Author name code: nagendra
ADS astronomy entries on 2022-09-14
author:"Nagendra, K.N."
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Title: Fast Iterative Techniques for Polarized Radiative Transfer
in Spherically Symmetric Moving Media
Authors: Megha, A.; Sampoorna, M.; Nagendra, K. N.; Anusha, L. S.;
Sankarasubramanian, K.
Bibcode: 2020ApJ...903....6M
Altcode:
For a more precise modeling of polarized spectral lines formed in
extended and expanding stellar atmospheres, the solution of the
radiative transfer equation for the Stokes vectors must be obtained
in a spherical geometry rather than in a planar geometry. In this
paper, we present the modern iterative techniques based on operator
perturbation to solve the spherically symmetric polarized radiative
transfer equation with velocity fields. We consider scattering on a
two-level atom and account for partial frequency redistribution. An
accurate numerical solution to such problems requires the use of
spatial grids with higher resolution. Consequently, Jacobi-based
methods lead to slower convergence rate. The convergence rate can be
improved by a factor of 2 or more when fast iterative schemes based
on Gauss-Seidel (GS) and successive overrelaxation (SOR) methods are
used over the Jacobi-based method. Here we present the Jacobi, GS,
and SOR iterative techniques for solving the abovementioned problem,
and discuss their convergence behavior.
Title: Importance of Angle-dependent Partial Frequency Redistribution
in Hyperfine Structure Transitions Under the Incomplete Paschen-Back
Effect Regime
Authors: Nagendra, K. N.; Sowmya, K.; Sampoorna, M.; Stenflo, J. O.;
Anusha, L. S.
Bibcode: 2020ApJ...898...49N
Altcode: 2020arXiv200704044N
Angle-frequency coupling in scattering of polarized light on atoms is
represented by the angle-dependent (AD) partial frequency redistribution
(PRD) matrices. There are several lines in the linearly polarized
solar spectrum, for which PRD combined with quantum interference
between hyperfine structure states play a significant role. Here we
present the solution of the polarized line transfer equation including
the AD-PRD matrix for scattering on a two-level atom with hyperfine
structure splitting and an unpolarized lower level. We account for
the effects of arbitrary magnetic fields (including the incomplete
Paschen-Back effect regime) and elastic collisions. For exploratory
purposes we consider a self-emitting isothermal planar atmosphere and
use atomic parameters that represent an isolated Na I D2
line. For this case we show that the AD-PRD effects are significant
for field strengths below about 30 G, but that the computationally
much less demanding approximation of angle-averaged PRD may be used
for stronger fields.
Title: Resonance Line Polarization in Spherically Symmetric Moving
Media: a Parametric Study
Authors: Megha, A.; Sampoorna, M.; Nagendra, K. N.; Anusha, L. S.;
Sankarasubramanian, K.
Bibcode: 2020arXiv200106201M
Altcode:
In the present paper we consider the problem of resonance
line polarization formed in the spherically symmetric expanding
atmospheres. For the solution of the concerned polarized transfer
equation we use the comoving frame formulation, and apply the
Accelerated Lambda Iteration (ALI) method. We restrict ourselves to
the non-relativistic regime of velocities wherein mainly Doppler shift
effects are significant. For our studies, we consider the scattering
on a two-level atom, including the effects of partial frequency
redistribution (PFR). We present the dependence of linearly polarized
profiles on different atmospheric and atomic parameters.
Title: Polarized Line Formation in Arbitrary Strength Magnetic Fields:
The Case of a Two-level Atom with Hyperfine Structure Splitting
Authors: Sampoorna, M.; Nagendra, K. N.; Sowmya, K.; Stenflo, J. O.;
Anusha, L. S.
Bibcode: 2019ApJ...883..188S
Altcode: 2019arXiv191010913S
Quantum interference effects, together with partial frequency
redistribution (PFR) in line scattering, produce subtle signatures in
the so-called Second Solar Spectrum (the linearly polarized spectrum of
the Sun). These signatures are modified in the presence of arbitrary
strength magnetic fields via the Hanle, Zeeman, and Paschen-Back
effects. In the present paper we solve the problem of polarized line
formation in a magnetized atmosphere taking into account scattering in a
two-level atom with hyperfine structure splitting together with PFR. To
this end we incorporate the collisionless PFR matrix derived in Sowmya
et al. in the polarized transfer equation. We apply the scattering
expansion method to solve this transfer equation. We study the combined
effects of PFR and the Paschen-Back effect on polarized line profiles
formed in an isothermal one-dimensional planar atmosphere. For this
purpose, we consider the cases of D2 lines of Li I and Na I.
Title: Resonance Line Polarization in Spherically Symmetric Moving
Media: a Parametric Study.
Authors: Megha, A.; Sampoorna, M.; Nagendra, K. N.; Anusha, L. S.;
Sankarasubramanian, K.
Bibcode: 2019spw..confE..14M
Altcode:
No abstract at ADS
Title: Polarized Line Transfer in the Incomplete Paschen-Back Effect
Regime with Angle-dependent Partial Frequency Redistribution.
Authors: Nagendra, K. N.; Sowmya, K.; Sampoorna, M.; Stenflo, J. O.;
Anusha, L. S.
Bibcode: 2019spw..confE..13N
Altcode:
No abstract at ADS
Title: Admixture of Dipole and Quadrupole Atomic Transitions in the
Presence of External Magnetic Field.
Authors: Oo, Y. Y.; Thant, W. M.; Nagendra, K.; Ramachandran, G.
Bibcode: 2019spw..confE...8O
Altcode:
No abstract at ADS
Title: Solution of Polarized Radiative Transfer Equation with
Cross-redistribution.
Authors: Nagendra, K. N.; Sampoorna, M.
Bibcode: 2019spw..confE...1N
Altcode:
No abstract at ADS
Title: Effects of Angle-Dependent Partial Frequency Redistribution
on Polarized Line Profiles
Authors: Sampoorna, M.; Nagendra, K. N.; Frisch, H.; Stenflo, J. O.
Bibcode: 2019ASPC..519..109S
Altcode:
Scattering of the solar limb-darkened radiation field on atoms and
molecules produces linearly polarized spectrum of the Sun (Second Solar
Spectrum). Partial frequency redistribution (PFR) plays a fundamental
role in shaping the wings of linearly polarized profiles of strong
resonance lines. Here we present the effects of the angle-dependent
(AD) PFR on resonance polarization both in the presence and absence
of magnetic fields. We consider scattering on a two-level atom with
unpolarized lower level, and a one-dimensional isothermal atmosphere.
Title: Polarized Line Formation in Spherically Symmetric Expanding
Atmospheres
Authors: Megha, A.; Sampoorna, M.; Nagendra, K. N.; Anusha, L. S.;
Sankarasubramanian, K.
Bibcode: 2019ASPC..519...27M
Altcode:
We consider the problem of polarized line formation in the spherically
symmetric expanding atmospheres. The velocity fields in line forming
regions produce Doppler shift, aberration of photons and also gives
rise to advection. These in turn can modify the amplitudes and
shapes of the emergent Stokes profiles. However, here we consider
only non-relativistic regime, wherein mainly Doppler shift effects
are significant. Thus only Doppler shift terms are considered in
the polarized transfer equation. For the solution of the concerned
polarized transfer equation we use the comoving frame formulation,
and apply the Accelerated Lambda Iteration (ALI) method. We present the
results by considering the scattering on a two-level atom, including the
effects of partial frequency redistribution (PFR). The polarized line
profiles are shown for few velocity laws, representative of expanding
spherical atmospheres. It is shown that the degree of polarization in
the lines depends sensitively on the extendedness R of the spherical
atmosphere. We also present a comparison of polarized profiles computed
under complete frequency redistribution (CFR) and PFR in the case of
static as well as expanding atmospheres.
Title: Polarized Line Formation with Incomplete Paschen-Back Effect
and Partial Frequency Redistribution
Authors: Sampoorna, M.; Nagendra, K. N.; Sowmya, K.; Stenflo, J. O.;
Anusha, L. S.
Bibcode: 2019ASPC..519..113S
Altcode:
Quantum interference between the hyperfine structure states is
known to depolarize the cores of some of the lines in the linearly
polarized spectrum of the Sun (the Second Solar Spectrum). The
presence of external magnetic fields in the line forming regions
modify these signatures through the Hanle, Zeeman, and incomplete/
complete Paschen-Back effects (PBE), depending on the strength of the
magnetic field. In an earlier paper, Sowmya et al. (2014) derived the
relevant collisionless partial frequency redistribution (PFR) matrix
for scattering on a two-level atom with hyperfine structure splitting
(HFS) and in the presence of arbitrary strength magnetic fields
(including the PBE regime). In the present paper we solve the problem
of polarized line transfer in a magnetized atmosphere, including this
PFR matrix. For this purpose, we apply a scattering expansion method
which is based on orders of scattering approach. We present the results
on the combined effects of PBE and PFR on the polarized line profiles
using the atomic parameters relevant to the Na I D2 line.
Title: Polarized Line Scattering Theory With Applications in
Astrophysical Radiative Transfer: A Historical Perspective
Authors: Nagendra, K. N.
Bibcode: 2019ASPC..519...51N
Altcode:
Ivan Hubený did fundamental work in the theory of partial frequency
redistribution (PFR) and radiative transfer (RT). One of the most
elegant works is the “theory of generalized redistribution functions
(GRFs)” for multi-level atoms (Hubený, Oxenius, Simonneau: HOS
theory). This ground breaking work is the high point of his contribution
to this field. The magnum opus “Theory of Stellar Atmospheres: An
introduction to astrophysical non-equilibrium quantitative spectroscopic
analysis” by Hubený & Mihalas (2014) is one of the greatest
creations in Astrophysics. It remains evergreen as a standard source
book for a long time. The powerful codes TLUSTY and SYNSPEC that
Ivan Hubený developed with his colleagues and made available to
the community, decades ago, has already opened up new paths, and has
perfected the way spectroscopic analysis of astronomical spectra is
done, in several fields, and will continue to inspire generations
to come.
Title: Polarized Line Formation in Spherically Symmetric Atmospheres
with Velocity Fields
Authors: Megha, A.; Sampoorna, M.; Nagendra, K. N.; Anusha, L. S.;
Sankarasubramanian, K.
Bibcode: 2019ApJ...879...48M
Altcode:
The plane-parallel approximation of the stellar atmospheres cannot
be applied to model the formation of optically thick lines formed in
extended atmospheres. To a good approximation these atmospheres can be
represented by a spherically symmetric medium. Such extended stellar
atmospheres are dynamic, in general, due to the systematic motions
present in their layers. Macroscopic velocity fields in the spectral
line forming regions produce Doppler shift, aberration of photons,
and also give rise to advection. All of these effects can modify
the amplitudes and shapes of the emergent Stokes profiles. In the
present paper we consider the problem of polarized line formation
in spherically symmetric media, in the presence of velocity
fields. Solving the radiative transfer problem in the observer’s
frame is a straightforward approach to handle the presence of velocity
fields. This method, however, becomes computationally prohibitive when
large velocity fields are considered, particularly in the case of the
line formation with partial frequency redistribution (PFR). In this
paper we present a polarized comoving frame method to solve the problem
at hand. We consider nonrelativistic radial velocity fields, thereby
accounting only for Doppler shift effects and neglecting advection
and aberration of photons. We study the effects of extendedness,
velocity fields, and PFR on the polarized line profiles formed in
highly extended atmospheres.
Title: Comoving Frame Method for Polarized PRD Line Transfer with
Velocity Fields
Authors: Sampoorna, M.; Nagendra, K. N.
Bibcode: 2019ASPC..526..153S
Altcode:
The solution of the transfer equation in moving atmospheres is a
classical problem. While the low-velocity regime can be handled in
a simpler manner using the observer's frame method, the regime of
high-velocity requires the comoving frame technique to be applied. We
show that even in the low-velocity regime (like that prevailing in the
solar atmosphere) we require the comoving frame method when linear
polarization together with partial frequency redistribution (PRD)
in line scattering is considered. This situation arises because of
the numerical difficulties that we encounter in the observer's frame
method, namely, a heavy demand on the angle, frequency, and depth
grids when we consider polarized line formation with PRD and velocity
fields. These difficulties can be overcome by applying the comoving
frame technique. Here we present the details of this technique and
its applications to solar-like conditions.
Title: Polarized Line Formation: Methods and Solutions
Authors: Nagendra, K. N.
Bibcode: 2019ASPC..526...99N
Altcode:
In this paper I briefly review few methods that we have developed over
the past several years for solving polarized line transfer problems
of varying degrees of complexity. I describe the way in which the
complexity of the transfer problem increases as one introduces the
physics of partial frequency redistribution (PRD), magnetic field,
multi-level coupling, the velocity fields, and finally multi-D effects
in the line forming region. Examples are given to demonstrate how
the methods are applied after suitable modifications. The methods are
qualitatively compared in terms of their numerical performance. Some
sample results are shown to demonstrate the usefulness of these methods
in theoretical modeling of the solar polarimetric data. The results
discussed are Hanle effect, the J and F state quantum interference
effects, the effects of angle-dependent PRD, the effects of macroscopic
velocity fields on the spectral lines formed in the solar atmosphere,
and finally transfer effects in multi-D geometry.
Title: Polarized Scattering Matrix for Magnetic Dipole Transitions
Authors: Megha, A.; Sampoorna, M.; Nagendra, K. N.; Sankarasubramanian,
K.
Bibcode: 2019ASPC..526..207M
Altcode:
Forbidden emission lines, produced by magnetic dipole (M1) transitions,
are difficult to observe in the laboratory, but naturally arise in the
highly ionized atoms present in the solar corona. The polarization of
these lines is the result of anisotropic excitation processes. The
polarization measurement of forbidden emission lines is the most
direct method of determining the magnetic field direction in the solar
corona. Here we consider the general case of M1 transitions in the
presence of magnetic fields of arbitrary strength. In particular,
we derive the scattering matrix for the M1 transitions using the
classical magnetic dipole oscillator model of Casini & Lin (2002)
and applying the scattering matrix approach of Stenflo (1998). The
derived scattering matrix covers, in a continuous way, saturated Hanle,
intermediate Hanle-Zeeman, and Zeeman regimes.
Title: Partial Frequency Redistribution Theory with Paschen-Back
Effect: Application to Li I 6708 Å Lines
Authors: Sowmya, K.; Nagendra, K. N.; Sampoorna, M.; Stenflo, J. O.
Bibcode: 2019ASPC..526...43S
Altcode:
The diagnostically important Li I D lines in the Second Solar Spectrum
are governed by the quantum interference processes that take place among
the magnetic substates belonging to different fine (J) and hyperfine (F)
structure states. This interference gets modified in the presence of
a magnetic field. The signatures of this interference in polarization
contain information on the nature of the vector magnetic field in
the solar atmosphere. With this motivation, we apply the polarized
redistribution matrix including Paschen-Back effect, derived based
on the Kramers-Heisenberg scattering matrix approach, to model the
polarization profiles of the Li lines observed in the quiet Sun. We
make use of the last scattering approximation which is based on the
concept that the polarization of the emergent radiation is generated
in the last scattering event, before the radiation escapes from the
atmosphere. We present a comparison of the quiet Sun observations of
the linear polarization profiles of Li I 6708 Å line system with the
theoretical profiles computed using our simple modeling approach. We
also present theoretical Stokes profiles in the Paschen-Back regime
of magnetic fields and compare them with the single scattered profiles.
Title: Effects of Lower-Level Polarization and Partial Frequency
Redistribution on Stokes Profiles
Authors: Supriya, H. D.; Nagendra, K. N.; Sampoorna, M.; Stenflo,
J. O.; Ravindra, B.
Bibcode: 2019ASPC..526...61S
Altcode:
The theory of polarized radiative transfer including the effects of
partial frequency redistribution (PRD) for a two-level and two-term atom
has been formulated in the scattering matrix approach. However there
exist several enigmatic features in the Second Solar Spectrum which
cannot be explained on the basis of said approach. The reason for this
lies in the approximations made in this approach. One such approximation
is the assumption that the lower level of the atom involved in the
scattering process is unpolarized. There are alternative approaches
based on the density matrix formalism to relax this assumption. It has
been shown that the inclusion of the polarization of all the atomic
levels involved in the scattering process is important. In our recent
studies, the collisionless redistribution matrix including the effects
of both PRD and lower-level polarization (LLP) was derived starting
from the Kramers-Heisenberg scattering formulation. We proposed a
simple numerical technique namely, the correction method, to solve
the problem of polarized radiative transfer with PRD and LLP. Here
we apply this technique to different atomic systems and discuss the
effects of PRD and LLP on the emergent Stokes profiles.
Title: Coronal magnetic field measurements using forbidden emission
lines
Authors: Megha, A.; Sampoorna, M.; Nagendra, K. N.; Sankarasubramanian,
K.
Bibcode: 2018IAUS..340...61M
Altcode:
The polarization measurement of coronal forbidden emission lines is the
most promising method of determining the direction of magnetic fields
in the corona. A classical theory for the forbidden lines was presented
in Megha et al. (2017) for the case of arbitrary strength magnetic
fields. Here we apply that theoretical formalism to study the effect
of density distributions, magnetic field configurations, and velocity
fields on the Stokes profiles formed in corona. For illustrations we use
the atomic parameters of the [Fe xiii] 10747 Å coronal forbidden line.
Title: Polarized Line Formation in Arbitrary Strength Magnetic Fields
Angle-averaged and Angle-dependent Partial Frequency Redistribution
Authors: Sampoorna, M.; Nagendra, K. N.; Stenflo, J. O.
Bibcode: 2017ApJ...844...97S
Altcode:
Magnetic fields in the solar atmosphere leave their fingerprints in the
polarized spectrum of the Sun via the Hanle and Zeeman effects. While
the Hanle and Zeeman effects dominate, respectively, in the weak
and strong field regimes, both these effects jointly operate in the
intermediate field strength regime. Therefore, it is necessary to
solve the polarized line transfer equation, including the combined
influence of Hanle and Zeeman effects. Furthermore, it is required
to take into account the effects of partial frequency redistribution
(PRD) in scattering when dealing with strong chromospheric lines with
broad damping wings. In this paper, we present a numerical method to
solve the problem of polarized PRD line formation in magnetic fields
of arbitrary strength and orientation. This numerical method is based
on the concept of operator perturbation. For our studies, we consider
a two-level atom model without hyperfine structure and lower-level
polarization. We compare the PRD idealization of angle-averaged
Hanle-Zeeman redistribution matrices with the full treatment of
angle-dependent PRD, to indicate when the idealized treatment is
inadequate and what kind of polarization effects are specific to
angle-dependent PRD. Because the angle-dependent treatment is presently
computationally prohibitive when applied to realistic model atmospheres,
we present the computed emergent Stokes profiles for a range of magnetic
fields, with the assumption of an isothermal one-dimensional medium.
Title: Hanle-Zeeman Scattering Matrix for Magnetic Dipole Transitions
Authors: Megha, A.; Sampoorna, M.; Nagendra, K. N.; Sankarasubramanian,
K.
Bibcode: 2017ApJ...841..129M
Altcode:
The polarization of the light that is scattered by the coronal ions is
influenced by the anisotropic illumination from the photosphere and
the magnetic field structuring in the solar corona. The properties
of the coronal magnetic fields can be well studied by understanding
the polarization properties of coronal forbidden emission lines that
arise from magnetic dipole (M1) transitions in the highly ionized
atoms that are present in the corona. We present the classical
scattering theory of the forbidden lines for a more general case of
arbitrary-strength magnetic fields. We derive the scattering matrix
for M1 transitions using the classical magnetic dipole model of Casini
& Lin and applying the scattering matrix approach of Stenflo. We
consider a two-level atom model and neglect collisional effects. The
scattering matrix so derived is used to study the Stokes profiles
formed in coronal conditions in those regions where the radiative
excitations dominate collisional excitations. To this end, we take
into account the integration over a cone of an unpolarized radiation
from the solar disk incident on the scattering atoms. Furthermore,
we also integrate along the line of sight to calculate the emerging
polarized line profiles. We consider radial and dipole magnetic field
configurations and spherically symmetric density distributions. For
our studies we adopt the atomic parameters corresponding to the [Fe
xiii] 10747 Å coronal forbidden line. We also discuss the nature of
the scattering matrix for M1 transitions and compare it with that for
the electric dipole (E1) transitions.
Title: Importance of Cross-redistribution in Scattering Polarization
of Spectral Lines: The Cases of 3P-3S Triplets
of Mg I and Ca I
Authors: Sampoorna, M.; Nagendra, K. N.
Bibcode: 2017ApJ...838...95S
Altcode:
Scattering on a multi-level atomic system has dominant contributions
from resonance and Raman scattering. While initial and final
levels are the same for resonance scattering, they are different
for Raman scattering. The frequency redistribution for resonance
scattering is described by the usual partial frequency redistribution
functions of Hummer, while that for Raman scattering is described
by cross-redistribution (XRD) function. In the present paper, we
investigate the importance of XRD on linear polarization profiles
of 3P-3S triplets of Mg I and Ca I formed in
an isothermal one-dimensional atmosphere. We show that XRD produces
significant effects on the linear polarization profiles when the
wavelength separations between the line components of the multiplet
are small, like in the cases of Mg I b and Ca I triplets.
Title: Polarized Line Formation in Non-monotonic Velocity Fields
Authors: Sampoorna, M.; Nagendra, K. N.
Bibcode: 2016ApJ...833...32S
Altcode:
For a correct interpretation of the observed spectro-polarimetric
data from astrophysical objects such as the Sun, it is necessary
to solve the polarized line transfer problems taking into account a
realistic temperature structure, the dynamical state of the atmosphere,
a realistic scattering mechanism (namely, the partial frequency
redistribution—PRD), and the magnetic fields. In a recent paper,
we studied the effects of monotonic vertical velocity fields on
linearly polarized line profiles formed in isothermal atmospheres
with and without magnetic fields. However, in general the velocity
fields that prevail in dynamical atmospheres of astrophysical objects
are non-monotonic. Stellar atmospheres with shocks, multi-component
supernova atmospheres, and various kinds of wave motions in solar and
stellar atmospheres are examples of non-monotonic velocity fields. Here
we present studies on the effect of non-relativistic non-monotonic
vertical velocity fields on the linearly polarized line profiles formed
in semi-empirical atmospheres. We consider a two-level atom model and
PRD scattering mechanism. We solve the polarized transfer equation in
the comoving frame (CMF) of the fluid using a polarized accelerated
lambda iteration method that has been appropriately modified for the
problem at hand. We present numerical tests to validate the CMF method
and also discuss the accuracy and numerical instabilities associated
with it.
Title: Polarized Line Formation with Lower-level Polarization and
Partial Frequency Redistribution
Authors: Supriya, H. D.; Sampoorna, M.; Nagendra, K. N.; Stenflo,
J. O.; Ravindra, B.
Bibcode: 2016ApJ...828...84S
Altcode:
In the well-established theories of polarized line formation with
partial frequency redistribution (PRD) for a two-level and two-term
atom, it is generally assumed that the lower level of the scattering
transition is unpolarized. However, the existence of unexplained
spectral features in some lines of the Second Solar Spectrum points
toward a need to relax this assumption. There exists a density matrix
theory that accounts for the polarization of all the atomic levels,
but it is based on the flat-spectrum approximation (corresponding to
complete frequency redistribution). In the present paper we propose a
numerical algorithm to solve the problem of polarized line formation
in magnetized media, which includes both the effects of PRD and the
lower level polarization (LLP) for a two-level atom. First we derive a
collisionless redistribution matrix that includes the combined effects
of the PRD and the LLP. We then solve the relevant transfer equation
using a two-stage approach. For illustration purposes, we consider
two case studies in the non-magnetic regime, namely, the J
a = 1, J b = 0 and J a = J b
= 1, where J a and J b represent the
total angular momentum quantum numbers of the lower and upper states,
respectively. Our studies show that the effects of LLP are significant
only in the line core. This leads us to propose a simplified numerical
approach to solve the concerned radiative transfer problem.
Title: Polarized Scattering of Light for Arbitrary Magnetic Fields
with Level-crossings from the Combination of Hyperfine and Fine
Structure Splittings
Authors: Sowmya, K.; Nagendra, K. N.; Sampoorna, M.; Stenflo, J. O.
Bibcode: 2015ApJ...814..127S
Altcode: 2015arXiv151207736S
Interference between magnetic substates of the hyperfine structure
states belonging to different fine structure states of the same term
influences the polarization for some of the diagnostically important
lines of the Sun's spectrum, like the sodium and lithium doublets. The
polarization signatures of this combined interference contain
information on the properties of the solar magnetic fields. Motivated
by this, in the present paper, we study the problem of polarized
scattering on a two-term atom with hyperfine structure by accounting
for the partial redistribution in the photon frequencies arising due
to the Doppler motions of the atoms. We consider the scattering atoms
to be under the influence of a magnetic field of arbitrary strength
and develop a formalism based on the Kramers-Heisenberg approach to
calculate the scattering cross section for this process. We explore the
rich polarization effects that arise from various level-crossings in
the Paschen-Back regime in a single scattering case using the lithium
atomic system as a concrete example that is relevant to the Sun.
Title: Modeling the center-to-limb variation of the Ca i 4227 Å
line using FCHHT models
Authors: Supriya, H. D.; Smitha, H. N.; Nagendra, K. N.; Stenflo,
J. O.; Bianda, M.; Ravindra, B.; Ramelli, R.; Anusha, L. S.
Bibcode: 2015IAUS..305..381S
Altcode:
The Ca i 4227 Å is a chromospheric line exhibiting the largest degree
of linear polarization near the limb, in the visible spectrum of the
Sun. Modeling the observations of the center-to-limb variations (CLV)
of different lines in the Second Solar Spectrum helps to sample the
height dependence of the magnetic field, as the observations made
at different lines of sight sample different heights in the solar
atmosphere. Supriya et al. (2014) attempted to simultaneously model
the CLV of the (I, Q/I) spectra of the Ca i 4227 Å line using the
standard 1-D FAL model atmospheres. They found that the standard FAL
model atmospheres and also any appropriate combination of them, fail
to simultaneously fit the observed Stokes (I, Q/I) profiles at all the
limb distances (μ) satisfying at the same time all the observational
constraints. This failure of 1-D modeling approach can probably be
overcome by using multi-dimensional modeling which is computationally
expensive. To eliminate an even wider choice of 1-D models, we attempt
here to simultaneously model the CLV of the (I, Q/I) spectra using the
FCHHT solar model atmospheres which are updated and recent versions
of the FAL models. The details of our modeling efforts and the results
are presented.
Title: Paschen-Back effect involving atomic fine and hyperfine
structure states
Authors: Sowmya, K.; Nagendra, K. N.; Sampoorna, M.; Stenflo, J. O.
Bibcode: 2015IAUS..305..154S
Altcode:
The linear polarization in spectral lines produced by coherent
scattering is significantly modified by the quantum interference between
the atomic states in the presence of a magnetic field. When magnetic
fields produce a splitting which is of the order of or greater than the
fine or hyperfine structure splittings, we enter the Paschen-Back effect
(PBE) regime, in which the magnetic field dependence of the Zeeman
splittings and transition amplitudes becomes non-linear. In general,
PBE occurs for sufficiently strong fields when the fine structure
states are involved and for weak fields in the case of hyperfine
structure states. In this work, we apply the recently developed theory
of PBE in the atomic fine and hyperfine structure states including the
effects of partial frequency redistribution to the case of Li i 6708
Å doublet. We explore the signatures of PBE in a single scattering
event and their applicability to the solar magnetic field diagnostics.
Title: Study of resonance scattering polarization in O i 130 nm lines
Authors: Anusha, L. S.; Nagendra, K. N.; Uitenbroek, Han
Bibcode: 2015IAUS..305..234A
Altcode:
Here we address the importance of frequency cross-redistribution on
the scattering polarization of the O i line at 130.2 nm. We compute
the polarized profiles of this line with ordinary partial frequency
redistribution and cross-redistribution using a two-dimensional
radiative transfer code.
Title: A revisit to model the Cr i triplet at 5204-5208 Å and the
Ba ii D2 line at 4554 Å in the Second Solar Spectrum
Authors: Smitha, H. N.; Nagendra, K. N.; Stenflo, J. O.; Bianda, M.;
Sampoorna, M.; Ramelli, R.
Bibcode: 2015IAUS..305..372S
Altcode:
In our previous attempt to model the Stokes profiles of the Cr i triplet
at 5204-5208 Å and the Ba ii D2 at 4554 Å, we found it
necessary to slightly modify the standard FAL model atmospheres to fit
the observed polarization profiles. In the case of Cr i triplet, this
modification was done to reduce the theoretical continuum polarization,
and in the case of Ba ii D2, it was needed to reproduce the
central peak in Q/I. In this work, we revisit both these cases using
different standard model atmospheres whose temperature structures
closely resemble those of the modified FAL models, and explore the
possibility of synthesizing the line profiles without the need for
small modifications of the model atmosphere.
Title: Polarized Line Formation in Moving Atmospheres with Partial
Frequency Redistribution and a Weak Magnetic Field
Authors: Sampoorna, M.; Nagendra, K. N.
Bibcode: 2015ApJ...812...28S
Altcode:
The dynamical state of the solar and stellar atmospheres depends
on the macroscopic velocity fields prevailing within them. The
presence of such velocity fields in the line formation regions
strongly affects the polarized radiation field emerging from these
atmospheres. Thus it becomes necessary to solve the radiative transfer
equation for polarized lines in moving atmospheres. Solutions based
on the “observer’s frame method” are computationally expensive
to obtain, especially when partial frequency redistribution (PRD)
in line scattering and large-amplitude velocity fields are taken into
account. In this paper we present an efficient alternative method of
solution, namely, the comoving frame technique, to solve the polarized
PRD line formation problems in the presence of velocity fields. We
consider one-dimensional planar isothermal atmospheres with vertical
velocity fields. We present a study of the effect of velocity fields
on the emergent linear polarization profiles formed in optically
thick moving atmospheres. We show that the comoving frame method is
far superior when compared to the observer’s frame method in terms
of the computational speed and memory requirements.
Title: On the importance of partial frequency redistribution in
modeling the scattering polarization
Authors: Nagendra, K. N.
Bibcode: 2015IAUS..305..351N
Altcode:
It is well-known that partial frequency redistribution (PRD) is the
basic physical mechanism to correctly describe radiative transfer
in spectral lines. In the case of polarized line scattering, the PRD
becomes particularly important to describe the line-wing polarization,
instead of the well-known mechanism of complete redistribution
(CRD). Historically, the two-level atom PRD scattering matrices
for polarized line scattering were first derived in the 1970's,
and later generalized to the case of arbitrary fields in 1997. The
latter formulation of the PRD matrices have subsequently been used in
the solution of the line transfer equation to successfully model the
non-magnetic (resonance scattering) and the magnetic (Hanle scattering)
polarization observations. In recent years, using the Kramers-Heisenberg
approach, we formulated PRD matrices for various physical mechanisms
like quantum interference involving fine- and hyperfine-structure
states in a two-term atom. The effect of collisions is included in
an approximate way. We have used these PRD matrices to model the
observed linear polarization in several interesting lines of the
Second Solar Spectrum. In this paper I present a few results which
highlight the importance of PRD in the interpretation of the polarized
Stokes profiles.
Title: Partial Redistribution Effects on Polarized Lines Formed in
Moving Media in the Presence of a Weak Magnetic Field
Authors: Sampoorna, M.; Nagendra, K. N.
Bibcode: 2015IAUS..305..387S
Altcode:
Macroscopic velocity fields in stellar atmospheres significantly
affect the shapes of the emergent Stokes profiles. The inextricable
coupling between the angle and frequency variables becomes more
complex in a moving medium when compared to a static medium. In this
paper we consider both complete frequency redistribution (CRD) and
partial frequency redistribution (PRD) in the line scattering of a
two-level atom in the presence of an external weak magnetic field. For
simplicity we consider empirical velocity laws to represent motion of
the atmospheric layers. We present emergent Stokes profiles computed
with CRD, angle-averaged PRD, and angle-dependent PRD. We show that
angle-dependent PRD effects are important both in non-magnetic and
magnetized scattering when vertical velocity gradients are present
in the atmosphere. The results are presented for simple atmospheric
models. They are expected to be of relevance to polarized line formation
in slowly expanding chromospheric layers.
Title: Polarimetry
Authors: Nagendra, K. N.; Bagnulo, Stefano; Centeno, Rebecca; Jesús
Martínez González, María.
Bibcode: 2015IAUS..305.....N
Altcode:
Preface; 1. Solar and stellar surface magnetic fields; 2. Future
directions in astrophysical polarimetry; 3. Physical processes;
4. Instrumentation for astronomical polarimetry; 5. Data analysis
techniques for polarization observations; 6. Polarization diagnostics
of atmospheres and circumstellar environments; 7. Polarimetry as a tool
for discovery science; 8. Numerical modeling of polarized emission;
Author index.
Title: Electron Scattering Redistribution Effect on Atomic Line
Polarization
Authors: Supriya, H. D.; Nagendra, K. N.; Ravindra, B.; Sampoorna, M.
Bibcode: 2014ASPC..489..117S
Altcode:
The scattering of line photons on free electrons modifies the
polarization of the atomic spectral lines. Hence it is important to
treat scattering by electrons as a redistribution process and to study
in detail its effect on line formation. The numerically difficult
problem of evaluation and the use of angle-dependent atomic and
electron scattering redistribution functions in the line transfer
equation is considered. Two numerical methods, namely approximate
lambda iteration and scattering expansion method, are used to solve
the relevant polarized transfer problem. A study of the polarized
line formation in a standard two-level atom picture including an exact
treatment of electron scattering redistribution shows the importance
of the latter in the analysis of polarized line profiles emitted by
solar and stellar atmospheres. The effect of electron scattering turns
out to be extremely important in the interpretation of very far wing
line polarization of solar and stellar spectral lines.
Title: Multi-Dimensional Polarized Radiative Transfer: Methods and
Solar Applications
Authors: Anusha, L. S.; Nagendra, K. N.
Bibcode: 2014ASPC..489..225A
Altcode:
Observations of the Sun using modern telescopes as well as numerical
simulations of the Sun reveal existence of enormous inhomogeneous
structures in the solar atmosphere. The polarized spectrum of the Sun
(Second Solar Spectrum), produced due to anisotropic scattering helps
to infer the temperature structure, magnetic fields, and other physical
properties of the solar atmosphere more accurately. Analysis of the
Second Solar Spectrum requires solution of the polarized radiative
transfer equation. To take spatial inhomogeneities into account, a
solution of the transfer equation in multi-dimensional geometries is
necessary. In this paper we present a specialized review of recent
developments in the methods to solve multi-dimensional polarized
radiative transfer equation and an application of these methods to
analyze the observations.
Title: Partial Redistribution Theory and its Applications in the
Interpretation of the Second Solar Spectrum
Authors: Nagendra, K. N.
Bibcode: 2014ASPC..489..179N
Altcode:
In this review we first describe briefly the techniques to formulate
partial redistribution (PRD) matrices for polarized line scattering on
atoms. These matrices have to be formulated by either semi-classical
or quantum mechanical approach for the physical process under
consideration. We describe the semi-classical approach that we have
been following in our derivations. We then consider the polarized
transfer equation for line formation in three types of atomic systems
and briefly present two numerical methods to solve them. Finally,
a modeling procedure is described to analyze the polarimetric
observations, taking the examples of Ca I 4227 Å line and the Cr
I triplet at 5204-5208 Å. With these examples we demonstrate that
our treatment of PRD is able to correctly account for the scattering
polarization that is caused by various physical processes.
Title: Solar Polarization 7
Authors: Nagendra, K. N.; Stenflo, J. O.; Qu, Z. Q.; Sampoorna, M.
Bibcode: 2014ASPC..489.....N
Altcode:
No abstract at ADS
Title: Effect of Cross-redistribution on the Resonance Scattering
Polarization of O I Line at 1302 Å
Authors: Anusha, L. S.; Nagendra, K. N.; Uitenbroek, H.
Bibcode: 2014ApJ...794...17A
Altcode: 2014arXiv1407.8456A
Oxygen is the most abundant element on the Sun after hydrogen
and helium. The intensity spectrum of resonance lines of neutral
oxygen, namely, O I (1302, 1305, and 1306 Å), has been studied
in the literature for chromospheric diagnostics. In this paper,
we study the resonance scattering polarization in the O I line at
1302 Å using two-dimensional (2D) radiative transfer in a composite
atmosphere constructed using a 2D magneto-hydrodynamical snapshot in
the photosphere and columns of the one-dimensional FALC atmosphere in
the chromosphere. The methods developed by us recently in a series
of papers to solve multi-dimensional polarized radiative transfer
have been incorporated in our new code POLY2D, which we use for our
analysis. We find that multi-dimensional radiative transfer including
XRD effects is important in reproducing the amplitude and shape of
scattering polarization signals of the O I line at 1302 Å.
Title: The Role of Quantum Interference and Partial Redistribution
in the Solar Ba <font size=2>II D2 4554 Å Line
Authors: Smitha, H. N.; Nagendra, K. N.; Stenflo, J. O.; Sampoorna, M.
Bibcode: 2014ASPC..489..213S
Altcode: 2014arXiv1409.0465S
The Ba <font size=2>II D2 line at 4554 Å is a good
example, where the F-state interference effects due to the odd isotopes
produce polarization profiles, which are very different from those
of the even isotopes that do not exhibit F-state interference. It is
therefore necessary to account for the contributions from the different
isotopes to understand the observed linear polarization profiles of
this line. In this paper we present radiative transfer modeling with
partial frequency redistribution, which is shown to be essential to
model this line. This is because complete frequency redistribution
cannot reproduce the observed wing polarization. We present the observed
and computed Q/I profiles at different limb distances. The theoretical
profiles strongly depend on limb distance (μ) and the model atmosphere
which fits the limb observations fails at other μ positions.
Title: Intrinsically Polarized Blend Lines
Authors: Sowmya, K.; Nagendra, K. N.; Sampoorna, M.
Bibcode: 2014ASPC..489..125S
Altcode:
The Second Solar Spectrum formed by coherent scattering processes in
the Sun, is highly structured. It is characterized by numerous blend
lines, both intrinsically polarizing and depolarizing, superposed on the
background continuum. These blend lines play an important role in the
interpretation of the Second Solar Spectrum. Since blend lines affect
the shapes of the neighboring spectral lines they have to be treated in
a sophisticated manner in order to efficiently model a given spectral
line of interest. The depolarizing blend lines - mostly considered to be
formed under LTE conditions - depolarize the background continuum and
thereby affect the absolute scale of the polarization measurement. An
understanding of the influence of the blend lines leads to a proper
determination of the zero-point of the polarization scale. With this
motivation we extend a previously developed framework to include many
blend lines formed under NLTE conditions, in the radiative transfer
equation. The results are shown for the particular case of two blend
lines situated on either side of the main spectral line.
Title: Polarized Light Scattering with the Paschen-Back Effect,
Level-crossing of Fine Structure States, and Partial Frequency
Redistribution
Authors: Sowmya, K.; Nagendra, K. N.; Sampoorna, M.; Stenflo, J. O.
Bibcode: 2014ApJ...793...71S
Altcode: 2014arXiv1407.5457S
The quantum interference between the fine structure states of an atom
modifies the shapes of the emergent Stokes profiles in the second solar
spectrum. This phenomenon has been studied in great detail both in the
presence and absence of magnetic fields. By assuming a flat-spectrum
for the incident radiation, the signatures of this effect have been
explored for arbitrary field strengths. Even though the theory which
takes into account the frequency dependence of the incident radiation
is well developed, it is restricted to the regime in which the magnetic
splitting is much smaller than the fine structure splitting. In the
present paper, we carry out a generalization of our scattering matrix
formalism including the effects of partial frequency redistribution
for arbitrary magnetic fields. We test the formalism using available
benchmarks for special cases. In particular, we apply it to the Li
I 6708 Å D1 and D2 line system, for which
observable effects from the Paschen-Back regime are expected in the
Sun's spectrum.
Title: The Quantum Interference Effects in the Sc II 4247 Å Line
of the Second Solar Spectrum
Authors: Smitha, H. N.; Nagendra, K. N.; Stenflo, J. O.; Bianda, M.;
Ramelli, R.
Bibcode: 2014ApJ...794...30S
Altcode: 2014arXiv1408.4247S
The Sc II 4247 Å line formed in the chromosphere is one of the lines
well known, like the Na I D2 and Ba II D2,
for its prominent triple-peak structure in Q/I and the underlying
quantum interference effects governing it. In this paper, we try
to study the nature of this triple-peak structure using the theory
of F-state interference including the effects of partial frequency
redistribution (PRD) and radiative transfer (RT). We compare our
results with the observations taken in a quiet region near the
solar limb. In spite of accounting for PRD and RT effects, it has
not been possible to reproduce the observed triple-peak structure in
Q/I. While the two wing PRD peaks (on either side of central peak)
and the near wing continuum can be reproduced, the central peak is
completely suppressed by the enhanced depolarization resulting from
the hyperfine structure splitting. This suppression remains for all
the tested widely different one-dimensional model atmospheres or for
any multi-component combinations of them. While multidimensional RT
effects may improve the fit to the intensity profiles, they do not
appear capable of explaining the enigmatic central Q/I peak. This
leads us to suspect that some aspect of quantum physics is missing.
Title: Center-to-limb Observations and Modeling of the Ca I 4227
Å Line
Authors: Supriya, H. D.; Smitha, H. N.; Nagendra, K. N.; Stenflo,
J. O.; Bianda, M.; Ramelli, R.; Ravindra, B.; Anusha, L. S.
Bibcode: 2014ApJ...793...42S
Altcode: 2014arXiv1407.5461S
The observed center-to-limb variation (CLV) of the scattering
polarization in different lines of the Second Solar Spectrum can be used
to constrain the height variation of various atmospheric parameters, in
particular the magnetic fields, via the Hanle effect. Here we attempt
to model the nonmagnetic CLV observations of the Q/I profiles of the
Ca I 4227 Å line recorded with the Zurich Imaging Polarimeter-3
at IRSOL. For modeling, we use the polarized radiative transfer
with partial frequency redistribution with a number of realistic
one-dimensional (1D) model atmospheres. We find that all the standard
Fontenla-Avrett-Loeser (FAL) model atmospheres, which we used, fail
to simultaneously fit the observed (I, Q/I) at all the limb distances
(μ). However, an attempt is made to find a single model which can
provide a fit to at least the CLV of the observed Q/I instead of a
simultaneous fit to the (I, Q/I) at all μ. To this end we construct a
new 1D model by combining two of the standard models after modifying
their temperature structures in the appropriate height ranges. This
new combined model closely reproduces the observed Q/I at all μ but
fails to reproduce the observed rest intensity at different μ. Hence
we find that no single 1D model atmosphere succeeds in providing a
good representation of the real Sun. This failure of 1D models does
not, however, cause an impediment to the magnetic field diagnostic
potential of the Ca I 4227 Å line. To demonstrate this we deduce the
field strength at various μ positions without invoking the use of
radiative transfer.
Title: Polarized Scattering with Paschen-Back Effect, Hyperfine
Structure, and Partial Frequency Redistribution in Magnetized
Stellar Atmospheres
Authors: Sowmya, K.; Nagendra, K. N.; Stenflo, J. O.; Sampoorna, M.
Bibcode: 2014ApJ...786..150S
Altcode: 2015arXiv151207731S
F-state interference significantly modifies the polarization
produced by scattering processes in the solar atmosphere. Its
signature in the emergent Stokes spectrum in the absence of magnetic
fields is depolarization in the line core. In the present paper,
we derive the partial frequency redistribution (PRD) matrix that
includes interference between the upper hyperfine structure states
of a two-level atom in the presence of magnetic fields of arbitrary
strengths. The theory is applied to the Na I D2 line that
is produced by the transition between the lower J = 1/2 and upper J =
3/2 states which split into F states because of the coupling with the
nuclear spin Is = 3/2. The properties of the PRD matrix for
the single-scattering case is explored, in particular, the effects of
the magnetic field in the Paschen-Back regime and their usefulness as
a tool for the diagnostics of solar magnetic fields.
Title: Line-interlocking Effects on Polarization in Spectral Lines
by Rayleigh and Raman Scattering
Authors: Sampoorna, M.; Nagendra, K. N.; Stenflo, J. O.
Bibcode: 2013ApJ...770...92S
Altcode:
The polarized spectrum of the Sun and stars is formed from the
scattering of anisotropic radiation on atoms. Interpretation of
this spectrum requires the solution of polarized line transfer in
multilevel atomic systems. While sophisticated quantum theories of
polarized line formation in multilevel atomic systems exist, they are
limited by the approximation of complete frequency redistribution
in scattering. The partial frequency redistribution (PRD) in line
scattering is a necessary component in modeling the polarized spectra
of strong lines. The polarized PRD line scattering theories developed so
far confine themselves to a two-level or a two-term atom model. In this
paper, we present a heuristic approach to the problem of polarized line
formation in multilevel atoms taking into account the effects of PRD
and a weak magnetic field. Starting from the unpolarized PRD multilevel
atom approach of Hubeny et al., we incorporate the polarization state
of the radiation field. However, the lower level polarization is
neglected. Two iterative methods of solving the polarized PRD line
transfer in multilevel atoms are also presented. Taking the example
of a five-level Ca II atom model, we present illustrative results for
an isothermal one-dimensional model atmosphere.
Title: Modeling the Quantum Interference Signatures of the Ba II
D2 4554 Å Line in the Second Solar Spectrum
Authors: Smitha, H. N.; Nagendra, K. N.; Stenflo, J. O.; Sampoorna, M.
Bibcode: 2013ApJ...768..163S
Altcode: 2013arXiv1303.7304S
Quantum interference effects play a vital role in shaping the linear
polarization profiles of solar spectral lines. The Ba II D2
line at 4554 Å is a prominent example, where the F-state interference
effects due to the odd isotopes produce polarization profiles,
which are very different from those of the even isotopes that have
no F-state interference. It is therefore necessary to account for the
contributions from the different isotopes to understand the observed
linear polarization profiles of this line. Here we do radiative
transfer modeling with partial frequency redistribution (PRD) of such
observations while accounting for the interference effects and isotope
composition. The Ba II D2 polarization profile is found
to be strongly governed by the PRD mechanism. We show how a full PRD
treatment succeeds in reproducing the observations, while complete
frequency redistribution alone fails to produce polarization profiles
that have any resemblance to the observed ones. However, we also find
that the line center polarization is sensitive to the temperature
structure of the model atmosphere. To obtain a good fit to the line
center peak of the observed Stokes Q/I profile, a small modification
of the FALX model atmosphere is needed, by lowering the temperature
in the line-forming layers. Because of the pronounced temperature
sensitivity of the Ba II D2 line it may not be a suitable
tool for Hanle magnetic-field diagnostics of the solar chromosphere,
because there is currently no straightforward way to separate the
temperature and magnetic-field effects from each other.
Title: Multi-dimensional Radiative Transfer to Analyze Hanle Effect
in Ca II K Line at 3933 Å
Authors: Anusha, L. S.; Nagendra, K. N.
Bibcode: 2013ApJ...767..108A
Altcode: 2013arXiv1308.3437A
Radiative transfer (RT) studies of the linearly polarized spectrum of
the Sun (the second solar spectrum) have generally focused on line
formation, with an aim to understand the vertical structure of the
solar atmosphere using one-dimensional (1D) model atmospheres. Modeling
spatial structuring in the observations of the linearly polarized
line profiles requires the solution of multi-dimensional (multi-D)
polarized RT equation and a model solar atmosphere obtained by
magnetohydrodynamical (MHD) simulations of the solar atmosphere. Our
aim in this paper is to analyze the chromospheric resonance line Ca
II K at 3933 Å using multi-D polarized RT with the Hanle effect and
partial frequency redistribution (PRD) in line scattering. We use an
atmosphere that is constructed by a two-dimensional snapshot of the
three-dimensional MHD simulations of the solar photosphere, combined
with columns of a 1D atmosphere in the chromosphere. This paper
represents the first application of polarized multi-D RT to explore
the chromospheric lines using multi-D MHD atmospheres, with PRD as the
line scattering mechanism. We find that the horizontal inhomogeneities
caused by MHD in the lower layers of the atmosphere are responsible for
strong spatial inhomogeneities in the wings of the linear polarization
profiles, while the use of horizontally homogeneous chromosphere
(FALC) produces spatially homogeneous linear polarization in the line
core. The introduction of different magnetic field configurations
modifies the line core polarization through the Hanle effect and can
cause spatial inhomogeneities in the line core. A comparison of our
theoretical profiles with the observations of this line shows that
the MHD structuring in the photosphere is sufficient to reproduce the
line wings and in the line core, but only line center polarization
can be reproduced using the Hanle effect. For a simultaneous modeling
of the line wings and the line core (including the line center),
MHD atmospheres with inhomogeneities in the chromosphere are required.
Title: An efficient decomposition technique to solve angle-dependent
Hanle scattering problems
Authors: Supriya, H. D.; Sampoorna, M.; Nagendra, K. N.; Ravindra,
B.; Anusha, L. S.
Bibcode: 2013JQSRT.119...67S
Altcode: 2013arXiv1304.5321S
Hanle scattering is an important diagnostic tool to study weak solar
magnetic fields. Partial frequency redistribution (PRD) is necessary
to interpret the linear polarization observed in strong resonance
lines. Usually angle-averaged PRD functions are used to analyze
linear polarization. However, it is established that angle-dependent
PRD functions are often necessary to interpret polarization profiles
formed in the presence of weak magnetic fields. Our aim is to present
an efficient decomposition technique, and the numerical method to solve
the concerned angle-dependent line transfer problem. Together with the
standard Stokes decomposition technique, we employ Fourier expansion
over the outgoing azimuth angle to express in a more convenient form,
the angle-dependent PRD function for the Hanle effect. It allows the
use of angle-dependent frequency domains of Bommier to solve the Hanle
transfer problem. Such an approach is self-consistent and accurate
compared to a recent approach where angle-averaged frequency domains
were used to solve the same problem. We show that it is necessary to
incorporate angle-dependent frequency domains instead of angle-averaged
frequency domains to solve the Hanle transfer problem accurately,
especially for the Stokes U parameter. The importance of using
angle-dependent domains has been highlighted by taking the example
of Hanle effect in the case of line transfer with vertical magnetic
fields in a slab atmosphere. We have also studied the case of polarized
line formation when micro-turbulent magnetic fields are present. The
difference between angle-averaged and angle-dependent solutions is
enhanced by the presence of micro-turbulent fields.
Title: Quantum interference with angle-dependent partial frequency
redistribution: solution of the polarized line transfer in the
non-magnetic case
Authors: Supriya, H. D.; Smitha, H. N.; Nagendra, K. N.; Ravindra,
B.; Sampoorna, M.
Bibcode: 2013MNRAS.429..275S
Altcode:
Angle-dependent partial frequency redistribution (PRD) matrices
represent the physical redistribution in the process of light scattering
on atoms. For the purpose of numerical simplicity, it is a common
practice in astrophysical literature to use the angle-averaged versions
of these matrices, in the line transfer computations. The aim of this
paper is to study the combined effects of angle-dependent PRD and the
quantum interference phenomena arising either between the fine structure
(J) states of a two-term atom or between the hyperfine structure (F)
states of a two-level atom. We restrict our attention to the case
of non-magnetic and collisionless line scattering on atoms. A rapid
method of solution based on Neumann series expansion is developed to
solve the angle-dependent PRD problem including quantum interference in
an atomic system. We discuss the differences that occur in the Stokes
profiles when angle-dependent PRD mechanism is taken into account.
Title: Polarized line formation with J-state interference in the
presence of magnetic fields: A Heuristic treatment of collisional
frequency redistribution
Authors: Smitha, H. N.; Nagendra, K. N.; Sampoorna, M.; Stenflo, J. O.
Bibcode: 2013JQSRT.115...46S
Altcode: 2012arXiv1209.0243S
An expression for the partial frequency redistribution (PRD) matrix
for line scattering in a two-term atom, which includes the J-state
interference between its fine structure line components is derived. The
influence of collisions (both elastic and inelastic) and an external
magnetic field on the scattering process is taken into account. The
lower term is assumed to be unpolarized and infinitely sharp. The
linear Zeeman regime in which the Zeeman splitting is much smaller than
the fine structure splitting is considered. The inelastic collision
rates between the different levels are included in our treatment. We
account for the depolarization caused by the collisions coupling the
fine structure states of the upper term, but neglect the polarization
transfer between the fine structure states. When the fine structure
splitting goes to zero, we recover the redistribution matrix that
represents the scattering on a two-level atom (which exhibits only
m-state interference—namely the Hanle effect). The way in which the
multipolar index of the scattering atom enters into the expression
for the redistribution matrix through the collisional branching
ratios is discussed. The properties of the redistribution matrix are
explored for a single scattering process for a L=0→1→0 scattering
transition with S=1/2 (a hypothetical doublet centered at 5000 Å and
5001 Å). Further, a method for solving the Hanle radiative transfer
equation for a two-term atom in the presence of collisions, PRD, and
J-state interference is developed. The Stokes profiles emerging from
an isothermal constant property medium are computed.
Title: Polarized Line Transfer with F-state Interference in a
Non-magnetic Medium: Partial Frequency Redistribution Effects in
the Collisionless Regime
Authors: Smitha, H. N.; Sowmya, K.; Nagendra, K. N.; Sampoorna, M.;
Stenflo, J. O.
Bibcode: 2012ApJ...758..112S
Altcode: 2012arXiv1208.6369S
Quantum interference phenomena manifest themselves in several ways
in the polarized solar spectrum formed due to coherent scattering
processes. One such effect arises due to interference between the fine
structure (J) states giving rise to multiplets. Another effect is that
which arises due to interference between the hyperfine structure (F)
states. We extend the redistribution matrix derived for the J-state
interference to the case of F-state interference. We then incorporate
it into the polarized radiative transfer equation and solve it for
isothermal constant property slab atmospheres. The relevant transfer
equation is solved using a polarized approximate lambda iteration (PALI)
technique based on operator perturbation. An alternative method derived
from the Neumann series expansion is also proposed and is found to be
relatively more efficient than the PALI method. The effects of partial
frequency redistribution and the F-state interference on the shapes
of the linearly polarized Stokes profiles are discussed. The emergent
Stokes profiles are computed for hypothetical line transitions arising
due to hyperfine structure splitting of the upper J = 3/2 and lower J =
1/2 levels of a two-level atom model with nuclear spin Is
= 3/2. We confine our attention to the non-magnetic scattering in the
collisionless regime.
Title: Polarized Partial Frequency Redistribution in Subordinate
Lines. II. Solution of the Transfer Equation with Rayleigh Scattering
Authors: Nagendra, K. N.; Sampoorna, M.
Bibcode: 2012ApJ...757...33N
Altcode:
It is quite common in line formation theory to treat scattering
in subordinate lines under the assumption of complete frequency
redistribution (CRD). The partial frequency redistribution (PRD) in
subordinate lines cannot always be approximated by CRD, especially
when the polarization state of the line radiation is taken into
account. Here we investigate the PRD effects in subordinate lines
including scattering polarization. The line formation is described by
a polarized non-LTE line transfer equation based on a two-level atom
model. We use the recently derived subordinate line redistribution
matrix. We devise polarized approximate lambda iteration methods to
solve the concerned transfer problem. The linear polarization profiles
of subordinate lines formed in non-magnetic (Rayleigh) scattering
atmospheres are discussed. We consider one-dimensional isothermal
planar model atmospheres. We show that in the polarized line transfer
calculations of subordinate lines, PRD plays as important of a role
as it does in the case of resonance lines. We also study the effect
of collisions on linear polarization profiles of subordinate lines.
Title: The effect of electron scattering redistribution on atomic
line polarization
Authors: Supriya, H. D.; Nagendra, K. N.; Sampoorna, M.; Ravindra, B.
Bibcode: 2012MNRAS.425..527S
Altcode: 2012MNRAS.tmp.3423S
The polarization of spectral lines is generated by the scattering
of angularly anisotropic incident radiation field on the atoms
in the stellar atmosphere. This atomic scattering polarization is
modified by frequency non-coherent scattering of line photons on free
electrons. With modern spectropolarimeters of high sensitivity, it
is possible to detect such changes in the spectral line polarization
caused by scattering on electrons. We present new and efficient
numerical techniques to solve the problem of line radiative transfer
with atomic and electron scattering frequency redistribution in planar
media. The evaluation and use of angle-dependent partial frequency
redistribution functions (both atomic and electron scattering type) in
the transfer equation require a lot of computing effort. In this paper,
we apply a decomposition technique to handle this numerically difficult
problem. This recently developed technique is applied for the first time
to the electron scattering partial redistribution. This decomposition
technique allows us to devise fast iterative methods of solving the
polarized line transfer equation. An approximate lambda iteration
(ALI) method and a method based on Neumann series expansion of the
polarized source vector are proposed. We show that these numerical
methods can be used to obtain a solution of the problem, when both
atomic and electron scattering partial frequency redistribution are
considered together. This is in contrast with the classical numerical
methods which require a great amount of computing time. We show the
importance of electron scattering redistribution in the far wing
line polarization, which has practical implications in the analysis
of polarized stellar or solar spectra, where non-coherent electron
scattering controls the line wing transfer.
Title: Blend lines in the polarized spectrum of the Sun
Authors: Sowmya, K.; Nagendra, K. N.; Sampoorna, M.
Bibcode: 2012MNRAS.423.2949S
Altcode: 2012MNRAS.tmp.3077S; 2015arXiv151207728S
Blend lines form an integral part of the theoretical analysis and
modelling of the polarized spectrum of the Sun. Their interaction with
other spectral lines needs to be explored and understood before we
can properly use the main spectral lines to diagnose the Sun. They are
known to cause a decrease in the polarization in the wings of the main
line on which they superpose, or in the polarization of the continuum,
when they are assumed to be formed either under the local thermodynamic
equilibrium (LTE) conditions or when their intrinsic polarizability
factor is zero. In this paper, we describe the theoretical framework
to include the blend lines formed under non-LTE conditions, in the
radiative transfer equation, and the numerical techniques to solve
it. The properties of a blend line having an intrinsic polarization
of its own and its interaction with the main line are discussed. The
results of our analysis show that the influence of the blend lines
on the main spectral lines, though small in the present context, is
important and needs to be considered when interpreting the polarized
spectral lines in the second solar spectrum.
Title: Forward-scattering Hanle effect in the solar Ca I 4227 Å line
Authors: Frisch, H.; Anusha, L. S.; Bianda, M.; Holzreuter, R.;
Nagendra, K. N.; Ramelli, R.; Sampoorna, M.; Smitha, H. N.; Stenflo,
J. O.
Bibcode: 2012EAS....55...59F
Altcode:
High sensitivity spectropolarimetric observations of the four Stokes
parameters of the solar Ca I 4227 Å line have been performed in
October 2010 at IRSOL with the ZIMPOL polarimeter, near the disk center,
outside an active region (Bianda et al. 2011). They were analyzed in
Anusha et al. 2011 with a combination of detailed radiative transfer
modelling of the Hanle effect for the linear polarization and weak
field Zeeman approximation for the circular polarization. This approach
made possible a unique determination of the magnetic field vector at
various positions along the slit of the spectrograph. A summary of
the observations and of their analysis is presented here.
Title: J-state interference signatures in the second solar
spectrum. Modeling the Cr i triplet at 5204-5208 Å
Authors: Smitha, H. N.; Nagendra, K. N.; Stenflo, J. O.; Bianda, M.;
Sampoorna, M.; Ramelli, R.; Anusha, L. S.
Bibcode: 2012A&A...541A..24S
Altcode: 2012arXiv1203.4934S
The scattering polarization in the solar spectrum is traditionally
modeled with each spectral line treated separately, but this is
generally inadequate for multiplets where J-state interference
plays a significant role. Through simultaneous observations of all
the 3 lines of a Cr i triplet, combined with realistic radiative
transfer modeling of the data, we show that it is necessary to include
J-state interference consistently when modeling lines with partially
interacting fine structure components. Polarized line formation theory
that includes J-state interference effects together with partial
frequency redistribution for a two-term atom is used to model the
observations. Collisional frequency redistribution is also accounted
for. We show that the resonance polarization in the Cr i triplet is
strongly affected by the partial frequency redistribution effects in
the line core and near wing peaks. The Cr i triplet is quite sensitive
to the temperature structure of the photospheric layers. Our complete
frequency redistribution calculations in semi-empirical models of the
solar atmosphere cannot reproduce the observed near wing polarization or
the cross-over of the Stokes Q/I line polarization about the continuum
polarization level that is due to the J-state interference. When
however partial frequency redistribution is included, a good fit to
these features can be achieved. Further, to obtain a good fit to the
far wings, a small temperature enhancement of the FALF model in the
photospheric layers is necessary.
Title: Polarized Line Formation in Multi-dimensional Media. V. Effects
of Angle-dependent Partial Frequency Redistribution
Authors: Anusha, L. S.; Nagendra, K. N.
Bibcode: 2012ApJ...746...84A
Altcode: 2013arXiv1308.3443A
The solution of polarized radiative transfer equation with
angle-dependent (AD) partial frequency redistribution (PRD) is a
challenging problem. Modeling the observed, linearly polarized strong
resonance lines in the solar spectrum often requires the solution of
the AD line transfer problems in one-dimensional or multi-dimensional
(multi-D) geometries. The purpose of this paper is to develop an
understanding of the relative importance of the AD PRD effects and the
multi-D transfer effects and particularly their combined influence on
the line polarization. This would help in a quantitative analysis of the
second solar spectrum (the linearly polarized spectrum of the Sun). We
consider both non-magnetic and magnetic media. In this paper we reduce
the Stokes vector transfer equation to a simpler form using a Fourier
decomposition technique for multi-D media. A fast numerical method
is also devised to solve the concerned multi-D transfer problem. The
numerical results are presented for a two-dimensional medium with a
moderate optical thickness (effectively thin) and are computed for
a collisionless frequency redistribution. We show that the AD PRD
effects are significant and cannot be ignored in a quantitative fine
analysis of the line polarization. These effects are accentuated by the
finite dimensionality of the medium (multi-D transfer). The presence
of magnetic fields (Hanle effect) modifies the impact of these two
effects to a considerable extent.
Title: Radiative transfer with J-state interference in a two-term
atom. Partial frequency redistribution in the non-magnetic case
Authors: Smitha, H. N.; Nagendra, K. N.; Sampoorna, M.; Stenflo, J. O.
Bibcode: 2011A&A...535A..35S
Altcode:
Context. Quantum interference phenomena play a fundamental role in
the formation of linear polarization that arises from scattering
processes in multiplets of the solar spectrum. In particular,
the J-state interference between different line components of a
multiplet (arising from transitions in a two-term atom) produces
significant effects in the linearly polarized spectra.
Aims:
We aim to solve the polarized radiative transfer equation for a
two-term atom with the unpolarized lower term in isothermal slabs,
including the effect of the interference between the upper J-states
and partial frequency redistribution (PRD). We consider only the case
of non-magnetic scattering.
Methods: The PRD matrix for the
J-state interference derived in previous works is incorporated into
the polarized transfer equation. The standard form of the two-level
atom transfer equation is extended to a two-term atom. The transfer
problem is then solved using a traditional polarized approximate lambda
iteration method.
Results: We show how the PRD and the J-state
interference together affect the shapes of the (I,Q/I) profiles. We
present the benchmark solutions for isothermal, constant-property
slabs of a given optical thickness. We consider a hypothetical doublet
produced by an L = 0 → 1 → 0 scattering transition with spin S =
1/2. We present the results in the form of Stokes (I,Q/I) profiles for
different values of (i) the line separation, (ii) optical thickness,
(iii) thermalization parameter, and (iv) the continuum opacity.
Title: Spectral line polarization with angle-dependent partial
frequency redistribution. IV. Scattering expansion method for the
Hanle effect
Authors: Nagendra, K. N.; Sampoorna, M.
Bibcode: 2011A&A...535A..88N
Altcode:
Context. The partial frequency redistribution (PRD) effects in line
scattering are necessary ingredients for interpreting the linear
polarization observed in strong resonance lines. It is a common
practice to use angle-averaged PRD functions for simplicity (obtained
by averaging over all scattering angles). It has been established that
the use of angle-dependent PRD functions instead of angle-averaged
functions is essential for weak fields.
Aims: Here we present
an efficient iterative method to solve the polarized line radiative
transfer equation in weak magnetic fields using angle-dependent
PRD functions.
Methods: Based on the theory of Stokes vector
decomposition for the Hanle effect combined with the Fourier azimuthal
expansion of the angle-dependent PRD function, we try to formulate an
efficient numerical method of solving the concerned transfer problem
in one-dimensional media. This iterative method (referred to as the
scattering expansion method, SEM) is based on a series expansion of
the polarized source vector in mean number of scatterings (Neumann
series expansion). We apply the SEM approach to handle both the exact
and various approximate forms of the Hanle scattering redistribution
matrix.
Results: The SEM is shown to be an efficient method
to solve angle-dependent PRD problems involving the Hanle effect. We
show that compared to the earlier methods such as the perturbation
methods, the SEM is stable and faster. We find that angle-dependent
PRD significantly affects the Stokes U parameter.
Title: Polarized Line Formation in Multi-dimensional Media. IV. A
Fourier Decomposition Technique to Formulate the Transfer Equation
with Angle-dependent Partial Frequency Redistribution
Authors: Anusha, L. S.; Nagendra, K. N.
Bibcode: 2011ApJ...739...40A
Altcode: 2013arXiv1308.3447A
To explain the linear polarization observed in spatially resolved
structures in the solar atmosphere, the solution of polarized radiative
transfer (RT) equation in multi-dimensional (multi-D) geometries is
essential. For strong resonance lines, partial frequency redistribution
(PRD) effects also become important. In a series of papers, we have been
investigating the nature of Stokes profiles formed in multi-D media
including PRD in line scattering. For numerical simplicity, so far we
have restricted our attention to the particular case of PRD functions
which are averaged over all the incident and scattered directions. In
this paper, we formulate the polarized RT equation in multi-D media
that takes into account the Hanle effect with angle-dependent PRD
functions. We generalize here to the multi-D case the method for Fourier
series expansion of angle-dependent PRD functions originally developed
for RT in one-dimensional geometry. We show that the Stokes source
vector S = (SI , SQ , SU ) T
and the Stokes vector I = (I, Q, U) T can be expanded
in terms of infinite sets of components \tilde{S}^(k), \tilde{I}^(k),
respectively, k in [0, +∞). We show that the components \tilde{S}^(k)
become independent of the azimuthal angle (phiv) of the scattered ray,
whereas the components \tilde{I}^(k) remain dependent on phiv due to the
nature of RT in multi-D geometry. We also establish that \tilde{S}^(k)
and \tilde{I}^(k) satisfy a simple transfer equation, which can be
solved by any iterative method such as an approximate Lambda iteration
or a Bi-Conjugate Gradient-type projection method provided we truncate
the Fourier series to have a finite number of terms.
Title: Polarized Line Formation in Multi-dimensional Media. III. Hanle
Effect with Partial Frequency Redistribution
Authors: Anusha, L. S.; Nagendra, K. N.
Bibcode: 2011ApJ...738..116A
Altcode: 2011arXiv1105.4052A
In two previous papers, we solved the polarized radiative transfer
(RT) equation in multi-dimensional (multi-D) geometries with partial
frequency redistribution as the scattering mechanism. We assumed
Rayleigh scattering as the only source of linear polarization (Q/I,
U/I) in both these papers. In this paper, we extend these previous
works to include the effect of weak oriented magnetic fields (Hanle
effect) on line scattering. We generalize the technique of Stokes
vector decomposition in terms of the irreducible spherical tensors
{T}^K_Q, developed by Anusha & Nagendra, to the case of RT with
Hanle effect. A fast iterative method of solution (based on the
Stabilized Preconditioned Bi-Conjugate-Gradient technique), developed
by Anusha et al., is now generalized to the case of RT in magnetized
three-dimensional media. We use the efficient short-characteristics
formal solution method for multi-D media, generalized appropriately to
the present context. The main results of this paper are the following:
(1) a comparison of emergent (I, Q/I, U/I) profiles formed in
one-dimensional (1D) media, with the corresponding emergent, spatially
averaged profiles formed in multi-D media, shows that in the spatially
resolved structures, the assumption of 1D may lead to large errors in
linear polarization, especially in the line wings. (2) The multi-D RT
in semi-infinite non-magnetic media causes a strong spatial variation
of the emergent (Q/I, U/I) profiles, which is more pronounced in the
line wings. (3) The presence of a weak magnetic field modifies the
spatial variation of the emergent (Q/I, U/I) profiles in the line core,
by producing significant changes in their magnitudes.
Title: Analysis of the Forward-scattering Hanle Effect in the Ca I
4227 Å Line
Authors: Anusha, L. S.; Nagendra, K. N.; Bianda, M.; Stenflo, J. O.;
Holzreuter, R.; Sampoorna, M.; Frisch, H.; Ramelli, R.; Smitha, H. N.
Bibcode: 2011ApJ...737...95A
Altcode:
Coherent scattering of limb-darkened radiation is responsible for the
generation of the linearly polarized spectrum of the Sun (the Second
Solar Spectrum). This Second Solar Spectrum is usually observed near the
limb of the Sun, where the polarization amplitudes are largest. At the
center of the solar disk the linear polarization is zero for an axially
symmetric atmosphere. Any mechanism that breaks the axial symmetry (like
the presence of an oriented magnetic field, or resolved inhomogeneities
in the atmosphere) can generate a non-zero linear polarization. In the
present paper we study the linear polarization near the disk center
in a weakly magnetized region, where the axisymmetry is broken. We
present polarimetric (I, Q/I, U/I, and V/I) observations of the Ca
I 4227 Å line recorded around μ = cos θ = 0.9 (where θ is the
heliocentric angle) and a modeling of these observations. The high
sensitivity of the instrument (ZIMPOL-3) makes it possible to measure
the weak polarimetric signals with great accuracy. The modeling of
these high-quality observations requires the solution of the polarized
radiative transfer equation in the presence of a magnetic field. For
this we use standard one-dimensional model atmospheres. We show that the
linear polarization is mainly produced by the Hanle effect (rather than
by the transverse Zeeman effect), while the circular polarization is due
to the longitudinal Zeeman effect. A unique determination of the full
\bm {B} vector may be achieved when both effects are accounted for. The
field strengths required for the simultaneous fitting of Q/I, U/I, and
V/I are in the range 10-50 G. The shapes and signs of the Q/I and U/I
profiles are highly sensitive to the orientation of the magnetic field.
Title: Observations of the forward scattering Hanle effect in the
Ca I 4227 Å line
Authors: Bianda, M.; Ramelli, R.; Anusha, L. S.; Stenflo, J. O.;
Nagendra, K. N.; Holzreuter, R.; Sampoorna, M.; Frisch, H.; Smitha,
H. N.
Bibcode: 2011A&A...530L..13B
Altcode: 2011arXiv1105.2157B
Chromospheric magnetic fields are notoriously difficult to measure. The
chromospheric lines are broad, while the fields are producing
a minuscule Zeeman-effect polarization. A promising diagnostic
alternative is provided by the forward-scattering Hanle effect, which
can be recorded in chromospheric lines such as the He i 10 830 Å
and the Ca i 4227 Å lines. We present a set of spectropolarimetric
observations of the full Stokes vector obtained near the center of the
solar disk in the Ca i 4227 Å line with the ZIMPOL polarimeter at the
IRSOL observatory. We detect a number of interesting forward-scattering
Hanle effect signatures, which we model successfully using polarized
radiative transfer. Here we focus on the observational aspects, while
a separate companion paper deals with the theoretical modeling.
Title: Polarized Line Formation with J-state Interference in the
Presence of Magnetic Fields. I. Partial Frequency Redistribution in
the Collisionless Regime
Authors: Smitha, H. N.; Sampoorna, M.; Nagendra, K. N.; Stenflo, J. O.
Bibcode: 2011ApJ...733....4S
Altcode:
Quantum interference phenomena play a fundamental role in astrophysical
spectra that are formed by coherent scattering processes. Here we
derive a partial frequency redistribution (PRD) matrix that includes
J-state interference in the presence of magnetic fields of arbitrary
strength. The paper focuses on PRD in the collisionless regime, which
in the traditional PRD terminology is referred to as Hummer's type-II
scattering. By limiting the treatment to the linear Zeeman regime,
for which the Zeeman splitting is much smaller than the fine-structure
splitting, it is possible to formulate analytical expressions for
the PRD matrices. In the special case of non-magnetic scattering we
recover the redistribution matrix derived from an independent quantum
electrodynamic formulation based on the metalevel theory.
Title: Linear Polarization of the Solar Ca I 4227 Å Line: Modeling
with Radiative Transfer and Last Scattering Approximation
Authors: Anusha, L. S.; Stenflo, J. O.; Frisch, H.; Bianda, M.;
Holzreuter, R.; Nagendra, K. N.; Sampoorna, M.; Ramelli, R.
Bibcode: 2011ASPC..437...57A
Altcode:
To model the Ca I 4227 Å line polarization, radiative transfer
effects with partial frequency redistribution (PRD) must be taken into
account. The numerical solution of the relevant polarized radiative
transfer (RT) equations is computationally very demanding. The
“last scattering approximation” (LSA) is a concept allowing
faster methods to be devised. It is based on the remark that a single
scattering of the radiation field is sufficient for creating most
of the polarization. Its key ingredient is the anisotropy of the
radiation field. If the anisotropy is extracted from the observed
center to limb variation of the intensity profile, only the wings
of the Q/I spectrum can be modeled (Sampoorna et al. 2009). We show
here that the core region may be modeled as well if one takes into
account the depth variation of the anisotropy which is obtained from
an unpolarized multilevel RT (Anusha et al. 2010). After a validation
of the LSA approach by comparison with a polarized RT calculation, we
apply both approaches to model recent observations of the Ca I 4227 Å
line polarization taken on the quiet Sun. Apart from a global scaling
factor, both approaches give a very good fit to the Q/I spectrum for
all the wavelengths. As the LSA is 8 times faster than the RT approach,
we can recommend it as an efficient method to analyze other strong
resonance lines in the second solar spectrum.
Title: Observations of the Solar Ca I 4227 Å Line
Authors: Bianda, M.; Ramelli, R.; Stenflo, J. O.; Anusha, L. S.;
Nagendra, K. N.; Sampoorna, M.; Holzreuter, R.; Frisch, H.
Bibcode: 2011ASPC..437...67B
Altcode:
Our aim is to understand some interesting polarization features
observed in the solar Ca I 4277 Å line. Here we only discuss the
observational aspects. Observations have also been made in other
chromospheric lines within a few hours of those in the Ca I 4227 Å
line, in the same region near the north solar limb, to illustrate the
potential of simultaneous observations in different lines.
Title: Spectral line polarization with angle-dependent partial
frequency redistribution. II. Accelerated lambda iteration and
scattering expansion methods for the Rayleigh scattering
Authors: Sampoorna, M.; Nagendra, K. N.; Frisch, H.
Bibcode: 2011A&A...527A..89S
Altcode:
Context. The linear polarization of strong resonance lines observed in
the solar spectrum is created by the scattering of the photospheric
radiation field. This polarization is sensitive to the form of
the partial frequency redistribution (PRD) function used in the
line radiative transfer equation. Observations have been analyzed
until now with angle-averaged PRD functions. With an increase in
the polarimetric sensitivity and resolving power of the present-day
telescopes, it will become possible to detect finer effects caused
by the angle dependence of the PRD functions.
Aims: We devise
new efficient numerical methods to solve the polarized line transfer
equation with angle-dependent PRD, in plane-parallel cylindrically
symmetrical media. We try to bring out the essential differences
between the polarized spectra formed under angle-averaged and the
more realistic case of angle-dependent PRD functions.
Methods:
We use a recently developed Stokes vector decomposition technique to
formulate three different iterative methods tailored for angle-dependent
PRD functions. Two of them are of the accelerated lambda iteration
type, one is based on the core-wing approach, and the other one on
the frequency by frequency approach suitably generalized to handle
angle-dependent PRD. The third one is based on a series expansion in
the mean number of scattering events (Neumann series expansion).
Results: We show that all these methods work well on this difficult
problem of polarized line formation with angle-dependent PRD. We
present several benchmark solutions with isothermal atmospheres to
show the performance of the three numerical methods and to analyze the
role of the angle-dependent PRD effects. For weak lines, we find no
significant effects when the angle-dependence of the PRD functions is
taken into account. For strong lines, we find a significant decrease in
the polarization, the largest effect occurring in the near wing maxima.
Title: Polarized Line Formation in Multi-dimensional Media. II. A
Fast Method to Solve Problems with Partial Frequency Redistribution
Authors: Anusha, L. S.; Nagendra, K. N.; Paletou, F.
Bibcode: 2011ApJ...726...96A
Altcode:
In the previous paper of this series, we presented a formulation of
the polarized radiative transfer equation for resonance scattering
with partial frequency redistribution (PRD) in multi-dimensional
media for a two-level atom model with unpolarized ground level, using
the irreducible spherical tensors {T}^K_Q(i,Ω) for polarimetry. We
also presented a polarized approximate lambda iteration method to
solve this equation using the Jacobi iteration scheme. The formal
solution used was based on a simple finite volume technique. In this
paper, we develop a faster and more efficient method which uses the
projection techniques applied to the radiative transfer equation (the
Stabilized Preconditioned Bi-Conjugate Gradient method). We now use a
more accurate formal solver, namely the well-known two-dimensional (2D)
short characteristics method. Using the numerical method developed in
Paper I, we can consider only simpler cases of finite 2D slabs due to
computational limitations. Using the method developed in this paper,
we could compute PRD solutions in 2D media in the more difficult
context of semi-infinite 2D slabs also. We present several solutions
which may serve as benchmarks in future studies in this area.
Title: Polarized Line Formation in Multi-dimensional
Media. I. Decomposition of Stokes Parameters in Arbitrary Geometries
Authors: Anusha, L. S.; Nagendra, K. N.
Bibcode: 2011ApJ...726....6A
Altcode:
The solution of the polarized line radiative transfer (RT) equation in
multi-dimensional geometries has been rarely addressed and only under
the approximation that the changes of frequencies at each scattering
are uncorrelated (complete frequency redistribution). With the
increase in the resolution power of telescopes, being able to handle
RT in multi-dimensional structures becomes absolutely necessary. In
the present paper, our first aim is to formulate the polarized RT
equation for resonance scattering in multi-dimensional media, using the
elegant technique of irreducible spherical tensors {T}^K_Q(i, Ω). Our
second aim is to develop a numerical method of a solution based on the
polarized approximate lambda iteration (PALI) approach. We consider both
complete frequency redistribution and partial frequency redistribution
(PRD) in the line scattering. In a multi-dimensional geometry, the
radiation field is non-axisymmetrical even in the absence of a symmetry
breaking mechanism such as an oriented magnetic field. We generalize
here to the three-dimensional (3D) case, the decomposition technique
developed for the Hanle effect in a one-dimensional (1D) medium which
allows one to represent the Stokes parameters I, Q, U by a set of six
cylindrically symmetrical functions. The scattering phase matrix is
expressed in terms of {T}^K_Q(i, Ω) (i=0,1,2, K=0,1,2, -K ≤ Q ≤
+K), with Ω being the direction of the outgoing ray. Starting from the
definition of the source vector, we show that it can be represented
in terms of six components SK Q independent of
Ω. The formal solution of the multi-dimensional transfer equation
shows that the Stokes parameters can also be expanded in terms of
{T}^K_Q(i, Ω). Because of the 3D geometry, the expansion coefficients
IK Q remain Ω-dependent. We show that each
IK Q satisfies a simple transfer equation with a
source term SK Q and that this transfer equation
provides an efficient approach for handling the polarized transfer in
multi-dimensional geometries. A PALI method for 3D, associated with
a core-wing separation method for treating PRD, is developed. It is
tested by comparison with 1D solutions, and several benchmark solutions
in the 3D case are given.
Title: Joint Discussion 10: 3D views on cool stellar atmospheres -
theory meets observation
Authors: Nagendra, K. N.; Bonifacio, P.; Ludwig, H. -G.
Bibcode: 2010HiA....15..331N
Altcode:
Much of what we know about the chemical composition of the Universe
actually stems from the chemical composition of stars, which is often
deciphered from the spectra emerging from their atmospheres. Cool,
low-mass and long-living stars allow to study the evolution of
the Universe's chemistry from a time shortly after the big bang
until today. The observation and interpretation of stellar spectra
is a classical field in astronomy but is still undergoing vivid
developments. The enormous increase in available computational
resources opened-up possibilities which led to a revolution in the
degree of realism to which modelers can mimic Nature. High-resolution,
high-stability, high-efficiency spectrographs are now routinely
providing stellar spectra whose full information content can only be
exploited if a very much refined description of a stellar atmosphere
is at hand.
Title: Generalization of the Last Scattering Approximation for the
Second Solar Spectrum Modeling: The Ca I 4227 Å Line as a Case Study
Authors: Anusha, L. S.; Nagendra, K. N.; Stenflo, J. O.; Bianda, M.;
Sampoorna, M.; Frisch, H.; Holzreuter, R.; Ramelli, R.
Bibcode: 2010ApJ...718..988A
Altcode:
To model the second solar spectrum (the linearly polarized spectrum
of the Sun that is due to coherent scattering processes), one needs
to solve the polarized radiative transfer (RT) equation. For strong
resonance lines, partial frequency redistribution (PRD) effects must be
accounted for, which make the problem computationally demanding. The
"last scattering approximation" (LSA) is a concept that has been
introduced to make this highly complex problem more tractable. An
earlier application of a simple LSA version could successfully model
the wings of the strong Ca I 4227 Å resonance line in Stokes Q/I
(fractional linear polarization), but completely failed to reproduce
the observed Q/I peak in the line core. Since the magnetic field
signatures from the Hanle effect only occur in the line core, we need
to generalize the existing LSA approach if it is to be useful for the
diagnostics of chromospheric and turbulent magnetic fields. In this
paper, we explore three different approximation levels for LSA and
compare each of them with the benchmark represented by the solution of
the full polarized RT, including PRD effects. The simplest approximation
level is LSA-1, which uses the observed center-to-limb variation of
the intensity profile to obtain the anisotropy of the radiation field
at the surface, without solving any transfer equation. In contrast,
the next two approximation levels use the solution of the unpolarized
transfer equation to derive the anisotropy of the incident radiation
field and use it as an input. In the case of LSA-2, the anisotropy
at level τλ = μ, the atmospheric level from which an
observed photon is most likely to originate, is used. LSA-3, on the
other hand, makes use of the full depth dependence of the radiation
anisotropy. The Q/I formula for LSA-3 is obtained by keeping the
first term in a series expansion of the Q-source function in powers
of the mean number of scattering events. Computationally, LSA-1 is 21
times faster than LSA-2, which is 5 times faster than the more general
LSA-3, which itself is 8 times faster than the polarized RT approach. A
comparison of the calculated Q/I spectra with the RT benchmark shows
excellent agreement for LSA-3, including good modeling of the Q/I
core region with its PRD effects. In contrast, both LSA-1 and LSA-2
fail to model the core region. The RT and LSA-3 approaches are then
applied to model the recently observed Q/I profile of the Ca I 4227
Å line in quiet regions of the Sun. Apart from a global scale factor
both give a very good fit to the Q/I spectra for all the wavelengths,
including the core peak and blend line depolarizations. We conclude
that LSA-3 is an excellent substitute for the full polarized RT and
can be used to interpret the second solar spectrum, including the
Hanle effect with PRD. It also allows the techniques developed for
unpolarized three-dimensional RT to be applied to the modeling of the
second solar spectrum.
Title: Recent Developments in Polarized Line Formation in Magnetic
Fields
Authors: Nagendra, K. N.; Sampoorna, M.; Anusha, L. S.
Bibcode: 2010ASSP...17..139N
Altcode: 2010rast.conf..139N; 2010rasp.book..139N
The nature of solar surface magnetism has been an open problem in
solar physics. In this paper we address three frontline problems
of spectropolarimetry of the Sun. We first review the theoretical
formulation and numerical solutions of Zeeman absorption and then the
Hanle scattering phenomena in 'turbulent magnetic fields'. We show that
the mean emergent Stokes profiles cannot be obtained by simply averaging
the scattering and absorption opacities, respectively, over a given
distribution of the random field (except when the micro-turbulence
prevails). A new formulation of the transfer equation is necessary
to study the astrophysically interesting meso-turbulence case. Such
formulations of the stochastic polarized radiative transfer problems for
absorbing and scattering media are developed only in recent years. We
review them and show some results computed by our new formulations.Until
recent years the solution of the polarized line radiative transfer
equation in LTE (Zeeman absorption in strong fields), and its NLTE
counterpart (Hanle scattering in weak fields), were treated as two
disparate problems. The reason for this artificial division was more
due to the theoretical and numerical difficulties encountered in the
solution of the combined Hanle-Zeeman radiative transfer equation. A
very general form of the transfer equation was formulated only a
decade ago, for the case of complete frequency redistribution. A
more difficult case of partial frequency redistribution is explored
by us recently. We review these developments through a study of the
Hanle-Zeeman effect in arbitrary strength magnetic fields.
Title: The Hanle Effect as Diagnostic Tool for Turbulent Magnetic
Fields
Authors: Anusha, L. S.; Sampoorna, M.; Frisch, H.; Nagendra, K. N.
Bibcode: 2010ASSP...19..390A
Altcode: 2010mcia.conf..390A
The Hanle effect is calculated for a random magnetic field characterized
by a finite correlation length and a probability density function
of the magnetic field vector. It is shown that linear polarization
is essentially independent of the magnetic field correlation length,
but strongly depends on the distribution of the field strength.
Title: Preconditioned Bi-conjugate Gradient Method for Radiative
Transfer in Spherical Media
Authors: Anusha, L. S.; Nagendra, K. N.; Paletou, F.; Léger, L.
Bibcode: 2009ApJ...704..661A
Altcode: 2009arXiv0906.2926A
A robust numerical method called the Preconditioned Bi-Conjugate
Gradient (Pre-BiCG) method is proposed for the solution of the radiative
transfer equation in spherical geometry. A variant of this method called
Stabilized Preconditioned Bi-Conjugate Gradient (Pre-BiCG-STAB) is also
presented. These are iterative methods based on the construction of a
set of bi-orthogonal vectors. The application of the Pre-BiCG method
in some benchmark tests shows that the method is quite versatile,
and can handle difficult problems that may arise in astrophysical
radiative transfer theory.
Title: The Hanle effect in a random magnetic field. Dependence of
the polarization on statistical properties of the magnetic field
Authors: Frisch, H.; Anusha, L. S.; Sampoorna, M.; Nagendra, K. N.
Bibcode: 2009A&A...501..335F
Altcode:
Context: The Hanle effect is used to determine weak turbulent magnetic
fields in the solar atmosphere, usually assuming that the angular
distribution is isotropic, the magnetic field strength constant, and
that micro-turbulence holds, i.e. that the magnetic field correlation
length is much less than a photon mean free path.
Aims: To
examine the sensitivity of turbulent magnetic field measurements
to these assumptions, we study the dependence of Hanle effect on
the magnetic field correlation length, its angular, and strength
distributions.
Methods: We introduce a fairly general random
magnetic field model characterized by a correlation length and a
magnetic field vector distribution. Micro-turbulence is recovered
when the correlation length goes to zero and macro-turbulence when it
goes to infinity. Radiative transfer equations are established for
the calculation of the mean Stokes parameters and they are solved
numerically by a polarized approximate lambda iteration method.
Results: We show that optically thin spectral lines and optically
very thick ones are insensitive to the correlation length of the
magnetic field, while spectral lines with intermediate optical depths
(around 10-100) show some sensitivity to this parameter. The result is
interpreted in terms of the mean number of scattering events needed to
create the surface polarization. It is shown that the single-scattering
approximation holds good for thin and thick lines but may fail for lines
with intermediate thickness. The dependence of the polarization on the
magnetic field vector probability density function (PDF) is examined
in the micro-turbulent limit. A few PDFs with different angular and
strength distributions, but equal mean value of the magnetic field,
are considered. It is found that the polarization is in general quite
sensitive to the shape of the magnetic field strength PDF and somewhat
to the angular distribution.
Conclusions: The mean field derived
from Hanle effect analysis of polarimetric data strongly depends on
the choice of the field strength distribution used in the analysis. It
is shown that micro-turbulence is in general a safe approximation.
Title: Origin of Spatial Variations of Scattering Polarization in
the Wings of the Ca I 4227 Å Line
Authors: Sampoorna, M.; Stenflo, J. O.; Nagendra, K. N.; Bianda, M.;
Ramelli, R.; Anusha, L. S.
Bibcode: 2009ApJ...699.1650S
Altcode: 2009arXiv0906.1184S
Polarization that is produced by coherent scattering can be modified
by magnetic fields via the Hanle effect. This has opened a window to
explorations of solar magnetism in parameter domains not accessible to
the Zeeman effect. According to standard theory the Hanle effect should
only be operating in the Doppler core of spectral lines but not in the
wings. In contrast, our observations of the scattering polarization
in the Ca I 4227 Å line reveal the existence of spatial variations
of the scattering polarization throughout the far line wings. This
raises the question whether the observed spatial variations in wing
polarization have a magnetic or nonmagnetic origin. A magnetic origin
may be possible if elastic collisions are able to cause sufficient
frequency redistribution to make the Hanle effect effective in the wings
without causing excessive collisional depolarization, as suggested by
recent theories for partial frequency redistribution (PRD) with coherent
scattering in magnetic fields. To model the wing polarization we bypass
the problem of solving the full polarized radiative transfer equations
and instead apply an extended version of the technique based on the
"last scattering approximation." It assumes that the polarization
of the emergent radiation is determined by the anisotropy of the
incident radiation field at the last scattering event. We determine
this anisotropy from the observed limb darkening as a function of
wavelength throughout the spectral line. The empirical anisotropy
profile is used together with the single-scattering redistribution
matrix, which contains all the PRD, collisional, and magnetic field
effects. The model further contains a continuum opacity parameter,
which increasingly dilutes the polarized line photons as we move away
from the line center, and a continuum polarization parameter that
represents the observed polarization level far from the line. This
model is highly successful in reproducing the observed Stokes Q/I
polarization (linear polarization parallel to the nearest solar
limb), including the location of the wing polarization maxima and the
minima around the Doppler core, but it fails to reproduce the observed
spatial variations of the wing polarization in terms of magnetic field
effects with frequency redistribution. This null result points in the
direction of a nonmagnetic origin in terms of local inhomogeneities
(varying collisional depolarization, radiation-field anisotropies,
and deviations from a plane-parallel atmospheric stratification).
Title: Theory of Polarized Scattering in the Mixed Hanle-Zeeman Regime
Authors: Sampoorna, M.; Nagendra, K. N.; Stenflo, J. O.
Bibcode: 2009ASPC..405...93S
Altcode:
We present the theory of polarized partial frequency redistribution
(PRD) in the presence of arbitrary magnetic fields. Starting from the
classical theory of \citet{sampoorna_bom99} we derive the laboratory
frame PRD matrices, which we call `Hanle-Zeeman redistribution
matrices', since they cover the partially overlapping weak and strong
field regimes. We show that for the simplest case of a triplet (J=0 →
1 → 0) transition, the classical and quantum \citep{sampoorna_bom97}
treatments give identical results.
Title: Phase Matrices for Higher Multipoles of Scattering in External
Magnetic Fields
Authors: Oo, Y. Y.; San, P. P.; Sampoorna, M.; Nagendra, K. N.;
Ramachandran, G.
Bibcode: 2009ASPC..405..119O
Altcode:
Scattering phase matrices for forbidden lines are derived, in
the presence of an external magnetic field, using the quantum
electrodynamical approach. The particular case of 0 → 2 → 0 M2
transitions are considered and Stokes profiles are shown in the strong
field (Zeeman) and weak field (Hanle) limits, covering also the regime
of intermediate field strengths.
Title: Solar Polarization 5: In Honor of Jan Stenflo
Authors: Berdyugina, S. V.; Nagendra, K. N.; Ramelli, R.
Bibcode: 2009ASPC..405.....B
Altcode:
No abstract at ADS
Title: Numerical Methods in Polarized Line Formation Theory
Authors: Nagendra, K. N.; Sampoorna, M.
Bibcode: 2009ASPC..405..261N
Altcode:
We review some numerical methods and provide benchmark solutions for
the polarized line formation theory with partial redistribution (PRD)
in the presence of magnetic fields. The transfer equation remains
non-axisymmetric when written in the `Stokes vector basis'. It
is relatively easier to develop numerical methods to solve the
transfer equation for axisymmetric radiation fields. Therefore
for non-axisymmetric problems it would be necessary to expand the
azimuthal dependence of the scattering redistribution matrices in a
Fourier series. The transfer equation in this so called `reduced form'
becomes axisymmetric in the Fourier domain in which it is solved, and
the reduced intensity is then transformed into the Stokes vector basis
in real space. The advantage is that the reduced problem lends itself
to be solved by appropriately organized PALI (Polarized Approximate
Lambda Iteration) methods. We first dwell upon a frequency by frequency
method (PALI7) that uses non-domain based PRD for the Hanle scattering
problem, and then compare it with a core-wing method (PALI6) that uses
a domain based PRD. The PALI methods use operator perturbation and
involve construction of a suitable procedure to evaluate an `iterated
source vector correction'. Another important component of PALI methods
is the `Formal Solver' (for example Feautrier, short characteristic,
DELOPAR etc.). The PALI methods are extremely fast on a computer and
require very small memory. Finally, we present a simple perturbation
method to solve the Hanle-Zeeman line formation problem in arbitrary
strength magnetic fields.
Title: Commission 36: Theory of Stellar Atmospheres
Authors: Landstreet, John D.; Asplund, Martin; Spite, Monique;
Balachandran, Suchitra B.; Berdyugina, Svetlana V.; Hauschildt, Peter
H.; Ludwig, Hans G.; Mashonkina, Lyudmila I.; Nagendra, K. N.; Puls,
Joachim; Randich, M. Sofia; Tautvaisiene, Grazina
Bibcode: 2009IAUTA..27..222L
Altcode:
Commission 36 covers the whole field of the physics of stellar
atmospheres. The scientific activity in this large subject has been very
intense during the last triennium and led to the publication of a large
number of papers, which makes a complete report quite impractical. We
have therefore decided to keep the format of the preceding report:
first a list of areas of current research, then Web links for obtaining
further information.
Title: Polarization : Proving ground for methods in radiative
transfer.
Authors: Nagendra, K. N.; Anusha, L. S.; Sampoorna, M.
Bibcode: 2009MmSAI..80..678N
Altcode:
Polarization of solar lines arises due to illumination of radiating
atom by anisotropic (limb darkened/brightened) radiation. Modelling
the polarized spectra of the Sun and stars requires solution of the
line radiative transfer problem in which the relevant polarizing
physical mechanisms are incorporated. The purpose of this paper is to
describe in what different ways the polarization state of the radiation
`complicates' the numerical methods originally designed for scalar
radiative transfer. We present several interesting situations involving
the solution of polarized line transfer to prove our point. They are
(i) Comparison of the polarized approximate lambda iteration (PALI)
methods with new approaches like Bi-conjugate gradient method that
is faster, (ii) Polarized Hanle scattering line radiative transfer in
random magnetic fields, (iii) Difficulties encountered in incorporating
polarized partial frequency redistribution (PRD) matrices in line
radiative transfer codes, (iv) Technical difficulties encountered
in handling polarized specific intensity vector, some components of
which are sign changing, (v) Proving that scattering polarization is
indeed a boundary layer phenomenon. We provide credible benchmarks in
each of the above studies. We show that any new numerical methods can
be tested in the best possible way, when it is extended to include
polarization state of the radiation field in line scattering.
Title: 3D views on cool stellar atmospheres: theory meets observation
Authors: Nagendra, K. N.; Bonifacio, P.; Ludwig, H. -G.
Bibcode: 2009MmSAI..80..601N
Altcode:
No abstract at ADS
Title: Projection methods for line radiative transfer in spherical
media.
Authors: Anusha, L. S.; Nagendra, K. N.
Bibcode: 2009MmSAI..80..631A
Altcode:
An efficient numerical method called the Preconditioned Bi-Conjugate
Gradient (Pre-BiCG) method is presented for the solution of radiative
transfer equation in spherical geometry. A variant of this method called
Stabilized Preconditioned Bi-Conjugate Gradient (Pre-BiCG-STAB) is also
presented. These methods are based on projections on the subspaces of
the n dimensional Euclidean space mathbb {R}n called Krylov
subspaces. The methods are shown to be faster in terms of convergence
rate compared to the contemporary iterative methods such as Jacobi,
Gauss-Seidel and Successive Over Relaxation (SOR).
Title: Zeeman line formation in solar magnetic fields. Studies with
empirical probability distribution functions
Authors: Sampoorna, M.; Nagendra, K. N.; Frisch, H.; Stenflo, J. O.
Bibcode: 2008A&A...485..275S
Altcode:
Context: Numerical simulations of magneto-convection and analysis
of solar magnetograms provide probability distribution functions
(PDFs) for the magnetic field strength.
Aims: In the paper,
we explore the effects of these PDFs on Zeeman line formation.
Methods: We calculate the mean Stokes parameters for a Milne-Eddington
atmosphere in the limit of optically thin (micro-turbulent) and thick
(macro-turbulent) magnetic structures and also the dispersion around
the mean profiles in the optically thick limit. Several types of PDFs
are considered: (a) Voigt function and stretched exponential type
PDFs for fields with fixed direction but fluctuating strength; (b)
a cylindrically symmetrical power law for the angular distribution of
magnetic fields with given field strength; (c) composite PDFs accounting
for randomness in both strength and direction obtained by combining a
Voigt function or a stretched exponential with an angular power law. For
optically thin structures, explicit expressions are given for the mean
values of the Zeeman absorption matrix elements. We also describe how
the averaging technique for a normal Zeeman triplet may be generalized
to the more common case of anomalous Zeeman splitting patterns.
Results: We show that, for magnetic field rms fluctuations of
the order of 6 G, consistent with observational data, Stokes I is
essentially independent of the shapes of the PDFs but Stokes Q, U, and
V and also the dispersion around the mean values are quite sensitive
to the tail behavior of the PDF. We confirm a previous result that
Stokes V is less sensitive to the scale of the magnetic structures
than Stokes Q and U. The composite PDF proposed for the fluctuations of
the magnetic field vector has an angular distribution peaked about the
vertical direction for strong fields, and is isotropically distributed
for weak fields; it can be used to mimic solar surface random fields.
Title: Some aspects of polarized line formation in magneto-turbulent
media
Authors: Sampoorna, M.; Frisch, H.; Nagendra, K. N.
Bibcode: 2008NewA...13..233S
Altcode:
Observations and numerical simulations of magneto-convection show a
highly variable solar magnetic field. Using a statistical approach,
we analyze the effects of random magnetic fields on Stokes profiles of
spectral lines. We consider the micro and macro-turbulent regimes,
which provide bounds for more general random fields with finite
scales of variations. The mean Stokes parameters are obtained in the
micro-turbulent regime, by first averaging the Zeeman propagation
matrix Φ^ over the probability distribution function P( B) of the
magnetic field and then solving the concerned radiative transfer
equation. In the macro-turbulent regime, the mean solution is obtained
by averaging the emergent solution over P( B). It is assumed that B has
a Gaussian distribution defined by its mean field B0, angular
distribution and dispersion. Fluctuations parallel and perpendicular
to B0 are considered. Spectral lines are parameterized by
their strength β, which is varied over the range 1-10 4. A
detailed comparison of micro and macro-turbulent limit with mean field
solution shows that differences are important for β ⩾ 10. When β
increases, the saturation behavior of micro-turbulent profiles are
significantly different from that of mean field profiles. The Stokes
profiles shapes are explained in terms of the non-linear β-dependence
of the Unno-Rachkovsky solution using approximate expressions for the
mean absorption coefficients. These expressions when inserted in the
Unno-Rachkovsky solution can predict Stokes profiles that match with
the numerical result to a good approximation.
Title: Hanle-Zeeman Redistribution Matrix. III. Solution of the
Polarized Line Formation Problem
Authors: Sampoorna, M.; Nagendra, K. N.; Stenflo, J. O.
Bibcode: 2008ApJ...679..889S
Altcode:
The polarized radiative transfer equation is solved numerically,
taking into account both the Zeeman absorption matrix and the
Hanle-Zeeman redistribution matrix, to obtain line profiles for
arbitrary magnetic field strengths, partial frequency redistribution,
and scattering-dominated line transitions. The limiting cases of
weak-field Hanle scattering and strong-field Zeeman true absorption
are retrieved. The intermediate regime, where both Zeeman absorption
and scattering effects are important, is studied in some detail. The
numerical method is applied to various test cases to illustrate aspects
of partial frequency redistribution on line scattering in magnetic
fields of arbitrary strength and direction.
Title: Polarization phase matrices for radiation scattering on
atoms in external magnetic fields: The case of forbidden transitions
in astrophysics
Authors: Oo, Yee Yee; Phyu San, Phyu; Sampoorna, M.; Nagendra, K. N.;
Ramachandran, G.
Bibcode: 2008arXiv0805.3860O
Altcode:
Using a quantum electrodynamical approach, we derive the scattering
phase matrices for polarized radiation involving forbidden line
transitions and in the presence of an external magnetic field. The
case of (J=0->2->0) scattering is considered as an example. The
non-magnetic Rayleigh scattering phase matrix is also presented. The
Stokes profiles in a single scattering event are computed for the
strong field (Zeeman) and weak field (Hanle) limits, covering also
the regime of intermediate field strengths (Hanle- Zeeman).
Title: Commission 36: Theory of Stellar Atmospheres
Authors: Spite, Monique; Landstreet, John D.; Asplund, Martin; Ayres,
Thomas R.; Balachandran, Suchitra C.; Dravins, Dainis; Hauschildt,
Peter H.; Kiselman, Dan; Nagendra, K. N.; Sneden, Christopher;
Tautvaišiené, Grazina; Werner, Klaus
Bibcode: 2007IAUTB..26..160S
Altcode:
The business meeting of Commission 36 was held during the General
Assembly in Prague on 16 August. It was attended by about 15
members. The issues presented included a review of the work made
by members of Commission 36, and the election of the new Organising
Committee. We note that a comprehensive report on the activities of
the commission during the last triennium has been published in Reports
on Astronomy, Transactions IAU Volume XXVIA. The scientific activity
of the members of the commission has been very intense, and has led
to the publication of a large number of papers.
Title: Hanle-Zeeman Redistribution Matrix. II. Comparison of Classical
and Quantum Electrodynamic Approaches
Authors: Sampoorna, M.; Nagendra, K. N.; Stenflo, J. O.
Bibcode: 2007ApJ...670.1485S
Altcode:
The Hanle-Zeeman redistribution matrix accounts for the intricately
coupled correlations in frequency, angle, and polarization between
the incoming and outgoing radiation and embodies the physics of
the scattering process. We show explicitly for a J=0-->1-->0
scattering transition the equivalence between the Hanle-Zeeman
redistribution matrix that is derived through quantum electrodynamics
and the one derived through classical, time-dependent oscillator
theory. This equivalence holds for all strengths and directions of the
magnetic field. Several aspects of the Hanle-Zeeman redistribution
matrix are illustrated, and explicit algebraic expressions are
given, which are of practical use for the polarized line transfer
computations. While the efficiency of the Hanle effect is usually
confined to the line core, we show how elastic collisions can produce a
``wing Hanle effect'' as well under favorable conditions in the solar
atmosphere.
Title: Scattering polarization in the presence of magnetic and
electric fields
Authors: Oo, Yee Yee; Sampoorna, M.; Nagendra, K. N.; Ananthamurthy,
Sharath; Ramachandran, G.
Bibcode: 2007JQSRT.108..161O
Altcode: 2007astro.ph..2312O
The polarization of radiation by scattering on an atom embedded in
combined external quadrupole electric and uniform magnetic fields
is studied theoretically. Limiting cases of scattering under Zeeman
effect, and Hanle effect in weak magnetic fields are discussed. The
theory is general enough to handle scattering in intermediate magnetic
fields (Hanle Zeeman effect) and for arbitrary orientation of magnetic
field. The quadrupolar electric field produces asymmetric line shifts,
and causes interesting level-crossing phenomena either in the absence
of an ambient magnetic field, or in its presence. It is shown that the
quadrupolar electric field produces an additional depolarization in
the Q/I profiles and rotation of the plane of polarization in the U/I
profile over and above that arising from magnetic field itself. This
characteristic may have a diagnostic potential to detect steady-state
and time-varying electric fields that surround radiating atoms in
solar atmospheric layers.
Title: Hanle-Zeeman Redistribution Matrix. I. Classical Theory
Expressions in the Laboratory Frame
Authors: Sampoorna, M.; Nagendra, K. N.; Stenflo, J. O.
Bibcode: 2007ApJ...663..625S
Altcode:
Polarized scattering in spectral lines is governed by a 4×4 matrix
that describes how the Stokes vector is scattered and redistributed
in frequency and direction. Here we develop the theory for this
redistribution matrix in the presence of magnetic fields of arbitrary
strength and direction. This general magnetic field case is called
the Hanle-Zeeman regime, since it covers both of the partially
overlapping weak- and strong-field regimes in which the Hanle
and Zeeman effects dominate the scattering polarization. In this
general regime, the angle-frequency correlations that describe the
so-called partial frequency redistribution (PRD) are intimately
coupled to the polarization properties. We develop the theory for
the PRD redistribution matrix in this general case and explore its
detailed mathematical properties and symmetries for the case of a
J=0-->1-->0 scattering transition, which can be treated in terms
of time-dependent classical oscillator theory. It is shown how the
redistribution matrix can be expressed as a linear superposition of
coherent and noncoherent parts, each of which contain the magnetic
redistribution functions that resemble the well-known Hummer-type
functions. We also show how the classical theory can be extended to
treat atomic and molecular scattering transitions for any combinations
of quantum numbers.
Title: Commission 36: Theory of Stellar Atmospheres
Authors: Spite, Monique; Landstreet, John; Asplund, M.; Ayres, T.;
Balachandran, S.; Dravins, D.; Hauschildt, P.; Kiselman, D.; Nagendra,
K. N.; Sneden, C.; Tautvaišiené, G.; Werner, K.
Bibcode: 2007IAUTA..26..215S
Altcode:
Commission 36 covers all the physics of stellar atmospheres. The
scientific activity in this large field has been very intense during
the last triennium and led to the publication of a large number of
papers which makes an exhaustive report practically not feasible. As
a consequence we decided to keep the format of the preceding report:
first a list of areas of current research, then web links for obtaining
further information.
Title: Turbulent magnetic field averages for the Zeeman effect .
Authors: Frisch, H.; Sampoorna, M.; Nagendra, K. N.
Bibcode: 2007MmSAI..78..142F
Altcode:
Stokes parameters measured in the Solar atmosphere are in general
time or space averages over a magnetic field probability distribution
function. Here we show how to write the Zeeman propagation matrix in a
reference frame defined with respect to the direction of a mean magnetic
field and how to average over a random magnetic field distribution. We
concentrate on the case of a normal Zeeman triplet but indicate how
to treat general Zeeman patterns. Numerical results are presented
for Gaussian distributions having cylindrical symmetry about a mean
field. Different models of probability distribution functions (PDF),
are compared.
Title: Polarized Spectral Line Formation in Turbulent Magnetic Fields:
The Zeeman and Hanle Effects
Authors: Frisch, H.; Sampoorna, M.; Nagendra, K. N.
Bibcode: 2006ASPC..358..126F
Altcode:
We present a short summary of work carried out on the effects of
random magnetic fields with finite correlation length on spectral line
polarization. The magnetic field is modeled by a step-wise Markovian
random process defined by a probability distribution and a correlation
length. Micro- and macro-turbulent limits are recovered when this
length goes to zero and infinity, respectively. For the Zeeman effect,
explicit expressions have been obtained for the mean emergent Stokes
parameters and for their r.m.s. fluctuations. Examples illustrate the
dependence of the mean Zeeman propagation matrix on the magnetic field
distribution, and the dependence of mean Stokes parameters and their
r.m.s. fluctuations on the correlation length of the magnetic field. For
the Hanle effect, explicit expressions have also been obtained for the
mean Stokes parameters. We outline the approach and give an explicit
expression for the mean value of Stokes Q.
Title: Stochastic polarized line formation. II. Zeeman line transfer
in a random magnetic field
Authors: Frisch, H.; Sampoorna, M.; Nagendra, K. N.
Bibcode: 2006A&A...453.1095F
Altcode:
Context: .The Zeeman effect produced by a turbulent magnetic field
or a random distribution of flux tubes is usually treated in the
microturbulent or macroturbulent limits where the Zeeman propagation
matrix or the Stokes parameters, respectively, are averaged over
the probability distribution function of the magnetic field when
computing polarized line profiles.
Aims: .To overcome these
simplifying assumptions we consider the Zeeman effect from a random
magnetic field which has a finite correlation length that can be
varied from zero to infinity and thus made comparable to the photon
mean free-path.
Methods: .The vector magnetic field is modeled
by a Kubo-Anderson process, a piecewise constant Markov process
characterized by a correlation length and a probability distribution
function for the random values of the magnetic field. The micro and
macro turbulent limits are recovered when the correlation goes to zero
or infinity.
Results: .An integral equation is constructed for
the mean propagation operator and explicit expressions are obtained
for the mean values and second-order moments of the Stokes parameters
at the surface of a Milne-Eddington type atmosphere. The expression
given by Landi Degl'Innocenti (1994) for the mean Stokes parameters
is recovered. Mean values and rms fluctuations around the mean values
are calculated numerically for a random magnetic field with isotropic
Gaussian fluctuations. The effects of a finite correlation length
are discussed in detail. Various extensions of the Milne-Eddington
and magnetic field model are considered and the corresponding
integral equations for the mean propagation operator are given.
Conclusions: .The rms fluctuations of the Stokes parameters are
shown to be very sensitive to the correlation length of the magnetic
field. It is suggested to use them as a diagnostic tool to determine
the scale of unresolved features in the solar atmosphere.
Title: Stochastic polarized line formation. I. Zeeman propagation
matrix in a random magnetic field
Authors: Frisch, H.; Sampoorna, M.; Nagendra, K. N.
Bibcode: 2005A&A...442...11F
Altcode:
This paper considers the effect of a random magnetic field on Zeeman
line transfer, assuming that the scales of fluctuations of the random
field are much smaller than photon mean free paths associated to
the line formation (micro-turbulent limit). The mean absorption and
anomalous dispersion coefficients are calculated for random fields with
a given mean value, isotropic or anisotropic Gaussian distributions
azimuthally invariant about the direction of the mean field. Following
Domke & Pavlov (1979, Ap&SS, 66, 47), the averaging process
is carried out in a reference frame defined by the direction of the
mean field. The main steps are described in detail. They involve the
writing of the Zeeman matrix in the polarization matrix representation
of the radiation field and a rotation of the line of sight reference
frame. Three types of fluctuations are considered : fluctuations along
the direction of the mean field, fluctuations perpendicular to the
mean field, and isotropic fluctuations. In each case, the averaging
method is described in detail and fairly explicit expressions for the
mean coefficients are established, most of which were given in Dolginov
& Pavlov (1972, Soviet Ast., 16, 450) or Domke & Pavlov (1979,
Ap&SS, 66, 47). They include the effect of a microturbulent velocity
field with zero mean and a Gaussian distribution. A detailed numerical
investigation of the mean coefficients illustrates the two effects
of magnetic field fluctuations: broadening of the σ-components by
fluctuations of the magnetic field intensity, leaving the π-components
unchanged, and averaging over the angular dependence of the π and σ
components. For longitudinal fluctuations only the first effect is at
play. For isotropic and perpendicular fluctuations, angular averaging
can modify the frequency profiles of the mean coefficients quite
drastically with the appearance of an unpolarized central component
in the diagonal absorption coefficient, even when the mean field is in
direction of the line of sight. A detailed comparison of the effects of
the three types of fluctuation coefficients is performed. In general
the magnetic field fluctuations induce a broadening of the absorption
and anomalous dispersion coefficients together with a decrease of their
values. Two different regimes can be distinguished depending on whether
the broadening is larger or smaller than the Zeeman shift by the mean
magnetic field. For isotropic fluctuations, the mean coefficients can
be expressed in terms of generalized Voigt and Faraday-Voigt functions
H(n) and F(n) introduced by Dolginov &
Pavlov (1972, Soviet Ast., 16, 450). These functions are related to
the derivatives of the Voigt and Faraday-Voigt functions. A recursion
relation is given in an Appendix for their calculation. A detailed
analysis is carried out of the dependence of the mean coefficients
on the intensity and direction of the mean magnetic field, on its
root mean square fluctuations and on the Landé factor and damping
parameter of the line.
Title: Scattering of Polarized Radiation by Atoms in Magnetic and
Electric Fields
Authors: Oo, Yee Yee; Nagendra, K. N.; Ananthamurthy, Sharath;
Ramachandran, G.
Bibcode: 2005astro.ph..9775O
Altcode:
The polarization of radiation by scattering on an atom embedded in
combined external quadrupole electric and uniform magnetic fields is
studied theoretically. Analytic formulae are derived for the scattering
phase matrix. Limiting cases of scattering under Zeeman effect, and
Hanle effect in weak magnetic fields are discussed.
Title: An operator perturbation method for polarized line
transfer. VI. Generalized PALI method for Hanle effect with partial
frequency redistribution and collisions
Authors: Fluri, D. M.; Nagendra, K. N.; Frisch, H.
Bibcode: 2003A&A...400..303F
Altcode:
A generalized iteration method is presented to solve the polarized line
transfer equation for a two-level-atom in an arbitrarily oriented,
weak magnetic field. The polarized redistribution matrix employed
accounts self-consistently for collisions as well as the presence of
a weak magnetic field responsible for the Hanle effect. The proposed
numerical method of solution is based on a Polarized Approximate Lambda
Iteration (PALI) method. A Fourier decomposition of the radiation field
and of the phase matrix with respect to the azimuthal angle reduces
the complexity of the problem. A generalized core-wing technique is
proposed, which permits an efficient implementation of the frequency
domain structure inherent in the polarized redistribution matrix. The
numerical method is tested for its accuracy and efficiency by comparing
with the existing methods.
Title: Numerical Methods for Solving the Polarized Line Transfer
Equations with Partial Frequency Redistribution
Authors: Nagendra, K. N.; Frisch, H.; Fluri, D. M.
Bibcode: 2003ASPC..307..227N
Altcode:
No abstract at ADS
Title: Numerical Solutions of Polarized Line Transfer Equations
Authors: Nagendra, K. N.
Bibcode: 2003ASPC..288..583N
Altcode: 2003sam..conf..583N
Recent developments in the NLTE polarized line formation theory in
Astrophysics is discussed. Attention is focussed on pure theoretical
aspects of the problem. A conventional method of solving the line
transfer equation is described briefly, in order to give a perspective
to the modern Polarized Approximate Lambda Iteration (PALI) methods,
which are developed only in recent years. Sample results computed
using this old finite difference method for the polarized line
transfer in planar and spherical media are presented. These examples
include polarized resonance scattering in spherical media, polarized
line formation in expanding atmospheres, and the role of collisional
frequency redistribution in polarized line scattering. Further,
the basic characteristics of a PALI method are described using a
prototype resonance polarization problem, keeping the assumption of
CRD. Directions are given about the manner in which the presence of an
external weak magnetic field can be incorporated in a PALI method. This
is basically, the well known problem of Hanle effect in weak magnetic
fields. The PALI method for Hanle effect is then described, first with
CRD, and then with PRD scattering mechanism. Finally the generalization
of PALI to the most difficult problem we have attempted until now,
namely the study of partial frequency redistribution in the presence
of collisions and an external weak magnetic field, is presented. Once
again, sample results are shown to illustrate the essentials of PALI
method, and the nature of solutions computed using this method. A
comparison of the conventional and the PALI approaches is made, in
some cases.
Title: Some Equations of Discrete Space Method (DSM) for Polarized
Line Transfer
Authors: Nagendra, K. N.
Bibcode: 2003ASPC..288..611N
Altcode: 2003sam..conf..611N
No abstract at ADS
Title: Hanle effect with angle-dependent partial redistribution
Authors: Nagendra, K. N.; Frisch, H.; Faurobert, M.
Bibcode: 2002A&A...395..305N
Altcode:
The polarized line transfer equation for the Hanle effect is solved in
the framework of an exact partial frequency redistribution (PRD) theory
developed by Bommier (1997a,b). In that theory the effect of collisions
on the Hanle effect is considered self-consistently. We follow that
approach in the line transfer computations presented here. The theory
formulated by Bommier clearly recognizes two levels of approximations
for exact PRD, in order to facilitate the solution of the line transfer
equation. The second level employs angle-dependent redistribution
functions, and numerically represents a more difficult problem compared
to the third level, which involves only the use of angle-averaged
frequency redistribution functions. We present a method which can
solve the problem in both the levels of approximation. The method
is based on a perturbative approach to line polarization. Although
computationally expensive, it offers the only practical means of solving
the angle-dependent Hanle PRD problem. We discuss the numerical aspects
of assembling the so called ``frequency domain dependent redistribution
matrices'', and also an efficient way of computing the scattering
integral. Some examples are presented to illustrate the interesting
aspects of the Hanle-PRD problem with angle-dependent frequency
redistribution. A comparison of the emergent profiles computed under
angle-averaged and angle-dependent redistribution is carried out, and
the effect of collisions is investigated. We show that it is necessary
to incorporate an angle-dependent redistribution mechanism especially
in the computation of the Stokes U parameter. We demonstrate that the
use of simple frequency domains is good enough in practical applications
of the Hanle PRD theory.
Title: Scattering Polarization and Hanle Effect: On the Importance
of Angle-Dependent Frequency Redistribution
Authors: Faurobert, M.; Frisch, H.; Nagendra, K. N.
Bibcode: 2001ASPC..248..145F
Altcode: 2001mfah.conf..145F
No abstract at ADS
Title: The Hanle Effect with Angle Dependent Redistribution Functions
Authors: Frisch, H.; Faurobert, M.; Nagendra, K. N.
Bibcode: 2001ASPC..236..197F
Altcode: 2001aspt.conf..197F
No abstract at ADS
Title: An Operator Perturbation Method of Polarized Line Transfer
V. Diagnosis of Solar Weak Magnetic Fields
Authors: Nagendra, K. N.; Frisch, H.; Faurobeet-Scholl, M.; Paletou, F.
Bibcode: 2000JApA...21..255N
Altcode:
No abstract at ADS
Title: Fast Numerical Methods for Polarized Line Radiative Transfer
in the Presence of Hanle Effect
Authors: Faurobert, M.; Frisch, H.; Nagendra, K. N.
Bibcode: 1999ASPC..184...28F
Altcode:
The Hanle effect provides a diagnostic tool for weak magnetic fields
which do not give rise to a measurable Zeeman effect, such as turbulent
fields or magnetic canopies. The lines which are sensitive to the
Hanle effect are formed under non-LTE conditions, by scattering of
photons. Inversion methods for such diagnostics require to solve the
non-LTE polarized transfer equation for a large number of magnetic
configurations. Fast numerical methods are thus highly required. We
present an Approximate Lambda Iteration method to treat the Hanle effect
for lines formed with complete frequency redistribution. Referred to
as PALI-H, this method is an extension of ALI methods first developed
for non polarized line transfer. The starting point is to recast the
polarized transfer equation into a vectorial integral equation for a
6-component source function. We show that the convergence of the method
is independent of the strength and direction of the magnetic field. The
method is very fast and allows to handle any type of depth-dependent
magnetic field.
Title: Solar polarization
Authors: Nagendra, K. N.; Stenflo, J. O.
Bibcode: 1999ASSL..243.....N
Altcode: 1999sopo.conf.....N
No abstract at ADS
Title: An operator perturbation method for polarized line transfer IV:
Applications to the Hanle effect with partial frequency redistribution
Authors: Nagendra, K. N.; Paletou, F.; Frisch, H.; Faurobert-Scholl, M.
Bibcode: 1999ASSL..243..127N
Altcode: 1999sopo.conf..127N
No abstract at ADS
Title: An operator perturbation method for polarized line
transfer. III. Applications to the Hanle effect in 1D media
Authors: Nagendra, K. N.; Frisch, H.; Faurobert-Scholl, M.
Bibcode: 1998A&A...332..610N
Altcode:
In this paper we present an Approximate Lambda Iteration method to
treat the Hanle effect (resonance scattering in the presence of
a weak magnetic field) for lines formed with complete frequency
redistribution. The Hanle effect is maximum in the line core and
goes to zero in the line wings. Referred to as PALI-H, this method is
an extension to non-axisymmetric radiative transfer problems of the
PALI method presented in Faurobert-Scholl et al. (1997), hereafter
referred to as Paper I. It makes use of a Fourier decomposition of the
radiation field with respect to the azimuthal angle which is somewhat
more general than the decomposition introduced in Faurobert-Scholl
(1991, hereafter referred to as FS91). The starting point of the
method is a vector integral equation for a six-component source vector
representing the non-axisymmetric polarized radiation field. As
in Paper I, the Approximate Lambda operator is a block diagonal
matrix. The convergence rate of the PALI-H method is independent
of the polarization rate and hence of the strength and direction
of the magnetic field. Also this method is more reliable than the
perturbation method used in FS91. The PALI-H method can handle any
type of depth-dependent magnetic field. Here it is used to examine
the dependence of the six-component source vector on the co-latitude,
azimuthal angle and strength of the magnetic field. The dependence of
the surface polarization on the direction of the line-of-sight and on
the magnetic field is illustrated with polarization diagrams showing
Q/I versus U/I at line center. The analysis of the results show that
the full six-dimension problem can be approximated by a two-component
modified resonance polarization problem, producing errors of at most
20% on the surface polarization at line center.
Title: An operator perturbation method for polarized line
transfer. I. Non-magnetic regime in 1D media.
Authors: Faurobert-Scholl, M.; Frisch, H.; Nagendra, K. N.
Bibcode: 1997A&A...322..896F
Altcode:
In this paper we generalize an Approximate Lambda Iteration (ALI)
technique developed for scalar transfer problems to a vectorial transfer
problem for polarized radiation. Scalar ALI techniques are based on a
suitable decomposition of the Lambda operator governing the integral
form of the transfer equation. Lambda operators for scalar transfer
equations are diagonally dominant, offering thus the possibility to
use iterative methods of the Jacobi type where the iteration process
is based on the diagonal of the Lambda operator (Olson et al., 1986,
JQSRT 35, 431). Here we consider resonance polarization, created by
the scattering of an anisotropic radiation field, for spectral lines
formed with complete frequency redistribution in a 1D axisymmetric
medium. The problem can be formulated as an integral equation for a
2-component vector (Rees, 1978PASJ...30..455R) or, as shown by Ivanov
(1995A&A...303..621I), as an integral equation for a (2x2) matrix
source function which involves the same generalized Lambda operator as
the vector integral equation. We find that this equation holds also in
the presence of a weak turbulent magnetic field. The generalized Lambda
operator is a (2x2) matrix operator. The element {11} describes the
propagation of the intensity and is identical to the Lambda operator of
non-polarized problems. The element {22} describes the propagation of
the polarization. The off-diagonal terms weakly couple the intensity and
the polarization. We propose a block Jacobi iterative method and show
that its convergence properties are controlled by the propagator for
the intensity. We also show that convergence can be accelerated by an
Ng acceleration method applied to each element of the source matrix. We
extend to polarized transfer a convergence criterion introduced by
Auer et al. (1994A&A...292..599A) based on the grid truncation
error of the converged solution.
Title: Green's matrix for Compton reflection of polarized radiation
from cold matter
Authors: Poutanen, Juri; Nagendra, K. N.; Svensson, Roland
Bibcode: 1996MNRAS.283..892P
Altcode:
The Compton-reflected spectrum from cold matter for incident X-rays
and gamma-rays with arbitrary angular, spectral and polarization
properties can be determined by a simple folding of Green's matrix
with the incident spectrum. We calculate Green's matrix by numerically
solving the polarized radiative transfer equation for an optically
thick planar slab of neutral matter using a method based on discrete
space theory. We account for both angular and polarization properties
of the fully relativistic Compton scattering cross-section as well as
photoelectric absorption and the generation of a fluorescent Fe line. We
describe through simple models the basic characteristics of polarized
spectra produced by Rayleigh and Compton scattering. The emphasis is
on exact computations of Compton-reflected spectra. For the first time,
the Fe line equivalent width can be calculated self-consistently for an
arbitrary angular distribution of the incident spectrum using Green's
matrix formalism.
Title: Models of highly extended dust shells around R Coronae Borealis
Authors: Nagendra, K. N.; Leung, Chun Ming
Bibcode: 1996MNRAS.281.1139N
Altcode:
Radiation transport models are constructed for the dust shells of the
hydrogen deficient supergiant star R Coronae Borealis (R CrB). IRAS
observations of R CrB are used as constraints in selecting the model
parameters. Based on suggestions from earlier work, a double-shell
model is employed as a standard configuration for R CrB. The first
shell is a hot inner shell of radius 5 arcsec surrounding the central
star. The second shell is a cold remnant shell which is highly extended,
with a radius of 10 arcmin (derived assuming a source distance of 1.6
kpc). The two shells can be spatially separated, and can have completely
different mechanisms for the heating of their dust grains. A detailed
parametric study of the system is undertaken using simple power law
density and opacity for single sized grains. The models of R CrB clearly
point to the configuration of a large well-separated cold dust shell
surrounding a tiny hot dust shell, which supports the existing view
that this extended shell is a fossil shell. However, the separation
between the two shells cannot be determined purely from the radiative
transfer modelling. This can still be treated as a free parameter
of the system for modelling purposes. The interstellar radiation
field (ISRF) incident on the outer boundary of the system plays a
considerable role in explaining the 60- and 100-μm surface brightness
of R CrB measured by IRAS. The observed constant temperature of the
fossil shell, however, poses serious theoretical problems. We propose
models in which the density varies as r^-gamma (gamma~=1.0-1.5). The
source of dust heating in our models is a combination of central star
radiation and ISRF. The amorphous carbon grains seem to give better
fits to the IRAS data than the crystalline grains do. A comparison of
both the analytic models and the radiative transfer models is made,
to clarify their usefulness in modelling the density and the surface
brightness data of R CrB. The models are schematic in nature, and are
not intended as best fitting models for the R CrB fossil shell.
Title: Preface
Authors: Stenflow, J. O.; Nagendra, K. N.
Bibcode: 1996SoPh..164D...9S
Altcode:
No abstract at ADS
Title: Partial Redistribution Effects on Line Polarization in the
Presence of Velocity Fields
Authors: Nagendra, K. N.
Bibcode: 1996SoPh..164...67N
Altcode:
Velocity fields in line formation regions strongly affect the line
polarization. The conventionally used observer's frame method of solving
the polarized transfer equation becomes expensive and inaccurate for
partial redistribution problems, when large amplitude velocity fields
have to be considered in the observer's frame. An alternative method of
solution is the comoving frame method. Partial redistribution problems
are solved using comoving frame formalism for line polarization caused
by resonance scattering.
Title: Solar polarization. Proceedings. International Workshop on
Solar Polarization, St. Petersburg (Russia), 8 - 12 May 1995.
Authors: Stenflo, J. O.; Nagendra, K. N.
Bibcode: 1996SoPh..164.....S
Altcode:
The following topics were dealt with: Solar physics, radiative transfer,
solar spectra, polarization, Hanle effect, coronal refraction, Stokes
profile inversion, solar prominences, solar atmosphere, imaging
polarimetry method, magnetic inclination of plages, field azimuth
disambiguation, solar pore polarimetry, sunspots, solar flares,
magnetographs, VUV polarization measurement.
Title: Solar polarization
Authors: Stenflo, J. O.; Nagendra, K. N.
Bibcode: 1996sopo.conf.....S
Altcode: 1996QB539.M23S66...
No abstract at ADS
Title: Collisional redistribution effects on line polarization in
spherical atmospheres
Authors: Nagendra, K. N.
Bibcode: 1995MNRAS.274..523N
Altcode:
The polarization of resonance and subordinate lines formed in
spherically symmetric media is studied. The effect of collisions is
shown by taking a combination of partial redistribution mechanisms
represented by the functions R_II-A(x,x'') and R_III-A(x,x''), which
are well-known redistribution functions that describe interaction of
line photons with a two-level model atom. A large number of models
are presented, which clearly show the effect of collisions on line
polarization, when basic model parameters of a two-level atom are
varied one at a time. Thus the grid of models generated serves to
establish upper and lower limits on the impact of collisions on line
polarization. Some results employing a more accurate treatment of
collisional redistribution are also shown. The purpose of this paper
is to understand the quantitative behaviour of line polarization when
collisional redistribution plays an important role in the transfer of
line radiation.
Title: Resonance Line Polarization in Spherical Atmospheres: Partial
Redistribution Effects Studied with the Domke-Hubeny Redistribution
Matrix
Authors: Nagendra, K. N.
Bibcode: 1994ApJ...432..274N
Altcode:
The problem of resonance line polarization including collisional
redistribution is studied using simple theoretical models. The medium is
assumed to be a static finite spherical shell atmosphere. The purpose
of this paper is to understand the behavior of line polarization when
elastic and inelastic collisions cause frequency redistribution upon
scattering. All the redistribution mechanisms that are astrophysically
relevant are considered in a simple two-level atom picture. The
dependence of resonance line polarization on physical parameters of the
two-level atom formulation and on the collisional parameters is shown in
detail. The Domke-Hubeny generalized collisional redistribution matrix
for polarized radiation is employed in most of the examples shown in
this paper. It is shown that different types of collisions that affect
photon redistribution in both resonance and subordinate lines can be
distinguished through the characteristic changes they produce in the
intensity and linear polarization profiles. The simple asymptotic
expressions of partial redistribution theory are shown to be useful
in the interpretation of model intensity and polarization profiles.
Title: The combined effect of partial redistribution and non-coherent
electron scattering on polarized resonance line transfer
Authors: Nagendra, K. N.; Rangarajan, K. E.; Rao, D. M.
Bibcode: 1993MNRAS.262..855N
Altcode:
The combined effect of partial frequency redistribution by atoms
(PRD) and noncoherent electron scattering (NCES) on line formation is
studied, taking account of the polarization state of the radiation
field in resonance line scattering. For the purpose of comparison,
a study is also made of redistribution by other mechanisms, namely
complete redistribution and coherent scattering (CS) combined with
NCES. A static medium stratified into plane-parallel homogeneous
layers is considered. A conventional two-level atom approximation is
employed as the basic model. The emergent linear polarization profile
exhibits interesting characteristics in its variation across the line
profile. Atomic redistribution plays an important role at the line
core and the near wings of the resonance line. Electron scattering,
however, makes significant contributions in the far wings (x is greater
than 10). For resonance lines with high optical thickness, NCES leads
to measurable changes of polarization in the far wings irrespective
of the atomic scattering mechanism employed.
Title: An analysis of the IRAS data of supergiant R Coronae Borealis.
Authors: Nagendra, K. N.; Leung, C. M.
Bibcode: 1991ASNYN...3...34N
Altcode:
No abstract at ADS
Title: Models of Highly Extended Dust Shells Around the Supergiants
R Coronae Borealis and W Hydrae
Authors: Nagendra, K. N.; Leung, C. M.
Bibcode: 1990BAAS...22.1248N
Altcode:
No abstract at ADS
Title: A model for the extended dust shell around the supergiant R
Coronae Borealis.
Authors: Nagendra, K. N.; Leung, C. M.
Bibcode: 1990BAAS...22..749N
Altcode:
No abstract at ADS
Title: A Model for the Extended Dust Shell around the Supergiant R
Coronae Borealis
Authors: Nagendra, K. N.; Leung, C. M.
Bibcode: 1990BAAS...22R.749N
Altcode:
No abstract at ADS
Title: Polarization of Resonance Lines Formed in Extended Spherical
Atmospheres
Authors: Nagendra, K. N.
Bibcode: 1989Ap&SS.154..119N
Altcode:
The problem of polarization of the resonance lines formed
in extended spherical atmospheres is studied in detail. In this
paper, the atmosphere is assumed to be at rest. The basic problem of
resonance line polarization in spherical atmospheres as compared to the
conventionally used plane-parallel atmospheres, is studied in Nagendra
(1988). Our main interest in this paper is to understand the behaviour
of polarized radiation fields in extended model spherical atmospheres
so that some constraints can be placed on the model parameters in
the modelling work conected with observations of polarization across
resonance lines. A comparison of polarized lines formed under three
kinds of line-scattering mechanisms is also made. They are CS=coherent
scatteirng, CRD=complete redistribution, and PRD=partial frequency
redistribution which, in the increasing order of generality, provide
a good approximation in the two-level atom approach, to the resonance
line polarization. The dependence of polarization on the opacity
laws, extendedness and on optical depth is studied in detail. The
distribution of line intensity and polarization across the visible
disk of an extended model stellar atmosphere is studied, in view of
the possible disk-resolved observations in future, of the extended
atmospheres of the stars.
Title: Resonance Line Polarization in Spherical Atmospheres
Authors: Nagendra, K. N.
Bibcode: 1988ApJ...335..269N
Altcode:
The problem of resonance line polarization is studied using simple
theoretical models. The medium is assumed to be a finite slab or a
finite spherical shell-like atmosphere. A comparison of lines formed
under complete redistribution and partial redistribution in frequency is
presented, for both plane-parallel and spherically symmetric media. The
dependence of the resonance line polarization of physical parameters of
the two-level atom formulation, and on the global parameters such as
extendedness, the opacity law in a spherical medium, and the boundary
conditions are studied. The resonance line polarization by coherent
scattering is also discussed.
Title: Some physical processes influencing the polarization of
continuum and line radiation
Authors: Nagendra, K. N.; Peraiah, A.
Bibcode: 1987A&A...181...71N
Altcode:
Some physical mechanisms which affect the continuum and line
polarization are studied. The physical conditions of the plasma
selected for this purpose represent different astrophysical situations
of interest, particularly the magnetic stars. The pure absorption
polarization transfer equation is solved individually taking these
effects into account.
Title: Polarization line radiative transfer in the atmospheres of
magnetic white dwarfs
Authors: Nagendra, K. N.
Bibcode: 1987AN....308..303N
Altcode:
Some physical mechanisms which can affect the Zeeman line profiles of
magnetic white dwarfs are studied. The pure absorption polarization
transfer equation is solved including these mechanisms. The broadening
of lines in magnetic white dwarfs is briefly discussed.
Title: Some Aspects of the Solution of Vector Transfer Equation in
a Magnetized Medium
Authors: Nagendra, K. N.; Peraiah, A.
Bibcode: 1985Ap&SS.117..121N
Altcode:
A simplification of the numerical method of solving the vector transfer
equation, given earlier by Nagendra and Peraiah (1985a), is described
for problems which involve only absorption. This allows us to attempt
to solve under realistic conditions and with reduced computing efforts,
the important problems of polarization of light emerging from magnetized
stars. For the purpose of illustration, the equations described are
used for solving the continuum and Zeeman line transfer problems.
Title: Numerical solution of the radiative transfer equation in a
magnetized medium
Authors: Nagendra, K. N.; Peraiah, A.
Bibcode: 1985MNRAS.214..203N
Altcode:
A numerical method of solution based on the discrete space theory of
radiative transfer as applied to the transfer problems in an anisotropic
medium is discussed. Two simple applications, namely the scattering
in the atmosphere of a hot magnetic white dwarf and in a plasma slab
immersed in a superstrong magnetic field are discussed. The normal wave
transfer equations for the scattering and absorption of radiation are
used for this purpose. The solutions are compared with those obtained
for the non-magnetic Thomson scattering in the same media. A comparative
study is made of the normal wave and Stokes vector equations for a
Zeeman active gas.
Title: Polarization of Continuum Radiation in Magnetic Atmospheres
Authors: Nagendra, K. N.; Peraiah, A.
Bibcode: 1984Ap&SS.104...61N
Altcode:
A numerical solution is presented for the problem of continuum radiative
transfer in a magnetoactive medium. The continuum opacities are
calculated in the presence of a strong magnetic field (H=107
G) typical of magnetic white dwarfs. The L.T.E. pure absorption model
is assumed for calculating the polarized radiation field emitted by a
realistic model atmosphere in the plane parallel approximation. The
wavelength dependence of the linear and circular polarizations are
calculated for both uniform and dipole field configurations.
Title: Effects on partial prequency redistributionR II on the level
population ratios in a resonance line
Authors: Peraiah, A.; Nagendra, K. N.
Bibcode: 1983Ap&SS..90..237P
Altcode:
Angle-averaged partial frequency redistributionR II has been employed
in obtaining a simultaneous solution of radiative transfer equation
in the comoving frame and the statistical equilibrium equation for
a non-LTE two level atom. We have obtained the ratios of population
densities of the upper and lower levels of the resonance line of PV by
utilizing the data given in Bernacca and Bianchi (1979). Line source
functions are also obtained for different types of variations of density
and velocity of the expanding gases. We have considered the atmosphere
to be 11 times as thick as the stellar radius. The first iteration was
started by putting the density of the upper level (N 2) equal
to zero. However, the convergent solution shows a substantial increase
inN 2 although it is still much less than the equilibrium
value. The line source function and the ratio of the densities of the
particles in the upper and lower levels fall sharply from a maximum at
τ=τmax to minimum at τ=0. We have studied the scattering
integralint {_{ - infty }^{ + infty } J_x φ _x } dx and found that
this quantity also varies quite similar to the ratioN 2/N
1 and the line source functionS L.
Title: Optical depth effects on the formation of spectral lines in
rotating and expanding spherical atmospheres
Authors: Peraiah, A.; Raghunath, G.; Nagendra, K. N.
Bibcode: 1981JApA....2..277P
Altcode:
The effect of high optical depths and large rotational velocities on
the spectral lines in rotating and radial expanding spherical shells is
investigated. It is assumed that the outer radius of the shell is three
times the inner radius and that there are no velocity gradients in the
shell; the radial optical depths are 10, 50, 100, and 500. The shell
rotates with velocities varying as 1/p, where p is the perpendicular
distance from the axis of rotation. Two expansion (radial) velocities,
V = 0 and V = 10 mean thermal units, are considered, with maximum
rotational velocities of 0, 5, 10, and 20. In the shell with no radial
motions, symmetric lines with emission in the wings were obtained for
rotational velocities = 0 and 5, while for rotational velocities greater
than or equal to 10, symmetric absorption lines were formed. Lines
with central emission were obtained in the case of an expanding shell.
Title: Lines formed in a slowly expanding thin spherical shell.
Authors: Peraiah, A.; Raghunath, G.; Nagendra, K. N.
Bibcode: 1980KodOB...3...30P
Altcode:
The authors have investigated how an optically thin spherical shell
with small velocities change the profiles and equivalent widths of
the lines. They have employed several types of variations in density,
velocity of expansion and source functions. In all the cases they
find that the line centres are shifted to the blue side almost in
proportion to the velocity of expansion. The shells moving with constant
velocities shift the line centre the most, irrespective of the density
variation. The velocity gradients shift the line centre the least.