A DZ white dwarf with a 30 MG magnetic field

ABSTRACT

1 INTRODUCTION

The first magnetic white dwarf was discovered by Kemp et al. (1970), through the detection of circularly polarized light from GJ 742. Since then, many hundreds of magnetic white dwarfs have been discovered (Kawka et al. 2007; Kepler et al. 2013), with observed field strengths spanning a few 10 kG up to about 1000 MG. For fields ranging from a few 100 kG to a few 10 MG, magnetic DA white dwarfs (i.e. those with spectra dominated by hydrogen absorption lines) are easy to identify in intensity spectra and their field strengths are simple to measure, as many hydrogen lines split into three components, where the degree of splitting is proportional to field strength. For smaller fields, where such splitting is unresolved, spectropolarimetry can be used instead (Bagnulo & Landstreet 2018, 2019, 2021; Landstreet & Bagnulo 2019). However, due to reduced throughput, spectropolarimetry is limited to only the brightest white dwarfs.

For higher fields, particularly those beyond 100 MG, identification is often still straightforward, though measuring the field strength is no longer trivial. The diamagnetic term in the Hamiltonian of the hydrogen atom (Wickramasinghe & Ferrario 2000) (resulting in the quadratic Zeeman effect due to its B² dependence), quickly exceeds the interaction strength of the paramagnetic term (linear Zeeman effect), and eventually even the electrostatic potential. This results in large shifts in wavelength, which ostensibly appear chaotic in their field strength dependence. Due to the n⁴ dependence on the quadratic Zeeman effect (where n is the principle quantum number; Wickramasinghe & Ferrario 2000), the shifts are first observed in the higher order Balmer lines, but beyond a few 10 MG also causes the wavelengths of the H α components to become chaotic. Because the size of the diamagnetic term in the Hamiltonian becomes comparable to the other terms, and overall the magnetic field is no longer a small perturbation to the system, the energies (and hence transition wavelengths), cannot be determined using perturbation theory, and instead must be determined numerically.

For hydrogen, the first detailed atomistic calculations were performed in the 1980s (Roesner et al. 1984; Forster et al. 1984; Henry & O’Connell 1985; Wunner 1987). The results of these calculations quickly found application to assignment of lines in strongly magnetic white dwarf spectra (Greenstein, Henry & Oconnell 1985; Angel, Liebert & Stockman 1985; Schmidt et al. 1986). More recent calculations have refined the atomic data for hydrogen in strong fields (Schimeczek & Wunner2014a, 2014b).

Even at these early stages, however, the magnetic white dwarf GD 229 was found to defy assignment of hydrogen spectral lines, leading to speculation that it may instead have a helium dominated atmosphere (Green & Liebert 1981; Schmidt, Latter & Foltz 1990; Schmidt et al. 1996). This hypothesis was proved correct when the first calculations of He i by Jordan et al. (1998) were matched to lines in the spectrum of GD 229, implying a surface field varying between 300 and 700 MG. The calculations themselves relied on finite-field full configuration interaction (ff-FCI) theory, a variational technique providing near-exact solutions to the time-independent electronic Schrödinger equation. Such a description is needed due to electro–electron repulsion term in the Hamiltonian. Similar calculations for He i had also been performed by Becken, Schmelcher & Diakonos (1999).

Calculations using variational approaches have been performed for systems with more electrons such as Li i (Zhao 2018), however for systems with more than three to four electrons, ff-FCI becomes numerically intractable due to the factorial scaling in computation time.

Fortunately, while white dwarfs with heavy elements in their atmospheres have been known for more than a century, those with magnetic fields have hitherto not been observed with field strengths exceeding ∼10 MG, where atoms are safely in the Paschen–Back regime. White dwarfs with heavier elements fall into two main classes: the DQs containing spectral features from carbon, and the DZs containing features from heavier metals (Sion et al. 1983) such as calcium and iron.

DQ white dwarfs, those with spectral features from carbon in their atmospheres (detected from C₂ Swan bands at low T_eff and C i/ii at higher T_eff) are generally understood to originate from convective dredge up of carbon from the core into the surrounding helium envelope (Fontaine et al. 1984; Pelletier et al. 1986; MacDonald, Hernanz & Jose 1998), though a separate population of massive DQs are thought to originate as the product of mergers (Dufour et al. 2007; Dunlap & Clemens 2015; Williams et al. 2016; Kawka, Vennes & Ferrario 2020; Hollands et al. 2020). Of these hot suspected merged DQs, a moderate fraction is also magnetic, showing Zeeman split C i/ii lines – some with field strengths of a few MG (e.g. Dufour et al. 2008). At lower T_eff some peculiar DQs (such as LHS 2229) show highly distorted and shifted Swan bands which have previously been hypothesized to arise from strong (100s of MG) magnetic fields. However, Kowalski (2010) demonstrated that the distorted molecular bands primarily result from pressure-effects occurring in high-density, low T_eff, helium-dominated white dwarf atmospheres. To date, no predictions for the wavelengths of atomic or molecular carbon transitions in strong magnetic fields have been performed.

White dwarfs with metals in their atmospheres are denoted with a Z in their spectral type, e.g. DAZ, DBZ, or DZ, depending which other lines are visible in their spectra. DZs specifically (the subject of this work) usually have helium dominated atmospheres, though are too cool to exhibit He i lines (⁠|${T_\mathrm{eff}}\lt 11,000$| K), although for |${T_\mathrm{eff}}\lt 5000$| K hydrogen lines are also diminished in strength, and so in some cases hydrogen atmosphere white dwarfs can also be classed DZ. Unlike the carbon in DQs, the metals observed in DZs (and DAZs/DBZs, etc.) require an external source, as gravitational settling should deplete white dwarf atmospheres of metals on time-scales that are always much shorter than white dwarf ages (Paquette et al. 1986) – specifically in the case of cool DZs, sinking time-scales are on the order of 10^6–7 yr, whereas their ages range from 10^9–10 yr (see Wyatt et al. 2014, Fig. 1).

Figure 1.

SDSS BOSS and Gemini GMOS spectra of SDSS J1143+6615 (G = 20.1 mag). The SDSS spectrum is shifted upwards by 4 × 10⁻¹⁷erg s⁻¹ cm⁻² Å⁻¹. Behind the Gemini spectrum, we show the SDSS spectrum again (light grey), but convolved to a resolving power of R = 1100 for direct comparison, demonstrating the virtually unchanged spectrum over two years. The zero-field air wavelengths of Ca i, Mg i, and Na i are shown by the solid vertical lines.

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A vast array of evidence now supports accretion of exoplanetesimals from an accompanying planetary system as the source of this metal pollution. Many metal-rich white dwarfs are observed with infrared excesses resulting from circumstellar debris discs (Zuckerman & Becklin 1987; Jura 2003; Rocchetto et al. 2015; Swan, Farihi & Wilson 2019a), with a sub-population of those also exhibiting gaseous emission from the sublimated part of the disc (Gänsicke et al. 2006; Gänsicke, Marsh & Southworth 2007; Dennihy et al. 2016; Manser et al. 2020, 2021). In a few cases, when the disc is viewed edge on, irregular transits are observed, demonstrating the tidal disruption of exoplanetesimals close to the white dwarf Roche radius (Vanderburg et al. 2015; Vanderbosch et al. 2020, 2021; Guidry et al. 2021; Farihi et al. 2022). In two cases the presence of planets themselves has been directly inferred, first from the accretion of an evaporating gas giant by WD J091+1914 (Gänsicke et al. 2019), and secondly from planetary transits at WD 1856+534 (Vanderburg et al. 2020). Despite these various sources of evidence for white dwarf planetary systems, white dwarf spectra containing metal lines remains the most common observable, and can be used to infer the composition of the accreted exoplanetesimals (Zuckerman et al. 2007; Klein et al. 2010; Gänsicke et al. 2012; Dufour et al. 2012; Farihi, Gänsicke & Koester 2013; Xu et al. 2014; Wilson et al. 2015; Hollands et al. 2017, 2018b; Blouin et al. 2019; Doyle et al. 2019; Swan et al. 2019b; Hoskin et al. 2020; Izquierdo et al. 2021; Hollands et al. 2022). A sub-population of DZs have also been found to exhibit magnetism.

The first discovered magnetic DZ (spectral type DZH) was LHS 2534 (Reid, Liebert & Schmidt 2001), which was found to have a 1.9 MG field strength from Zeeman split lines of Na i, Mg i, and blended Zeeman components from Ca i/ii. The field strength of LHS 2534 was recently revised to 2.1 MG by Hollands et al. (2021) along with the detection of Zeeman splitting of Li i and K i. Since this initial discovery, additional DZHs were identified by Schmidt et al. (2003) and Dufour et al. (2006) (WD 0155+003 and G 165−7, respectively). With the advent of data release 10 (DR10) of the Sloan Digital Sky Survey (SDSS), Hollands, Gänsicke & Koester (2015) identified a further seven objects, bringing the known sample to ten, and finding a high magnetic incidence of 13 ± 4 per cent for DZs. With SDSS DR12, Hollands et al. (2017) measured the fields of an additional 15 DZs¹, with the range of surface averaged field strengths, B_s, spanning 0.57 ± 0.04 MG to 10.70 ± 0.07 MG. Like LHS 2534, most of these DZs were identified from Zeeman triplets arising from the Na i resonance doublet (λ ≃ 5890 Å), and the Mg i triplet (λ ≃ 5180 Å). Several magnetic DAZ white dwarfs have also been identified, i.e. those with hydrogen dominated atmospheres, though their field strengths are typically below 1 MG (Kawka & Vennes 2011; Farihi et al. 2011; Zuckerman et al. 2011; Kawka & Vennes 2014; Kawka et al. 2019). With none of the objects published so far demonstrating fields exceeding 11 MG, calculations of metals in ultra-strong magnetic fields have thus far not been essential for the analysis of DZH spectra.

In this work, we investigate SDSS J114333.48+661531.83 (hereafter SDSS J1143+6615), a faint (G = 20.1 mag) magnetic DZ white dwarf with a peculiar spectrum arising from a sufficiently strong magnetic field that spectral features are almost entirely unrecognizable. In Section 2, we present our observations as well as public data on SDSS J1143+6615. In Section 3, we discuss our finite-field coupled-cluster calculations for metals in strong magnetic fields. In Section 4, we make use of our atomic data calculations to identify the spectral lines of SDSS J1143+6615 while simultaneously measuring the strength of its magnetic field. In Section 5, we attempt to model the field structure of SDSS J1143+6615, while in Section 6, we discuss the applicability of our atomic data to higher field strengths and use in model atmospheres, with our conclusions presented in Section 7.

2 OBSERVATIONS

2.1 SDSS

SDSS J1143+6615 was originally observed in SDSS using the BOSS spectrograph (Baryon Oscillation Spectroscopic Survey), first published in SDSS Data Release 12 (plate-MJD-fiberID 7114-56748-0973). The SDSS spectrum is shown at the top of Fig. 1. This spectrum was first classified as a candidate DZH white dwarf by Kepler et al. (2016). This object also appeared in the DZ sample of Hollands et al. (2017), where it was suggested to have a magnetic field exceeding 20 MG.

The overall slope of the spectrum appears consistent with a cool white dwarf with effective temperature (T_eff) in the range 5000–7000 K, but is otherwise highly unusual, exhibiting a myriad of unidentified features. In particular, several bands of broad features are seen near 4700, 5500, and 6400 Å. However, two sharper absorption features stand out as resembling atomic lines. One of these appears at about 5890 Å, and so could be from the Na i-D resonance doublet (which in the absence of a magnetic field would appear blended here). The other sharp feature is located at ≃5125 Å, and due to its asymmetry resembles the Mg i-b triplet which is commonly observed in cool DZ white dwarfs where the asymmetry arises from neutral broadening by helium atoms in a dense helium dominated atmosphere (Allard et al. 2016; Hollands et al. 2017; Blouin 2020). However, while the asymmetry appears qualitatively similar, the wavelength is bluer by about 50 Å than should be the case for the Mg triplet. While the SDSS spectrum does extend to 10 400 Å, we see no evidence for other absorption features beyond what is shown in Fig. 1. With none of the spectral features firmly identified, we speculated that SDSS J1143+6615 is a strongly magnetic DZ white dwarf, where the quadratic Zeeman effect is no longer negligible, causing additional shifts of Zeeman-split spectral lines, and resulting in the appearance of many unidentified features in the spectrum.

The SDSS spectrum itself is composed of four sub-spectra, each taken with 900 s exposure times. While these individual spectra are extremely noisy, owing to the faintness of SDSS J1143+6615, smoothing the data and down-sampling hinted at possible variability between exposures. Because magnetic white dwarfs are known to have rotation periods of minutes to days (Brinkworth et al. 2013; Kilic et al. 2021), we considered the possibility of spectral line shapes/positions evolving with rotational phase. We therefore sought to obtain higher quality spectra of SDSS J1143+6615 in order to confirm this rotation, as well potentially identify spectral lines.

2.2 Gemini

We obtained additional spectra using the GMOS (Gemini Multi Object Spectrograph) instrument on the Gemini North telescope on 2016 April 1st (exactly two years after the SDSS spectrum was taken). The instrumental set-up used the B600_G5307 grating with a 0.75 arcsec slit, giving us a resolving power of about 1100 at 4600 Å. In total, we took 17 exposures lasting 628 s each, separated by 15 s of readout time. The GMOS detector uses three CCDs which covered 4100–7000 Å with our instrumental set-up. This results in two ≃25 Å gaps between each CCD with no spectral coverage, though these did not cover any important features identified from the SDSS spectrum (Fig. 1).

We reduced the GMOS spectra using the Starlink distribution of software for bias-subtraction, flat-fielding, and optimal-extraction (Horne 1986; Marsh 1989) of the spectral trace. Wavelength-calibration was performed using molly.² For flux-calibration, we initially used our observed flux standard, EG 131, but found this gave unsatisfactory results, since it was observed at the end of the night, whereas our science observations were observed at the start. We instead made use of the SDSS spectrum from Section 2.1, as the SDSS flux calibrations are typically accurate to 1 per cent. For each chip, we took the ratio of the spectra (in units of counts) and the already flux-calibrated SDSS spectrum, rebinned onto the same wavelengths as the GMOS spectra. We then fitted third-order polynomials to these ratios to define a calibration function, which we then used to rescale the Gemini spectra into flux units. Note that fluxes redwards of 6700 Å were dominated by telluric absorption and so data beyond this wavelength were ignored and are not shown in Fig. 1.

Our initial goal for these time-resolved spectra was to search for variability, which may arise from rotation of a magnetic white dwarf, bringing different parts of the magnetic field structure into view, and thus causing Zeeman components to change in shape and wavelength. We show the trailed, normalized Gemini spectra in Fig. 2 for chip-2 of GMOS. This chip has the largest spectral signal-to-noise ratio (S/N), and contains many of the unassigned spectral features, including the proposed Mg i and Na i lines. In the bottom panel, we show a zoom-in of the suggested Mg i line, which because of the large shift from the rest-wavelength, should be particularly sensitive to changes in the magnetic field (if it is indeed Mg). We do not detect variability in any of the spectral features, suggesting a lack of rotation on time-scales of a few hours.

Figure 2.

Trailed continuum-normalized spectra for our Gemini observations of SDSS J1143+6615. The top panel shows the entirety of chip-2, which contains both of the sharp features suggested to be from Mg and Na. The bottom panel shows a zoom-in of the suggested Mg line, demonstrating an absence of spectral variability on a 3 h time-scale.

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Given the lack of variability between our 17 spectra, we chose to co-add these into a single high S/N spectrum. We show this in the bottom of Fig. 1 (dark grey). This is compared with the SDSS spectrum (light grey) which has been convolved to the same spectral resolution as our Gemini data. Almost all features appear unchanged, with perhaps only minor differences in the core strengths of the 5400 and 5500 Å features, and a slight change in wavelength of the feature at 4650 Å. This comparison demonstrates a lack of variability on a time-scale of two years.

With the higher S/N spectrum, the proposed Na i and Mg i lines are seen to be blue shifted by 5.5 and 52 Å respectively. The asymmetric nature of the latter (discussed in Section 2.1), is also much clearer. For the proposed Na i line, this could be plausibly explained as a ≃300 km s⁻¹ blue shift (not including any gravitational redshift from the white dwarf) if SDSS J1143+6615 is a halo object. That being said, the much slower 18 ± 2 km s⁻¹ tangential velocity from Gaia EDR3 (see Section 2.3) argues against this explanation. Furthermore, such an explanation is effectively ruled out by the proposed Mg i line, since its much larger wavelength shift would correspond to a velocity shift of about 3000 km s⁻¹. Therefore, magnetism remains a more likely hypothesis for explaining the spectrum of SDSS J1143+6615. In addition to the lines observed from the SDSS spectrum, the Gemini spectrum also reveals the possible presence of the Ca i resonance line (Fig. 1, purple), with a small blue shift of 1.6 Å.

2.3 Gaia

Despite its curious spectrum containing many anomalous features precluding obvious spectroscopic classification, the measured non-zero proper-motion by SDSS confirms that SDSS J1143+6615 is a Galactic object. However, without knowing the absolute brightness of this star, SDSS J1143+6615 could not be claimed to be a white dwarf with certainty.

In 2018 April, the second data release (DR2) from Gaia space mission made public approximately 1 billion parallaxes (Gaia Collaboration et al. 2018). This included SDSS J1143+6615 which had a measured parallax of 7.79 ± 0.68 mas, confirming the location of this star along the white dwarf cooling track within the Hertzsprung–Russel diagram (HR-diagram). In 2020 December, a refined parallax of 7.24 ± 0.46 mas was made available from Gaia EDR3 (early data release 3; Gaia Collaboration et al. 2021) corresponding to a distance of 138.8 ± 9.0 pc. The EDR3 HR-diagram is shown in Fig. 3. SDSS J1143+6615 is indicated by the red point, and is compared against a background of white dwarfs selected from Gentile Fusillo et al. (2021) with |${\tt {PWD}} \gt 0.75$| and |${\tt {parallax\_over\_error}} \gt 20$|⁠. From its location in the HR-diagram, it is clear that SDSS J1143+6615 is a cool white dwarf with a typical mass. Therefore, Gentile Fusillo et al. (2021) found |${T_\mathrm{eff}}=5810\pm 460$| K and log g = 8.17 ± 0.33 fitting the Gaia photometry with pure hydrogen atmosphere models, and |${T_\mathrm{eff}}=5680\pm 470$| K and log g = 8.08 ± 0.33 for pure helium atmosphere models. Interestingly, if Fig. 3 is recreated using Gaia DR2 data, SDSS J1143+6615 appears to be offset from the white dwarf sequence towards higher masses, with Gentile Fusillo et al. (2019) finding |${T_\mathrm{eff}}=6990\pm 710$| K and log g = 8.73 ± 0.29 for hydrogen atmosphere models, and |${T_\mathrm{eff}}=6870\pm 750$| K and log g = 8.67 ± 0.33 for helium atmosphere models. That said, these parameter shifts amount to only 1.4σ changes at most and so are in statistical agreement.

Figure 3.

Gaia EDR3 Hertzsprung–Russel diagram showing the location of SDSS J1143+6615 (red) compared with the white dwarf cooling sequence (grey histogram). The error bars represent 1σ uncertainties.

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3 ATOMIC DATA CALCULATIONS

The predicted transition wavelengths and oscillator strengths can be found in Tables A1–A5. Additionally, the obtained B − λ curves are shown in Fig. 4. The intensity of the transitions, i.e. oscillator strengths, are indicated via the opacities of the curves. As all of the investigated transitions are of ns → np or np → (n + 1)s type, where n is the main quantum number of the orbital (without field), there is in all cases a splitting into three components, i.e. the central π (transition from/into a p₀ orbital) as well as the two σ (transition from/into p₊₁ and p₋₁) components.⁵ As can be seen here, the splitting is only linear for fields below about 5–10 MG while for higher field strengths, the form of the B − λ curves becomes much more complicated. The distortion from a simple Zeeman behaviour is transition dependent: For the np → (n + 1)s transitions (Mg and Ca i 6142), the influence of the magnetic field on the central π component is much more pronounced than for the ns → np transitions (Na, Ca ii, Ca i 4227). The principal reason for this behaviour is that in the former case, the transitions are between orbitals of different main quantum numbers. The orbitals hence experience a different amount of polarization through the magnetic field, i.e. those of higher main quantum number are polarized more strongly due to their more diffuse nature. Effectively this means that the s and p₀ orbitals and the respective electronic states, do not evolve in a parallel manner with increasing magnetic field. Hence, in contrast to the simple perturbative picture, the central π component is no longer constant with increasing magnetic field strength. In addition, the transitions with decreasing energy difference in the magnetic field, i.e. ns → np₋₁ and np₊₁ → (n + 1)s become less relevant for observations, as they decrease in intensity (see equation 8). In addition, small changes in the magnetic field lead to large changes in the transition wavelength and hence such transitions will be blurred out in the spectra for strong fields. A more detailed discussion on the form of the energy levels and the resulting for of the B − λ curve of the Mg transition can be found in Kitsaras & Stopkowicz (in preparation). As noted in Hampe et al. (2020), high-accuracy predictions are required as even the prediction for the transition least affected by the magnetic field, i.e. the central π component of Na can vary by up to 100 Å depending on the level of theory and basis set used.⁶

Figure 4.

Calculated transition wavelengths as a function of field strength. For each Zeeman triplet, the line opacities are scaled to the oscillator strengths.

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4 LINE IDENTIFICATION

With the wavelengths and oscillator strengths calculated in Section 3, we were able to compare these with the spectrum of SDSS J1143+6615. With no immediate indication of which spectral features could correspond to the σ-components of the calculated transitions, we began by restricting ourselves to the π-components only. In Section 2, we identified possible π-components of Na i, Mg i, and Ca i in the SDSS and GMOS spectra, based on the sharpness of the lines, rough proximity in wavelengths to the field-free values, and characteristic asymmetry in the case of Mg.

We compare these lines to our calculated wavelengths as a function of field strength in the top panels of Fig. 5. From the bottom-right-hand panel, it is clear that the Na line shift could be explained by either a relatively small field of ≃30 MG or much larger field of ≃410 MG, owing to a turnaround in wavelength at ≃240 MG. This degeneracy is entirely resolved by the large shift of the Mg line which has only one wavelength solution and is also consistent with a field of ≃30 MG. Thus, to our surprise, the peculiar spectrum of SDSS J1143+6615 (Fig. 1) can not result from a field in the regime of 100s of MG, but is best explained by a field strength an order of magnitude lower, though notably still a factor three higher than all previously identified DZH white dwarfs (Hollands et al. 2015, 2017; Dufour et al. 2015).

Figure 5.

Top row: Spectral regions covering the suspected π-components of Ca i, Mg i, and Na i. Bottom row: Predicted wavelengths for the corresponding π-components as a function of field strength. In all panels, the black dashed lines indicate the field-free vacuum wavelengths for each line, whereas the dotted lines indicate the wavelengths expected for a 30 MG field.

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For Ca i, the match in wavelength is quite poor, though thus far we have neglected wavelength shifts that may arise from radial motion and gravitational redshift, the latter of which could be on the order of 100 km s⁻¹ if SDSS J1143+6615 is particularly massive, which is typically the case for magnetic white dwarfs (Liebert 1988; Kawka et al. 2020; Ferrario, Wickramasinghe & Kawka 2020). Additionally, the absent treatment of relativistic effects may here play a role in the quality of the prediction. It is also clear that at 30 MG, the predicted wavelength for Mg is a similar amount bluer than the line centre (though with greater relative accuracy). To account for this, we fitted the field strength and radial velocity simultaneously. We measured the line centres for all three π-components by simply fitting parabolas to the central few pixels (five for Ca and seven for Mg and Na), constraining them with uncertainties of 0.1–0.3 Å. Performing a least squares fit to the three line centres, we found a magnetic field strength of 29.92 ± 0.05 MG and a redshift of 74 ± 8 km s⁻¹. With these best-fitting values, the residuals are −0.7, 0.0, and 1.8 Å for the Ca, Mg, and Na lines, respectively. This was a clear improvement for Ca i and Mg i, though it provides a somewhat worse result for the Na i line.

With the field strength established from the π-components, we could then determine the expected wavelengths of the σ-components. We make this comparison in Fig. 6. We first investigated the components of Na and Mg, with their σ-components identified with relative ease. In particular, the large broad feature at ≃6350 Å is established as the σ⁺ component of Na, which does not appear blended with any of the other nearby features. Near 5500 Å both the Na σ⁻ and Mg σ⁺ components are observed, though notably the order of their wavelengths has swapped due to the components crossing at a field strength of ≃25 MG. The Mg σ⁻ component is inferred to be the broad asymmetric feature at ≃4800 Å. The asymmetry appears more extreme than for the π-component, which itself is more asymmetric than the σ⁺ component. This may imply that the degree of neutral broadening affects each component differently, which perhaps is not surprising given that both the perturbations from neutral helium atoms and the magnetic field both alter the energy levels of Mg.

Figure 6.

Line identification diagram for SDSS J1143+6615. The Zeeman triplets from our finite-field coupled-cluster calculations are shown by the solid curves, with the naïve wavelengths from the linear Zeeman effect indicated by the dotted lines. These are plotted over the spectrum of SDSS J1143+6615 (grey), where black dashed lines match Zeeman components to features in the spectrum for a field strength of approximately 30 MG (light grey horizontal band).

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Having identified all components from Na i and Mg i, we proceeded with classifying transitions from Ca. For the Ca i resonance line, we had already identified the π-component (rest wavelength at 4227 Å; see Fig. 5, left). As our Gemini GMOS spectrum does not go bluer than about 4090 Å, the σ⁻-component is not covered, and so we were only able to search for the σ⁺ component which, at 30 MG, has an expected wavelength of 4475 Å. Indeed, a spectral feature was found at this wavelength which we attribute to the σ⁺ component (Fig. 6).

The final Ca transitions are less certain, though we still make some attempt at their classification. For the Ca ii Zeeman triplet (H + K resonance doublet in the absence of an external magnetic field), only the σ⁺ component is expected to be covered by our GMOS spectrum at a field strength of 30 MG. While we detect a feature at the expected wavelength of 4160 Å (Fig. 6), the signal-to-noise ratio is somewhat poor at this end of the spectrum, making this assignment less secure. However, it is worth noting that for a T_eff between 5000 and 7000 K, both Ca i and Ca ii resonance lines are typically observed together in non-magnetic DZs (Hollands et al. 2017).

Finally, we consider the Ca i 4p → 5s transition, which in the absence of an external magnetic field appears as a triplet (due to the spin–orbit interaction) centred on 6142 Å. In the presence of a strong magnetic field, this transition appears as a Zeeman triplet exhibiting the strongest quadratic shift of all the transitions calculated in Section 3. Nevertheless, weak transitions are observed at all of the expected wavelengths. Whether this assignment is correct is debatable: the identified central component at around 6060 Å shows some asymmetry, as is observed in the field-free case (see SDSS J0916+2540 in fig. 10 of Hollands, Gänsicke & Koester 2018a). On the other hand, the 6142 Å triplet is typically much weaker than the Ca i 4227 Å resonance line, and is only usually visible for extremely large calcium abundances. Yet, in the case of SDSS J1143+6615, the established components of the 4227 Å Ca i Zeeman triplet are not particularly strong, suggesting that the 6142 Å components would likely be too weak to be visible. Given the sheer number of unknown features in the spectrum of SDSS J1143+6615, it is probable that our assignments to the 6142 Å triplet in Fig. 6 might also originate from some other source.

Many anomalous features in the spectrum of SDSS J1143+6615 remain unclassified. In particular, two strong and broad features are observed at wavelengths of ≃4570 and ≃4660 Å, between the σ⁺-component of the Ca i resonance line, and the σ⁻-component of Mg i. The strength of these features suggest they originate from another element commonly observed in DZ spectra. With the strongest Na, Mg, and Ca lines already accounted for, the most likely candidate is therefore Fe. In the field-free case, a large number of Fe i lines can be found between 4000 and 4500 Å (see Hollands et al. 2018a, Fig.7). Among the strongest transitions in this range are the ³F → ⁵G and ³F → ³G multiplets, which share the same lower level. We therefore suggest that the unidentified features at ≃4570 and ≃4660 Å arise from these iron transitions. Additional unidentified features include broad absorption around 4300 Å (between the π- and σ⁺-components of Ca i), sharp features at ≃5200/5330/5580 Å, and several other features at ≃6030/6450/6530/6620 Å (some of which we were unable to conclusively assign to the Ca i 6142 Å multiplet). We note that the feature near 5200 Å is close to the field-free wavelength of the Cr i 4s → 4p triplet (5208 Å, vacuum), and so that feature could plausibly correspond to the π-component of the Cr i transition. Firmly establishing the origin of these remaining features necessarily will require additional finite-field coupled-cluster calculations in the future, with the above Fe and Cr transitions as the highest priority. For these systems, treatment of field-dependent relativistic effects and a robust treatment of multireference character in the electronic structure will be important.

Figure 7.

Left: Visualization of the field structure of SDSS J1143+6615 modelled with an offset-dipole. Right: The simulated absorption spectrum of SDSS J1143+6615 (red) using data from our finite-field coupled cluster calculations.

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5 MAGNETIC FIELD MODELLING

With several of the spectral features of SDSS J1143+6615 identified, we finally sought to model the magnetic field distribution across its surface. For a purely dipolar magnetic field, the field strength spans a factor of two between the equator and poles. This results in broadened spectral lines, particularly the σ-components due to their stronger wavelength dependence of the field strength. It is clear from the width of the Na i σ⁺ component that the range of magnetic field strengths on the visible hemisphere of SDSS J1143+6615 spans a much narrower field range, with Fig. 6 suggesting about 24–31 MG. Thus, it is necessary to invoke a field structure more complex than a centred dipole.

5.1 The offset dipole model

We chose to use the offset-dipole model, which has been commonly used in the analysis of magnetic white dwarf field structures (Achilleos & Wickramasinghe 1989). This model is similar to a centred-dipole, but allows for the origin of the field to be shifted within the white dwarf. In principle, this shift can be applied in three dimensions, but typically it is only applied along the magnetic field axis by a fractional amount of the white dwarf radius, a_z. The offset-dipole model has been successfully applied to many different white dwarfs (Achilleos et al. 1992; Putney & Jordan 1995; Külebi et al. 2009; Hollands et al. 2015) leading to much improved fits with only a single additional free parameter, which is advantageous compared to a more general multipole expansion.

5.2 Application to SDSS J1143+6615

We applied the offset-dipole model to SDSS J1143+6615 initially focusing on the Na triplet. From analysing the π-components of Mg and Na in Section 4, we established a surface averaged field of ≃30 MG, and hence located the features corresponding to the σ-components. Due to the asymmetry of the Mg components, we decided to begin our focus on the Na triplet. However, the σ⁻ component of Na and the σ⁺ component of Mg are somewhat overlapping (≃5500 Å), and so we chose to restrict ourselves to the π and σ⁺ components of Na (≃6400 Å). Overall, we therefore had five parameters to adjust: the polar field strength B_d, the dipole inclination, and the dipole-offset a_z, which controlled the field distribution; plus the Lorentzian line strength (A_j in Section 5.1) and width which are most easily inferred by the relatively static π-component.

Unsurprisingly, the Lorentzian profiles used provide a poor fit for the asymmetric π- and σ⁻-components of Mg i, though reasonable agreement is found for the σ⁺-component. As discussed previously, this may indicate that the degree of neutral broadening is field-dependent, and affects the bluer components more strongly. For the Ca i 4227 Å resonance line, when the width and strength parameters are adjusted to match the π-component, the strength and shape of the σ⁺-component (≃4090 Å) also agree well with the observations. This demonstrates that the values of B_d, a_z, and the inclination found from the Na lines are also appropriate for this transition. For the Ca ii triplet, the width of the σ⁺ component is also seen to be in agreement with the data, though the signal-to-noise ratio in this part of the spectrum is too poor to compare the shape of the line with the data. Finally, for the Ca i 6142 Å Zeeman-triplet, only the shape of the σ⁺-component in is reasonable agreement with the data, furthering the argument from Section 4 that these transitions may originate from another source.

6 DISCUSSION

6.1 DZs with much stronger fields

In Section 5, we constructed a toy-model for generating simplified magnetic DZ spectra, including atomic data from Section 3. While it turned out that SDSS J1143+6615 has only a 30 MG field, in principle our model allows us to generate synthetic spectra for much larger fields, with 470 MG covering all the transitions we calculated in Section 3. Since ongoing/upcoming spectroscopic surveys such as WEAVE, DESI, SDSS V, and 4MOST, are expected to yield hundreds of thousands of white dwarf spectra in the next decade, we investigate which transitions ought to be focused on for identifying even higher field DZ stars in the future.

In Fig. 8, we show models with average surface fields spanning 25–400 MG against the same curves from Fig. 4. For all five models, we used the same inclination and dipole offset as found for SDSS J1143+6615, i.e. 15 deg and −0.15 respectively. Note that the B_s values are the surface averaged field strengths whereas the dipole field strength, B_d, is approximately 52 per cent larger (see equation 13).

Figure 8.

Simulated magnetic DZ spectra for five different surface averaged field strengths (B_s), with each spectrum offset from one another by 1 in normalized flux. The inclination- and dipole-offset parameters are fixed to the values found for SDSS J1143+6615 (i.e. 15 deg and −0.15, respectively). The background Zeeman triplets have the same meaning as in Fig. 4, with the field strength scale given on the right-hand axis.

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The bottom model has a field B_s = 25 MG, similar in strength to that found for SDSS J1143+6615, and thus shows a similar spectrum. Despite the relatively uniform field for an inclination of 15 deg and a_z = −0.15, as the field increases, the σ-components still become washed out, and for most of the transitions are almost invisible at fields of around 100 MG and above. For Mg i, the σ⁺ component still remains visible above 100 MG due to its increase in oscillator strength.

On the other hand, most of the π-components remain relatively steady in wavelength. For the Na π-component, as already noted in Hampe et al. (2020), the wavelength changes very little below 100 MG, leaving this line similarly sharp as for a 25 MG field. The Na line reaches a maximum in blue-shift at 240 MG (100 Å bluer than the field free wavelength), before rapidly turning around and moving to redder wavelengths. Therefore, for B_s = 400 MG, the line becomes much broader, but remains clearly visible. Therefore, this transition ought to be used as a primary marker for identifying cool magnetic DZ white dwarfs with >100 MG fields.

Similarly, the Ca i π-component remains relatively stationary up to 100 MG, but becomes more washed out for larger fields due to the quadratic Zeeman effect, and becoming broadened to a width of 100 Å at 400 MG. Therefore, this line is likely to be less reliable than the Na π-component for identifying the highest field DZs, but will still remain reliable up to 200 MG.

The Ca ii π-component is also near stationary, and should still be recognizable even at 400 MG, making this a more obvious choice for identifying warmer high field DZs where the Na i and Ca i lines may be too weak to identify. Note that at 300 MG, the Ca i and Ca ii π-components overlap producing a blended spectral feature.

Finally, the Mg i π-component experiences a much larger quadratic shift than the other transitions considered here. Therefore, at 400 MG, the line appears broad and asymmetric though is notably still visible, in part due to the increased oscillator strength for this component, which is close to four times larger than in the field-free case, thereby also showcasing the importance of considering field-dependent intensities. Note that this toy-model does not consider the intrinsic asymmetry caused by neutral broadening, which itself could be a function of field strength.

A final consideration is that we have not yet identified all the features in the spectrum of SDSS J1143+6615. Therefore, at very high field strengths of 100s of MG, these unclassified features will also appear shifted into other parts of the spectrum further complicating the identification of the transitions discussed above. Furthermore, other strong lines outside the optical such as the Mg i and Mg ii resonance lines (field free wavelengths at 2853 and 2799 Å, respectively), may find some of their Zeeman split components shifted into the optical, providing other atomic features requiring identification.

6.2 Use in model atmospheres

Ideally, the atomic data we have presented in Section 3 can be utilized in white dwarf model atmospheres for more detailed analyses of magnetic DZ stars. As we have shown in this work, however, this is not necessary for a basic assessment. For simply determining the surface-average field strength, B_s, and which ions are present in the atmosphere, it is sufficient to simply compare our atomic data with the spectrum in question, as was demonstrated in Section 4 for SDSS J1143+6615. Furthermore, determining the field structure of a white dwarf can be achieved with a simple model such as the toy-model we demonstrated in Section 5. Importantly, our toy-model is computationally efficient, taking only a few seconds to produce Fig. 7.

Of course, much can still be learned from incorporating our atomic data into model atmospheres. In our toy-model from Section 5, the strength and widths of the Lorentzian profiles we used have no physical basis, and are simply adjusted to give acceptable agreement with the data. In a model atmosphere, the strengths and widths of the features seen in the spectrum of SDSS J1143+6615 can be investigated by adjusting the abundances and |${T_\mathrm{eff}}$| (and to some extent the surface gravity) of the model, allowing these atmospheric properties to be measured in a physically meaningful way.

The main challenge of such an approach is the computation time required. In the field-free case, the final model spectrum is integrated over the stellar disc from spectra calculated at different angles between the surface-normal and the observer. For finite-fields, however, the synthetic spectra must also be calculated over a grid of field strengths and angles between the field and observer. In particular, the field strength axis of the grid must be computed at sufficiently fine steps so that artefacts from undersampling are not present when integrating over the stellar disc. Therefore, depending on the range of field strengths required, computation may take hundreds to thousands of times longer than in the field-free case. If the T_eff, log g, or abundances require refinement when comparing against a particular spectrum, the grid must then be recomputed with updated atmospheric parameters, leading to an even larger amount of computation time.

For that reason, we have refrained from including our atomic data within model atmospheres at the present time, and also because it exceeds the scope of our primary goals of classifying the spectral features of SDSS J1143+6615 and measuring its field strength. However, future work should perform a detailed atmospheric analysis of SDSS J1143+6615 utilizing the atomic data presented here to measure its T_eff, log g, and abundances.

7 CONCLUSIONS

We have identified SDSS J1143+ 6615 as DZ white dwarf with strong magnetic field resulting in its unique spectrum. Using finite-field, coupled-cluster calculations we were able to identify lines from Na i, Mg i, and Ca i–ii that were split and shifted by the linear and quadratic Zeeman effects. This also allowed us to establish a field strength of ≃30 MG, demonstrating that DZ spectra are challenging to interpret at only a few 10 MG, due to multiple overlapping transitions from a variety of chemical elements, which is not the case for magnetic DAs or DBs at the same field strength. Using the offset-dipole model, we were able to obtain an adequate fit to the spectral features of Na with an almost pole-on observation angle, and the dipole offset away from the observer.

Despite our success in elucidating some of the peculiar features in the spectrum of SDSS J1143+6615, many transitions still lack classification at the present time. Giving consideration to the elements and lines most commonly encountered in non-magnetic cool DZ stars, future atomic data calculations should concentrate on Fe and Cr lines, as well as additional transitions of Ca. Because SDSS J1143+6615 is currently the only available test for these calculations, and only samples the relatively low-field end, searching for additional high-field DZs within ongoing and future spectroscopic surveys (such as SDSS V, WEAVE, and DESI) is imperative to test the accuracy of our atomic data further at field strengths of many 100 MG.

ACKNOWLEDGEMENTS

MAH was supported by grant ST/V000853/1 from the Science and Technology Facilities Council (STFC). SS acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) grant number STO 1239/1-1 S.S. and within project B5 of the TRR 146 (Project No. 233 630 050). Based on observations obtained at the international Gemini Observatory, a program of NSF’s NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini Observatory partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigación y Desarrollo (Chile), Ministerio de Ciencia, Tecnología e Innovación (Argentina), Ministério da Ciência, Tecnologia, Inovações e Comunicações (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea).

DATA AVAILABILITY

All observations of SDSS J1143+6615 are either public (SDSS, Gaia) or no longer proprietary (Gemini). The Gemini data can be be obtained from the Gemini Observatory Archive, using program ID GN-2015B-FT-29. Atomic data calculations presented in this work are available in the appendix tables.

Footnotes

REFERENCES

APPENDIX A: ATOMIC DATA TABLES