CONSTRAINTS ON THE LIFETIMES OF DISKS RESULTING FROM TIDALLY DESTROYED ROCKY PLANETARY BODIES

The IRAC image photometry was performed on the individual BCD frames downloaded from the Spitzer archive and reduced using S18.7.0. These were corrected for array-location dependence as described in the IRAC Data Handbook.⁶ We used the point source extraction package apex multiframe within MOPEX (Makovoz et al. 2006) to perform point-spread function (PSF) fitting photometry on the individual BCDs. We do not apply a color correction because we use the isophotal wavelengths in our analysis. The magnitude of the correction is therefore negligible compared to our uncertainties. The pixel phase correction for channel 1 data is also minimal in comparison to the uncertainties and therefore was not applied. A minimum 5% uncertainty on the flux density was assigned to account for the systematic uncertainties in this method (Reach et al. 2005).

The Infrared Spectrometer (IRS; Houck et al. 2004) Peak-up Imaging was analyzed according to the method of Brinkworth et al. (2012). The sky background was found to be variable over the array (see Figure 1), so the individual BCDs were initially median-combined to make both a flat and sky, using 3σ rejection. This was then scaled to the overall median of the post-BCD mosaic and subtracted from each of the individual BCDs. To completely reduce the sky level to zero, the edges of the image were temporarily discarded and a median of the center of the image was found and removed. The sky can therefore be assumed to be zero and an infinite sky annulus is effectively used during photometry. The flat-fielded and median-subtracted BCDs were mosaicked using MOPEX with a dual-outlier rejection. The pixel scale for the mosaic was set to the default 18 pixel⁻¹. The photometry was performed with apphot within IRAF using a 3 pixel aperture radius, but no sky subtraction. The aperture correction to the calibration aperture sizes, as provided by the Spitzer Science Center, was performed and the fluxes were converted from MJy sr⁻¹ to mJy. As an estimate of the error, a series of other apertures were placed around the target on the mosaic and the standard deviation of fluxes was taken as the uncertainty. All measured Spitzer fluxes are listed in Table 3.

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Supplemental near-infrared photometry for most target stars was obtained on the 2011 March 23 with the William Herschel Telescope using the Long-Slit Intermediate Resolution Infrared Spectrograph (LIRIS; Manchado et al. 1998), and on 2011 August 10–12 with the New Technology Telescope using Son of Isaac (SOFI; Moorwood et al. 1998). Images taken in a nine-point dither pattern were obtained with the J, H, and K_s-band filters with typical total exposure times of 270 s in clear conditions; three standard star fields were observed in a similar manner for photometric zero-point calibration. These data were reduced in the standard manner, by subtracting a median sky from each image in the dithered stack, flat fielding (using sky flats), then averaging and recombining frames.

LIRIS suffers from what is known as a detector reset anomaly, which appears in certain frames as a discontinuous jump (in dark current) between the upper and the lower two quadrants. To remove this unwanted signal, after flat fielding and sky subtraction, the detector rows were collapsed into a median column (with real sources rejected) and subsequently subtracted from the entire two dimensional image. The resulting fully reduced frames exhibit smooth backgrounds, free of the anomalous gradient. Analogously, SOFI exhibits significant cross-talk between quadrants, which becomes apparent for bright sources. The IRAF script crosstalk was employed on each of the raw frames prior to processing as described above, effectively removing any unwanted artifacts.

Aperture photometry of standard stars and relatively bright targets was performed using r ≈ 4'' aperture radii and sky annuli of ranging between 5'' and 8''. For relatively faint targets or those with neighboring sources, smaller apertures were employed with corrections derived from several brighter stars within the same image field and filter. In a few cases, PSF-fitting photometry (i.e., daophot) was used as a second method in addition to photometry with small apertures. All data taken in the K_s-band filter were flux-calibrated using ARNICA (Hunt et al. 1998) K-band standard star photometry. The measured photometry can be found in Table 4.

The IRAC mosaicked images of GD 61, CBS 127, and HE 1349−2305 all reveal nearby background objects that have the potential to contaminate aperture photometry (see Figures 2 and 3 for the images of GD 61 and HE 1349−2305, respectively). The background object near GD 61 is separated by 19 at P.A. 25° in the images taken with LIRIS (epoch 2011.2), and is readily resolved in the H and K bands. The near-infrared brightness of this neighbor is (J, H, K) = (17.7,17.0,16.7) mag and never approaches the apparent magnitude of the white dwarf. Based on this fact and the colors of the neighbor, it must be a background object. This source contributes weakly in the 4.5 μm image, where it is well separated by apex, and is no longer seen at 7.9 μm. CBS 127 and HE 1349−2305 both have neighbors 28 distant (north and west, respectively); these are satisfactorily separated by PSF fitting with apex.

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Figures 4–7 illustrate the spectral energy distributions (SEDs) of the 15 metal-rich white dwarfs listed in Table 1. In addition to the supplementary near-infrared photometry described above, we include shorter wavelength photometry from a variety of literature and catalog sources (e.g., GALEX; Martin et al. 2005, SDSS DR7; Abazajian et al. 2009, DENIS; Epchtein et al. 1999, and CMC; Copenhagen University Obs. et al. 2006). In cases where independent JHK observations were not taken for this program, data from 2MASS (Skrutskie et al. 2006) and UKIDSS (Lawrence et al. 2007) were used. Where possible, a comparison of the IRAC photometry was made to data available at similar wavelengths from the Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010).

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For flux scaling of the spectral models in the figures, the most reliable photometry was used, which was primarily SDSS ugriz and the JHK data taken specifically for this program. Specifically, the GALEX fluxes were not used to constrain the white dwarf atmospheric models because (1) metal lines may significantly suppress these fluxes compared with a pure helium (or hydrogen) model atmosphere and (2) the interstellar reddening to these objects is not well known.

Table 1 lists the best available stellar parameters from the literature for each of the science targets. We used these parameters as an initial estimate for the white dwarf spectral models. A grid of synthetic white dwarf atmospheric spectra from Koester (2010) was interpolated to these temperature and surface gravity estimates. After comparing the fluxes from the models with the most reliable optical and near-infrared data, in some cases we found that a slightly different effective temperature provided a superior fit to the data (shown in the upper right corner of the relevant figure). These parameters thus used in the SED fits do not reflect an independent parameter determination, but rather our attempt to best constrain the infrared photospheric emission.

We model stars with an infrared excess as a combination of the white dwarf photosphere and an optically thick, geometrically thin disk with a temperature profile T_disk∝r^β, with β = −3/4 (Adams et al. 1988; Chiang & Goldreich 1997; Jura 2003). The ratio of stellar radius to distance sets the absolute scale of the white dwarf photospheric flux, and is fixed in our modeling based on the best available parameters. The free parameters in the disk model are the inner disk radius (R_in), the outer disk radius (R_out) and the disk inclination (i). A grid of disk models was calculated with inner (T_in) and outer disk temperatures (T_out; each corresponding to a value of R_in and R_out) ranging from 100 to 1800 K in steps of 50 K, and inclination ranging from 0 to 90° in steps of 5°. T_out was also fixed to be cooler than T_in. A least χ² method was used to fit the H-, K-, and IRAC-band infrared fluxes and provide an estimate of the uncertainties.

In Figure 5, the panels show a slice through the χ² cube at the best-fit solution. The upper panel of each pair displays a slice at the best-fitting T_out, whilst the best-fit inclination defines the slice in the lower panel. Regions of high χ² are shown as dark areas and the least χ² solution is marked as a red circle and its flux is displayed in Figure 4. Solid black lines show the 1σ contours around the minimum, while 2σ and 3σ contours are shown as dashed gray lines. The upper left of each lower panel is excluded by the condition T_in > T_out.

The flat disk models naturally possess a modest degree of uncertainty. The three free parameters (T_in, T_out, and i) are somewhat degenerate in how they combine to determine the viewed solid angle of the disk. Even in the case when longer wavelength data (such as 24 μm photometry) are available, this degeneracy is not broken (Jura et al. 2007). Broadly speaking, the inner disk edges are fairly well constrained by their 2.2 and 3.6 μm emission. The 4.5 to 7.9 μm flux, however, can be reproduced by a relatively wide temperature range and a higher inclination, or a narrower temperature range and a lower inclination. Importantly, the newly detected disks mimic the emission seen at more than one dozen dusty white dwarfs observed with Spitzer IRAC and MIPS (e.g., Farihi et al. 2009) whose outer disk radii are consistently within the Roche limit of d ≳ 1 km solid bodies (∼1 R_☉). This corresponds to an outer disk temperature near 500 K, which is a likely lower limit for T_out when interpreting the χ² surfaces. All together, the infrared emissions are precisely that expected for dust particles resulting from the tidal destruction of a large planetary body, whose constituent elements now rain onto and pollute the surfaces of the host stars.

HE 0110−5630. The available photometry for HE 0110−5630 consists of GALEX far- and near-UV, DENIS I-band, and J, H, and K measurements taken with SOFI. The photospheric emission is relatively well constrained by the J- and H-band fluxes and, therefore, so is the extrapolation to longer infrared wavelengths. The measured IRAC excess for HE 0110−5630 is consistent with warm circumstellar dust and is shown in Figure 4. The best-fit disk model has inner and outer temperatures of 1000 and 900 K, respectively, at an inclination of i = 60°. This corresponds to a narrow ring; however, as can be seen in the χ² panels, the solution is degenerate with respect to inner disk temperatures above 800 K, and inclination, and the outer temperature is not at all constrained. The χ² surfaces shown in Figure 5 show that a vast range in disk temperatures could provide a fit to the excess. We therefore cannot confidently estimate any of the disk parameters.

GD 61. From analysis of IRAC images and Peak-Up Imaging, this object was found to have an excess in Farihi et al. (2011). Here we include the JHK data taken with LIRIS to better constrain the stellar flux. These images resolve a nearby (background) source (Figure 2) that almost certainly contaminates the 2MASS photometry used in previous fits. From its spectral energy distribution and brightness, this object is not associated with the white dwarf. In the Spitzer IRAC images, the PSF of the white dwarf and the contaminant overlap slightly; however, the apex PSF-fitting photometry cleanly separates the two objects. We can therefore confirm the presence of a disk from the IRAC and IRS fluxes.

When performing χ² fitting of the infrared excess, we exclude the 15.6 μm flux due to a possible increase owing to silicate emission, and therefore the disk continuum emission would be overestimated. Such emission has been detected in all eight dusty white dwarfs observed spectroscopically with IRS (Reach et al. 2005; Jura et al. 2009). A disk model with inner- and outer-disk temperatures of 1450 and 800 K, respectively, and an inclination i = 85° is the best fit to the excess (Figures 4 and 5). The disk model temperature and inclination are again degenerate, and a large range of parameters would fit the infrared fluxes. We therefore find that the values reported previously (T_in = 1, 300 K, T_out = 1, 000 K and i = 79°; Farihi et al. 2011) are not inconsistent with ours, given the uncertainties in temperature and inclination.

HE 1349−2305. After deconvolving the white dwarf and neighboring background source (Section 3.1), a significant infrared excess is still found over the stellar model (Figure 4). The white dwarf photosphere is well constrained by short-wavelength and near-infrared photometry shown in Table 4. WISE 3.4 and 4.6 μm fluxes are also available for this object; however, the nearby source seen in Figure 3 is within the WISE beam width and therefore significantly contaminates the WISE fluxes. We therefore do not include these fluxes in the disk modeling. The excess is best fitted with a disk having T_in = 1, 700 K, T_out = 550 K and an inclination of i = 85°. The uncertainties are rather modest in this case.

NLTT 51844. This was one of two white dwarfs bright enough for us to attempt blue Peak-Up image using IRS. The 15.6 μm flux measurement is in excess over the white dwarf model by slightly more than 3σ, similar to the MIPS 24 μm photometric excesses detected at G238-44 and G180-57 (Farihi et al. 2009). Spitzer galaxy counts at these wavelengths are 6 × 10⁻⁴ galaxies per square arcsecond (Marleau et al. 2004) for the brightness needed to produce an apparent excess in these cases. Therefore, the probability that such a source falls within a radius of two IRS Blue Peak-Up FWHM (38) can be as high as 10%. Thus, the probability that three sources have excesses at roughly 20 μm (and not at shorter wavelengths) due to background objects, out of 30 stars surveyed at these longer wavelengths, can be as high as 65%.

While we cannot yet rule out a chance alignment in these cases, the astrometric position of NLTT 51844 in the IRS image mosaic coincides within 2'' of its position in the IRAC images based on the absolute Spitzer astrometry provided in the processed image files. Both NLTT 51844 and G180-57 are cool, helium-rich stars where metals can reside for Myr timescales, and hence their accretion history is unconstrained. In any case, dust that emits only at such long wavelengths must be sufficiently far from the white dwarf that it cannot be the immediate source of the atmospheric metals. Still, if the data for NLTT 51844 (as well as G238-44 and G180-57) are not spurious, we may be seeing an important clue to the nature of numerous polluted white dwarfs without obvious infrared excesses.

The remaining white dwarfs (Figures 6 and 7) all show no evidence for infrared excess emission. They are relatively well constrained in the optical and near-infrared and therefore we can extrapolate with some certainty to IRAC wavelengths. Below we provide notes on a few particular stars.

HE 0446−2531. This star is perhaps the most highly polluted white dwarf observed by Spitzer and yet does not have an infrared excess. If the analysis of Friedrich et al. (2000) is correct, this star contains 8 × 10²⁴ g of metals in its atmosphere—a mass approaching that of Pluto. The long diffusion timescale for heavy elements in this star allows for the possibility that a disk has been fully dissipated, but the very large mass involved suggests we are still within at most a few diffusion timescales of the accretion event. Two possibilities are: (1) accretion is ongoing from gaseous debris, or (2) accretion has ended but total mass involved was at least an order of magnitude higher. It is noteworthy that this star and HS 2253+8023 are both highly polluted (Friedrich et al. 2000) yet lack infrared excess, with HE 0446−2531 having nearly ten times more metal mass in its convection zone. However, Klein et al. (2011) found that HS 2253+8023 has an order of magnitude less calcium than originally reported by Friedrich et al. (2000), and therefore the same may be true of HE 0446−2531.

HE 0449−2554. A partly resolved extension is seen in the IRAC 7.9 μm image, and the combined source is offset slightly from the position of the white dwarf in the 4.5 μm image. While the SOFI JHK images of this star do not reveal any sources within a few arcseconds of the science target, the marginal, apparent excess seen in the SED (Figure 6) is probably caused by a background object. Better data are needed to rule out contamination in IRAC.

CBS 127. As mentioned above, a background object lies within a few arcseconds of the white dwarf, and could potentially contaminate moderate-size apertures used for IRAC photometry. We used PSF-fitting routines within apex to ensure both stars were photometrically disentangled. There is no infrared excess at this white dwarf.

HE 1350−1612 was observed in both the Cycle 5 and 6 programs and images were thus taken at 3.6, 4.5 (twice), and 7.9 μm. The average 4.5 μm flux is shown in Table 1 and Figure 7. The white dwarf was not detected in the 7.9 μm image, and thus a 3σ upper limit is given. Nevertheless, all infrared data are consistent with the predicted white dwarf photosphere.

PHL 131 is a 40,000 K white dwarf with interstellar calcium absorption, and hence not metal-rich. We do not analyze this star but its IRAC fluxes are reported in Table 3.

Of the 15 metal-polluted stars surveyed, three have an infrared excess consistent with circumstellar dust disk, and indicative of a remnant planetary system. Including the objects presented here, the total number of metal-polluted white dwarfs observed with Spitzer in Cycles 1 through 7 is 92. This includes the 52 objects from Table 4 of Farihi et al. (2009), 8 from Table 1 of Farihi et al. (2010b), 3 white dwarfs with gas disks from Gänsicke et al. (2006, 2007, 2008), 14 DBZ white dwarfs from Xu & Jura (2012), and the 15 objects studied here. Of these 92 objects, DAZ- and DBZ-type white dwarfs are represented in proportions of 39 and 53, respectively.

Our expanded study corroborates the previous finding (Kilic et al. 2008; Farihi et al. 2009, 2010b) that the detection of infrared excess is less frequent among DBZ white dwarfs than among their DAZ counterparts. 11 of 37 (30%) surveyed DAZ stars have circumstellar dust, whereas the same fraction for surveyed DBZ-type white dwarfs is only 7 of 52 (13%).⁷ Because one would not expect disks to preferentially form around stars based on their atmospheric properties, the difference must be due to the longer diffusion timescales (Paquette et al. 1986; Koester & Wilken 2006). We will return to this point in more detail below.

Figure 8 illustrates the time-averaged metal accretion rate versus cooling age for the 88 metal-rich white dwarfs now observed by Spitzer with published abundances (four of the Xu & Jura 2012 stars, including one with a detected infrared excess, do not have published abundances), updating Figure 10 from Farihi et al. (2010b). All accretion rates were calculated based on the accumulated metal abundances, using Equation 2 of Koester & Wilken (2006), as performed in Farihi et al. (2009). One difference between the method used in Farihi et al. (2009) and here is that in previous studies instantaneous and time-averaged accretion rates were calculated assuming the in-falling material either had solar calcium abundances (1: 43; Koester & Wilken 2006), or 1% solar (1: 109; i.e., metals only; Jura et al. 2007; Farihi et al. 2009) by mass fraction, based on the measured calcium abundance.

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While the latter approach is still likely to be broadly correct, recent progress in the field enables a more accurate estimation. There are now of order 10 metal-polluted white dwarfs with measured Mg, Si, Ca, and Fe abundances (Zuckerman et al. 2003, 2007, 2010, 2011), and a handful also with O abundances (Dufour et al. 2010; Klein et al. 2010; Vennes et al. 2010; Klein et al. 2011; Melis et al. 2011; Vennes et al. 2011). Based on these data, it is increasingly clear that the accreted material has a composition similar to that of rocky, terrestrial material of the solar system. Zuckerman et al. (2010) find that Ca represents, on average, close to 1 part in 60, by mass, of all the accreted heavy elements. For comparison, this ratio is 1 part in 62.5 for the Earth (Allègre et al. 1995). Based on these facts, Figure 8 and Table 1 employ accretion rates and convective envelope masses, assuming Ca is 1.6% (1/62.5) of the total mass. To convert between the accretion rates quoted in Farihi et al. (2009, 2010b) and those shown in Figure 8, one must multiply by a factor of 109/62.5 for DAZ white dwarfs and 43/62.5 for DBZ white dwarfs.

Compared to the same figures shown in previous (related) papers, the accretion rates for the DAZ white dwarfs are therefore slightly decreased, while their DBZ counterparts are slightly increased.

The mass of metals in the convective envelope of DBZ-type white dwarfs provides a lower limit on the total mass of the destroyed parent body. From the DAZ stars, we can infer the instantaneous accretion rates because one can safely assume accretion-diffusion equilibrium. There is no a priori reason that disks around hydrogen and helium-rich stars should be different (Jura et al. 2007). Therefore, assuming the inferred accretion rates of DAZ stars represent an accurate cross section over disk lifetimes, and similarly for the observed metal masses contained within DBZ stellar envelopes, we can obtain a typical disk lifetime by combining the average of these two observed quantities, i.e.,

$$\begin{equation} t_{\rm disk}\sim \frac{\langle M_z \rangle _{\rm DBZ,disk}}{\langle dM_z/dt \rangle _{\rm DAZ,disk}}. \end{equation} \tag{ 1 }$$

Figure 9 demonstrates that the mass of metals in the convective envelope of DBZ white dwarfs is not a strong function of temperature, and hence cooling age. Therefore, calculating the average convective envelope metal mass across cooling ages should be robust. In contrast, the convective envelope depth increases substantially in this temperature range for DAZ stars (Koester 2009), as can be seen by the large increase in convective envelope metal masses at longer cooling ages in Figure 9.

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Among 13 DAZ white dwarfs observed to have infrared excess with Spitzer, the average metal accretion rate is 9.7 × 10⁸ g s⁻¹. Among eight DBZ-type stars with dust detected by Spitzer⁸, the average metal content of the convection zone is 4.1 × 10²² g, and we thus estimate t_disk ≃ 1.3 × 10⁶ yr. However, Figure 8 shows that the accretion rates and convection zone metal content vary by orders of magnitude, and suggests that using the logarithmic mean and standard deviation, 〈log [dM_z/dt (g s⁻¹)]〉 = 8.8 ± 0.4 and 〈log [M_z (g)]〉 = 21.9 ± 1.1, may be a more appropriate choice. The resulting estimate for the disk lifetime is log [t_disk (yr)] = 5.6 ± 1.1. Because the metal masses in the DBZ stars are lower limits on the total parent body masses, it is reasonable to presume the same for the disk lifetimes.

At least 1% of all white dwarfs with cooling ages less than 0.5 Gyr have dusty disks (Farihi et al. 2009; Girven et al. 2011; Steele et al. 2011). Assuming that all white dwarfs host (remnants of) planetary systems and go through intermittent phases of debris, one would expect any given star to exhibit detectable infrared excess for 0.01 × 0.5 × 10⁹ yr =5 × 10⁶ yr, which is broadly consistent with our estimate above.

Different possible scenarios for the evolution of dust disks, and corresponding estimates for their lifetimes have been discussed in more detail by Jura (2008), Rafikov (2011a, 2011b), and Bochkarev & Rafikov (2011).

A main motivation of the work of Jura (2008) was to explain the existence of metal-polluted white dwarfs without infrared excess detection by the continuous accretion from a gaseous disk that is replenished by the repeated tidal destruction of multiple small asteroids. In this scenario, dust disks are associated with the disruption of massive asteroids, and Jura (2008) estimated the lifetimes of the gaseous and dust disks to be ∼0.5 × 10⁴ yr and ∼1.5 × 10⁵ yr, respectively, though he underlined the uncertainty of these estimates. Jura (2008) discussed his scenario in the context of a much smaller sample of metal-polluted white dwarfs, in which the majority of high dM/dt stars were displaying infrared excess identifying the presence of dusty disks. It is unlikely that the much larger accretion rates that are now known for some stars without infrared excess can be explained by multiple impacts of small asteroids.

Rafikov (2011a) showed that Poynting–Robertson drag can explain accretion rates of ∼10⁸ g s⁻¹, and estimated the lifetime of massive (∼10²² g) dust disks to several Myr. The higher accretion rates observed in a number of white dwarfs require the presence of gas that increases the viscosity, and Rafikov (2011b) shows that sublimation of the inner disk may be sufficient to lead to a runaway evolution which produces accretion rates of 10¹⁰–10¹¹ g s⁻¹, exhausting the disk in ∼10⁵ yr. Rafikov (2011b) explicitly excluded the generation of gas by multiple impacts of additional asteroids. Extending the analysis to the global evolution of the debris disks, Bochkarev & Rafikov (2011) confirm that massive disks have expected life times of several Myr if subject to Poynting–Robertson drag only, but that sublimation may substantially shorten these estimates.

Our disk lifetime estimate in the previous section is broadly compatible with all the estimates summarized above. However, our approach assumed that accretion rates do not change significantly as a function of disk age, and that the Spitzer sample is representative of the potential changes in the accretion rate. Zuckerman et al. (2010) suggested that accretion onto a white dwarf could proceed through three distinct phases: a build-up, a steady state, and a decline, where the steady state lifetime should be orders of magnitude longer than the beginning and ending episodes. If correct, and disk lifetimes are typically ≃ 10⁶ yr, then it will be very unlikely for an arbitrary white dwarf with circumstellar dust to be observed in anything other than a steady state phase.

Inspection of Figure 8 shows, as already mentioned in Section 4.1, that the fraction of DBZ with Spitzer infrared excess is substantially smaller than that of DAZ. The detection of dust seems to be associated with a minimum accretion rate, and if we only consider white dwarfs with dM/dt ⩾ 10⁸ g s⁻¹, we find that 10/21 (48%) of DAZ have infrared excess, compared to only 7/30 (23%) for the DBZ.⁹ This difference is, however, subject to small number statistics and future observations may show that both DAZ and DBZ are equally likely to show an infrared excess. If there is a real disparity, this fact is most likely related to the one significant difference between DBZ and DAZ—their diffusion timescales—and allows some independent gauge of the disk lifetime. Assuming a typical lifetime of the disks that is similar to or greater than the diffusion timescale of the DBZ stars (t_disk ≳ t_diff ≃ 10⁵ − 10⁶), we would expect a similar ratio of DAZ and DBZ stars exhibiting infrared excess. By contrast, if t_disk ≪ t_diff, it would be very unlikely to detect the disks around any given DBZ. Thus, the small but non-negligible fraction of DBZ with observed infrared excess leads us to conclude that the typical disk lifetime is somewhat shorter than the diffusion timescales in DBZ—or that the disk lifetimes vary by substantial amounts (which is in fact consistent with the large uncertainty in our logarithmic estimate of the life time). It appears unlikely that a large fraction of the highly polluted DBZ white dwarfs without infrared excess are currently accreting from a gaseous disk, as the lifetimes of these gaseous disks are short (Jura 2008).

Figure 8 also reveals a fairly well defined upper limit in dM/dt for DAZ white dwarfs. Based on our updated calculation of the accretion rates, there are no DAZ white dwarfs observed with Spitzer with accretion rates above 2 × 10⁹ g s⁻¹. The most extremely metal-polluted DAZ white dwarf star currently known, GALEX 1931+0117¹⁰ (Vennes et al. 2010), still only accretes at a rate of 3–4 × 10⁹ g s⁻¹ (Melis et al. 2011; Vennes et al. 2011). In contrast, there are several DBZ white dwarfs with metal accretion rates one to two orders of magnitude larger than observed in any DAZ, and substantially exceeding the rates that can be explained by Poynting–Robertson drag (Rafikov 2011a, 2011b). For instance, HE 0446–2531, HE 0449–2554, and HE 1350–162 all have accreted several 10²³ g to nearly 10²⁵ g within the last diffusion timescale, i.e., ≃ 10⁶ yr (or correspondingly larger amounts of material if the accretion ended several diffusion timescales ago). This is comparable to the extremely metal-polluted DZ white dwarf, SDSS J0956+5912, which has at least 1.5 × 10²³ g of metals in its atmosphere (Koester et al. 2011). Assuming the planetary bodies and physical mechanisms do not depend on the type of white dwarf, this substantial difference in the maximum accretion rates found for DAZ and DBZ white dwarfs forces us to conclude that either the calculated metal accretion rates for DBZ white dwarfs suffer from systematic errors, or the extremely high metal accretion rates found among DBZ white dwarfs simply have not been seen yet in DAZ white dwarfs.

One may speculate that both DAZ and DBZ white dwarfs can undergo a short-lived phase of very high rate accretion. Whereas the long diffusion timescales of DBZ provide an efficient "memory" to such events, the opposite is true for the DAZ, where the large amount of metals would be cleared out of the atmosphere within days to at most years—reducing the probability of witnessing such an event. A number of possible scenarios that would lead to such phases are conceivable. Secondary impacts of asteroids on an existing massive dust disk could generate sufficient amounts of gas to lead to runaway accretion (Jura 2008). Alternatively, a collisional cascade in the initially highly eccentric disk may generate sufficient amounts of gas to cause runaway accretion, effectively preventing the formation of a longer lived disk. In either scenario, the perturbed debris disks are expected to evolve on relatively short timescales, and Gänsicke et al. (2008) have shown that the structure of the gaseous disk in SDSS J084539.17+225728.0 changes on timescales of a few years. An initial spike in the accretion rate could also be related to the direct accretion of a substantial fraction of the disrupted asteroid, with the possible subsequent formation of a debris disk feeding the white dwarf at a lower rate. Another alternative comes from the fact that asteroid accretion events are, by definition, produced from eccentric orbits. One can therefore envisage that a fraction of disks are therefore born in an unstable configuration. The disk would, similarly to the above scenario, have many disk–disk interactions, producing a significant amount of gas. Therefore, the disk could accrete at a very high rate, leaving a heavily polluted white dwarf without a dust disk.