INFRARED SIGNATURES OF DISRUPTED MINOR PLANETS AT WHITE DWARFS

Figure 2 displays the Spitzer photometry for GD 16, which reveals a prominent mid-infrared excess indicative of T ∼ 1000 K circumstellar dust. The 2MASS data on this star appear to indicate an infrared excess at H and K bands, but it is likely that these data are unreliable; independent near-infrared photometry at high S/N (J. Farihi et al. 2009, in preparation) indicate an excess only at K band, as seen in Figure 2. No published optical photoelectric photometry exists, but photographic data indicate V ≈ 15.5 mag from both USBO-B1 and the original discovery paper (Monet et al. 2003; Giclas et al. 1965); this V-band flux is plotted and agrees well with the measured near-infrared fluxes for a white dwarf of the appropriate temperature.

A detailed optical spectral analysis of GD 16 by Koester et al. (2005b) yielded atmospheric parameters T_eff = 11, 500 K, [H/He] = −2.9, and M_V = 12.05 mag, with the assumption of log g = 8.0. Using the measured J = 15.55 mag together with the model-predicted V − J = 0.09 color for the relevant effective temperature, surface gravity, and helium-rich composition, the white dwarf is expected to have M_J = 12.14 mag and hence a nominal photometric distance of d = 48 pc (Bergeron et al. 1995a, 1995b). Figure 9 shows a fit to the thermal dust emission using the flat ring model of Jura (2003). This model does an excellent job of reproducing all infrared data from 2.2 μm onward. For an 11,500 K star, Figure 9 inner and outer dust temperatures correspond to 12 and 30 stellar radii, or 0.15 and 0.39 R_☉, respectively (Chiang & Goldreich 1997). The fractional disk luminosity, τ = L_IR/L = 0.02, is about 2/3 of the disks at G29-38, GD 362, and GD 56 (Farihi et al. 2008b; Jura et al. 2007a).

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Spitzer photometry for PG 1457−086 is shown in Figure 5 and reveals flux excess just over 3σ at both 3.6 and 4.5 μm, relative to the expected photospheric flux. The 2MASS photometry for this white dwarf at H and K bands has large errors, and independent near-infrared photometry at high S/N (J. Farihi et al. 2009, in preparation) indicates a probable, slight excess at K band. Overall, the infrared excess is mild but indicative of very warm, T ∼ 1500 K emission. Koester et al. (2005a) and Liebert et al. (2005) derive T_eff = 20, 400 K and 21,500 K, respectively, for PG 1457−086, while both find a surface gravity log g ≈ 8.0. The 20,400 K value was employed for the figures, because it is more conservative, predicting slightly less infrared excess than 21,500 K and because the model of Koester et al. (2005a) accounts for the measured high-calcium abundance, while the Liebert et al. (2005) model does not.

Figure 10 shows a narrow ring model fitted to the infrared excess. To match the relatively low fractional luminosity of the disk, τ = 0.0006, the model ring is narrow in radial extent and highly inclined. For a 20,400 K white dwarf, the inner and outer dust temperatures, in an optically thick disk, correspond to 19 and 21 stellar radii, or 0.25 and 0.28 R_☉, respectively (Chiang & Goldreich 1997).

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Due to the shape of the mild infrared excess at PG 1457−086, it is the only metal-rich white dwarf Spitzer target where a substellar companion might be considered as viable. If the measured J = 16.07 mag is solely due to the white dwarf photosphere, then the difference between the measured and model-predicted K-band flux for such a 20,000 K hydrogen white dwarf is ΔK = 0.24 mag or an excess of K = 17.8 mag, which corresponds to M_K = 12.6 mag at the photometric distance of 110 pc (Liebert et al. 2005). If a self-luminous, substellar companion of this K-band brightness were orbiting PG 1457−086, it would have an effective temperature near 1500 K and a spectral type around L7 (Vrba et al. 2004; Dahn et al. 2002). It is worth noting that the white dwarf GD 1400 has an unresolved L7 brown dwarf companion, and the ratio of the 3.6 to 2.2 μm excess there is 1.3 (Farihi et al. 2005b), whereas for PG 1457−086 this value is 0.7, although these ratios are consistent within the uncertainties.

As mentioned in the Appendix, winds from nearby M dwarf companions can, and occasionally do, pollute the atmospheres of white dwarfs. But for PG 1457−086, an M dwarf companion is ruled out by the absence of significant near-infrared excess (Farihi et al. 2005a). As described in the Appendix, there is no known connection between brown dwarf companions and the presence of metal pollution in white dwarf photospheres. Therefore, for PG 1457−086 to be metal-rich and to possess an L dwarf companion would require, simultaneously, two low probability events. For this reason, it is highly unlikely that the excess infrared emission at PG 1457−086 is due to a brown dwarf companion.

von Hippel et al. (2007) reported the discovery of Spitzer IRAC 4.5 and 7.9 μm excess due to warm dust at LTT 8452. Figure 7 shows the SED of this metal-rich white dwarf, now plotted with its IRAC 3.6, 5.7, and MIPS 24 μm data, which better constrain the inner and outer temperatures of the disk. Figure 11 shows a flat ring model fitted to the mid-infrared emission of LTT 8452, with inner edge temperature T_in = 1000 K, outer temperature T_out = 600 K, and a modest inclination angle of i = 53°, whereas von Hippel et al. (2007) derived a temperature range of 900–550 K and i = 80°. From the flat disk model, the fractional infrared luminosity of LTT 8452 is τ = 0.008.

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Figure 4 includes all available Spitzer data for G238-44, one of the most highly contaminated, nearest, and brightest DAZ white dwarfs (Holberg et al. 1997). The aperture laid down for MIPS 24 μm photometry was extrapolated to the expected location of the star at the epoch 2007.3 observation, using the Hipparcos measured J2000 position and proper motion (Perryman et al. 1997). The expected position on the MIPS 24 μm array is in excellent agreement with the measured position of G238-44 in the epoch 2004.9 IRAC images, after accounting for proper motion, and coincides with a faint MIPS 24 μm source. However, the potential for source confusion is high.

If associated with the white dwarf, the apparent 24 μm excess at the location of G238-44 would be just slightly greater than 0.04 mJy (2.0σ). Based on Spitzer MIPS 24 μm source counts from deep imaging, the expected number of background galaxies of brightness 0.04 ± 0.02 mJy is around 8000 deg⁻² (Marleau et al. 2004). Hence, within an area of diameter equal to one full width at half-maximum (FWHM) intensity at this wavelength (about 2.3 pixels or approximately 25 arcsec²), there should be 0.015 galaxies of the right brightness to contaminate the MIPS aperture. Using a binomial probability, the chance of finding at least one in 20 metal-rich white dwarf targets confused with a background source in this manner is then 26%; therefore the 24 μm flux may not originate from G238-44.

Further, if the potential contamination area is widened to encompass two FWHM, the probability of source confusion is then 71%, although such a large area is perhaps overly conservative. The expected position of G238-44 is offset from the centroid position of the MIPS source by 0.35 pixels or 09, the error of which is hard to estimate due to the relatively low S/N.

Turning to the white dwarf G180-57 (Figure 4), it has a 0.06 mJy (or 1.5σ excess) 24 μm source at its expected position on the array. All the previous arguments and caveats apply, and the chance that at least two in 20 MIPS targets are contaminated in this way is still relatively high, somewhere between 4% and 34%.

In any event, cold dust alone at this 24 μm luminosity level cannot explain the source of external metals in these white dwarfs. Blackbody dust which reveals itself at 24 μm but not at 8 μm would be around 200 K or cooler, and located too far away (at or beyond 10 and 70 R_☉ for G180-57 and G238-44m, respectively) to be the source of accreted metals in the photospheres of either white dwarf. Average-sized grains of 10 μm with density 2.5 g cm⁻³ at these distances would imply (minimum) dust masses around 10¹⁵–10¹⁷ g for τ = 10⁻⁵, appropriate for these detections, if real (Farihi et al. 2008b). This mass is insufficient to sustain an accretion rate of 3 × 10⁸ g s⁻¹ for more than 10 years at G238-44 (Koester & Wilken 2006).

The MIPS observations of vMa 2, plotted in Figure 1, suggest a 24 μm flux significantly lower than expected for a simple Rayleigh–Jeans extrapolation from its IRAC fluxes. Rather than an excess, the SED of vMa 2 appears deficient at 24 μm, at the 4σ level; the measured 24 μm flux is 0.11 ± 0.03 mJy, while the predicted flux is 0.23 mJy (Figure 1). Blackbody models are essentially no different than pure helium atmosphere white dwarf models at these long wavelengths (Wolff et al. 2002, see their Figure 1), hence the apparent deficit rests on the validity of the relatively low S/N photometry rather than model predictions. Still, this intriguing possibility requires confirmation with superior data, and if confirmed, vMa 2 would become by far the highest temperature white dwarf to display significant infrared flux suppression due to collision-induced absorption (B. Hansen 2007, private communication; Farihi 2005).

0108 + 277. NLTT 3915 belongs to an optical pair of stars separated by roughly 3'' on the sky in a 1995.7 POSS II plate scan. The northeast star has proper motion μ = 022 yr⁻¹ at θ = 222° (Lepine & Shara 2005) which can be readily seen between the POSS I and POSS II epochs. The IRAC images reveal two overlapping sources with a separation of 24 as determined by ${\sf daophot}$ at all four wavelengths. Using this task to deconvolve the pair of stars photometrically reveals that the object to the southwest is likely a background red dwarf, based on its 2MASS and IRAC photometry, while the northeast star has colors consistent with a very cool white dwarf. Kawka & Vennes (2006) identified this star as a 5200 K DAZ white dwarf whose spectrum appears to exhibit sodium but not calcium—an anomalous combination for a metal-rich degenerate. Keck/HIRES observations confirm this object is a white dwarf, but with a cool DA spectrum and no metal features (C. Melis 2008, private communication). The data tables include fluxes, and the Appendix gives limits on substellar companions for this target, but it is excluded from analyses for metal-rich white dwarfs.

0208 + 396. G74-7 is the prototype DAZ star. IRAC observations of this white dwarf, previously reported in Debes et al. (2007) were taken during a solar proton event, and individual frames are plagued by legion cosmic rays, making the reduction and photometric analysis difficult, especially at the longer wavelengths. Only the 4.5 and 7.9 μm fluxes are plotted and tabled; some data were irretrievably problematic.

1202−232. This bright and nearby white dwarf is located about 8'' away from a luminous infrared galaxy in the IRAC 8 μm images, and the white dwarf location is swamped by light from the galaxy at MIPS 24 μm, where the diffraction limit is 6''. Hence, the upper limit on the white dwarf 24 μm flux is about a factor of 5 worse than for a typical star.

1334 + 039. LHS 46 has been (mistakenly) identified as spectral-type DZ in several papers over the years (Greenstein 1984; see discussion in Liebert 1977). It was first (correctly) reclassified as a DC star by Sion et al. (1990), and modern, high resolution spectroscopy confirms the white dwarf is featureless in the region of interest (Zuckerman & Reid 1998). Unfortunately, some relatively current literature (Holberg et al. 2008, 2002; McCook & Sion 1999) still contains the incorrect spectral type for this star, and it was included in the Spitzer observations. The data tables include fluxes, and the Appendix gives limits on substellar companions for this target, but it is excluded from analyses for metal-rich white dwarfs.

The fraction of white dwarfs cooler than about 20,000 K that display photospheric metals depends on the stellar effective temperature and on whether its photospheric opacity is dominated by hydrogen or helium. For DA white dwarfs, it is easier to detect metals in cooler stars and, in a survey that focused primarily on DA white dwarfs cooler than 10,000 K, Zuckerman et al. (2003) found that of order 25% displayed a calcium K line. In the extensive SPY survey, more focused on warmer white dwarfs with T_eff ≳ 10, 000 K, Koester et al. (2005a) found that only 5% show a calcium K line. To produce a detectable optical wavelength line at such high temperatures (and high opacities) typically requires a greater fractional metal abundance than at low temperatures. Because extensive ground and, especially, Spitzer surveys have failed to reveal infrared excess emission at any single white dwarf that lacks photospheric metals (Farihi et al. 2008b; Mullally et al. 2007; Hoard et al. 2007; Hansen et al. 2006; Farihi et al. 2005a), one can regard the above percentages as upper limits to the fraction of DA white dwarfs that possess dusty disks—at least until more sensitive metal abundance measurements can be achieved. Perhaps more instructive is the number DA stars without metals observed with Spitzer; 121 such targets are reported between Mullally et al. (2007) and Farihi et al. (2008a), none of which show evidence for circumstellar dust, an upper limit of 0.8%.

Lower limits to the fraction of DA white dwarfs with dust disks can be estimated in the following way. Consider four large surveys of white dwarfs: the Palomar–Green Survey (347 DA stars; Liebert et al. 2005), the Supernova Progenitor Survey (478 DA stars; Koester et al. 2005a); the 371 white dwarfs from Farihi et al. (2005a), and the 1321 nonbinary stars present in both the 2MASS (Hoard et al. 2007) and McCook & Sion (1999) catalogs. The presently known frequency of dust disks for white dwarfs in these four surveys, all of which tend to find warmer white dwarfs (due to luminosity), are 1.4, 1.5, 1.1, and 0.8%, respectively. Thus, because not all stars in these surveys have been searched for dust disks, one can say that, at a minimum, 1% of all warm white dwarfs have dust disks.

Spitzer surveys of cool metal-rich white dwarfs have now targeted 53 stars at 3–8μm using IRAC (a few at 4.5 and 7.9 μm only, but most at all wavelengths), while 31 of these targets have also been observed at 24 μm with MIPS (see Table 4). Including the unlikely exceptions discussed above, there does not exist a clear-cut case of MIPS 24 μm detection at any white dwarf without a simultaneous and higher IRAC flux—a telling result by itself. Hence, targets observed with IRAC only should accurately reflect the frequency of dusty degenerates.

Of the 53 contaminated white dwarfs surveyed specifically for circumstellar disks, 21% have mid-infrared data consistent with warm dust, regardless of atmospheric composition. With more data, this estimate is somewhat larger than the result of Kilic et al. (2008) that at least 14% of polluted white dwarfs have an infrared excess. As shown in Figure 12, the likelihood of a white dwarf displaying an infrared excess is strongly correlated with effective temperature and/or its measured calcium pollution. Only two of 34 stars with T_eff ⩽ 10, 000 K have an excess, while that fraction is nine of 19 stars with T_eff > 10, 000 K. Alternatively, nine (or 10) of 17 stars with [Ca/H(e)] ⩾−8.0 possess an excess, whereas that fraction is only one (or two) of 38 stars with [Ca/H(e)] <−8.0.

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4.1.1. The Role of Cooling Age

Cooling ages for the white dwarfs in Table 4 were calculated with the models of P. Bergeron (2002, private communication; Bergeron et al. 1995a, 1995b). The white dwarf masses used as input for the cooling ages were taken from the literature (see Table 4 references), or log g = 8.0 was assumed for those stars with no estimates available. Because cooling age is sensitive to mass and within the literature discrepancies exist among white dwarf mass estimates, the values here should not be considered authoritative, but preference was given to values based on trigonometric parallaxes (Holberg et al. 2008) and those with multiple, independent, corroborating determinations. Still, it should be the case that a typical uncertainty in cooling age is between 10% and 20% due to an error in white dwarf mass.

Figure 12 indicates that the most metal-polluted white dwarfs are the warmest, and Figure 13 illustrates the same phenomenon, but now cast in terms of white dwarf cooling age. In the context of a model of tidal destruction of minor planets, as described below, it is plausible that younger (i.e., shorter cooling time) white dwarfs would be the most polluted. As indicated in the figure and tables, circumstellar dust disks are found at: eight of 17 stars (47%) with t_cool < 0.5 Gyr, two of 12 stars (17%) with 0.5 Gyr <t_cool < 1.0 Gyr, and only one of 24 (4%) stars with t_cool > 1.0 Gyr. The three dusty white dwarfs with cooling ages beyond 0.5 Gyr are: GD 362 and LTT 8452, both with cooling ages near 0.8 Gyr; and G166-58 with cooling age 1.29 Gyr.

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To estimate the percentage of white dwarfs with cooling ages less than 0.5 Gyr that might have dusty disks, one can consider each of the four large surveys mentioned in the previous section. For example, the SPY survey contains 14 metal-contaminated DA white dwarfs with likely cooling ages less than 0.5 Gyr (T_eff ≳ 11, 000 K). Of these 14, eight have been observed with IRAC and six have dust disks, while the remaining six, unobserved stars are prime candidates for dust disks (see Section 4.2). The SPY sample contains approximately 400 DA stars in this range of cooling ages and therefore, based on these data, it is likely that between 2% and 3% of white dwarfs with t_cool < 0.5 Gyr have detectable dust disks. This estimate is also consistent with the 72 DB white dwarfs (all with T_eff ≳ 11, 000 K) observed with SPY (Voss et al. 2007; Koester et al. 2005b), two of which are metal-contaminated and possess dust disks.

In a similar manner, the SPY survey contains 10 metal-contaminated DA white dwarfs with likely cooling ages between 0.5 and 1.5 Gyr (7000 K ≲ T_eff ≲ 11, 000 K). Of these 10, eight have been observed with IRAC and one has a dust disk, while the remaining two, unobserved stars both have [Ca/H] > − 8.0, and hence one of them may have a disk. The SPY sample contains around 70 DA stars in this range of cooling ages, and therefore these data suggest between 1% and 2% white dwarfs with 0.5 Gyr < t_cool < 1.5 Gyr may have detectable dust disks. However, the smaller number statistics for these cooler white dwarfs make this estimate somewhat uncertain.

Assuming only metal-bearing white dwarfs can possess warm dust disks, the Spitzer observations suggest similar percentages. The fraction of younger, t_cool < 0.5 Gyr white dwarfs with dust disks can be estimated by observing the fraction of these with IRAC excess is 0.47, and the fraction of SPY DA white dwarfs with metals in this cooling age range is 0.05, leading to a frequency between 2% and 3%, consistent with the above estimate. For somewhat older white dwarfs where 0.5 Gyr <t_cool < 1.5 Gyr, the same fractions are 0.10 with IRAC excess, and 0.13 with metal lines among SPY DA targets, yielding a frequency of 1%. For white dwarfs with t_cool > 1.5 Gyr, there are not enough stars in appropriate age bins to make meaningful estimates.

Holberg et al. (2008) have compiled a nearly (estimated at 80%) complete catalog of white dwarfs within 20 pc of the Sun. In this local volume, there are 24 white dwarfs with 10,000 K <T_eff ≲ 20, 000 K, and at least one white dwarf (G29-38), i.e., 4%, has an infrared excess. If the true percentage of warm dusty white dwarfs is 2.5%, then out to 50 pc there should be nine such stars with infrared excess. Currently, only GD 16 and GD 133 meet these criteria; either this expectation is incorrect or several white dwarfs within 50 pc of the Sun with an infrared excess remain to be discovered.

4.1.2. Implications for Disk Lifetimes and Pollution Events

It has been previously argued that DAZ white dwarfs with the highest calcium abundances also have an infrared excess and, thus, a circumstellar disk (Kilic et al. 2006). This argument has been slightly recast (Jura et al. 2007b; Jura 2008) to suggest that those DAZ white dwarfs with the highest metal-accretion rates are the stars with infrared excess. However, put into this proper context, both DAZ and DBZ white dwarfs should exhibit this correlation, if correct. This connection between the disk frequency and metal-accretion rate is now re-evaluated for all 53 metal-polluted white dwarfs observed with the IRAC camera.

Assuming accretion–diffusion equilibrium (i.e., a steady state), mass-accretion rates were calculated using Equation 2 of Koester & Wilken (2006) with the best available photospheric calcium abundance determinations, together with the solar calcium abundance relative to either hydrogen or helium, as appropriate. Settling times for various metals are usually within a factor of 2 (Koester & Wilken 2006; Dupuis et al. 1993), and calcium is employed here because it is the best studied element. Where available for DAZ stars, convective envelope masses and calcium diffusion timescales were taken from Table 3 of Koester & Wilken (2006). Otherwise log g = 8 was assumed and diffusion timescales were taken directly from their Table 2, while convective envelope masses were interpolated using their Table 3 values for stars of similar effective temperature and surface gravity (log g = 8 was also assumed for G166-58 for reasons discussed in Farihi et al. 2008b). For DBZ stars, convective envelope masses were read from Figure 1 of Paquette et al. (1986), and calcium diffusion times were interpolated using their Table 2 values; all assuming M = 0.6 M_☉.

In actuality, steady state accretion is likely for the warmer DAZ stars, but much less so for the cooler DAZ and DBZ stars, which have relatively long metal dwell times. Based on this work and the results of Kilic et al. (2008), it is likely that disks at white dwarfs typically dissipate within 10⁴–10⁵ yr, a timescale at which photospheric metals persist in the cooler DAZ and DBZ stars. Hence, the calculated accretion rates listed in Table 4 should be considered time-averaged over a single diffusion timescale, and may not accurately describe stars where detectable metals may dwell beyond 10³ yr. An additional factor of 1/100 was introduced to "correct" the Koester & Wilken (2006) Equation 2 rates to reflect that only accreted heavy elements are of interest, this factor being the typical dust-to-gas ratio in the interstellar medium. Figure 14 plots these accretion rates for all Spitzer observed stars versus effective temperature. Figure 15 plots the same metal-accretion rates versus white dwarf cooling age.

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The results show that the implied metal-accretion rates are quite similar between the two atmospheric varieties of metal-rich white dwarf, strengthening the argument that their externally polluted photospheres are caused by a common phenomenon; namely circumstellar material. In fact, at accretion rates dM/dt≳ 3 × 10⁸ g s⁻¹, the fraction of metal-rich white dwarfs with circumstellar dust—regardless of atmospheric composition—is over 50%. If one associates accretion rate with circumstellar disk mass (a modest assumption), then this picture is consistent with white dwarf dust disks being strongly linked to tidal disruption events involving fairly large minor planets; while less massive disrupted asteroids give rise to more tenuous (possibly shorter-lived, possibly gaseous) disks, and modest accretion rates.

While the white dwarfs with an infrared excess are likely accreting from tidally disrupted minor planets, the origin of the pollution in high accretion rate white dwarfs without obvious evidence for a disk is uncertain. Jura (2008) has proposed that if multiple, small asteroids are tidally disrupted, their debris self-collides so the dust is efficiently destroyed, and matter then accretes onto the white dwarf from an undetected gaseous disk. Given that the typical timescale for collisions within a white dwarf dust disk is hours, with disk velocities at hundreds of km s⁻¹, it is also conceivable that a single, modest-sized planetesimal could grind itself down to gas-sized material via self-collisions (Farihi et al. 2008b). Another possibility is that the dwell time of the photospheric metals in the white dwarf is longer than the characteristic disk dissipation time (Kilic et al. 2008).

Two competing models to explain the metal-contaminated photospheres of white dwarfs involve interstellar accretion and circumstellar accretion (Sion et al. 1990; Alcock et al. 1986). The SEDs of the mid-infrared excess at GD 16 and PG 1457−086 are well explained by a circumstellar dust disk arising from a tidally disrupted asteroid or similar planetesimal, but are not consistent with interstellar accretion. The same argument applies to the other metal-rich white dwarfs with mid-infrared excess; simply stated, the compact size and olivine composition of the dust are at odds with expectations for disks formed through interstellar accretion (Jura et al. 2007a, 2007b; Reach et al. 2005).

With Equation 3 of Jura et al. (2007a) and following their method, predicted 24 μm fluxes were calculated for all 32 metal-rich white dwarfs observed with MIPS, assuming interstellar Bondi–Hoyle-type grain accretion followed by Poynting–Robertson drag onto the star. Figure 16 displays these predicted infrared fluxes from interstellar accretion versus the MIPS 24 μm upper limits for 25 white dwarfs; stars with fluxes consistent with circumstellar disks were excluded, as were targets with contaminated photometry. The plot shows that the predicted fluxes are typically more than an order of magnitude greater than the observed 3σ upper limits. With few possible exceptions, this simple interstellar accretion model cannot account for the observed infrared data.

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Furthermore, there is now a growing number of DB white dwarfs with evidence for external pollution via carbon deficiency, relative to metals such as iron (Desharnais et al. 2008; Zuckerman et al. 2007; Jura 2006; Wolff et al. 2002). They have accreted rocky material, independent of any evidence for or against circumstellar dust. Table 5 lists the seven known helium- and metal-rich white dwarfs with low carbon abundances or upper limits. Of these stars, only two of six observed by Spitzer have strong evidence in favor of circumstellar dust, GD 40 and GD 362. This raises the possibility that the remaining dust-poor and metal-rich stars are accreting gaseous heavy elements.

Figure 14 reveals that the DBZ HS 2253 + 803 has one of the highest implied (time-averaged) accretion rates among all polluted white dwarfs, yet has no dust disk as evidenced by the Spitzer photometry presented here. This white dwarf is spectacularly carbon-poor relative to its metal content, and is highly likely to have accreted rocky material (Jura 2006). In this particular case, its near pure helium nature gives it a likely diffusion timescale of 10⁶ yr, and hence a dissipated disk is quite possible; yet its atmospheric metal composition should nonetheless provide a measure of the extrasolar planetary material it has accreted.

The preponderance of the evidence firmly favors the interpretation that heavily metal-contaminated white dwarfs are currently accreting, or have in their recent history accreted, rocky planetesimal material in either dusty or gaseous form. The origin of the metals in white dwarfs at the low end of [Ca/H(e)] ratios and metal-accretion rates, such as a number of DAZs in the survey of Zuckerman et al. (2003) remains unclear.

The circumstellar disks around white dwarfs likely arise from the tidal disruption of minor planets, with inferred metal-accretion rates dM/dt≳ 3 × 10⁸ g s⁻¹. In the solar system, the dust production rate in the zodiacal cloud is near 10⁶ g s⁻¹ (Fixsen & Dwek 2002). Collisions among parent bodies result in dust production rates around main-sequence A-type stars in excess of 10¹⁰g s⁻¹ (Chen et al. 2006). Thus, the rate estimated for the erosion of minor planets around white dwarfs is within the range inferred for main-sequence stars. Because A-type stars evolve into white dwarfs, it appears there is an ample population of parent bodies among the main-sequence progenitors of white dwarfs to account for the observed pollutions.

The model in which an infrared excess around a white dwarf results from a tidally disrupted minor planet requires that the orbit of an asteroid is perturbed substantially (Debes & Sigurdsson 2002). It is plausible that a Jupiter-mass planet is such a perturber. Cumming et al. (2008) find that 10.5% of solar-type stars have gas giant planetary companions with periods between 2 and 2000 days. Clearly, many of these planets will be destroyed when the main-sequence star becomes a first ascent and then asymptotic giant (Farihi et al. 2008a), but half of these planets have periods longer than 1 year and may survive. At present, the known fraction of main-sequence stars with massive planets that could persist beyond the post-main-sequence evolution of their host is comparable to the estimated fraction of white dwarfs with an infrared excess. The upper mass limits for self-luminous companions to white dwarfs, as found by this and previous Spitzer studies, are consistent with the scenario that asteroid orbits are perturbed by a typical gas giant planet (Farihi et al. 2008a, 2008b).

In addition to circumstellar dust, IRAC observations are sensitive to unresolved substellar companions at white dwarfs, via photometric excess (Farihi et al. 2008a; Mullally et al. 2007; Debes et al. 2007; Hansen et al. 2006; Farihi et al. 2005b). Moreover, there exist a few metal-enriched white dwarfs which reside in close, detached, binaries with unevolved M dwarfs; in such systems, the source of the photospheric pollutants is thought to be the stellar wind of the main-sequence companion (Zuckerman et al. 2003). Therefore, it is reasonable to ask if previously unseen, close, substellar companions might be responsible for the heavy elements in some contaminated white dwarfs.

Externally polluted white dwarfs observed with IRAC at 4.5 μm, but without obvious photometric excess due to circumstellar dust, and not previously analyzed for substellar companions, were selected from this study as well as from Jura et al. (2007a) and Mullally et al. (2007), yielding a total of 23 targets. All data were analyzed independently as described in Section 2. Following the technique of Farihi et al. (2008b), the absolute magnitude of a substellar companion with a putative 3σ photometric excess (above the observed flux) at 4.5 μm was determined using their Equation 2. This magnitude was translated into a mass based on a calculated total (main sequence plus cooling) age for each white dwarf, utilizing appropriate substellar evolutionary models (I. Baraffe 2007, private communication; Baraffe et al. 2003).

Table 6 lists the main-sequence lifetime, t_ms, cooling age, t_cool, distance from Earth, absolute 4.5 μm magnitude brightness limit and corresponding companion upper mass limit, for each metal-rich white dwarf. Combined with previous results for 20 similar IRAC targets (16 from Farihi et al. 2008a; four from Debes et al. 2007), there are now 43 cool contaminated white dwarfs observed with IRAC and analyzed for substellar companions; none show evidence for closely orbiting, potentially polluting secondaries down to an average mass of 19 ± 8 M_J. The hypothesis that unseen, brown dwarf companions are the source of heavy elements in some white dwarf atmospheres is essentially ruled out by these observations.

Again, following the methodology of Farihi et al. (2008a), upper mass limits were established for spatially resolved (widely bound) T and sub-T dwarf companions for eight white dwarfs within d ≈ 20 pc. These upper mass limits (M_J, Table 7) are considerably higher (average of 60 M_J) than those for unresolved companions at the same stars, due to the criteria that they must be well detected at all four IRAC channels (Farihi et al. 2008a).