Constraining the Surface Inhomogeneity and Settling times of Metals on Accreting White Dwarfs

Due to the short settling times of metals in DA white dwarf atmospheres, any white dwarfs with photospheric metals must be actively accreting. It is therefore natural to expect that the metals may not be deposited uniformly on the surface of the star. We present calculations showing how the temperature variations associated with white dwarf pulsations lead to an observable diagnostic of the surface metal distribution, and we show what constraints current data sets are able to provide. We also investigate the effect that time-variable accretion has on the metal abundances of different species, and we show how this can lead to constraints on the gravitational settling times. Subject headings: accretion, accretion disks — diffusion — convection — stars: oscillations — stars: variables: other — white dwarfs 1. ASTROPHYSICAL CONTEXT There are two main classes of white dwarf stars: those with hydrogen-rich atmospheres (spectral type DA) and those with helium-rich atmospheres (non-DA spectral types). The reason for this is that the high surface gravities of white dwarfs lead to efficient gravitational settling, with the lightest elements rising to the surface. In addition, some DA white dwarfs have spectra showing metal lines of elements such as Ca and Mg (Zuckerman et al. 2003), and these stars are referred to as DAZs; about 20% of all DAs fall into this category. Recently, Dufour et al. (2007) have announced a new class of white dwarf with carbon-dominated atmospheres, the “hot DQ” stars, several examples of which have been found in the Sloan Digital Sky Survey (Liebert et al. 2003). The presence of metals in the DAZs is intriguing since the settling time scale for the metals may be many orders of magnitude shorter than the evolutionary age of these objects. Indeed, for DAZs with Teff ∼ 12,000 K, the settling time scale can be on the order of days or weeks, meaning that these objects are experiencing ongoing accretion (Koester & Wilken 2006). This ongoing accretion is consistent with the fact that nearly a dozen of these objects have detected dust disks (“debris disks,” see Tokunaga, Becklin, & Zuckerman 1990; Becklin et al. 2005; Kilic et al. 2005; Reach et al. 2005; Kilic et al. 2006; von Hippel et al. 2007; Jura, Farihi, & Zuckerman 2007; Jura et al. 2007; Farihi, Zuckerman, & Becklin 2008) and these disks are assumed to be the sources of the metal lines seen in these white dwarf atmospheres. The best studied object of this DAZ class with an observed disk, G29-38, is also a multi-periodic variable white dwarf (DAV), pulsating in non-radial g-modes with periods of a few hundred to one thousand seconds. The technique of asteroseismology uses the observed pulsation modes of a star to infer and constrain the interior 1 Department of Astronomy, University of Texas at Austin, Austin, TX, USA; [email protected] 2 Department of Physics and Astronomy, University of Delaware, Newark, DE 3 Delaware Asteroseismic Research Center, Mt. Cuba Observatory, Greenville, DE 4 Department of Physics, Siena College, Loudonville, NY structure of the star, thus obtaining information on the star’s structure as a function of radius (Bradley & Winget 1994; Kawaler & Bradley 1994; Metcalfe, Salaris, & Winget 2002; Montgomery, Metcalfe, & Winget 2003). In contrast to this, our approach in this paper is to use the different angular dependence of the pulsations to constrain the accretion process. Since the accretion is most likely occurring through a disk, it is natural to suppose that the metals may not be uniformly distributed across the star’s surface. We present calculations showing how we can place constraints on the non-uniformity of the accretion process. 2. GRAVITATIONAL SETTLING AND HORIZONTAL DIFFUSION For all our calculations, we use the DAZ G29-38 as our template, since, given its brightness, it has the greatest potential for successful measurement of a surface inhomogeneity. In addition, we have archival data on this star appropriate to this application. In this section, we will therefore attempt to quantify the importance of gravitational settling and horizontal diffusion assuming a model with parameters similar to those of G29-38. FIG. 1.— Possible surface metal distributions for accretion centered on the poles (dashed curve) or the equator (solid curve) as a function of the polar angle (co-latitude). 2 Montgomery, Thompson, & von Hippel FIG. 2.— The diagnostic R as a function of inclination angle θi. The left panel shows the results for the polar distribution of metals shown in Figure 1 and the right panel shows the results for the equatorial distribution. The different curves are labeled by the |lm〉 values of the relevant pulsation modes. Bergeron et al. (2004) find Teff = 11,820 K and logg = 8.14 for this star, while Koester, Provencal, & Shipman (1997) find Teff = 11,600 K and logg = 8.05. Interpolating in the tables of Koester & Wilken for Ca yields a settling time of ∼ 13 days for the first set of parameters5 and a settling time of ∼ 23 days for the second. We note that while unseen helium in the atmosphere could lengthen these settling times considerably (e.g., García-Berro et al. 2007), its presence is inconsistent with the pulsation results for this star: such an amount would imply a much deeper surface convection zone, in conflict with that found by Montgomery (2005). Since these stars should have surface convection zones, the dominant form of horizontal transport of Ca will be due to the turbulent viscosity. We can estimate the size of this diffusion coefficient as D ∼ vClh, where vC is a typical convective velocity and lh is the assumed “mixing length” for convection. From our white dwarf evolution code (e.g., see Montgomery et al. 1999) we find for both sets of stellar parameters that D ≈ 1.5 ·1010cm2/sec. For these simple exploratory calculations we assume azimuthal symmetry for both the accretion and the surface metal distribution, i.e., Z = Z(θ, t) and S = S(θ, t), where Z(θ, t) is the metal abundance, S(θ, t) is the source function of metals accreting onto the white dwarf, θ is the “co-latitude” of a point on the star’s surface, and t is time. Since convection will uniformly mix material vertically, we can treat the convective region as a single “zone” and write an equation for the time evolution of Z as a function of θ and t: