A cool, magnetic white dwarf accreting planetary debris

We present an analysis of spectroscopic data of the cool, highly magnetic and polluted white dwarf 2MASS J0916-4215. The atmosphere of the white dwarf is dominated by hydrogen, but numerous spectral lines of magnesium, calcium, titanium, chromium, iron, strontium, along with Li I, Na I, Al I, and K I lines, are found in the incomplete Paschen-Back regime, most visibly, in the case of Ca II lines. Extensive new calculations of the Paschen-Back effect in several spectral lines are presented and results of the calculations are tabulated for the Ca II H&K doublet. The abundance pattern shows a large lithium and strontium excess, which may be viewed as a signature of planetary debris akin to Earth's continental crust accreted onto the star, although the scarcity of silicon indicates possible dilution in bulk Earth material. Accurate abundance measurements proved sensitive to the value of the broadening parameter due to collisions with neutral hydrogen (Gamma_HI), particularly in saturated lines such as the resonance lines of Ca I and Ca II. We found that Gamma_HI if formulated with values from the literature could be overestimated by a factor of 10 in most resonance lines.


INTRODUCTION
The spectroscopic analysis of magnetic white dwarfs covers a wide range of field strengths from 10 −3 to 10 3 MG.The hydrogen and helium line spectra have been extensively modelled over the whole range of field strength (Garstang & Kemic 1974;Schimeczek & Wunner 2014) but difficulties remain in modelling the field geometry.With few exceptions, magnetic white dwarf are assumed to harbor offset dipole fields.The study of trace elements, from lithium to iron and beyond, in magnetic white dwarf spectra is in its infancy.
Recent studies have tackled the problem of line formation in the full Paschen-Back regime in high magnetic fields (Zhao 2018;Hollands et al. 2023), while Kemic (1975) initiated the first study of the Ca ii H&K doublet in the incomplete Paschen-Back regime (see also Hardy, Dufour, & Jordan 2017), i.e., between the anomalous regime and the full Paschen-Back regime where the lines assume the shape of a simple polarization triplet ( and  ± ) with energy separations simply proportional to the magnetic field strength.
Meaningful results with abundances of various elements from sodium to iron were obtained at relatively low field in the anomalous Zeeman regime (Kawka & Vennes 2011, 2014;Kawka et al. 2019).In their study of low-field DZ white dwarfs Hollands et al. (2021) encountered the line spectrum of potassium in the incomplete Paschen-Back regime and the lithium spectrum in the full regime.
★ E-mail: svennes@iinet.net.au The behavior of spectral lines in magnetic white dwarfs is not yet fully understood and new calculations in fields ranging from 1 to 100 MG are clearly warranted.
Heavy element pollution in white dwarf atmospheres is well documented.Restricting the discussion to cool white dwarfs, accretion of external material, irrespective of the source, is the essential mechanism (Dupuis, Fontaine, & Wesemael 1993).In hot white dwarfs, where selective radiation pressure plays an important role (Chayer et al. 1995), the composition of the atmosphere is dictated by local physical conditions.However, in cool convective white dwarfs the composition of the atmosphere offers clues to the nature and composition of the source itself.Debris material from disrupted planetary bodies is the most likely source (Zuckerman et al. 2007;Jura & Young 2014;Zuckerman & Young 2018;Veras 2021) but variations over this theme rely heavily on the accuracy of abundance measurements.
Based on a large sample of cool, polluted white dwarfs Hollands, Gänsicke, & Koester (2018) observed a range of likely sources, from (Earth's) crust-like to core-like based on key abundance proportions: calcium, iron or magnesium divided by a weighted sum of the three, which typify, respectively, the Earth's crust, core, and mantle.Interestingly, most objects point towards a mixture of sources which is labelled bulk Earth.Further, studies of oxygen-polluted white dwarfs have demonstrated evidence of the stoichiometric balance of elements in rock-forming oxides in the accreted material, bearing a close resemblance to rocky bodies in the solar system (Klein et al. 2010;Xu et al. 2014;Doyle et al. 2023).The detection of the light elements, beryllium and lithium, has raised the possibility that such polluted white dwarfs might be used as tracers of spallation environments (Klein et al. 2021;Doyle, Desch, & Young 2021), differentiation into planetary-crust material (Hollands et al. 2021), and/or windows into the early universe and Big Bang Nucleosynthesis (Kaiser et al. 2021).Overall, the study of abundance patterns in white dwarf stars reveals a complicated history of past and present interaction with their circumstellar environment.
In this context, we present new echelle spectra of the magnetic white dwarf 2MASS J0916−4215 (Section 2) revealing several spectral lines formed in the incomplete Paschen-Back regime in a 11-12 MG magnetic field (Section 3).2MASS J0916−4215 is a new cool, magnetic white dwarf with trace elements in a hydrogen-rich atmosphere.The abundance pattern shows a two orders of magnitude excess in lithium and an overall distribution, with the notable exception of silicon, pointing out to a primordial origin for the material analogous to Earth's continental crust.A discussion and summary are presented in Section 4. We describe in Appendix A our new calculations of the Paschen-Back effect for all spectral lines found in 2MASS J0916−4215.

OBSERVATIONS
The star 2MASS J0916−4215 appears as a white dwarf candidate in the catalogue of Gentile Fusillo et al. (2019).We first observed 2MASS J0916−4215 with the MIKE double echelle spectrograph attached to the Magellan 2 -Clay telescope at Las Campanas Observatory on UT 2021 December 18 and 19.We set the slit width to 1 arcsec to provide a resolution  = /Δ = 22 000 and 28 000 on the red and blue sides, respectively (Bernstein et al. 2003).We obtained an additional spectrum with the MagE (Magellan echellette) spectrograph (Marshall et al. 2008) attached to the Magellan 1 -Baade telescope on UT 2022 March 23.We set the slit width to 0.5 arcsec to provide a resolution  = 8000.The spectra were corrected for telluric absorption using a template provided by the TAPAS database (Bertaux et al. 2014) and using the telluric routine within the Image Reduction and Analysis Facility (IRAF). 1 We employ air wavelengths throughout this work.
We collected optical and infrared photometric measurements from the SkyMapper survey (Onken et al. 2019), Two Micron All Sky Survey (2MASS: Skrutskie et al. 2006) and Wide-field Infrared Survey Explorer (WISE: Cutri et al. 2013;Marocco et al. 2021).These are listed in Table 1 together with photometric and astrometric data from the Gaia Data Release 3 (Gaia Collaboration et al. 2022).2MASS J0916−4215 is a nearby white dwarf well within the 40 pc sample (Gentile Fusillo et al. 2019;O'Brien et al. 2023), also known as WD J091600.94−421520.68 (WD J0916−4215) or Gaia DR3 5427528254746168192, and was tentatively classified as a magnetic white dwarf (O'Brien et al. 2023

ANALYSIS
The high-dispersion spectra show heavy element lines in a cool hydrogen-rich atmosphere.Nominally, the spectrum is classified as DZAH owing to the presence of strong, magnetic-displaced magnesium, calcium and iron lines and a much weaker hydrogen line.
The presence of a weak, Zeeman-split H line could imply a temperature slightly above 5 000 K in a hydrogen-dominated atmosphere but it could also imply a higher temperature, e.g., ≈ 8 000 K, in a helium-dominated atmosphere.The temperature and gravity may be constrained further with an analysis of the spectral energy distribution (Section 3.1).All spectral lines display a pattern revealing the presence of a surface average magnetic field of approximately 11.3 MG in strength.With other closely related polluted, hydrogenrich, magnetic white dwarfs (Kawka et al. 2019) that were classified as DAZH white dwarfs, 2MASS J0916−4215 is part of a class of objects accreting material from their circumstellar environment.Many spectral lines in 2MASS J0916−4215 follow an apparent triplet pattern which at this field strength implies the onset of the full Paschen-Back regime.Closer examination reveals a more complex pattern, particularly in the case of the Ca ii H&K doublet.The  and  ± components follow a pattern already described by Kemic (1975) which transitions between the low-field anomalous Zeeman and the high-field full Paschen-Back triplet pattern with dominant components shifted by Δ = 0 and ± 0  and vanishing components at ±2 0 , where  0 = ℏ/(2  ) is the Bohr magneton and  is the magnetic field strength (see Appendix A).In addition, each polarization component is itself a doublet.
This pattern, which can be described as belonging to the incomplete Paschen-Back regime (Landi Degl'Innocenti & Landolfi 2004), shows split  and  components leaving six strong line components that are characteristic of2 S -2 P o spectral lines (Fig. 1).For example, sitting close to the  components of the Ca ii H&K line, split  components of the Al i 3955 line are evident.In the Paschen-Back regime the individual polarization components develop a doublet appearance for fine-structure doublets, e.g., Ca ii H&K or Al i 3955, or a triplet appearance for fine-structure triplets, e.g., Mg i 5178. 2 This pattern deviates strongly from the anomalous Zeeman pattern explored by Kawka et al. (2019) in the lower field (≲ 0.3 MG) DAZH white dwarf NLTT 7547.The Ca ii and Al i lines in a field of ≈11.3 MG belong to the incomplete Paschen-Back regime.

Model atmospheres and stellar parameters
We computed a series of convective model atmospheres containing trace abundance of heavy elements.The electron density is computed self-consistently with the ionization equilibrium of all constituents of the atmosphere including most associated metal monohydrides (Sauval & Tatum 1984).
We note that Bédard, Bergeron, & Fontaine (2017) found that convective energy transfer may be suppressed in a ≈10 000 K hydrogenrich white dwarf following their spectroscopic analysis of the lowfield magnetic white dwarf WD 2105−820.Indeed, Tremblay et al. (2015) predicted that even a weak field (≈ 5 kG) would partially suppress convective energy transfer in the line forming region of an ≈10 000 K hydrogen-rich white dwarf, although vertical mixing may still take place (see also Cunningham et al. 2021).Similar 3D radiation magnetohydrodynamic calculations applicable to cooler, predominantly neutral atmosphere of cool (≈ 5 000 K) DAZ white dwarfs (Kawka et al. 2019) are not currently available, but setting the plasma- parameter to unity in the line-forming region allows to estimate a critical field of the order of ≈ 40 kG (Cunningham et al. 2021).Although mixing may not be suppressed, the maximum extent of the mixing region under the influence of a strong magnetic field is uncertain but it will be assumed identical to that of a non-magnetic hydrogen envelope (see Section 3.4).As indicated above, we adopted convective model atmosphere structures.
We first estimated the temperature and surface gravity using all available photometric measurements and the Gaia parallax (Table 1).
Figure 2 shows a model atmosphere analysis of these measurements: tiplet wavelengths obtained from the weighted average (by statistical weight) of the upper and lower energy levels.1.We include the Wise W3 band at 12  as an upper limit.2MASS J0916−4215 is a cool white dwarf with  eff = 5 250±250 K, and log  = 8.13 ± 0.13, where the surface gravity  is expressed in cm s −2 .A weak H line confirms its low temperature and its hydrogen-rich composition.The strength of the H  component restricts further the range of acceptable parameters with error bars ( eff ) = 100 K and (log ) = 0.06.We conclude that  eff = 5 250 ± 100 K, and log  = 8.13 ± 0.06 Using Gaia data, Gentile Fusillo et al. (2019) estimated a temperature  eff = 5056 and 5142 K and a surface gravity of log  =7.99 and 8.05, based on pure-helium and pure-hydrogen models, respectively, in agreement with our own measurements.The stellar parameters correspond to a white dwarf mass of 0.66 ± 0.04  ⊙ and cooling age of 4.8 ± 0.8 Gyr.We used the evolutionary mass-radius relations of Benvenuto & Althaus (1999).Note that a slightly longer cooling age of 6 Gyr is obtained with calculations generated by the Montréal White Dwarf Database (MWDD, Bédard et al. 2020) 3 .2MASS J0916−4215 belongs to a class of polluted, cool magnetic white dwarfs originally identified in a high proper-motion surveys such as G 77-50 (Farihi et al. 2011), NLTT 10480 (Kawka & Vennes 2011), NLTT 43806 (Zuckerman et al. 2011), and NLTT 7547 (Kawka et al. 2019).2MASS J0916−4215 extends the distribution towards higher fields which now covers average surface fields from 70 kG to 11.3 MG.

Field strength and structure, and the Paschen-Back effect
The wavelength extent of the triplet line pattern in some lines, e.g., Mg i 5178, is given by and corresponds to an average surface field of ≈11.3 MG.Despite large wavelength shifts most lines appear narrow.The  components are relatively stable in the Paschen-Back regime resulting in minimal broadening when integrating over a dipole field distribution.However the appearance of the  components is sensitive to the field distribution and the offset dipole is a common feature in modelling magnetic white dwarf spectra as it tends to homogenize the field and narrow the spectral lines as observed in 2MASS J0916−4215.We computed the field geometry following Martin & Wickramasinghe (1981), Martin & Wickramasinghe (1984), and Achilleos & Wickramasinghe (1989).An approximate relationship between the dipole field strength  d and its surface average  S is given by where   is the offset along the  axis expressed as a fraction of the stellar radius.A centered dipole is located at   = 0. Adopting   = −0.Vennes et al. (2018).
Other than for the Ca ii H&K doublet (Kemic 1975) Paschen-Back calculations are not available in the range of magnetic field reached in 2MASS J0916−4215.The Landi Degl' Innocenti & Landolfi (2004) theory includes only the linear Zeeman effect and generally applies to low-field stars.In higher field objects such as this one, quadratic field effects as discussed by Kemic (1975) in the case of the Ca ii H&K doublet remain to be explored.As noted earlier by Kawka et al. (2019), both sets of calculations, Kemic (1975) and Landi Degl' Innocenti & Landolfi (2004), are in agreement at lower fields (<<10 MG) where quadratic effects are negligible, although Kawka et al. (2019) incorrectly stated that Landi Degl'Innocenti & Landolfi (2004) included the effect of quadratic terms.For a general application to spectral lines observed in 2MASS J0916−4215 it was therefore necessary to develop further the theory elaborated by Landi Degl' Innocenti & Landolfi (2004) and include the quadratic terms when computing atomic energy levels embedded in an external magnetic field (Appendix A).
Figure 3 shows a series of Ca ii H&K synthetic spectra illustrating the Paschen-Back effect with increasing dipole field strength for a model appropriate for the white dwarf 2MASS J0916−4215 (see Section 3.3).The pattern greatly varies with field strength and a model with a dipole field of 20 MG closely resembles the observed calcium spectrum (Fig. 1).
Figure 4 shows predicted line positions as a function of magnetic field strength for the singlet Li i 6707 and the triplet Ca i 6142.Adopting a field of 11.3 MG corresponding to the average surface field in 2MASS J0916−4215, we find a quadratic shift of Δ = −672.4km s −1 in the  component of 6142 while it retains a triplet structure, but this shift is only Δ = 10.4 km s −1 in the  component of 6707.Our stellar radial velocity measurement for 2MASS J0916−4215 is based on a set of these narrow  line components.All lines are affected with varying degrees by linear and quadratic shifts.We initially selected Li i 6707 as the primary velocity indicator because of its small predicted velocity shift.This single line sets the observed heliocentric radial velocity at   = 21.3 km s −1 .The gravitational redshift correction will be considered in Section 3.5.Next, we selected 8 additional lines with increasing Paschen-Back shifts (Δ in km s −1 ) in order to evaluate the reliability of our Paschen-Back models in predicting line shifts: Ca ii3945 (Δ = 5.8), K i 7676 (Δ = −33.1),Na i 5891  and 𝜆5178) we find Δ = 20.6±3.7 km s −1 .We conclude, based on 7 good lines, that   = 21 ± 4 km s −1 .Figure 4 compares the Paschen-Back calculations to the observed line positions.The predicted 6142 line position is in error by −13 km s −1 , or ≈2% of the total shift relative to the observed line position (assuming   = 21 km s −1 ), while the predicted position of the 6707 line is in error by ≈0.7 km s −1 , or ≈7%.Inclusion of the quadratic shift is not only essential for radial velocity measurements but also in securing correct line identifications.

Line profile and spectral analysis
The opacity of individual line components is calculated using normalized Lorentzian profiles, (3) convoluted with Doppler profiles.Here,   is the frequency of the shifted line centre, and Γ is the sum of the natural width and the broadening parameter, i.e, the full-width at half-maximum expressed in rad s −1 .In cool hydrogen-rich atmosphere, line broadening is dominated by collisions with neutral hydrogen atoms (Barklem, Piskunov, & O'Mara 2000b).The hydrogen lines were modelled following the quadratic Zeeman calculations of Schimeczek & Wunner (2014) and using selfbroadening parameters of Barklem, Piskunov, & O'Mara (2000a).Table 2. Calcium broadening parameters at full-width half-maximum Γ/ H i (rad s −1 cm 3 ),  = 5 250 K, and population departures. (1) (2) We included the linear and quadratic energy corrections in the calculation of the Boltzmann factors for all energy levels.The spectra show numerous Ca i excited lines along with the resonance line Ca i 4226.The excited lines emerge between 1.9 and 2.5 eV above the ground-state and should only be weakly populated at temperatures near 5 250 K as the Boltzmann factor for the excited states ranges between 3 × 10 −3 and 10 −2 relative to the ground-state.Abundance measurements in saturated lines, e.g., the Ca ii H&K doublet in the spectrum of 2MASS J0916−4215, are very sensitive to broadening parameter values.On the other hand, abundance measurements are relatively insensitive to broadening parameters in the linear part of the curve-of-growth, e.g., the excited line Ca i 6142 line in 2MASS J0916−4215.Detailed line shape measurements obtained with echelle spectra also help in constraining broadening parameter values, e.g., in the Li i 6707 spectral line.For a simple formalism we estimated the broadening parameter for collision with neutral hydrogen following Castelli (2005): where  4 =  ()/10 4 , ⟨ 2  ⟩ and ⟨ 2  ⟩ are the mean square radii (atomic units) for the lower and upper levels of the transition, respectively, and  H i is the density of neutral hydrogen.We used as a reference the Γ parameters for collisions with neutral hydrogen formulated in Barklem, Piskunov, & O'Mara (2000b) and compared them to values obtained with the Castelli (2005) formula, both at a temperature of 5 250 K (3rd and 2nd columns in Table 2).We used mean square radii listed in Appendix A (Tables A1 and A2).Values tabulated by Barklem, Piskunov, & O'Mara (2000b) are systematically higher than estimated using the Castelli (2005) formula but only by a factor of ≈2.The width of the strong Ca i 4300 line calculated using the Castelli (2005) formula is exceedingly narrow Adopting a model atmosphere at  eff = 5 250 K and log  = 8.13, we computed the line profiles in the Paschen-Back regime described in Appendix A using the broadening parameters for collision with neutral hydrogen listed in Barklem, Piskunov, & O'Mara (2000b), when available, or calculated using the Castelli (2005) formula, and fitted them to the MagE and MIKE spectra with a varying abundance.Two problems arose: narrow spectral lines such as Li i 6707 appeared excessively broadened in the models, and the calcium abundance varied by up to a factor of 100 between measurements based on ground-state lines and measurements based on excited lines (column 4 in Table 2).Reductions in Γ values of up to 1.5 orders of magnitude (−1.5 dex; column 5) were necessary to reconcile calcium abundance measurements based on individual lines (column 4).As expected, saturated ground-state lines required large reductions in Γ values while unsaturated excited lines required no corrections.The resulting abundance measurements are 1.3 dex higher, i.e., log Ca/H = −6.7,than estimated using un-corrected Γ values, i.e., log Ca/H = −8.0,and are mutually consistent (column 6).
Similar difficulties arose in two separate measurements of the sodium abundance: adopting the reference parameters Γ the abundance based on the strong resonance line Na i 5891 is one order of magnitude lower than the abundance measured using the weak excited line Na i 8190.By adjusting the broadening parameter Γ for the 5891 line by −1.7 dex we obtained consistent abundance measurements at log Na/H = −6.1.
Finally, to resolve difficulties in matching the synthetic line profiles to the observed ones, literature-based Γ values for the narrow groundstate lines Li i 6707, Al i 3955, K i 7676, and Sr i 5607 were reduced by 1.0 dex.Similarly a modest correction of −0.5 dex was required in the case of Mg i 5178 to match its narrow width.The resulting abundance measurements based on unsaturated spectral lines were not affected by this procedure.We adopted a correction of −1.0 dex for the remaining line broadening parameters.Our echelle spectra exposed the need for considerable revisions in broadening parameter values.Hollands et al. (2021) and Elms et al. (2022) also concluded that broadening parameters by collisions with neutral atoms required large correction factors.
Interestingly, Kawka & Vennes (2011), Kawka & Vennes (2014), and Kawka et al. (2019) have already noted that calcium abundance measurements based on Ca i 4226, Ca ii H&K, and the Ca ii 8578 were often inconsistent.Based on our new results we propose to revise upward the calcium abundance measurement in NLTT 7547 that was presented by Kawka et al. (2019) from log Ca/H = −10.1We show again the profiles at full width but with an increased abundance (×2).The K i  doublet appearance is a signature of the Paschen-Back effect at a field of 11.3 MG while Li i  achieves a singlet appearance.
to −9.3, i.e, adjusted to the measurement based on the unsaturated Ca ii 8578 line.
Figure 5 shows our analysis of the K i 7676 and Li i 6707 line profiles.The narrow line shapes were matched by synthetic line profiles including, as noted above, reduced broadening parameters (log Γ − 1.0).The line profiles computed at the same abundance but with the original line broadening parameters obtained from Barklem, Piskunov, & O'Mara (2000b) are clearly too broad and shallow with line wings extending far from the line centre.To show this effect more clearly we also show the full profiles at twice the nominal abundance.
The line positions are accurately matched with our Paschen-Back models including the quadratic effect.Their singlet (Li i) and doublet (K i) appearances follow directly from their respective fine-structure energy separation constants  which is much larger in the case of the K i 7676 upper level ( 2 P o ) relative to the Li i 6707 upper level,  =38.5 versus 0.2 cm −1 , or the Na i 5891 upper level (see Appendix A2).
Table 3 lists the resulting abundance measurements by number and Figure 6 shows the corresponding spectral synthesis along with the Magellan echellette (MagE) spectrum (see also Appendix B).We estimate the individual abundance uncertainties at a factor of 2 (±0.3 dex).They are dominated by uncertainties in the effective temperature of the star (±100 K) and to a lesser extent the uncertainties in the broadening parameters, and the line profile fitting.Therefore, because systematic errors dominate the error budget, errors in abundance ratios should be lower than in individual measurements.
All Ca ii lines belong to the incomplete Paschen-Back regime.The ultraviolet Ca ii lines show the close doublet pattern () as well as split  ± patterns ≈ ±85Å on both sides and, as expected at ≈ 11.3 MG (see Equation ( 1)), a weak vanishing component at Δ ≈ +170Å ( ≈ 4120Å).The infrared Ca ii sextet shows a corresponding number of  components (five are clearly visible in the spectrum and a sixth one is merged with another).The  ± components of Mg i 5178 and Ca i 4445 (a sextet at zero field) show a complex triplet structure well matched by the models.Several narrow lines do not have obvious identifications and are shown with "?" marks.A possible identification of a feature near 4480 Å with the excited Mg ii 4482 line is problematic because of its high excitation energy and vanishing Boltzmann factor at low temperature.
Figure 7 shows our new abundance measurements listed in Table 3.The abundance measurements are normalized to the magnesium abundance and divided by the corresponding abundance measurements in various bodies and the resulting ratios are plotted for each element.Excluding upper limits (e.g., silicon), the mean and standard deviation of the ratios relative to Earth's continental crust are 0.05 ± 0.51, while these values increase to 1.20 ± 0.73 relative to the CI-chondrites and 1.36 ± 0.94 relative to the bulk Earth.
The lithium over-abundance in 2MASS J0916−4215 approaches +2 dex relative to the bulk Earth (McDonough 2003) or CI-chondrites Figure 6.Model spectrum (red line) fitted to the blue side of the MagE echelle spectrum (grey line) using air wavelengths.All spectral lines modelled in this work are marked with the corresponding element and polarization state (  or  ± ).The complete spectral coverage is presented in Appendix B. (Lodders 2019), but it appears considerably closer to the lithium content of Earth's (bulk) continental crust (Rudnick & Gao 2003).The strontium abundance would also point towards a parent body composed of crust-like material.However, the upper limit on the abundance of silicon would imply a large silicon deficit in the parent body assuming Earth's crust composition.On the other hand, the upper limits on the abundance of vanadium, manganese, and nickel do not help discriminate between the three scenarios.The standard deviation in Δ cc,Mg measurements is ≈ 0.5 showing an uncertainty in individual measurements of the order of a factor of 3, or some degrees of variations in the composition of the actual accreted material relative to the estimated composition of crust-like material (Rudnick & Gao 2003).Oxygen is not detectable in cool white dwarfs such as 2MASS J0916−4215, so possible oxide-balance of the accreted material cannot be ascertained (Klein et al. 2010).

Effect of diffusion: build-up and steady-state regimes
Before any chemical separation takes effect, i.e., at a time following an accretion event much shorter than the diffusion time scale,  << , the abundance pattern (relative to Mg) in the atmosphere replicates the pattern in the accretion flow: The comparisons with the full red line depicted in Figure 7 follow this assumption.However, in a steady-state regime at a time  >>  when equilibrium is established between diffusion losses at the bottom of the convection zone and the surface resupply, the abundance pattern (relative to Mg) is given by: where (X) is the diffusion time scale for a given element X (or specifically Mg) obtained from the MWDD (Bédard et al. 2020).We note that the mass of the convection zone is calculated assuming that the strong magnetic field has no effect on the depth of the mixed layers in a magnetic white dwarf.The dashed red line in Figure 7 shows mild deviations in the abundance pattern due to diffusion.Assuming either early build-up or steady-state regimes, accretion from a source with crust-like material appears more likely.However, the absence of silicon remains problematic.Moreover, sodium, aluminium and chromium deviate from expected (either in build-up phase or steady-state phase) abundance ratios by more than 0.5 dex.Such large deviations exceed mere statistical errors and may reflect systematic errors in adopted broadening parameter values.The mass of individual elements accreted onto the star per units of time is given as its mass fraction  acc of the accretion flow and is expressed in  ⊙ yr −1 or g s −1 : where  identifies an element by its atomic weight,  acc is the total mass accreted by units of time,  atm,A is the mass fraction of element  in the atmosphere,  ≈  , where  is the abundance by number (X/H in Table 3),  cvz is the mass of the convection zone (log M cvz /M ⊙ = −6.33),and  A is the diffusion time-scale of element  at the bottom of the convection zone.Figure 8 shows the diffusion time-scales employed in the calculation of individual accretion rates onto 2MASS J0916−4215.The resulting accretion rates for individual elements are compared to similar rates onto the polluted, magnetic white dwarf NLTT 43806 (Zuckerman et al. 2011).The rates found in 2MASS J0916−4215 are a factor of 6 larger than in NLTT 43806 but, otherwise, follow a very similar trend.Zuckerman et al. (2011) concluded that the material accreted onto NLTT 43806 belong to lithosphere-like material (crust and upper mantle).Although the overall abundance pattern in 2MASS J0916−4215 would suggest a similar conclusion, the scarcity of aluminium and the absence of silicon indicate that the silicon-rich crust material should be diluted with other types of material.We find that, depending on the adopted scenario, i.e., early build-up phase (equation ( 6)) or late steady-state phase (equation ( 7)), individual abundance measurements would vary by at most a factor of 3 due to variations in the diffusion time scales, particularly between lighter and heavier elements.Such mild variations are not readily detectable in our measurements.Therefore, we cannot recover the timeline of accretion events onto 2MASS J0916−4215.
Complete or partial suppression of convective mixing could noticeably shorten diffusion time-scales in cool white dwarfs.Following an accretion event the fully convective envelope of a 5 000 K hydrogen-rich white dwarf would retain heavy elements over a timescale of ≈ 10 6 years, while a radiative photosphere without mixing would retain its composition for merely ≈ 10 −3 years.Unless they are in the process of accreting material, the cool, polluted magnetic white dwarfs would be relatively rare compared to their non-magnetic counterparts.However, an examination of the large sample of polluted, magnetic white dwarfs analyzed by Hollands et al. (2017) and Hollands, Gänsicke, & Koester (2018) shows that the abundance patterns of magnetic and non-magnetic white dwarfs do not differ significantly and that magnetic fields are common among polluted white dwarfs which implies that deep mixing remains effective even among magnetic white dwarfs.

Photometric variations
We could not detect any significant photometric variations in 2MASS J0916−4215 in the TESS data.Several other cool, magnetic white dwarfs show variations of several hundredth of a magnitude, e.g., NLTT 8435 (Vennes et al. 2018), possibly due to surface field variations or surface elemental abundance variations.However, no surface element abundance changes other than for hydrogen and helium (see, e.g., Caiazzo et al. 2023), as measured with absorption line equivalent widths, have ever been convincingly demonstrated for any white dwarf (e.g., see Section 3.2 in Johnson et al. 2022).With its large element abundances and complex but well-modelled line spectrum, 2MASS J0916−4215 would be the ideal candidate for such a study.

DISCUSSION AND SUMMARY
The intermediate field white dwarf 2MASS J0916−4215 is among the first of its class to show spectral lines in the incomplete Paschen-Back regime.Previously, this pattern was observed in the K i spectrum of the lower field DZ white dwarf LHS 2534 (Hollands et al. 2021).The  components of several spectral lines show doublet, e.g., Al i 3955, K i 7676, and Ca ii 3945, or triplet appearances characteristic of the incomplete Paschen-Back regime.The infrared Ca ii 8578 triplet shows a complex multi-components core structure that falls well into the incomplete Paschen-Back regime.The pattern clearly departs from the anomalous Zeeman effect displayed in several lower-field white dwarfs (Kawka & Vennes 2011, 2014;Kawka et al. 2019) but it shows split components and residuals of line components that should vanish entirely in the full Pashen-Back regime.We have presented new calculations of the incomplete Paschen-Back regime and listed sample results for the Ca ii H&K doublet.Although we developed a reliable method to adjust broadening parameters in the high density atmosphere of cool white dwarf stars, ab initio calculations of these parameters are needed to help eliminate potential systematic errors in abundance measurements.
To date, lithium has been detected in only seven white dwarfs (Kaiser et al. 2021;Hollands et al. 2021;Elms et al. 2022, this work), including 2MASS J0916−4215, all of them cooler than ≈ 5000 K with the exception of 2MASS J0916−4215.Three of them are magnetic which suggests a high incidence of magnetism in cool, polluted white dwarfs (Kawka & Vennes 2014; Hollands, Gänsicke, & Koester  (Bédard et al. 2020). cvz is the adopted mass of the convection zone expressed as a fraction of the solar mass.The diffusion time scale for strontium ( = 87.6 u) was estimated by extrapolating available data (full line).
2018; Kawka et al. 2019).The lithium to sodium abundance ratio varies from log Li/Na = −0.1 to −3.2 (in 2MASS J0916−4215) but with the five other measurements clustering around −1.8 ± 0.3.The material accreted on 2MASS J0916−4215 may not be as "differentiated" as the material analyzed by Hollands et (2021).The low ionization potential of neutral lithium precludes a detection in warmer objects but a search for this element in objects with temperatures up to 6 000 K should be attempted.2MASS J0916−4215 joins a class of cool, polluted hydrogenrich white dwarfs as its highest field member ( S ≈ 11.3 MG,  p = 24 MG).Pending a definitive model atmosphere analysis, SDSS J1143+6615 (Hollands et al. 2023) could constitute an even higher field member of this class although a hydrogen-rich composition appears unlikely.The spectral energy distribution of 2MASS J0916−4215 does not show an excess in the infrared, however the WISE upper limit for the 12m flux measurement provides for the possibility of an infrared excess in the mid-infrared range.JWST mid-infrared imaging is needed to investigate the presence of a dusty circumstellar environment.The overall abundance pattern indicates a lithium-and strontium-rich source of material similar to the Earth's crust but the scarcity of aluminium and silicon argues against this simple interpretation and qualitatively different sources of material are also required.strength) for the Ca ii H&K doublet are presented in Tables A3 and  A4.Kemic (1975) wrote the Hamiltonian for an atom embedded in an external magnetic field as the sum of the spin-orbit, linear Zeeman and quadratic terms:

A1 The Ca II H&K doublet
where  =  0  and  0 = ℏ/(2  ) is the Bohr magneton acting at a magnetic field strength , and  =  2 /(8   2 ) is a constant factor applied to the quadratic term H Q , where all constants have their usual meaning.
In the matrix below, we show the linear Paschen-Back (H B ) matrix elements added to the spin-orbit (H so ) elements for the upper energy levels (4p 2 P) of the CaII H&K doublet.The matrix elements due to   are    , where   =   (, , ) is the Lande  value and  the magnetic quantum number, while the matrix elements due to  so describe the level fine-structure.The formulation recovers the anomalous Zeeman effect in the low field limit  << , e.g.,  ≲ 0.5 MG for CaII H&K.Here,  is the energy separation constant for a given multiplet, e.g.,  = 148.593cm −1 for the CaII 4p configuration.The resulting tri-diagonal matrix structure, (, ) on the diagonal and, when allowed, (,  − 1) and ( − 1, ) off the diagonal, shows the mixed  levels with a common  number at  = +1/2 (lines 3 and 4) and  = −1/2 (lines 5 and 6) which alters the level structure at higher field: These matrix elements are ordered following the state vectors | ⟩, or excluding  : where the numbers  are in common to all states, e.g.,  = 1/2 and  = 1 for the Ca ii 4p level.The quadratic matrix elements are obtained following (Kemic 1975): where

(𝑘 )
are operators acting on the state vectors which we solved 4 4 We recovered the general expression for the matrix elements of the Ca ii 4p level as written in equation( 8) of Kemic (1975), but we retained the more general formulation for the sign of the expression, i.e, (−1) +++ ′ − , which would be applicable to other cases involving different  values.However, as shown in the text, we could not recover all matrix elements as written in equations (9,11,12a,12b) of Kemic (1975).

Figure A1.
Fractional line strength as a function of magnetic field in the present work and compared to values tabulated in Kemic (1975).The curves are labelled, from top to bottom in order of appearance, with the corresponding line transition components.The top six line components survive into the full Paschen-Back regime while the lower four components vanish.
following Racah (1942) and Cowan (1981).As demonstrated by Kemic (1975) the non-zero matrix elements of the H Q matrix follow the same tri-diagonal structure described above.Therefore, the following matrix elements are added to the Zeeman and linear Paschen-Back elements: The angular part as described by Kemic (1975) includes factors involving the 3-j and 6-j symbols described by Racah (1942) and more recently by Cowan (1981) and Landi Degl' Innocenti & Landolfi (2004), and the matrix elements of spherical harmonics ⟨ || ( ) || ′ ⟩ (Racah 1942; Cowan 1981).We solved the 3-j and 6-j symbols using fortran subroutines W3JS and W6JS supplied in Landi Degl 'Innocenti & Landolfi (2004).The radial part involves the ⟨ 2 ⟩ expectation value (mean square radius) for a given configuration: where  nl is a normalized radial wave function.Although distinct radial wave functions and ⟨ 2 ⟩ expectation values are suggested by Kemic (1975) for the 4p levels P 3/2 and P 1/2 we adopted a single value for the entire upper level configuration, ⟨ 2 ⟩ 4p = 22.04 atomic units (a.u.), while we adopted ⟨ 2 ⟩ 4s = 14.84 a.u.for the lower level configuration.These and other required ⟨ 2 ⟩ values for various elements and configurations were computed by us using Cowan et al.'s fortran code RCN following the Hartree-Fock (non-relativistic) scheme.We note that our H Q coefficients differ by up to a factor of 2 from those of Kemic (1975) which also appeared inconsistent with each other.The differences do not add up to large deviations in the resulting line wavelengths or strengths due to the relative dominance of the linear Paschen-Back effect in the range of magnetic field considered here (≲ 50 MG).
Applying the selection rule Δ = 0, ±1 we recover 10 line components listed in the header of Table A3.After solving for the eigenvalues and eigenvectors of the H so +H B +H Q matrix using the subroutine jacobi supplied in Press, Flannery, & Teukolsky (1986), the wavelength of each component listed in Table A4 was obtained using the calculated energy values: where  c is given in Å and  designates any of the individual line components ( )− > ( ′  ′ ),  up,c is any eigenvalue of the upper level matrix,  4s = 0 cm −1 and  4p = 25340.10cm −1 are the average configuration energy values recommended by NIST for the lower and upper level of the Ca ii H&K doublet.The energy  low,c belongs to one of the two lower energy levels: The wavelengths are converted from vacuum to air.The method for calculating the strength of each line component is described by Landi Degl' Innocenti & Landolfi (2004).The factors listed in Table A3 are to be applied to the full line strength   , e.g.,   = 27 a.u.for the CaII H&K doublet.Furthermore, the sum of these factors is normalized to unity within each polarization component ( = 0, ±1), i.e.,  0 =  −1 =  +1 =   , so that the sum of the polarization components weighted by the geometrical factors is equal to the total line strength   : where  is the angle between the local magnetic field and the line-ofsight.For the purpose of computing the line opacity, the line strength of each component is converted into an oscillator strength following the relation: where   c is the product of the statistical weight and oscillator strength and  c is the shifted wavelength for a given component ( )− > ( ′  ′ ) and provided in Table A4. Figure A1 compares the results of our line strength calculations to values tabulated in Kemic (1975), but renormalized to  =0,±1 = 1, for fields ranging from 0 to 50 MG showing an excellent agreement for all 10 line components although some tabulated values in Kemic (1975) suffer from rounding errors.Figure A2 shows our calculated line position and illustrated line strength for the CaII H&K doublet showing the transition from the anomalous Zeeman regime through the incomplete Paschen-Back regime.

A2 Other line transitions in the incomplete Paschen-Back regime
The formulation adopted for the Ca ii H&K doublet can be directly applied to other spectral lines with 2 S-2 P o terms such as the Na i 5891 doublet.Table A1 lists the atomic data employed to extend the Paschen-Back calculations to these spectral lines.We also employed the formalism of Kemic (1975) to calculate the quadratic Paschen-Back effect in spectral lines with other types of configurations from  = 0 to 4 (S, P, D, F, H) and total electronic spin from  = 0 to 3 (Table A2).The linear Paschen-Back matrix elements applicable to these configurations were obtained following the method described by Landi Degl' Innocenti & Landolfi (2004).
In the presence of multiple terms in mixed configurations we computed, when available, the term-dependent ⟨ 2 ⟩ values following Condon & Shortley (1935).For example, compare the ⟨ 2 ⟩ values in the calcium 3d4p 3 F o , 3 D o , and the 3 P o levels (Table A2).In addition, ⟨ 2 ⟩ values proved strongly correlated with the calculated level energy values.Therefore we adjusted the correlation potential factor (Cowan 1981) to achieve a better match between the calculated energy values and the energy values tabulated at NIST (Kramida et al. 2022).
In the main body of the text we reviewed the accuracy of the theory when confronting our models to high-resolution spectroscopy of the intermediate field white dwarf 2MASS J0916−4215.The results were satisfactory in most cases, particularly the 2 S -2 P o lines.Comparing the line "center-of-gravity" with the full Paschen-Back calculations of Hollands et al. (2023) shows increasing deviations beyond 20 MG which for the spectral lines considered here would constitute a practical limit to the accuracy of the theory employed in our own calculations.However, the morphology of the Ca ii lines in particular remains far from a simple triplet structure.Since they were computed in zero-field conditions, the mean square radii (⟨ 2 ⟩) remain a major source of uncertainties in computations of the quadratic effect within the present scheme.In addition, failure of the LS-coupling in more complex or mixed electron configurations and uncertainties in Lande-g values also affect the accuracy of the linear Paschen-Back model predictions.Table A2.Reference data for Paschen-Back calculations of the multiplet or singlet lower () and upper () levels.Along with the electronic configurations and associated terms, we include the average lower (  ) and upper (  ) energy levels, the total line strength (  ), and the mean square radii for the lower and upper levels employed in the quadratic Paschen-Back calculations.We indicate whether a given singlet or multiplet is observed in the spectrum of 2MASS J0916−4215.

Figure 1 .
Figure 1.MIKE (black line) and MagE (grey line) echelle spectra showing both Ca ii 3945 and Al i 3955 split  components in the incomplete Paschen-Back regime.Wavelengths in all figures are in air.

Figure 2 .
Figure 2. Spectral energy distribution of the cool white dwarf 2MASS J0916−4215 (blue squares) and best fit synthetic colors (open red circles).The photometric data points are listed in Table1.We include the Wise W3 band at 12  as an upper limit.

Figure 3 .
Figure 3. Morphology of the Ca ii H&K doublet with increasing dipole field strength ( d ) from 0 to 40 MG (from bottom to top).The field geometry (inclination and offset) and stellar parameters ( eff = 5 250K and log  = 8.13) of the models are appropriate for the subject of this study, 2MASS J0916−4215.The transition from the anomalous Zeeman regime (depicted at 2 MG) to the full Paschen-Back regime (depicted at 40 MG) sees the number of visible line components drop from 10 to 6, with the resulting pattern falling in three groups, the central  components and the  ± components resembling a simple Zeeman triplet.The spectra are shifted vertically by one unit for clarity.

Figure 4 .
Figure 4. Linear (short dashed lines) and the sum of linear and quadratic (full lines) Paschen-Back calculations compared to observed line positions in the stellar rest frame (open squares, see text).Assuming an average surface field of 11.3 MG, the Ca i 6142 line is shifted by ≈ −14Å by the quadratic effect in agreement with the observed line positions, while the Li i 6707 is shifted by only +0.2Å.The mean zero-field wavelength of the Ca i multiplet is shown with a long dashed line.

Figure 5 .
Figure 5. Telluric-corrected MIKE echelle spectrum (red line) for the Li i line (top panel) and K i line (bottom panel) compared to model spectra with broadening parameters at full value (grey line) and divided by 10 (black line).We show again the profiles at full width but with an increased abundance (×2).The K i  doublet appearance is a signature of the Paschen-Back effect at a field of 11.3 MG while Li i  achieves a singlet appearance.

Figure 7 .
Figure 7. Abundance set in the photosphere of 2MASS J0916−4215.The abundance measurements (by number) are normalized to magnesium (X/Mg * ) and divided by the same measurements in Earth's continental crust (cc), CI-chondrites (CI), and the bulk Earth composition (⊕) as listed in Table 3. Silicon, vanadium, manganese, and nickel abundance upper limits are marked with down arrows.The full red line (= 1) follows the early build-up assumption, while the dashed red line assumes accretion-diffusion equilibrium (see text).

Figure 8 .
Figure 8.Estimated mass accretion rate (top panel) onto the photosphere of 2MASS J0916−4215 (black squares) and NLTT 43806 (red squares, Zuckerman et al. 2011) assuming steady-state equilibrium.Rate upper limits are marked with down arrows.Diffusion time scales (open squares in bottom panel) for a set of elements with atomic weight from  = 6.9 u (Li) and  = 63.6 u (Cu) obtained from the MWDD(Bédard et al. 2020). cvz is the adopted mass of the convection zone expressed as a fraction of the solar mass.The diffusion time scale for strontium ( = 87.6 u) was estimated by extrapolating available data (full line).

Figure A2 .
Figure A2.Shifted wavelength positions as a function of magnetic field of the CaII H&K doublet from zero field through the incomplete Paschen-Back regime and towards full Paschen-Back.The line strength is shown from full strength in black through declining strengths using lighter shades of grey towards the full Paschen-Back regime.At this stage, the polarization components  and  ± retain doublet structures.

Figure B1 .
Figure B1.Model spectrum (top black line offset by +1.0), MIKE echelle spectrum (middle grey line offset by +0.5) and MagE echelle spectrum (bottom grey line) using air wavelengths.All spectral lines modelled in this work are marked with the corresponding element and polarization state (  or  ± ).The question marks ("?") locate unknown features that would not belong to any of the elements identified in the spectrum.Regions affected by telluric absorption are marked with the ⊕ symbol.

Figure B2 .
Figure B2.Same as Figure B1 and towards the blue end of the spectra.
3 would approximately correspond to an offset dipole field strength of 22 MG close to our dipole field model.The line positions and shapes in the MagE and MIKE spectra are matched with models at  d = 24 MG and   = −0.3inclined at  = 70 • .Other examples of offset dipole modelling are presented by

Table 3 .
(Rudnick & Gao 2003)s (log X/H and log X/Mg) by number in 2MASS J0916−4215 and relative to the bulk Earth, CI-chondrites, and the Earth's continental crust, and estimated mass accretion rate for a given element.Individual errors are estimated at ±0.3 dex.Bulk Earth(McDonough 2003).Abundance relative to the bulk Earth: log X/Mg − log X/Mg ⊕ .CI-chondrites(Lodders 2019).Abundance relative to CI-chondrites: log X/Mg − log X/Mg CI .Earth's (bulk) continental crust(Rudnick & Gao 2003). Abundance relative to Earth's (bulk) continental crust: log X/Mg − log X/Mg cc . Mass accretion rate onto the star for a given element in steady-state equilibrium (see Section 3.4).

Table A1 .
Reference data for Paschen-Back calculations of the 2 S and 2 P o terms.The configurations  and  designate the lower and upper levels, respectively, while  is the fine-structure energy separation constant,   is the total line strength, and ⟨ 2 ⟩  and ⟨ 2 ⟩  are the mean square radii for the lower and upper levels, respectively.

Table A3 .
Fractional line strength  for each Ca H&K component as a function of the magnetic field strength in the incomplete Paschen-Back regime.Each component is labeled with the quantum numbers (  ,  ) for the lower level ( 1 S) and (  ′ ,  ′ ) for the upper level ( 2 P), and its membership to the H or K spectral line and grouped within the  ( = 0) or  ±

Table A4 .
Same as TableA3but for the air wavelength (in Å) of each Ca H&K component as a function of the magnetic field strength in the incomplete Paschen-Back regime and towards the full regime.