ASTEROID BELTS IN DEBRIS DISK TWINS: VEGA AND FOMALHAUT

To avoid saturation, we only used data taken in the IRS high-resolution (R ∼ 600) modules (short–high (SH): 9.9–19.6 μm and long–high (LH): 18.7–37.2 μm). The sizes of the slits are 47 × 113 and 111 × 223 for the SH and LH modules, respectively; a significant fraction of the light from a point source is outside the slit. Standard slit loss correction for a point source was applied to the extracted spectra; therefore, the part of the spectrum where a point source dominates the emission has the correct spectral shape.

IRS SH and LH spectral mapping data centered at the position of Vega were obtained through IRS calibration programs (PID 1406, 1409, 1411, and 1413) in 2004. Here, we present six sets of observations where the slit was placed on the Vega position based on PCRS pointing information (no IRS peakup). These data were taken in the spectral mapping mode with two positions parallel to the slit (747 step⁻¹) and three positions perpendicular to the slit direction (1''step⁻¹); we only used the ones taken at the center perpendicular position (two positions along the slit center). We used the SMART software (v.8.2.5; Higdon et al. 2004) to reduce the BCD products from the Spitzer Science Center (SSC) pipeline version of S18.18. Each of the spectra was extracted with the full-slit mode without sky subtraction, flux calibrated to a point source using the S18.18 calibration, and shown in Figure 1. We then pinned all of the SH spectra to 35.03 Jy at 10.6 μm according to the absolute calibration scale defined in Rieke et al. (2008), and shifted the corresponding LH spectra (a scale factor of 0.975) to join smoothly with the SH ones. The final spectrum was obtained as a weighted average of these six AORs (black dashed line in Figure 1).

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The Fomalhaut system was observed with the Spitzer IRS instrument in program PID 90 in 2004 June and November. Again, we only report here the observations with the IRS SH and LH modules, due to saturation of the signal in the low-resolution modules. In IRS SH, we obtained a standard staring-mode nodded observation, with an on-target integration time of 503 s. This observation was preceded by a high-accuracy pointing peakup on a nearby star with no infrared excess, HD 216922, using the IRS Blue (13–19 μm) peakup camera. With IRS LH, we obtained spectral-mapping observations with a strip of nine slit positions separated in the dispersion direction by 48 and centered on Fomalhaut. The integration time per slit position was 122 s. The spectral mapping exercise was preceded by a moderate-accuracy IRS pointing peakup using the red (19–26 μm) camera and HD 216922. Spectra at the extremes of the spectral map indicated that sky emission is negligible compared to that by Fomalhaut, so no sky subtraction was performed on the center-position spectrum we discuss here. Data reduction began with basic calibrated data products of the IRS data pipeline, version S11. From these data, we removed, by interpolation in the spectral direction, permanently bad and "rogue" pixels identified in the IRS dark-current data for all observing campaigns up to and including the one in which each Fomalhaut spectrum was taken. Again, we used SMART for full-slit extractions of spectra from the two-dimensional data. Similar observations were made of α Lac (A1 V) in IRS SH and γ Dra (K1 III). Along with template spectra for these stars provided by M. Cohen (2004, private communication), we used these observations to construct relative spectral response functions (RSRFs) for each spectrometer, and in turn used these RSRFs to calibrate the Fomalhaut spectra. We estimate the resulting spectrophotometric accuracy to be approximately 5%.

The final combined, smoothed spectra for both systems are shown in Figure 2. For comparison, the stellar photospheric models (details see Section 3.1) are also shown. Given the geometry of the outer cold rings (radius of 20'' in the Fomalhaut system (Acke et al. 2012) and radius of 11'' in the Vega system (Sibthorpe et al. 2010)) and the sizes of the point spread functions (PSFs) in the IRS wavelengths, the spectral flux beyond ∼30 μm is partially contaminated by the cold ring. The spectral shape and flux level shortward of ∼30 μm are mostly from the star and any unresolved inner warm component. This is consistent with the fact that the observed spectrum between 10 and 13 μm agrees well with the expected stellar photosphere. The excess spectrum (after stellar photospheric subtraction) is also shown in the lower panel for both systems in Figure 2.

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Herschel PACS 70 and 160 μm data for Vega and Fomalhaut were obtained by the GT program and published in an early reduction by Sibthorpe et al. (2010) and Acke et al. (2012), respectively. We retrieved the archival data and reduced them with the Herschel Interactive Processing Environment (HIPE, V9.0 user release; Ott 2010). We applied the standard processing steps up to the level 1 stage. During this process, we applied 2nd level deglitching to remove outliers with "timeordered" option and 20σ threshold.¹¹ This is very effective for data with high levels of coverage. After producing level 1 data, we selected the science frames from the time-line by applying spacecraft-speed selection criteria (between 8'' s⁻¹ and 12'' s⁻¹ for the slow scan, and 18'' s⁻¹ and 22'' s⁻¹ for the median scan). The final level 2 mosaics were generated using highpass filtering with the script "photProject" and a pixel scale of 1'' at 70 μm and 2'' at 160 μm. To avoid flux loss in the highpass filtering process, we applied a circular mask of 60'' radius centered on the position of the target. Since the PACS data on Vega and Fomalhaut were obtained with different scan rates (10'' s⁻¹ and 20'' s⁻¹, respectively), PSF observations matched to the observing parameters should be used for comparison. We used PACS data on α Boo (ObsId 1342247634 and 1342247635) and α Tau (ObsId 1342214211 and 1342214212) as our PSF reference and reduced them using the same reduction procedure including the masking radius. These two stars are ones of the PACS primary calibrators, where their fluxes and PSF behaviors are characterized by Müller et al. (2011). We have made sure that these PSF stars have the consistent encircled energy fraction as a function of circular aperture radius derived by the PACS point-source calibration (Müller et al. 2011). To illustrate the major features seen in the PACS images, we show the final 70 μm mosaics of Vega and Fomalhaut in Figure 3 along with the comparison PSFs.

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The emission of the stellar photosphere from optical to mid-IR (∼8 μm) was determined in a number of steps. Most infrared photometry measurements (like Spitzer/IRAC and Akari) are referred to Vega, although Vega is unsatisfactory as a standard (i.e., fast-rotating, infrared excess). However, these missions also measured Sirius using the same technique, so we have taken the measurements with nominal uncertainties of 0.01 mag directly compared to Sirius in terms of magnitude differences. Sirius (A1V) is very similar in spectral type to Vega (A0V) and Fomalhaut (A4V), and is well behaved in the infrared with no evidence for an infrared excess (Price et al. 2004).

All three stars are severely saturated in the Two Micron All Sky Survey (2MASS) data. Therefore, for accurate measurements at 2.2 μm, we used data from the DIRBE instrument on COBE. The reduction of these data is described by Price et al. (2010) for analyzing stellar variability. Due to the large DIRBE beam (42'× 42'), the contribution from stars in the field surrounding the target of interest needs to be removed. We evaluated this effect using 2MASS data, making Sirius fainter by 0.014 mag and Fomalhaut fainter by 0.003 mag while the contribution in the Vega field is negligible. Both Vega and Fomalhaut are reported to have K band excess at 1.29% ± 0.19% (Absil et al. 2006) and 0.88% ± 0.12% (Absil et al. 2009) above the photosphere using interferometry. After accounting for these K-band excesses of Vega and Fomalhaut, the K-band magnitude difference for photospheres between Fomalhaut and Sirius is 2.35 mag, and 1.39 mag between Vega and Sirius. For measurements in the IRAC bands, we adopted the results from Marengo et al. (2009) who used the PSF fitting technique to recover accurate photometry for saturated sources. The magnitude differences between Fomalhaut and Sirius are 2.36 mag, 2.36 mag, 2.36 mag, and 2.33 mag at 3.6, 4.5, 5.8, and 8 μm, respectively (to be discussed further by Espinoza et al. in preparation). The magnitude differences between Vega and Sirius are 1.38 mag, 1.38 mag, 1.38 mag, and 1.35 mag at 3.6, 4.5, 5.8, and 8 μm, respectively. Furthermore, Midcourse Space Experiment (MSX) also measured both Vega and Sirius. The measured 8 μm flux of Vega is lower by 1% compared to the MSX predicted flux based on the measurement of Sirius (Price et al. 2004). We adopt a magnitude difference of 1.36 mag at 8 μm between Vega and Sirius.

With the nominal uncertainty for these measurements (0.01 mag), there is no evidence for excesses from 2.2 to 8 μm (after excluding the K-band excesses from interferometric measurements) for both systems. We used appropriate Kurucz atmospheric models for photospheric predictions in other wavelengths by normalizing the model fluxes from 2.2 to 8 μm to the measured photometry that has been transferred to the absolute calibration scale proposed by Rieke et al. (2008). The parameters in the Kurucz model are T_eff = 9500 K and log g = 4.0 for Vega, and T_eff = 8750 K and log g = 4.0 for Fomalhaut. Given the accurate Hipparcos parallax measurements (van Leeuwen 2007), the integrated luminosity is 17.45 L_☉ for Fomalhaut and 56.05 L_☉ for Vega (viewing from pole-on).

Using these normalized Kurucz models, we can then predict the photospheric level at the wavelengths of interest and determine the excess near the stars by subtracting off the stellar contribution from the IRS spectra presented in Section 2.1. Note that the Kurucz models in the infrared wavelengths are basically in the Rayleigh–Jeans regime.

The Vega system is viewed pole-on, and its disk has previously been resolved at various wavelengths (Holland et al. 1998; Heinrichsen et al. 1998; Su et al. 2005; Marsh et al. 2006; Sibthorpe et al. 2010). As shown in Figure 3, the PACS 70 μm image appears to be centrally peaked with a smoothed, axis-symmetric extended halo. The size of the Vega cold disk observed in the higher resolution Herschel data (Sibthorpe et al. 2010) agrees with the one observed in the Spitzer data (Su et al. 2005). The stellar photosphere is about 0.81 Jy at 70 μm, consistent with the centrally peaked morphology seen at that wavelength. The FWHM of the central source is 56 × 53 measured on a field of 21'' × 21'' centered on the star, which is slightly more extended than the measured FWHM of the PSF star, α Boo (55 × 52). To minimize the influence of the cold ring (peaked at radius of ∼11''–14'', see Figure 4) in estimating the photometry of the central source, small aperture sizes with appropriate aperture corrections should be used. On the other hand, the aperture has to be large enough to contain most of the flux and to minimize the centroid uncertainty. We tried several aperture settings from 3'' to 55 (FWHM) with and without a sky annulus and used the PSF star α Boo to derive the values of aperture correction. The resultant fluxes range from 1.00 Jy to 1.62 Jy with a median value of 1.01 Jy (using an aperture of 5'' and sky annulus between 5''and 7''). At 160 μm, the beam (FWHM) is 116 × 101 (measured from α Boo), making it very difficult to estimate the photometry of the central source alone without significant contamination from the flux of the cold ring. A 5'' aperture without sky annulus gives a total flux of 0.56 Jy, which should be considered as an upper limit since it contains some fraction of the cold ring contribution. The photosphere of Vega is 0.15 Jy at 160 μm. Therefore, the excess flux at the star position, based on small aperture photometry, is 0.2 Jy at 70 μm and <0.4 Jy at 160 μm.

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To illustrate our PSF-subtraction approach, cuts were made along a PA of 120° (corresponding to the horizontal axis of the displayed image) of the Vega system, with the NW side on the negative side of the x-axis. These cuts, shown in Figure 4, were centered at the stellar position at 70 μm and represented the mean value over a width of 5 pixels (i.e., 5'' and 10'' at 70 and 160 μm, respectively). No significant asymmetry in disk brightness nor center offset were seen in these cuts. After subtracting a photospheric scaled α Boo PSF, an additional point-like source is clearly present in the resultant cut (blue dashed line in Figure 4) at 70 μm. The maximum of the point-source contribution (star and unresolved disk) can be estimated by forcing the peak values (single pixel) matched between the Vega and α Boo data. A flux density of 1.19 Jy for such a PSF subtraction is required and its resultant profile is shown as the thin solid blue line in Figure 4, suggesting a maximum brightness of <0.38 Jy for this unresolved disk at 70 μm. Using the flux obtained with the small aperture photometry (0.2 Jy for the unresolved disk), the resultant profile is shown as the blue dotted-dashed line in Figure 4, relatively flat in the central ±5'' region as expected. Therefore, the brightness of the unresolved disk in the Vega system is ∼0.2 Jy with a maximum value <0.38 at 70 μm. Due to the uncertainty of the PSF (as much as 10% at 70 μm; Kennedy et al. 2012) and that of flux calibration, we simply assume a conservative lower-bound error of 20% (i.e., a minimum flux of 0.16 Jy at 70 μm for the central unresolved disk). At 160 μm, the maximum scaled PSF is ∼0.45 Jy by forcing the peak value to zero, suggesting a maximum brightness of <0.3 Jy for this unresolved disk component.

Since the Fomalhaut debris system is inclined by 67°, it is very difficult to separate the point source from the cold ring, even using very small apertures. One can still estimate the maximum brightness of the unresolved component by assuming that there is no contamination from other sources inside a very small aperture. For an aperture radius of 3'', the encircled flux is 0.95 Jy at 70 μm and 0.77 Jy at 160 μm after applying appropriate aperture corrections. Taking out the contribution of the stellar photosphere (0.368 Jy and 0.07 Jy at 70 and 160 μm, respectively, based on our stellar photospheric model), the unresolved disk component has a maximum flux of <0.58 Jy and <0.7 Jy at 70 and 160 μm, respectively.

Similar to Vega, cuts were made along the major-axis of the disk and shown in Figure 5. These cuts were centered at the stellar position at 70 μm and represented the mean value over a width of 4 pixels (i.e., 4'' and 8'' at 70 and 160 μm, respectively). The center of the outer cold ring (the dashed, vertical line in Figure 5) was estimated by the mid-point of the two bright peaks (marked as dotted, vertical lines for 70 μm and dot-dashed, vertical lines for 160 μm in Figure 5) at both wavelengths. The 160 μm ansae peak closer to the ring center than the ones at 70 μm, which was first reported by Acke et al. (2012) and suggested to be due to blurring in the large PSF at 160 μm. Although the peak positions at 70 and 160 μm are not the same, the cold ring centers at the same position at both 70 and 160 μm. We used a PSF star, α Tau, for photospheric subtraction by scaling it to match the expected photospheres (0.368 Jy and 0.07 Jy at 70 and 160 μm). The photosphere-subtracted cuts are shown in long-dashed lines in Figure 5. At 70 μm, it is clear that an additional source of emission is present at the center of the disk. We tried two different PSF subtractions to estimate the brightness of this additional component. First, we arbitrarily increased the scaling of α Tau, fixed in the stellar photosphere position, until the central region (from −5'' to +5'') has a relatively flat distribution in the cut. The resultant cut is shown as a thin solid blue line with an additional scale of 0.17 Jy. Second, we subtract a second α Tau PSF by adjusting both the position and the scaling until the resultant cut is relatively flat (dot-dashed line in Figure 5). The second method gives a scale of 0.165 Jy for this additional source. Combining both methods, we conclude that the central unresolved component is 0.17 Jy at 70 μm. There is no easy way to estimate the error associated with this number since it depends on the detailed structures of various components; we simply assume a 20% error for further analysis. At 160 μm, it is not obvious that an additional source is required in the system. We estimated the upper limit of such a component by increasing the scaling factor of α Tau fixed at the star position (any potential offset between the star and this inner component is washed out by the large beam size at 160 μm). An upper limit of 0.08 Jy at 160 μm was inferred.

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In summary, the central unresolved disk component in the Fomalhaut system is about 0.17 Jy at 70 μm, with upper- and lower-bound fluxes of 0.58 Jy and 0.136 Jy (assuming 20% uncertainty), and <0.08 Jy at 160 μm, based on the analysis of the data only (without invoking any assumption of modeling). Our values are consistent with the best-fit model presented in Acke et al. (2012). Their model estimates that the fluxes of the central point source (star + unresolved disk component) are 0.54 Jy (5% of the total flux) and 0.124 Jy (2% of the total flux) at 70 and 160 μm, respectively, implying that the unresolved disk accounts for flux of 0.172 Jy at 70 μm and 0.054 Jy at 160 μm.

The Vega Spitzer 24 μm observation (Su et al. 2005) was severely saturated, making it difficult to constrain the brightness of this unresolved component without invoking some modeling assumptions. Therefore, we seek other relevant measurements in the mid-infrared to validate the excess levels seen in the IRS spectrum. Tokunaga (1984) measured a handful of nearby A-type stars at 20 μm using the IRTF bolometer with a beam size of 5'' to define the 20 μm magnitude system (relative to Vega). Only two stars, α CMa (Sirius) and γ UMa, in his list do not have a 24 μm excess. We used the color V −[20] of these two stars to extract the excess of Vega at 20 μm. We adopt V of −1.40 and 2.40 mag for α CMa and γ UMa, suggesting a photosphere color V −[20] of −0.04 mag in Tokunaga's system. Based on the observed V −[20] color of 0.03, the excess of Vega is ∼7% above the photosphere (i.e., 0.7 Jy). It is difficult to assess the errors associated with the 20 μm measurement; therefore, we simply assume a maximum 50% error at this wavelength. Another source of mid-IR measurements for Vega comes from MSX photometry where the excesses (relative to Sirius) at 14.65 and 21.34 μm have been reported by Price et al. (2004) to be 4% and 17% above the photosphere (i.e., excesses of 0.7 Jy at 14.65 μm and 1.5 Jy at 21.34 μm), respectively. The contamination from the cold ring needs to be taken into account given the large beam size of MSX (a resolution of ∼20''). Assuming a typical dust temperature of 60 K for the cold ring and normalizing the cold ring flux observed in the far-infrared (Su et al. 2005), the flux contamination from the cold ring is less than 0.1% of the photosphere at 14.65 μm, and ∼4% of the photosphere at 21.34 μm. The nominal error for the MSX measurements is ∼1.5% (Rieke et al. 2008), which includes the errors from stellar photospheric prediction. Including the possible contamination from the cold ring, the final errors are 0.31 Jy and 0.38 Jy at 1σ for the excesses at 14.65 and 21.34 μm. The Vega system was recently observed by the Submillimeter Array (SMA) at 880 μm (Hughes et al. 2012). Interestingly, the spatially unresolved 880 μm flux within 5'' of the Vega photosphere lies slightly above the predicted photospheric value (see Figure 6), although only at the less than 2σ level. Combined with our estimates from the PACS images and the IRS spectrum, the excess SED of the central component in the Vega system is shown in Figure 6. We note that Liu et al. (2004) also report an upper limit (∼250 mJy) of 0.7% (1σ) of the photospheric level at 10.6 μm, using nulling interferometry (BLINC/MMT) that probes a region less than 15 (12 AU) from the star.

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The spectral shape (λ < 30 μm) of the inner component is well represented by a blackbody of 170 K, just above the temperature at which water ice sublimates in vacuum. This temperature corresponds to a distance (∼2.7 AU) in the middle of our asteroid belt in the solar system, but ∼14 AU in the Vega system (using an average stellar luminosity of 37 L_☉ viewed by the dust along the equator of the star; Aufdenberg et al. 2006) assuming blackbody-like emitters.

The existence of a close (unresolved) warm component around the Fomalhaut disk was first suggested by the Spitzer 24 μm observation where an additional point-like source (0.6 ± 0.2 Jy) along with the expected stellar photosphere is required to fit the resolved disk image at 24 μm (Stapelfeldt et al. 2004). The central component of the Fomalhaut system was also tentatively detected by the ALMA observation at 850 μm (Boley et al. 2012). A flux density of ∼4.4 mJy at the star position was estimated after the primary beam correction, which is quite uncertain at the star position because the ALMA observation was centered at Fomalhaut b. The expected photosphere at 850 μm is ∼2 mJy (not 3 mJy as stated in Boley et al. 2012). Therefore, the excess at 850 μm could be as high as 2.4 mJy. We simply took this at face value and assumed a 50% error (as maximal). The excess SED for the Fomalhaut central unresolved component is shown in Figure 7 along with the excess spectrum measured by IRS.

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We note that Acke et al. (2012) suggested ionized gas (free–free emission from a hot stellar wind) as being responsible for the unresolved central excess from K band to 70 μm, and derived F_ν∝ν^{0.8 ± 0.1} power law. With their derived power law, the excess at 850 μm (350 GHz) is expected to be 18–23 mJy, which is clearly not consistent with the ALMA observation. In addition, this power-law spectrum is inconsistent with the 2.2–5.8 μm colors discussed in Section 3.1. Instead, we suggest that the excess emission detected longward of ∼13 μm arises from thermal dust emission. Similar to the warm component in the Vega system, the spectral shape (λ < 30 μm) of the inner component in Fomalhaut is also well represented by a blackbody of 170 K, suggesting a radial distance of ∼11 AU from the star for blackbody-like grains.

The warm excesses detected in both systems are restricted to the vicinity of the star, i.e., unresolved at various wavelengths. The amount of warm excess emission is derived either from resolved images where the central component is (mostly) separated from the cold ring, or from the spectra taken centered at the star position. Given the sizes of the IRS slits, some amount of the flux from the cold rings is expected to fall into the slit, especially at wavelengths longer than ∼30 μm. It is difficult to estimate the exact flux contamination without further modeling because it depends on the geometry of the cold ring at different wavelengths. One can estimate the maximum flux contamination in the worst scenario case by assuming that the warm and cold components are spatially coincident. Therefore, the maximum flux contamination is ≲1% of the photosphere for λ < 20 μm and could be up to 50% at 33 μm given the typical dust temperature of the cold ring (∼70–50 K). Thus, the derived dust temperature (∼170 K in both systems) based on the data shortward of 30 μm has very little contamination from the cold ring. In other words, the excess arises from material less than 3'' (unresolved with an FWHM ∼ 6'') from the star in both cases. Similar to the definition of habitable zones around stars, debris disk structures should be identified and characterized in terms of dust temperatures rather than physical distances so that the heating power of different spectral types of stars is taken into account and common features in disks can be discussed and compared directly. The characteristic temperature of ∼170 K for the excesses suggests that they are asteroid-belt analogs. It corresponds to a distance of 2'' (∼14 AU) from Vega (using the lower luminosity viewed by the dust in the equatorial direction) or of 15 (∼11 AU) from Fomalhaut assuming blackbody radiators. The resultant location of the asteroid belt given a dust temperature depends greatly on grain properties. Using astronomical silicates (Laor & Draine 1993), the dust can be as close as 13 (10 AU) from Vega, and 1'' (8 AU) from Fomalhaut using grains with a radius of 10 μm. Furthermore, the IRAM PdBI observations at 1.3 mm by Piétu et al. (2011) do not find any excess within 25 while the SMA data by Hughes et al. (2012) suggest a tentative excess within 5'' from the star, suggesting that the warm excess around Vega likely arises from emission outside a radius of 13 (10 AU) from the star. The fractional luminosity is 7 × 10⁻⁶ for the Vega warm component and 2 × 10⁻⁵ for the Fomalhaut warm component, much more luminous than our current zodiacal cloud (10⁻⁸ to 10⁻⁷; Dermott et al. 2002).

The origin of the warm excess in the vicinity of the star is unknown. One hypothesis is that the warm excess originates from dust grains in the cold belt, which are transported inward by P-R and/or stellar wind drag, as suggested by Reidemeister et al. (2011) for the outer warm belt of the Eri debris system. In the Fomalhaut system, the presence of an inner disk, extending inward up to ∼35 AU (45) and composed of grains dragged-in from the cold planetesimal belt, has been suggested by Acke et al. (2012) to explain the PACS 70 μm profile. Conventionally, a P-R-transported flow has a constant surface density and is expected to extend to the star, resulting in a surface brightness profile that is centrally peaked at the star. However, Acke et al. (2012) find that the inner disk in Fomalhaut has a clear truncation (outside the unresolved warm component). Therefore, neither the warm nor the hot excess observed in the Fomalhaut system is unlikely to have arisen from the inward transport of grains by P-R drag. On the other hand, we cannot rule out such a possibility (either solely or partially due to the dragged-in grains) in the Vega system (it is centrally peaked at 70 μm) without detailed modeling for the whole system at multiple wavelengths (K. Y. L. Su et al. in preparation). Nevertheless, the observed dust temperature (170 K) of the warm excess suggests that this component does not extend all the way to the star; instead, an inner truncation (>1'') is required.

Furthermore, the tentative detection of the central component at submillimeter wavelengths in both systems argues for the presence of large grains, suggesting that the warm emission likely arises from a full spectrum of particle sizes similar to the cold planetesimal belt, i.e., a form of planetesimal belt like our asteroid belt. The wavelength-dependent power index at long wavelengths (F_ν ∼ λ^−l) is a measure of the grain size distribution in a collision-dominated debris disk (Wyatt & Dent 2002; Gáspár et al. 2012). The cold planetesimal belts observed in bright debris disks have much steeper slopes compared to Rayleigh–Jeans slope (e.g., Gáspár et al. 2012). Similar behavior is seen in the warm component in the Vega and Fomalhaut systems (see Figures 6 and 7), implying the presence of large grains. Limited by the uncertainties in the current observations, the exact slope of the warm component at long wavelengths cannot be determined accurately. Future high-resolution observations in the submillimeter and millimeter will help shed light on the nature of the warm excesses.

One interesting note on the nature of these warm excesses is that the observed levels of dust (f_d ∼ 7 × 10⁻⁶ for Vega and f_d ∼ 2 × 10⁻⁵ for Fomalhaut) are consistent with being the in situ, steady-state collisional evolution of large parent bodies for the lifetime of the stars (∼400 Myr). The maximum fraction luminosity (f_max) for such a system can be estimated using Equation (18) from Wyatt et al. (2007), where f_max is on the order of (2–3) × 10⁻⁵ for both systems, assuming an asteroid belt at ∼10 AU with a width of ∼1 AU and consisting of ∼10 km (diameter) planetesimals in collisional cascades. Due to the uncertainties of some parameters in the model, Wyatt et al. (2007) suggest that a transient event producing the observed dust is only required when f_obs ≫ 10³f_max. Thus, the observed dust in the warm component is consistent with it being generated through collisional grinding in an asteroid belt in both systems.

In our solar system, the locations of the minor bodies that failed to form planets are elegantly arranged and sculpted by planetary perturbations over the course of 4.5 Gyr evolution. The inner edge of the Kuiper belt's dusty disk is thought to be maintained by massive planets (Liou & Zook 1999), whereas the more tenuous asteroid-belt dust (i.e., zodiacal cloud) has a structure also determined by gravitational perturbations via both the giant and terrestrial planets (Dermott et al. 1994; Murray et al. 1998). It has been suggested that the dominant source of dusty debris inside ∼5 AU in our solar system results from the breakup of asteroids (e.g., Dermott et al. 2002). However, a recent dynamical model incorporating multiple sources (asteroids and short- and long-period comets) by Nesvorný et al. (2010) suggests that particles originating from Jupiter-family (short-period) comets dominate the mid-infrared emission in the zodiacal cloud while the contribution of asteroidal dust is <10%. The origin of our own zodiacal cloud is still a matter of considerable debate, making an understanding of the warm dust around nearby stars (i.e., exo-zodi) even more valuable.

Dust locations solely based on temperatures derived from ex-solar debris disk SEDs are ambiguous. However, based on resolved images at multiple wavelengths, four systems—Vega, Fomalhaut, Eri,¹² and HR 8799—clearly have two dust belts. The disk SEDs of these four systems are shown together in Figure 8 for easy comparison. As discussed in Section 4.3, the dust observed in the warm components for both Vega and Fomalhaut is consistent with being generated in a steady-state collisional cascade in a planetesimal belt. A similar conclusion is obtained for the other two systems. For HR 8799, the warm dust component has a temperature ∼150 K and fractional luminosity f_d ∼ 2 × 10⁻⁵ (Su et al. 2009); the latter is much less than the maximum fractional luminosity for steady state collisional dust production, f_max ≈ 3 × 10⁻⁴, estimated for a belt at 8 AU at HR 8799's age of ∼30 Myr. For the Eri system, the warm component has f_d ∼ 3 × 10⁻⁵ and a temperature of ∼150 K (Backman et al. 2009); the latter is compatible with f_max ≈ 8 × 10⁻⁶ for a belt at 3 AU at an age of ∼800 Myr. Based on the similarity of the observed SEDs and resolved disk structures, it is very likely that the warm components in these four systems originate in well-separated planetesimal belts (remnants of planet formation) resembling the configuration of our own solar system, which has two left-over planetesimal belts: an asteroid belt near the water-frost line, between Mars and Jupiter, with a characteristic temperature of ∼170–150 K, and a Kuiper belt at ∼30–55 AU with dust emission peaked at ∼50 K.

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There are two aspects of the two-belt systems that must be explained. The first one is related to how these two separate belts were created, and the other one addresses how the system maintains such a large gap without dust filling it in. Mechanisms to explain both the formation and evolution of the two belt systems are required to fully account for the observed pattern. Planetesimal belts can be expected in regions where there was not enough material or not enough time to form a planet, because once formed, a planet would scatter or accrete the surrounding planetesimals. Therefore, the stable locations of left-over planetesimals are governed by where the giant planets form and their migration history, which may include strong dynamical instabilities. Consequently, the location of the observed dust is not expected to show much order among systems since it greatly depends on the numbers and positions of the giant planets and their migration history. In addition, it is an observational fact that higher mass stars harbor a more massive protoplanetary disk (Natta et al. 2000; Williams & Cieza 2011) and more giant planets (Johnson et al. 2010). Generally, one does not expect to find similarity in planetary configurations between low-mass (i.e., low luminosity) and high-mass (i.e., high luminosity) stars.

Among the four systems shown in Figure 8, only Eri is a solar-like star and the rest are early-type stars. The luminosity (the dust heating power) difference ranges roughly two orders of magnitude among them, and yet the characteristic dust temperatures of the warm and cold belts are very similar. It is interesting to note that many unresolved systems also have a similar two-belt configuration in the dust distribution based on their temperatures derived from unresolved excess emission (Chen et al. 2009; Morales et al. 2011). The fact that the temperatures of the warm belts peak at similar temperatures (∼170–190 K) between early-type and solar-like systems (Morales et al. 2011), however, suggests that temperature-sensitive mechanisms play a major role in determining planetesimal configuration in the warm belts. Giant planets are expected to form right outside the water-frost line where the amount of solid material and dynamical timescales favor the formation process (Kretke & Lin 2007).

The young HR 8799 system has four giant planets (Marois et al. 2008, 2010) separating the inner and outer dust belts (Su et al. 2009). These four planets are very massive (∼7–10 M_J), so that they dominate the dynamics in the HR 8799 system. Dust particles spiraling inward from the cold belt under P-R drag are likely to be dynamically scattered and ejected by one of the planets before they reach the inner region. Thus, the warm component in the HR 8799 system is unlikely to arise from grains generated in the cold planetesimal belt. Comets on plunging orbits originating from the cold belt could possibly cross over and then break-up or sublimate to deliver dust interior to the giant planets. However, it is difficult to estimate the inward flux of cometary bodies inside the orbits of the giant planets without detailed numerical simulations such as the one done by Bonsor et al. (2012). Furthermore, while the comet delivery scenario proposed by Nesvorný et al. (2010) might work for our solar system at its current age, it is not clear that this scenario is directly applicable for younger systems (such as the ones we discussed here) where in situ dust generation in the younger, more massive planetesimal belts is expected to dominate.

Of the remaining three systems, Fomalhaut is the only other one that harbors a directly detected planet, Fomalhaut b (Kalas et al. 2008). Although the reality of Fomalhaut b has been questioned because the spectrophotometry of Fomalhaut b cannot be reconciled with models for thermal emission from a young giant planet, a problem originally acknowledged in Kalas et al. (2008) and further demonstrated by Spitzer non-detections (Marengo et al. 2009; Janson et al. 2012), a perturbing planet is required to account for the observed disk asymmetry (Kalas et al. 2005; Quillen 2006; Chiang et al. 2009). Nevertheless, recent re-analyses on the public HST data by Currie et al. (2012) and Galicher et al. (2012) confirm the detection of Fomalhaut b, with its position found to be just interior to the ring and comoving with the star, making it a candidate to maintain the sharp inner boundary of the Fomalhaut cold ring. The orbit of Fomalhaut b has not yet been fully demonstrated to be consistent with the range of orbits required for the perturber, and its mass is also quite uncertain due to the lack of detections at other wavelengths, ranging from <a few M_J (from the constraint of cold belt as detected in scattered light; Chiang et al. 2009) to a few M_⊕ (from the constraint derived from the properties of the cold belt in the submillimeter; Boley et al. 2012). Furthermore, ground-based high-contrast observations have also set a mass limit of <2 M_J for any planet between ∼10–40 AU (Kenworthy et al. 2009) and of <12–20 M_J from 4 to 10 AU (M. A. Kenworthy et al. 2012, submitted) in the Fomalhaut system.

The clumpy structures claimed from millimeter and submillimeter imaging of the Vega cold Kuiper-belt analog were often taken as signatures of gravitational perturbations by planets (Holland et al. 1998; Wilner et al. 2002; Wyatt 2003; Marsh et al. 2006). Recent observations at higher resolution and sensitivity have failed to detect the clumps and instead are consistent with a smooth, broad, and axisymmetric disk (Piétu et al. 2011; Hughes et al. 2012). In any case, Vega has been the target of several deep searches for planets through direct imaging, and strong limits have been placed at the H, L', and M bands for planets with masses ≳ 3–4 M_J between ∼20 and ∼70 AU (Marois et al. 2006; Heinze et al. 2008). For Eri, planets with masses ≳2–3 M_J in a radial distance of 6–35 AU are ruled out at H, L', and M bands (Lafrenière et al. 2007; Heinze et al. 2008) and Spitzer IRAC bands (Marengo et al. 2006, 2009). Using radial velocity (RV) and astronometry techniques, a close, eccentric planet, Eri b, with a semimajor distance of ∼3–4 AU was identified by Hatzes et al. (2000) and Benedict et al. (2006), although the discovery of the inner warm dust belt (∼3 AU) casts doubt on the existence of a high eccentricity planet (Backman et al. 2009). Furthermore, a recent analysis of all available RV data of Eri by Anglada-Escudé & Butler (2012) found a significantly different orbital solution and suggested that the long-term RV variability is likely due to stellar activity cycles rather than a putative planet. Nonetheless, Eri b, if it exists just outside the warm belt, is a candidate to shepherd the warm dust belt (Backman et al. 2009).

Although the exact locations of the warm components in these disks are unknown (unresolved), the orbital ratios (R_cold/R_warm) between the outer cold belt (mostly resolved in the submillimeter and millimeter wavelengths) and the warm belt estimated from the dust temperature are roughly ≳10 in all four systems,¹³ similar to the ratio in our own solar system (the asteroid belt at ∼3 AU and the Kuiper belt at ∼35 AU). A significant deficit of large grains (best tracers for the unseen parent bodies) in the region between the belts are evident due to the fact that high-resolution submillimeter observations do not detect such a filled-in component (Boley et al. 2012), although small grains can drift inward from the cold belt (Acke et al. 2012). The zone between the warm and cold belts that is mostly free of dust is very large. If the observed dust is being generated in both belts through collisional cascades of large parent bodies, then we need a cleaning mechanism to maintain such a large dust-free zone.

The large gap between the warm and cold belts may be maintained by one or multiple planet-mass objects in the gap. If these planets have dynamical influence over the entire gap, then they are likely to scatter any material that is either generated in the gap or drifts into it. The dynamical influence of a planet is given approximately by the extent of the overlap of first-order resonances which creates an unstable "chaotic zone" in the vicinity of a planetary perturber (Wisdom 1980; Duncan et al. 1989; Mustill & Wyatt 2012). Applying this criterion, we find that in the Vega system, the large gap could be maintained by a single hypothetical object with a mass of a few 100 M_J (i.e., a star/brown dwarf) in a circular orbit; for Fomalhaut, a similar mass estimate of a few 100 M_J is obtained for a single hypothetical object in an eccentric orbit (e = 0.1, based on the observed eccentricity of its cold dust ring). One can also imagine a lower mass perturber in a very eccentric orbit being responsible for maintaining such a large gap; the required eccentricity can be estimated by assuming that its pericenter and apocenter are near the locations of the warm and cold belts. We find that a large eccentricity (e  ∼  0.8) is required, implying an object with a mass of ∼50 M_J in both cases. Our simple estimates show that the gaps in both Vega and Fomalhaut are too large to be explained by a single circular/eccentric perturber, without contradicting the current upper limits of a few M_J based on non-detections of planetary companions in the gaps.

The minimum number of planets residing between the two belts is two—one inner planet outside the warm dust belt to shepherd the inner planetesimal belt, and one outer planet interior to the cold dust belt to scatter large grains that drift inward from the cold belt. We can then estimate the mass of two equal-mass planets in this case. This is illustrated in Figure 9, where we plot the orbital distances of the two equal-mass planets and their associated chaotic zone widths as a function of the mass ratio between the planet and the host star. In the case of Vega (M_* = 2.5 M_☉ and we assume e = 0 for the hypothetical planets), we find that two ∼40 M_J planets are needed to maintain the dust free zone. In the case of Fomalhaut (M_* = 2.0 M_☉ and we assume e = 0.1 for the hypothetical planets), we find that two ∼63 M_J planets are needed. Although these masses are likely overestimates because the single-planet chaotic zone formulae do not account for the strong secular perturbations that can extend the unstable zones in multiple planet systems (e.g., Moro-Martín et al. 2010), it is evident that just two planets in low eccentricity orbits with mass ∼M_J are inadequate for explaining the gaps.

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Our simplified approach, estimating the masses for single planetary perturbers and for two equal-mass perturbers in order to explain the large gaps in Vega and Fomalhaut, gives limits that are more than an order of magnitude higher than the ∼few M_J planet mass limits for these systems based on non-detections of planetary companions in the gaps. It is, therefore, probable that multiple, lower mass planets are responsible for clearing the gaps. We can estimate the numbers and orbital radii of multiple, Jupiter-mass planets that would clear this region by assuming that they are separated such that their chaotic zones just fill the regions between them. We find that four or five 1 M_J mass planets are required for the large gap in Vega and in Fomalhaut. (This is reminiscent of the HR 8799 system in which four planets separate the two dust belts.) Upcoming ground-based high-contrast, direct imaging surveys using facilities like LBT/LBTI, Gemini/GPI, VLT/SPHERE should be able to find and/or place tighter mass limits on the planets in the large gap. Stars that have a debris disk with two distinct dust belts separated by a large gap are attractive targets for future searches for exoplanets.

The presence of hot dust in the close vicinity of both systems is revealed by ground-based interferometric observations. For Fomalhaut, 0.88% ± 0.12% of excess emission in the K band was reported by Absil et al. (2009). For Vega, a 1.29% ± 0.19% excess in the K band (Absil et al. 2006) and a 1.23% ± 0.45% excess in the H band (Defrère et al. 2011) were reported. Generally, these studies have ruled out other possible sources such as low-mass companions or stellar winds as the cause of excess emission. Mennesson et al. (2011) also put a very tight constraint on the location of the hot dust (within 0.2 AU) by combining all interferometric measurements. Defrère et al. (2011) further modeled the properties of the hot excess of Vega (both spatial visibility and SED) and reached several conclusions: the dust responsible for this hot excess (1) has very steep density and particle size distributions (density power index less than −3 and particle size power index ∼ − 5), (2) resides from 0.1 AU (dust sublimation radius with temperature of 1700 K) to less than 0.2 AU from the star, and (3) has a minimum particle size of 0.01–0.2 μm with a significant fraction of carbonaceous composition (due to the fact that carbonaceous grains have a much higher sublimation temperature and lack prominent mineralogical features). Although their model does include large grains (with maximum size of 1000 μm) in the calculation, there is basically no grain with sizes larger than 0.3 μm (the average size particle is 0.27 μm) with such a steep size distribution. In other words, the only explanation for such a hot excess is a population of sub-μm carbonaceous grains located in a narrow-ring-like region at the dust sublimation radius (0.1 AU). In the case of Vega, this hot excess is very different from what we know about the zodiacal cloud in our solar system, which has a relatively flat density distribution (density power index of −0.34; Kelsall et al. 1998) and mostly contains large particles with sizes of ∼10–100 μm (Fixsen & Dwek 2002). The use of exozodi in this context, as has been referred to broadly in the literature for Vega, is misleading.

The origin of the hot dust population is unknown. One scenario that has been discussed frequently in the literature is that the hot dust arises from evaporating comets dynamically perturbed from the outer cold disks. In the case of Vega, a total mass of ∼10⁻⁹ M_⊕ is required to account for the observed excess emission in near-IR (Defrère et al. 2011). The radiation blow-out timescale at 0.1 AU is on the order of a year, suggesting a very high dust replenishing rate. This high replenishing rate implies that this phenomenon is unlikely to be in a static state. If not in steady-state, then the hot dust may be created in transient dynamical events similar to the late heavy bombardment in our solar system, which was caused by a dynamical instability of the asteroid belt (Strom et al. 2005). However, this kind of hot excess has also been found around stars that have no detectable infrared excess indicative of the presence of a planetesimal population (Absil et al. 2008), making this scenario unsatisfactory to explain the hot excess phenomenon.

Kobayashi et al. (2009) present an analytical model to form a narrow ring due to sublimation of dust grains drifting radially inward due to P-R drag. This scenario could work in Vega and Fomalhaut if the dust drifting inward is refractory, i.e., the asteroid belt discussed in this paper is the source of the particles. This hypothesis only requires a P-R dominated disk; no dynamical perturbation is required. The replenishing rate implies that a total mass of >0.4 M_⊕ in the Vega asteroid belt is required to retain such a rate over the age of the system (∼400 Myr). However, the range of distances and particle sizes from this drag-in component would result in a much higher excess flux in the mid-IR (see Figure 5 in Kobayashi et al. 2011). Furthermore, one would expect that the amount of hot dust should be proportional to the amount of dust in the source region, i.e., the asteroid belt. The fact that the hot dust in both systems has a similar fractional luminosity, 5 × 10⁻⁴ (Absil et al. 2006, 2009), while the warm dust in the Fomalhaut system is ∼3 times more than that of the Vega system, argues against this non-stochastic transport hypothesis.

There may be an alternative direction for models of the hot dust component. The basic requirements are (1) a mechanism that prevents the radiation-pressure blow-out of very small dust grains at very small astrocentric distances (∼10 stellar radii, i.e., the ∼1500 K dust that is creating the K-band excess); (2) a grain composition that yields a featureless, roughly Rayleigh–Jeans (or steeper than Rayleigh Jeans) spectrum between 2 and 10 μm (for consistency with the SED constraints in Section 3.1); and (3) a mechanism for the generation grains of the appropriate composition. Although a detailed analysis is beyond the scope of this paper, these requirements evoke the behavior of nano dust grains (with sizes of ten to a few tens of nm) that are charged by the solar wind and trapped by the solar magnetic field near the Sun. The underlying hypothesis is that sungrazing planetesimals deliver silicate-rich grains to the stellar neighborhood, but when the silicates break down they yield metal oxides (Mann & Murad 2005). Because of the high abundance of Mg and Fe in silicates, MgO and FeO are produced profusely in this process through reactions such as

$$\begin{equation*} {\rm MgSiO_3 \rightarrow MgO + SiO + \frac{1}{2}O_2} \end{equation*}$$

and

$$\begin{equation*} {\rm Fe_2SiO_4 \rightarrow 2FeO + SiO + \frac{1}{2}O_2}. \end{equation*}$$

The resulting nano grains are very refractory: for example, MgO has a melting temperature of 3100 K and boiling temperature of 3873 K, nearly as high as the sublimation temperature of carbon, 3900 K; FeO boils at 3687 K. Consequently, MgO, FeO, and composite grains of the same elements can survive close to the star. We use the hot excess around Vega as an example to illustrate the resultant SED since the location for such a hot excess is best constrained among all the K-band excess sources. We adopted the optical constants for a mixed iron/magnesium oxide, Mg_0.6Fe_0.4O (Henning et al. 1995) and computed their absorption and scattering efficiencies using the Mie theory. We computed a model SED (Figure 10) for such a narrow ring located between 0.18 and 0.2 AU from the star, and composed of grains with radii of 5–20 nm (0.005–0.02 μm) in a size-distribution power index of −3.5 using the stellar model computed by Aufdenberg et al. (2006; for dust viewed from the stellar equator). The dust temperatures for these gains in the region of 0.18–0.2 AU from Vega reach ∼2400 K. With a grain density of 4.8 g cm⁻³, a total mass of 2.3 × 10⁻⁹ M_⊕ produces the 1.29% excess of the star at 2.2 μm. Toward longer wavelengths to 10 μm, the output falls roughly as ν^2.8, where Rayleigh–Jeans falls as ν², that is, the spectrum is somewhat steeper than Rayleigh–Jeans (see Figure 10). There is a prominent feature at ∼18 μm in our model SED due to the crystalline form of the material used to determine the optical constants, which might not represent the actual form of material resulting from sublimation of silicate-rich planetesimals. The mass in nano dust is equivalent to about 60 Halley Comet nuclei. Although the conversion of a comet nucleus to nano dust will not be fully efficient, it should be reasonably high since the process involves erosion of large grains, and so most of the solid material (which is believed to constitute more than half the mass of a typical comet nucleus) may be converted into oxides. Thus, it appears that a plausible number of planetesimals (perhaps no more than a couple of hundred) would suffice to provide the nano grain population. Finally, we would like to emphasize that the model SED shown in Figure 10 is one example that satisfies the requirements (as listed previously) in our hypothesis. Our proposed scenario does not depend on the exact nature of the material as long as they meet these requirements.

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We now consider the magnetic trapping of these dust grains. Vega has a magnetic field strength of ∼1.2 G, with about a quarter in a dipolar component and the rest in higher order terms (Petit et al. 2010). To first order, this is similar to that of the Sun, with a surface field of about 1 G dominated by a dipole but with complex components due to activity. Nano grains in the vicinity of Vega will acquire electric charge through the photoelectric effect. This process has been modeled by Pedersen & Gómez de Castro (2011), who find that a 30 nm grain illuminated by an A0 star will reach a level of about 200 e⁻ or charge per mass ratio (Q/m) ∼2 × 10⁻⁶. Smaller grains will reach higher values of Q/m. Assuming a field similar to that of the Sun, these values are in the range where trapping occurs (Czechowski & Mann 2010). In summary, the hypothesis that the 2 μm excess is emitted by nano grains trapped in the magnetic field of Vega does satisfy the three conditions listed above. A better understanding of the behavior of the magnetic field of the star is required to improve our understanding of whether it is indeed the process that accounts for this emission. Furthermore, this magnetically trapped, nano dust model can apply to other stars that show K-band hot excesses through interferometry as long as these stars possess some magnetic field and can charge these nano particles through the photoelectric effect (for early-type star) or stellar wind (for late-type stars).