star HD 83443. The inner and more massive planet, HD 83443b, has the smallest semimajor axis among all currently known exoplanets. Unlike other short-period exoplanets, it maintains a substantial orbital eccentricity, e1=0.079+/-0.008, in spite of efficient tidal damping. This is a consequence of its secular interactions with HD 83443c, whose orbital eccentricity e2=0.42+/-0.06. Dissipation, associated with tides the star raises in the inner planet, removes energy but not angular momentum from its orbit, while secular interactions transfer angular momentum but not energy from the inner to the outer planet's orbit. The outward transfer of angular momentum decreases the tidal decay rate of the inner planet's orbital eccentricity while increasing that of the outer planet. The alignment of the apsides of the planets' orbits is another consequence of tidal and secular interactions. In this state the ratio of their orbital eccentricities, e1/e2, depends on the secular perturbations the planets exert on each other and on additional perturbations that enhance the inner planet's precession rate. Tidal and rotational distortions of the inner planet along with general relativity provide the most important of these extra precessional perturbations, each of which acts to reduce e1/e2. Provided the planets' orbits are coplanar, the observed eccentricity ratio uniquely relates sini and C≡(k2/k2J)(R1/RJ)5, where the tidal Love number, k2, and radius, R1, of the inner planet are scaled by their Jovian equivalents. Title: Compressible Magnetohydrodynamic Turbulence in Interstellar Plasmas Authors: Lithwick, Yoram; Goldreich, Peter Bibcode: 2001ApJ...562..279L Altcode: 2001astro.ph..6425L Radio wave scintillation observations reveal a nearly Kolmogorov spectrum of density fluctuations in the ionized interstellar medium. Although this density spectrum is suggestive of turbulence, no theory relevant to its interpretation exists. We calculate the density spectrum in turbulent magnetized plasmas by extending the theory of incompressible magnetohydrodynamic (MHD) turbulence given by Goldreich & Sridhar to include the effects of compressibility and particle transport. Our most important results are as follows:1. Density fluctuations are due to the slow mode and the entropy mode. Both modes are passively mixed by the cascade of shear Alfvén waves. Since the shear Alfvén waves have a Kolmogorov spectrum, so do the density fluctuations.2. Observed density fluctuation amplitudes constrain the nature of MHD turbulence in the interstellar medium. Slow mode density fluctuations are suppressed when the magnetic pressure is less than the gas pressure. Entropy mode density fluctuations are suppressed by cooling when the cascade timescale is longer than the cooling timescale. These constraints imply either that the magnetic and gas pressures are comparable or that the outer scale of the turbulence is very small.3. A high degree of ionization is required for the cascade to survive damping by neutrals and thereby to extend to small length scales. Regions that are insufficiently ionized produce density fluctuations only on length scales larger than the neutral damping scale. These regions may account for the excess of power that is found on large scales.4. Provided that the thermal pressure exceeds the magnetic pressure, both the entropy mode and the slow mode are damped on length scales below that at which protons can diffuse across an eddy during the eddy's turnover time. Consequently, eddies whose extents along the magnetic field are smaller than the proton collisional mean free path do not contribute to the density spectrum. However, in MHD turbulence eddies are highly elongated along the magnetic field. From an observational perspective, the relevant length scale is that transverse to the magnetic field. Thus, the cutoff length scale for density fluctuations is significantly smaller than the proton mean free path.5. The Alfvén mode is critically damped at the transverse length scale of the proton gyroradius and thus cascades to smaller length scales than either the slow mode or the entropy mode. Title: Incompressible MHD Turbulence Authors: Goldreich, Peter Bibcode: 2001Ap&SS.278...17G Altcode: The inertial range of incompressible MHD turbulence is most conveniently described in terms of counter propagating waves. Shear Alfvén waves control the cascade dynamics. Slow waves play a passive role and adopt the spectrum set by the shear Alfvén waves. Cascades composed entirely of shear Alfvén waves do not generate a significant measure of slow waves. MHD turbulence is anisotropic with energy cascading more rapidly along k ⊥ than along k ∥. Anisotropy increases with k ⊥ such that the excited modes are confined inside a cone bounded by k ∥∝ k perp 2/3. The opening angle of the cone, θ( k ⊥)∝ k ⊥ -1/3, defines the scale dependent anisotropy. MHD turbulence is generically strong in the sense that the waves which comprise it are critically damped. Nevertheless, deep inside the inertial range, turbulent fluctuations are small. Their energy density is less than that of the background field by a factor θ2( k ⊥)≪. MHD cascades are best understood geometrically. Wave packets suffer distortions as they move along magnetic field lines perturbed by counter propagating wave packets. Field lines perturbed by unidirectional waves map planes perpendicular to the local field into each other. Shear Alfvén waves are responsible for the mapping's shear and slow waves for its dilatation. The former exceeds the latter by θ-1( k ⊥)≫ 1 which accounts for dominance of the shear Alfvén waves in controlling the cascade dynamics. Title: Simulations of Incompressible Magnetohydrodynamic Turbulence Authors: Maron, Jason; Goldreich, Peter Bibcode: 2001ApJ...554.1175M Altcode: 2000astro.ph.12491M We simulate incompressible MHD turbulence using a pseudospectral code. Our major conclusions are: (1) MHD turbulence is most conveniently described in terms of counterpropagating shear Alfvén and slow waves. Shear Alfvén waves control the cascade dynamics. Slow waves play a passive role and adopt the spectrum set by the shear Alfvén waves. Cascades composed entirely of shear Alfvén waves do not generate a significant measure of slow waves. (2) MHD turbulence is anisotropic, with energy cascading more rapidly along k⊥ than along k∥, where k⊥ and k∥ refer to wavevector components perpendicular and parallel to the local magnetic field, respectively. Anisotropy increases with increasing k⊥ such that excited modes are confined inside a cone bounded by k∥~kγ⊥, where γ<1. The opening angle of the cone, Θ(k⊥)~k-(1-γ)⊥, defines the scale-dependent anisotropy. (3) The one-dimensional inertial range energy spectrum is well fitted by a power law, E(k⊥)~k-α⊥, with α>1. (4) MHD turbulence is generically strong in the sense that the waves that comprise it suffer order unity distortions on timescales comparable to their periods. Nevertheless, turbulent fluctuations are small deep inside the inertial range. Their energy density is less than that of the background field by a factor of Θ(α-1)/(1-γ)<<1. (5) MHD cascades are best understood geometrically. Wave packets suffer distortions as they move along magnetic field lines perturbed by counterpropagating waves. Field lines perturbed by unidirectional waves map planes perpendicular to the local field into each other. Shear Alfvén waves are responsible for the mapping's shear and slow waves for its dilatation. The amplitude of the former exceeds that of the latter by 1/Θ(k⊥), which accounts for dominance of the shear Alfvén waves in controlling the cascade dynamics. (6) Passive scalars mixed by MHD turbulence adopt the same power spectrum as the velocity and magnetic field perturbations. (7) Decaying MHD turbulence is unstable to an increase of the imbalance between the fluxes of waves propagating in opposite directions along the magnetic field. Forced MHD turbulence displays order unity fluctuations with respect to the balanced state if excited at low k⊥ by δ(t)-correlated forcing. It appears to be statistically stable to the unlimited growth of imbalance. (8) Gradients of the dynamic variables are focused into sheets aligned with the magnetic field whose thickness is comparable to the dissipation scale. Sheets formed by oppositely directed waves are uncorrelated. We suspect that these are vortex sheets, which the mean magnetic field prevents from rolling up. (9) Items 1-6 lend support to the model of strong MHD turbulence put forth by Goldreich & Sridhar (GS). Results from our simulations are also consistent with the GS prediction γ=2/3, as are those obtained previously by Cho & Vishniac. The sole notable discrepancy is that one-dimensional energy spectra determined from our simulations exhibit α~3/2, whereas the GS model predicts α=5/3. Further investigation is needed to resolve this issue. Title: Compressible Turbulence in Interstellar Plasmas Authors: Lithwick, Yoram; Goldreich, P. Bibcode: 2001AAS...198.9003L Altcode: 2001BAAS...33..919L Radio-wave scintillation observations reveal a nearly Kolmogorov spectrum of interstellar electron density fluctuations. Although this spectrum is suggestive of turbulence, there is little theoretical understanding of compressible turbulence in magnetized plasmas. We calculate the spectrum of density fluctuations by extending the theory of incompressible magnetohydrodynamic (MHD) turbulence given by Goldreich & Sridhar (1995) to include the effects of compressibility and particle transport. Our most important results are as follows. (1) Density fluctuations are due to the slow mode and the entropy mode. Both modes are passively mixed by the cascade of shear Alfvén waves. Since the shear Alfvén waves have a Kolmogorov spectrum, so do the density fluctuations. (2) Density fluctuation amplitudes constrain the nature of MHD turbulence in the interstellar medium. Slow mode density fluctuations are inversely proportional to β , the ratio of the gas pressure to the magnetic pressure. Entropy mode density fluctuations are suppressed by cooling when the cascade timescale is longer than the cooling timescale. These constraints suggest that either β is of order unity or the outer scale of the turbulence is very small. (3) A high degree of ionization is required for the cascade to survive damping by neutrals and extend to small lengthscales. Regions that are insufficiently ionized only produce density fluctuations on lengthscales larger than the neutral damping scale. They may account for the excess of power that is found on large scales. (4) Both the entropy mode and the slow mode are damped on lengthscales below that at which protons can diffuse across an eddy during the eddy's turnover time. Consequently, eddies whose extents along the magnetic field are smaller than the proton collisional mean free path do not contribute to the density spectrum. However, in MHD turbulence eddies are highly elongated along the magnetic field. From an observational perspective, the relevant lengthscale is that transverse to the magnetic field. Thus the cut-off lengthscale for density fluctuations is significantly smaller than the proton mean free path. (5) The Alfvén mode is critically damped at the transverse lengthscale of the proton gyroradius, and thus cascades to smaller lengthscales than either the slow mode or the entropy mode. Title: Gravity Modes in ZZ Ceti Stars. IV. Amplitude Saturation by Parametric Instability Authors: Wu, Yanqin; Goldreich, Peter Bibcode: 2001ApJ...546..469W Altcode: 2000astro.ph..3163W ZZ Ceti stars (also known as DAV stars) exhibit small-amplitude photometric pulsations in multiple gravity modes. As the stars cool, their dominant modes shift to longer periods. We demonstrate that parametric instability limits overstable modes to amplitudes similar to those observed. In particular, it reproduces the trend that longer period modes have larger amplitudes. Parametric instability is a form of resonant three-mode coupling. It involves the destabilization of a pair of stable daughter modes by an overstable parent mode. The three modes must satisfy exact angular selection rules and approximate frequency resonance. The lowest instability threshold for each parent mode is provided by the daughter pair that minimizes (δω2+γ2d)/ κ2, where κ is the nonlinear coupling constant, δω is the frequency mismatch, and γd is the energy damping rate of the daughter modes. Parametric instability leads to a steady state if |δω|>γd and to limit cycles if |δω|<γd. The former behavior characterizes low radial order (n<=3) parent modes, and the latter those with n>=5. In either case, the overstable mode's amplitude is maintained at close to the instability threshold value. Although parametric instability defines an upper envelope for the amplitudes of overstable modes in ZZ Ceti stars, other nonlinear mechanisms are required to account for the irregular distribution of amplitudes of similar modes and the nondetection of modes with periods longer than 1200 s. Resonant three-mode interactions involving more than one excited mode may account for the former and Kelvin-Helmholtz instability of the mode-driven shear layer below the convection zone for the latter. Title: Apse Alignment of Narrow Eccentric Planetary Rings Authors: Chiang, E. I.; Goldreich, P. Bibcode: 2000ApJ...540.1084C Altcode: 2000astro.ph..4193C The boundaries of the Uranian ɛ, α, and β rings can be fitted by Keplerian ellipses. The pair of ellipses that outline a given ring share a common line of apsides. Apse alignment is surprising because the quadrupole moment of Uranus induces differential precession. We propose that rigid precession is maintained by a balance of forces due to ring self-gravity, planetary oblateness, and interparticle collisions. Collisional impulses play an especially dramatic role near ring edges. Pressure-induced accelerations are maximal near edges because there (1) velocity dispersions are enhanced by resonant satellite perturbations and (2) the surface density declines steeply. Remarkably, collisional forces felt by material in the last ~100 m of a ~10 km wide ring can increase equilibrium masses up to a factor of ~100. New ring surface densities are derived that accord with Voyager radio measurements. In contrast to previous models, collisionally modified self-gravity appears to allow for both negative and positive eccentricity gradients; why all narrow planetary rings exhibit positive eccentricity gradients remains an open question. Title: Interstellar Scintillation and the Inertial Range of MHD Turbulence Authors: Goldreich, Peter Bibcode: 2000astu.progE...8G Altcode: No abstract at ADS Title: Growth of Perturbations in Gravitational Collapse and Accretion Authors: Lai, Dong; Goldreich, Peter Bibcode: 2000ApJ...535..402L Altcode: 1999astro.ph..6400L When a self-gravitating spherical gas cloud collapses or accretes onto a central mass, the inner region of the cloud develops a density profile ρ~r-3/2 and the velocity approaches free fall. We show that in this region nonspherical perturbations grow with decreasing radius. In the linear regime, the tangential velocity perturbation increases as r-1, while the Lagrangian density perturbation, Δρ/ρ, grows as r-1/2. Faster growth occurs if the central collapsed object maintains a finite multiple moment, in which case Δρ/ρ increases as r-l, where l specifies the angular degree of the perturbation. These scaling relations are different from those obtained for the collapse of a homogeneous cloud. Our numerical calculations indicate that nonspherical perturbations are damped in the subsonic region and that they grow and approach the asymptotic scalings in the supersonic region. The implications of our results to asymmetric supernova collapse and to black hole accretion are briefly discussed. Title: Discovery of Solar Atmospheric Motions Authors: Goldreich, Peter Bibcode: 1999ApJ...525C.962G Altcode: 1999ApJC..525..962G No abstract at ADS Title: Gravity Modes in ZZ Ceti Stars. III. Effects of Turbulent Dissipation Authors: Goldreich, Peter; Wu, Yanqin Bibcode: 1999ApJ...523..805G Altcode: 1998astro.ph.10038G We investigate dynamical interactions between turbulent convection and g-mode pulsations in ZZ Ceti variables. Since our understanding of turbulence is rudimentary, we are compelled to settle for order-of-magnitude results. A key feature of these interactions is that convective response times are much shorter than pulsation periods. Thus the dynamical interactions enforce near uniform horizontal velocity inside the convection zone. They also give rise to a narrow shear layer in the region of convective overshoot at the top of the radiative interior. Turbulent damping inside the convection zone is negligible for all modes, but that in the region of convective overshoot may be significant for a few long-period modes near the red edge of the instability strip. These conclusions are in accord with those reached earlier by Brickhill. Our major new result concerns nonlinear damping arising from the Kelvin-Helmholtz instability of the aforementioned shear layer. Amplitudes of overstable modes saturate where dissipation due to this instability balances excitation by convective driving. This mechanism of amplitude saturation is most effective for long-period modes, and it may play an important role in defining the red edge of the instability strip. Title: Spectral Energy Distributions of Passive T Tauri Disks: Inclination Authors: Chiang, E. I.; Goldreich, P. Bibcode: 1999ApJ...519..279C Altcode: 1998astro.ph.12194C We compute spectral energy distributions (SEDs) for passive T Tauri disks viewed at arbitrary inclinations. Semianalytic models of disks in radiative and hydrostatic equilibrium are employed. Over viewing angles for which the flared disk does not occult the central star, the SED varies negligibly with inclination. For such aspects, the SED shortward of ~80 μm is particularly insensitive to orientation, since short wavelength disk emission is dominated by superheated surface layers, which are optically thin. The SED of a nearly edge-on disk is that of a class I source. The outer disk occults inner disk regions, and emission shortward of ~30 μm is dramatically extinguished. Spectral features from dust grains may appear in absorption. However, millimeter-wavelength fluxes decrease by at most a factor of 2 from face-on to edge-on orientations. We present illustrative applications of our SED models. The class I source 04108+2803B is considered a T Tauri star hidden from view by an inclined circumstellar disk. Fits to its observed SED yield model-dependent values for the disk mass of ~0.015 Msolar and a disk inclination of ~65° relative to face-on. The class II source GM Aur represents a T Tauri star only slightly obscured by its circumstellar disk. Fitted parameters include a disk mass of ~0.050 Msolar and an inclination of ~60°, where the viewing angle is chosen to reproduce the observed visual extinction of AV=0.5 mag. Title: Gravity Modes in ZZ Ceti Stars. II. Eigenvalues and Eigenfunctions Authors: Wu, Yanqin; Goldreich, Peter Bibcode: 1999ApJ...519..783W Altcode: We report on numerical calculations of nonadiabatic eigenvalues and eigenfunctions for g-modes in ZZ Ceti variables. The spectrum of overstable l=1 modes delineates the instability strip. Its blue edge occurs where ωτc~1 for the n=1 mode. Here ω is radian frequency and τc is about 4 times the thermal time at the bottom of the surface convection zone. As a ZZ Ceti cools, its convection zone deepens, longer period modes become overstable, but the critical value of ωτc separating overstable and damped modes rises. The latter is a consequence of enhanced radiative damping for modes that propagate immediately below the convection zone. The critical value of ωτc is of observational significance, because modes with the smallest value of ωτc are most observable photometrically. Maximum periods for overstable modes predicted for our cooler model envelopes are about a factor of 2 longer than the observational upper limit of 1200 s. We assess a number of plausible resolutions for this discrepancy among which convective overshoot and nonlinear saturation look promising. The nonadiabatic eigenfunctions enable us to predict relative amplitudes and phases of photospheric variations of flux and velocity, quantities made accessible by recent observations. We also present asymptotic formula for damping rates of high-order modes, a result of consequence for future investigations of nonlinear saturation of the amplitudes of overstable modes. Title: The infrared excess - age relationship: debris, a NASA key project Authors: Spangler, C.; Silverstone, M. D.; Becklin, E. E.; Hare, J.; Zuckerman, B.; Sargent, A. I.; Goldreich, P. Bibcode: 1999ESASP.427..405S Altcode: 1999usis.conf..405S The DEBRIS project is primarily a survey of infrared radiation from nearby solar-like stars to establish how many display infrared emission in excess of that from the stellar photosphere and the timescale over which the excess persists. Such excess infrared emission indicates associated circumstellar material that could be the debris of planet formation. This survey used the ISOPHOT C100 detector ( te{lem96}) and the 60 and 90 μm filters to search for infrared flux around approximately 300 stellar targets with a variety of ages, masses and multiplicities. Here we present a summary of our results. The preliminary description of a drop of infrared excess as a function of age reported earlier by te*{bec98} is substantiated. Using a larger data set, better reduced data, and new age estimates, we find τpropto{age}-2. Several main sequence stars with newly discovered excesses are also discussed. Title: Gravity Modes in ZZ Ceti Stars. I. Quasi-adiabatic Analysis of Overstability Authors: Goldreich, Peter; Wu, Yanqin Bibcode: 1999ApJ...511..904G Altcode: 1998astro.ph..4305G We analyze the stability of g-modes in white dwarfs with hydrogen envelopes. All relevant physical processes take place in the outer layer of hydrogen-rich material, which consists of a radiative layer overlaid by a convective envelope. The radiative layer contributes to mode damping, because its opacity decreases upon compression and the amplitude of the Lagrangian pressure perturbation increases outward. The convective envelope is the seat of mode excitation, because it acts as an insulating blanket with respect to the perturbed flux that enters it from below. A crucial point is that the convective motions respond to the instantaneous pulsational state. Driving exceeds damping by as much as a factor of 2 provided ωτc>=1, where ω is the radian frequency of the mode and τc~4τth, with τth being the thermal time constant evaluated at the base of the convective envelope. As a white dwarf cools, its convection zone deepens, and lower frequency modes become overstable. However, the deeper convection zone impedes the passage of flux perturbations from the base of the convection zone to the photosphere. Thus the photometric variation of a mode with constant velocity amplitude decreases. These factors account for the observed trend that longer period modes are found in cooler DA variables. Overstable modes have growth rates of order γ~1/(nτω), where n is the mode's radial order and τω is the thermal timescale evaluated at the top of the mode's cavity. The growth time, γ-1, ranges from hours for the longest period observed modes (P~20 minutes) to thousands of years for those of shortest period (P~2 minutes). The linear growth time probably sets the timescale for variations of mode amplitude and phase. This is consistent with observations showing that longer period modes are more variable than shorter period ones. Our investigation confirms many results obtained by Brickhill in his pioneering studies of ZZ Cetis. However, it suffers from two serious shortcomings. It is based on the quasiadiabatic approximation that strictly applies only in the limit ωτc>>1, and it ignores damping associated with turbulent viscosity in the convection zone. We will remove these shortcomings in future papers. Title: Gravity-Modes in ZZ Ceti Stars III. Eigenvalues and Eigenfuctions Authors: Wu, Yanqin; Goldreich, Peter Bibcode: 1998astro.ph.12085W Altcode: We report on numerical calculations of nonadiabatic eigenvalues and eigenfunctions for g-modes in ZZ Ceti variables. The spectrum of overstable $l=1$ modes delineates the instability strip. Its blue edge occurs where $\omega \tau_c \approx 1$ for the $n=1$ mode. Here $\omega$ is radian frequency and $\tau_c$ is about four times the thermal timescale at the bottom of the surface convection zone. As a ZZ Ceti cools, its convection zone deepens, longer period modes become overstable, but the critical value of $\omega\tau_c$ separating overstable and damped modes rises. The latter is a consequence of enhanced radiative damping for modes which propagate immediately below the convection zone. The critical value of $\omega\tau_c$ is of observational significance because modes with the smallest value of $\omega\tau_c$ are most observable photometrically. Maximum periods for overstable modes predicted for our cooler model envelopes are about a factor two longer than the observational upper limit of $1,200\s$. We assess a number of plausible resolutions for this discrepancy among which convective overshoot and nonlinear saturation look promising. The nonadiabatic eigenfunctions enable us to predict relative amplitudes and phases of photospheric variations of flux and velocity, quantities made accessible by recent observations. We also present asymptotic formula for damping rates of high order modes, a result of consequence for future investigations of nonlinear saturation of the amplidues of overstable modes. Title: Sources of ISS Authors: Goldreich, Peter Bibcode: 1998AAS...192.4603G Altcode: 1998BAAS...30..878G Interstellar scintillations (ISS) of small angular diameter radio sources have been studied for 30 years. They arise from fluctuations of the interstellar electron density. These exhibit an anisotropic Kolmogorov spectrum over scales ranging from about 10(9) cm to 10(15) cm, with the power in these fluctuations varying by orders of magnitude from place to place. The optics of this phenomenon is fairly well understood. The same cannot be said for the physical process responsible for producing the electron density fluctuations. In recent work, Goldreich and Sridhar argue that incompressible MHD turbulence will produce a power law velocity spectrum of shear Alfven waves that matches the ISS electron density spectrum. The velocity spectrum arises from a critical balance between the linear wave periods and nonlinear wave interactions. Its connection to the electron density spectrum remains a puzzle. One might speculate that the turbulent velocity field mixes specific entropy as a passive contaminant thereby giving rise to isobaric temperature and density fluctuations whose spectra mimic that of the velocity field. This is the manner in which the atmospheric scintillations of optical stars arise. However, as a consequence of its rapid of cooling, specific entropy is not conserved in large scale motions of ionized interstellar gas. This severely limits the outer scale of the electron density spectrum. Implications of this limitation to potential sources and sites of ISS will be explored in this lecture. Title: Oscillating White Dwarfs Authors: Goldreich, Peter Bibcode: 1998AAS...192.7501G Altcode: 1998BAAS...30Q.931G As a hydrogen (DA) white dwarf cools, it passes through an instability strip centered at T_eff ~ 12,000 K having width Delta T_eff ~ 1,000 K. Within this temperature range it exhibits photometric variations associated with excited g-modes of low angular degree. The pulsations undergo a pronounced evolution as the star travels across the instability strip. The periods of the dominant modes increase, and the modes display larger photometric amplitudes and greater temporal variability. These trends are consequences of the mechanism of linear overstability which excites the modes together with the nonlinear interactions that saturate their amplitudes. Linear overstability is due to convective driving, a novel mechanism proposed by Brickhill. Convective driving relies on the ability of the convective motions to respond to the instantaneous pulsational state. It destabilizes modes whose periods are comparable to or shorter than the thermal timescale at the base of the star's surface convection zone. As a white dwarf cools, its convection zone deepens and modes of longer period become overstable. Longer period modes have smaller effective mass and thus faster growth and decay rates than modes of shorter period. This accounts for their increased variability. Amplitude saturation is due to parametric instability which involves the destabilization of pairs of linearly damped daughter modes by an overstable parent mode. The frequencies and angular degrees of parent and daughter modes satisfy resonance relations. Parametric instability requires the amplitude of the parent mode to exceed a critical threshold. It thereby sets the envelope for the amplitudes of overstable modes. However, it cannot account for the uneven distribution of energy among modes which have similar linear growth rates. This feature is due to nonlinear interactions such as parametric up and down conversion which couple two excited modes to one damped mode. Title: Spectral Energy Distributions of T Tauri Stars with Passive Circumstellar Disks Authors: Chiang, E. I.; Goldreich, P. Bibcode: 1997ApJ...490..368C Altcode: 1997astro.ph..6042C We derive hydrostatic, radiative equilibrium models for passive disks surrounding T Tauri stars. Each disk is encased by an optically thin layer of superheated dust grains. This layer reemits directly to space about half the stellar energy it absorbs. The other half is emitted inward and regulates the interior temperature of the disk. The heated disk flares. As a consequence, it absorbs more stellar radiation, especially at large radii, than a flat disk would. The portion of the spectral energy distribution contributed by the disk is fairly flat throughout the thermal infrared. At fixed frequency, the contribution from the surface layer exceeds that from the interior by about a factor 3 and is emitted at more than an order of magnitude greater radius. Spectral features from dust grains in the superheated layer appear in emission if the disk is viewed nearly face-on. Title: Magnetohydrodynamic Turbulence Revisited Authors: Goldreich, P.; Sridhar, S. Bibcode: 1997ApJ...485..680G Altcode: 1996astro.ph.12243G In 1965, Kraichnan proposed that MHD turbulence occurs as a result of collisions between oppositely directed Alfvén wave packets. Recent work has generated some controversy over the nature of nonlinear couplings between colliding Alfvén waves. We find that the resolution to much of the confusion lies in the existence of a new type of turbulence, intermediate turbulence, in which the cascade of energy in the inertial range exhibits properties intermediate between those of weak and strong turbulent cascades. Some properties of intermediate MHD turbulence are the following: (1) in common with weak turbulent cascades, wave packets belonging to the inertial range are long-lived (2) however, components of the strain tensor are so large that, similar to the situation in strong turbulence, perturbation theory is not applicable; (3) the breakdown of perturbation theory results from the divergence of neighboring field lines due to wave packets whose perturbations in velocity and magnetic fields are localized, but whose perturbations in displacement are not; (4) three-wave interactions dominate individual collisions between wave packets, but interactions of all orders n >= 3 make comparable contributions to the intermediate turbulent energy cascade; (5) successive collisions are correlated since wave packets are distorted as they follow diverging field lines; (6) in common with the weak MHD cascade, there is no parallel cascade of energy, and the cascade to small perpendicular scales strengthens as it reaches higher wavenumbers; (7) for an appropriate weak excitation, there is a natural progression from a weak, through an intermediate, to a strong cascade. Title: Globally asymmetric supernova. Authors: Goldreich, P.; Lai, D.; Sahrling, M. Bibcode: 1997upa..conf..269G Altcode: Asymmetries in type II supernova explosions are the most plausible cause of the high space velocities of radio pulsars. The origin of these asymmetries is unknown. Recent work has stressed the potential importance of local instabilities which occur subsequent to the collapse of the stellar core. By contrast, relatively little attention has been paid to the possibility that substantial asymmetry might already be present during core collapse. The presupernova star supports g-modes in which the core oscillates with respect to the stellar envelope. These modes gain excitation by modulating the release of nuclear energy. Asymmetries associated with these modes could then be amplified during core collapse. The most likely place for this to occur is in the supersonically collapsing outer core. Title: The Debris ISO Key Project: Early Release Observations Authors: Becklin, E. E.; Silverstone, M.; Zuckerman, B.; Spangler, C.; Sargent, A.; Goldreich, P.; Mannings, V. Bibcode: 1996AAS...188.5207B Altcode: 1996BAAS...28..902B The ISO Dust Debris key project is a collaboration between UCLA and Caltech to study the physical nature and evolutionary history of dust debris clouds around solar mass stars. The sample objects include F and G stars within 20 pc of the Sun, as well as members of nearby open clusters and weak-line T Tauri stars. The 60 and 100 micron observations, made with ISOPHOT, consist of broadband single- element photometry plus far-infrared imaging with a small array. Approximately 15 nearby and cluster stars were released during February 1996 for early observations to assess the performance of ISO. The results of these early release observations, and their effect upon the Debris program, will be discussed. Issues concerning signal-to-noise and calibration will be addressed. Title: Single sided shepherding. Authors: Goldreich, Peter; Rappaport, Nicole; Sicardy, Bruno Bibcode: 1995Icar..118..414G Altcode: Narrow ringlets are observed to form at isolated Lindblad resonances in recent simulations (J. Hänninen and H. Salo, Icarus 108, 325-346 (1994) and 117, 435-438 (1995)). Our remarks are directed toward the interpretation of this phenomenon. Ringlet formation is a consequence of the negative angular momentum luminosity promoted by satellite perturbations of the streamlines of particle flow. Contraction halts once the surface density reaches a value such that the angular momentum luminosity vanishes. Our estimate for the formation time, tf ∼ (Mp/Ms)1/2Ω,-1, for low order resonances in optically thin rings is consistent with the results of the simulations. Ringlets drift across the width of resonance, W ∼ (Ms/Mp)1/2a, as a result of the unbalanced satellite torque. This occurs on the much longer timescale td ∼ (Mp/Ms)Ω-1 and cannot be observed in current simulations. Title: Ringlet Formation at Isolated Lindblad Resonance Authors: Rappaport, N.; Goldreich, P.; Sicardy, B. Bibcode: 1995DPS....27.2903R Altcode: 1995BAAS...27.1136R No abstract at ADS Title: Toward a Theory of Interstellar Turbulence. II. Strong Alfvenic Turbulence Authors: Goldreich, P.; Sridhar, S. Bibcode: 1995ApJ...438..763G Altcode: We continue to investigate the possibility that interstellar turbulence is caused by nonlinear interactions among shear Alfven waves. Here, we restrict attention to the symmetric case where the oppositely directed waves carry equal energy fluxes. This precludes application to the solar wind in which the outward flux significantly exceeds the ingoing one. All our detailed calculations are carried out for an incompressible magnetized fluid. In incompressible magnetohydrodynamics (MHD), nonlinear interactions only occur between oppositely direct waves. We present a theory for the strong turbulence of shear Alfven waves. It has the following main characteristics. (1) The inertial-stage energy spectrum exhibits a critical balance between linear wave periods and nonlinear turnover timescales. (2) The 'eddies' are elongated in the direction of the field on small spatial scales; the parallel and perpendicular components of the wave vector, kz and k(perpendicular) are related by kz approximately equals kperpendicular to2/3 L-1/3, where L is the outer scale of the turbulence. (3) The 'one-dimensional' energy spectrum is proportional to kperpendicular-5/3-an anisotropic Kolmogorov energy spectrum. Shear Alfvenic turbulence mixes specific entropy as a passive contaminant. This gives rise to an electron density power spectrum whose form mimics the energy spectrum of the turbulence. Radio wave scattering by these electron density fluctuations produces anisotropic scatter-broadened images. Damping by ion-neutral collisions restricts Alfvenic turbulence to highly ionized regions of the interstellar medium. Title: Toward a Theory of Interstellar Turbulence. I. Weak Alfvenic Turbulence Authors: Sridhar, S.; Goldreich, P. Bibcode: 1994ApJ...432..612S Altcode: We study weak Alfvenic turbulence of an incompressible, magnetized fluid in some detail, with a view to developing a firm theoretical basis for the dynamics of small-scale turbulence in the interstellar medium. We prove that resonant 3-wave interactions are absent. We also show that the Iroshnikov-Kraichnan theory of incompressible, magnetohydrodynamic turbulence -- which is widely accepted -- describes weak 3-wave turbulence; consequently, it is incorrect. Physical arguments, as well as detailed calculations of the coupling coefficients are used to demonstrate that these interactions are empty. We then examine resonant 4-wave interactions, and show that the resonance relations forbid energy transport to small spatial scales along the direction of the mean magnetic field, for both the shear Alfven wave and the pseudo Alfven wave. The three-dimensional inertial-range energy spectrum of 4-wave shear Alfven turbulence guessed from physical arguments reads E(kz, kperpendicular) approximately VAvLL-1/3kperpendicular-10/3, where VA is the Alfven speed, and vL is the velocity difference across the outer scale L. Given this spectrum, the velocity difference across lambdaperpendicular approximately kperpendicular exp -1 is vlambda (sub perpendicular) is approximately vL(lambdaperpendicular/L)2/3. We derive a kinetic equation, and prove that this energy spectrum is a stationary solution and that it implies a positive flux of energy in k-space, along directions perpendicular to the mean magnetic field. Using this energy spectrum, we deduce that 4-wave interactions strengthen as the energy cascades to small, perpendicular spatial scales; beyond an upper bound in perpendicular wavenumber, kperpendicularL is approximately (VA/vL)3/2, weak turbulence theory ceases to be valid. Energy excitation amplitudes must be very small for the 4-wave inertial-range to be substantial. When the excitation is strong, the width of the 4-wave inertial-range shrinks to zero. This seems likely to be the case in the interstellar medium. Title: Excitation of Neutron Star Normal Modes during Binary Inspiral Authors: Reisenegger, Andreas; Goldreich, Peter Bibcode: 1994ApJ...426..688R Altcode: As a compact binary inspirals due to the emission of gravitational waves, its orbital period decreases continuously down to approximately 1 ms, its value at coalescence. During the last part of the inspiral, the two stars are close together, and their tidal interactions become strong. Neutron stars have many normal modes (core g-modes, crustal discontinuity modes, shear modes, etc.) whose periods lie in the range (approximately several ms) swept by the orbital period. Some of these modes are resonantly excited by the tidal force. The amount of energy a mode absorbs is proportional to the square of the overlap integral between its displacement field and the tidal force field. For all modes of interest, this overlap is poor, resulting in relatively weak excitation. For the best case, the absorbed energy is only a small fraction (approximately 10-6) of the orbital energy, so the orbital phase shift is too weak to be detected by observations of the gravitational wave signal emitted by the inspiraling binary. However, with displacement amplitudes of excited quadrupole modes ranging up to 0.5% of the stellar radius, the possibility of a detectable electromagnetic signature cannot be dismissed. Both the periods of the modes and the energy they absorb depend quite strongly on the internal structure of the star. Their observation could shed light on the correct high-density equation of state. Title: Effect of Nonlinear Interactions on p-Mode Frequencies and Line Widths Authors: Kumar, Pawan; Goldreich, Peter; Kerswell, Richard Bibcode: 1994ApJ...427..483K Altcode: We calculate the effect of nonlinear interactions among solar acoustic modes upon the modal frequencies and energy loss rates (or line widths). The frequency shift for a radial p-mode of frequency 3 mHz is found to be about -0.5 microHz. The magnitude of nonlinear frequency shift increases more rapidly with frequency than the inverse mode mass (mode mass is defined as the ratio of energy in the mode to its surface velocity amplitude squared). This frequency shift is primarily due to nonresonant three-mode interactions and is dominated by high l surface gravity waves (f-modes) and p-modes. The line width of a radial p-mode of frequency 3 mHz, due to resonant nonlinear interactions, is about 0.3 microHz. This result is consistent with that of Kumar and Goldreich (1989). We also find, in agreement with these authors, that the most important nonlinear interactions of trapped p-modes involve f-modes and high-frequency p-modes (frequency greater than about 5 mHz) which propagate in the solar photosphere. Thus, using the arguments advanced by Kumar & Goldreich (1989), we conclude that nonlinear couplings cannot saturate the overstable solar p-modes at their small observed amplitudes. Both the nonlinear frequency shifts and line widths, at a fixed frequency, are proportional to the inverse of mode mass which for modes of degree greater than about 100 is approximately l0.8. Therefore, the frequency of an f-mode of l = 1000, due to nonlinear interactions, is decreased by approximately 0.4%. Title: The Effects of Scattering on Solar Oscillations Authors: Goldreich, Peter; Murray, Norman Bibcode: 1994ApJ...424..480G Altcode: Acoustic modes are scattered by turbulent velocity fluctuations in the solar convection zone. The strongest scattering occurs near the top of the acoustic cavity where the mode changes character from propagating to evanescent. This layer is located at depth z1 approximately g/omega2 below the photosphere. The scattering optical depth taus is of order M12, where M1 is the Mach number of the energy-bearing eddies at z1. The corresponding contribution to the line width is gammas is approximately (omega) M12/(pi)(n+1), where n is the mode's radial order. At the top of the acoustic cavity the correlation time of energy-bearing eddies is much longer than omega-1. Also, the pressure scale height H and the eddy correlation length Lambda are comparable to omega/c, where c is the sound speed. Thus scattering couples modes of similar omega and all l and has little effect on the sum of their energies. Observations show that mode energies decline with decreasing n (increasing l) at fixed omega. Consequently, scattering damps p-modes and excites f-modes. Title: Excitation of Solar p-Modes Authors: Goldreich, Peter; Murray, Norman; Kumar, Pawan Bibcode: 1994ApJ...424..466G Altcode: We investigate the rates at which energy is supplied to individual p-modes as a function of their frequencies nu and angular degrees l. The observationally determined rates are compared with those calculated on the hypothesis that the modes are stochastically excited by turbulent convection. The observationally determined excitation rate is assumed to be equal to the product of the mode's energy E and its (radian) line width Gamma. We obtain E from the mode's mean square surface velocity with the aid of its velocity eigenfuction. We assume that Gamma measures the mode's energy decay rate, even though quasi-elastic scattering may dominate true absorption. At fixed l, E(Gamma) arises as nu7 at low nu, reaches a peak at nu approximately equal 3.5 mHz, and then declines as nu4.4 at higher nu . At fixed nu, E(Gamma) exhibits a slow decline with increasing l. To calculate energy input rates, Palpha, we rely on the mixing-length model of turbulent convection. We find entropy fluctuations to be about an order of magnitude more effective than the Reynolds stress in exciting p-modes . The calculated Palpha mimic the nu7 dependence of E(Gamma) at low nu and the nu-4.4 dependence at high nu. The break of 11.4 powers in the nu-dependence of E(Gamma) across its peak is attributed to a combination of (1) the reflection of high-frequency acoustic waves just below the photosphere where the scale height drops precipitously and (2) the absence of energy-bearing eddies with short enough correlation times to excite high-frequency modes. Two parameters associated with the eddy correlation time are required to match the location and shape of the break. The appropriate values of these parameters, while not unnatural, are poorly constrained by theory. The calculated Palpha can also be made to fit the magnitude of E(Gamma) with a reasonable value for the eddy aspect ratio. Our resutls suggest a possible explanation for the decline of mode energy with increasing l at fixed nu. Entropy fluctuations couple to changes in volume associated with the oscillation mode. These decrease with decreasing n at fixed nu, becoming almost zero for the f-mode. Title: Excitation of Neutron Star Oscillation Modes During Binary Inspiral Authors: Reisenegger, A.; Goldreich, P. Bibcode: 1994AIPC..308..311R Altcode: 1994exrb.conf..311R As a compact binary inspirals due to the emission of gravitational waves, its orbital period decreases continuously down to ∼1 ms, its value at coalescence. During the last part of the inspiral, the two stars are close together and their tidal interactions become strong. Neutron stars have many normal modes whose periods lie in the range swept by the orbital period. Some of these modes are resonantly excited by the tidal force. The amount of energy a mode absorbs is proportional to the square of the overlap integral between its displacement field and the tidal force field. For all modes of interest, this overlap is poor, resulting in relatively weak excitation. The absorbed energy is only a small fraction (≲10-6) of the orbital energy, so the orbital phase shift is too weak to be detected by observations of the gravitational wave signal emitted by the inspiraling binary. However, with displacement amplitudes of excited quadrupole modes ranging up to 0.5% (or more) of the stellar radius, the possibility of a detectable electromagnetic signature cannot be dismissed. Both the periods of the modes and the energy they absorb depend quite strongly on the internal structure of the star. Their observation could shed light on the correct high-density equation of state. Title: Interactions among convection, magnetic fields and p-mode oscillations in the sun Authors: Goldreich, Peter Bibcode: 1993STIN...9414479G Altcode: Two papers on different aspects of the excitation and damping of solar oscillations were accepted for publication in the Astrophysical Journal. The first paper evaluates the rate at which turbulent convection feeds energy into individual p-modes. It is shown that stochastic excitation by turbulent convection provides a satisfactory fit to the product of the mode energies and linewidths. A somewhat surprising conclusion is that entropy fluctuations are about an order of magnitude more significant than are fluctuations of the Reynolds stress in exciting p-modes. However, entropy fluctuations cannot excite f-modes. This may account for the relatively low energies of the f-modes compared to those of the p-modes. The second paper explores the role of scattering of acoustic modes by turbulent velocity fluctuations. Scattering of a mode is concentrated near the top of its acoustic cavity. Because the turbulence has low Mach number, scattering couples modes having similar frequencies. Its net effects are to increase the linewidths of all modes and to transfer energy from p-modes to f-modes. Scattering is likely to be the dominant source for the linewidths of p-modes. In particular, it may account for the unexpectedly large linewidths measured for low frequency modes. Copies of preprints of the two papers referred to above are attached. The remainder of the report is devoted to a description of unpublished results. Title: Magnetic Field Decay in Isolated Neutron Stars Authors: Goldreich, Peter; Reisenegger, Andreas Bibcode: 1992ApJ...395..250G Altcode: Three mechanisms that promote the loss of magnetic flux from an isolated neutron star - Ohmic decay, ambipolar diffusion, and Hall drift - are investigated. Equations of motions are solved for charged particles in the presence of a magnetic field and a fixed background of neutrons, while allowing for the creation and destruction of particles by weak interactions. Although these equations apply to normal neutrons and protons, the present interpretations of their solutions are extended to cover cases of neutron superfluidity and proton superconductivity. The equations are manipulated to prove that, in the presence of a magnetic force, the charged particles cannot be simultaneously in magnetostatic equilibrium and chemical equilibrium with the neutrons. The application of the results to real neutron stars is discussed. Title: A New Class of g-Modes in Neutron Stars Authors: Reisenegger, Andreas; Goldreich, Peter Bibcode: 1992ApJ...395..240R Altcode: Because a neutron star is born hot, its internal composition is close to chemical equilibrium. In the fluid core, this implies that the ratio of the number densities of charged particles (protons and electrons) to neutrons is an increasing function of the mass density. This composition gradient stably stratifies the matter giving rise to a Brunt-Vaisala frequency N of about 500/s. Consequently, a neutron star core provides a cavity that supports gravity modes (g-modes). These g-modes are distinct from those previously identified with the thermal stratification of the surface layers and the chemical stratification of the crust. We compute the lowest-order, quadrupolar, g-modes for cold, Newtonian, neutron star models with M/solar M = 0.581 and M/solar M = 1.405, and show that the crustal and core g-modes have similar periods. We also discuss damping mechanisms and estimate damping rates for the core g-modes. Particular attention is paid to damping due to the emission of gravitational radiation. Title: Constraints on Gravity Waves in Upper Planetary Atmospheres from Occultation Observations Authors: Sicardy, B.; Goldreich, P.; Roques, F. Bibcode: 1992DPS....24.1607S Altcode: 1992BAAS...24..963S No abstract at ADS Title: Solar Phonons Authors: Goldreich, P. Bibcode: 1992DPS....24.3301G Altcode: 1992BAAS...24.1004G No abstract at ADS Title: Puzzles and Prospects in Planetary Ring Dynamics (lecture) Authors: Goldreich, P. Bibcode: 1992IAUS..152...65G Altcode: No abstract at ADS Title: Thermal and Mechanical Damping of Solar p-Modes Authors: Goldreich, Peter; Kumar, Pawan Bibcode: 1991ApJ...374..366G Altcode: Nonadiabatic effects associated with the transfer of energy and with turbulent stresses add small imaginary parts, omega-i(1) and omega-i(2), to solar p-mode eigenfrequencies. Numerical calculations have shown that these quite different processes make comparable contributions to omega-i at frequencies well below the acoustic cutoff at omega-ac. Analytic expressions are derived which reveal the connection between omega-i(1) and omega-i(2). The estimates yield omega-i proportional to omega exp 8 for omega much less than omega-ac in good agreement with the numerical calculations. However, the observed line width is proportional to omega exp 4.2 at low frequencies. It is suspected that there is an unmodeled component of perturbed convective energy transport or of turbulent viscosity that makes an important contribution to omega-i at omega much less than omega-ac. Title: Implications of Solar p-Mode Frequency Shifts Authors: Goldreich, Peter; Murray, Norman; Willette, Gregory; Kumar, Pawan Bibcode: 1991ApJ...370..752G Altcode: An expression is derived that relates solar p-mode frequency shifts to changes in the entropy and magnetic field of the sun. The frequency variations result from changes in path length and propagation speed. Path length changes dominate for entropy perturbations, and propagation speed changes dominate for most types of magnetic field peturbations. The p-mode frequencies increased along with solar activity between 1986 and 1989; these frequency shifts exhibited a rapid rise with increasing frequency followed by a precipitous drop. The positive component of the shifts can be accounted for by variations of the mean square magnetic field strength in the vicinity of the photosphere. The magnetic stress perturbation decays above the top of the convection zone on a length scale comparable to the pressure scale height and grows gradually with depth below. The presence of a resonance in the chromospheric cavity means that the transition layer maintains enough coherence to partially reflect acoustic waves even near cycle maximum. Title: Wave Generation by Turbulent Convection Authors: Goldreich, Peter; Kumar, Pawan Bibcode: 1990ApJ...363..694G Altcode: Wave generation by turbulent convection in a plane parallel, stratified atmosphere lying in a gravitational field is studied. The turbulent spectrum is related to the convective energy flux via the Kolmogorov scaling and the mixing length hypothesis. Efficiencies for the conversion of the convective energy flux into both trapped and propagating waves are estimated. Title: The Dominant Local Instabilities in an Einstein--de Sitter Universe Authors: Blaes, O. M.; Goldreich, P. M.; Villumsen, J. V. Bibcode: 1990ApJ...361..331B Altcode: The evolution of compensated aspherical perturbations in an Einstein-de Sitter universe with cold collisionless matter is investigated using both N-body simulations and quasi-analytic similarity solutions to treat the evolution of compensated axisymmetric structures. The solutions consist of a central void surrounded by an inhomogeneous shell. It is assumed that the structures are much smaller than the horizon scale so that Newtonian gravity is a good approximation. The results obtained from the two approaches are compared, and the implications are discussed. Title: Neutron Starquake Models for Gamma-Ray Bursts Authors: Blaes, O.; Blandford, R.; Goldreich, P.; Madau, P. Bibcode: 1989ApJ...343..839B Altcode: The component parts of gamma-ray burst models based on neutron starquakes are assessed. The requirements for, and the properties of, neutron starquakes are reviewed. The coupling of seismic waves to Alfven waves is evaluated and the behavior of Alfven waves in the magnetosphere is investigated. An attempt is made to identify the principal sources of free energy in the interiors of old neutron stars. Title: Neptune's Story Authors: Goldreich, P.; Murray, N.; Longaretti, P. Y.; Banfield, D. Bibcode: 1989Sci...245..500G Altcode: It is conjectured that Triton was captured from a helio-centric orbit as the result of a collision with what was then one of Neptune's regular satellites. The immediate post-capture orbit was highly eccentric with a semimajor axis a ~ 103RN and a periapse distance rp that oscillated periodically above a minimum value of about 5RN. Dissipation due to tides raised by Neptune in Triton caused Triton's orbit to evolve to its present state in lesssim 109 years. For much of this time Triton was almost entirely molten. While its orbit was evolving, Triton cannibalized most of the regular satellites of Neptune and also perturbed Nereid, thus accounting for that satellite's highly eccentric and inclined orbit. The only regular satellites of Neptune that survived were those that formed well within 5RN and they move on inclined orbits as the result of chaotic perturbations forced by Triton. Neptune's arcs are confined around the corotation resonances of one of these inner satellites. The widths and lengths of the arcs imply that the satellite's radius is at least 30/(sin i)2/3 kilometers for i lesssim 1, where i is the angle of inclination. Title: The formation of sharp edges in planetary rings by nearby satellites Authors: Borderies, N.; Goldreich, P.; Tremaine, S. Bibcode: 1989Icar...80..344B Altcode: Sharp edges, boundaries between regions of high and low optical depth in planetary rings, are maintained by shepherd satellites which transfer angular momentum to and from the ring particles. We derive equations that govern the shapes of the perturbed streamlines near such a boundary. These equations are solved for a simple numerical model whose parameters are chosen to resemble those of the Encke division and its associated satellite. The results of our calculation faithfully reproduce the sharp edges which bound the division and imply that the ring thickness in the unperturbed regions far from the edges is of order 10 m. In particular, the angle-averaged surface density is found to vary on a much shorter radial length scale than that over which the satellite torque is applied. We demonstrate that this striking feature is related to the local reversal of the viscous transport of angular momentum in the most strongly perturbed regions. Title: Tides in Rotating Fluids Authors: Goldreich, Peter; Nicholson, Philip D. Bibcode: 1989ApJ...342.1075G Altcode: The tidal disturbance forced in a differentially rotating fluid by a rigidly rotating external potential is investigated. In the study, the fluid is assumed to be inviscid, insulated, and self-gravitating, and to have laminar unperturbed and perturbed velocity fields. Under the steady torque of second-order strength exerted by the external potential, no secular changes in the angular momenta of fluid particles are observed, except possibly at corotation where the angular velocity is equal to the pattern speed of the potential. Except at corotation, all of the angular momentum is found to be transported away by internal stresses. Title: Nonlinear Interactions among Solar Acoustic Modes Authors: Kumar, Pawan; Goldreich, Peter Bibcode: 1989ApJ...342..558K Altcode: The rates at which nonlinear interactions transfer energy among the normal modes of a plane-parallel, stratified atmosphere are evaluated. It is shown that every p-mode in the 5-minute band is involved in many near-resonant triplets, and, as a consequence, the energy transfer rates are independent of the mode line widths. It is also found that nonlinear mode coupling cannot limit the growth of overstable p-modes, which favors the hypothesis that the sun's p-modes are stochastically excited by turbulent convection. Title: Tidal Friction in Early-Type Stars Authors: Goldreich, Peter; Nicholson, Philip D. Bibcode: 1989ApJ...342.1079G Altcode: Theoretical and observational results related to tides in early-type stars are summarized. The results suggest that the tidal torque on an early-type star is concentrated near the boundary between the convective core and the radiative envelope, and that a train of gravity waves which is excited at this boundary outwardly transports the angular momentum removed from the fluid by the torque. Tidal despinning to synchronous rotation is shown to proceed from the outside to the inside of the star. The present model can account for the previous finding that Zahn's (1975, 1977) theory for tidal evolution in early-type close binaries is compatible with the observed rates of orbit circularization, while significantly underestimating the observed rates of spin synchronization. Title: Neptune's Satellite System, Theory and Observations Authors: Banfield, D.; Goldreich, P.; Murray, N.; Longaretti, P. Y. Bibcode: 1989BAAS...21Q.911B Altcode: No abstract at ADS Title: Neutron starquakes and the nature of gamma-ray bursts. Authors: Madau, P.; Blaes, O.; Blandford, R. D.; Goldreich, P. Bibcode: 1989plap.work..247M Altcode: The authors investigate the possibility that gamma-ray bursts originate from quakes deep in the solid crust of a neutron star. Seismic waves are radiated if shear stress is relieved by brittle fracture. However they cannot propagate directly to the surface but are temporarily trapped below a reflecting layer. The shaking of the stellar surface couples the seismic waves to Alfvén waves which propagate out into the magnetosphere. The crust-magnetosphere transmission coefficient strongly increases with wave frequency and magnetic field strength. Alfvén wave luminosities sufficient to power galactic gamma-ray bursts are possible if magnetic fields ⪆1011G cover at least part of the stellar surface. As the Alfvén waves propagate out into the low density magnetosphere, they become increasingly charge starved, thereby accelerating particles to relativistic energies. Title: Neutron Starquake Models of Gamma-Ray Bursts Authors: Blaes, O.; Blandford, R.; Goldreich, P.; Madau, P. Bibcode: 1988BAAS...20.1054B Altcode: No abstract at ADS Title: Nonlinear Calculations of 3-mode Coupling in One Dimension Authors: Kochanek, C. S.; Blaes, O. M.; Goldreich, P. M. Bibcode: 1988BAAS...20.1010K Altcode: No abstract at ADS Title: Research at Palomar Observatory in planetary astronomy Authors: Soifer, B. T.; Goldreich, P. Bibcode: 1988plas.rept..119S Altcode: A wide range of observational studies are carried out to improve our understanding of the bodies of the outer solar system. Using the 200-inch Hale telescope, near-infrared observations are made of Uranus, Neptune, and the Pluto-Charon system. High time resolution occultation observations of the Uranus Ring system are used to study in detail the dynamics of this system. Occultation studies of Neptune are probing this intriguing ring-arc system. Occulation observations of the Pluto-Charon system probe the surface properties of these distant bodies. In addition, the plate material of the PSSII servey is being used to search for new comets and asteroids. Researchers observed one Neptune stellar occultation in July 1987 and completed the analysis of a series of seven separate Neptune occultation observations in conjunction with Nicholson et al., of Cornell. The analysis has shown that minimum of three ring arcs, at radii ranging from 54,000 km - 67,000 km are required to account for the high quality ring events. Current theoretical models can account for these data. Of two observations scheduled of Pluto-Charon mutual occulations scheduled for the 200-inch, the Charon eclipse event was successfully observed (the other was clouded out). Title: Lunar and planetary studies Authors: Muhleman, Duane O.; Goldreich, P.; Ingersoll, A. P.; Westphal, J. A. Bibcode: 1988plas.rept...95M Altcode: This grant supports the core program in planetary astronomy at Caltech. The research includes observations in the IR, sub-mm, mm and cm wavelengths at national and Caltech observatories with a strong emphasis on integrating the observations with spacecraft data and with models of atmospheric structure, dynamics and chemistry. Muhleman's group made extensive observations of Saturn, Uranus and Neptune which are being interpreted in terms of deep atmospheric structures which are obvious in the 2 and 6 cm maps of Saturn and Uranus. The microwave measurements are one of the few sources of information below the 2 bar level. Goldreich is investigating the dynamics of narrow rings with postdoctoral fellow, Pierre-Yves Longaretti. Their work has focused on the role of collisional stresses on the precession of the rings, since the Voyager radio science results imply that the previous model based on the ring's self-gravity is not the entire story. In addition Borderies, Goldreich and Tremaine have completed an investigation of the dynamics of the Encke division in Saturn's A ring. Title: Distribution Functions for the Time-averaged Energies of Stochastically Excited Solar p-Modes Authors: Kumar, Pawan; Franklin, Joel; Goldreich, Peter Bibcode: 1988ApJ...328..879K Altcode: The excitation of a damped harmonic oscillator by a random force is studied as a model for the stochastic excitation of a solar p-mode by turbulent convection. An extended sequence of observations is required to separate different p-modes and thus determine the energies of individual modes. Therefore, the observations yield time-averaged values of the energy. The theory of random differential equations is applied to calculate distribution functions for the time-averaged energy of the oscillator. The instantaneous energy satisfies a Boltzmann distribution. With increasing averaging time, the distribution function narrows and its peak shifts toward the mean energy. Numerical integrations are performed to generate finite sequences of time-averaged energies. These are treated as simulated data from which approximate probability distributions for the time-averaged energy are obtained. Title: The Interaction of Acoustic Radiation with Turbulence Authors: Goldreich, Peter; Kumar, Pawan Bibcode: 1988ApJ...326..462G Altcode: The authors derive expressions for the spectral emissivity and absorptivity of acoustic radiation by low Mach number turbulent fluids. They consider three types of turbulence. The first is free turbulence, that is, turbulence which is not subject to external forces. The second and third examples are special cases of forced turbulence, turbulence maintained by stirring with spoons and turbulent pseudoconvection. The resulting formulae are used to estimate the equilibrium energies and quality factors of the acoustic modes in a box which contains turbulent fluid. The scattering of acoustic radiation by the turbulent velocity and pressure fluctuations is treated and the rate at which nonlinear interactions transfer energy among the acoustic modes is evaluated. This work is a first step in the attempt to relate the excitation of the Sun's acoustic modes to the turbulence in the solar convection zone. Title: Waves and normal modes in spiral galaxies. Authors: Goldreich, Peter Bibcode: 1988oseg.proc..127G Altcode: This lecture is an elementary introduction to some properties of linear waves in rotating fluids, with special attention paid to spiral waves in disk galaxies. To keep the lecture simple and self-contained, the author quotes results obtained for gaseous disks with the aid of the WKBJ approximation, wherever reasonable. Also, he makes many other oversimplifying assumptions. He also restricts attention to rotation laws for which Ω(r) > 0 and dΩ(r)/dr < 0. Title: The Interaction of Acoustic Radiation with Turbulence Authors: Goldreich, Peter Bibcode: 1988amfm.conf..399G Altcode: No abstract at ADS Title: Physics of modes in a differentially rotating system - Analysis of the shearing sheet Authors: Narayan, Ramesh; Goldreich, Peter; Goodman, Jeremy Bibcode: 1987MNRAS.228....1N Altcode: A model compressible two-dimensional fluid system with constant density, constant shear, and Coriolis force is investigated to study the linear nonvortical modes of the shearing sheet. The analysis leads to an eigenvalue problem based on the Parabolic Cylinder differential equation. Each mode has an associated corotation radius, and a particular conserved action is identified that is positive for fluid on one side of corotation and negative on the other side. In the shearing sheet, no instability is possible unless there is positive feedback introduced into the amplifier at the boundaries. When tunneling is small, most of the modes are neutral, while when tunneling is large, both neutral and unstable modes are common. Slight perturbation of the equilibrium shearing sheet by the introduction of density and/or velocity perturbations results in a corotation resonance. Title: Self-Similar Quadrupole Perturbations in an Expanding Universe Authors: Blaes, O. M.; Villumsen, J. V.; Goldreich, P. Bibcode: 1987BAAS...19Q1115B Altcode: No abstract at ADS Title: The stability of accretion tori. II - Non-linear evolution to discrete planets Authors: Goodman, J.; Narayan, R.; Goldreich, P. Bibcode: 1987MNRAS.225..695G Altcode: Hawley has shown through two-dimensional computer simulations that a slender torus in which a linear Papaloizou-Pringle (PP) instability with azimuthal wavenumber m, is excited evolves non-linearly to a configuration with m nearly disconnected "planets". The authors present an analytical fluid equilibrium that they believe represents his numerical planets. They analyse the linear modes of the analytical planet and find that there are numerous instabilities, though they are not as violent as the PP instability in the torus. The authors also discuss the energy and vorticity of neutral modes, and they argue that when the torus breaks up into planets, neutral modes with negative energy and non-zero vorticity are excited in order to conserve total energy and specific vorticity. The authors speculate that the fluid in Hawley's simulations may be approaching two-dimensional turbulence. Title: Shepherding of the Uranian Rings. II. Dynamics Authors: Goldreich, Peter; Porco, Carolyn C. Bibcode: 1987AJ.....93..730G Altcode: We explore the dynamical significance of the orbital resonances, identified in Paper I (Porco and Goldreich 1987), involving the satellites l986U7 and 1986U8, and the ɛ, δ, and γ rings. We demonstrate that these satellites are capable of exerting torques on the ɛ ring which supply at the inner edge, and remove at the outer edge, the unperturbed angular-momentum luminosity transported by viscous stresses outward across the ring, provided that the ring has a mass which is comparable to the predicted value Mɛ = 6.1 x 1018 g, and that it is not more than a few meters thick. Thus there is no compelling reason to question the applicability of the standard theory of shepherding to the ɛ ring. However, the standard theory does place rather stringent requirements on the ring's properties, suggesting that confinement might be due to the reduction of the angular-momentum luminosity by flux reversal. Flux reversal could be associated either with the ring's eccentricity gradient or with nonlinear density waves in its interior. The drag due to the planet's extended neutral hydrogen atmosphere probably has only a minor effect on the dynamics of the ɛ, δ, and γ rings. However, it poses a severe problem for the shepherding of the α and β rings unless their masses have been seriously underestimated. This problem, and the large s-band optical depths of these rings, lead us to question the proposal that self-gravity is responsible for enforcing rigid precession in narrow rings. Title: Shepherding of the Uranian Rings. I. Kinematics Authors: Porco, C. C.; Goldreich, P. Bibcode: 1987AJ.....93..724P Altcode: The possible kinematical relationships between the Uranian rings and all ten newly discovered satellites of Uranus are examined. Observational evidence is presented for interactions between 1986U7 and 1986U8 and the epsilon, delta, and gamma rings. The likelihood that these two satellites are shepherding the epsilon ring and are responsible for some of its internal structure is assessed, and the results are used to propose a reduction in the ring radius scale. It is also proposed that 1986U7 and 1986U8 are the inner and outer shepherds for the epsilon ring, that 1986U7 is the outer shepherd for the delta ring, and that 1986U8 is an outer shepherd for the gamma ring. The results of theoretical investigations of planetary ring dynamics are then applied to these associations, and their dynamical significance is evaluated. The resonant torques which the satellites exert on each ring are calculated and compared to the viscous torque which transports angular momentum outward through the ring and to the atmospheric-drag torque which causes the ring's orbit to decay. Title: Nonlinear density waves in planetary rings Authors: Borderies, Nicole; Goldreich, Peter; Tremaine, Scott Bibcode: 1986Icar...68..522B Altcode: We discuss the steady-state structure of the nonlinear density waves generated in a planetary ring at the Lindblad resonances of a satellite. We show that strong density waves lead to an enhancement of the background surface density in the wave zone. Title: The Interaction of Particles and Bending Waves in Disks of Finite Thickness Authors: Goldreich, P.; Chakrabarti, S. K. Bibcode: 1986BAAS...18Q1017G Altcode: No abstract at ADS Title: Excitation and Damping of Solar p-modes Authors: Kumar, P.; Goldreich, P. Bibcode: 1986BAAS...18.1011K Altcode: No abstract at ADS Title: Research in planetary astronomy at Palomar Observatory Authors: Soifer, B. T.; Goldreich, P. Bibcode: 1986plas.rept...60S Altcode: The goals of this program are three fold. The ring systems in the outer solar system were studied. New comets and asteroids in the solar system were surveyed. The volatile surface coverings of the primitive bodies in the outer solar system were studied via infrared spectroscopy. Title: Towards a theory for Neptune's arc rings Authors: Goldreich, P.; Tremaine, S.; Borderies, N. Bibcode: 1986AJ.....92..490G Altcode: It is proposed that the incomplete rings of Neptune consist of a number of short arcs centered on the corotation resonances of a single satellite. The satellite must have a radius of the order of 100 km or more and move on an inclined orbit. Corotation resonances are located at potential maxima. Thus, mechanical energy dissipated by interparticle collisions must be continually replenished to prevent the arcs from spreading. It is shown that each corotation resonance is associated with a nearby Lindblad resonance, which excites the ring particles' orbital eccentricity, thus supplying the energy required to maintain the arc. The ultimate energy reservoir is the satellite's orbital energy. Therefore, interaction with the arcs damps the satellite's orbital inclination. The self-gravity of the arcs limits their contraction and enforces a relation between arc length and mass. The estimated arc masses are so small, of the order of 10 to the 16th g, that the satellite's orbital inclination suffers negligible decay over the age of the solar system. The inferred surface mass densities are comparable to those found in the major rings of Saturn and Uranus. Title: The stability of accretion tori. I - Long-wavelength modes of slender tori Authors: Goldreich, P.; Goodman, J.; Narayan, R. Bibcode: 1986MNRAS.221..339G Altcode: The principal branch of the dispersion relation of a slender torus is considered, and it is shown that the long-wavelength unstable modes of the two-dimensionally incompressible ribbon are coupled edge waves. Height-averaged fluid equations are used to estimate the growth rate as a function of q, n, and beta, and the governing equation for linear modes is reduced from a partial to an ordinary differential equation. Two-dimensional modes when n = 1/2 are solved, and the underlying physical mechanism of the instability modes of Papaloizou and Pringle (1985) is identified. The presence of corotation resonance is reflected by a singularity in the differential equation of the height-integrated system whenever there is a nonzero gradient of vorticity per unit surface density, and regions of parameter space are identified where both principal modes decay. For the special case q = 2, n = 0, the system is found to be unstable. Title: An Explanation for Neptune's Arc Rings Authors: Borderies, N.; Goldreich, P.; Tremaine, S. D. Bibcode: 1986BAAS...18R.778B Altcode: No abstract at ADS Title: Dirk Brouwer Memorial Lecture "Planetary Rings" Authors: Goldreich, Peter Bibcode: 1986BAAS...18R.838G Altcode: No abstract at ADS Title: New Ideas on the Confinement of Narrow Rings Authors: Borderies, N.; Goldreich, P.; Tremaine, S. Bibcode: 1986ESASP.242...95B Altcode: 1986sbos.conf...95B The confinement of narrow rings by shepherd satellite has been predicted by Goldreich and Tremaine in 1979. Detailed studies of how the confinement is achieved revealed new important phenomena. They are presented in this paper. Title: Initial Results on Ring/Satellite Dynamics from the Voyager Uranus Encounter Authors: Porco, C. C.; Goldreich, P. Bibcode: 1986BAAS...18..769P Altcode: No abstract at ADS Title: Evolution of the Janus-Epimetheus coorbital resonance due to torques from Saturn's ring Authors: Lissauer, J. J.; Goldreich, P.; Tremaine, S. Bibcode: 1985Icar...64..425L Altcode: We analyze the interactions between Saturn's coorbital satellites, Janus and Epimetheus, and the outer edge of the A ring, which is presumably maintained by these moons at their 7:6 resonance. Using two distinct but conceptually related methods, we show that ring torques are driving these satellites into a tighter lock. Unless there is a counterbalancing force which we have neglected, their orbital configuration will evolve from the current horseshoe-type lock to one of tadpole orbits around a single Lagrange point in ∼20 myr. This finding adds an additional member to the list of short time scale problems associated with the interactions between Saturn's rings and its inner moons Title: A granular flow model for dense planetary rings Authors: Borderies, N.; Goldreich, P.; Tremaine, S. Bibcode: 1985Icar...63..406B Altcode: We study the viscosity of a differentially rotating particle disk in the limiting case where the particles are densely packed and their collective behavior resembles that of a liquid. The pressure tensor is derived from the equations of hydrodynamics and from a simple kinetic model of collisions described by Haff (1983). We find that density waves and narrow circular rings are unstable if the liquid approximation applies. The resulting development of nonlinear perturbations may give rise to "splashing" of the ring material in the vertical direction. These results may help in understanding the origin of the ellipticities of ringlets, the nonaxisymmetric features near the outer edge of the Saturnian B ring, and the unexplained residuals in kinematic models of the Saturnian and Uranian rings. Title: Concluding Remarks Authors: Goldreich, P. Bibcode: 1985ASSL..117..313G Altcode: 1985mlrg.proc..313G No abstract at ADS Title: Non-axisymmetric instability in thin discs Authors: Goldreich, P.; Narayan, R. Bibcode: 1985MNRAS.213P...7G Altcode: Theoretical evidence is presented for the inevitability of the appearance of instability in dynamically rotating thin disks. The disk is treated as a parallel shear flow of a thin, compressible, uniform density gas sheet with a constant velocity gradient. A parabolic-cylinder differential equation is used to express the perturbations, which include corotation and Lindblad resonances. Eigenmodes are found to arise in forbidden regions between the resonant turning points. A maximum growth rate is characterized for the eigenmodes (WKB modes). A reflecting boundary is determined necessary for formation of a self-sustained oscillation, and will appear if the density at the inner or outer edge of the disk cuts off on a scale shorter than the radial eigenmode wavelength. Title: Erratum: "Saturn's nonaxisymmetric ring edges at 1.95 Rs and 2.27 Rs" [Icarus, Vol. 60, No. 1, p. 17 - 28 (1984)]. Authors: Porco, C.; Danielson, G. E.; Goldreich, P.; Holberg, J. B.; Lane, A. L. Bibcode: 1985Icar...61..173P Altcode: No abstract at ADS Title: The eccentric Saturnian ringlets at 1.29 Rs and 1.45 Rs Authors: Porco, C.; Nicholson, P. D.; Borderies, N.; Danielson, G. E.; Goldreich, P.; Holberg, J. B.; Lane, A. L. Bibcode: 1984Icar...60....1P Altcode: The shapes and kinematics of the two major eccentric ringlets in Saturn's C ring are studied in data acquired by four Voyager experiments: imaging science (ISS), radio science (RSS), ultraviolet spectrometer (UVS), and photopolarimeter (PPS). It is found that the ringlets have mean widths of ∼25 km (Titan, 1.29 Rs) and ∼64 km (Maxwell, 1.45 Rs), eccentricities of order 10 -4, sharp edges on a scale of ∼1 km, normal optical depths τ ∼ 1-2, and are embedded in essentially empty gaps ( τ < 0.05). In addition, they exhibit positive linear width-radius relations, suggesting that differential precession across the ringlets is being prevented by the self-gravity of the ring particles. The kinematics of the Maxwell ringlet are determined solely by Saturn's nonspherical gravity field; the kinematics of the Titan ringlet are apparently determined by its interaction with Titan. Masses, mean surface mass densities, and mass extinction coefficients have been calculated. The comparatively large optical depths and mass extinction coefficients in these features suggest an environment and particle size distribution different from the remainder of the C ring and presumably caused by the mechanism responsible for ring confinement. Title: Saturn's nonaxisymmetric ring edges at 1.95 Rs and 2.27 Rs Authors: Porco, C.; Danielson, G. E.; Goldreich, P.; Holberg, J. B.; Lane, A. L. Bibcode: 1984Icar...60...17P Altcode: The outer edges of Saturn's A and B rings, at 2.27 Rs and 1.95 Rs, have been examined using data acquired by four Voyager experiments. The shapes and kinematics of these features are influenced by their proximity to strong low-order Lindblad resonances. The data for the A-ring edge are consistent with a seven-loded radial distortion of amplitude 6.7 ± 1.5 km which rotates with the mass-weighted mean angular velocity of the coorbital satellite system. The B-ring edge has essentially a double-lobed figure of radial amplitude 74 ± 9 km which rotates with the mean motion of Mimas, though there is an indication that it is not completely described withe a simple Saturn-centered ellipse. An upper limit of 10 m has been placed on the vertical thickness in the unperturbed region of the B ring. Title: Excitation of inclinations in ring-satellite systems Authors: Borderies, N.; Goldreich, P.; Tremaine, S. Bibcode: 1984ApJ...284..429B Altcode: Resonant gravitational interactions between a ring and a satellite produce secular variations of their orbital inclinations. Interactions at vertical resonances, analogous to Lindblad resonances but involving inclinations instead of eccentricities, excite inclinations. There is no inclination analog of the corotation resonance. An equatorial ring changes the inclination of a nearby satellite in qualitatively the same way that a satellite in an equatorial orbit changes the inclination of a nearby ring. Viscous dissipation in a ring leads to an equilibrium value of its inclination. These results provide a basis for discussing the origins of the inclinations of planetary rings. Title: Self-similar gravitational collapse in an expanding universe Authors: Fillmore, J. A.; Goldreich, P. Bibcode: 1984ApJ...281....1F Altcode: The authors derive similarity solutions which describe the collapse of cold, collisionless matter in a perturbed Einstein-de Sitter universe. They obtain three classes of solutions, one each with planar, cylindrical, and spherical symmetry. The solutions can be computed to arbitrary accuracy, and they follow the development of structure in both the linear and nonlinear regimes. Title: Erratum - the Dynamics of Elliptical Rings Authors: Borderies, N.; Goldreich, P.; Tremaine, S. Bibcode: 1984AJ.....89..727B Altcode: No abstract at ADS Title: The Shepherding of Narrow Rings Revisited Authors: Borderies, N.; Goldreich, P.; Tremaine, S. Bibcode: 1984BAAS...16Q.676B Altcode: No abstract at ADS Title: Self-similar spherical voids in an expanding universe Authors: Fillmore, J. A.; Goldreich, P. Bibcode: 1984ApJ...281....9F Altcode: The authors derive similarity solutions which describe the evolution of spherically symmetric voids in a perturbed Einstein-de Sitter universe filled with cold, collisionless matter. The character of a solution depends upon the profile of the initial density deficit. Gradual perturbations give rise to holes within which the density rises smoothly to the background value. Steep perturbations result in voids bounded by overdense shells with sharp edges, i.e. collisionless gravitational shocks. Title: A Simple Derivation of Capture Probabilities for the J+1:J and J+2:J Orbit-Orbit Resonance Problems Authors: Borderies, N.; Goldreich, P. Bibcode: 1984CeMec..32..127B Altcode: We present a simplified analytic derivation of the capture probabilities for the j+1∶j and j+2∶j orbital resonances. We apply Henrard's method which is based on an extension of the theory of adiabatic invariants and recover the results originally obtained by Yoder. Title: Perturbed Particle Disks Authors: Borderies, N.; Goldreich, P.; Tremaine, S. Bibcode: 1984plri.coll..339B Altcode: 1984prin.conf..339B; 1984IAUCo..75..339B No abstract at ADS Title: Spoke Electrodynamics Authors: Acterberg, A.; Blandford, R.; Goldreich, P. Bibcode: 1984plri.coll..549A Altcode: 1984prin.conf..549A; 1984IAUCo..75..549A No abstract at ADS Title: Sharp Edges of Planetary Rings Authors: Borderies, N.; Goldreich, P.; Tremaine, S. Bibcode: 1984plri.coll..345B Altcode: 1984prin.conf..345B; 1984IAUCo..75..345B The ring systems of Saturn and Uranus exhibit a number of sharp edges across which the optical depth drops from order unity to near zero. At least two and perhaps all of these features are associated with the location of orbital resonances between a satellite and a ring particle. The authors illustrate with qualitative argument the results of a numerical simulation concerning the outer edge of the B ring of Saturn. Title: Solved and Unsolved Problems in Planetary Ring Dynamics Authors: Borderies, N.; Goldreich, P.; Tremaine, S. Bibcode: 1984plri.coll..327B Altcode: 1984prin.conf..327B; 1984IAUCo..75..327B The authors discuss a number of new ideas related to planetary ring dynamics and apply them to Saturn's rings. For simplicity, the ring particles are taken to be identical, smooth, spheres composed of dense and chemically pure ice. Title: The Eccentric Ringlet at 1.29 RS Authors: Porco, C.; Borderies, N.; Danielson, G. E.; Goldreich, P.; Holberg, J. B.; Lane, A. L.; Nicholson, P. D. Bibcode: 1984plri.coll..259P Altcode: 1984IAUCo..75..259P; 1984prin.conf..259P The authors present preliminary results of an examination of the eccentric ringlet near 1.29 Rs in Saturn's inner C ring. Situated near the Titan 1:0 apsidal resonance, this feature provides an opportunity to study the behavior of a ring under the influence of both the gravity field of a major planet and the potential of an external satellite. Title: Dynamics of Eccentric and Inclined Ringlets Authors: Borderies, N.; Goldreich, P.; Tremaine, S. Bibcode: 1984plri.coll..341B Altcode: 1984IAUCo..75..341B; 1984prin.conf..341B No abstract at ADS Title: Unsolved problems in planetary ring dynamics Authors: Borderies, N.; Goldreich, P.; Tremaine, S. Bibcode: 1984prin.conf..713B Altcode: Voyager photographic data are used for microscopic and macroscopic analyses of the dynamics of the Saturn rings. Attention is given to particle collisions, local velocity dispersion, the particle size distribution, mechanisms governing the ring structure and interactive effects with Saturn satellite orbits. The mechanical properties and collision dynamics of the particles are discussed, along with models for perturbed and unperturbed disks. The forces involved and true scale of particle erosion are considered, as are instabilities in the rings, satellite torques, resonances, and the time scale for the evolution of the satellite orbits. Finally, the possibility that angular momentum exchanges are occurring between individual rings and satellite orbits is examined. Title: The dynamics of elliptical rings. Authors: Borderies, N.; Goldreich, P.; Tremaine, S. Bibcode: 1983AJ.....88.1560B Altcode: The paper investigates the evolution of eccentric rings under the influence of (1) differential precession due to the planetary quadrupole moment; (2) self-gravity; (3) viscous forces due to interparticle collisions; and (4) eccentricity excitation by shepherd satellites. The principal conclusions are that: (1) uniform precession can be enforced by self-gravity; the resulting configuration is both dynamically and secularly stable. (2) due to viscous forces the line of apsides at the inner ring edge is not exactly aligned with the line of apsides at the outer edge; the apse shift may be detectable in the alpha and beta rings of Uranus; (3) the mean eccentricity is determined by a balance between viscous damping and excitation by shepherds; (4) and the dimensionless eccentricity gradient is expected to be positive and of order unity in most eccentric rings, as observed. Title: Eccentric Ringlet in the Maxwell Gap at 1.45 Saturn Radii: Multi-Instrument Voyager Observations Authors: Esposito, L. W.; Borderies, N.; Goldreich, P.; Cuzzi, J. N.; Holberg, J. B.; Lane, A. L.; Pomphrey, R. B.; Terrile, R. J.; Lissauer, J. J.; Marouf, E. A.; Tyler, G. L. Bibcode: 1983Sci...222...57E Altcode: The Voyager spacecraft observed a narrow, eccentric ringlet in the Maxwell gap (1.45 Saturn radii) in Saturn's rings. Intercomparison of the Voyager imaging, photopolarimeter, ultraviolet spectrometer, and radio science observations yields results not available from individual observations. The width of the ringlet varies from about 30 to about 100 kilometers, its edges are sharp on a radial scale < 1 kilometer, and its opacity exhibits a double peak near the center. The shape and width of the ringlet are consistent with a set of uniformly precessing, confocal ellipses with foci at Saturn's center of mass. The ringlet precesses as a unit at a rate consistent with the known dynamical oblateness of Saturn; the lack of differential precession across the ringlet yields a ringlet mass of about 5 × 1018 grams. The ratio of surface mass density to particle cross-sectional area is about five times smaller than values obtained elsewhere in the Saturn ring system, indicating a relatively larger fraction of small particles. Also, comparison of the measured transmission of the ringlet at radio, visible, and ultraviolet wavelengths indicates that about half of the total extinction is due to particles smaller than 1 centimeter in radius, in contrast even with nearby regions of the C ring. However, the color and brightness of the ringlet material are not measurably different from those of nearby C ring particles. We find this ringlet is similar to several of the rings of Uranus. Title: Erratum - Precession of Inclined Rings Authors: Borderies, N.; Goldreich, P.; Tremaine, S. Bibcode: 1983AJ.....88.1074B Altcode: No abstract at ADS Title: Magnetic focusing in the Sco X-1 radio source Authors: Achterberg, A.; Blandford, R. D.; Goldreich, P. Bibcode: 1983Natur.304..607A Altcode: There has been much theoretical discussion of the confinement of the radio jets associated with extragalactic radio sources1. There are now several examples of sources where the minimum pressure inferred in the jets appears to exceed the external gas pressure, which suggests that magnetic pinching may be playing an important part in the confinement2-4. This issue has been highlighted by recent remarkable observations, using the Very Large Array, of the radio lobes associated with the galactic X-ray source Sco X-1 by Fomalont et al.5. We argue here that these results necessitate magnetic focusing and that this also strengthens the case for magnetic focusing in the extragalactic sources. We show that a sufficient overpressure with respect to the ambient interstellar medium can be achieved if the radius of the jet is reduced by a factor 10-100, and the converging flow becomes dissipative. The radio lobes which form at that point quickly die out again due to electron expansion losses in the now rapidly diverging jet. On the basis of this model it is predicted that the Faraday rotation changes sign across the radio lobes, and that the shape of the lobes should be conical with the apex pointing towards Sco X-1. Title: Perturbed particle disks Authors: Borderies, N.; Goldreich, P.; Tremaine, S. Bibcode: 1983Icar...55..124B Altcode: The Boltzmann moment equations are solved to determine the velocity ellipsoid in a particle disk near an isolated satellite resonance. In a coordinate frame which rotates with the pattern speed of the perturbation potential, the solutions are stationary functions of the azimuthal angle. From the velocity ellipsoid we obtain the stress tensor due to particle collisions and consequently, the viscous angular momentum flux. We show that the magnitude of the rate of deformation tensor in a perturbed particle disk is bounded from above by KΩ(1 + τ 2) 1/2, where Ω is the orbital angular velocity, τ is the optical depth, and K is a dimensionless constant of order unity. It is also found that in sufficiently perturbed regions there are ranges of azimuthal angle within which the radial component of the angular momentum flux is negative. It is even possible for the angular momentum luminosity, the radial flux integrated over azimuth, to be negative. These results are important for understanding sharp edges and the decay of density waves in planetary rings. They are also relevant to the damping of differential precession and eccentricity in narrow ringlets. Title: Precession of inclined rings Authors: Borderies, N.; Goldreich, P.; Tremaine, S. Bibcode: 1983AJ.....88..226B Altcode: It is proposed that node alignment across an inclined ring is maintained by the self-gravity of the ring, which cancels the tendency for differential node precession due to the planet's multiple moments. The analysis is confined to circular rings, which is a self-consistent approximation because the interactions among circular and inclined ringlets do not generate secular perturbations of eccentricity. Title: The variations in eccentricity and apse precession rate of a narrow ring perturbed by a close satellite Authors: Borderies, N.; Goldreich, P.; Tremaine, S. Bibcode: 1983Icar...53...84B Altcode: We derive a Hamiltonian which describes the first-order perturbations of orbital eccentricity and apse precession rate of a narrow ring due to a close satellite whose orbit is also eccentric. Our treatment covers cases in which the satellite crosses the ring. The level curves of the Hamiltonian are displayed for several values of the parameters. We apply our results to the interaction of Saturn's F ring with its inner shepherd satellite. Title: Sharp edges of planetary rings Authors: Borderies, N.; Goldreich, P.; Tremaine, S. Bibcode: 1982Natur.299..209B Altcode: The ring systems of Saturn and Uranus exhibit several sharp edges, across which the optical depth drops from order unity to essentially zero. At least two and perhaps all of these features are associated with the location of orbital resonances between a satellite and the ring particles. It is remarkable that the optical depth varies on a distance scale which is much finer than that over which angular momentum can be transferred between a satellite and the ring material. The important features of this phenomenon are: (1) a perturbed band of width Delta a/a of about (satellite mass/planetary mass) to the 1/2 power adjacent to the edge within which the angular momentum transfer occurs, (2) streamlines perturbed such that the angular momentum luminosity decreases smoothly across the band to zero at the edge even though the optical depth remains constant, and (3) dynamical equilibrium requires a relation between the random velocity, the rate of deformation and the optical depth. Title: Radial widths, optical depths, and eccentricities of the Uranian rings Authors: Nicholson, P. D.; Matthews, K.; Goldreich, P. Bibcode: 1982AJ.....87..433N Altcode: Observations of the stellar occultation by the Uranian rings of 15/16 August 1980 are used to estimate radial widths and normal optical depths for segments of rings 6, 5, 4, alpha, beta, eta, gamma, and delta. Synthetic occultation profiles are generated to match the observed light curves. A review of published data confirms the existence of width-radius relations for rings alpha and beta, and indicates that the optical depths of these two rings vary inversely with their radial widths. Masses are obtained for rings alpha and beta, on the assumption that differential precession is prevented by their self-gravity. A quantitative comparison of seven epsilon-ring occultation profiles obtained over a period of 3.4 yr reveals a consistent structure, which may reflect the presence of unresolved gaps and subrings. Title: Linear polarization of radio frequency lines in molecular clouds and circumstellar envelopes Authors: Goldreich, P.; Kylafis, N. D. Bibcode: 1982ApJ...253..606G Altcode: It is predicted that interstellar lines possess a few percent linear polarization provided that the optical depth in the source region is both anisotropic and of order unity and the radiative rates are at least comparable to the collision rates. These conditions are expected to be met in many sources which emit radio and far-infrared line radiation. Under circumstances in which the Zeeman splitting exceeds both the radiative and collisional rates the linear polarization is aligned either parallel or perpendicular to the projection of the magnetic field on the plane of the sky. This 'strong magnetic field' limit is expected to apply to all radio frequency lines and to many of those far infrared lines which form between levels whose magnetic moments are comparable to the Bohr magneton. The 'weak magnetic field' limit is relevant to most far-infrared lines formed between levels with magnetic moments of order the nuclear magneton. In this limit the polarization direction is determined by the orientation of the propagation direction with respect to the anisotropic optical depth. Title: The dynamics of planetary rings Authors: Goldreich, P.; Tremaine, S. Bibcode: 1982ARA&A..20..249G Altcode: The physical processes that occur in planetary rings are discussed. The theoretical arguments leading to the conclusion that Saturn's rings are solid particles in nonuniform rotation are summarized, and the optical depth, thickness, and particle size of the rings are discussed. The influence of nearby satellites on the rings is analyzed, and asymmetries in the rings are briefly discussed. What is known of the rings of Uranus and of Jupiter's ring is summarized. Some of the dynamical processes and influences that are expected to be incorporated in more advanced theories of planetary rings are reviewed in detail, including radiation drag, plasma drag, interparticle collisions and viscosity, resonances with external satellites, shepherd satellites and moonlets. Finally, the orbital evolution of the shepherd satellites caused by the rings is estimated. Title: Resonances and rings in the solar system Authors: Goldreich, P. Bibcode: 1982coas.conf..121G Altcode: Observations of the structure and dynamics of planetary ring systems are examined for the effects of satellite resonances. The evidence that indicates the presence of solar system resonances includes the preferences for low order commensurabilities in the orbital periods of Jovian and Saturn satellites, the resonant state of Mercury's spin, and the Kirkwood gaps in the number density versus semi-major axis distribution of the asteroids. The influences of collisions and resonance torques in ring systems are discussed. The morphology of planetary rings is determined by satellites exerting torques on the ring material at positions of low order resonances. The actions of Mimas on the Cassini division in Saturn's rings is described, and the existence of small, undiscovered satellites within the ring is predicted. Title: The dynamics of planetary rings Authors: Goldreich, P. Bibcode: 1982asme.proc...41G Altcode: Recent ground observations, and high-resolution photographs and occultation profiles derived from Voyager radio, photopolarimeter, and ultraviolet spectrometer instruments, have provided information about the dynamics of planetary ring systems. Saturn's rings are found to contain particles ranging in size from centimeters to tens of meters and are thought to be composed primarily of ice. Groundbased occulation profiles have revealed nine rings with an eccentric orbit. Circular velocities of Saturnian ring particles were measured at 20 km/s, and random velocities on the order of 0.1 to 1mm/s. The orbital resonances of satellites were found to drive density waves in Saturn's and to account for the observed abrupt ring boundraries. It is proposed that a pair of 'shepherding satellites' accompanies each ring of one side, and that gravitational forces between the satellites keep ring material in an equilibrium orbit. The effect of the rings' own self-gravity is shown to cancel the differential precession observed in the rings. Explanations are offered as to why Saturnian rings reveal structure down to 300 m, but it is conceded that more work is necessary. Title: The Uranus occultation of 10 June 1979. I - The rings Authors: Nicholson, P. D.; Matthews, K.; Goldreich, P. Bibcode: 1981AJ.....86..596N Altcode: Observations and analysis of a stellar occultation by the rings of Uranus on 10 June 1979 are presented. Occultations by rings 4, α, β, γ, δ, and ɛ are identified, and radii and azimuths of the occulting segments in the plane of the rings calculated. Results for rings γ and δ are consistent with the hypothesis (Elliot et al. 1978; Nicholson et al. 1978) that these two rings are circular and coplanar, and an approximate upper limit of 8 × 10-5 is placed on the eccentricity of either ring. Coplanar elliptical models are presented for rings α and β with eccentricities of (6.0±0.3) × l0-4 and (4.9±0.5) × 10-4, respectively. For ring 4 two possible elliptical models are obtained, with eccentricities of (1.2±0.4) × 10-3 and (6.0±0.3) × 10-4, the former being preferred. The width-radius relation established previously for the E ring is confirmed, and the elliptical model for this ring is slightly revised. An improved estimate for Uranus's J2 of (3.390±0.005) × l0-4, based on the apsidal precession of the ɛ ring, and an upper limit for |J4| of ∼1 × 10-4, based on the precession of rings 4 and β are obtained. Title: INVITED TALK - Planetary Rings Authors: Goldreich, Peter Bibcode: 1981BAAS...13..525G Altcode: No abstract at ADS Title: The origin of the eccentricities of the rings of Uranus Authors: Goldreich, P.; Tremaine, S. Bibcode: 1981ApJ...243.1062G Altcode: The effect of gravitational perturbations from a nearby satellite on the eccentricity e of a narrow particulate ring is considered. The perturbations near a resonance in an eccentric ring may be divided into corotation and Lindblad terms. For small e, the corotation terms damp e, whereas the Lindblad terms excite e. In the absence of saturation the corotation terms win by a small margin, and e damps. However, if the perturbations open gaps at the strongest resonances, then the Lindblad terms win, and e grows. This result offers an explanation for the existence of both circular and eccentric rings around Uranus. It is also shown that eccentricity changes induced by circular rings on eccentric satellite orbits are similar to those induced by satellites with circular orbits on eccentric rings. Title: On mapping the magnetic field direction in molecular clouds by polarization measurements Authors: Goldreich, P.; Kylafis, N. D. Bibcode: 1981ApJ...243L..75G Altcode: It is predicted that interstellar radio-frequency lines possess a few percent linear polarization, provided that (1) the radiative transition rate is at least comparable to the collision rate, (2) the optical depth is moderate and anisotropic, and (3) the number of extrema of the velocity component along the line of sight through the source is small. If the Zeeman splitting exceeds both the collisional frequency and the radiative transition rate, then the polarization is aligned either perpendicular to or parallel to the projection of the magnetic field on the plane of the sky. Title: Disk-satellite interactions. Authors: Goldreich, P.; Tremaine, S. Bibcode: 1980ApJ...241..425G Altcode: The rate at which angular momentum and energy are transferred between a disk and a satellite which orbit a central mass is calculated. It is shown that the angular momentum and energy transfer at Lindblad resonances tends to increase the satellite's orbit to lowest order in eccentricity, whereas the transfer at corotation resonances tends to decrease it. The results are applied to the interaction between Jupiter and the protoplanetary disk. The angular momentum transfer is shown to be so rapid that substantial changes in both the structure of the disk and the orbit of Jupiter must have taken place on a time scale of a few thousand years. Title: Theoretical studies of solar oscillations Authors: Goldreich, P. Bibcode: 1980cait.rept.....G Altcode: Possible sources for the excitation of the solar 5 minute oscillations were investigated and a linear non-adiabatic stability code was applied to a preliminary study of the solar g-modes with periods near 160 minutes. Although no definitive conclusions concerning the excitation of these modes were reached, the excitation of the 5 minute oscillations by turbulent stresses in the convection zone remains a viable possibility. Theoretical calculations do not offer much support for the identification of the 160 minute global solar oscillation (reported by several independent observers) as a solar g-mode. A significant advance was made in attempting to reconcile mixing-length theory with the results of the calculations of linearly unstable normal modes. Calculations show that in a convective envelope prepared according to mixing length theory, the only linearly unstable modes are those which correspond to the turbulent eddies which are the basic element of the heuristic mixing length theory. Title: Homologously collapsing stellar cores Authors: Goldreich, P.; Weber, S. V. Bibcode: 1980ApJ...238..991G Altcode: The paper investigates the collapse of nonrotating gas spheres with a polytropic equation of state for n = 3, corresponding to gamma = 4/3. Such polytropes provide a reasonable approximation to collapsing stellar cores during the early phase before nuclear density is reached. A family of exact homologously collapsing configurations is found. Homologous collapse of the entire core is possible if the pressure at a given density is reduced by up to 3% from the value for a marginally stable static core. For a greater pressure reduction, an inner core can collapse homologously, the mass of which varies as the 3/2 power of the reduced pressure at the onset of the collapse. Title: Interpretation of circumstellar masers Authors: Goldreich, P. Bibcode: 1980IAUS...87..551G Altcode: Consideration is given to problems in the theoretical interpretation of masers operating in the circumstellar envelopes about oxygen-rich Mira variables and supergiants. Following a brief review of the dynamics, thermodynamics and chemistry of circumstellar envelopes, current knowledge of the location, directionality, input energy, saturation state and nature of the pumping of circumstellar SiO, H2O and OH masers is indicated. Specific theoretical models of 1612-MHz OH masers, main line OH masers, H2O masers and SiO masers are then considered. Title: The excitation of density waves at the Lindblad and corotation resonances by an external potential. Authors: Goldreich, P.; Tremaine, S. Bibcode: 1979ApJ...233..857G Altcode: The linear response of a differentially rotating two-dimensional gas disk, both with and without self-gravity, to a rigidly rotating external potential is calculated on the assumptions that the speed of sound is much smaller that the orbital velocity and that the external potential varies on the scale of the disk radius. The results show that: (1) the external potential exerts torques on the disk only at the Lindblad and corotation resonances; (2) the torque is positive at the outer Lindblad resonance and negative at the inner Lindblad resonance; (3) the torque at corotation has the sign of the radial vorticity gradient; and (4) the torques are of the same order of magnitude at both types of resonance and independent of the speed of sound in the disk. It is found that the external potential also excites density waves in the vicinity of the Lindblad and corotation resonances, that the long trailing wave is excited at a Lindblad resonance, and that short trailing waves are excited at the corotation resonance. The behavior of particle disks is briefly discussed, and the external torques on particle disks are proven to be identical to those on gas disks Title: Precession of the epsilon ring of Uranus. Authors: Goldreich, P.; Tremaine, S. Bibcode: 1979AJ.....84.1638G Altcode: It is noted that the outer and inner boundaries of the epsilon ring of Uranus can be fitted by aligned Keplerian ellipses. Four possible mechanisms for maintaining uniform precession in the epsilon ring are considered: the ring's self-gravity, precession due to a satellite, smooth pressure gradients, and shocklike phenomena. It is proposed that apse alignment is maintained by the self-gravity of the ring. In this case, a ring mass of approximately 5 x 10 to the 18th g and a mean surface density at quadrature of about 25 g/sq cm are estimated. Title: The rings of Saturn and Uranus. Authors: Goldreich, P.; Tremaine, S. Bibcode: 1979ASIC...47..129G Altcode: No abstract at ADS Title: The Rings of Saturn and Uranus Authors: Goldreich, P. Bibcode: 1979IAUS...81..191G Altcode: Saturn has bright, broad rings separated by narrow gaps, while the rings of Uranus are dark, narrow and widely spaced. Presumably, both sets of rings lie inside the Roche limit, which is why the ring material has not condensed into satellites. This paper briefly reviews what is known about each ring system, with emphasis on properties of significance to dynamical astronomy. Title: The Absorption of Trapped Line Photons by Dust Authors: Goldreich, Peter.; Kwan, John Bibcode: 1979ApJ...227..150G Altcode: We derive the rate at which photons in an optically thick line are absorbed by cold dust. This rate is approximately equal to the dust optical depth to the cloud center times the rate at which the photons escape from the cloud. Our derivation is in response to a recent article by Strel'nitskii in which he incorrectly criticized our previous application of this result to models for the pumping of cosmic masers. Strel'nitskii now agrees that his criticism was unjustified. Subject headings: masers - nebulae: general - radiative transfer Title: Solar System Studies - Jupiter Imaging at 5-MICRONS - Jupiter Magnetosphere - Infrared Observations of Planets - Diameter of Pluto Authors: Kowal, C.; Goldreich, P.; Terrile, R. J.; Westphal, J.; Trauger, J.; Munch, G.; Roesler, F. L.; Soifer, B. T.; Matthews, K.; Neugebauer, G.; Sargent, W. L. W.; Boksenberg, A.; Arnold, S. J. Bibcode: 1979haob.rept..721K Altcode: No abstract at ADS Title: HII Regions in Galaxies - Luminosity Classification - Nonequilibrium Companions of Spiral Galaxies - Dwarf Irregular Galaxies in Nearby Groups Authors: Kennicutt, R.; Sulentic, J. W.; Arp, H. C.; di Tullio, G.; Lo, K. Y.; Sargent, W. L. W.; Sancisi, R.; Goldreich, P.; Kowal, C.; Thuan, T. X. Bibcode: 1979haob.rept..752K Altcode: No abstract at ADS Title: Towards a theory for the Uranian rings Authors: Goldreich, P.; Tremaine, S. Bibcode: 1979Natur.277...97G Altcode: Interparticle collisions, radiation drag, and differential precession all tend to disrupt the rings of Uranus. The first two effects lead to radial spreading which would disrupt a free ring in less than or approximately 100,000,000 yr. It is proposed that the rings are confined in radius by gravitational torques from a series of small satellites that orbit with the ring system. Differential precession tends to destroy the apse alignment of the elliptical epsilon ring. It is suggested that apse alignment is maintained by the self-gravity of the ring. The resulting mass of the epsilon ring is approximately 5 times 10 to the 18th power g. Its radial confinement requires (for example) a pair of satellites of mass approximately 10 to the 19th power g, in circular orbits roughly 500 km away on either side of the ring Title: The rings of Uranus: results of the 10 April 1978 occultation. Authors: Nicholson, P. D.; Persson, S. E.; Matthews, K.; Goldreich, P.; Neugebauer, G. Bibcode: 1978AJ.....83.1240N Altcode: Observations of the April 10, 1978, stellar occultation by the rings of Uranus are presented. Nine rings were observed, and their radii and widths are calculated. Rings eta, gamma, and delta are found to be most likely circular and coplanar, in agreement with previous analyses; the remaining rings are either noncircular or slightly inclined. The width of the epsilon ring is a linear function of its radius from the center of Uranus, projected onto the satellites' orbital plane; this suggests that it forms one continuous noncircular ring. The optical-depth profile of the epsilon ring has not changed significantly since March 1977. A model of this ring which fits all available observations adequately is that of a uniformly precessing Keplerian ellipse coplanar with the satellites' orbits. This model permits predictions of the radius and width of the epsilon ring for future occultations. The precession rate is used to determine J2 for Uranus, on the assumption that precession is caused solely by the planetary oblateness and not by satellite-ring interactions. Title: The excitation and evolution of density waves. Authors: Goldreich, P.; Tremaine, S. Bibcode: 1978ApJ...222..850G Altcode: The behavior of density waves in a simple analog of a rotating disk is analyzed. The model considered is a thin gas sheet whose unperturbed velocity field is a parallel shear flow; the effects of rotation are simulated by introducing a Coriolis acceleration; the model exhibits Lindblad resonances and includes both long- and short-wavelength density waves among its linear perturbations. The linear perturbation equations are set up in a form appropriate to the model and rewritten in terms of the comoving coordinates of the unperturbed flow. The resulting equations are Fourier transformed and then solved in the tight-winding limit. The excitation of a wave packet by an external potential is calculated along with the packet's subsequent propagation, and analytical solutions are obtained which are valid everywhere, including the Lindblad and corotation resonances. It is found that a barlike potential excites long-wavelength trailing density waves at the Lindblad resonances. The amplification of a density wave incident on the forbidden zone surrounding corotation is also examined. Title: The Rings of Uranus (invited). Authors: Goldreich, P. Bibcode: 1978BAAS...10..581G Altcode: No abstract at ADS Title: The formation of the Cassini division in Saturn's rings Authors: Goldreich, P.; Tremaine, S. D. Bibcode: 1978Icar...34..240G Altcode: The satellite Mimas excites a trailing spiral density wave in Saturn's rings at the position of the 2:1 resonance. The density wave carries negative angular momentum and propagates outward. The wave is damped by a combination of nonlinear and viscous effects, and its negative angular momentum is transferred to the ring particles. Consequently, the particles just outside the 2:1 resonance spiral inward, opening a gap. The inner edge of the gap is close to the resonance position in agreement with the location of the inner edge of the Cassini division. Despite its tiny mass, Minas is able to clear a gap as wide as the Cassini division. We estimate the ability of Saturn's satellites to open other gaps in the rings. The upper limit to the width of Encke's division implies that the velocity dispersion of the ring particles is <10 -2 cm sec -1. Title: The velocity dispersion in Saturn's rings Authors: Goldreich, P.; Tremaine, S. D. Bibcode: 1978Icar...34..227G Altcode: The velocity dispersion in a differentially rotating disk of particles such as Saturn's rings is determined by the details of the collision process. Collisions give rise to a viscous stress that converts orbital energy into random motions. Since the collisions are not perfectly elastic, the energy in random motions is dissipated as heat. With increasing velocity dispersion the latter process becomes more important relative to the former because the collisions become less elastic. The velocity dispersion adjusts so that the effects of these two processes balance. The condition for this balance fixes the coefficient of restitution ɛ of the colliding particles as a function of the disk's optical depth τ. We solve the Boltzmann moment equations to determine ɛ(τ). If the rings are about as old as the solar system then their radial width implies that the velocity dispersion of the ring particles is less than 0.2 cm sec -1. The corresponding vertical thickness is then less than 10 m. We discuss the effects of collisions on the particles in Saturn's rings. If the particles are made of ice they are eroded by collisions and accrete the collisional debris. The time scale for erosion and accretion is probably shorter than the age of the solar system. Furthermore, for ice particles ɛ is likely to be substantially less than unity even at impact velocities as low as 10 -3 cm sec -1. Thus, a ring of ice particles would be a monolayer thick. Title: Occultations by Uranian Rings Authors: To, J.; Persson, E.; Nicholson, P.; Matthews, K.; Goldreich, P.; Neugebauer, G.; Elliot, L.; Churms, J.; Marsden, B. G. Bibcode: 1978IAUC.3215....3T Altcode: Further analysis, by the undersigned, of the occultation of SAO 158687 on 1977 Mar. 10 yields the following radii for the rings, assumed to lie in Uranus' equatorial plane (cf. IAUC 3051, 3061): Ring Distance Ring Distance alpha 44 835 km delta 48 408 km beta 45 788 epsilon1 51 697 gamma 47 732 epsilon2 51 030 These results (uncertainty +/- 30 km) are very similar to those by Wasserman et al. (1977, Bull. Am. Astron. Soc. 9, 498). As the result of observations of another predicted occultation on 1977 Dec. 23 (IAUC 3108; Astron. J. 1977, 82, 849, 929, 1036), Millis and Wasserman (1978, Astron. J. to be published) confirm the existence of the gamma, delta and epsilon rings and possibly also of the alpha ring; these measurements, obtained at a wavelength of 0.88 um at Cabezon Observatory, Tenerife, were difficult to make in the dawn sky. E. Persson, Hale Observatories; P. Nicholson, K. Matthews, P. Goldreich and G. Neugebauer, California Institute of Technology, report that successful observations at a wavelength of 2.2 um were made with the 250-cm du Pont telescope at the Las Campanas Observatory on 1978 Apr. 10 of a third occultation by the Uranian rings (only the occultation by Uranus itself could be detected in the case of a fainter star on 1978 Apr. 4). All five 'original' rings were detected, and assuming the above values for the alpha, beta, gamma and delta rings the radii of further rings were determined as follows: Ring Distance Ring Distance kappa1 42 029 km kappa2 42 148 km iota1 42 394 iota2 42 304 theta1 42 660 theta2 42 696 eta1 47 290 eta2 47 289 epsilon1 50 848 epsilon2 51 402 As before (IAUC 3051, 3061), subscripts refer to the western and eastern edges of the rings, respectively. The kappa ring is uncertain. The iota1 and theta1 events are undoubtedly identical with the alpha3 and beta3 events previously described by Millis et al. (IAUC 3051). The eta events were also subsequently reported by L. Elliot (unpublished) in the case of the SAO 158687 occultation. The more recent results for epsilon suggest that this is indeed a complete ring but that its width is relatively substantial, its shape non-circular and its structure complex. The possible zeta ring of J. Churms (IAUC 3051) has not been confirmed, and there appear to be no confirmed rings of any consequence outside epsilon. Title: On the radiative acceleration of quasar absorption line clouds. Authors: Goldreich, P. Bibcode: 1978PhyS...17..225G Altcode: Radiation may expel dense gas clouds from quasar emission line regions. Small clouds can be accelerated to velocities approaching c if they are confined so as to maintain high densities. The kinetic energies, inferred from the spectra, of low to medium excitation quasar absorption clouds are too large for radiation pressure to have played a dominant role in their acceleration. If these clouds are physically associated with the quasars, they are probably accelerated by the same stresses that are responsible for their confinement. Models for absorption clouds based on radiatively driven instabilities in expanding quasar envelopes predict column densities that are much larger than those deduced from observation. Title: Revenge of tiny Miranda Authors: Goldreich, P.; Nicholson, P. Bibcode: 1977Natur.269..783G Altcode: DERMOTT and Gold1 have proposed a resonance model for the rings of Uranus. They assume the rings are, in fact, arcs composed of small particles librating about stable resonances determined by pairs of satellites, either Ariel and Titania or Ariel and Oberon. Dermott and Gold dismiss as insignificant resonances involving tiny Miranda. We report here that, by a wide margin, the strongest resonances are all associated with Miranda. Furthermore, we show that the hypothesis that the rings are made up of librating particles, while original and ingenious, is incorrect. Title: The Azimuthal Asymmetry in the Brightness of Saturn's Rings Authors: Alcock, C.; Goldreich, P. Bibcode: 1977BAAS....9..462A Altcode: No abstract at ADS Title: A Dynamical Explanation for the Azimuthal Brightness Asymmetry of Saturn's A Ring Authors: Colombo, G.; Goldreich, P.; Harris, A. W. Bibcode: 1977BAAS....9Q.462C Altcode: No abstract at ADS Title: Turbulent Viscosity and Jupiter's Tidal Q. Authors: Goldreich, P.; Nicholson, P. D. Bibcode: 1977BAAS....9..474G Altcode: No abstract at ADS Title: Solar seismology. I. The stability of the solar p-modes. Authors: Goldreich, P.; Keeley, D. A. Bibcode: 1977ApJ...211..934G Altcode: The stability of the radial p-modes of the sun is investigated by computing nonadiabatic eigenvalues and eigenfunctions for a solar envelope model which extends from an inner radius of about 0.3 solar radius out to an optical depth of about 0.0003. The calculations take into account in a crude fashion the response of the convective flux to the oscillation. The dynamical effect of turbulence in the convection zone is parametrized in terms of a turbulent shear viscosity. The results show that if damping by turbulent viscosity is neglected, all modes with periods longer than 6 minutes are unstable. The familiar kappa-mechanism, which operates in the H ionization-H(-) opacity region, is the dominant source of driving of the oscillations. Modes with periods shorter than 6 minutes are stabilized by radiative damping in the solar atmosphere. When turbulent dissipation of pulsational energy is included, all modes are predicted to be stable. However, the margin of stability is very small. In view of the large uncertainty that must be assigned to the estimate of turbulent damping, it is concluded that theoretical calculations cannot unequivocally resolve the question of the stability of the solar p-modes. Title: Solar seismology. II. The stochastic excitation of the solar p-modes by turbulent convection. Authors: Goldreich, P.; Keeley, D. A. Bibcode: 1977ApJ...212..243G Altcode: We test the hypothesis that the solar p-modes are stabilized by damping due to turbulent viscosity in the convective zone. Starting from the assumption that the modes are stable, we calculate expectation values for the modal energies. We find that the interaction between a p-mode and the turbulent convection is such that the modal energy tends toward equipartition with the kinetic energy of turbulent eddies whose lifetimes are comparable to the modal period. From the calculated values of the modal energies, we compute rms surface velocity amplitudes. Our predicted rms surface velocities range from 0.01 cm/sec for the fundamental radial mode to 0.6 cm/sec for the radial mode whose period is approximately 5 minutes. The predicted surface velocities for the low order p-modes are much smaller than the velocities inferred from recent observations. Title: Turbulent Viscosity and Jupiter's Tidal Q Authors: Goldreich, P.; Nicholson, P. D. Bibcode: 1977Icar...30..301G Altcode: A recent estimate of tidal dissipation by turbulent viscosity in Jupiter's convective interior predicts that the current value of the planet's tidal Q ∼ 5 × 106. We point out a fundamental error in this calculation, and show that turbulent dissipation alone implies that at present Q ∼ 5 × 1013. Our reduced estimat for the rate of tidal dissipation shows conclusively that tidal torques have produced only negligible modifications of the orbits of the Galilean satellites over the age of the solar system. Title: Solar oscillations Authors: Goldreich, P.; Keeley, D. A. Bibcode: 1977ComAp...7...35G Altcode: The main observational results and related theoretical investigations concerning solar oscillations are reviewed. The normal modes of oscillation of the sun are classified according to their angular eigenvalues and the number of their radial nodes. Observations of excited normal modes are discussed, particularly in relation to five-minute oscillations, nonlinearly coupled unstable g-modes, oscillations of the sun's apparent diameter, and reported oscillations of the solar surface with a period of 2 hr 40 min. Linear stability calculations are briefly examined, and stochastic excitation of p-modes by turbulent convection is considered. The five-minute oscillations are described as the superposition of many excited nonradial p-modes. Title: A perspective of physics. Volume 1. Selections from 1976 Comments on Modern Physics. Authors: Peierls, R.; Okun, L. B.; Zel'dovich, Ya. B.; Lee, B. W.; Quigg, C.; Harari, H.; Jacob, M.; Cline, D.; Mann, A. K.; Lewin Keller, Q.; Negele, J. W.; Stephens, F. S.; Leggett, A. J.; Keyes, R. W.; Tauc, J.; Nagel, S. R.; Lovesey, S. W.; Thompson, A. H.; Paczynski, B.; Goldreich, P.; Sargent, W.; Larson, R. B.; Grindlay, J. E.; Spitzer, L.; Brodsky, S. J.; Karl, G.; Picqué, J. L.; Stroke, H. H.; Dose, V.; Ambartzumian, R. V.; Letokhov, V. S.; Harris, F. M.; Bekefi, G.; Deutsch, C.; Furth, H. P.; Tsytovich, V. N.; Callen, J. D. Bibcode: 1977ppv..book.....P Altcode: No abstract at ADS Title: The evolution of the universe. Authors: Goldreich, P. Bibcode: 1977fost.conf..215G Altcode: No abstract at ADS Title: Spiral structure as an explanation for the asymmetric brightness of Saturn's A ring Authors: Colombo, G.; Goldreich, P.; Harris, A. W. Bibcode: 1976Natur.264..344C Altcode: No abstract at ADS Title: Quasar Absorption Lines Authors: Goldreich, P.; Sargent, W. Bibcode: 1976ComAp...6..133G Altcode: Observations of quasar absorption lines are reviewed, noting that systems of lines in some objects have been convincingly identified with absorption redshifts that are different from the emission redshifts. Evidence for and against line locking between the emission and absorption redshifts in the same object is discussed, and it is suggested that the hypothesis of intervening objects cannot be dismissed on the basis of existing statistical redshift data. Physical properties of absorbing clouds are considered, the role of radiation pressure at the present location of the absorbing clouds is examined, and it is inferred that the clouds should have the form of thin sheets. A direct observational test for the location of the absorbing material is outlined. Title: OH-IR stars. II. A model for the 1612 MHz masers. Authors: Elitzur, M.; Goldreich, P.; Scoville, N. Bibcode: 1976ApJ...205..384E Altcode: A model is presented for the 1612 MHz OH masers associated with infrared stars. Its principal conclusions are as follows. The central stars are losing 3 x l0- M0 yr -1, and the masers operate in the outer regions (r > 1016 cm) of the circumstellar envelopes. The maser radiation is narrowly beamed in the radial direction, both inward and outward. Thus the two maser emission features originate in the near and the far sides of the expanding circumstellar gas. The 1612 MHz maser is powered by the absorption of 35 photons which excite the OH molecules from the 211312, J = 312 ground state to the 211112, J = 512 state. The excited OH molecules return to the ground state by a series of radiative decays. In most cases, the radiative cascade proceeds directly down the 211112 ladder. The final and the most important step in the pump cycle is the radiative decay from the 211112, J = 112 state to the 211312, J = 3/2 state. If the transitions which link these two states are optically thick, a strong inversion of the F = 1 F = 2 1612 MHz transition is produced. Subject headings: infrared: sources - masers - stars: circumstellar shells Title: OH-IR stars. I. Physical properties of circumstellar envelopes. Authors: Goldreich, P.; Scoville, N. Bibcode: 1976ApJ...205..144G Altcode: A theoretical model of the circumstellar envelope which surrounds a OH-IR star is developed. The circumstellar gas is ejected by radiation pressure which acts on dust grains that condense in the atmosphere of the central star. The dust grains transfer momentum to the gas by collisions with the gas molecules. These collisions are the dominant source of heat input to the circumstellar gas. The major sources of cooling are the emission of radiation by H20 molecules and adiabatic expansion. The gas temperature decreases from T 2 x l0 K near the stellar surface at r 6 x 1013 cm, to T 8 x 102 K at r = 1015 cm and to T 102 K at r = 1016 cm. The OH molecule abundance in the circumstellar envelope is controlled by chemical exchange reactions and by the dissociation of H20 molecules. The reaction OH + H2 H20 + H + 0.69 eV, which has an activation energy of 0.3 eV, rapidly converts OH molecules into H20 molecules in the warm (T > 5 x 102 K) inner (r <% 2 x 1015 cm) region of the circumstellar envelope. Beyond r 2 x 1015 cm, T is so low that the exchange reaction is very slow and the mean lifetime of an OH molecule is greater than the expansion time scale for the circumstellar envelope. In the outer region of the circumstellar envelope, OH molecules are produced from the photodissociation of H20 molecules by the interstellar ultraviolet radiation and from the dissociation of H20 molecules by collisions with dust grains. These processes are capable of producing OH number densities greater than 1 at r 1016 cm. The predicted values of the gas temperature, T, and the OH abundance, o , depend upon the rate of mass loss from the central star, (t) The results quoted above are based on a calculation with = 3 x l0- M0 yr-1. In general, T varies inversely and o varies directly with (I). Subject headings: infrared: sources - stars: circumstellar shells Title: Interstellar masers. Authors: Goldreich, P. Bibcode: 1975ampi.proc..409G Altcode: 1975ampi.conf..409G Observations of interstellar OH, H2O, and SiO masers are discussed along with some theoretical topics related to their interpretation. Energy-level diagrams are presented for all three molecules, regions associated with the masers are identified, and estimates are given for the luminosities, angular sizes, spatial arrangements, brightness temperatures, time variations, and polarization of the strongest OH and H2O masers. Physical conditions in the masers are analyzed on the assumptions that the interaction between maser radiation and active molecules can be described by the ordinary equation of radiative transfer and that simple rate equations determine the populations of the maser levels. A simple pump mechanism is outlined for a three-level scalar molecule, and determination of the actual sizes of interstellar masers is considered. Thermodynamic constraints on strong H2O maser pumps are examined along with a model for 1612-MHz OH masers associated with M-type Mira variables and some effects due to the trapping of IR line radiation between the maser levels and other rotational levels. The problem of polarization of the OH maser lines from sources associated with H II regions is discussed. Title: Astrophysical Masers. V. Pump Mechanism for H2O Masers Authors: Goldreich, Peter; Kwan, John Bibcode: 1974ApJ...191...93G Altcode: The problem of determining the pump mechanism for strong H2O masers is considered. It is shown that for every class of pump models, very general constraints may be placed on the maser size and luminosity. A careful evaluation of these constraints indicates that collisional pumps and external radiative pumps are unlikely to account for the strong H2O masers associated with HII regions. It is proposed that these sources are powered by a hot-dust, cool-gas, internal radiative pump. Specifically, the pump mechanism involves the absorption by the H2O molecules of 6.3- photons which are emitted by the hot dust grains. These absorptions produce transitions from the rotational levels of the ground vibrational state to the rotational levels of the first-excited vibrational state. The pump action results because there are more absorptions per degenerate sublevel from the 523 level than from the 616 level. The heat sink in this pump model is provided by the cool H2 molecules which collisionally de-excite the H2O molecules from the first-excited vibrational state to the ground vibrational state. The hot-dust, cool-gas, pump model can operate only in the vicinity of a highly luminous, variable star. The variation in temperature of the dust grains which are heated by the radiation from the star is essential to prevent the dust and gas from coming into thermal equilibrium. One feature of this pump model is that at any point the dust temperature must vary from below to above the gas temperature. Thus, the strong intensity variations observed in H2O masers find a natural explanation in the model. Title: Molecular Clouds Authors: Goldreich, Peter; Kwan, John Bibcode: 1974ApJ...189..441G Altcode: It is proposed that molecular clouds are in a state of gravitational collapse. The coupled equations of statistical equilibrium and radiative transfer from diatomic molecules in a collapsing cloud are solved for arbitrary optical depths in the rotational lines. It is shown that most of the observed CS and SiO lines and the stronger CO lines are optically thick. In this limit the emitted intensities are independent of the molecular dipole moments. The rate at which energy is radiated in the CO lines is found to exceed the rate at which work is done by the adiabatic compression of the collapsing gas. This result implies the existence of an energy source which maintains the temperature of the gas against the cooling due to radiative energy losses. It is suggested that collisions between gas molecules and warm dust grains transfer energy to the gas. The dust grains are heated by radiation from H ii regions and protostars in the center of the molecular cloud. This picture is supported by the detection of copious far infrared fluxes from many molecular clouds. The rate of energy transfer from the dust to the gas is calculated to be sufficient to maintain the gas at temperatures deduced from observations of CO lines if NH2 > 10 . Subject headings: molecules, interstellar - nebulae Title: Astrophysical Masers.IV. Line Widths Authors: Goldreich, Peter; Kwan, John Bibcode: 1974ApJ...190...27G Altcode: The standard theory of maser line widths predicts line narrowing during unsaturated amplification and rebroadening to the full Doppler width during saturated amplification. These predictions conflict with the observations of narrow maser lines from sources which appear to be saturated. in an attempt to resolve this conflict, a new theory of maser line widths is developed which takes into account effects arising from infrared line radiation trapped between the molecular levels. it is shown that if a series of infrared lines connects the upper and lower maser levels, line rebroadening during saturated amplification does not occur provided that each infrared transition satisfies A![exp (hv/kT) - 1] R, where A and v are respectively the spontaneous emission rate and transition frequency of the infrared line, T is the kinetic temperature, and R is the stimulated emission rate across the maser levels. in fact, under these conditions, the maser line continues to narrow during saturated amplification since the gain profile is not affected by saturation. Subject headings: line profiles masers Title: A model for the 1612_MHz masers in OH-IR stars. Authors: Elitzur, M.; Goldreich, P.; Scoville, N. Bibcode: 1974mode.book.....E Altcode: 1974STIN...7520184E Shklovsky (1966) first suggested that OH masers might be pumped by the absorption of infrared radiation. Subsequently, the physics of the pump process was investigated by Litvak (1969) and by Litvak and Dickenson (1972). The present paper attempts to bring into sharper focus the essential features responsible for the inversion of the 1612 MHz transition in OH-IR stars. Title: The Formation of Planetesimals Authors: Goldreich, Peter; Ward, William R. Bibcode: 1973ApJ...183.1051G Altcode: Four stages in the accretion of planetesimals are described. The initial stage is the condensation of dust particles from the gaseous solar nebula as it cools. These dust particles settle into a thin disk which is gravitationally unstable. A first generation of planetesimals, whose radii range up to 10-1 km, form from the dust disk by direct gravitational collapse to solid densities on a time scale of the order of 1 year. The resulting disk, composed of first-generation planetesimals, is still gravitationally unstable, and the planetesimals are grouped into clusters containing approximately 10 members. The contraction of these clusters is controlled by the rate at which gas drag damps their internal rotational and random kinetic energies. On a time scale of a few thousand years, the clusters contract to form a second generation of planetesimals having radii of the order of 5 km. Further coalescence of planetesimals proceeds by direct collisions which seem capable of producing growth at a rate of the order of 15 cm per year at 1 a.u. The final stage of accretion during which planet-sized objects form is not considered here. Subject headings: planets - solar system Title: Astrophysical Masers. 111. Trapped Infrared Lines and Cross-Relaxation Authors: Goldreich, Peter; Keeley, Douglas A.; Kwan, John Y. Bibcode: 1973ApJ...182...55G Altcode: Infrared line radiation trapped between a maser level and other rotational levels produces a rapid relaxation among the degenerate substates of the maser level. The rate of this relaxation is comparable to the spontaneous decay rates of the infrared transitions. This cross-relaxation has important effects on the apparent source sizes and the polarization properties of interstellar masers. It also affects the relative amplification of the components of a hyperfine-split maser line. The effect on apparent source size is pronounced when the cross-relaxation rate y exceeds the decay rate of the maser levels. In this limit, cross-relaxation enables maser radiation directed in a narrow solid angle to saturate the population excess in all magnetic sublevels. This property is essential to the arguments which suggest that the apparent sizes of interstellar OH and H2O masers are much smaller than their physical sizes. Cross-relaxation has an important effect on the polarization of radiation emitted by saturated masers if the relaxation rate y is greater than the stimulated emission rate R. For cases in which the Zeeman splitting g is greater than the maser line width Aw the maser amplifies the Zeeman pattern. In the presence of rapid cross-relaxation (y > R), the rates of amplification of the a- and n-components of the Zeeman pattern are unequal and depend upon the angle between the propagation direction and the magnetic field. For R < g < Aw, the limiting maser polarization is linear. However, cross-relaxation suppresses the growth of linear polarization until and unless the stimulated emission rate becomes as large as the relaxation rate. Subject headings: masers - molecules, interstellar - polarization - Zeeman effect Title: Astrophysical Masers. 11. Polarization Properties Authors: Goldreich, Peter; Keeley, Douglas A.; Kwan, John Y. Bibcode: 1973ApJ...179..111G Altcode: The equations governing the transfer of polarized radiation in astrophysical masers are derived. It is found that the magnetic field and the plasma in maser sources play a central role in determining the polarization of the emitted radiation. The character of the polarization depends upon the relative sizes of the decay constant of the maser levels, F; the stimulated-emission rate R; the Zeeman splitting, g and the bandwidth of the amplified radiation, Aw. Unsaturated masers (R < F) emit unpolarized radiation unless g > Aw. For g > A they amplify the Zeeman pattern if the Faraday rotation per gain length in the source is small. If the Faraday rotation per gain length is large, then the a components of the Zeeman pattern are 100 percent circularly polarized and the n component is unpolarized. Saturated masers (R > F) emit unpolarized radiation unless g > (RF)"2. If the Faraday rotation across the region of saturated amplification is small, the emitted radiation is partially linearly polarized for (RF)"2 < g Aw whereas for g > Aw it is just the amplified Zeeman pattern. If the Faraday rotation across the saturated region is large, all linear polarization is destroyed. For g > Aw, the a components of the Zeeman pattern are again 100 percent circularly polarized. Subject headings: masers - polarization - radiative transfer - Zeeman effect Title: The Case Against Planet X Authors: Goldreich, Peter; Ward, William R. Bibcode: 1972PASP...84..737G Altcode: The dynamical consequences of the hypothetical trans-Plutonian planet suggested by Brady (1972) are considered. It is concluded that the combination of large mass (9 x ) and unusual orbital inclination (120 ) would have two serious effects on the solar system: (a) the angle between the solar axis and the normal to the ecliptic would suffer large variations ( 2ir) with a period of a few times 1O years, and (b) the coplanar configuration of the outer solar system would be disrupted on a time scale of 106 years. The large residuals in the orbit of Halley's comet which prompted the suggestion of a trans-Plutonian planet can be explained in terms of nongravitational forces and the weak orbital binding energy of this object. Key words: comet - Halley's comet - celestial mechanics - trans-Plutonian planet Title: On Parametric Down-Conversion in Astrophysical Masers Authors: Goldreich, Peter; Kwan, John Y. Bibcode: 1972ApJ...176..345G Altcode: The mechanism of parametric down-conversion proposed by Litvak cannot explain the observed preference for circular polarization in OH maser emission because the nonlinear interaction between oppositely circularly polarized microwaves is too weak. Title: Astrophysical Masers. I. Source Size and Saturation Authors: Goldreich, Peter; Keeley, Douglas A. Bibcode: 1972ApJ...174..517G Altcode: The relation between the apparent and actual sizes of maser sources is important in determining their internal physical conditions. This relation is investigated for models of homogeneous maser clouds having spherical and tube-shape geometries. It is found that the apparent linear size of a spherical OH or H2O maser may be as much as two orders of magnitude smaller than the physical extent of the emitting cloud. On the other hand, the observed size of a maser having the shape of a thick tube is given by its transverse dimensions. The treatment of maser radiation uses rate equations for the level populations and the ordinary equation of transfer. Approximate analytic calculations are presented for all degrees of saturation. The theory is developed in terms of a multilevel scalar atom in which only two levels are coupled by the maser radiation. Thus, the effects of hyperfine splitting and polarization are ignored. Title: Tides and the Earth-Moon System Authors: Goldreich, Peter Bibcode: 1972SciAm.226d..42G Altcode: 1972SciAm.226...42G No abstract at ADS Title: Magnetosphere Theory of Pulsar Electrodynamics Authors: Goldreich, P. Bibcode: 1972phpu.conf..151G Altcode: No abstract at ADS Title: Pulsar Theory II. Radiation Mechanisms Authors: Goldreich, P.; Pacini, F.; Rees, M. J. Bibcode: 1972CoASP...4...23G Altcode: 1972ComAp...4...23G No abstract at ADS Title: History of the Lunar Orbit Authors: Goldreich, P. Bibcode: 1972NASSP.300..485G Altcode: 1972poss.conf..485G; 1972pss..conf..485G No abstract at ADS Title: Coherent Synchrotron Radiation Authors: Goldreich, Peter; Keeley, D. A. Bibcode: 1971ApJ...170..463G Altcode: A simple model consisting of a distribution of charges constrained to move on a ring is the basis of an investigation of coherent synchrotron radiation. The radiation produced as a result of a nonrandom particle distribution on the ring is examined from the viewpoint of the interaction of individual particles with the total electric field of the system A linear stability analysis shows that, under reasonable conditions, a uniform distribution of particles is unstable to clumping. The model is applied to pulsars, in which the high brightness temperatures suggest that a cooperative emission mechanism is responsible for the radiofrequency radiation. The application to circular accelerators and storage rings is discussed briefly. Title: Pulsar Theory I. Dynamics and Electrodynamics Authors: Goldreich, P.; Pacini, F.; Rees, M. J. Bibcode: 1971CoASP...3..185G Altcode: 1971ComAp...3..185G No abstract at ADS Title: Charged Particle Motion Near Pulsars Authors: Goldreich, Peter Bibcode: 1971PASP...83..599G Altcode: Key words: pulsar models - charge bunching - synchrotron radiation Title: Cooling effects on interstellar shock waves from supernovae. Authors: Straka, W. C.; Goldreich, Peter; Sargent, W. L. W. Bibcode: 1971BAAS....3..451S Altcode: No abstract at ADS Title: History of the Lunar Orbit Authors: Goldreich, P. Bibcode: 1970pss..conf..491G Altcode: No abstract at ADS Title: Stellar Winds Authors: Goldreich, Peter; Julian, William H. Bibcode: 1970ApJ...160..971G Altcode: We present relativistic analytic solutions for the model stellar-wind problem posed by Weber and Davis. These solutions are consistent with the hypothesis of minimum torque proposed by Michel, but do not require it as an additional assumption. The equation of radial momentum has three critical points which occur where the radial flow velocity is equal to the propagation speed of infinitesimal disturbances. A unique continuous solution is determined by the requirement that it pass through all these critical points. Title: The Obliquity of Venus Authors: Goldreich, P.; Peale, S. J. Bibcode: 1970AJ.....75..273G Altcode: Solar gravitational tides tend to reduce the obliquity of Venus from its present value near 1800 whether or not Venus is locked in the synodic spin resonance with the Earth. Both a thermally driven atmospheric tidal torque and dissipation of energy at the boundary between a rigid mantle and a differentially rotating liquid core are possible mechanisms for maintaining the retrograde spin. The latter mechanism, where precession of the spin vector about the orbit normal induces the differential rotation, is almost certaintly capable of driving the obliquity to 1800 from values greater than 900 for a wide range of reasonable core viscosities and spin angular velocities. Title: Neutron Star Crusts and Alignment of Magnetic Axes in Pulsars Authors: Goldreich, Peter Bibcode: 1970ApJ...160L..11G Altcode: The electromagnetic torque which brakes the rotation of a magnetic neutron star also tends to align the magnetic axis with the rotation axis. For fluid stars the time scale for alignment is comparable to the time scale for rotational braking. However, the presence of a crystalline mantle impedes the alignment of the magnetic axis with the rotation axis. Actual rates of alignment are probably determined by creeping or cracking of the solid mantle. Title: Pulsar Electrodynamics Authors: Goldreich, Peter; Julian, William H. Bibcode: 1969ApJ...157..869G Altcode: Gold has suggested that pulsars are rotating magnetic neutron stars which formed in supernova explo- sions. We have investigated the simplest such model, one in which the magnetic dipole moment is aligned with the rotation axis. Our conclusions are as follows: In spite of its intense surface gravity, the star must possess a dense magnetosphere. The particles in the region threaded by those field lines which close within the light cylinder (of radius 5 X 1O~ p cm, where P sec is the stellar rotation period) rotate with the star. In the corotating zone the space-charge den- sity is 7 X 102 BZ/P electronic charges per cm3, where B~ (in gauss) is the component of magnetic field parallel to the rotation axis. The field lines which extend beyond the light cylinder close in a boundary zone near the supernova shell. Charged particles escape along these lines and are electrostatically accelerated up to energies of 3 X 1O'~ ZR63 B12 p-2 eV in the boundary zone. (Here, the stellar radius is R6 X 106 cm, and the magnetic field at the polar surface is B12 X 1012 gauss) Beyond the light cylinder the magnetic field becomes pre- dominantly toroidal. Its strength is 6 X 1O~ R63 B12 P~ ~ gauss at a distance of r~0 parsecs from the central star. The magnetic torque on the star causes its rotation period to lengthen at the rate P~ dP/dt = 10-8 B122 R64 p-2 M-' yr1 for an 211 solar-mass star. The rotational energy lost by the star is trans- ported out by the electromagnetic field and is then transmitted to the particles in the boundary zone. We compare our model with the observed properties of the Crab pulsar (NP 0332) and CP 1919 Title: Pulsar Electrodynamics Authors: Goldreich, Peter; Julian, William H. Bibcode: 1969BAAS....1R.242G Altcode: No abstract at ADS Title: Some remarks on polar wandering. Authors: Goldreich, P.; Toomre, A. Bibcode: 1969JGR....74.2555G Altcode: This paper lends fresh support to the hypothesis that large angular displacements of the earth's rotation axis relative to the entire mantle have occurred on a geological time scale, owing to the gradual redistribution (or decay or manufacture) of density inhomogeneities within the earth by the same convective processes that are responsible for continental drift. The first of our three contributions is a pedagogic theorem that rigorously illustrates this mechanism of polar wandering for a `quasi-rigid body'. That theorem states that any slow changes in shape of such a body preserve as an adiabatic invariant the solid angle traced out by its angular momentum vector as viewed from its principal axes. Thus, if the body were once set spinning about the axis with the greatest moment of inertia, it would always continue to spin almost exactly about the same principal axis no matter how that axis moves through the deforming body. The second and main contribution is our refutation of the widely accepted notion that the earth's figure shows unmistakable signs of the faster spin rate of the past. If correct, the degree of permanence of the rotation bulge so inferred by G. J. F. MacDonald (1963, 1965) and D. P. McKenzie (1966) would have been an effective impediment against any significant polar wandering of the earth as a whole. However, we show here that, after subtraction of the hydrostatic flattening, the remaining or nonhydrostatic part of the earth's inertia ellipsoid is distinctly triaxial. Such a triaxial shape, as well as the coincidence of the present rotation axis with the principal axis having the largest of the nonhydrostatic moments of inertia, is indeed to be expected of any randomly evolving, nearly spherical object without too much `memory' for its past axis of rotation. Finally, we discuss briefly some statistical aspects of polar wandering on the assumption that the earth is such an object. Title: Io, a jovian unipolar inductor Authors: Goldreich, P.; Lynden-Bell, D. Bibcode: 1969ApJ...156...59G Altcode: We show that To may be considered to be a unipolar generator which develops an emf of 7 X 1O~ volts across its radial diameter (as seen from a coordinate frame fixed to Jupiter). This voltage difference is transmitted along the magnetic flux tube which passes through Jo. The induced charge separation on the surface of the flux tube causes the plasma within it to rotate with Jo's orbital angular velocity. A current of about 106 amp is driven across each foot of the flux tube in the Jovian ionosphere. The current flows up along the half-surface of the flux tube which faces Jupiter, crosses the magnetic field in Jo, and then flows down along the opposite half of the flux tube's surface. Because the number density of charge carriers in the magnetosphere is low (we use fi = 0.5 cm3 in our calculations) the current must be carried by keV electrons which are electrostatically accelerated at lo and at the top of Jupiter's ionosphere. We argue that beam instabilities in the current sheets are responsible for the Jo-induced decametric bursts. The geometry of the beaming of the bursts strongly suggests coherent cyclotron radiation as the emission mechanism. In addition, we are led to predict that Europa should modulate decametric bursts whose max- imum frequencies are below 8 Mc/s. Bigg was the first to demonstrate that the position of lo in its orbit was strongly correlated with the reception of Jovian decametric bursts. Previously it had been established that the detection of the bursts was influenced by the position of Jupiter's magnetic dipole. Our aim is to explain the origin of To's great influence. I. THE INTERACTION OF 10 WITH JUPITER'S MAGNETOSPHER Title: The physics of rotating magnetic neutron stars Authors: Goldreich, P. Bibcode: 1969PASA....1..227G Altcode: 1969PASAu...1..227G No abstract at ADS Title: A Theoretical Upper Bound to the Solar Oblateness Authors: Goldreich, Peter; Schubert, Gerald Bibcode: 1968ApJ...154.1005G Altcode: In an earlier paper we investigated the stability of differentially rotating solar models. We concluded that in regions of constant mean molecular weight the angular momentum per unit mass could not significantly decrease away from the rotation axis (in the absence of toroidal magnetic fields). In the current investigation we show that it is conceivable for sufficiently large gradients of rotational angular velocity to exist in the Sun's core to account for the solar oblateness which Dicke and Goldenberg claim to have measured. However, we argue that it is more likely that turbulent mixing destroyed the gradients of mean molecular weight during the time when most of the Sun's angular momentum was lost and that, in reality, only very little angular momentum remains hidden in the solar interior Title: Stimulated Inverse Compton Scattering in Radio Sources Authors: Goldreich, Peter; McCary, Richard; Rees, Martin J. Bibcode: 1968Natur.217..931G Altcode: OSTER1 has claimed that energy losses due to the inverse Compton effect can be considerably larger than was previously thought, when stimulated scattering is taken into account. If true, this would seriously hinder the understanding of compact radio sources. We wish to point out that Oster's discussion is erroneous and grossly exaggerates the importance of stimulated inverse Compton scattering. Title: The Dynamics of Planetary Rotations Authors: Goldreich, Peter; Peale, Stanton J. Bibcode: 1968ARA&A...6..287G Altcode: No abstract at ADS Title: On the controversy over the effect of tidal friction upon the history of the Earth-Moon system a reply to comments by H. Gerstenkorn Authors: Goldreich, Peter Bibcode: 1968Icar....9..391G Altcode: In a recent paper Gerstenkorn claims that discrepancies between the results of his integrations and those of Goldreich are due to an error in the latter's calculations. We show, contrary to this claim, that the source of these discrepancies is a fundamental error in Gerstenkorn's technique of averaging his equations over a processional period of the Earth-Moon system. As a result, Gerstenkorn's calculations are not quantitatively correct. However, the qualitative conclusions of his pioneering work on the past history of the Earth-Moon system are not affected by the error. Title: Differential Rotation in Stars Authors: Goldreich, Peter; Schubert, Gerald Bibcode: 1967ApJ...150..571G Altcode: The stability of differential rotation in the radiative zones of stars is investigated. For sufficiently large x/ (x is the thermal diffusivity and the kinematic viscosity),it is shown that a necessary condition for stability in regions of homogeneous chemical composition is that the angular momentum per unit mass be an increasing function of distance from the rotation axis. In cylindrical coordinates (i:', `o,z), this condition is given by a( 2 )/a > 0 and 3 /az = 0, where is the angular velocity. The condition is also a sufficient one when applied to axisymmetric perturbations. The stable thermal stratification which exists in the radiative zone cannot prevent the instability since, in stars, the thermal diffusivity is much greater than the kinematic viscosity. The turbulent diffusion of angnlar momentum, which arises when the stability condition is violated, is so rapid that it would appear to preclude the fast rotation of the Sun's interior which has been proposed by Dicke * In the absence of the instability associated with thermal diffusion, i.e,if x = 0, Dicke's solar model is found to be stable. Another means whereby angular momentum might be brought up from the solar interior is by the mechanism of spin-down associated with the formation of an Ekman boundary layer just below the solar convective envelope. The transport of angular momentum, either by spin-down or by turbulent diffusion, would result in the mixing of material below the convective zone of solar type stars if an external torque were applied to the stellar surface. Thus, the depletion of lithium and beryllium would be an inevitable consequence of the loss of a significant fraction of the star's initial angular momentum. Title: Spin-orbit coupling in the solar system. II. The resonant rotation of Venus Authors: Goldreich, Peter; Peale, Stanton Bibcode: 1967AJ.....72..662G Altcode: In earlier papers we examined the stability of a resonant rotation rate for Venus. In the resonant rotation state the planet presents the same face to the earth at each inferior conjunction. The present investigation is concerned with the capture probability at this resonance. It is shown that trapping at the resonance can be understood if Venus possesses a fluid core similar to the earth's. Maximum capture probability occurs if the core responds to changes in angular velocity of the mantle with a time lag of about 3 X 10 yr. If Venus is in the resonant rotation state, then mapping of its gravitational field will determine the direction of its primordial rotation and may also permit an estimate of its magnitude. Title: Precession of the Moon's Core Authors: Goldreich, Peter Bibcode: 1967JGR....72.3135G Altcode: The coupling between an assumed lunar liquid core and the moon's mantle is investigated. It is found that a core of low viscosity would not share the observed 18.6-year precession of the mantle. The possibility of detecting such a core is briefly discussed. Title: Rotation of the Sun Authors: Goldreich, Peter; Schubert, Gerald Bibcode: 1967Sci...156.1101G Altcode: Dicke has interpreted his recent measurement of the sun's oblateness as implying a fast (1.8-day period) rotation of the solar radiative interior. We find that differentially rotating solar models, such as the one proposed by Dicke, are unstable. The rate of turbulent diffusion in the unstable regions of these models is so rapid that it appears to preclude a fast-spinning solar interior. As a corollary of the stability analysis, we conclude that the loss of a significant fraction of a star's angular momentum must be accompanied by the mixing of material below its convective zone. Such mixing inevitably leads to the depletion of lithium in the star's photosphere. Title: The Interaction of Supernova Remnants with the Interstellar Medium Authors: Goldreich, P.; Sargent, W. L. W. Bibcode: 1967LIACo..14..289G Altcode: 1967LIACo...6..289G; 1967MSRSL..15..289G No abstract at ADS Title: The history of the lunar orbit Authors: Goldreich, P. Bibcode: 1967metp.conf..221G Altcode: No abstract at ADS Title: Spin-orbit coupling in the solar system Authors: Goldreich, P.; Peale, S. Bibcode: 1967metp.conf..219G Altcode: No abstract at ADS Title: Spin-orbit coupling in the solar system Authors: Goldreich, Peter; Peale, Stanton Bibcode: 1966AJ.....71..425G Altcode: Two possible types of resonant spin rates for planets and satellites are investigated. The first occurs in eccentric orbits at rotation rates which are commensurate with the orbital mean motion. A resonant spin state exists at each half-integer multiple of the mean motion, the simplest case being the well-known syn- chronous rotation. The second class of resonant spins involves the presence of another planet or satellite. A planet (or satellite) with such a resonant spin always aligns the same axis toward the second planet (or satellite) at each conjunction. Averaged equations of motion are derived, and stability criteria are formulated for both types of resonance. Probabilities of capturing a planet (or satellite) into one of the commensurate rotation states as it is being despun by tidal friction are calculated. Application ot the results to Mercury reveals that the very small value of (B-A)/ -0 would suffice to stabilize Mercury's rotation period at -22 of its orbital period. The probability that Mercury would be cap- tured at this resonance is calculated for several assumed forms of tidal torques. Venus may be in a resonant spin state of the second kind. A sidereal rotation period of 243.16 days retrograde would be commensurate with its synodic motion. However, a large value of (B-A)/C( > 10- ) seems to be required to stabilize this rotation. In addition, the capture probability at this resonance appears to be small. Title: On the Origin of Planetary Nebulae Authors: Abell, G. O.; Goldreich, P. Bibcode: 1966PASP...78..232A Altcode: No abstract at ADS Title: Resonant Rotation for Venus? Authors: Goldreich, Peter; Peale, Stanton J. Bibcode: 1966Natur.209.1117G Altcode: CAN the rotation period of Venus be commensurate with its synodic period? If the rotation period is 243.16 days retrograde, the axis of Venus which points toward the Earth at one inferior conjunction will point toward the Earth at all subsequent inferior conjunctions. This value for the rotation period of Venus lies within the errors of the measured rotation period1-3. If the moment of inertia of Venus about this axis is minimal, it is possible that the planet is locked into this synodic commensurability with its permanent bulge or longest axis pointed toward the Earth at each inferior conjunction. In other words, the presence of the Earth may have stabilized the sidereal rotation period of Venus at the value of 243.16 days retrograde. To investigate the stability of such a commensurate spin angular velocity for Venus, we shall use a method similar to that used to determine the stability of the spin angular velocities of Mercury which are commensurate with its orbital angular velocity4. Title: Final spin states of planets and satellites Authors: Goldreich, Peter Bibcode: 1966AJ.....71....1G Altcode: The spin of a planet or satellite which is losing angular momentum through tidal friction may approach one of at least two distinct final states. We derive a criterion which determines whether or not the final state will be one of synchronous rotation. If the tidal phase lag is independent of the amplitude and frequency of the tide, then synchronous rotation will result when [3 (B - A )/C] >1 (9 Sire2). If this inequality is not satisfied, the body will end up spinning with a mean angular velocity which is somewhat larger than its orbital mean motion. This criterion is only slightly altered if the phase lag varies with amplitude and/or fre- quency. It is easily seen that the moon fails to satisfy the condition for synchronous rotation with the present value of its orbital eccentricity. However, the moon could have attained synchronous rotation if its mean orbital eccentricity was less than 0.041 at some time in the past. No analytical treatment of commensurate spin states other than synchronous rotation is included in the present investigation. However, the results of a separate study of these commensurate spins are briefly described. Title: Interaction of Supernova Remnants with the Interstellar Medium. Authors: Goldreich, Peter; Sargent, Wallace L. W. Bibcode: 1966AJ.....71Q.162G Altcode: We investigate the way in which the shock-wave produced in the interstellar medium by a supernova explosion slows. The shock path and the motion of the gas behind the shock are computed numerically using von Neumann and Richtmyer's method of artificial viscosity. We take into account the effects of radiative cooling behind the shock front. The cooling has an important effect on the motion for old remnants and leads to thermal instabilities in obj ects as old as the Cygnus Loop. We conclude that the visible filaments in the Loop are regions of thermal instability where the density rises and the temperature falls by large factors from their values immediately behind the shock front in a distance of the order of 0.01 PC. A shell of cool gas having a density about 102 times that in the undisturbed interstellar medium must lie behind the filaments in the Cygnus Loop. The mass in this shell should be greater than 100 M0 and is perhaps as great as 1000 M0. We also investigate the motion taking into account the interstellar magnetic field. In this case, as has been shown by van der Laan, the compression of the magnetic field limits the density rise in the thermal instability behind the shod~. Title: More on spin-orbit coupling in the solar system Authors: Goldreich, P.; Peale, S. J. Bibcode: 1966AJ.....71R.856G Altcode: No abstract at ADS Title: Near-commensurate satellite orbits in the solar system Authors: Goldreich, P. Bibcode: 1966IAUS...25..268G Altcode: No abstract at ADS Title: History of the Lunar Orbit Authors: Goldreich, Peter Bibcode: 1966RvGSP...4..411G Altcode: 1966RvGeo...4..411G No abstract at ADS Title: Q in the Solar System Authors: Goldreich, Peter; Soter, Steven Bibcode: 1966Icar....5..375G Altcode: Secular changes brought about by tidal friction in the solar system are reviewed. The presence or absence of specific changes is used to bound the values of Q (the specific dissipation function) appropriate for the planets and satellites. It is shown that the values of Q separate sharply into two groups. Values in the range from 10 to 500 are found for the terrestrial planets and satellites of the major planets. On the other hand, Q for the major planets is always larger than 6 × 10 4. Estimates of tidal dissipation in the atmospheres of the Jovian planets lead to values of Q which are consistent with those we have calculated on the basis of secular changes in the satellites' orbits. However, it is difficult to reconcile these large Q's with the much smaller values obtained in laboratory tests of solids. Lyttleton's hypothesis that Pluto is an escaped satellite of Neptune is critically examined. Using the Q's we obtain for the major planets and their satellites, we show that any eccentricity that Triton's orbit may have possessed after a near encounter with Pluto would have been subsequently damped, thus accounting for its present near-circular orbit. Title: Tidal De-spin of Planets and Satellites Authors: Goldreich, Peter Bibcode: 1965Natur.208..375G Altcode: RECENTLY, Peale and Gold1 have shown that the non-synchronous rotation of Mercury is likely to be a consequence of tidal friction. They point out that in an eccentric orbit the spin of an axially symmetric planet will not relax to the orbital mean motion, but instead will approach a final value which is somewhat larger. The final spin rate will be somewhere between the mean orbital angular velocity and the orbital angular velocity at perihelion. The precise value for the final spin is determined by the condition that the net tidal torque on the planet around each orbit be equal to zero. The spin rate at which this condition is satisfied is determined by the frequency and amplitude dependence of the planet's `Q' (1/Q is the specific dissipation function2). According to Peale and Gold: ``The condition discussed here is based on the supposition that the solar torque exerted on the tidal bulge exceeds that exerted on any permanent deformation from axial symmetry. In the converse case a period of 88 days for the rotation would indeed result.'' This last statement, if true, would imply that Mercury's principal moments of inertia, in the plane perpendicular to its spin axis, differ by less than a few parts in 107. This value is very small when compared with the values known for the Moon-another solid, slowly rotating body. Title: Inclination of satellite orbits about an oblate precessing planet Authors: Goldreich, Peter Bibcode: 1965AJ.....70....5G Altcode: Equations are derived which govern the rate of change of the inclination of a satellite orbit to the equator of an oblate precessing planet. It is shown that if the motion of the satellite's ascending node on the equator plane has a period which is short when compared with the planet's precession period then the satellite's orbit will maintain a constant inclination to the planet's equator. This criterion is satisfied by many satel- lite orbits in the solar system. These include all satellites whose orbit planes lie in their planets' equator planes. In addition, it is shown that changes in the planet's obliquity, which are slow when compared with the satellite's nodal period, will also not affect a satellite's inclination to the equator plane of its planet. Finally a theory of the origin of natural satellites is proposed which accounts for their inclinations at the present time. Title: I. Gravitational stability of uniformly rotating disks Authors: Goldreich, P.; Lynden-Bell, D. Bibcode: 1965MNRAS.130...97G Altcode: The stability of a stratified self-gravitating rotating medium is discussed. It is found that in all cases pressure effects stabilize short waves while long waves are stabilized by rotation. when is greater than a number between i 8 and (depending on the polytropic index) there is an intermediate range of unstable wave-lengths centred at several times the thickness of the sheet. These instabilities would lead to the break-up of the sheet into masses of definite size. This paper serves as an introduction to the following one on the stability of differentially rotating media and the formation of spiral arms. Title: II. Spiral arms as sheared gravitational instabilities Authors: Goldreich, P.; Lynden-Bell, D. Bibcode: 1965MNRAS.130..125G Altcode: This paper treats examples of gravitational instability in differentially rotating media. The particular cases dealt with include polytropic, stratified sheets of gas, as well as the infinite homogeneous media. Application of the results is made to the formation of spiral arms in a differentially rotating disk galaxy. Relations are derived which connect the thickness of the galactic disk, its density, and the velocity dispersion perpendicular to the galactic plane with Oort's differential rotation parameters A and B. Sections 1 and 2 discuss requirements of any theory of spiral arms. Sections 3 to 8 give a mathematical treatment of gravitational instability in a sheared rotating stratified medium. Section 9 discusses these results qualitatively and proposes a theory of spiral arm formation based on them. Section 10 gives the observational tests and consequences of the theory. Finally Section Ii gives a very brief discussion of barred spirals and points to the many problems left unsolved by the present work. Title: An explanation of the frequent occurrence of commensurable mean motions in the solar system Authors: Goldreich, P. Bibcode: 1965MNRAS.130..159G Altcode: In this paper a tidal origin of commensurable satellite mean motions is proposed. It is shown that special cases of commensurate mean motions are not disrupted by tidal forces. Furthermore, at least four, and probably seven, 9f the best examples of commensurabilities have this stability. The significance of these stable configurations to the evolution of satellite systems is discussed and some inferences are drawn about tidal dissipation in Jupiter and Saturn. Title: On the eccentricity of satellite orbits in the solar system Authors: Goldreich, P. Bibcode: 1963MNRAS.126..257G Altcode: In this paper the secular changes in the eccentricities of satellite orbits in the solar system are investigated. Two mechanisms which affect the eccentricities are considered. One of them is the tide raised on the planet by the satellite, which has been the subject of discussion in the past; the other is the tide raised on the satellite by the planet. It is seen that cases arise in the solar system in which each of these tide's effect on eccentricity is dominant.