# «Analysis of a Very Massive DA White Dwarf via the Trigonometric Parallax and Spectroscopic Methods Conard C. Dahn US Naval Observatory, Flagstaﬀ ...»

Accepted for Ap.J.

Analysis of a Very Massive DA White Dwarf via the

Trigonometric Parallax and Spectroscopic Methods

Conard C. Dahn

US Naval Observatory, Flagstaﬀ Station, P.O. Box 1149, Flagstaﬀ, AZ 86002-1149

P. Bergeron

D´partement de Physique, Universit´ de Montr´al, C.P. 6128, Succ. Centre-Ville,

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Montr´al, Qu´bec, Canada H3C 3J7

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James Liebert

Steward Observatory, University of Arizona, Tucson AZ 85726

Hugh C. Harris US Naval Observatory, Flagstaﬀ Station, P.O. Box 1149, Flagstaﬀ, AZ 86002-1149 and S. K. Leggett UKIRT, Joint Astronomy Centre, 660 North A’ohoku Place, Hilo, HI 96720 ABSTRACT –2– By two diﬀerent methods, we show that LHS 4033 is an extremely massive white dwarf near its likely upper mass limit for destruction by unstable electron captures. From the accurate trigonometric parallax reported herein, the eﬀective temperature (Teﬀ = 10, 900 K) and the stellar radius (R = 0.00368 R ) are directly determined from the broad-band spectral energy distribution — the parallax method. The eﬀective temperature and surface gravity are also estimated independently from the simultaneous ﬁtting of the observed Balmer line proﬁles with those predicted from pure-hydrogen model atmospheres — the spectroscopic method (Teﬀ = 10, 760 K, log g = 9.46). The mass of LHS 4033 is then inferred from theoretical mass-radius relations appropriate for white dwarfs. The parallax method yields a mass estimate of 1.310–1.330 M, for interior compositions ranging from pure magnesium to pure carbon, respectively, while the spectroscopic method yields an estimate of 1.318–1.335 M for the same core compositions.

This star is the most massive white dwarf for which a robust comparison of the two techniques has been made.

Subject headings: stars: fundamental parameters — stars: individual (LHS 4033) — white dwarfs

1. INTRODUCTION LHS 4033 (WD 2349−031) is a white dwarf discussed in a recent paper by Salim et al. (2004). The star has also been part of the Luyten Half Second (LHS) survey µ ≥ 0.6 yr−1 white dwarf sample of C. C. Dahn, H. C. Harris, S. K. Leggett, & J. Liebert (2004, in preparation), virtually all of which have been targeted for accurate trigonometric parallaxes at the U.S. Naval Observatory, for purposes of estimating the luminosity function of cool white dwarfs.

In the H-R diagram, this object lies to the left of the diagonal sequence of white dwarfs, indicating that its mass is larger and its radius is smaller than normal. In this paper we show that its mass is, indeed, extraordinarily large for a white dwarf. A few other stars of mass similar to the value we present in this paper for LHS 4033 have been found – i.e.

the low-ﬁeld magnetic star PG 1658+441 (?)M ∼ 1.31 M ]schmidt92,dupuis03, and the highly-magnetic white dwarf RE J0317−853 (?)M = 1.35 M ]barstow95,ferrario97. Note that the application of the Balmer line ﬁtting procedure to these two magnetic stars of high mass is impossible for a ∼300 MG magnetic white dwarf like RE J0317−853, and the mass of this star has been determined indirectly from the companion. Also, the modelling of the Zeeman triplet components of PG 1658+441 by Schmidt et al. (1992) is only approximate, –3– and it is inherently less accurate due, among other things, to the lack of a rigorous theory of Stark broadening in the presence of the 3.5 MG ﬁeld. Hence, both mass estimates are quite uncertain.

Since the spectral type of LHS 4033 is DA and non-magnetic, the mass may be estimated by ﬁts to the Balmer lines (?, see, e.g.,)]BSL in a much more rigorous fashion. The surface gravity used with suitable evolutionary models yields independent determinations of the mass and radius. The eﬀective temperature may also be estimated from broad-band photometry once the dominant atmospheric constituent is known. This, along with an accurate trigonometric parallax, permits a diﬀerent estimate of the luminosity, radius, and mass (Bergeron et al. 2001). While it has been possible to compare the parameter determinations of these methods for limited samples of white dwarfs, it is particularly interesting to do so for a massive star.

** 2. OBSERVATIONAL DATA**

Photometry with BV I ﬁlters was obtained three times with the USNO 1.0 m telescope generally during 1997-1998. JHK data were obtained on 1998 October 12 using the IRCAM camera outﬁtted with the UKIRT system ﬁlters and calibrated using UKIRT standards (Hawarden et al. 2001). Colors are reduced to the Johnson system for B–V, the Cousins system for V –I, and the CIT system for J–H and H–K. Errors are 0.02 mag in BV I and 0.05 in JHK. Our optical and infrared photometry for LHS 4033 is given in Table

1. Salim et al. (2004) also report CCD photometry for LHS 4033, on the Johnson-Cousins system, from the Lick Observatory 1 m Nickel telescope. They obtain B = 17.162 ± 0.020, V = 16.992 ± 0.017, R = 16.987 ± 0.030, and I = 16.936 ± 0.036, based on 2-3 observations per band. The corresponding color indices, B–V = 0.17 and V –I = 0.056, thus agree with our measurements within the uncertainties. In the model atmosphere analysis presented below, we rely on our own photometric measurements only.

Trigonometric parallax observations were carried out over a 6.05 year interval (1997.76 – 2003.81) using the USNO 1.55 m Strand Astrometric Reﬂector equipped with a Tek2K CCD camera (Dahn 1997). The absolute trigonometric parallax and the relative proper motion and position angle derived from the 150 acceptable frames are given in Table 1. The parallax and apparent V magnitude then yield an absolute magnitude, also included in Table

1. Further details regarding the astrometry for LHS 4033 will appear in a paper on white dwarf parallaxes (Dahn et al. 2004, in preparation).

Finally, optical spectroscopy was secured on 2003 October 1 using the Steward Obser4– vatory 2.3-m reﬂector telescope equipped with the Boller & Chivens spectrograph and a UV-ﬂooded Texas Instrument CCD detector. The 4.5 arcsec slit together with the 600 lines mm−1 grating blazed at 3568 ˚ in ﬁrst order provided a spectral coverage of 3120–5330 ˚ at A A an intermediate resolution of ∼ 6 A ˚ FWHM. The 3000 s integration yielded a signal-to-noise ratio around 55 in the continuum. Our optical spectrum for LHS 4033 is contrasted in Figure 1 with that of G61−17, a DA white dwarf with an eﬀective temperature comparable to that of LHS 4033, but with a normal surface gravity and mass (Teﬀ = 10, 680 K, log g = 8.06, M = 0.64 M ) according to the spectroscopic analysis of the DA stars from the PG sample by J. Liebert, P., Bergeron, & J. B. Holberg (2003, in preparation). The strong decrement of the high Balmer lines already indicates that LHS 4033 is a massive white dwarf.

We ﬁrst proceed to ﬁt the optical and infrared photometry using the technique described in Bergeron et al. (1997, 2001). Broadband magnitudes are ﬁrst converted into observed ﬂuxes using Eq. [1] of Bergeron et al. (1997) with the appropriate zero points. The resulting energy distribution is then compared with those predicted from our model atmosphere calculations, with the monochromatic ﬂuxes properly averaged over the same ﬁlter bandpasses.

Our model atmospheres are hydrogen-line blanketed LTE models, and assume a pure hydrogen composition. Convection is treated within the mixing-length theory, with the ML2/α = 0.6 formulation following the prescription of Bergeron et al. (1995). The calculations of theoretical spectra are described at length in Bergeron et al. (1991b), and include the occupation probability formalism of Hummer & Mihalas (1988). This formalism allows a detailed calculation of the level populations in the presence of perturbations from neighboring particles, and also provides a consistent description of bound-bound and bound-free transitions.

m m The observed ﬂuxes fλ and Eddington model ﬂuxes Hλ — which depend on Teﬀ and log g — for a given bandpass m are related by the equation

1986), which is based on a steepest descent method. The value of χ2 is taken as the sum over all bandpasses of the diﬀerence between both sides of Eq. [1], properly weighted by the corresponding observational uncertainties. Only Teﬀ and the solid angle π (R/D)2 are considered free parameters, while the uncertainties are obtained directly from the covariance matrix of the ﬁt.

We ﬁrst assume log g = 8.0 and determine the eﬀective temperature and the solid angle, which, combined with the distance D obtained from the trigonometric parallax measurement, yields directly the radius of the star R. The latter is then converted into mass using an appropriate mass-radius relation for white dwarf stars. Here we ﬁrst make use of the massradius relation of Hamada & Salpeter (1961) for carbon-core conﬁgurations. This relation is preferred to the evolutionary models of Wood (1995) or those of Fontaine et al. (2001), which extend only up to 1.2 and 1.3 M, respectively. Uncertainties due to ﬁnite temperature eﬀects and core composition will be discussed below. In general, the value of log g obtained from the inferred mass and radius (g = GM/R2 ) will be diﬀerent from our initial assumption of log g = 8.0, and the ﬁtting procedure is thus repeated until an internal consistency in log g is achieved. The parameter uncertainties are obtained by propagating the error of the photometric and trigonometric parallax measurements into the ﬁtting procedure.

Our best ﬁt to the optical BV I and infrared JHK photometry of LHS 4033 is displayed Figure 2. The monochromatic ﬂuxes from the best ﬁtting model are shown here as well, although the formal ﬁt is performed using only the average ﬂuxes (ﬁlled dots). The solution at Teﬀ = 10, 900 ± 290 K and R = 0.00368 ± 0.00013 R implies a stellar mass of M =

1.330 ± 0.004 M and a value of log g = 9.43 ± 0.02. The parameters of both methods are summarized in Table 2. The predicted absolute visual magnitude obtained from the values of Teﬀ and log g is MV = 14.63, in perfect agreement with the value derived from the parallax given in Table 1.

** 3.2. Spectroscopic Analysis**

The optical spectrum of LHS 4033 is ﬁtted with the same grid of model atmospheres following the procedure described in Bergeron et al. (1992) and Bergeron et al. (1995). The spectrum is ﬁrst ﬁtted with several pseudo-Gaussian proﬁles (Saﬀer et al. 1988) using the nonlinear least-squares method of Levenberg-Marquardt described above. Normal points deﬁned by this smooth function are then used to normalize the line ﬂux to a continuum set to unity at a ﬁxed distance from the line center. The comparison with model spectra, which are convolved with the appropriate Gaussian 6 ˚ instrumental proﬁle, is then carried out in A terms of these normalized line proﬁles only. Our minimization technique again relies on the –6– Levenberg-Marquardt method using the Hβ to H8 line proﬁles. Our best ﬁt is displayed in Figure 3.

Remarkably, our spectroscopic solution Teﬀ = 10, 760 ± 150 K and log g = 9.46 ± 0.04, which translates into M = 1.335 ± 0.011 and R = 0.00358 ± 0.00019 R using the HamadaSalpeter mass-radius relation for carbon-core conﬁgurations, is in excellent agreement with the solution obtained with the photometry and trigonometric parallax method. This is arguably the most massive white dwarf subjected to a rigorous mass determination (?, see, e.g., Table 3 of)]dupuis02. Note that despite the extreme surface gravity of LHS 4033, the Hummer-Mihalas formalism used in the line proﬁle calculations remains perfectly valid, since the density at the photosphere remains low (ρ ∼ 10−5 g cm−3 ) as a result of the high opacity of hydrogen at these temperatures.

** 3.3. Mass-Radius Relation**

In a venerable paper, Hamada & Salpeter (1961) ﬁrst employed an equation-of-state (EOS) including coulomb “corrections” to the pressure and energy of a degenerate Fermi gas (Salpeter 1961) to calculate the mass-radius-central density relations for models composed of helium through iron. These corrections to the classic Chandrasekhar EOS for degenerate matter are more important at high mass. It may also be noted that, especially at the relatively low eﬀective temperature of LHS 4033, neglect of the internal energy of the ions (“zero-temperature” modelling) is likely to be a reasonable assumption.

Since LHS 4033 may have a core composed of material much heavier than carbon, we must explore the eﬀects of core composition on the results of our analysis. We compare in Figure ?? the mass-radius relation obtained from the detailed evolutionary carbon- and carbon/oxygen-core models of ?)][see also Bergeron et al. 2001]fon01 with the HamadaSalpeter zero-temperature conﬁgurations for carbon and magnesium at a mass of 1.3 M, the highest mass of the Fontaine et al. models. At the eﬀective temperature and mass of LHS 4033, the carbon- or carbon/oxygen-core models of Fontaine et al. reveal that ﬁnite temperature eﬀects are extremely small, and account for an increase in radius of only ∼ 0.5 % (i.e. by comparing the radius at 10,000 K with the value at 3500 K where it becomes constant). Moreover, at the temperature of LHS 4033, the carbon-core models of Fontaine et al. and Hamada-Salpeter diﬀer by only 2.7 % in radius, or 0.007 M in mass. Details of the equation-of-state are thus also negligible in the present context. Finally, the Mg and C conﬁgurations of Hamada-Salpeter diﬀer by 7.4 % in radius, or 0.02 M in mass. Indeed, the parallax method with the Mg conﬁgurations yields a mass of 1.310 M (instead of 1.330 when C conﬁgurations are used), while the spectroscopic method yields a mass of 1.318 M –7– (instead of 1.335 M ). These results are also reported in Table 2. We thus argue that our mass estimates are uncertain by 0.02 M at the most.