11institutetext: Instituto de Ciencias Astronómicas, de la Tierra y del
Espacio (CONICET-UNSJ), Av. España 1512 (sur), 5400 San Juan, Argentina

# A numerical code for the analysis of magnetic white-dwarf spectra that
includes field effects on the chemical equilibrium

Matías Vera-Rueda and René D. Rohrmann

We present a new magnetic-atmosphere model code for obtaining synthetic
spectral fluxes of hydrogen-rich magnetic white dwarfs. To date, observed
spectra have been analyzed with models that neglect the magnetic field’s
effects on the atomic populations. In this work, we incorporate state-of-art
theory in the evaluation of numerical densities of atoms, free electrons, and
ions in local thermodynamical equilibrium under the action of a magnetic
field. The energy distribution of atoms is rigorously evaluated for arbitrary
field strength. This energy pattern includes going from tightly bound states
to metastable or truly bound, highly excited states embedded in the continuum,
that is, over the first Landau level. Finite nuclear mass effects and the
coupling between the internal atomic structure and the motion of the atom
across the magnetic field are also considered. Synthetic fluxes are generated
with integrations of numerical solutions of polarized radiative transfer over
the visible stellar disk using a spherical $t$-design method. The atmosphere
code is tested with observations from the Sloan Digital Sky Survey for a group
of known magnetic white dwarfs. Physical stellar parameters are obtained from
least-squares fits to the observed energy distribution and compared with
results of previous works. We show that the use of zerofield ionization
equilibrium in spectral analyses can lead to underestimated effective
temperatures for highly magnetic white dwarfs.

###### Key Words.:

magnetic fields — white dwarfs — stars: atmospheres — radiative transfer —
atomic processes — opacity

##  1 Introduction

Magnetic white dwarfs (MWDs) represent a very challenging field of research in
astrophysics. Blackett (1947) suggested their existence several decades ago,
but no evidence was found for a number of years. The first known MWD,
Grw$+70^{\circ}8247$, was identified by its strong circular polarization in
the continuum spectrum (Kemp et al. 1970). By the late 1970s, sixteen other
MWDs, including twelve isolated objects and four in binary systems, were
recognized (Angel 1978). Around the mid 1990s, about 40 MWDs with field
strengths greater than 1 MG ($10^{6}$ gauss) were studied (Schmidt & Smith
1995). With the arrival of larger surveys, mainly the Sloan Digital Sky Survey
(SDSS), the number of known MWDs has grown to over 600 isolated stars and
about 200 objects in interacting binary systems (Ferrario et al. 2020). About
800 MWDs have recently been analyzed in a comprehensive study (Amorim et al.
2023).

Evidence of magnetic fields on the surface of white dwarfs has been deduced
from the detection of broad-band circular polarization (e.g. Kemp et al. 1970;
Angel & Landstreet 1971; West 1989; Berdyugin et al. 2022), cyclotron features
observed in optical and UV spectra (e.g. Visvanathan & Wickramasinghe 1979;
Green & Liebert 1981; Bailey et al. 1991), Zeeman splittings of spectral lines
(e.g. Angel et al. 1974; Liebert et al. 1975; Wickramasinghe & Bessell 1976;
Kepler et al. 2013), and spectropolarimetric measurements (e.g. Aznar Cuadrado
et al. 2004; Afanas’ev et al. 2018; Landstreet & Bagnulo 2019). In some cases,
the observation of a stable oscillation period may also suggest a rotating
magnetic star with localized magnetic structures on its surface (Katz 1975;
Landi Degl’Innocenti 1976; Dupuis et al. 2000; Kilic et al. 2015). Signatures
of spot-like components have been found in a number of white dwarfs (e.g.
Shtol’ et al. 1997; Maxted et al. 2000; Valyavin et al. 2008; Vornanen &
Berdyugin 2012; Brinkworth et al. 2013) including the first candidate MWD in
any globular cluster (Pichardo Marcano et al. 2023).

Studies reveal a field distribution with surface strengths in the
$10^{3}$–$10^{9}$ G range (Kawka 2020). Most MWDs have been discovered via
Zeeman identifications in the SDSS (Gänsicke et al. 2002; Schmidt et al. 2003;
Vanlandingham et al. 2005; Kepler et al. 2013; Amorim et al. 2023) and present
field strengths exceeding 1 MG due to the limit of detectability at the
spectral resolutions used and because the Zeeman splitting is greater than the
Stark broadening for this field range (Chanmugam 1992).111Although the field
strength changes across the stellar surface and smears the spectral features
(e.g., line wavelength $\lambda$), its mean value can be deduced from the so-
called stationary lines ($d\lambda/dB\approx 0$). The Zeeman splitting becomes
undetectable below $\approx 50$ kG (Bagnulo et al. 2018). Detection of weak
fields, below 1 MG and near a few kilogauss, is possible with the observation
of polarization in spectral line wings through spectropolarimetry surveys
(Schmidt & Smith 1995; Aznar Cuadrado et al. 2004; Kawka et al. 2007; Valyavin
et al. 2006; Landstreet et al. 2012).

Surface magnetic fields are present in roughly 10% of the total white dwarf
population (Liebert et al. 2003; Hollands et al. 2015). This proportion is
also found in MWDs with weak fields (Jordan et al. 2007; Landstreet et al.
2012). A higher incidence of magnetism was found in studies of nearly complete
13 pc (around $20$%) and 20 pc ($12$%) volume-limited samples (Kawka et al.
2007; Holberg et al. 2016). Nevertheless, magnetic properties of about 80% of
white dwarfs are still unknown (Valyavin 2015), and MWDs with fields under 0.1
MG remain to be discovered (Kawka et al. 2007).

The origin of magnetic fields in white dwarfs is not well understood. They may
arise from (i) fossil fields retained during the evolution of magnetic
progenitors (Fontaine et al. 1973; Angel et al. 1981; Wickramasinghe &
Ferrario 2005), (ii) a preceding evolutionary stage through a dynamo process
or a convective mechanism (Ruderman & Sutherland 1973; Levy & Rose 1974;
Kissin & Thompson 2015; Cantiello et al. 2016), (iii) a binary system
evolution through different mechanisms (Wickramasinghe & Ferrario 2000; Tout
et al. 2008; García-Berro et al. 2012; Briggs et al. 2015), or (iv) the
cooling of an already formed white dwarf (Isern et al. 2017). Mergers (García-
Berro et al. 2012; Briggs et al. 2015) or accretion (Tout et al. 2008;
Nordhaus et al. 2011) could explain why magnetic white dwarfs tend to be
significantly more massive (mean mass $\approx 0.8$ $M_{\sun}$, Liebert 1988;
Kawka et al. 2007; Kepler et al. 2013) than nonmagnetic degenerates ($\approx
0.6$ $M_{\sun}$, Kleinman et al. 2013). A considerable amount of work is
required to understand the origin and properties of the fields and their
incidence on the structure and evolution of these stars.

The interpretation of MWDs requires detailed atmosphere modeling. The first
attempts to explain the radiation of MWDs were based on a graybody
magnetoemission model (Kemp 1970; Shipman 1971; Chanmugam et al. 1972).
However, an appropriated analysis of the radiation spectrum emitted at the
surface of such stars demands numerical approaches for solving the transfer of
polarized radiation through a magnetized medium (Unno 1956; Beckers 1969;
Hardorp et al. 1976; Martin & Wickramasinghe 1979; nagendra1985) and improved
evaluations of the emission and absorption processes of gases in the presence
of magnetic fields (e.g., Garstang 1977; Henry & Oconnell 1984; Roesner et al.
1984; Ruder et al. 1994; Merani et al. 1995; Zhao & Stancil 2007). With
advances in the required input physics, considerable effort has been made to
progressively obtain more realistic model atmospheres (Martin & Wickramasinghe
1979; Wickramasinghe & Martin 1979; O’Donoghue 1980; nagendra1985; Schmidt et
al. 1986; Wickramasinghe & Ferrario 1988; Jordan 1989, 1992; Euchner et al.
2002).

Although reasonable fits to the MWDs’ spectra of light emission have been
achieved, including comprehensive analyses of a great number of objects
(Külebi et al. 2009; Amorim et al. 2023; Hardy et al. 2023), a number of
simplifications and rough approximations remain to be addressed. In
particular, the direct effects of the magnetic field on the hydrostatic
structure of the atmosphere are neglected (Wickramasinghe & Ferrario 1988;
Jordan 1992). Consequently, zerofield models are used to provide the pressure
and temperature distributions of magnetic-atmosphere models, ignoring induced
Lorentz forces (Landstreet 1987) and likely deformations of the atmospheric
geometry (Stepien 1978; Fendt & Dravins 2000). Furthermore, detailed and
rigorous evaluations of a number of radiative processes are still required. In
particular, continuum opacities arising from bound-free and free-free
transitions are poorly approximated, and an appropriate theory of Stark
broadening of spectral lines for arbitrary magnetic field is not available
(although some efforts are in progress; e.g., Kieu et al. 2017; Rosato 2023).

Similarly, detailed calculations of the ionization equilibrium and occupation
numbers of atomic levels in magnetic fields have not been included in any
model of magnetic white dwarfs. In fact, until recently, no reliable
evaluation of chemical equilibrium for the intermediate range of magnetic
field strengths (the realm of the MWDs) existed, even for hydrogen gas.
However, we recently conducted a comprehensive evaluation of hydrogen
ionization balance in arbitrary magnetic fields, and we showed that field
effects are significant in the conditions found in the atmospheres of the
strongest magnetic white dwarfs (Vera Rueda & Rohrmann 2020). Although our
chemical model considers the most basic chemical species (neutral atoms,
protons, free electrons) and does not include complexes such as molecules,
particle chains and negative ions, this represents a step toward a
comprehensive physical representation of the gas in the atmospheres of
magnetic white dwarfs.

The present work is aimed at solving the inconsistency of current model
atmospheres of hydrogen-rich magnetic white dwarfs (DAH stars), which use
field-dependent opacities but ignore the magnetic effects on the particle
abundances. Detailed evaluations of atomic hydrogen populations in magnetic
fields involve changes in the structure of atoms beyond the Zeeman
perturbative approach. These changes have consequences on the gas partition
function. Its determination requires taking into account finite temperatures
and, hence, the thermal motion of particles, with the particularity that the
motion of an atom across the magnetic field affects its internal energies
(Pavlov & Meszaros 1993). This demands the use of the so-called pseudomomentum
to separate the center-of-mass motion of the atom from the relative electron-
proton motion (Gor’kov & Dzyaloshinskiǐ 1968).

Here, we apply a chemical model that utilizes fits of accurately evaluated
energy levels of atoms at rest (Schimeczek & Wunner 2014b). These evaluations
are complemented by center-of-mass effects on the internal atomic structure,
which arise from the finite proton mass and thermal motions across the
magnetic field. For sufficiently large values of the pseudomomentum transverse
to the magnetic field, our gas model considers the formation of the so-called
decentered states, i.e., atomic states where the electronic wave function is
shifted from the Coulomb center to a magnetic well (Burkova et al. 1976).
Furthermore, particle interaction effects on the chemical equilibrium are
included through the usual occupation probability approach with a few updates
(effective atomic sizes depending of the field strength). Beyond the chemical
equilibrium calculation, all other constitutive physics in our MWD atmosphere
models (opacity sources, radiative transfer of polarized light) follow the
state-of-art theory as detailed below.

The structure of this paper is as follows. Basic approximations for modeling
atmospheres of magnetic white dwarfs are presented in Section 2. In Section 3,
we provide the chemical model that determines the occupation numbers of atomic
states for a magnetized gas at ionization equilibrium. Section 4 describes the
method used to solve the radiative transfer equations for polarized light
represented by the four Stokes parameters, while Section 5 lists the relevant
opacity sources and magnetooptical parameters used in the transfer
calculation. In Section 6, we discuss the integration method of the Stokes
intensities over the observed stellar hemisphere. In Section 7, we compare
SDSS observed spectra to the theoretical spectra as a check of the new
numerical code. We analyze how the improved constitutive physics used in our
code affects the determination of physical parameters of strong magnetic white
dwarfs (Sect. 7.1). Additionally, predictions of our spectral energy
distribution fits for a group of magnetic white dwarfs are compared with
previous studies in the literature (Sect. 7.2 and Sect. 7.3). A final
discussion and conclusions follow.

##  2 Basic assumptions

The models presented here are based on pure hydrogen local thermodynamics
equilibrium (LTE), plane-parallel atmospheres. Since there are no calculations
to date that fully account for the effects of magnetic fields on the
hydrostatic structure of MWDs (Stepien 1978; Landstreet 1987; Ferrario et al.
2020), we assume that magnetic pressure is negligible in the outer layers of
the atmosphere where the spectrum of a star originates (Wickramasinghe &
Ferrario 1988; Jordan 1992). Therefore, we used zero-field models to compute
the temperature and pressure distributions throughout the atmospheric layers
(Rohrmann et al. 2012). We also imposed the null-convective-flux condition
according to results suggesting that convection is suppressed by magnetic
fields (Jordan 2001; Tremblay et al. 2015; Gentile Fusillo et al. 2018). In
addition, surface gravity is fixed at $\log g=8$, a typical value for white
dwarfs, as applied in other MWD studies (Euchner et al. 2002; Külebi et al.
2009; Amorim et al. 2023). This choice seems reasonable, given that mass
determination from spectral line fitting is not fully reliable in the absence
of an appropriate theory for combined Stark and magnetic broadening.

On the other hand, the field distribution on the stellar surface is assumed to
be generated by a dipole that is centered or offset with respect to the
barycenter of the star, for which we assume a spherical shape. Specifically,
we chose Cartesian coordinates centered in the star, with the $z$-axis along
the magnetic dipole and the line of sight forming an angle of $i$ on the $xz$
plane (see Achilleos & Wickramasinghe 1989). Thus, the pole-on and equator-on
views of the star are given by $i=0^{\circ}$ and $i=90^{\circ}$, respectively.
With distances measured in stellar radius units,
$r=(x^{2}+y^{2}+z^{2})^{1/2}=1$ defines the stellar surface. The dipole center
is located at the position $\mathbf{a}=(a_{x},a_{y},a_{z})$. Consequently, the
magnetic field is described by

$\mathbf{B}=\frac{B_{d}}{2r^{\prime
5}}\left(3x^{\prime}z^{\prime},3y^{\prime}z^{\prime},3z^{\prime 2}-r^{\prime
2}\right),\hskip 10.84006ptr^{\prime}=(x^{\prime 2}+y^{\prime 2}+z^{\prime
2})^{1/2},$ (1)

where $w^{\prime}=w-a_{w}$ ($w=x,y,z$) and $B_{\text{d}}$ equal the polar
field strength at the stellar surface for the case of a centered dipole
($\mathbf{a}=0$). If the dipole is off-center, the field strengths at the
poles are $B_{\text{d}}(1\pm a)^{-3}$. Through the parameters $B_{\text{d}}$,
$i$, and $\mathbf{a}$, Eq. (1) yields a nonuniform distribution over the
stellar surface. The dipole configuration is the simplest and most important
term in a multipolar expansion of the magnetic potential. Such an
approximation resulted to be adequate for calculating emerging flux from
magnetic white dwarfs (Martin & Wickramasinghe 1984; Achilleos &
Wickramasinghe 1989; Külebi et al. 2009), especially if one takes into account
that in some cases it is not easy to distinguish between an offset dipole and
a combination of multipoles (Martin & Wickramasinghe 1984; Putney & Jordan
1995).

##  3 Ionization equilibrium and occupation numbers

We give special attention to the determination of occupation numbers of
hydrogen atoms in a magnetic field. The details of evaluations used in the
present work are described in depth by Vera Rueda & Rohrmann (2020). Here,
only a short synopsis is provided.

The inner state of a hydrogen atom is specified by a set $\kappa$ of quantum
numbers, for which one usually chooses the asymptotic ones corresponding to
the Coulomb approximation at the weak-field limit, $\kappa=\\{n,l,m,m_{s}\\}$
($\beta\ll 1$, $\beta=B/B_{0}$, $B_{0}\approx 4.70103\times 10^{9}$ G), or
those of the high-field Landau regime, $\kappa=\\{N,\nu,m,m_{s}\\}$ ($\beta\gg
1$), where $n$, $l$, $m,$ and $m_{s}$ are, respectively, the principal,
azimuthal, magnetic and spin quantum numbers, whereas $N$ is the number of the
Landau level and $\nu$ is the usually called the longitudinal quantum number
associated with atomic excitations in the magnetic-field direction. Both sets
share $m$ and $m_{s}$, while the remaining quantum numbers are connected by
mathematical relations first given in Vera Rueda & Rohrmann (2020).

The traslational state of an atom is labeled by the eigenvalue,
$\mathbf{k}=(k_{\perp},k_{z}),$ of the so-called pseudomomentum operator, with
$k_{\perp}$ and $k_{z}$ being the components transversal and parallel to the
field, respectively. The pseudomomentum is associated with the translational
invariance of the Hamiltonian of the atom in a magnetic field (Gor’kov &
Dzyaloshinskiǐ 1968). The binding energy of a magnetized atom and its motion
perpendicular to the field $\mathbf{B}$ are generally not separable (Pavlov &
Meszaros 1993). The total energy of the atom is expressed by

$E=\mathcal{E}_{\kappa}(k_{\perp})+\frac{k_{z}^{2}}{2m_{\text{H}}},$ (2)

where $m_{\text{H}}$ is the atom mass. Explicit analytical approximations of
$\mathcal{E}_{\kappa}(k_{\perp})$ are given in Vera Rueda & Rohrmann (2020)
for any combination of inner and translational states. These evaluations
include energy data for atoms at rest (Schimeczek & Wunner 2014b) and the
formation of the so-called decentered states that arise in atoms with high
pseudomomentum transverse to the field (Potekhin et al. 2014).

The number densities of atoms ($n_{\text{H}}$), electrons ($n_{\text{e}}$),
and protons ($n_{\text{p}}$) are obtained from LTE conditions for an
electrically neutral hydrogen gas ($n_{\text{e}}=n_{\text{p}}$) at a given
temperature, $T$. The ionization equilibrium for hydrogen in a magnetic field
is given by

$\frac{n_{\text{H}}}{n_{\text{e}}n_{\text{p}}}=\frac{\lambda_{e}^{3}Z_{\text{H}}}{2}f(\eta),$
(3)

with

$f(\eta)=\frac{\tanh(\eta)(1-e^{-q\eta})}{q\eta^{2}},\hskip
21.68121ptq=\frac{2m_{\text{e}}}{m_{\text{p}}},$ (4)
$\lambda_{\text{e}}=\hbar\sqrt{\frac{2\pi}{k_{\mathrm{B}}Tm_{\text{e}}}},\hskip
21.68121pt\eta=\frac{\hbar\omega_{\text{e}}}{2k_{\mathrm{B}}T},\hskip
21.68121pt\omega_{\text{e}}=\frac{eB}{m_{\text{e}}c},$ (5)

where $Z_{\text{H}}$ is the internal partition function of the atoms,
$\lambda_{\text{e}}$ the electron thermal wavelength, $\omega_{\text{e}}$ the
cyclotron frequency, $\hbar=h/2\pi$, $h$ the Planck constant, $k_{\mathrm{B}}$
the Boltzmann constant, $c$ the speed of light, $e$ the electron charge, and
$m_{\text{e}}$ and $m_{\text{p}}$ the electron and proton masses. The factor
$f(\eta)$ comes from the excess of chemical potential from free electrons and
ionized atoms. The internal partition function contains nonideal effects and
the coupling of internal and transverse kinetic energies,

$Z_{\text{H}}=\sum_{\kappa}\frac{1}{m_{\text{H}}k_{\mathrm{B}}T}\int
w_{\kappa}(k_{\perp})e^{-\mathcal{E}_{\kappa}(k_{\perp})/(k_{\mathrm{B}}T)}k_{\perp}dk_{\perp},$
(6)

with $w_{\kappa}(k_{\perp})$ being the so-called occupational probability of
the state $(\kappa,k_{\perp})$. The quantity $w_{\kappa}(k_{\perp})$
represents a reduction of the phase space available in an atom due to
interactions with other particles. The number density of atoms in an inner
state, as required in opacity evaluations, is determined by

$n_{\kappa}=\frac{n_{\text{H}}}{m_{\text{H}}k_{\mathrm{B}}TZ_{\text{H}}}\int
w_{\kappa}(k_{\perp})e^{-\mathcal{E}_{\kappa}(k_{\perp})/(k_{\mathrm{B}}T)}k_{\perp}dk_{\perp}.$
(7)

In the zero-field limit, $\eta\rightarrow 0$, $f(\eta)\rightarrow 1$, and the
usual Saha function is recovered from Eq. (3), with

$Z_{\text{H}}\rightarrow\sum_{\kappa}w_{\kappa}e^{-\epsilon_{\kappa}/(k_{\mathrm{B}}T)},\hskip
14.45377pt\mathcal{E}_{\kappa}(k_{\perp})\rightarrow\epsilon_{\kappa}+\frac{k_{\perp}^{2}}{2m_{\text{H}}},\hskip
14.45377pt(\beta\rightarrow 0),$ (8)

where $w_{\kappa}$ is the occupational probability and $\epsilon_{\kappa}$ the
binding energy of the state $\kappa$, both uncoupled from the translational
motion. Furthermore, in field-free conditions the pseudomomentum of an atom is
reduced to the usual canonical momentum. 222Actually, only the transversal
component is affected by the field as a motion integral associated with the
translational invariance of the atom.

Free electron dynamics shows a continuum of energy due to movements along the
field direction and quantized contributions (units of the cyclotron energy)
from the perpendicular direction. In the absence of particle perturbations, it
is given by

$E=\hbar\omega_{\text{e}}\left(N+m_{s}+\frac{1}{2}\right)+\frac{k_{z}^{2}}{2m_{\text{e}}},\hskip
7.22743ptN=0,1,2,\dots,\hskip 7.22743ptm_{s}=\pm\frac{1}{2},$ (9)

where a giromagnetic ratio of $g_{\text{e}}=2$ has been assumed. These energy
values are required for evaluating the ionization equilibrium and
photoionization thresholds.

##  4 Radiative transfer

Radiative transfer of polarized light in a plane-parallel atmosphere with an
LTE source function is described by the following four coupled differential
equations over the four Stokes parameters $\\{I,Q,U,V\\}$ (Beckers 1969;
Hardorp et al. 1976):

$\displaystyle\mu\frac{dI}{d\tau}$ $\displaystyle=$
$\displaystyle\eta_{I}\left(I-B_{\text{p}}\right)+\eta_{Q}Q+\eta_{V}V,$ (10)
$\displaystyle\mu\frac{dQ}{d\tau}$ $\displaystyle=$
$\displaystyle\eta_{Q}\left(I-B_{\text{p}}\right)+\eta_{I}Q+\rho_{R}U,$ (12)
$\displaystyle\mu\frac{dU}{d\tau}$ $\displaystyle=$
$\displaystyle\rho_{R}Q+\eta_{I}U-\rho_{W}V,$ (14)
$\displaystyle\mu\frac{dV}{d\tau}$ $\displaystyle=$
$\displaystyle\eta_{V}\left(I-B_{\text{p}}\right)+\rho_{W}U+\eta_{V}V,$ (16)

with $B_{\text{p}}$ being the source function assumed to be the Planck
function, $\mu=\cos\theta$, $\theta$ the angle between the direction of light
propagation and the normal of the stellar surface ($z^{\prime}$ axis in a
local coordinate system), $\tau$ the optical depth,
$d\tau=-\kappa_{p}dz^{\prime}$, $\kappa_{p}$ the Rosseland mean opacity
evaluated with the unpolarized continuum absorption coefficient, $\rho_{R}$
and $\rho_{W}$ magneto-optical parameters, and $\eta_{I}$, $\eta_{Q}$, and
$\eta_{V}$ combinations of the absorption coefficients given by

$\displaystyle\eta_{I}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\eta_{p}\sin^{2}\psi+\frac{1}{4}\left(\eta_{l}+\eta_{r}\right)\left(1+\cos^{2}\psi\right),$
(17) $\displaystyle\eta_{Q}$ $\displaystyle=$
$\displaystyle\left[\frac{1}{2}\eta_{p}-\frac{1}{4}\left(\eta_{l}+\eta_{r}\right)\right]\sin^{2}\psi,$
(19) $\displaystyle\eta_{V}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\eta_{r}-\eta_{l}\right)\cos\psi.$ (21)

Here, $\psi$ is the angle between the direction of light propagation and the
direction of the magnetic field, while $\eta_{l}$, $\eta_{p}$, and $\eta_{r}$
are the monochromatic absorption coefficients (normalized to $\kappa_{p}$)
originating from atomic transitions due to radiation with left-circular
($\Delta m=-1$), linear ($\Delta m=0$), and right-circular ($\Delta m=+1$)
polarizations, respectively.

Our code solves Eqs. (10) using the semi-analytical method of Martin &
Wickramasinghe (1979) for a grid of optical depths $\\{\tau_{j}\\}$, where it
is assumed that

$X=X_{a}+X_{b}\tau+\sum_{i=1}^{4}X_{c,i}\exp\left(a_{i}\tau\right)$ (22)

between two successive layers ($\tau_{j}\leq\tau\leq\tau_{j+1}$) for the
Stokes intensities $X=I,Q,U,V$, with $X_{a}$, $X_{b}$, and $X_{c,i}$
constants. The evaluation method starts from an initial condition of Unno
(1956) at the inner boundary and then follows an iterative procedure from the
deepest layer to the outermost one. Within the limit of a field-free medium
($B\rightarrow 0$), $\rho_{R}=\rho_{W}=0$, $\eta_{r}=\eta_{l}=\eta_{p}$,
$\eta_{Q}=\eta_{V}=0$, and the usual single radiative-transfer equation on $I$
is obtained.

##  5 Opacity sources

Line absorption cross-sections were calculated with the h2db database of
Schimeczek & Wunner (2014a), which constitutes the most recent and complete
ones for computing energies and oscillator strengths of an isolated hydrogen
atom in arbitrary magnetic fields. Bound-bound cross-sections are represented
by

$\sigma_{\text{bb}}=\frac{\pi
e^{2}f}{m_{\text{e}}c}\frac{\Re\left[W\left(Z_{l}\right)\right]}{\sqrt{2\pi}\Delta_{D}},$
(23)

where $f$ is the oscillator strength of the transition, $\omega_{l}$ the line-
center frequency, and $\Delta_{D}$ the thermal broadening

$\Delta_{D}=\left(\frac{2k_{B}T}{m_{\text{H}}c^{2}}\right)^{1/2}\omega_{l}.$
(24)

In Eq. (23), $\Re\left[W(Z_{l})\right]$ represents a Voigt profile (Faddeyeva
& Terent’ev 1961; Armstrong 1967), with

$W(Z_{l})=e^{-Z_{l}^{2}}\left(1+\frac{2i}{\sqrt{\pi}}\int_{0}^{Z_{l}}e^{t^{2}}\text{d}t\right)$
(25)

and

$Z_{l}=\frac{\omega-\omega_{l}+\frac{1}{2}\Delta_{S}i}{\sqrt{2}\Delta_{D}}.$
(26)

In the last equation, $\Delta_{S}$ is the Stark broadening, which was
calculated according to Jordan (1992):

$\Delta_{S}=0.0192cF_{0}\overline{n_{k}}C,$ (27)

with $F_{0}$ being the Holtsmark normal field strength, $\overline{n_{k}}$ an
average value calculated from the lower and upper $n$ quantum numbers (Unsöld
1968; Rauch & Werner 1991), and $C$ a free parameter that was set equal to 0.1
(see Putney & Jordan 1995). Eq. (27) is just a rough approximation, since no
comprehensible data about the effects of simultaneous arbitrary magnetic and
electric fields on the hydrogen atom have been published so far.

The knowledge of the bound-free opacity of atoms at magnetic fields is still
fragmentary. Its calculation demands considerable work since it involves
numerous transition channels (Jordan 1989; West 1989), many initial states,
and more sophisticated wave functions than those present in the zero-field
photoionization process. A number of rigorous evaluations have been performed,
but they are limited in terms of the field strength domain, wavelength range,
and number of bound states and/or light polarizations (e.g., Kara & McDowell
1981; Bhattacharya & Chu 1985; Delande et al. 1991; Wang & Greene 1991; Merani
et al. 1995; Zhao & Stancil 2007; Zhao et al. 2016; Zhao 2021). Therefore, we
followed the usual treatment based on transition probabilities calculated at
the rigidity approximation for the wave functions (Lamb & Sutherland 1974),
where the allowed transitions ($\Delta l=\pm 1$, $\Delta m=0,\pm 1$) from
bound states to Landau states are distributed according the Wigner-Eckart
theorem, and the wavelengths of all bound-free edges are calculated with
accurate energies of bound and free states as functions of the field strength
(Eqs. (2) and (9)). The precision of this method has been discussed by Jordan
& Merani (1995).

A cyclotron absorption process is produced by $\Delta m=+1$ free-free
transitions (i.e., right hand circularly polarized light)333Transitions with
$\Delta m=0$ and $-1$ are forbidden by conservation laws in the kinematics
(Lamb & Sutherland 1972). of electrons from a lower Landau state into a more
energetic one, with a peak near the cyclotron resonance due to transitions
between adjacent levels. The corresponding cross-section is calculated as

$\sigma_{\text{cy}}=\frac{\Re\left[W\left(Z_{c}\right)\right]}{\sqrt{2\pi}\Delta_{\text{e}}}\sigma_{+},$
(28)

with

$Z_{c}=\frac{\omega-\omega_{\text{e}}+\nu_{\text{eff}}i}{\sqrt{2}\Delta_{\text{e}}}$
(29)

and

$\Delta_{\text{e}}=\left(\frac{2k_{B}T}{m_{\text{e}}c^{2}}\right)^{1/2}\left|\cos\left(\psi\right)\right|\omega_{\text{e}},$
(30)

where $\sigma_{+}$ is the frequency-integrated cross-section calculated
quantum-mecanically as (Lamb & Sutherland 1974)

$\sigma_{+}\approx\frac{e^{2}}{\hbar c}\left(\frac{2\pi
c}{\omega_{\text{e}}}\right)^{2}\left(\frac{Be\hbar}{m_{e}^{2}c^{3}}\right)\frac{\omega_{\text{e}}}{1-e^{-\hbar\omega_{\text{e}}/k_{B}T}},$
(31)

and $\nu_{\text{eff}}$ is half the half-width of a Lorentzian profile
describing collisions by electrons (Bekefi 1966).

Figure 1: Dependence of frequency of electron-electron collisions
($\nu_{\text{eff}}$) with field strength as predicted by Eq. (32) (continuum
line for $B<6.3T^{3/2}$, dash-dotted line for $B>6.3T^{3/2}$) and Eq. (33)
(dashed line). The selected physical conditions (gas temperature $T$, mass
density $\rho$) correspond to different depths in a model atmosphere with
$T_{\text{eff}}=15000$ K.

This is given by (Zheleznyakov et al. 1999)

$\nu_{\text{eff}}=\left\\{\begin{array}[]{ll}\displaystyle\frac{2\sqrt{2\pi}}{5\exp(1)}\frac{m_{e}^{1/6}c^{4/3}(k_{\mathrm{B}}T)^{1/2}n_{p}}{B^{4/3}},&B>6.3T^{3/2},\\\\[12.91663pt]
\displaystyle\frac{8\sqrt{2\pi}}{15}\frac{e^{4}n_{p}}{m_{e}^{1/2}\left(k_{\mathrm{B}}T\right)^{3/2}}\ln\left(\frac{m_{e}^{1/2}c(k_{\mathrm{B}}T)^{3/2}}{e^{3}B}\right),&B\leq
6.3T^{3/2}.\end{array}\right.$ (32)

For low enough magnetic fields, Eq. (32) gives unphysical values (logarithmic
divergence). To avoid that, we adopted the zero-field expression of Ginzburg
(1967) as the upper limit for the $\nu_{\text{eff}}$ value (see Fig. 1):

$\nu_{\text{eff}}=\frac{2\sqrt{2\pi}}{3}\frac{e^{4}n_{p}}{m_{e}^{1/2}\left(k_{\mathrm{B}}T\right)^{3/2}}\ln\left(\frac{k^{3}T^{3}}{4\pi
n_{p}e^{6}}\right).$ (33)

Finally, the code includes the calculation of magneto-optical parameters,
Faraday rotation ($\rho_{R}$), and Voigt effect ($\rho_{W}$), which describe
anomalous dispersion of light. While $\rho_{R}$ arises from the inequality
between refractive indexes for right and left circularly polarized light in
the presence of a magnetic field, $\rho_{W}$ comes from a phase shift between
linear polarized components of the electric-field vector, both parallel and
perpendicular to the magnetic field. Magneto-optical parameters have
contributions from both lines and continuum, as described by Martin &
Wickramasinghe (1981, 1982), and Jordan et al. (1991). Specifically, line
contributions are given by

$\rho_{R}=-\left(\sum_{i}\eta_{ri}F_{ri}-\sum_{j}\eta_{lj}F_{lj}\right)\cos\psi,$
(34)
$\rho_{W}=-\left(\sum_{i}\eta_{ri}F_{pi}-\frac{1}{2}\sum_{j}\eta_{lj}F_{lj}\frac{1}{2}\sum_{k}\eta_{rk}F_{rk}\right)\sin^{2}\psi,$
(35)

with

$F=\frac{1}{a}\left(\frac{1}{2}xV+\frac{1}{4}\frac{\partial V}{\partial
x}\right)$ (36)

being the dispersion function (Wittmann 1974; Martin & Wickramasinghe 1981),
where $a=\Im\left(Z_{l}\right)$, $x=\Re\left(Z_{l}\right)$,
$V=\Re\left[W\left(Z_{l}\right)\right]$, and

$\frac{\partial V}{\partial x}=-\frac{\Re\left[Z_{l}\cdot
W\left(Z_{l}\right)\right]}{\sqrt{\pi}\Delta_{D}^{2}},$ (37)

$W$ and $Z_{l}$ being given by Eqs. (25) and (26), respectively. In Eqs. (34)
and (35), the sums extend over all allowed transitions. On the other hand,
continuum contributions to $\rho_{R}$ and $\rho_{W}$ were computed as they
were by Külebi (2010), which featured a self-consistent calculation from
kinetic theory of plasmas to determine refractive and absorptive properties of
magnetic atmospheres, such that

$\rho_{R}=-\frac{\sqrt{\pi}}{2c}\frac{\omega_{p}^{2}}{\sqrt{2}\Delta_{D}}\Im\left[W\left(Z_{c}\right)\right]\cos\psi,$
(38)
$\rho_{W}=-\frac{\sqrt{\pi}}{4c}\frac{\omega_{p}^{2}}{\sqrt{2}\Delta_{D}}\Im\left[W\left(Z_{c}\right)\right]\sin^{2}\psi,$
(39)

where $\omega_{p}=\sqrt{4\pi n_{e}e^{2}/m_{e}}$ is the plasma frequency.

Figure 2: Absorption coefficients for bound-bound (red), bound-free (blue;
dark and light lines for $\Delta m=1,-1$ transitions), and cyclotron (green)
processes for various magnetic field strengths.

In Fig. 2, we compare the different contributions of hydrogen opacity arising
from transitions $\Delta m=\pm 1$ in a gas with $T=20000$ K for different
magnetic field strengths. For low enough magnetic strengths ($B\la 1$ MG),
opacities from different light polarizations tend to be coincidental and
resemble those of a field-free gas. As magnetic field strength increases,
photoionization continua and spectral lines split in a number of components.
Besides this, cyclotron absorption is dominant around cyclotron wavelength and
becomes negligible far away it, with the peak moving from far- to near-
infrared wavelengths as the field strength increases.

##  6 Surface integration method

Equation (1) provides the magnetic-field value in each point of the stellar
atmosphere affecting particle populations, opacities and radiative energy
transfer. Because of the high surface gravity in white dwarfs, their
atmospheres are compact (thickness much lower than the stellar radius), and
the depth dependence of the magnetic field as given by Eq. (1) can be
neglected. Individual model atmospheres are computed for a selection of $M$
points centered in cells that cover the visible hemisphere of the star.
Transfer equations (10) are solved in these points, and the resulting Stokes
intensities are integrated with a quadrature scheme to obtain the emerging
stellar flux,

$X=\sum_{j=1}^{M}w_{j}X_{j},$ (40)

where $w_{j}$ is a quadrature weight and $X_{j}=I_{j}$, $Q_{j}$, $U_{j}$,
$V_{j}$ are the emerging intensities in the $j$th-cell.

Figure 3: Segmentations of stellar surface and their performances in energy-
flux integrations for a centered dipole ($B_{d}=10$ MG, $i=45^{\circ}$). The
magnetic-field strength in each cell is averaged using its values on the
center (black circle) and vertices (red circles). Upper panel: Segmentation
based on spherical $t$-design ($t=19$) and its associated Voronoi
tessellation. Middle panel: Segmentation constructed with equal steps in
latitude and longitude. Lower panel: Variation of mean squared error of
emerging flux as a function of the cell number for $t$-design (solid lines)
and constant latitude and longitude step (dashed lines) integrations and
different field strengths. Table 1: Stellar parameters of a selection of MWDs
derived from SDSS spectra fits in the present work and other recent studies.

|  | This work |  |  |  | Hardy+ |  |  |  | Amorim+ |  |
---|---|---|---|---|---|---|---|---|---|---|---|---
Star |  $T_{\text{eff}}$[K] |  $B_{\text{d}}$[MG] |  $i$[∘] | $a_{z}$ |  $T_{\text{eff}}$[K] |  $B_{\text{d}}$[MG] |  $i$[∘] | $a_{z}$ |  $T_{\text{eff}}$[K] |  $B_{\text{d}}$[MG] |  $i$[∘] | $a_{z}$
J0725+3214 | 26000 | 13.16 | 80 | $-0.21$ | 22240 | 13.82 | 52 | 0.06 | 24000 | 15.15 | 58 | $0.29$
J0805+2153 | 39000 | 7.00 | 85 | $-0.15$ | 30000 | 3.00 | 17 | 0.30 | 37141 | 6.77 | 66 | $-0.49$
J0931+3219 | 11000 | 8.46 | 75 | $0.19$ | 13476 | 9.04 | 52 | 0.30 | 18000 | 12.95 | 15 | $0.26$
J1018+0111 | 10500 | 108.12 | 60 | $0.1^{*}$ | 12845 | 76.07 | 16 | 0.05 | 11000 | 122.69 | 65 | 0.01
J1511+4220 | 12000 | 11.28 | 40 | $0.28$ | 11595 | 14.01 | 40 | 0.30 | 11500 | 12.83 | 35 | $-0.28$
J1516+2746 | 33000 | 3.25 | 5 | $0.25$ | – | – | – | – | 30000 | 3.03 | 75 | $-0.44$
J1603+1409 | 10000 | 47.01 | 70 | $0.15$ | 10547 | 48.88 | 47 | 0.23 | 9500 | 49.40 | 52 | $-0.24$
J2247+1456 | 19000 | 437.10 | 10 | $-0.15$ | – | – | – | – | 18000 | 515.09 | 29 | 0.21

444(*) Additional $a_{y}=0.07$ dipole offset.

Table 2: For the studied MWDs, S/N of spectra, and percentage error $\chi^{2}$ derived from our fits using centered and offset dipole configurations.  Star | S/N | centered | offset
---|---|---|---
J072540.8+321401.1 | 07.43 | 1.39 | 1.32
J080502.3+215320.5 | 18.52 | 1.99 | 1.96
J093126.1+321946.1 | 07.49 | 1.41 | 1.29
J101805.0+011123.5 | 49.54 | 52.11 | 40.42
J151130.2+422023.0 | 19.79 | 4.97 | 3.88
J151606.3+274647.0 | 14.23 | 2.33 | 2.10
J160357.9+140930.0 | 16.91 | 6.74 | 5.00
J224741.46+145638 | 29.37 | 36.12 | 29.23

We used a quadrature integration based on spherical $t$-designs. A spherical
$t$-design is a set of $N$ points distributed on the stellar surface
($\mathbb{S}^{2}$) for which the average value over these points of any
spherical polynomial with a degree of at most $t$, $p(\mathbf{r})$, is equal
to the average value of the polynomial over the sphere (Delsarte et al.
1977):555A spherical polynomial is a combination of the standing waves on the
surface $\mathbb{S}$.

$\frac{1}{N}\sum^{N}_{j=1}p(\mathbf{r}_{j})=\frac{1}{|\mathbb{S}^{2}|}\int_{\mathbb{S}^{2}}p(\mathbf{r})d(\mathbf{r}).$
(41)

In practice, we adopted $N$-point arrays as provided by Hardin & Sloane (1996)
for specific $t$ values in the $t$-design scheme. The $N$ points are uniformly
distributed on the stellar surface, and over them a Voronoi spherical
segmentation is calculated (upper panel in Fig. 3). The magnetic-field
distribution in the stellar disk is determined for a number $M$ ($<N$) of
Voronoi cells (all those that are in the visible hemisphere), where
$\mathbf{B}$ is averaged taking into account its values at the center and
vertices of each cell. The set of points in the $t$-design and the values on
the magnetic field on them are finally used in Eq. (40).

The example shown in Fig. 3 (upper panel) corresponds to a $t$-design
segmentation for $M=120$ points ($t=21$) on the visible hemisphere of a star
with a centered dipolar field ($B_{\text{d}}=10$ MG, $i=45^{\circ}$). For the
same conditions, the middle panel in the figure shows the segmentation
resulting when constant steps in latitude and longitude are taken. This
segmentation produces a higher density of points toward the poles, while the
$t$-design gives a uniform distribution over the whole stellar disk. The
bottom panel in Fig. 3 shows the accuracy in evaluating emergent energy flux
for both surface mappings as a function of the point number $M$ on the visible
disk. There, $\epsilon_{\text{ms}}$ represents the mean squared error in the
convergence of the emerging flux in the $380$ nm $\leq\lambda\leq 800$ nm
range (in practice, we used a reference model with $M=1000$). These
evaluations correspond to MWD models with $T_{\text{eff}}=20000$ K and three
magnetic dipole intensities $B_{\text{d}}/B_{0}=10^{-4}$, $10^{-3}$, and
$10^{-2}$. Convergence decreases as the field strength grows, but it clearly
demands fewer iterations when the $t$-design is used. Reliable results of
emerging fluxes and spectral fits are obtained with $t$-design quadratures for
a reasonable cell number (typically from $M\approx 40$ for $\beta\approx
10^{-4}$ to $M\approx 120$ for $\beta\approx 10^{-2}$) and CPU time. Appealing
to its fast convergence, we can omit the “magnetic broadening” sometimes used
to solve the finite discretization of the stellar atmosphere (Jordan et al.
1991; Külebi et al. 2009).

##  7 Model atmospheres and spectral fits

To check the new magnetic atmosphere code with improved constitutive physics,
we analyzed a sample of magnetic white dwarfs with observed spectra in the
SDSS survey and compared our best spectrum fits with the predictions of
preexisting models. The procedure to find the best model for each object is by
visual comparison of the observed spectral energy distribution with
predictions of a set of model atmosphere calculations. The free parameters
($T_{\text{eff}}$, $B_{\text{d}}$, $i$, $a_{z}$) are selected through an
error-reduction process using a least-squares method on the difference between
observed ($f^{\text{obs}}_{l}$) and synthetic ($f^{\text{cal}}_{l}$) fluxes,

$\chi^{2}=\frac{1}{N}\sum_{l}\frac{\left(f^{\text{obs}}_{l}-f^{\text{cal}}_{l}\right)^{2}}{\sigma_{l}^{2}},$
(42)

with $N$ being the flux-data number and $\sigma_{l}$ the data precision.

The selected sample of MWDs comprises eight stars with SDSS energy
distribution from 380nm to 800nm. Since our main goal is to test the
performance of the new code, the objects were chosen on the basis of their
variety of field strength and effective temperature. Table 1 summarizes the
results of the analyses of the MWDs and provides a comparison with other
recent theoretical studies. The first row in the table gives the object names
ordered by their right ascension, adopting the usual compact notation for
SLOAN targets based on epoch J2000.0 coordinates (for the sake of clarity,
full SDSS identifications are given in Table 2). Further rows specify the
best-fit stellar parameters calculated in the present work and those obtained
in recent studies of Hardy et al. (2023) and Amorim et al. (2023). The
goodness of each fit is represented by the $\chi^{2}$ value (Table 2). Small
errors are obtained for MWDs with weak or moderated field strength; however,
as usual, fit deviations increase for high field objects where radiative
transfer codes for magnetized white dwarf atmospheres have greater difficulty
reproducing the observed spectra (e.g., Euchner et al. 2006). Incidentally,
the high-field stars in our sample (J2247+1456 and J1018+0111) have SDSS
spectra with high signal-to-noise ratios (Table 2), which exacerbates their
$\chi^{2}$ values.

The present study focuses on the normal total intensity (Stokes parameter
$I$). Although the code can also provide light-polarization information (Fig.
4), this is not examined here since the data available in the literature are
limited and because opacity theory becomes inappropriate, especially for
evaluating the continuum polarization at high field strengths (e.g., Putney &
Jordan 1995); hence, current analyses are based in part on rather empirical
relationships (Bagnulo & Landstreet 2020; Berdyugin et al. 2022).

Figure 4: Observed wavelength dependence of circular polarization $V$ (gray
line, Liebert et al. 1983) showing Balmer line polarization for a low field
MWD (J1659+4401, also known as PG 1658+441). The blue line represents our fit
model with parameter values indicated on the plot.

###  7.1 The effect of magnetic field on populations

Using the chemical model described in Sect. 3, we studied the impact of the
magnetic field on the populations of electronic states and analyzed how it
affects the synthetic spectra of a magnetic white dwarf. For this purpose, we
chose the SLOAN spectrum of J2247+1456, a magnetic white dwarf with the
highest field of the eight stars analyzed in this paper.

####  7.1.1 J2247+1456

The magnetic field of this object was identified by Harris et al. (2003) and
the average strength of its surface field was evaluated as $300$ MG. The SDSS
spectrum shows features often found in strongly magnetic white dwarfs, and
despite the efforts made, models cannot reproduce it very well. The observed
flux was described by Schmidt et al. (2003) as originating from an atmosphere
of $T_{\text{eff}}=18000$ K and a centered dipole of 560 MG. Külebi et al.
(2009) used 421.15 MG and 469.52 MG centered and offset dipoles, respectively,
to fit the spectrum with an effective temperature of 50000 K. Kepler et al.
(2013) estimated a magnetic strength of 47.0 MG based on measurements of
H$\alpha$ splitting (although this visual field determination method is
inaccurate at $B\ga 100$ MG), whereas Amorim et al. (2023) fit the spectrum
with an offset dipole of 515.09 MG and $T_{\text{eff}}=18000$ K. These studies
show a range of varying results, most likely because the input physics at very
strong fields is not completely clear to date.

Figure 5: Theoretical fluxes fit to SDSS J2247+1456 spectrum (gray lines)
comparing the effects of the magnetic field on particle populations (specified
on the plot). Upper spectrum is displaced vertically for clarity.

In Fig. 5, we compare the synthetic spectrum computed using our magnetic
chemical model (green line) to the spectrum obtained with a standard
ionization equilibrium without magnetic effect on populations (red line). From
these fits, we find field strengths of $B_{\text{d}}=437$ MG and 428 MG,
respectively, which are relatively close to values determined from other
models. On the other hand, our evaluations of the effective temperature (19000
K and 18000 K, respectively) fall near to the mean value of previous works,
except for the high value obtained by Külebi et al. (2009). As can be
appreciated in Fig. 5, we obtain a lightly lower spectrum fit error when
magnetic effects are included in the chemical model ($\chi^{2}\approx 29\%$
compared to 45% in the nonmagnetic case).

The most important result of our fits is that the inclusion of magnetic
effects on the ionization equilibrium leads to a higher temperature
distribution in the atmosphere of a highly magnetic white dwarf
($T_{\text{eff}}=19000$ K compared to $18000$ K in the nonmagnetic model).
This can be understood by analyzing the particle abundances in both models.
Effects of magnetic field on occupation numbers are clearly seen in Fig. 6 for
a model atmosphere with $T_{\text{eff}}=19000$ K and $B=437$ MG.

Figure 6: Atomic hydrogen abundances as function of Rosseland mean optical
depth for an atmosphere with $T_{\text{eff}}=19000$ K and $B=437$ MG. Upper
panel: Fraction of neutral atoms calculated with (blue line) and without (red
line) magnetic-field effects on the ionization equilibrium. Lower panel:
Abundance of atoms in $n=2$ state in absence of magnetic field (red line) and
disaggregated into sublevels when the field is present (solid blue lines for
spin-down and dashed blue lines for spin-up states). Abundances of
$2p_{-1}(m_{s}=+\frac{1}{2})$ and $2p_{+1}(m_{s}=-\frac{1}{2})$ states are
equal (lowest solid line).

Fig. 6 shows the fraction of neutral atoms (upper panel) and those in
sublevels of the $n=2$ state (lower panel) using our ionization model (blue
lines) and a standard field-free calculation (red lines). The first thing to
note is that the magnetic field causes a significant rise in the population of
neutral atoms. This is mostly due to the binding energy of the ground state
$(n,l,m,m_{s})=(1,0,0,-\frac{1}{2})$ increasing monotonically with the field
strength, which favors the recombination process. The greatest deviations with
respect to the zero-field evaluation occur in outer layers of the atmosphere
where strong spectral lines are formed. Each $(n,l)$ level splits into
sublevels characterized by their magnetic ($m$) and spin ($m_{s}$) quantum
numbers. Energies of spin-up states and those with positive $m$ approach the
continuum, while the spin-down states with nonpositive $m$ become tighter for
conditions typical of a magnetic white dwarf atmosphere (Vera Rueda & Rohrmann
2020). Consequently, sublevel populations are significantly different than
those assuming a field-free gas, as can be seen in Fig. 6 for atoms in $n=2$.

The discussed changes in the ionization equilibrium affect the opacity and
thus the emerging energy distribution from the star. Specifically, the
increase in the abundance of neutral atoms in model atmospheres that include
magnetic-field effects on the particle distributions is compensated by a
reduction of the gas temperature in models with abundances evaluated in the
zero field. In conclusion, the use of a field-free chemical model leads to a
significant underestimation of the effective temperature (a thousand degrees
in the case analyzed here) for highly magnetic white dwarfs. This discrepancy
is expected to decrease for objects with weaker fields. However, we do not
repeat a similar analysis of this effect in low field stars due to the
inherent difficulties in finding optimal fits in MWD spectrum modeling. Due to
the proximity of minimum values of $\chi^{2}$ for different combinations of
free parameters (which is well known; e.g., Martin & Wickramasinghe 1984;
Euchner et al. 2006; Külebi et al. 2009), field effects on particle abundances
in weak magnetic stars become hidden by the search method of the best spectrum
fit.

Figure 7: Location of a group of MWDs in $B$–$T_{\text{eff}}$ diagrams, as
determined from spectrum fits (symbols) and spectroscopic analysis (arrows).
Vertical (horizontal) arrows mark effective temperature (field strength)
obtained from spectroscopic analysis. Open (filled) symbol for centered (off-
center) dipole model. Notation: a (Eisenstein et al. 2006), b (Kleinman et al.
2013), c (Dufour et al. 2017), d (Gentile Fusillo et al. 2021), e (Kepler et
al. 2013), f (Kilic et al. 2020), g (Leggett et al. 2018), h (Bédard et al.
2020), i/squares (Külebi et al. 2009), diamonds (Hardy et al. 2023), triangles
(Amorim et al. 2023), x (Vanlandingham et al. 2005), crosses (Schmidt et al.
2003), stars (Wickramasinghe & Cropper 1988), circles (this work).

###  7.2 Comparison with other fit studies

Figure 7 shows calculated values of dipole intensity and effective temperature
for the MWDs in Table 1, which were obtained from our best spectral fit and by
estimations of previous studies. Objects are ordered top to bottom and left to
right by decreasing the field intensity. Values
($B_{\text{d}},T_{\text{eff}}$) obtained from spectrum fits with magnetic
models are represented by symbols, where circles correspond to results
obtained in the present work. Vertical arrows indicate $T_{\text{eff}}$ values
derived in surveys with atmosphere models for nonmagnetic DA white dwarfs.
Horizontal arrows point out field-strength values resulting from Zeeman
splitting measurements and those obtained from flux fits with magnetic models
without $T_{\text{eff}}$ values reported. A brief description of each star is
summarized below, except J2247+1456, which was discussed in the previous
subsection.

To assess the accuracy of the new magnetic atmosphere code, below we compare
our calculated spectrum for J1018+0111, J1511+4220, and J0805+2153 to those
published by Külebi et al. (2009), in both centered and offset dipole
configurations. Later, in Section 7.3, we show results from flux fits of the
remaining MWDs in Table 1 for which no published synthetic spectra were found
for a direct comparison, but whose physical parameters have been derived in
other works.

Figure 8: Observed spectrum of J1018+0111 (gray lines) compared to best fits
obtained in this work (green and blue lines) and by Külebi et al. (2009) (red
lines), for both centered and offset dipoles (their intensity maps are shown
in the inserted plots).

####  7.2.1 J1018+0111

Also known as PG 1015+014, the first data concerning this object were
published in the Palomar-Green survey (Green et al. 1986). However, a previous
report by Angel (1978) gave some details about preliminary unpublished
observations of this star by the Steward Observatory and Kitt Peak National
Observatory groups, who observed variable circular polarization with an
amplitude of $1.5\%$ and a period of $98.75$ min, and undefined absorption
features in the spectrum. These two facts were attributed to the magnetic
nature of a rotating white dwarf with a surface magnetic field stronger than
$100$ MG.

Wickramasinghe & Cropper (1988) tested different magnetic-field geometries to
fit the star spectrum using a model with $T_{\text{eff}}=14000$ K and
concluded that a centered dipole is the most suitable configuration for the
magnetic field, with a strength of $B_{\text{d}}=120$ MG and inclination of
$i=70^{\circ}$–$110^{\circ}$ depending on the variability phase. However, some
spectral features could not be fit, which suggested a more complex field
structure. Schmidt et al. (2003) used a centered dipole approximation to fit
J1018+0111’s spectrum from SDSS DR1 with $B_{\text{d}}=120$ MG and
$T_{\text{eff}}=12500$ K. A detailed study of Euchner et al. (2006) using the
superposition of various multipoles identified contributions to Zeeman
features with fields in the range of 50–90 MG and an effective temperature of
10000 K, but it was not enough to appropriately fit to all phases. Subsequent
works based on offset-dipole fits (Külebi et al. 2009; Amorim et al. 2023;
Hardy et al. 2023) found $B_{\text{d}}$ values in the same range as previous
studies ($\approx 76$–$123$ MG). Our best fit, $B_{\text{d}}=108$ MG and
$T_{\text{eff}}=10500$ K (full circle in Fig. 7), falls within that range.

Figure 8 compares our best fits for centered (blue line) and offset dipole
(green line) models with the observed SDSS flux (gray lines) and synthetic
ones of Külebi et al. (2009) (red lines). Our centered dipole model fails to
reproduce the flux around the $\lambda 555$ region and overestimates it for
wavelengths smaller than $\lambda 480$, while the fit of Külebi et al.
underestimates the $\lambda 515$ and $\lambda 740$ surroundings and shows
other discrepancies in the bluer region. The use of an offset dipole improves
both models, but some discrepancies remain, especially below $\lambda 470$ and
for features redward of $\lambda 720$. However, most differences between the
observed flux and our synthetic spectrum are removed in the intermediate
region.

As other authors have pointed out, mentioned difficulties in achieving a good
fit of the observed spectrum may be due to an oversimplification of the dipole
model to describe the field distribution on the surface of J1018+0111.
However, it should be taken into account that an excessive number of free
parameters for the magnetic field representation can mask limitations in the
approximations used to evaluate the emitted energy in each grid point of the
stellar surface.

Figure 9: Same as Fig. 8, but for J1511+4220.

####  7.2.2 J1511+4220

This was first identified as a MWD by Vanlandingham et al. (2005); the authors
derived a centered dipolar magnetic field with $B_{\text{d}}=12$ MG and
$i=60^{\circ}$ through a geometric approach and an effective temperature of
$9750$ K from color-color diagrams. This object was included in various
studies based on zero-field models with temperature estimations into two
range: one close 11000 K and other around 32500 K (Fig. 7). Best spectrum fits
with dipole field geometry (Külebi et al. 2009; Kepler et al. 2013; Amorim et
al. 2023; Hardy et al. 2023) were reached with $T_{\text{eff}}$ at the first
domain and field strength in the range of $\approx 10$–$23$ MG. Our best
evaluations, $B_{\text{d}}=11.3$ MG and $T_{\text{eff}}=12000$ K, approach the
lowest values previously reported.

Hardy et al. (2023) categorized the SDSS spectrum of this object among those
not well reproduced with a simple dipole. Fig. 9 presents our best fits and
those of Külebi et al. (2009). Centered dipole models are not really
satisfactory, although our fit (with lower field strength and slightly higher
temperature, Fig. 7) shows some improvements, especially in H$\alpha$ wings
(to the sides of 656nm), H$\beta$ component positions (around 486nm; although,
component depths are underestimated), and the flux curve below $\lambda 480$.
Offset dipole configurations improve both fits. In the synthetic spectrum of
Külebi et al. (2009) ($B_{\text{d}}=8.37$ MG, $i=6^{\circ}$, $a_{z}=0.31$),
H$\alpha$ wings are corrected and the underestimation of the continuum below
$\lambda 480$ nm is partially reduced. In our model ($B_{\text{d}}=12.28$ MG,
$i=40^{\circ}$, $a_{z}=0.28$), the offset dipole improves H$\alpha$ wings,
provides a better distribution of H$\beta$ components, and fits the bluer
region of the spectrum slightly better. The magnetic-field parameters
determined in this work are similar to those obtained by Hardy et al. (2023)
and those of Amorim et al. (2023), except for the displacement in the $z$
direction (see Table 1).

Figure 10: Same as Fig. 9, but for J0805+2153.

####  7.2.3 J0805+2153

This is the hottest object we studied. Its magnetic nature was first
recognized by Vanlandingham et al. (2005) from SLOAN DR3. They fit the SDSS
spectrum with a centered magnetic dipole of $B_{\text{d}}=5$ MG,
$i=60^{\circ},$ and $T_{\text{eff}}=28000$ K. This object was also included in
the surveys of Eisenstein et al. (2006) and Kleinman et al. (2013), with
$T_{\text{eff}}=38211$ K and 37141 K, respectively. Dufour et al. (2017)
estimated its effective temperature to be relatively low (19713 K). Magnetic
properties of this object were estimated in $3\leavevmode\nobreak\
$MG$\leavevmode\nobreak\ <B_{\text{d}}<7$ MG, $17^{\circ}\leavevmode\nobreak\
<i<87^{\circ}$ MG, and $-0.49\leavevmode\nobreak\ <a_{z}<0.3$ (Külebi et al.
2009; Kepler et al. 2013; Amorim et al. 2023; Hardy et al. 2023). Our optimal
fit yields $B_{\text{d}}=7$ MG, $i=85^{\circ}$, $a_{z}=-0.15$, and
$T_{\text{eff}}=39000$ K. The variety of values obtained in different works
for the inclination angle of the field and the shift of the dipole center (as
in other objects analyzed here) suggests that these parameters (or the field
geometry in general) are not well constrained.

Observed and calculated fluxes for J0805+2153 are displayed in Fig. 10. The
centered dipole fit of Külebi et al. (2009) yields strong H$\beta$ components
and features below $\lambda 450$, which are mostly corrected with a
considerable shift of the dipole center ($a_{z}=0.39$) and a reduction of
$B_{\text{d}}$ from 6.1 MG (centered) to 3.1 MG (offset). Our synthetic
spectra with both centered and offset dipoles ($B_{\text{d}}\approx 7$ MG)
reasonably reproduce most of the observed spectrum.

###  7.3 Additional spectral fits

Figure 11: Comparison between emerging flux of MWD stars from SLOAN Digital
Sky Server (gray lines) and calculated by our code for a centered dipole model
(blue lines) and offset dipole (green lines, displaced vertically for
clarity).

Figure 11 shows emerging fluxes (gray lines) of four MWD classified as white
dwarfs by Kleinman et al. (2013) from SDSS DR7 and identified as magnetic
objects by Kepler et al. (2013). The magnetic nature of these MWDs was also
analyzed by Hardy et al. (2023) (except J1516+2746) and Amorim et al. (2023).
Results in Fig. 11 are sorted upward by ascending magnetic dipole strength
(comprising from $\sim 3\,\text{MG}$ to $\sim 40\,\text{MG}$), which shows the
overall effect of field strength on white-dwarf spectra. In fact, as the field
becomes stronger, hydrogen absorption lines are initially broadened (e.g.,
J1516+2746), split into different Zeeman components (J0725+3214), and finally
mix together (especially members from high Balmer lines) when the field is
strong enough (J1603+1409). Fig. 11 also displays our fits with centered (blue
lines) and offset (green lines) dipoles compared with observed fluxes (no
calculated emerging fluxes were found in the literature for these stars). We
make a few comments for each object in the following sections.

####  7.3.1 J1603+1409

This object was reported as a short-period variable star by Scholz et al.
(2018) with an estimated variability period of $110\pm 3$ min. Observed and
calculated spectra are shown in the top of Fig. 11. Magnetic strength in the
stellar surface is high enough to cause line mixing below $520$ nm, whereas
$\sigma_{-}$, $\pi$, and $\sigma_{+}$ components of H$\alpha$ ($\approx\lambda
600$, $\lambda 650,$ and $\lambda 700$, respectively) become clearly
separated. An overall agreement is obtained with a centered dipole of
$B_{\text{d}}=38.55$ MG at a near pole-on view ($i=10^{\circ}$). A better fit
results from a strongly inclined ($i=70^{\circ}$) offset dipole ($a_{z}=0.15$)
with a strength ($B_{\text{d}}=47.01$ MG) similar to those derived by Amorim
et al. (2023) and Hardy et al. (2023) (Table 1).

####  7.3.2 J0725+3214

This object represents an example of a warm MWD with moderate surface fields
(Fig. 11). Its SDSS spectrum is reasonably well represented with
$B_{\text{d}}\approx 13$ MG and $T_{\text{eff}}=26000$ K. The low fitting
error suggests that the field geometry is close to that of a dipole (Table 2).
Hardy et al. (2023) and Amorim et al. (2023) found values slightly higher for
$B_{\text{d}}$ and lower for $T_{\text{eff}}$ (Table 1). On the other hand,
the spectroscopic fitting of J0725+3214 with zero-field model atmospheres by
Kleinman et al. (2013) and Dufour et al. (2017) gave appreciably higher
effective temperatures, $T_{\text{eff}}=34711$ K and 28951 K, respectively
(Fig. 7).

####  7.3.3 J0931+3219

Strong absorption features are observed for this MWD, which can be reproduced
with an effective temperature of 11000 K according to our synthetic spectrum
(Fig. 11). Values of $T_{\text{eff}}=11526$ K (Dufour et al. 2017) and $16248$
K (Kleinman et al. 2013) were derived from nonmagnetic models. Our flux fits
revel moderate field strengths ($\ga 8$ MG), as that estimated through visual
inspection of Zeeman splitting (Kepler et al. 2013). A dipole shift along the
$z$-axis reduces the fit error slightly. Somewhat higher $T_{\text{eff}}$ and
$B_{\text{d}}$ values were found by Hardy et al. (2023) and Amorim et al.
(2023), as shown in Table 1 and Fig. 7.

####  7.3.4 J1516+2746

This object was studied as one of several hot white dwarfs from SDSS DR12 by
Bédard et al. (2020), which inferred an effective temperature of $40581$ K by
spectroscopic analysis. A higher value, $T_{\text{eff}}=67365$ K, was
determined from color-color diagrams (Gentile Fusillo et al. 2021). Other
estimates place $T_{\text{eff}}$ at lower values (Fig. 7). A Zeeman pattern of
H$\alpha$ and H$\beta$ suggests a mean field of $2.6<B<3.0$ MG (Kepler et al.
2013). Using magnetic synthetic spectrum fitting, Amorim et al. (2023)
estimated $B_{\text{d}}=3.03$ MG and $T_{\text{eff}}=30000$ K. We obtained
$B_{\text{d}}=3.25$ MG and $T_{\text{eff}}=33000$ K with an offset dipole
configuration that reduces the fit error of our best centered dipole model
(Fig. 11).

Table 3: Correlations on physical parameters derived from Amorim et al. (2023) ($=A$), Hardy et al. (2023) ($=H$) and this work ($=V$). Comparison between $H$ and $A$ corresponds to a common sample of 118 MWDs, most of them with moderate and low field strengths (115 stars with $B_{\text{d}}<50$ MG).  | $T_{\text{eff}}$ | $B_{\text{d}}$ | $i$ | $a_{z}$
---|---|---|---|---
$A$–$V$ | 0.97 | 0.99 | 0.35 | $-0.30$
$H$–$V$ | 0.99 | 0.98 | 0.04 | $+0.41$
$H$–$A$ | 0.69 | 0.96 | 0.15 | $-0.23$

###  7.4 Discussion

Several results emerge from the spectral fits and comparisons previously
analyzed. As stated in prior works, good fits to the observations are achieved
for stars with low field strengths. Specifically, dipole field geometries are
generally adequate to reproduce observed spectra in objects with mean field
strengths lower than 50 MG, with offset dipoles usually providing the best
fit. However, comparisons of physical parameters derived from different
numerical codes reveal some heterogeneous results. In particular, reasonable
agreement is found in the evaluation of the dipole strength, but significant
differences appear in the derived values for the inclination of the dipole
axis and its displacement from the stellar center.

Table 3 shows correlations between pairs of values derived from different
studies for common groups of stars. Although the number of analyzed stars in
the present work is small, some trends can be inferred by comparisons with
results of Amorim et al. (2023) and Hardy et al. (2023). Such trends are
confirmed by comparisons between results from Hardy et al. (2023) and Amorim
et al. (2023) for a sample of 118 stars studied in both works. Calculated
correlation coefficients close to unity show a systematic agreement in
determinations of $B_{\text{d}}$, indicating that the mean surface fields in
spectrum fitting appear to be reasonably well determined. On the contrary,
derived values for $i$ and $a_{z}$ present very poor correlations. This
suggests that the dipole inclination and offset in numerical codes regulate
the range of field strengths that effectively contribute to the spectral
shape, rather than describing the exact geometric distribution of the field on
the stellar surface.

On the other hand, spectrum fitting for magnetic white dwarfs with high field
strengths becomes sensitive to magnetic-field effects on the ionization
equilibrium in the atmosphere. A detailed search for the best spectrum fitting
for the highly magnetic star SDSS J2247+1456, with and without field effects
on the chemical equilibrium, showed a substantial change in the calculated
$T_{\text{eff}}$, it being 1000 K higher in the former case (magnetic gas
model). However, obtaining good spectrum fitting for highly magnetic white
dwarfs is very difficult, likely because these stars have more complex field
geometry than a dipole, or also because of the physical approximations used in
the atmosphere models. Therefore, the search for the best fit usually yields
several close solutions with $T_{\text{eff}}$, on the order of or greater than
the change originating from the chemical model used. In any case, it is clear
that the use of a field-free chemical model introduces a systematic error in
spectral fitting, which mainly implies an underestimation of surface
temperatures for highly magnetic objects.

##  8 Conclusions

We present a new code for synthetic spectrum calculation of pure-hydrogen MWD
atmospheres. To our knowledge, our code is currently the only one capable of
calculating a full solution to the radiative transfer equations, including the
effects of the magnetic field on both opacities and abundances of atomic
populations. We demonstrate that incorporating magnetic-field effects on the
chemical equilibrium of the gas leads to a substantial increase in the
temperature distribution in highly magnetized atmospheres. This impacts the
determination of the effective temperature of highly magnetic white dwarfs and
may consequently affect the characterization of other physical properties such
as radius and mass. Although this effect on $T_{\text{eff}}$ is significant,
its magnitude is on the same order as the current capability of magnetic
synthetic spectrum models to reproduce observed spectra in these stars.
Presently, effective temperature predictions for highly magnetic objects
resulting from different atmosphere codes show discrepancies of a similar or
even higher magnitude than changes arising from the use of a magnetic chemical
model.

Comparison with results of other spectrum-fitting codes in a sample of MWDs
using dipole geometry indicates reasonable agreement in the evaluation of the
surface field strengths. However, there is notable disagreement in the
identification of the dipole inclination relative to the observer and its
offset from the stellar center, even for weak magnetic stars. On the other
hand, differences found in the observed spectrum fits could indicate a
magnetic field geometry that is more complex than a simple shifted dipole and
uncertainties in the input physics of the atmosphere model. It remains to be
seen whether more detailed agreement can be achieved when better continuum
opacity data become available. Work in this regard is in progress.

###### Acknowledgements.

This publication makes use of data from the Sloan Digital Sky Survey (SDSS)
which is supported by the Alfred P. Sloan Foundation, the Heising-Simons
Foundation, the National Science Foundation, and the Participating
Institutions.

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