EXPLORATIONS BEYOND THE SNOW LINE: SPITZER/IRS SPECTRA OF DEBRIS DISKS AROUND SOLAR-TYPE STARS

We examined the shortest wavelength data (SL2 and the "bonus" order, SL3) using the same technique as described above. We adjusted the SL3 data to fit SL2 and then tied the short-wavelength end of SL2 to the photospheric model. The dispersion (1σ) around fits to the photospheric models is 1% in a photometric band defined between 6 and 6.5 μm and 2% in a photometric band defined between 7.5 and 8 μm. There was no evidence of any excess shortward of 8 μm above the 3σ level.

In performing the fitting, we found that there was a systematic offset between the Kurucz models and Spitzer spectra for stars later than K5. Figure 5 shows the fractional excess in the SL2/SL3 wavelength band relative to the Kurucz models pinned to the stellar emission at 5 μm, for four groups of spectral types: F, G, K0–K4, and K5 and later. F and G stars reproduce the Kurucz photospheres very clearly, while early K stars show small deviations (∼1%), and late K and early M stars show greater deviations (>3%). As reported by Bertone et al. (2004), both of the commonly used stellar atmosphere models, Kurucz and NextGen (Hauschildt et al. 1999a, 1999b), fail to accurately match later spectral-type stars. From our analysis here and from previous investigations which used MIPS 24 μm observations of nearby K and M stars (Beichman et al. 2006b; Gautier et al. 2007) and found redder K_s − [24] colors than predicted by theory for both Kurucz and NextGen, it appears that this deviation between model and actual spectra is most severe closer to the near-infrared. Examination of the longer wavelength emission for these later-type stars (IRS modules SL1 and LL1/2, and MIPS data when available) revealed no evidence for longer wavelength excess emission. Thus, we attribute this disagreement, resulting in a 5%–10% apparent excess, as due to problems with the photospheric models in the 2–10 μm portion of the spectrum, and not as real excess due to dust emission.

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While the focus of this paper is IRS spectra, for many of our sample stars there is corresponding MIPS photometry at both 24 and 70 μm. Most of these data have already been published; for consistency, we have re-reduced all of them with a uniform set of analysis parameters. Our analysis is similar to that previously described in Beichman et al. (2005a), Bryden et al. (2006), and Beichman et al. (2006b). At 24 μm, images are created from the raw data using software developed by the MIPS instrument team (Gordon et al. 2005), with image flats chosen as a function of scan mirror position to correct for dust spots and with individual frames normalized to remove large-scale gradients (Engelbracht et al. 2007). At 70 μm, images are also processed with the MIPS instrument team pipeline which includes corrections for time-dependent transients (Gordon et al. 2007). Aperture photometry is performed as in Beichman et al. (2005a) with aperture radii of 153 and 148, background annuli of 306–434 and 394–788, and aperture corrections of 1.15 and 1.79 at 24 and 70 μm, respectively. For three systems that are marginally resolved at 70 μm (HD 10647, HD 38858, and HD 115617; see Section 5.2.3), the small aperture fails to capture all of the extended emission; for these three cases the MIPS fluxes listed in Table 5 are based on model fits to each disk (G. Bryden et al. 2009, in preparation). While our procedure has changed little since Bryden et al. (2006) was published, note that improvements in the instrument calibration since then have increased the overall 70 μm flux conversion by 4%, from 15.8 to 16.5 mJy/arcsec²/MIPS_70_unit (MIPS_70_unit is an internally defined standard based on the ratio of the measured signal to that from the stimulator flash signal (Gordon et al. 2007). Overall, we find no qualitative disagreement between our results and those from earlier publications.

We detected IRS excess emission toward 16 stars. These excesses begin longward of ∼25 μm for 10 stars, and between ∼15 and 25 μm for the other six stars. Two of the excess detections are of borderline significance and are included because of the additional information of a MIPS 70 μm excess (see Section 4.2): HD 110897 and HD 117043, both with χ32 = 2.8. Out of the sample of 152 stars, these 16 stars correspond to a 30–34 μm excess detection rate of 10.5% ± 2.6%, which is consistent with the fraction of stars with excesses found in a previous IRS survey: 12% ± 5% (Beichman et al. 2006a). We must, however, correct these statistics for the sources that were not observed as part of this sample because they were claimed as part of other, earlier Spitzer programs. Comparing our initial selection of sources meeting our astrophysical criteria with early guaranteed time or legacy programs yields 51 additional stars, which we list in Table 6. With this correction, the success rate for long-wavelength IRS excesses is not 16/152 = 10.5% ± 2.6%, but 24/203 = 11.8% ± 2.4%, essentially the same as found in our earlier determination (Beichman et al. 2006a), but with much lower uncertainty.

HD 72905 from the FGK survey (Beichman et al. 2006a) presents an interesting example of the challenges in identifying a weak infrared excess, particularly around 8–14 μm where the stellar photosphere is bright. Using IRAC data from the FEPS program (Carpenter et al. 2008), we use our standard technique to fit Hipparcos visible photometry, partially saturated 2MASS observations at JHK_s, and IRAC 3.6 and 4.8 μm data to a Kurucz model for a 6000 K G0V star with [Fe/H] = −0.08. The resultant fit has a reduced χ² of 0.97. Pinning the SL2 data to a Kurucz photosphere using the 20 shortest wavelength SL2 points requires a ∼2% adjustment to the SSC pipeline data and reveals no fractional excess from 5 to 8 μm greater than 2%. A similar conclusion applies if we fit a solar photosphere (Rieke et al. 2008) to the IRAC 3.6 and 4.8 μm data. Extending the Kurucz photosphere to longer wavelengths yields a marginal excess of about 5% at IRAC 7.8 μm that carries through to IRS SL1 and MIPS 24 μm. The fractional excess has a significance at the ∼2σ level relative to the ∼2% uncertainties in the photospheric models. However, changing photospheric models makes the excess all but vanish. Fitting the Rieke et al. (2008) solar photosphere instead of the Kurucz model reduces the level of excess to 2% or less out to 25 μm (including MIPS 24). We conclude that we cannot claim any statistically significant excess at <25 μm. At longer wavelengths, the difference between photospheric models becomes less important, and the existence of a weak excess starting at λ>25 μm becomes evident.

None of our sample stars showed excesses in the short-wavelength 8.5–12 μm portion of the spectrum, giving a fractional incidence of <0.7% for these mature stars. Adding in stars with previous Spitzer observations, we find an overall excess detection rate of 2 stars (HD 69830 and HD 109085) out of 203 = 1.0% ± 0.7% for the 8.5–12 μm band. This confirms the rarity of detectable short-wavelength excesses compared with ones at longer wavelengths, as seen in Beichman et al. (2006a) and noted in earlier studies using the Infrared Astronomical Satellite (IRAS) and the Infrared Space Observatory (ISO).

The FEPS survey (Carpenter et al. 2008; Hillenbrand et al. 2008) used Spitzer to observe nearby sun-like stars with ages between 3 Myr and 3 Gyr. Using our criteria of >3σ above the photosphere (or >2σ with a known 70 μm excess), Hillenbrand et al. (2008) find excesses in the 30–34 μm band for 22 out of the sample of 328 stars, although not all stars in the sample have reported IRS spectra. Carpenter et al. (2008) measure excesses using colors rather than comparison with Kurucz models, and find 71 out of 314 stars (22.6% ± 2.7%) with excesses in the long-wavelength IRS band, and 2 out of 314 (0.6% ± 0.5%) in the short-wavelength IRS band. As these stars are on average younger than the stars in our sample, it is not surprising that there is a higher incidence of IRS-detected excesses.

Five stars in the sample were known to have planets as of May 2009: HD 4308 (Udry et al. 2006), HD 10647 (Mayor et al. 2003), HD 40307 (Mayor et al. 2009), HD 154345 (Wright et al. 2008), and HD 164922 (Butler et al. 2006). Of these five stars, only HD 10647 shows an excess at both 70 μm and IRS wavelengths. The other four planet-bearing systems have no detected excesses.

We have MIPS data from other programs for about half (78) of the sample stars, as noted in Table 4. Table 5 lists all of the stars in our sample with IRS and/or MIPS 70 μm excesses. Of the 16 stars with IRS excesses, 14 have excesses in both the 30–34 μm IRS band and the MIPS photometry at 70 μm; only HD 154577 has an IRS excess with no detectable MIPS excess (HD 190470 is in a particularly noisy field close to the galactic plane, so the error bars on the 70 μm flux are so large that nothing can be said about whether or not there is an excess). Including the stars with previous IRS observations (Table 6) gives 22 stars with IRS excesses, 20 of which also have strong or weak MIPS 70 μm excesses. HD 110897 was not originally considered to have an IRS excess because of a marginal χ32 value (2.8), but this can be considered a weak excess because of the additional information of a strong MIPS 70 μm excess. HD 117043 has a marginally significant IRS excess (χ32 = 2.8) and a marginally significant MIPS 70 μm excess (χ70 = 1.9), but because χ32 is close to 3, this star was also included as a weak excess detection at both wavelengths.

Out of the 78 stars with both MIPS 70 μm and IRS data, only three stars (HD 90089, HD 132254, and HD 160032) have excess MIPS 70 μm emission with no significant IRS excess. Hillenbrand et al. (2008) find a similar trend, with >80% of their stars with MIPS 70 μm excesses also possessing IRS 33 μm excesses, and no reported stars possessing an IRS excess with no corresponding MIPS 70 μm excess. This implies that there may be a lower limit to debris disk temperatures, with a corresponding upper limit on disk sizes. Kuiper Belt analogs appear to happen preferentially in regions with temperatures around 50 K, and not at lower temperatures.

All of the stars with MIPS 70 μm data also have MIPS 24 μm measurements. Only two stars have greater than 2σ 24 μm fractional excesses: HD 10647 has a 24 μm excess that agrees with its large IRS excess. HD 38392 has a large apparent 24 μm excess with no IRS excess. However, examination of the 2MASS and MIPS 24 μm image for this star shows a bright companion star, HD 38393 (K_s∼ 2.5 mag), about 15 away. Although the IRS slit does not cross the companion star, and therefore should not effect the spectrum, the uncertainty in the 24 μm photometry is inflated by the companion. Further, this star appears to be variable at the 5% level in a number of visible compilations (Hipparcos time series photometry and Nitschelm et al. 2000). An alternate explanation is the presence of an M dwarf companion. Such a companion could produce a 20% excess at 24 μm, and would be too faint to notice if the system's spectral type was measured using optical observations. Follow-up imaging using adaptive optics would be needed to test this hypothesis. Reinforcing the peculiarity of the MIPS 24 μm data point, the MIPS data do not show any 70 μm excess for HD 38392.

A useful metric for the limits on dust surrounding these stars is L_dust/L_⋆, which is related to the fractional flux limit of an excess relative to the Rayleigh–Jeans tail of the star's photosphere (Bryden et al. 2006; Beichman et al. 2006a):

$$\begin{equation} { {L_{\rm dust}} \over {L_*} } = { {F_{\rm dust}} \over {F_*} } { { e^{x_d}-1} \over {x_d} } \left({{T_d} \over {T_*}}\right) ^3, \end{equation} \tag{ 1 }$$

where F_dust = F_ν(Observed) − F_ν(Photosphere). At the peak of the blackbody curve, x_d ≡ hν/kT_d has a constant value of 3.91, corresponding to T_d = 367 K at 10 μm. At this wavelength, L_dust/L_⋆ = 3.5 × 10⁻³(T_*/5600 K)⁻³F_dust/F_⋆. At 30–34 μm, the corresponding equation is L_dust/L_⋆ = 1.3 × 10⁻⁴(T_*/5600 K)⁻³F_dust/F_⋆, assuming T_d = 115 K. (For comparison, the typical dust temperatures traced by the MIPS 24 and 70 μm data are 154 and 53 K, respectively.) In Table 4 and Figure 10, we evaluate L_dust/L_⋆ for each star using the appropriate effective temperature (listed in Table 1), luminosity (from our stellar photosphere models), and its measured fractional excess in each band, F_dust/F_*, or, in the case of an upper limit, 3σ_pop where σ_pop is the dispersion in fractional excess averaged over the whole sample (0.010 at 8.5–12 μm; 0.028 at 30–34 μm). This definition of L_dust/L_⋆ assumes that the emitting material is all at the location where the peak of the T_d blackbody matches the wavelength of observation such that for stars with excesses the given value of L_dust/L_⋆ is actually a minimum. More dust emission, and higher values of L_dust/L_⋆, would be required for material located substantially interior or exterior to this point.

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The 3σ limits on L_dust/L_⋆ at 8.5–12 μm and 30–34 μm have 2σ clipped average values of L_dust/L_⋆ = 11 ± 4 × 10⁻⁵ and 1.31 ± 0.49 × 10⁻⁵, respectively (Table 4). In comparison with our solar system, which has L_dust/L_⋆∼ 10⁻⁷ (Backman & Paresce 1993; Dermott et al. 2002), the IRS results set limits (3σ) on warm (360 K) dust peaking at 10 μm of ∼1000 times the level of dust emission in our solar system. For cooler dust (∼115 K) peaking at 30–34 μm, the 3σ limit corresponds to ∼100 times the nominal L_dust/L_⋆ of our zodiacal cloud. For objects with excesses in the IRS bands, we determine L_dust/L_⋆ explicitly by integrating over the data between 10 and 34 μm and using the models discussed below (Section 5.2.1) to extrapolate out to and beyond the MIPS 70 μm data point.

Figure 11 summarizes the rates of IR excess detection in IRS spectral surveys and compares them with MIPS photometric results. For two wavelengths in each instrument, the distribution of detection rates is shown as a function of the fractional dust flux (F_dust/F_⋆). Note that F_dust/F_⋆ can be easily translated to a fractional disk luminosity using Equation (1). It is clear from this figure that the dominant dust around solar-type stars tends to be colder than is optimal for detection at IRS wavelengths and generally exhibits higher F_dust/F_⋆ at longer wavelengths. Nevertheless, because we can detect excesses down to much smaller levels of F_dust/F_⋆ within the IRS spectra, the overall detection rate of IR excess for IRS at 32 μm is similar to that for MIPS at 70 μm. By comparison, IRS at 10 μm and MIPS at 24 μm have relatively few detections, but are both consistent with the overall trend from the other wavelengths. While it is difficult to extrapolate these distributions down to fainter values, the curves can be fit by log-normal distributions with median values of F_dust/F_⋆∼ 0.06 at 70 μm and F_dust/F_⋆∼ 0.003 at 32 μm. These fractional fluxes correspond to L_dust/L_⋆ ∼ 5 × 10⁻⁷ for a solar temperature star, consistent with estimates for our Kuiper Belt's emission (Stern 1996).

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While the individual spectra provide the best measure of the range of dust temperatures in each system (Section 5.1), Figure 11 provides a sense of the generalized disk characteristics. The separation between the 32 μm and 70 μm distributions in Figure 11, for example, can be translated to a representative dust temperature of ∼65 K. In reality, a range of temperatures are present and, as is found in Section 5.2.1, the dust in each system is often not well fit by a single emission temperature, but rather by a distribution. This is also apparent from the overall statistics, as evidenced by the inability of a single blackbody to fit the trends seen in Figure 11; the separation between the 24 and 70 μm curves is consistent with 75 K dust, while the separation between the 10 and 70 μm curves corresponds to dust temperatures >100 K. A similar trend is found by Hillenbrand et al. (2008), who find that >1/3 of their surveyed debris disks have evidence for multiple dust temperatures based on their colors at MIPS and IRS wavelengths.

The excesses found in this survey are in most cases weak and relatively featureless beyond a simple rise to longer wavelengths. A few objects are exceptional: HD 10647 stands out for having a very strong excess, L_dust/L_⋆ = 10^−3.9 rising up to 70 μm and continuing out to 160 μm (Tanner et al. 2009); this source also appears to be extended at 70 μm (G. Bryden et al., in preparation). HD 40136 and possibly HD 10647 show a bump around 20 μm which might be attributable to small grains. In addition to HD 10647, HD 38858 and HD 115617 both also show evidence for extended MIPS emission (G. Bryden et al. 2009, in preparation).

The IRS and MIPS excesses detected toward some of the 152 stars discussed here can be used to characterize the properties and spatial location of the emitting material. Unfortunately, even the simplest characterization cannot be unique given the wide variety of grain sizes and compositions as well as possible locations for these different species. The complexity of debris disks is evident as one attempts to model the most prominent debris disks for which high-quality IRS spectra and fully resolved maps are available (e.g., Vega; Su et al. 2005). In this section, we first apply a simple, single-component model that fits the majority of sources; we assume that uniform, large-grained (∼10 μm) dust is located in an annulus centered on the star. We then examine somewhat more sophisticated models for disks where additional complexity seems warranted.

5.2.1. Simple Dust Models

As a first step in analyzing these data, we fitted the IRS spectra and MIPS 70 μm photometry using a simple model of optically thin dust located within a single dust annulus centered around the star. As described in Beichman et al. (2006a), we calculated the power-law temperature profiles, T(r) = T₀(L/L_☉)^α(r/r₀)^β, for grains in radiative equilibrium with the central star. We use dust emissivities for 10 μm silicate grains (Draine & Lee 1984; Weingartner & Draine 2001), the minimal size suggested by the lack of significant features in most of the spectra. For the 10 μm silicate grains, we obtained the following numerical relationship: T(r) = 255 K(L/L_☉)^0.26(r/AU)^−0.49. These calculated coefficients and power-law constants closely follow analytical results (Backman & Paresce 1993). We then calculated the dust excess by integrating over the surface brightness of a disk between R₁ and R₂, with $F_\nu (\lambda)={ {2\pi } \over {D^2} } \int \tau _0(\lambda) (r/r_0)^{-p} B_\nu (T(r)) r dr$. The disk surface density distribution expected for grains dominated by Poynting–Robertson drag is roughly uniform with radius, i.e., p = 0 (Burns et al. 1979; Buitrago & Mediavilla 1985; Backman 2004). We examined a number of other cases with 0 < p < 1 that would reflect different dust dynamics, but did not find results that were substantially different from those for p = 0.

We fitted the excess emission from a single annulus to 83 data points longward of 21 μm (just longward of the last point used for flux normalization of the LL1 IRS module) for 10 stars, and to 160 data points longward of 14 μm (just longward of the last point used for the flux normalization of the LL2 IRS module) for nine stars, depending on whether there was any hint of an excess shortward of 21 μm. We included the MIPS 70 μm data, which were available for all 19 of the stars modeled. By varying τ₀, R₁, and R₂, we were able to minimize the reduced χ² to values between 0.6 and 1.2, except for HD 10647, which has a fit with a reduced χ² of 5.7, indicating a simple 10 μm dust grain model does not satisfactorily fit the infrared excess observed for this star (see Sections 5.2.2 and 5.2.3). Results of the model fitting are shown in Figure 12 and Table 7 and are discussed below.

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Table 7. Large Dust Models

Star	Excess	Spectral Type	R1 − R2 (AU)	T1 − T2 (K)	Reduced χ2	Optical Depth τ (10 μm)	Mgraina (M⊕)	(M⊕ < 10 km)	Ldust/L⋆b (10−7)
HD 870	IRS and MIPS 70 μm	K0V	12 − 12	58 − 57	0.8c	1.5 × 10−3	7.8 × 10−6	0.2	245 ± 91
HD 1461	IRS and MIPS 70 μm	G0V	19 − 26	62 − 54	0.9c	2.7 × 10−4	3.4 × 10−5	1.06	343 ± 45
HD 10647	IRS and MIPS 70 μm	F9V	16 − 29	69 − 52	5.7d	10.0 × 10−4	2.4 × 10−4	7.7	2627 ± 118e
HD 38858	IRS and MIPS 70 μm	G4V	12 − 29	69 − 44	1.0c	1.7 × 10−4	5.3 × 10−5	1.7	677 ± 141
HD 40136	IRS and MIPS 70 μm	F1V	1 − 16	353 − 92	0.8d	1.1 × 10−5	1.2 × 10−6	0.039	62 ± 14
HD 45184	IRS and MIPS 70 μm	G2IV	11 − 27	77 − 49	1.2d	2.0 × 10−4	5.3 × 10−5	1.68	810 ± 89
HD 76653	IRS and weak MIPS 70 μm	F6V	16 − 18	77 − 73	0.9c	4.7 × 10−4	1.0 × 10−5	0.33	164 ± 47
HD 90089	IRS limit and MIPS 70 μm	F2V	>15	<70	0.8c	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	167 ± 21
HD 110897	Weak IRS and MIPS 70 μm	G0V	17 − 31	67 − 49	0.9c	5.5 × 10−5	1.6 × 10−5	0.50	187 ± 34
HD 115617	IRS and MIPS 70 μm	G5V	4 − 25	120 − 47	0.7d	3.1 × 10−5	8.1 × 10−6	0.26	199 ± 39f
HD 117043	Weak IRS and weak MIPS 70 μm	G6V	1 − 2	257 − 170	1.1d	3.4 × 10−5	3.8 × 10−8	0.0012	69 ± 21
HD 132254	IRS limit and MIPS 70 μm	F7V	>15	<70	0.7c	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	208 ± 62
HD 135599	IRS and MIPS 70 μm	K0	11 − 12	61 − 56	0.7c	1.3 × 10−3	2.4 × 10−5	0.77	994 ± 84
HD 154577	IRS and MIPS 70 μm limit	K2V	10 − 10	58 − 57	0.7c	8.4 × 10−4	3.6 × 10−6	0.1	104 ± 70
HD 158633	IRS and MIPS 70 μm	K0V	11 − 13	61 − 55	0.8c	3.5 × 10−4	8.1 × 10−6	0.26	299 ± 53
HD 160032	IRS limit and MIPS 70 μm	F3IV	>15	<70	0.6c	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	320 ± 94
HD 190470	IRS, poor MIPS datag	K3V	2 − 6	129 − 70	0.8d	9.7 × 10−5	1.4 × 10−6	0.044	307 ± 43
HD 199260	IRS and MIPS 70 μm	F7V	19 − 20	68 − 66	1.0d	8.4 × 10−4	1.1 × 10−5	0.34	177 ± 34
HD 219623	IRS and MIPS 70 μm	F7V	0.4 − 22	449 − 63	1.1d	1.4 × 10−5	3.0 × 10−6	0.09	136 ± 22

Notes. ^aCalculated assuming 10 μm dust grains. ^bSee note in Section 5.2.1 for how L_dust/L_⋆ is calculated. ^cλ > 21 μm, 74 dof. ^dλ > 14 μm, 152 dof. ^eTanner et al. (2009) calculate L_dust/L_⋆ = 2770 × 10⁻⁷ including the MIPS 160 μm data point. ^fTanner et al. (2009) calculate L_dust/L_⋆ = 259 × 10⁻⁷ including the MIPS 160 μm datapoint. ^gThis star is located in a particularly noisy field, so the model is fit to the IRS data only, and L_dust/L_⋆ is calculated using data out to 35 μm only.

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Mass estimates are notoriously tricky to derive given uncertainties in grain sizes. Assuming a silicate grain density of 3.3 g cm⁻³, we calculate dust masses of 4 × 10⁻⁷–2.4 × 10⁻³ M_⊕. Extrapolating this estimate using the −3.5 index power law appropriate for a distribution of sizes from a collisional cascade (Dohnanyi 1969) up to a maximum size of 10 km yields total mass estimates as shown in Table 7. Submillimeter observations of all these sources would further constrain the dust size and distribution and thus the total mass of the emitting material.

L_dust/L_⋆ values were obtained by integrating the excess over frequency, including a power-law interpolation between 35 and 70 μm (if available). We used a simple blackbody curve to extrapolate beyond 70 μm, based on the middle of the temperature range found by the model and quoted in Table 7. For stars with a 70 μm excess only, we used Equation (1) at 70 μm.

The models match the spectra quite well (Figure 12), yielding dust temperatures between ∼50 and 450 K (Figure 13) and dust locations between ∼1 and 35 AU (Figure 14). The majority of disks are located between 10 and 30 AU from their stars with several (∼7/19) showing a single temperature fit perhaps indicative of a ring-like structure that may be found with higher resolution data. Since the IRS data place only a limit on the emission shortward of 34 μm for HD 90089, HD 132254, and HD 160032, the single grain model can be used to show that the inner edge of the disk seen at 70 μm must start beyond ∼15 AU corresponding to material cooler than ∼70 K. The mean value of the disk sizes shown in Figure 14 is 14 ± 6 AU.

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It is important to note, however, that these disk sizes are crucially dependent on the assumed grain size and that, as discussed below, smaller grains could dramatically increase the distance at which the emitting grains are actually located (e.g., G. Bryden et al., in preparation).

5.2.2. More Complex Dust Models

While models using single population of 10 μm dust grains reproduce the weak, featureless spectra of most of our stars with excesses, we tried to model some of the excesses using a more realistic mixture of grains of different sizes and material compositions. The compositional model applied to HD 69830 (Lisse et al. 2007) and HD 113766 (Lisse et al. 2008) utilizes a combination of water ice, amorphous and crystalline olivine, and amorphous and crystalline pyroxene. The mix used here contains roughly 50:50 rocky dust and water ice, similar to the abundances seen in the small icy bodies of the Kuiper Belt, but of the 152 program stars, only two (HD 10647 and HD 40136) show hints for spectral features around 20 μm and neither of these stars has a statistically significant IRS excess shortward of 18 μm, severely hampering the fitting due to the lack of emission features available to constrain the models. The remainder of the stars did not offer enough statistically significant data to merit more sophisticated modeling than the simple characterization described in Section 5.2.1.

Applying the compositional model to HD 40136, we were able to derive good, although relatively unconstrained fits to the 18–35 μm IRS data, finding evidence for crystalline olivine (50:50 Fe/Mg-rich), crystalline pyroxene, FeS and some water ice, with a reduced χ² ∼ 0.8 (Figure 15). Removing silicates worsened the fit to a reduced χ² ∼1.6, mostly due to a failure to fit the data around the 18–20 μm silicate feature. However, the S/N is poor shortward of 20 μm due to the bright stellar photosphere, making these identifications preliminary.

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The model for HD 10647 yields a similar mix of ices and silicates, with a reduced χ² ∼ 0.8 (Figure 15). Removing silicates from the fit gives a significantly worse reduced χ² ∼ 56, strongly supporting the inclusion of silicates in the model spectrum. This model is discussed further in Tanner et al. (2009).

Because of the low S/N shortward of 18 μm for both of these stars, identification of minerals will have to await future observations. The Herschel Space Observatory could prove especially useful to check for the evidence of a water ice feature near 62 μm (Figure 15).

5.2.3. Stars with Observed Extended Emission

Only three stars have convincing evidence for warm dust: HD 40136, HD 190470, and HD 219623. One star, HD 117043, has hints of warm excess but is too weak at both IRS and MIPS for further consideration. HD 40136 and HD 219623 have MIPS excesses as well, while HD 190470 has significant cirrus contamination so the MIPS limit is poor. The simple model (Section 5.2.1) for HD 40136 and HD 219623 shows material extending to within 1 AU (Figure 14), suggestive of disks with active reprocessing of material, given the short grain lifetimes at these small orbital radii (Wyatt 2008).

All of the remaining stars with excesses have their emitting material located in regions analogous to the Kuiper Belt in our solar system, typically beyond 10 AU, out to a maximum value of 30 AU (Figure 14). In the two cases where the signal-to-noise ratio is (barely) adequate for mineralogical analysis, HD 10647 (Section 5.2.2 and Tanner et al. 2009) and HD 40136 (Section 5.2.2), the suggestion of significant amounts of water ice is intriguing and is to be expected for regions that lie well beyond the snow line, where volatiles are predicted to be abundant (Pollack et al. 1996). Figure 14 shows the location of the snow line for 1 Myr old stars (using stellar models from Siess et al. 2000), an age when stellar luminosity and the volatile content of the outer disk should be stabilizing. The majority of our sample have material located at or well beyond the snow line.

The total quantity of material responsible for the observed excesses is poorly constrained by our data, because of uncertainties in grain properties and in the extrapolation up to maximum particle size. Table 7 lists total masses extrapolating a population of bodies with 3.3 g cm⁻³ and a N(a) ∝ a^−3.5 size distribution up to 10 km. The median value for 13 stars with strong IRS and MIPS excesses is 0.34 M_⊕. The average and standard deviation, 1.0 ± 2.1 M_⊕, are dominated by a few outliers with more massive disks: HD 1461, HD 38858, and HD 45184 all around 1–2 M_⊕, and especially HD 10647 with an extrapolated total mass of 7.7 M_⊕. These mass estimates can be compared to various estimates for the mass of our own Kuiper Belt or to models of the primitive solar nebula. In the Nice model of outer solar system formation, for example, the outer disk contains roughly 10–150 M_⊕ of material in bodies with densities of 1 g cm⁻³ in sizes up to 300 km (Alessandro et al. 2009). It is difficult to compare these values directly to ours, since we assumed a smaller maximum size, 10 km versus 300 km for Alessandro et al. (2009). However, this is offset by our different density assumptions: we assumed 3.3 g cm⁻³, while Alessandro et al. (2009) used 1 g cm⁻³. More importantly, our 18–70 μm data are probably missing significant emission from other populations of dust: more distant and/or larger grains would emit at longer wavelengths. On the other hand, the order of magnitude agreement between our measurements and an existing solar system model is encouraging.

It should be noted, however, that the stars with strong excesses are in the minority of our sample (<12%) and that the vast majority of the mature stars in our study (and other Spitzer studies) have Kuiper Belt disk masses less than 0.1 M_⊕, the approximate mass of the Kuiper Belt in our solar system (Gladman et al. 2001). This relatively strict upper limit must eventually be reconciled with the presence or absence of gas giant or icy giants in the outer reaches of planetary systems.