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FIRTEZ-dz: A Forward and Inverse solver of the polarized Radiative Transfer Equation under Zeeman regime in geometrical scale

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read FIRTEZ-dz is the first code to solve the polarized radiative transfer equation in both forward and inverse modes directly in geometrical height, with analytic response functions, and it shows that height-scale inferences are only as…

desk verdict A plausible first in z-space polarized RTE inversion with analytic response functions, a clean degeneracy analysis, and public code—but the printed Saha factor has a wrong temperature exponent and the Jacobian is never checked against finite differences; deserves review after fixes. read the letter →

arxiv 1908.08075 v1 pith:SQDZ2AKE submitted 2019-08-21 astro-ph.SR

A. Pastor Yabar , J. M. Borrero , B. Ruiz Cobo This is my paper · ORCID
classification astro-ph.SR
keywords radiativetransferpolarizedZeemaneffectStokesinversionresponsefunctionsgeometricalheightscalesolarphotospherespectropolarimetry
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper presents FIRTEZ-dz, a numerical code that solves the polarized radiative transfer equation in the Zeeman regime directly on a geometrical height scale, for both forward synthesis and Stokes inversion, and claims this is the first such code. The motivation is that standard inversions work in optical depth and convert to height using gas pressure and density that are not consistent with the full three-dimensional magnetohydrodynamic force balance, so the resulting height scale and its spatial derivatives are unreliable. The paper demonstrates that even when temperature, the three magnetic-field components, and line-of-sight velocity are correctly recovered in optical depth, their height-scale versions are only as accurate as the gas pressure or density used in the conversion. If the claims hold, the code opens a route to computing spatial derivatives like the Lorentz force and the divergence-free condition directly from inverted atmospheres, and the authors identify the thermodynamic degeneracies that must be resolved to do so.

What carries the argument

The load-bearing object is the slab-wise Stokes evolution operator in geometrical scale: the atmosphere is divided into $N$ slabs of constant properties, the formal solution of the polarized transfer equation is written as a concatenation of analytic evolution operators, and the response functions are obtained by differentiating that concatenation with respect to the physical parameter of each slab. For the thermodynamic parameters this differentiation requires the derivatives of the electron density with respect to $T$, $P_g$, and $\rho$, which the paper derives analytically from Saha ionization with at most three ionization stages per element and an ideal-gas equation of state. The inverse solver uses these analytical response functions as the Jacobian of a Levenberg-Marquardt optimizer, regularized through singular-value decomposition, with a sequential cycle that increases the number of perturbed slabs from a constant shift up to all 256 height levels.

What would settle it

Take a realistic penumbral atmosphere, perturb a single slab's temperature by a small amount, and compare the analytic response function from Eq. (6) with a finite-difference derivative obtained by re-running the forward solver on the perturbed atmosphere; a mismatch above numerical noise at the SVD threshold of $10^{-4}$ times the largest eigenvalue would show that the thermodynamic derivatives in Appendices A-C are the weak link.

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Extended reading notes

Core claim

The paper's central claim is that solving the polarized radiative transfer equation in geometrical height $z$ rather than optical depth $\tau$ is not only feasible but yields analytic response functions, and that this changes what can be inferred. The code computes the derivatives of the Stokes vector with respect to $T$, $P_g$, $\rho$, $B_x$, $B_y$, $B_z$, and $v_{\rm LOS}$ at each height slab analytically, by differentiating the slab-wise formal solution; the thermodynamic derivatives require the electron-density derivatives derived in Appendices A-C under Saha equilibrium and an ideal-gas equation of state. Tested on a penumbral pixel from a sunspot MHD simulation, the inverse solver recovers temperature, magnetic field, and velocity stratifications reliably in optical depth, but in height these stratifications are distorted whenever the assumed gas pressure and density are wrong. The paper also shows two intrinsic degeneracies of height-scale inversion: shifting the entire atmosphere vertically leaves spectra unchanged, and different combinations of $T$, $P_g$, and $\rho$ can produce the same absorption coefficient. Adding hydrostatic equilibrium removes the second degeneracy up to a vertical shift, but applying that assumption to MHD-simulated data biases the height scale while leaving the optical-depth inference accurate.

Load-bearing premise

The inversion's height-scale credibility rests on analytic derivatives of the electron density with respect to $T$, $P_g$, and $\rho$, derived under an ideal-gas equation of state and Saha ionization with at most three ionization stages per element; if those derivatives are wrong, the Jacobian is biased and the code's synthetic tests would not reveal it because the same forward model generates the data being fitted.

Editorial extensions

If this is right

  • Inversions in geometrical scale inherit two unavoidable degeneracies: a vertical shift of the whole atmosphere produces identical Stokes spectra, and distinct thermodynamic triples $(T, P_g, \rho)$ can yield the same absorption coefficients.
  • If the gas pressure and density used are not accurate, the height-scale stratifications of temperature, magnetic field, and velocity are inaccurate even when their optical-depth stratifications are perfectly recovered.
  • Enforcing hydrostatic equilibrium removes the thermodynamic degeneracy but leaves the vertical-origin degeneracy, so height maps remain ambiguous by an overall shift unless additional forces, such as magnetic pressure and tension, are included.
  • Solving in $z$ makes spatial derivatives such as $\nabla\times\mathbf{B}$ and the Lorentz force directly computable from the inverted atmosphere, which is the paper's stated route toward self-consistent three-dimensional inversions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A finite-difference check of the analytic response functions against numerical perturbations would isolate whether the electron-density derivatives are the limiting ingredient; the paper's synthetic tests cannot do this because the forward and inverse modules share the same derivatives.
  • The paper's thermodynamic degeneracy suggests a test it does not run: fitting spectra with two line pairs formed at very different heights, or with multi-wavelength continuum, should shrink the family of equivalent $(T, P_g, \rho)$ stratifications.
  • Because the Jacobian is computed analytically, coupling this solver to an update of gas pressure and density from the three-dimensional momentum equation looks computationally cheap; in that mode the missing absolute height origin would come from the force balance itself.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This manuscript presents FIRTEZ-dz, a parallelized one-dimensional code that solves the forward and inverse polarized radiative transfer equation in the Zeeman regime on a geometrical height grid. The forward solver uses a slab-wise formal solution with constant-property slabs, LTE source functions, an ideal-gas equation of state with variable molecular weight, a reduced NICOLE opacity package, and Zeeman line opacities. The inverse solver uses a Levenberg-Marquardt algorithm with analytically computed response functions (Eq. 6) and SVD regularization, together with a sequential cycle schedule that progressively increases the number of free parameters. The forward module is demonstrated on a Rempel (2012) sunspot MHD snapshot, and the inverse module is tested on noise-free synthetic Stokes profiles generated by the same forward model. The paper identifies two intrinsic degeneracies of the geometrical-scale formulation (a height-origin shift and a thermodynamic-parameter degeneracy) and shows that under hydrostatic equilibrium the thermodynamic stratification is recovered only up to a height shift. A final test on an MHD pixel shows accurate recovery in optical depth but systematic distortions in geometrical height when hydrostatic equilibrium is assumed, supporting the abstract claim that the reliability of the height-scale output depends on the accuracy of the gas-pressure and density inputs.

Significance. If the analytic response functions are correct, FIRTEZ-dz is a valuable new tool: it is the first publicly released code that computes analytic response functions and performs inversion directly in geometrical height, which is a necessary step toward incorporating Lorentz-force and solenoidality constraints into inversions. The paper's main physical finding—the two intrinsic degeneracies and the dependence of the inferred z-stratification on the accuracy of Pg and rho—is clearly demonstrated and is a property of the radiative transfer equations rather than of the specific fitting algorithm. The code is released on GitHub, which enables independent verification and application to large DKIST/EST datasets, and the parallelization is well matched to the intended per-pixel use. The main caveat is that the numerical tests are all self-consistent, with synthetic data generated by the same forward module, so the correctness of the analytic derivatives is not independently established; the printed thermodynamic-derivative formulas also contain a sign error that must be resolved.

major comments (2)
  1. [Appendix A, Eq. (A.5); Appendices B and C] Equation (A.5) defines alpha_{i,j} with a factor c1 T^{+3/2}, whereas the Saha-Boltzmann population ratio n_j/(n_e n_{j+1}) requires c1 T^{-3/2} with c1 = h^3/(2 pi m_e k_B)^{3/2}. Since Eqs. (A.11), (B.6), and (C.2), and through them the thermodynamic response functions in Eq. (6), all build on this definition, the printed formulas describe a Jacobian with the wrong temperature dependence unless the implementation uses the corrected exponent. The paper provides no finite-difference check of the response functions, so an implementation error of this kind would not be exposed by the synthetic inversions. Please correct the formula, re-derive the dependent expressions, and add a finite-difference validation against the forward code. Note also that the second term of Eq. (B.5) is printed as (partial n_e/partial T)_rho but should be (partial n_e/partial P_g)_T.
  2. [Section 3.2, Figs. 3-6] All inversion tests use noise-free synthetic profiles generated by the same forward module whose analytic response functions are being tested. This closed loop cannot reveal a biased Jacobian caused by an error in the thermodynamic derivatives, because both the synthetic data and the inversion share the same forward code. The authors should validate the response functions against high-accuracy finite-difference derivatives of their own forward code and, ideally, invert profiles synthesized by an independent code (e.g., SIR, NICOLE, or STIC) for a common model atmosphere, to establish that the recovery of T, Bx, By, Bz, and vLOS in optical depth is not an artifact of self-consistency.
minor comments (4)
  1. [Section 2.2.1] The word 'folloing' in the sentence introducing the derivatives of the Stokes evolution operator should be 'following'.
  2. [Section 3.1] In the paragraph discussing Fig. 2, the text 'unlike the response function to Bx, Bx, Bx' repeats 'Bx' three times; the intended quantity is likely Bx, By, or Bz.
  3. [Appendix A] The section title states 'Derivative of the number of electron density with respect to temperature at constant gas density', but the first sentence says 'at constant gas pressure'; both should refer to the same variable (rho).
  4. [Section 4] The GitHub repository link would be more reproducible if a specific version or commit identifier, and ideally a DOI, were provided for the code version used in the tests.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the code's forward and inverse derivations are self-contained, and the central degeneracy result is an analytic property of the RTE rather than a fitted prediction.

full rationale

The paper's load-bearing derivations are self-contained: the analytic response functions in Eq. (6) are obtained by direct differentiation of the slab-wise formal solution in Eq. (4), and the thermodynamic derivatives in Appendices A-C are solved analytically from the Saha equation and the ideal-gas equation of state. No parameter is fitted and then renamed as a prediction, and no uniqueness claim is imported from the authors' prior work. The inversion tests use synthetic spectra generated by the same forward model, which is a closed-loop validation rather than a circular derivation; the central degeneracy result (height-scale reliability depends on Pg/rho) is an analytic property of the RTE in geometrical scale and is also tested against a Rempel (2012) MHD snapshot, so it does not reduce to a fit. Self-citations to Ruiz Cobo & del Toro Iniesta (1992) and Puschmann et al. (2010) are methodological background and are not load-bearing for the paper's central novelty claim. The apparent T^{3/2} in Eq. (A.5) and the typo in Eq. (B.5) are correctness or typographical concerns, not circularity. Overall, the derivation chain does not exhibit any step where a result is equivalent to its inputs by construction.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new physical entities. The method rests on standard modeling assumptions (LTE, plane-parallel, Zeeman only, ideal gas EOS, Saha ionization) and on an analytic Stokes-evolution operator from the literature. The hand-set algorithmic parameters (SVD threshold, normalization constants, cycle schedule) are the main adjustable inputs that affect inversion output and are listed as free parameters.

free parameters (3)
  • SVD eigenvalue threshold = 10^-4 of the largest eigenvalue
    Section 2.3: controls the number of eigenvalues kept when inverting the dampened Hessian and determines the effective number of free parameters (~350 of 1280) and smoothing; no sensitivity test is reported.
  • Response function normalization constants = Tbar=6000 K, Pgbar=1e5 dyn/cm2, rhobar=3e-7 g/cm3, Bbar=1000 G, vbar=1e5 cm/s
    Section 3.1, Fig. 2: these hand-picked values make response functions dimensionless; they scale the Jacobian and can affect LM convergence, but the paper does not test this dependence.
  • Sequential cycle schedule = 1, 8, 64, 256 perturbations per physical parameter
    Section 3.2: the number of free parameters is progressively increased through cycles; the schedule is a hand-set regularization that influences the final solution.
assumptions (7)
  • domain assumption LTE holds, so the source function is the Planck function.
    Section 2.1: 'under the assumption of LTE'; underpins the source function in Eq. (3) and the Saha ionization in the appendices.
  • domain assumption Plane-parallel atmosphere and complete redistribution approximation.
    Section 2.1 states these assumptions at the start.
  • domain assumption Zeeman effect is the only magnetic polarization mechanism.
    Section 1: 'we focus on the solution of the polarimetric RTE under the Zeeman magnetic regime'; Hanle and Paschen effects are excluded.
  • domain assumption Ideal gas EOS with variable molecular weight, including free electrons.
    Section 2.1: 'an ideal gas equation with a variable molecular weight to account for free electrons'; used for the EOS and electron density derivatives.
  • standard math Analytic Stokes-evolution operator from Landi Degl'Innocenti and Landi Degl'Innocenti (1985, Eq. 10).
    Section 2.1, Eqs. (3) and (4) rely on this expression; the code uses it for forward synthesis and response functions.
  • domain assumption Saha ionization equilibrium with at most three ionization stages per element.
    Appendices A to C derive electron-density derivatives assuming Nion=3 and Saha populations.
  • domain assumption Reduced NICOLE opacity package is adequate for continuum opacity.
    Section 2.1: 'We follow a reduced scheme of the so-called NICOLE opacity package'; affects forward syntheses and the tau-to-z conversion.

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Cite this review

Pith. "Pith review of FIRTEZ-dz: A Forward and Inverse solver of the polarized Radiative Transfer Equation under Zeeman regime in geometrical scale." pith.science (2026). https://pith.science/paper/SQDZ2AKE

@misc{pith2026190808075,
  author       = {Pith},
  title        = {Pith review of: FIRTEZ-dz: A Forward and Inverse solver of the polarized Radiative Transfer Equation under Zeeman regime in geometrical scale},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SQDZ2AKE}},
  note         = {Machine review of arXiv:1908.08075}
}
read the original abstract

We present a numerical code that solves the forward and inverse problem of the polarized radiative transfer equation in geometrical scale under the Zeeman regime. The code is fully parallelized, making it able to easily handle large observational and simulated datasets. We checked the reliability of the forward and inverse modules through different examples. In particular, we show that even when properly inferring various physical parameters (temperature, magnetic field components, and line-of-sight velocity) in optical depth, their reliability in height-scale depends on the accuracy with which the gas-pressure or density are known. The code is made publicly available as a tool to solve the radiative transfer equation and perform the inverse solution treating each pixel independently. An important feature of this code, that will be exploited in the future, is that working in geometrical-scale allows for the direct calculation of spatial derivatives, which are usually required in order to estimate the gas pressure and/or density via the momentum equation in a three-dimensional volume, in particular the three-dimensional Lorenz force.

Figures

Figures reproduced from arXiv: 1908.08075 by the authors.

Figure 1
Figure 1. Examples of three simulated Stokes profiles as RTE is solved for a snapshot of the Rempel simulation of a sunspot (Rempel 2012). Top panel: Monochromatic continuum intensity map in which three pixels are highlighted in black, red, and light-blue squares as representatives of low magnetized, penumbral, and umbral areas, respectively; their spectral profiles are presented following this same color code in panels from … view at source ↗
Figure 2
Figure 2. Analytical dimensionless response functions for, from the second row to the last one, T(z), Pg(z), ρ(z), Bx(z), By(z), Bz(z), and vLOS (z) for the penumbral pixel highlighted in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Results of the inversion of three thermodynamic parameters. Three different inversions are presented, in each of which a single physical parameter is inverted: the temperature, T(z), (solid-purple line), the gas pressure, Pg(z), (dashed-red), and the density, ρ(z), (dash-dotted-blue). Left panels: Simulated spectral lines inverted (‰ syn(λ), in black) coming from the same penumbral pixel highlighted in [PITH_FULL_I… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Two different inversions of the gas pressure, Pg(z), for the penumbral pixel highlighted in [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Inversion of Stokes profiles in hydrostatic equilibrium for different values of the gas pressure top boundary condition, Pg(zmax = 2048 km). Left column: Stokes parameters, from top to bottom, I, Q, U, and V, for the simulated case, ‰ syn(λ) (solid-black lines), the in…
Figure 6
Figure 6. Figure 6: Inversion of a penumbral pixel (highlighted in [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: Difference between the Stokes profiles of the inversion results and the inverted ones for the example considered in [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

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Works this paper leans on

30 extracted references · 24 canonical work pages

  1. [1]

    Anstee, S. D. & O’Mara, B. J. 1995, MNRAS, 276, 859

    1995

  2. [2]

    Barklem, P. S. 1998, PhD thesis, The University of Queensland, Australia

    work page 1998

  3. [3]

    S., Anstee, S

    Barklem, P. S., Anstee, S. D., & O’Mara, B. J. 1998, PASA, 15, 336

    work page 1998

  4. [4]

    Barklem, P. S. & O’Mara, B. J. 1997, MNRAS, 290, 102

    1997

  5. [5]

    M., Lites, B

    Borrero, J. M., Lites, B. W., Lagg, A., Rezaei, R., & Rempel, M. 2014, A&A, 572, A54

    work page 2014

  6. [6]

    M., Rempel, M., & Solanki, S

    Borrero, J. M., Rempel, M., & Solanki, S. K. 2010, Astronomische Nachrichten, 331, 567

    work page 2010

  7. [7]

    M., Tomczyk, S., Kubo, M., et al

    Borrero, J. M., Tomczyk, S., Kubo, M., et al. 2011, Sol. Phys., 273, 267

    work page 2011

  8. [8]

    Cao, W., Gorceix, N., Coulter, R., Coulter, A., & Goode, P. R. 2010, in Proc. SPIE, V ol. 7733, Ground-based and Airborne Telescopes III, 773330

    work page 2010

Show all 30 references
  1. [9]

    2010, in Proc

    Collados, M., Bettonvil, F., Cavaller, L., et al. 2010, in Proc. SPIE, V ol. 7733, Ground-based and Airborne Telescopes III, 77330H

    2010

  2. [10]

    2008, A&A, 484, L17

    Danilovic, S., Gandorfer, A., Lagg, A., et al. 2008, A&A, 484, L17

    2008

  3. [11]

    Danilovic, S., Schüssler, M., & Solanki, S. K. 2010b, A&A, 513, A1 de la Cruz Rodríguez, J., Leenaarts, J., Danilovic, S., & Uitenbroek, H. 2019, A&A, 623, A74 del Toro Iniesta, J. C. & Ruiz Cobo, B. 2016, Living Reviews in Solar Physics, 13, 4 del Toro Iniesta, J. C., Ruiz Co...

    2019

  4. [12]

    2012, Journal of Computational Physics, 231, 919

    Freytag, B., Steffen, M., Ludwig, H.-G., et al. 2012, Journal of Computational Physics, 231, 919

    2012

  5. [13]

    K., Fligge, M., & Bruls, J

    Frutiger, C., Solanki, S. K., Fligge, M., & Bruls, J. H. M. J. 2000, A&A, 358, 1109 González-Morales, P. A., Khomenko, E., Downes, T. P., & de Vicente, A. 2018, A&A, 615, A67

    2000

  6. [14]

    2007, Sol

    Kosugi, T., Matsuzaki, K., Sakao, T., et al. 2007, Sol. Phys., 243, 3 Landi Degl’Innocenti, E. 1976, A&AS, 25, 379 Landi Degl’Innocenti, E. & Landi Degl’Innocenti, M. 1972, Sol. Phys., 27, 319 Landi Degl’Innocenti, E. & Landi Degl’Innocenti, M. 1985, Sol. Phys., 97, 239 Landi ...

    2007

  7. [15]

    1944, Quart

    Levenberg, K. 1944, Quart. Appl. Math., 2, 164

    1944

  8. [16]

    W., Rempel, M., Borrero, J

    Lites, B. W., Rempel, M., Borrero, J. M., & Danilovic, S. 2017, ApJ, 835, 14 Löptien, B., Lagg, A., van Noort, M., & Solanki, S. K. 2018, A&A, 619, A42

    2017

  9. [17]

    Marquardt, D. W. 1963, Journal of the Society for Industrial and Applied Math- ematics, 11, 431 Martinez Pillet, V . & Vazquez, M. 1993, A&A, 270, 494

    1963

  10. [18]

    K., Solanki, S

    Mathew, S. K., Solanki, S. K., Lagg, A., et al. 2004, A&A, 422, 693 Mili´c, I. & van Noort, M. 2018, A&A, 617, A24

    2004

  11. [19]

    Priest, E. R. 1987, Solar magneto-hydrodynamics

    1987

  12. [20]

    G., Ruiz Cobo, B., & Martínez Pillet, V

    Puschmann, K. G., Ruiz Cobo, B., & Martínez Pillet, V . 2010, ApJ, 720, 1417

    2010

  13. [21]

    Rachkovsky, D. N. 1967, Izvestiya Ordena Trudovogo Krasnogo Znameni Krymskoj Astrofizicheskoj Observatorii, 37, 56

    1967

  14. [22]

    H., & Knölker, M

    Rempel, M., Schüssler, M., Cameron, R. H., & Knölker, M. 2009, Science, 325, 171 Riethmüller, T. L., Solanki, S. K., Barthol, P., et al. 2017, ApJS, 229, 16

    2009

  15. [23]

    R., Wagner, J., Keil, S., et al

    Rimmele, T. R., Wagner, J., Keil, S., et al. 2010, in Proc. SPIE, V ol. 7733, Ground-based and Airborne Telescopes III, 77330G Ruiz Cobo, B. & del Toro Iniesta, J. C. 1992, ApJ, 398, 375

    2010

  16. [24]

    B., Bjelksjo, K., Korhonen, T

    Scharmer, G. B., Bjelksjo, K., Korhonen, T. K., Lindberg, B., & Petterson, B. 2003, in Proc. SPIE, V ol. 4853, Innovative Telescopes and Instrumentation for Solar Astrophysics, ed. S. L. Keil & S. V . Avakyan, 341–350

    2003

  17. [25]

    H., Schou, J., Bush, R

    Scherrer, P. H., Schou, J., Bush, R. I., et al. 2012, Sol. Phys., 275, 207

    2012

  18. [26]

    2012, Astronomische Nachrichten, 333, 796

    Schmidt, W., von der Lühe, O., V olkmer, R., et al. 2012, Astronomische Nachrichten, 333, 796

    2012

  19. [27]

    2015, A&A, 577, A7

    Socas-Navarro, H., de la Cruz Rodríguez, J., Asensio Ramos, A., Trujillo Bueno, J., & Ruiz Cobo, B. 2015, A&A, 577, A7

    2015

  20. [28]

    K., Walther, U., & Livingston, W

    Solanki, S. K., Walther, U., & Livingston, W. 1993, A&A, 277, 639

    1993

  21. [29]

    1955, Physik der Sternatmospharen, MIT besonderer Berucksichti- gung der Sonne

    Unsold, A. 1955, Physik der Sternatmospharen, MIT besonderer Berucksichti- gung der Sonne

    1955

  22. [30]

    1974, Sol

    Wittmann, A. 1974, Sol. Phys., 35, 11 Article number, page 13 of 16 A&A proofs: manuscript no. manuscript_pastoryabar Appendix A: Derivative of the number of electron density with respect to temperature at constant gas density We focus on the derivative of the density of elect...

    1974

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