HERSCHEL-RESOLVED OUTER BELTS OF TWO-BELT DEBRIS DISKS—EVIDENCE OF ICY GRAINS^*

Detected and characterized by Spitzer photometry and spectroscopy, a sample of 69 nearby stars with warm excess was compiled by Morales et al. (2009, 2011). We consider here 59 of these stars, 26 with evidence for two spatially separated belts of orbiting dust deduced from detailed SED fitting analysis. The remaining 10 stars are too faint for Herschel detection, or were observed non-optimally as part of a large-scale fast-scan program. Table 1 lists the basic characteristics of our 59 sample stars. Most of the stars are observed by two Herschel observing programs led by PI Morales, but a handful come from other programs for which the data are publicly available. All stars are observed using standard small scan maps with the PACS instrument, with simultaneous observations at 70 (or 100) and 160 μm. Each map consists of two cross-scans, separated in position angle by 40°, resulting in a central region of maximum exposure covering several square arcminutes. In general, the choice of either 70 or 100 μm as the primary observing wavelength depends on whether the system was detected already at 70 μm by Spitzer.

The Herschel Interactive Processing Environment, version 10.0.0 (HIPE; Ott et al. 2010) was used to reduce all of the data. Applying a high-pass filter to the images removes instrumental 1/f noise and background structure (i.e., galactic cirrus) on scales larger than the filter widths (66'' at 100 μm, and 102'' at 160 μm). To avoid any source removal from the filtering, we excluded a region within 15'' of each target. We used HIPE's second-level deglitching method to remove outlying flux measurements within each pixel. The final mosaics, oriented in the detector frame, have 1'' pixels for the 70 and/or 100 μm images, and 2'' for the 160 μm images (cf., the native detector pixel sizes of 32 and 64). The images of all well-resolved debris systems, rotated such that north is up, are presented in the first columns of Figures 1 (70 μm), 2 (100 μm), and 3 (160 μm).

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Fluxes are measured with aperture photometry. Each source flux is measured within a circular aperture of 5'' radius at 70/100 μm or 10'' at 160 μm, and aperture corrections of 1.77, 2.02, and 1.63, respectively.³ For Herschel/PACS images that are extended (in Section 2.4 below), a larger 10'' aperture is used at 70/100 μm, with aperture corrections of 1.31 and 1.39, respectively. Although this aperture is larger than the size for maximal signal-to-noise ratio (S/N) at 70/100 μm, a smaller (5'') aperture fails to capture all of the extended emission for the resolved systems studied here; the larger aperture yields fluxes that are, on average, 5% higher.

Seven targets (HD 61005, HD 70313, HD 71722, HD 92536, HD 93738, HD 104860, and HD 142139) were observed following a HIFI recovery event, during which time (2012 March 22–23) the Herschel observatory was kept at a relatively warm attitude toward the Sun. Every observation made during this window (2012 March 22–24) suffers from offsets of 159 and 437 in the detector's x- and y-coordinate frame. After accounting for this known problem, the net offsets of these seven stars are mostly less than ∼2'', consistent with pointing jitter observed within the Herschel/DUNES key program (Eiroa et al. 2013). Two exceptions remain unexplained—HD 10939 and HD 191174 (see Table 2). To account for Herschel pointing uncertainty, the target position at 160 μm is set to the 70 or 100 μm centroid.

We measured the uncertainty for each flux in Table 2 directly from the surrounding field variation, by convolving with the selected photometry aperture for each target and applying the corresponding aperture correction. Additionally, systematic calibration uncertainties of 2.64%, 2.75%, and 4.15% for PACS70, PACS100, and PACS160, respectively,⁴ are also included in the SED fitting, discussed in Section 4. Table 2 lists the resulting photometric flux measurements and uncertainties for the 57 debris systems detected of the 59 target stars in our sample.

Table 2.  Herschel Photometry

Name	70 μm			100 μm			160 μm
	Offseta	Fν (mJy)	Fν/F⋆	Offseta	Fν (mJy)	Fν/F⋆	Fν (mJy)	Fν/F⋆
HD 166b	09	98.0 ± 3.4	6.2	18	67.8 ± 1.8	8.8	26.5 ± 2.9	8.8
HD 1404	⋯	⋯	⋯	03	30.5  ± 2.4	4.0	14.7  ± 5.4	5.0
HD 10939b	42	381.1 ± 2.2	44.8	43	387.4 ± 2.7	92.9	239.8 ± 3.9	147
HD 12039	18	15.2 ± 1.2	6.4	⋯	⋯	⋯	6.6 ± 5.3	14.4
HD 13246	05	22.4 ± 1.1	6.8	⋯	⋯	⋯	11.9 ± 4.5	18.8
HD 19668	14	7.1 ± 0.8	3.6	⋯	⋯	⋯	7.7 ± 3.7	20.5
HD 23267	⋯	⋯	⋯	08	7.4 ± 1.4	10.3	−6.7 ± 6.6	−23.8
HD 23642	03	8.6 ± 1.5	4.6	⋯	⋯	⋯	14.8 ± 12.1	41.2
HD 23763c	07	4.1 ± 2.6	2.1				−10.5 ± 8.8	−27.9
HD 24141	23	14.8 ± 1.2	2.4	⋯	⋯	⋯	5.9 ± 9.5	5.0
HD 24817	11	8.3 ± 0.9	2.1	⋯	⋯	⋯	−4.1 ± 5.0	−5.5
HD 28355b	⋯	⋯	⋯	02	127.3 ± 14.0	20.0	45.5 ± 31.2	18.3
HD 30422b	14	64.1 ± 3.0	14.8	09	43.3 ± 2.1	20.4	17.6 ± 3.8	21.2
HD 32977	15	39.8 ± 1.2	4.7	⋯	⋯	⋯	38.7 ± 13.3	23.7
HD 37286	11	23.1 ± 1.6	6.1	⋯	⋯	⋯	−1.5 ± 6.2	−2.1
HD 38056	⋯	⋯	⋯	11	35.5 ± 2.1	32.0	9.4 ± 5.6	21.7
HD 38206b	06	365.1 ± 3.0	87.5	⋯	⋯	⋯	188.9 ± 6.5	236
HD 43989	04	14.2 ± 1.9	5.7	⋯	⋯	⋯	3.8 ± 6.7	7.9
HD 60737	⋯	⋯	⋯	13	21.4 ± 1.5	14.4	8.6 ± 3.8	14.9
HD 61005b	04	711.0 ± 3.0	337	02	707.2 ± 4.5	685	473.2 ± 7.2	1174
HD 70313b	⋯	⋯	⋯	03	181.3 ± 4.8	52.6	106.7 ± 3.9	79.2
HD 71722b	⋯	⋯	⋯	14	120.5 ± 4.1	64.1	46.9 ± 8.7	63.8
HD 72687	04	6.0 ± 1.3	2.7	⋯	⋯	⋯	−1.9 ± 4.9	−4.5
HD 74873b	15	33.8 ± 2.1	6.6	⋯	⋯	⋯	17.8 ± 6.1	18.1
HD 79108	⋯	⋯	⋯	25	60.9 ± 2.4	40.6	32.9 ± 5.5	56.1
HD 80950	⋯	⋯	⋯	04	31.6 ± 1.5	15.5	13.2 ± 4.2	16.5
HD 85301d	⋯	⋯	⋯	20	39.4 ± 2.0	22.6	28.0 ± 4.5	41.2
HD 87696	⋯	⋯	⋯	08	27.2 ± 1.6	2.7	10.3 ± 3.2	2.6
HD 90905d	⋯	⋯	⋯	11	24.0 ± 2.6	8.7	13.4 ± 5.8	12.4
HD 92536	14	15.7 ± 1.0	6.1	⋯	⋯	⋯	−8.6 ± 5.3	−17.4
HD 93738c	13	2.8 ± 1.0	1.1	⋯	⋯	⋯	−1.0 ± 10.9	−2.1
HD 98673	⋯	⋯	⋯	24	12.2 ± 2.4	7.6	16.3 ± 4.7	26.0
HD 104860b	⋯	⋯	⋯	15	277.1 ± 3.5	240	243.4 ± 5.2	540
HD 107146b	09	750.7 ± 3.6	135	⋯	⋯	⋯	763.4 ± 11.3	720
HD 110411b	17	257.3 ± 2.9	22.3	14	174.6 ± 3.9	30.8	81.9 ± 4.1	37.0
HD 115892	⋯	⋯	⋯	12	46.2 ± 2.4	1.3	13.1 ± 7.4	1.0
HD 118972	⋯	⋯	⋯	03	28.6 ± 2.5	6.5	8.0 ± 5.2	4.7
HD 125283	03	9.1 ± 1.0	2.1	⋯	⋯	⋯	−2.3 ± 4.5	−2.8
HD 126135	17	5.3 ± 1.2	4.2	⋯	⋯	⋯	2.1 ± 6.2	8.6
HD 128207	03	5.8 ± 1.4	1.7	⋯	⋯	⋯	9.8 ± 13.2	15.1
HD 132238	10	11.6 ± 1.0	6.6	⋯	⋯	⋯	0.7 ± 5.4	2.1
HD 136246d	⋯	⋯	⋯	02	33.6 ± 2.4	40.4	27.1 ± 5.7	83.4
HD 136482	⋯	⋯	⋯	11	7.6 ± 1.4	7.6	−1.5 ± 4.9	−3.8
HD 138965b	⋯	⋯	⋯	05	540.3 ± 4.6	356	290.2 ± 4.5	489
HD 141378b	⋯	⋯	⋯	02	192.1 ± 4.1	57.3	109.0 ± 8.7	83.2
HD 142139	12	27.1 ± 2.0	5.6	⋯	⋯	⋯	69.2 ± 20.0	74.9
HD 145229	⋯	⋯	⋯	03	56.7 ± 2.3	33.2	35.8 ± 5.0	53.7
HD 145964	16	9.2 ± 0.9	3.0	⋯	⋯	⋯	4.7 ± 5.1	7.9
HD 153053b	⋯	⋯	⋯	09	202.2 ± 5.0	57.8	124.9 ± 5.8	91.3
HD 159170	08	19.5 ± 0.9	2.5	⋯	⋯	⋯	2.8 ± 4.1	1.9
HD 159492b	⋯	⋯	⋯	06	137.6 ± 3.5	17.5	54.6 ± 4.7	17.8
HD 183216	⋯	⋯	⋯	07	21.0 ± 2.6	8.0	13.6 ± 4.6	13.3
HD 183324b	28	44.4 ± 1.9	8.5	11	35.2 ± 2.2	13.8	25.8 ± 3.6	25.9
HD 191174d	⋯	⋯	⋯	30	37.0 ± 4.8	20.7	36.6 ± 6.5	52.5
HD 192425b	⋯	⋯	⋯	15	228.0 ± 4.7	44.4	149.7 ± 5.7	74.7
HD 196544	⋯	⋯	⋯	09	51.3 ± 2.5	15.9	25.0 ± 8.0	19.9
HD 215766	⋯	⋯	⋯	03	16.3 ± 1.1	6.7	11.4 ± 3.1	12.0
HD 220825	07	33.5 ± 0.9	3.8	⋯	⋯	⋯	9.9 ± 2.7	5.8
HD 223352	⋯	⋯	⋯	03	29.8 ± 2.3	4.7	17.8 ± 5.1	7.2

Notes. Ten stars from the original Morales et al. (2011) sample were not observed by Herschel: HIP 345, HIP 682, HIP 30030, HIP 74752, HIP 74824, HIP 75476, HIP 76395, HIP 95560, HE 848, and HII 1101.

^aOffset between the observed and nominal target positions. ^bExcess emission is spatially resolved. ^cNot enough signal to noise ratio for a Herschel PACS detection at any waveband. ^dExcess emission is marginally resolved.

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Image analysis identical to that presented in Morales et al. (2013) is applied to the overall sample. By subtracting the instrument point source function (PSF) from each image, extended emission is readily seen. We use the Herschel observations of the bright calibrator star alf Cet as the reference PSF. The debris disk data and reference PSF were obtained with the same observing strategy (i.e., two cross-scans separated by 40°), and were reduced using the same pipeline script. When projected into the detector frame (and not in the northeast frame, as is standard), the PSF trefoil pattern is always aligned in the same direction. It is then used for analysis, and to produce each final mosaic. Figures 1–3, column two, show the PSF-subtracted images, where the PSF is artificially normalized to the brightest pixel in order to emphasize the extent of surplus disk emission. Extended emission on two sides of the star is expected, e.g., for an inclined disk. For unresolved sources, however, all emission is removed when performing PSF subtraction.

We now consider a simple model for each debris system, by assuming that the emission originates from an inclined ring of orbiting dust surrounding a star. To begin, we subtract the expected emission of the stellar photosphere, estimated from its near-infrared photometry, extrapolated to the Herschel wavebands; the resultant images of excess flux (third column of Figures 1–3) are essentially indistinguishable from the original images (first column), because the star is extremely faint relative to the dust emission.

Next, by varying three ring parameters—radius, inclination, and position angle—we search for the ring model that best matches the excess flux, minimizing the remaining residual emission. The best fit ring model, after convolution with the PACS instrument PSF, is shown in the fourth column of Figures 1–3, for each resolved debris system in our sample (note that the PSF trefoil pattern is evident for each system's image and ring model). Ring models before convolution are included as black ovals within each model image. In the majority of cases, the Herschel/PACS images are consistent with a thin-ring model, as demonstrated by clean residual maps (last column of Figures 1–3). In some cases, however, (e.g., HD 107146), the thin-ring model leaves significant residuals; a wider dust distribution appears necessary to fit the Herschel image, in that case.

Although 18 debris systems are resolved at 70 and/or 100 μm, only 10 disks have an outer edge that is also significantly resolved at 160 μm (Figure 3). Failure to also resolve the other eight debris disks at 160 μm is due to both the intrinsic lower angular resolution at a longer wavelength, and also because of the disks' lower S/N (relatively larger noise for fainter disks), revealing an outer edge to each PACS-160 μm detected dust system.

The best-fit model radius, inclination, and position angle are listed in Table 3 for each of the 18 (14 A-type and 4 solar-type) spatially resolved disks, and at each wavelength. To determine error bars, we vary parameters until the residuals increase by the equivalent of a 1σ point source.

To further illustrate extended sources detected by Herschel, the mean radial profiles, as seen by PACS at 70, 100, and/or 160 μm, are shown in Figure 4 for two representative extended debris system in our sample—HD 166 and HD 192425. The dust's measured emission around resolved systems like these is spatially extended and readily observed by comparing its profile to the normalized reference PSF from, e.g., alf Cet.

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In summary, of the 18 spatially resolved debris systems, nine are resolved with Herschel/PACS at 70 μm, 15 at 100 μm (where six systems are resolved both at 70 and 100 μm), and ten at 160 μm (all of which are also seen at 70 and/or 100 μm). Our sample break down is summarized in Table 4, and separated by spectral type. Of the 59 Spitzer debris disks originally considered, 57 are detected by Herschel, 26 are two-belt dusty systems, and 18 cold/outer belts are spatially resolved.

We explore the emissivity properties, (a, λ), of amorphous astronomical silicates (AstroSil), believed to be one of the fundamental constituents from which the solar system formed. Following the Morales et al. (2013) grain mixtures, AstroSil is combined with pure water ice and "dirty" ice (H₂O and NH₃ with 10% volume pollution of amorphous Carbon (aC) inclusions), to form icy grains. Table 5 summarizes the different complex optical constants and references used: Draine & Lee (1984) provide the optical constants for AstroSil, Warren & Brandt (2008) for water ice, and Preibisch et al. (1993) for aC and the "dirty" ice mixture.

Also as in Morales et al. (2013), we simulate two types of inhomogeneous icy grains: AstroSil particles with icy mantles (CMPs for core-mantle particles), and icy grains with AstroSil inclusions (IMP, here, stands for inclusions-matrix particles); see Figure 5 in Morales et al. (2013). Lastly, we examine the effect of porous spherical grains by adding inclusions of vacuum into an IMP. This "fluffy" material provides the opportunity to increase the minimum size of each grain, i.e., the blowout size, and keep the grain density low.

In the case of IMPs, we examine the effect of AstroSil (or vacuum) inclusions into an ice matrix, employing the effective medium theory (EMT; Bohren & Huffman 1983). For a specified volume fraction of embedded inclusions, the EMT results in an averaged dielectric function for a system of sub-grains with different electromagnetic properties.

A grain's absorption cross-section, as a function of wavelength, is computed using its average dielectric, from the scattering and extinction coefficients (Q_abs = Q_ext − Q_sca), in the framework of Mie theory (Bohren & Huffman 1983). Table 5 summarizes and Section 3.2 describes the different grain combinations attempted. Similarly, we compute the absorptivities (or emissivities) for CMPs using Mie theory for coated spheres—amorphous AstroSil cores with icy mantles, with volume fraction (core versus ice) of f = 0.5. Emissivity profiles, as a function of wavelength and grain size for the resultant (a, λ), are shown in our Appendix for porous grains, and for the rest of grain cases in Appendix A of Morales et al. (2013).

To model the thermal emission from dust in outer/cold belts, we consider the following variety of grain compositions, as explored in Morales et al. (2013), and add porosity as a forth possibility.

• 1.  
  Homogeneous AstroSil-only grains.
• 2.  
  Inhomogeneous IMP,

  • a.  
    AstroSil inclusions in a matrix of water ice, or
  • b.  
    AstroSil inclusions in a matrix of "dirty" ice.
• 5.  
  Inhomogeneous CMP,

  • a.  
    AstroSil cores with pure water ice mantles, or
  • b.  
    AstroSil cores with "dirty" ice mantles.
• 8.  
  Inhomogeneous Porous IMPs,

  • a.  
    Vacuum inclusions in a matrix combination of AstroSil +"dirty" ice.

The icy matrix of case 2, and the icy mantle of case 3 comprise 50% of the grain's volume (f = 0.5). In case 4, we test for porosity by considering the extreme case of a 90% volume fraction of vacuum inclusions in a matrix made of the 50/50 "dirty ice"/AstroSil IMP of case 2b above. We did not attempt vacuum inclusions in grains from case 2a (pure water matrix with AstroSil inclusions) because the model grains already showed a considerable deficiency in long (PACS) wavelength emission, due to the intrinsic properties of pure water. The resultant emissivities for the different compositions proposed above differ enough from each other to provide a significant, observable change in the shape of our SED models (see our Appendix and Appendix A of Morales et al. 2013 for emissivity curves).

In general, we find that grains of radius ≳3 μm no longer produce spectral features in the mid-infrared Spitzer/IRS spectrum. A large number of small (≲3 μm) grains, however, will generate spectral features at 9.8 and ∼20 μm for AstroSil, where ice features at 3.1 and ∼43 μm are produced by small icy grains. Although featureless for any composition, the larger grains do influence the otherwise Rayleigh–Jeans slope at the Herschel wavebands, by suppressing the flux. Accordingly, grain-sizes can be organized into three generalized regimes: small grains (≲3 μm), able to produce spectral features; middle-sized grains (3 μm ≲ a ≲ 30 μm) that necessarily suppress emission relative to a blackbody over the sampled Herschel wavelengths; and big grains (≳30 μm), blackbody-like in unit emissivity, expected to be present—but with a small contribution to the overall surface area.

Including the optical constants of dirty ice in complex grains to our uniform SED model approach is a step forward in exploring the effect of more realistic grain properties, but not definitive. More robust SED analyses require both thermal and scatter light imaging to pin down the grain populations, as well as allowing for grain size distribution (with power-law slope set to q = −3.5), to differ from the assumed steady state collisional cascade above. Such studies have been carried out for a few individual debris disk sources; e.g., HR 4796A by Rodigas et al. (2015), and β Pic by Ballering et al. (2016). Here, however, we acknowledge that our results from a uniform approach may be modified as more data is included for singular sources, and more realistic grain property calculations become available, i.e., fractal grains, etc.

Under the assumption of idealized spherical grains distributed in narrow dusty belts, ten resolved debris systems (HD 166, HD 10939, HD 28355, HD 61005, HD 70313, HD 71722, HD 110411, HD 138965, HD 141378, and HD 159492) are best fit (Δχ² > 3.0, Table 6) with a two-belt debris disk model where the inner/warm component is comprised of AstroSil compact grains, and the outer/cold belt of IMPs with a dirty ice matrix. The icy grain model fits (Figure 5, blue curves) best reproduce the observed SEDs, while matching the resolved sizes of the outer/cold dust.

Figure 6 compares the icy versus rocky quality-of-fit χ² values for photometry of λ ≥ 70 μm, which is dominated by the outer/cold dust emission. Sources above the diagonal dashed line in Figure 6, color coded in blue, have long wavelength SEDs best fit by icy grains, whereas the sources below the dashed line are best explained by rocky emissivities. For a representative source in our sample, the color–color diagram of Figure 7 shows the outer/cold expected behavior as a function of radius (AU) from the central star HD 110411. The measured fluxes of the resolved dust (with stars and warm excess subtracted) is best reproduced by icy grains (IMPs, case 2b in Section 3.2), suggesting a reservoir of icy planetesimals.

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For four star-disk systems (HD 30422, HD 74873, HD 104860, and HD 183324) in our resolved sample, using only the Spitzer and Herschel measurements, we are not able to select between AstroSil or dirty ice IMPs for the outer/cold belt, because their Δχ² ∼ 0 (Table 6 and black symbols near diagonal on Figure 6). Although the compositions of these outer/cold dust components remain inconclusive, we use the dirty icy IMPs to estimate minimum dust masses. Unlike most of the other systems in our resolved sample, we note that the excess emission around HD 104860 (F8) continues to rise from the Spitzer 70 μm measurement to the Herschel 100 and 160 μm photometry. If we adopt a wider outer/cold dust belt of ${\rm{\Delta }}r\sim 0.3{r}_{0}$ to model the emission, we find a slightly better icy-grain fit to the sub-millimeter data from SCUBA. Because this system is also significantly resolved at Herschel 160 μm—at a somewhat larger radii (∼166 AU versus 123 AU at 100 μm)—we conclude that the the icy-grain model with a wider dust region (${\rm{\Delta }}r\sim 0.3{r}_{0}$) is probably a better description of the outer/cold debris around HD 104860. Another example of a system with cold debris and inconclusive composition (i.e., it may turn out to be icy)  is HD 30422. We note that the Δχ² is driven by the inconsistency in relative calibration between Spitzer/MIPS 70 and Herschel/PACS 70 photometry. The blue curve (IMPs) in Figure 5, clearly best fits the shape of the SED provided by Spitzer/IRS (5–35 μm) and Herschel/PACS (70, 100, and 160 μm) data.

The last four debris systems in our resolved sample (HD 38206, HD 107146, HD 153053, and HD 192425) have Δχ² < 0 (Table 6, and magenta symbols under the dashed line of Figure 6). Although Δχ² < 0 favors rocky emissivities, we note that 1) the quality-of-fit test comparison (Figure 6) is not as clear in selecting AstroSil over the icy IMP grains (because the magenta symbols tend to fall close to the diagonal dashed line) when compared to the rest of our resolved sample, and 2) that some of these resolved systems, as is the case for HD 107146 (resolved in scatter light, with a broad radial outflow of dust particles smaller than the radiation pressure limit; Ardila et al. 2004), are more complex than the simple two-belt model approach we take here. Once (sub-)millimeter data are included, the dust emission is not only wider in wavelength range, it also implies a radially broader dust architecture that may require a third dust component, and perhaps different grain populations.

In general, for the sample of the 18 resolved systems studied here, the major problem with the AstroSil-only grains (case 1 in Section 3.2, and magenta curves in Figure 5) is that the "flat" shape of the SED, produced for the outer/cold dust component, poorly matches the long wavelength Spitzer/Herschel photometry; i.e., it severely over- or under-predicts the emission (see χ_AS and χ_IMP for 70, 100, and/or 160 μm in Table 6).

The simple blackbody fits from Morales et al. (2011) (green curves in Figure 5) offer a great assessment of the dust's characteristic temperatures of two-belt systems, but the radial locations of blackbody grains are generally closer to the parent star than those observed by Herschel. The Herschel imaging of resolved systems (Section 2.4) reveals that the outer/cold components can be significantly farther out, up to a factor of ∼4.5 in the case of HD 61005 (Figure 9 and Table 7).

Table 7.  Fitting Parameters—Resolved Systems, Part I

		Blackbody Fits				AstroSil-only Fits		AstroSil+IMP Fits
Name	Spectral	Twarm	Tcold	${R}_{\mathrm{warm},\mathrm{BB}}$	${R}_{\mathrm{cold},\mathrm{BB}}$	${R}_{\mathrm{warm},\mathrm{AS}}$	${R}_{\mathrm{cold},\mathrm{AS}}$	${R}_{\mathrm{warm},\mathrm{AS}}$	${R}_{\mathrm{cold},\mathrm{IMP}}$
	Type	(K)	(K)	(AU)	(AU)	(AU)	(AU)	(AU)	(AU)
HD 166	K0Ve	166	65	1.8	11.7	1.6	39.7	1.0	32.5
HD 10939	A1V	182	55	15.0	166.0	12.8	184.0	14.8	160.3
HD 28355	A7V	123	61	16.8	68.4	23.3	128.1	21.2	129.7
HD 30422	A31V	254	75	6.1	69.3	4.2	133.6	5.7	124.7
HD 38206	A0V	238	62	10.8	159.1	8.7	168.6	9.4	166.5
HD 61005	G8Vk C	124	53	3.6	19.8	3.3	94.4	3.1	66.9
HD 70313	A3V	184	57	11.6	121.8	6.6	150.6	8.0	145.9
HD 71722	A0V	216	68	13.1	131.0	9.1	144.8	9.4	139.7
HD 74873	A1V	177	56	15.7	155.2	8.8	119.0	11.1	126.3
HD 104860	F8	199	48	2.8	49.2	4.8	137.7	4.2	132.1
HD 107146	G2V	107	47	6.8	35.1	13.2	96.5	13.2	96.5
HD 110411	A0V	283	81	7.6	92.7	4.0	127.0	7.1	131.4
HD 138965	A1V	174	56	16.4	156.2	16.0	185.1	17.2	184.5
HD 141378	A5IV	182	58	9.5	92.4	10.0	110.2	6.6	127.9
HD 153053	A5IV-V	167	60	11.3	86.9	8.4	203.0	16.6	192.4
HD 159492	A5IV-V	175	78	10.3	51.6	9.8	117.9	10.3	125.2
HD 183324	A0V	137	57	32.3	187.7	29.0	193.7	28.4	179.8
HD 192425	A2V	208	58	10.3	133.3	13.3	274.1	13.0	231.4

Note. Table summarizing the best SED fitting results for the cases proposed in Section 4. Columns 3 and 4 are the characteristic temperatures of the grains (as in Morales et al. 2011); columns 5 and 6 are the radial locations, assuming blackbody grains in thermal equilibrium with the parent star. Columns 7 and 8 (case 1) are the radial locations obtained when homogeneous particles of AstroSil are used as the only composition, for both inner and outer dust belts. Columns 9 and 10 (case 2b) are the radial locations obtained when AstroSil is assumed to be the composition of the warm/inner dust, and the emissivities of the inhomogeneous inclusion-matrix particles (IMPs), using "dirty" ice, are used for the cold/outer dust component. All radial location estimates have an uncertainty of about ±5 AU.

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As in Morales et al. (2013), we discard grains containing pure water ice, i.e., cases 2a and 3a in Section 3.2. Whether in the form of IMPs or CMPs, water ice exceedingly suppresses infrared emission, due to its intrinsic inefficiency at absorbing ∼0.1 to ∼1 μm radiation and emitting at λ ≳ 70 μm. The distinct steepening of the long-wavelength emission suggests that the pure water ice particles considered here are not a likely explanation for the resolved systems being studied.

The CMPs of case 3 in Section 3.2 display emissivities strongly influenced by the mantles of ice, comparable to IMPs (Appendix A in Morales et al. 2013). CMPs with "dirty" ice mantles (case 3b) remain only slightly less efficient than the "dirty" ice IMPs (case 2b), and their influence on the resultant SEDs is small and about equally acceptable. Composition, rather than the internal structure of the compact solid grains modeled here, is a more important factor in reproducing the shape of an SED, so we proceed with "dirty" ice IMPs.

We mostly discard the porous grains of case 4 (Section 3.2), due to (1) the resulting severe flux inefficiency when fitting the SED shape of resolved debris systems, and (2) the disagreement with their spatially resolved dust locations. The exception is HD 71722 (Figure 5), for which the quality of fit from the porous grain profile is comparable to that of the "icy" grain IMP fit, resulting in an SED dust radial location centered around R ∼ 132 AU, in excellent agreement with the resolved measurement of 139 ± 27 AU. Although both fits, porous and "dirty" ice IMPs, for HD 71722 utilize the radiative blowout grain limit as their minimum grain size, a_min = a_BOS, the only significant difference between them is that the minimum size porous grain is a factor of ∼10 larger than the minimum icy IMP (${a}_{\mathrm{BOS},\mathrm{IMP}}\sim 4.5\,\mu {\rm{m}}$ for HD 71722).

We begin SED modeling, assuming minimum grain sizes are limited by what is expected from radiative blowout arguments for each star-disk system. This initial assumption is a good approach to fit unresolved systems, yet spatially resolved debris belts provide us with an opportunity to characterize the size distribution by varying the minimum grain size to match the grain's observed location. Although dust masses and fractional luminosities estimated from our model fits are comparable to previous results for our sample (Morales et al. 2011, 2013), the minimum grain sizes for most of the resolved systems in our sample have never been measured before. Now, with the disk sizes, we deduce the value a_min for 18 spatially resolved debris disks, and compare against the expectation from radiative blowout, a_BOS. In Table 8, we list both a_min and its ratio to blowout, f_MB ≡ a_min/a_BOS.

Table 8.  Fitting Parameters—Resolved Systems, Part II

Name	Spectral	Lwarm	Lcold	Mwarm	Mcold	${a}_{{\min },{AS}}$	${a}_{\min ,\mathrm{IMP}}$	${f}_{\mathrm{MB},\mathrm{AS}}$	${f}_{\mathrm{MB},\mathrm{IMP}}$
	Type	L⋆	L⋆	(MMoon)	(MMoon)	(μm)	(μm)
HD 166	K0Ve	4.7E−05	7.5E−05	7.1E−06	8.9E−03	0.3	0.4	1.0	1.0
HD 10939	A1V	1.7E−05	5.9E−05	1.1E−03	5.0E−01	5.6	15.8	1.0	2.0
HD 28355	A7V	8.0E−06	1.7E−05	1.2E−03	6.4E−02	5.0	5.6	1.0	1.0
HD 30422	A31V	1.0E−05	2.3E−05	6.1E−05	5.3E−02	2.2	3.2	1.0	1.0
HD 38206	A0V	1.1E−04	1.5E−04	1.5E−03	5.5E−01	5.0	7.1	1.0	1.0
HD 61005	G8Vk C	1.2E−04	2.9E−03	1.5E−04	2.3	1.6	2.0	5.5	5.0
HD 70313	A3V	2.0E−05	4.6E−05	3.2E−04	2.1E−01	3.2	4.5	1.0	1.0
HD 71722	A0V	2.8E−05	7.2E−05	3.4E−04	1.6E−01	4.0	5.0	1.0	1.0
HD 74873	A1V	2.6E−05	7.7E−06	4.5E−04	1.5E−02	3.2	4.5	1.0	1.0
HD 104860	F8	2.3E−05	6.4E−04	4.0E−05	1.9	3.5	4.5	6.5	6.0
HD 107146	G2V	8.9E−05	1.1E−03	2.1E−03	2.5	2.2	3.2	5.0	5.0
HD 110411	A0V	1.5E−05	3.6E−05	8.1E−05	5.6E−02	2.8	4.0	1.0	1.0
HD 138965	A1V	7.8E−05	3.2E−04	2.3E−03	1.3	5.6	7.9	2.0	2.0
HD 141378	A5IV	1.1E−05	4.5E−05	1.8E−04	1.2E−01	5.6	11.2	2.0	3.0
HD 153053	A5IV-V	8.0E−06	2.9E−05	1.1E−03	2.5E−01	2.8	12.6	1.0	3.0
HD 159492	A5IV-V	3.4E−05	2.7E−05	9.6E−04	4.6E−02	1.0	4.0	0.4	1.0
HD 183324	A0V	7.8E−06	6.0E−06	1.6E−03	4.0E−02	8.9	12.6	3.0	3.0
HD 192425	A2V	2.7E−05	2.2E−05	9.0E−04	3.4E−01	10.0	17.8	2.5	3.0

Note. Mass estimates (in lunar masses), assuming a population of grain sizes from a_min up to 1 mm. We estimate M_warm and M_cold for each star's best fit model (AstroSil for the warm/inner belts, and dirty ice IMPs for the cold/outer belts), where a_min is the minimum grain radius for a given composition (±0.5 μm), f_MB is the minimum-to-blowout grain-size factor, and f_MB = a_min/a_BOS.

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For the 18 stars considered here, the SED fits to the outer dust components result in minimum grain sizes, a_min, that range from ∼0.5 to 15 μm, with a median of ∼5 μm, and uncertainties of ≲0.5 μm. (see Table 8 and Figure 8). We also see considerable variation in grain size relative to blowout. For A-type stars, the SED fit to most outer belts using pure AstroSil or IMP grains have median a_min of ∼4.5 μm or ∼6.4 μm, respectively. These sizes are most often equal to the expected blowout sizes, given the density of AstroSil or "dirty ice" IMPs and the properties of each star, i.e., f_MB ≈ 1 (Figure 8). The resolved cold debris around three of the four solar-type stars, on the other hand (HD 61005, HD 104860, and HD 107146), tend to have much larger grains relative to what would be expected from blowout (i.e., f_MB ≈ 5, 6, and 5 respectively). Solar-type star HD 166, which is well fit with a_min = a_BOS = ∼0.4 μm, is an exception. Lastly, the SED for HD 159492 requires grains that are smaller than the blowout limit. It needs a factor of $\sim \tfrac{1}{3}$ of the blowout size for AstroSil-only (a_min ∼ 1.0 μm) to fit the SED of the warm-inner dust component.

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The minimum-to-blowout factor, f_MB, has also been deduced for other spatially resolved debris systems with Spitzer/IRS+MIPS 70 μm data of comparable quality. For example, Su et al. (2009) find f_MB ∼ 5 for the outer planetesimal belt around the A-type star HR 8799, assuming a grain population of AstroSil. Krist et al. (2010) find f_MB ∼ 3 for HD 207129 (G0V), assuming a somewhat steeper slope for the grain size distribution (q = −3.7). Also with a steeper slope, Golimowski et al. (2011) find a minimum grain size for HD 92945 (K1V) that is even larger compared to blowout, f_MB > 10.

The principal results (minimum grain size, dust luminosity, mass) of our SED model fits are listed in Table 8. We find that the dust luminosity ratios for the inner/warm and the outer/cold dust components, L_warm and L_cold, are generally in the range of 10⁻⁵ to 10⁻⁴, with the cold component a factor of ∼2.5 more luminous. The exceptions are: (a) HD 159492, HD 183324, and HD 192425, whose two components are comparable in brightness; (b) HD 74873, whose warm component is ∼3.4 times brighter than the cold; and (c) three of the four resolved solar-type stars, HD 61005, HD 104860, and HD 107146, along with two A-type stars, HD 38206 and HD 138965, whose cold components are about an order of magnitude brighter than any others (L_cold ≈ 3 × 10⁻³, 6 × 10⁻⁴, 1 × 10⁻³, 2 × 10⁻⁴, and 3 × 10⁻⁴, respectively).

We also calculate minimum dust masses for each warm/cold component, integrating over all dust sizes from a_min up to 1 mm, as in Morales et al. (2013). Dust masses for the warm components are ∼0.1% of a lunar mass (ranging from ∼7 × 10⁻⁶ to ∼4 × 10⁻³ M_moon; see Table 8). The outer disk always has more dust, typically about a factor of 100 or more. The bright outer disks of the five stars above (Solar-type: HD 61005, HD 104860, and HD 107146; A-type: HD 38206 and HD 138965) are again the outliers, with 2.0, 1.9, 2.2, 0.6, and 1.3 M_moon of dust. Our minimum dust mass for HD 104860 is comparable to an independent dust mass estimate, based solely on the SCUBA sub-millimeter flux (13 M_moon; Najita & Williams 2005).

For 41 unresolved debris systems, where the radial extent of the dust is unknown, we begin modeling each SED by assuming minimum grain sizes determined from radiative blowout, a_min = a_BOS. However, based on SED shape, nine warm single-belt debris systems (HD 24141, HD 24817, HD 37285, HD 87696, HD 132228, HD 159170, HD 215766, HD 220825, and HD 223352) display slightly broader profiles, compared to the AstroSil-only shape, that are closer to a blackbody curve (Figure 10). In our attempt to adopt more realistic grain properties, as opposed to simple blackbody curves, we successfully broaden the shape of the AstroSil curves by increasing the minimum grain size for these systems by a factor of about three in all cases, except for HD 132238. HD 132238 is a main-sequence B8V type star with a_min = a_BOS ∼ 22 μm—already a large grain, compared to the wavelengths explored—and increasing a_min does not make a significant difference in broadening the shape of the SED fit.

To broaden these nine SEDs, we also consider the possibility of a wider unresolved ring (${\rm{\Delta }}r\sim 0.3{r}_{0}$), and/or the existence of a faint Kuiper-like component, to reproduce the surplus in flux at the Herschel PACS wavelengths. However, we find that the wider ring fail to reproduce the shape of the SED. Although we do not discard the possibility of a faint outer/cold component, the increase in minimum grain size is an excellent solution to eight of these nine warm debris disk SEDs.

We pursue the overall trend of rocky AstroSil grain emissivities for all single-belt and inner/warm components of two-belt SEDs, and attempt both rocky and dirty ice grain emissivities for those with evidence for an outer belt. We find two exceptions to this trend: HD 80950, an 80 Myr old A0V star; and HD 136482, a 15 Myr old B8/B9V star. Both exhibit an SED with two warm components, ∼300 and ∼135 K, for the inner/warm and outer/warm dust belts, respectively. Thus, we fit these two-belt debris system with AstroSil only for the warm dust distributions (Figure 10).

Table 9 summarizes the results of our SED model fits; i.e., dust luminosity ratios and minimum dust mass estimates. The median value of the dust luminosity ratios for single-belt systems or inner/warm dust components are similar to the resolved sample; ∼10⁻⁵. For those debris systems with a second dust component, the dust luminosity ratios range from 10⁻⁶ to 10⁻⁴. Their median (∼10⁻⁵) is comparable to the inner/warm luminosity ratios, and generally fainter than the outer/cold dust belts of the resolved sample.

The estimated dust masses for the single-belt and inner/warm components are generally within the range of 10⁻⁵ to 10⁻² M_moon (see Table 9). Like the resolved two-belt debris disk sample, unresolved systems with a outer/cold component tend to have about a factor of 100 more dust in the outer/cold disk than the inner/warm disk, ranging here from ∼ 7 × 10⁻³ to ∼0.6 M_moon.

Several other debris disks have also been imaged by Herschel. Although most of these publications do not entail an SED analysis using icy grain emissivities, each does include a measurement of dust temperature from a blackbody fit to the SED. Combined with the measured disk size, this provides a rough indication of whether the disk emission is dominated by small grains. The key metric we consider is the ratio of the Herschel-observed disk size to that expected from the SED alone, assuming blackbody emission (R_disk/R_BB). If the disk is comprised of small grains, these grains will be hotter than blackbody or, equivalently, the disk size will be larger than expected from the SED fit with blackbody curves. Larger grains will behave more like blackbodies, (a, λ) ≈ 1, with R_disk/R_BB approaching unity.

Figure 9 shows a compilation from many of the published Herschel results (Liseau et al. 2010; Matthews et al. 2010; Marshall et al. 2011; Kennedy et al. 2012; Lestrade et al. 2012; Wyatt et al. 2012; Bonsor et al. 2013; Booth et al. 2013), where disk size relative to expectation from the SED (R_disk/R_BB) is shown as a function of stellar luminosity. There is a general trend apparent; the disks around lower luminosity stars are larger than the blackbody expectation. This is roughly consistent with expectation from blowout; higher-luminosity stars should have relatively large, blackbody-like dust grains, whereas lower luminosity stars should have small super-heated grains. The 18 resolved systems featured in this work (red squares) fall within the overall trend.

6.2.1. HD 166

6.2.2. HD 10939

6.2.3. HD 28355, HD 30422, HD 70313, HD 71722, and HD 110411

6.2.4. HD 38206, HD 153053, and HD 192425

6.2.5. HD 61005, HD 104860, and HD 107146

6.2.6. HD 74873 and HD 183324

6.2.7. HD 138965

6.2.8. HD 141378

6.2.9. HD 159492