1 Introduction.

Runaway accretion of metals from compact debris disks onto white dwarfs.

Abstract

It was recently proposed that metal-rich white dwarfs (WDs) accrete their metals from compact debris disks found to exist around more than a dozen of them. At the same time, elemental abundances measured in atmospheres of some WDs imply vigorous metal accretion at rates up to g s, far in excess of what can be supplied solely by Poynting-Robertson drag acting on such debris disks. To explain this observation we propose a model, in which rapid transport of metals from the disk onto the WD naturally results from interaction between this particulate disk and spatially coexisting disk of metallic gas. The latter is fed by evaporation of debris particles at the sublimation radius located at several tens of WD radii. Because of pressure support gaseous disk orbits WD slower than particulate disk. Resultant azimuthal drift between them at speed m s causes aerodynamic drag on the disk of solids and drives inward migration of its constituent particles. Upon reaching the sublimation radius particles evaporate, enhancing the density of metallic gaseous disk and leading to positive feedback. Under favorable circumstances (low viscosity in the disk of metallic gas and efficient aerodynamic coupling between the disks) system evolves in a runaway fashion, destroying debris disk on time scale of yr, and giving rise to high metal accretion rates up to g s, in agreement with observations.

Subject headings:
White dwarfs — Accretion, accretion disks — Protoplanetary disks

1. Introduction.

Recent detections of near-infrared excesses around a number of metal-rich white dwarfs (WD) imply existence of warm circumstellar material reprocessing stellar radiation (Zuckerman & Becklin 1987; Graham et al. 1990; Farihi et al. 2010). Spectral modeling suggests that this material resides in an extended, compact, optically thick and geometrically thin disk (Jura 2003; Jura et al. 2007), similar to the Saturn’s rings (Cuzzi et al. 2010). Disks have inner edges at several tens of the WD radii , roughly consistent with being set by particle sublimation at these locations. Their outer radii lie close to the Roche radius of the WD R, supporting the suggestion by Jura (2003) that such compact debris disks are produced by tidal disruption of asteroid-like bodies scattered into low-periastron orbits by gravitational perturbations of massive planets, which have survived the AGB phase of stellar evolution.

Availability of large reservoir of high-Z elements in the form of debris disk in the immediate vicinity of some WDs naturally led to the suggestion (Jura 2003) that metal enrichment of these stars is caused by accretion from such disks. This scenario provides a promising alternative to the previously proposed interstellar accretion model of metal pollution of WDs (Dupuis et al. 1993), which is not consistent with observations of WDs with He atmospheres.

Theoretical estimates of settling time of heavy elements in WDs imply that their observed atmospheric abundances can be maintained against gravitational settling by accretion of metals at rates g s (Farihi et al. 2009, 2010). If the circumstellar accretion hypothesis is correct, an evolving disk of debris must be able to supply such to the WD.

The actual transfer of metals from the disk of solids truncated at sublimation radius to the WD must be accomplished in this picture via the gas disk extending from the WD surface to and beyond. Observational evidence of such gaseous component around several metal-rich WDs hosting compact debris disks has been found by Gänsicke et al. (2006, 2007, 2008) in the form of double-peaked emission lines of Ca II and Fe II. These spectroscopic signatures are naturally explained as arising in a disk of metallic gas (no H or He emission lines have been detected around these WDs) in Keplerian rotation around WDs and spatially coincident with dusty disks (Melis et al. 2010).

Even though metals are passed to the WD through the gaseous disk the rate of mass transfer is ultimately controlled by evolution of the particulate disk. By analogy with Saturn’s rings one expects that the collisional viscosity in debris disk is too low to drive the non-negligible . However, radiation of the WD can be quite important and Rafikov (2011; hereafter R11) demonstrated that Poynting-Robertson (PR) drag on the disk naturally drives mass accretion at the rate g s. While this is high it still falls short of explaining the highest observed g s.

Here we propose a new picture of the debris disk evolution, which naturally combines several physical ingredients present in close vicinity of disk-bearing WDs.

2. Description of the model.

Our model of the WD-disk system is described below and is schematically illustrated in Figure 1. We assume a disk of solid particles to extend from the Roche radius all the way in to the sublimation radius

(1)

where is the sublimation temperature of solid particles (which we take to be K if particles are silicate), K) is the normalized stellar temperature , is the WD radius, and is the ratio of particle emissivities for starlight and for its own thermal radiation (in the following we assume macroscopic particles and set ). For (Ehrenreich et al. 2011) one finds R, in agreement with observations (Jura et al. 2007, 2009).

At present the size of particles constituting the disk is rather poorly constrained. Graham et al. (1990) suggested cm. At the same time high-resolution infrared spectroscopy with IRS onboard of Spitzer reveals strong m emission feature in disk spectra indicative of the existence of a population of small, micron-sized silicate dust particles (Jura et al. 2009). But the fraction of total disk mass locked up in such fine dust is unknown. The actual particle size is not very important for our treatment as long as the disk is optically thick. Then it is similar to Saturn’s rings and most likely behaves as a granular flow. In the following we will treat particulate disk as if it were a solid plate.

Solid particles brought to sublimate feeding a disk of metallic gas at this location. Viscous torques cause it to spread all the way to the WD surface providing means of metal transport from to the star. However, because of angular momentum conservation part of the newly produced metallic gas has to move outwards of (Lynden-Bell & Pringle 1974) naturally explaining the existence of gaseous disks spatially coexisting with dusty debris disks in some systems (Melis et al. 2010). Jura (2008) proposed collisional sputtering of small asteroids as another source of metallic gas; for the sake of clarity we do not consider this mechanism here.

We emphasize here that the two disks have an overlap in radial distance, while in vertical direction gaseous disk is much more extended than the disk of particles, see Figure 1. The latter must be very thin because inelastic collisions between particles rapidly damp any vertical random motions.

External gas disk outside has temperature different from (higher than) that of the dust disk at the same radii (Melis et al. 2010) because of the different balance of heating and cooling for the two disks. This keeps gas temperature above the sublimation point even outside , although some condensation of metallic gas on the particle surfaces may be happening there (we neglect it in this work).

Simultaneous existence of the two disks of high-Z elements in different phases drives their mutual evolution in the following way. Any coupling between the outer portion of the gas disk and particulate disk acts to transfer angular momentum from the faster rotating particulate disk to the slower spinning gaseous disk, causing inward motion of particles in the disk of solids. If the coupling is strong enough positive feedback becomes possible in the system: increasing mass of gaseous disk leads to the increase of through the disk of solids (as described below), which in turn reinforces evaporation at and increases gaseous mass even further.

Figure 1.— Schematic picture of two interacting disks (black - particulate, gray - gaseous) around the WD. Disk of solid particles is confined between the sublimation radius (where solids evaporate feeding gas disk) and Roche radius . Flow of high-Z material is indicated by arrows (white - solid particles, black - gas). Inward migration of solids is induced by drag exerted on them by the gas disk, which orbits WD slower than the disk of particles.

Coupling between the gaseous and particulate high-Z disks arises because gaseous disk orbits WD at angular speed slightly lower than the Keplerian speed at which the disk of solid particles rotates. This difference is caused by the pressure support present in gaseous disk, , and is known to cause a variety of important effects in protoplanetary disks, such as inward migration of solids (Weidenschilling 1977). Relative azimuthal velocity between the gaseous and particulate disks at distance from the WD is

(2)

where is a constant, is the normalized WD mass , is the mean molecular weight of the metallic gas normalized by (value of for pure Si), K) is the normalized gas temperature , and km s is the gas sound speed (clearly ).

This azimuthal drift gives rise to aerodynamic drag between the disks. Azimuthal drag force per unit surface area of a particulate disk with surface mass density causes inward radial migration of the disk material at speed . This inward particle drift gives rise to mass transport at the rate

(3)

Note that if is independent of then the same is true for .

External force per unit area can be generally represented in the form

(4)

where the first term describes the coupling between the gas disk with surface density and particulate disk, and constant determines the strength of coupling. It is natural to expect that drag scales with gas density ; this expectation is confirmed in §4.

Second term represents forces acting even in the absence of coupling to gas disk. PR drag is an example of such force and one can easily show (R11) that for PR drag , where is the speed of light, is the efficiency of radiative momentum absorption by the disk surface, and (Friedjung 1985) is the incidence angle of stellar radiation. From equation (3) the rate of mass transport due to PR drag alone (when ) at is (R11)

(5)

In the following we will assume that .

3. Coupled evolution of the particulate and gaseous disks.

Particle sublimation at increases the mass of gaseous disk at the rate . Assuming that surface density of gaseous disk varies on scale we can write that sublimation increases at the rate . At the same time, viscous spreading reduces at the rate , where is the characteristic viscous time in the disk

(6)

assuming -parametrization of viscosity (Shakura & Sunyaev 1973). This timescale can be very short if is not very small.

We can now describe the evolution of in the vicinity of with the following heuristic equation:

(7)

A solution of this equation satisfying the initial condition (no gas disk initially) is

(8)

where we used equation (5) and

(9)

is the sublimation time on which the mass of gaseous disk increases if . From equations (3), (4) and (8) the rate at which particles sublimate is

(10)

and the total mass lost by the debris disk to sublimation can be easily obtained by integrating this expression.

The behavior of and is shown in Figure 2 for different values of , which clearly demonstrates that for (i.e. as long as ) debris disk evolution is insensitive to and . This is because initially the disk of metallic gas is not dense enough for the drag it produces on the debris disk to compete with the PR drag — it takes certain time to accumulate enough gas mass by particle sublimation.

This means, in particular, that if the debris disk starts with mass , which is lower than the critical mass , then its evolution is determined only by the PR drag and coupling to the gaseous disk is never effective — the debris disk is eroded before gas disks grows massive enough. Lifetime of the debris disk is then

(11)

independent of the relation between and .

In the opposite case of a massive initial disk —  — evolution does depend on . Whenever the action of viscosity is so effective at removing gas released by particle sublimation at that the gas does not accumulate there. Then simply saturates at the constant low level, accretion rate tends to for , and disk gets exhausted on timescale given by equation (11). This evolutionary path clearly cannot explain the highest observed values of .

Figure 2.— Time evolution of (a) and (b) total sublimated mass . Evolutionary tracks are shown for different values of the feedback parameter and demonstrate runaway behavior for and saturation of for . Note that for the behavior of both and is independent of .

However, in the case of a massive disk and sublimation supplies gas to the region faster than viscous diffusion removes it, and grows exponentially on timescale (as long as particle disk has enough mass to provide the source). This means that also increases exponentially for (see Figure 2). This runaway stops and gaseous disk wanes only when the mass of particulate disk is exhausted. The latter happens on the runaway timescale

(12)

determined by the condition . Clearly, if .

Thus, the ratio

(13)

is the critical parameter determining the strength of positive feedback for massive disks: if a particulate disk gets rather rapidly (within several ) converted into metallic gas at , with its subsequent viscous accretion onto the WD on timescale of several . Whereas for feedback is not strong enough to reinforce gas supply at and disk slowly evolves on timescale due to PR drag. As equation (13) shows, runaway evolution and high require (1) strong coupling between the gaseous and particulate disks and (2) low viscosity.

4. Aerodynamic coupling.

Relative azimuthal motion between gaseous and particulate disks at speed induces aerodynamic drag between them. Drag produces force per unit area of the disk (Schlichting 1979)

(14)

where is the gas density and Re is a proportionality constant. There is a significant spread of opinions regarding the value of Re characterizing drag by the turbulent flow on a smooth solid plate, with numbers ranging between Re (Dobrovolskis et al. 1999) to Re (Goldreich & Ward 1973). It is also likely that a smooth plate approximation underestimates drag (overestimates the value of Re) for the particulate debris disk, which does not have continuous surface and may interact with gas more like a rough plate (Schlichting 1979) or even as a combination of individual particles, in which case smaller Re is more appropriate.

According to equation (14) aerodynamic drag force can be written in the form with

(15)

where equation (2) was used. From equation (9) sublimation time at is

(16)

where km s. The critical mass separating low- and high-mass disks (see §3) is

(17)

The feedback parameter for aerodynamic coupling is

(18)

and depends on both the viscosity parameter and the strength of aerodynamic coupling, parameterized by Re.

Equation (18) shows that runaway evolution with requires rather low viscosity in the gaseous disk, at the level of , and for higher Re smaller is needed. Viscosity is most likely provided by the magnetorotational instability (MRI), which requires certain level of ionization to operate effectively. In the ideal MHD limit simulations with no net flux typically produce (Hawley et al. 1995). However, in our scenario gaseous disk exists in immediate contact with the particulate disk, which is observationally known to contain a population of micron size dust grains (Jura et al. 2009). MRI-driven turbulence will mix some of this fine dust with the gas, lowering ionization fraction (small grains have large surface area and are very efficient charge absorbers), and giving rise to the non-ideal MHD effects, e.g via the increased resistivity (Balbus 2009). The latter are known from simulations to decrease effective substantially, down to or lower (Fleming et al. 2000; Bai & Stone, in preparation). As a result, it is conceivable that in our model can be much lower than in the ideal MHD limit of MRI.

Thus, presence of dusty debris disk in close contact with the gaseous disk can quite naturally lengthen viscous timescale and facilitate gas accumulation, making possible runaway evolution due to aerodynamic drag. We note that after the particulate disk completely sublimates the value of in the resultant gaseous disk will go up since the source of small dust particles lowering ionization has disappeared. Closer to the WD, in the dust-free region , ionization and can always be higher than in the external disk. Faster viscous evolution in this region (resulting in lower ) may explain lack of line emission from the inner part of the gas disk (Gänsicke et al. 2006, 2007, 2008; Melis et al. 2010).

5. Discussion.

Our calculations clearly demonstrate that compact debris disks around WDs are self-destructive whenever there is strong coupling between the gaseous and solid components. Adopting for illustration Re and setting all other dimensionless constants to unity we find from equations (16)-(18) yr, g and that is needed for .

Then the typical lifetime of a massive disk with g (mass of a 200 km asteroid) in the runaway scenario (if the viscosity is low) is on the order of several sublimation timescales, i.e. about yr. According to equations (10) and (12) in this case the maximum for this , achieved right before the debris disk completely disappears, is max g s. These numbers agree with observational inferences (Farihi et al. 2009, 2010) quite well.

However, if positive feedback is not strong enough to drive runaway (e.g. because or higher Re), then and it takes 3 Myr for the same disk to be exhausted, as equation (11) demonstrates. In this case gas surface density at saturates at the low level g cm, see equation (8).

Equation (11) also shows that a low-mass disk with g (mass of a 20 km asteroid) is destroyed by PR drag very rapidly, within several thousand years (as long as the disk is optically thick to incoming stellar radiation, see R11).

Our model of debris disk evolution naturally explains coexisting gaseous and particulate debris disks reported in Melis et al. (2010). Systems with reported IR excesses but lacking emission lines of high-Z elements in gas phase may simply possess lower mass gaseous disks, which have not had enough time to develop by sublimation of solids at . Compositional variations between different WDs may also explain such systems.

Calculations presented in this work are rather simple and local in nature. We studied only one coupling mechanism — aerodynamic drag, while other possibilities may be available as well (e.g. induction interaction (Drell et al. 1965; Gurevich et al. 1978) between the MRI-generated B-field in the gaseous disk and the debris disk particles). Also, here we did not consider low surface density disks (we use only the thin plate approximation), non-trivial initial radial distribution of debris surface density, fate of angular momentum lost by debris disk and absorbed by gas disk, and so on. Future global models of coupled evolution of gaseous and particulate debris disks (Bochkarev & Rafikov, in preparation) will take these issues into account to provide a more accurate description of WD pollution with circumstellar high-Z material.

RRR is grateful to Xue-Ning Bai and Konstantin Bochkarev for useful discussions. The financial support of this work is provided by the Sloan Foundation, NASA via grant NNX08AH87G, and NSF via grant AST-0908269.

Footnotes

  1. affiliation: Department of Astrophysical Sciences, Princeton University, Ivy Lane, Princeton, NJ 08540; rrr@astro.princeton.edu
  2. affiliation: Sloan Fellow

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