White Dwarf Stars

White dwarf stars are the final stage of most stars, born single or in multiple systems. We discuss the identification, magnetic fields, and mass distribution for white dwarfs detected from spectra obtained by the Sloan Digital Sky Survey up to Data Release 13 in 2016, which lead to the increase in the number of spectroscopically identified white dwarf stars from 5000 to 39000. This number includes only white dwarf stars with log g>= 6.5 stars, i.e., excluding the Extremely Low Mass white dwarfs, which are necessarily the byproduct of stellar interaction.


Introduction
White dwarf stars are the final evolutionary state of stars with initial masses up to 8.5-10.6 M ⊙ 1 , corresponding to 95 -97 % of all stars. The fraction depends on the stellar metallicity, which affects both the Initial-Mass-Function and the Initialto-Final-Mass Relation. For single stars, the minimum mass of a present day white dwarf is around 0.30-0.45 M ⊙ 2 , because progenitors that would become lower mass white dwarfs have main sequence evolution time larger than the age of the Universe. Such masses correspond, considering the mass-radius relation of white dwarfs, to a minimal log g ≃ 6.5. Evolutionary models e.g. by Ref. 3 indicate that the maximum surface gravity for main sequence A stars, which have similar optical spectra to DA white dwarfs, corresponds to log g ≤ 4.75, even for very low metallicity. There is therefore a gap between low mass white dwarfs and main sequence stars, 4.75 ≤ log g ≤ 6.5. Most white dwarfs do not generate energy from nuclear fusion, but radiate due to residual gravitational contraction. Because of the degenerate equation of state, contraction is accompanied by a loss of thermal energy instead of increase as in This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 3.0 (CC-BY) License. Further distribution of this work is permitted, provided the original work is properly cited. the case of ideal gases; the evolution of white dwarfs is therefore often simply described as cooling. The radius of an average white dwarf star is of the same order of the Earth's radius, which implies that they have small surface area, resulting in very large cooling times; it takes approximately 10 10 years for the effective temperature of a normal mass white dwarf to decrease from 100 000 K to near 5 000 K. Consequently, the cool normal mass ones are still visible and among the oldest objects in the Galaxy 4 . Therefore, studying white dwarfs is extremely important to comprehend the processes of stellar formation and evolution in the Milky Way 5,6 .
The progenitors of white dwarfs lose most of their envelope in the giant phases, where mass loss depends on metallicity. If the remainder H mass were above ≃ 10 −4 M * , or the He mass above ≃ 10 −2 M * there would be observable nuclear burning in the white dwarf phase. The limits depend on the mass of the white dwarf. As shown in Table 1, most white dwarfs have atmospheres dominated by H, and the remainder by He. All other elements are only small traces, much less abundant than in the Sun, due to separation in the strong gravitational field 7 . The lightest elements float to the surface once the white dwarf cools below effective temperature T eff ≃ 100 000 K. Except for the H or He composition, the atmosphere of the white dwarfs in their later cooling evolution has thus lost all memory of the previous evolutionary phases.
We selected candidates to white dwarf stars from the 4.5 million SDSS spectra through their colors, automated spectra fitting by the SDSS pipeline, and targeted search of template spectra. After complete, consistent human identifications of each candidate white dwarf spectrum, we fitted the optical spectra to local thermodynamic equilibrium (LTE) grids of synthetic non-magnetic spectra derived from model atmospheres 8 . Our more recent DA model grid uses the ML2/α = 0.8 convection approximation, and for the DBs we use the ML2/α = 1.25 approximation. The DA grid extends up to T eff = 100 000 K, but NLTE effects are not included. Ref. 9 concluded pure hydrogen atmospheres of DA white dwarfs are well represented by LTE calculations for effective temperatures up to 80 000 K, but when traces of helium are present, non-local thermodynamic equilibrium (NLTE) effects on the Balmer lines occur, down to effective temperatures of 40 000 K. Ref. 9 concluded LTE models should exclude traces of helium for the consistent analysis of hydrogen dominated white dwarf spectra. We fitted the spectral lines and photometry separately 8 , selecting between the hot and cool solutions using photometry as an indicator. We include corrections to T eff and log g based on three-dimensional convection calculations from Ref. 10.
At the cool end of our sample, log g = 6.5 corresponds to a mass around 0.2 M ⊙ , well below the single mass evolution in the lifetime of the Universe. The He-core white dwarf stars in the mass range 0.2 − 0.45 M ⊙ , referred to as low-mass white dwarfs, are usually found in close binaries, often double degenerate systems 11 , being most likely a product of interacting binary star evolution. More than 70% of those studied by Ref. 12 with masses below 0.45 M ⊙ and all but a few with masses below 0.3 M ⊙ show velocity variations 13,14 . Ref. 2 suggests single low-mass white dwarfs result from the evolution of old metal-rich stars that truncate evolution before the helium flash due to severe mass loss. They also conclude all white dwarfs with masses below ≃ 0.3 M ⊙ must be a product of binary star evolution involving interaction between the components, otherwise the lifetime of the progenitor on the main sequence would be larger than the age of the Universe.

Spectral fits
In Fig. 1 we show the results of our effective temperature and surface gravity determinations for all candidates. We calculated the single star evolutionary models shown in the figure with the MESA 15 evolutionary code, including diffusion. In Fig. 2 the evolutionary models are those with rotation and diffusion of Ref. 16.

Spectral Classification
We looked for the following features to aid in the classification for each specified white dwarf subtype:

DZs
Of the white dwarfs in our sample, 3% have spectra with metal lines, probably due to accretion of rocky material surrounding the stars 23-26 . Calcium  Most stars that produced white dwarfs are born in binaries or multiple systems. Ref. 58 demonstrates that while around 70% of stars more massive than the Sun are in binaries, two-thirds of the most common stars, M type dwarf stars, are single. More than 10% of the spectroscopically identified white dwarfs in SDSS have red companions 49,59 . Refs. 60, 61 show that nearly 25% of all main sequence binaries are close enough that mass transfer interactions occur when the more massive star becomes a red giant or an asymptotic giant star. If mass transfer exceeds the Eddington limit, the secondary star is not able to accrete the transferred material and the system evolves through a common envelope phase, i.e., the core of the giant and the main sequence companion orbit within the outer layers of the giant star, leading to the shrinkage of the orbit and the release of orbital energy. The orbital energy deposited into the envelope eventually ejects it. Therefore a close binary is formed by the core of the giant star and a main sequence companion, later a close white dwarf-main sequence binary. An ELM will be formed if the envelope is ejected before the helium-flash, which would happen if the star has initial mass too low, i.e., M 2M ⊙ , to reach conditions to fuse helium in the core before it becomes degenerate.
Systems in which a white dwarf is receiving mass from a nearby low mass star are classified as Cataclysmic Variables 62 . Accretion of mass by a white dwarf, or the merger of two white dwarfs, may lead to a supernova (SNIa) explosion, if the system grows over the Chandrasekhar mass limit. The binary white dwarf system SDSS J065133.338+284423.37, with a 12.75 m orbital period, has been identified by Ref. 63, showing clear rapid orbital decay consistent with general relativity prediction due to gravitational wave radiation. White dwarfs have been identified as companions to millisecond pulsars 64-76 . Ref. 77 uses the Gaia parallaxes for 6 directly observed and 46 white dwarf stars in binary system to estimate their mass-radius relation. By Gaia second data release, in September 2017, the distance to thousand of white dwarf stars should be known, allowing a precise determination of the mass-radius relation for white dwarfs and clarifying the nature of the Extremely Low Mass white dwarfs and subdwarfs.

Mass Distribution
We estimated the masses of all DA white dwarfs found by Ref. 78, 79 and 49. There were no new optical stellar spectra in SDSS Data Release 13. For the DA mass distribution, we only consider spectra with S/N≥ 15 to have reliable mass determinations. We use the mass-radius relations of Refs. 80, 81 and 3 to calculate the mass of our stars from the T eff and log g values obtained from our fits, after correcting to 3D convection following Ref. 82. Figure 4 shows the mass distribution by number for DAs with T eff ≥ 13 000 K, where convection is unimportant, and for DBs with T eff ≥ 16 000 K reported by Ref. 83. Because our surface gravities show an unexplained decrease below T eff = 10 000 K, Figure 5 shows the mass distribution for different cutoff temperatures.
Considering white dwarfs with larger mass have smaller radius, and therefore can only be seen to smaller distances in a magnitude limited survey as SDSS, we calculated the density by correcting the visible volume with the 1/V max method of Ref. 84, up to a maximum g=19 magnitude, shown in Figure 5. The distribution shows that the DA and DB distributions have very different shapes. The DA's has a tail to larger masses, while the DB's is extended to lower masses. This is probably reflecting some limitation in the progenitors that can undergo very-late thermal pulses and become DBs.

Magnetic Fields
Ref. 85 presents a review on magnetic fields in white dwarf stars. When examining each white dwarf candidate SDSS spectrum by eye, we found 822 stars with Zeeman splittings indicating magnetic fields above 2 MG -the limit where the line splitting becomes too small to be identified at the SDSS spectral resolution 86 .
If the line splitting and magnetic fields were not recognized, the spectral fittings of DA and DB models would have rendered too high log g determinations due to magnetic broadening being misinterpreted as pressure broadening. We estimated the mean fields for the new DAHs following Ref. 87 as being from 2 MG to 700 MG. We caution that stars with large fields are difficult to identify because fields above around 30 MG intermixes subcomponents between different hydrogen series components so much that, depending on effective temperature and signal-tonoise, it becomes difficult to identify the star as containing hydrogen at all, and affecting even the colors significantly. Additionally, white dwarf stars with fields above 100 MG (see Fig. 6) represent the intermediate regime in which the spectra have very few features, except for a few stationary transitions that have similar wavelengths for a reasonable distribution of magnetic fields over the surface of the star. We found significant Zeeman splittings in the spectra of around 4% of all white dwarfs. Both the low field limit and the high field limit are totally dominated by systematic effects, not the real limits. Polarization is hard to detect because it re-   (Table 2), assuming dipole magnetic distribution, obtaining B≃10-19 MG from the Hα and Hβ splittings. A fit to the optical SDSS u,g,r colors with non-magnetic models results in T eff = 12 200 ± 300 K, log g = 8.2 ± 0.1, and mass M = 0.72 ± 0.06M ⊙ , all unreliable because the large Zeeman splittings affect the colors, specially u, substantially. A fit to the r,i,z colors results in T eff = 13 500 ± 500 K, but the log g went to the high border of the grid. GALEX observations nuv=18.11 ± 0.04, fuv=18.62 ± 0.07 are consistent with T eff ≃ 12 000 K, but the spectra does not fit from the red to the blue (see Fig.7).
The Catalina Real-time Transient Survey has 311 measurements with a few measurements significantly brighter, with a mean of V CSS = 17.76 ± 0.05 and a change from 17.89 ± 0.11 to 17.18 ± 0.16 in 14 minutes. The timescale of change is of the order of a few hours, but there is no periodic variation found. The 32 SDSS photometric measurements also show variations in color.

Observations
To test for short timescale variations and obtain a higher signal-to-noise average spectrum, we obtained 3 × 1800 s exposures for SDSS J030407. on timescale from hours to several years, probably associated with rotation. Our spectra from SDSS and SOAR is too sparse to estimate the timescale of variability.

Analysis
The field structure over the surface of the white dwarf becomes accessible to measurement via the profiles of the photospheric Zeeman-broadened Balmer absorption lines, an approach dubbed Zeeman tomography 96  ture of a model atmosphere (T eff , log g) and a given magnetic field vector with respect to the line of sight and the normal on the surface of the star, it calculates theoretical flux and polarization spectra 100,101 . The model spectra were calculated assuming a surface gravity of log g = 8, considering DAs show a sharp peak in the mass distribution around this value 103,104,10 , and we do not have a clear way to estimate the mass of the star. Considering no polarization information is available for this star, our analysis is limited to the flux spectra (Stokes parameter I).
Limb darkening is accounted for by a simple linear scaling law 96 . We determine the magnetic field geometry with the code described in Ref. 87. This code calculates the total flux (and circular polarization) spectra for an arbitrary magnetic field topology by adding up appropriately weighted model spectra for a large number of surface elements and then evaluating the goodness of fit. Magnetic field geometries are accounted for by multi-pole expansions of the scalar magnetic potential. The individual multi-pole components may be independently oriented with respect to the rotation axis of the white dwarf and offset with respect to its center, allowing in principle for rather complex surface field topologies. Additional free parameters are the white dwarf effective temperature and the inclination of the rotation axis with respect to the line of sight. The observed spectra was fitted using an evolutionary algorithm 105 with a least-squares quality function. Additionally to the Zeeman effect, Stark broadening is considered. For the case when the electric and magnetic fields are parallel, Ref. 106 estimated the effect on stationary line components, which are transitions that vary slowly in wavelength for large intervals of magnetic field strengths. Stationary lines are more pronounced than non-stationary lines, as they are not smeared out extensively due to the variation of the magnetic field strength over the stellar surface. However, no atomic data for hydrogen in the presence of both a magnetic and electric field are available for arbitrary strengths and arbitrary angles between two fields. Therefore, only a crude approximation 100 is used in the model and systematic uncertainties are unavoidable, particularly in the lowfield regime (≤ 5 MG) where the Stark effect dominates. Consequently, effective temperatures and surface gravities derived from fitting the Balmer lines alone are less reliable than in the case of non-magnetic white dwarfs. This may also result in disagreements with temperature estimates derived from the continuum slope. Time-resolved analysis for rotating single magnetic white dwarfs is instrumental in determining rather complex field structures [e.g. VLT observations 96,97,98 ]. However this usually relies on the preliminary knowledge of period which is usually derived via photometry, separately. Although the individual SDSS fiber spectra exists with 15 minute exposure time, due to the lack of information on spin period, we constrained ourselves to the co-added spectra which includes 3 to 6 individual spectra with total exposure time of at least 45 minutes (see Table 2). For white dwarfs, the dipole magnetic field Ohmic decay timescale is 10 10 yr and even the higher multi-poles can live for such a long period of time 107 . Figures 9 and 10 shows two graphic forms to present the field distribution across the surface of the star: (i) the B × cos φ diagram that depict the frequency distribution of field vectors over the surface of the star corresponding to the spectrum; and (ii) a synthetic picture of the field distribution. The latter include (a) the field strength B, (b) cos = B l /B with B l the field component along the line of sight.
To extend the calculations of the splittings for hydrogen levels higher than n=7 for these fields, where perturbation theory is no longer applicable 100,108 , Refs. 109, 110 extended the calculations to n=11. This was achieved by using a two-dimensional finite element expansion of the wave functions in terms of B-splines in the directions parallel and perpendicular to the magnetic field, instead of using asymptotically valid basis expansions in terms of spherical harmonics or Landau orbitals, as has been previously done in the literature, which fails either in the highfield or the low-field regime, respectively. In this way energies could be calculated for 300 states with an accuracy of better than 10 −6 Rydberg across several symmetry subspaces over the entire regime of magnetic field strengths. The wealth of the resulting wavelength information is the basis for the analysis of the spectra presented in this paper. Details of the computational method can be found in Ref. 111, and results from 0 to 800 MG are shown in Fig. 6.  The combined effect of electric and magnetic fields on the spectral lines is very complicated and has been studied only for special cases of the geometry 106,87 . Detailed model grids, which include also the effect of the magnetic field on the radiative transfer are not yet available.
We demonstrate the Zeeman splitting of the magnetic DA white dwarf SDSS J030407.40-002541.92 shows complex field distribution across the surface of the star, with a mean value of B ≃ 13 MG, and the Zeeman components change from different spectra, probably due to rotation of the star with the magnetic axis not aligned to the rotation axis.
Ref. 112 discuss how magnetic field can suppress convection in white dwarfs, resulting in magnetically generated spots on the stellar surface that can cause the observed variability, both spectral and photometric, when the star rotates. They also discuss that a magnetic field of the order of the observed in SDSS J030407.40-02541.74 can totally suppress convection, affecting the cooling of the white dwarf. The fast and irregular variations we see in SDSS J030407.40-02541.74 are different from the stable ones they report for the weaker magnetic field WD 1953-011.

Rotation
In general the measured rotation period for single white dwarfs ranges from 1 h to 18 d, with a median around 1 d 113 . The fastest single white dwarf rotator from asteroseismological measurements (Table 3)  Differential rotation in white dwarfs was studied by Refs. 120 -123, using the change in rotation splitting of non-radial pulsations, agreeing with the evidence of substantial loss of angular momentum on the red giant branch from asteroseismology of red giant stars 124,125 .

Axions and Dark Mass
Axions are the best candidates for dark mass 126 . Refs.127 -132 show white dwarf pulsations and luminosity function are consistent with extra cooling caused by axions of masses around 17 ± 4 mev.

Population synthesis
We did a population synthesis analysis and computed a theoretical mass distribution through a Monte Carlo simulation -fitting single star initial mass functions, initialto-final mass relations for masses 0.45 M ⊙ ≤ M < 1.0 M ⊙ -to obtain a history of star formation for the DAs. With our calculated lifetimes from ZAMS to the present cooling stage, all white dwarfs born 12 Gyr ago are below T eff ≃ 10 000K, most are expected to be around 4500 K. For our sample with T eff ≥ 13 000 K, Figure 11 shows the mean mass around 0.64 M ⊙ requires a burst of star formation in the last 2 Gyr, as a white dwarf with such mass has a short lived progenitor mass with a mass around 2.5 M ⊙ . This is in contrast with the uniform star formation estimated by Ref. 133 from the ML2/α = 0.6 mass distribution of Ref. 103. These results indicate most white dwarfs are cooler than our present sample and we must develop ways to disentangle them from red dwarfs. Such cool helium atmosphere white dwarfs show no lines in the spectra, while hydrogen atmosphere ones show only a weak Hα line. Proper motions and parallaxes are therefore the key separators, as cool white dwarfs are much fainter than red dwarfs at similar temperatures. Therefore GAIA will be crucial for their identification.