The Inner 25 au Debris Distribution in the Eri System

The pipeline-produced Level 3 data products (i.e., nod-subtracted, dithers aligned, flux-calibrated merged data) were the basis for further analysis (coadding and custom background subtraction). The Eri observations consist of 10 Level 3 images. Visual inspection of these images found one of the Level 3 products (#6, with the shortest total on-source integration time) has elongated image shapes. We did not use these data for the final coadd. We coadded the good images at sub-pixel levels with two different registration methods. The first method was to define the centroid of the source by fitting a 2D Gaussian profile⁶ to the 10 × 10 pixel area centered on the source. The measured FWHM of the source in each of the images is given in Table 1. The average FWHM is 41 × 34 ± 04 × 03; on average, there is 9% variation in source FWHM among the nine good images. The second method was to determine the sub-pixel shifts by cross correlating the central 10 × 10 pixel area centered around the source. In both methods, each of the images was registered in sub-pixel levels, and then coadded with weights determined by its integration time.

The flatness of the background in the vicinity of the target is an important factor to assess the extension of the Eri disk. We used a custom sky subtraction program to take out the large-scale background structure in the final coadded data by fitting a low-power, two-dimensional polynomial on the coadded image with the source region (central 384 region) masked out. The final sky-subtracted coadded data are slightly different depending on the registration methods. We will discuss the subtle difference in Section 3.2.

The SOFIA PSF is not as stable as space-based observatories given that it is an airborne facility. Therefore, it is important to evaluate the instrument PSF using point-source observations like Ceres and stellar calibrators. Ceres was observed eight times with the F348 filter as a low-temperature flux calibrator during FORCAST flights in 2015. In addition, stellar calibration observations with the same filter in 2015 are also included in our analysis: six α Boo, two α Tau, two β UMi, and one γ Dra. We determined the PSF variation by comparing the FWHM of these data using the same method as in Eri. The average FWHM of the Ceres data is 361 × 342 (±028) with a variation of ∼8%. The average FWHM of the stellar calibrator data is 337 × 315 (±020) with a variation of 6%. This suggests that the typical PSF variation in the SOFIA data is 6%–8% for the central core (bright) region. We then combined the stellar observations to build a high signal-to-noise-ratio (S/N) PSF to assess the variation in the wing (faint) part of the PSF. Since the stars have various brightnesses, and these data were obtained at various altitudes (not flux calibrated), each individual Level 3 mosaic was first normalized before combining. The normalization is based on the core flux within a small aperture (radius of 2.5 pixels = 192). Since the S/N in each individual image is relatively high for these bright targets, the centroiding methods make no difference in the final coadded data. We generated two final coadded PSFs: one for Ceres and one for all stellar calibrators together. We also applied an additional sky subtraction as in the Eri data for both coadded PSF data sets. Since both PSFs and our Eri data are coadded from many individual observations, the variation in the PSF should average out. Based on the final combined PSF data, it appears that Ceres is slightly broader than the stellar calibrator (as judged by the measured FWHM). A detailed comparison between the stellar and Ceres PSFs is given in Section 3.1.

To evaluate the FORCAST 35 μm PSF, we generated a theoretical PSF using a custom IDL code with inputs of a wavelength and jitter value to mimic the actual observations. We generated 50 PSFs at wavelengths across the F348 filter, and coadded them with weightings according to the filter transmission to get a composite PSF. We found that a jitter value of 175 and a boxcar smooth factor of 3.4 pixels produce a good match in terms of measured FWHM with the observed stellar PSF.

We compared the two observed PSFs and the theoretical PSF in terms of azimuthally averaged radial profiles computed as follows. We first created a series of concentric rings with a width of 1 pixel (0768) about the source centroid (determined by the 2D Gaussian fit), and computed the average value of all the pixels that fall in each ring. The measurement error at each radius is the standard deviation of all pixels in that ring, divided by the square root of the number of pixels in the ring. Since the FORCAST data are in the background-limited regime, the background noise also contributes to the average flux-measurement error. The background noise per pixel is found by computing the standard deviation of the pixels on a part of the blank region away from the source. The background noise per ring (i.e., background-noise error) is estimated by taking the background noise per pixel and dividing it by the square root of the number of pixels in the ring. The total error in the average flux measurement per ring, therefore, is the measurement error and the background-noise error added in quadrature. Figure 1 shows the normalized radial profiles for the PSF characterization. The high S/N of the Ceres data enable us to track the radial profile up to 25'' from the center, achieving a dynamical range of 10⁴. Overall, the observed PSFs (both Ceres and calibrators) match the theoretical one very well, except for the region at radii of 4''–10''. The exact reason for the mismatch is unknown, but probably related to how the data were taken and combined. As shown in Section 2.2, the Ceres PSF is slightly broader than the stellar PSF by 8% in the measured FWHM. Being a red source, Ceres is expected to be slightly larger due to diffraction across the bandwidth of the filter. However, the color difference can only account for a 2% difference between 5000 and 200 K sources. Ceres had an apparent diameter of 07 around the time of observations, which can account for an additional 2% difference in measured FWHM. The rest of this discrepancy is most likely due to telescope tracking errors unique to the Ceres observations. The SOFIA telescope uses a different technique to track on non-sidereal targets and the tracking can be slightly less accurate than sidereal tracking. Despite its high S/N, for this reason we cannot simply use Ceres as our PSF standard for model convolution (see Section 5). Instead, we constructed a hybrid PSF with the core (inner 10'') high S/N) region using the observed stellar PSF and the wing (outer 10'') region using the (noiseless) theoretical PSF. We used this hybrid PSF for all of the following analysis.

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The subtle difference between the two registration methods in combining the Eri data is best shown in the measured FWHM and the azimuthally averaged radial profiles for the image of the star (Figure 2). The left panel of Figure 2 shows the central 12'' region, and the right panel shows the full range up to 30''. The centroiding method gives a slightly sharper image by ∼3.5% (FWHM = 363 × 348), but the surface brightness agrees within 1σ as shown in the radial profiles. The profile outside 10'' is within 3σ of zero and is consistent with no signal within the expected uncertainties in flat fielding. Therefore, we only concentrate our further analysis for the region inside a radius of 10'' around Eri.

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Compared to the profile of the stellar calibrators (i.e., the hybrid PSF), the Eri profile is slightly extended in the range of 4''–10'' from the star.⁷ The slight extension is mostly evident at radii of 3''–5'' (10–16 au at the distance of Eri), and is independent of the registration methods. For simplicity, we adopt the centroiding coadded image for further analysis. The final coadded image of the Eri system is shown in Figure 3. The stellar photosphere of Eri is estimated to be 0.81 Jy at 34.8 μm (Section 4.1). The total flux within 10'' is 1.30 ± 0.09 Jy, suggesting that we detect an inner (<30 au) excess that is slightly more extended than the PSF.

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We estimated the photospheric output of Eri as follows. Since the star is only slightly cooler than the Sun, we used the carefully determined SED of the Sun as a starting point (Rieke et al. 2008). We took the effective temperature of the Sun to be 5780 K and adjusted the overall shape of the assumed SED of Eri by the ratio of blackbodies, one at the temperature of the Sun and the other at a temperature assigned for Eri. We left the latter temperature as a free parameter and varied it to minimize χ² as determined relative to photometry of the star at H, K_S, Spitzer IRAC1, IRAC3, IRAC4, WISE W3, and W4.⁸ We omitted the IRAC2 band because it contains the CO fundamental absorption, which we expect to be deeper in Eri than in the Sun. We found a sharp minimum in χ² at a temperature of 5127 K, which is in satisfactory agreement with the nominal temperature for a K2V star (the type of Eri, Di Folco et al. 2004) of 5090K. The photospheric flux densities indicated by this procedure are 0.81 Jy at 34.8 μm (FORCAST F348 band), and 1.74 Jy at 23.68 μm (MIPS 24 μm band).

To characterize the excess emission at 34.8 μm near the star, PSF subtraction is necessary. We scaled the hybrid PSF to match the photosphere of Eri by normalizing its total flux within an aperture of 12'' to be 0.81 Jy without sky annulus (the sky is zero in the hybrid PSF). To account for the absolute flux calibration uncertainty, which includes (1) the photospheric prediction, 2%, and (2) the FORCAST flux calibration 6%,⁹ the PSF subtraction was also performed after scaling the hybrid PSF within ±6.3% of the nominal photospheric value, allowing us to set the lower and upper boundaries of the uncertainty in the PSF-subtracted image. The nominal photospheric subtracted image is shown in Figure 3(b), and the excess-only radial profiles are shown in Figure 4. The resultant peak flux in the excess-only profiles varies by 30%–40%, depending sensitively on the exact scales of the PSF fluxes. However, these differences decrease significantly for the region outside the FWHM of the beam (i.e., outside a radius of 2''). We also evaluated the impact from the variation of the PSF FWHM (8%, see Section 2.2) in the photospheric subtraction. Using a narrower PSF, the resultant disk flux near the core is expected to be lower while the flux outside the core region would be slightly higher. Using a broader PSF, the resultant disk profile should have an opposite effect (i.e., higher flux near the core and slightly lower flux outside the core). We tested the changes by artificially broadening and sharpening the scaled PSF by 8%, and found that the resultant disk profiles are still within the uncertainty boundary set by the absolute flux calibration. Therefore, the extension (compared to the PSF profile) beyond 2'' is robust, not subject to the uncertainties in absolute flux calibration nor to the PSF subtraction.

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The excess emission is resolved at 34.8 μm by ≳2 beam widths (i.e., the emission region is extended beyond 10 au). The excess flux at 34.8 μm within 10'' is 0.49 ± 0.09 Jy, 60% of the stellar photospheric output. The observed profile is consistent with (1) a broad Gaussian structure peaked at the star, with a width of 18 au (green line in Figure 4), or (2) an unresolved source at the center plus a Gaussian-profile ring peaked at 10 au, with a width of 10 au. In summary, the SOFIA data confirm the excess emission near the star within 20 au, but cannot differentiate whether the emission region is one broad ring or composed of two separate structures (e.g., an unresolved source plus a ring).

We searched the Spitzer archive and found unpublished MIPS 24 μm data (AOR 8969984), which account for an additional 50% of integration depth in addition to the published data (AOR 4888832) presented in Backman et al. (2009). We used the MIPS instrument team in-house pipeline (Gordon et al. 2005; Engelbracht et al. 2007) to reprocess these data to correct for instrument artifacts, and combined all data into a final mosaic. The combined data appear to be point-like, but with a FWHM of 571 × 563, slightly extended compared with a typical point source (550 × 542). We subtracted the photospheric contribution by scaling a blue calibration PSF to the estimated photospheric value (1.74 Jy). The photospheric subtracted image has a FWHM of 717 × 689, much broader than that of a point source.

To characterize the excess emission at 24 μm further, we computed the azimuthally averaged radial profiles of the excess emission, shown in Figure 5. As at 35 μm, the uncertainty in the photospheric profiles was estimated by repeating the analysis with adjustments of ±2.8% in the nominal photospheric value (2% from the absolute flux calibration and 2% from the photospheric extrapolation). The new 24 μm surface brightness profile is very similar to the one published by Backman et al. (2009) (Figure 5), but the improved reduction substantially reduces the errors at larger radii (enhanced by $1/\sqrt{N}$ where N is the number of pixels in each of the annuli). Compared to the profile of a point source, the photospheric subtracted image has the first and second dark Airy rings (corresponding to 20 and 65 au in Eri) partially filled, supporting the evidence that the excess emission is slightly resolved in the MIPS 24 μm band.

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To gain insights into the spatial distribution of the excess emission, we constructed a simple geometric two-ring model. The first ring is fixed at the star position with a specified width and represents the emission inside 20 au, and the second ring represents the emission from the cold Kuiper-belt-like ring, with a specified peak position and width. A final, high-resolution synthesized image is the sum of these two rings with a given relative flux ratio and a fixed total flux; i.e., there are four free parameters in this two-ring model: the width of the central ring, the peak position of the outer ring, the width of the outer ring, and the relative flux ratio. These high-resolution model images were then inclined to view at 30° from face-on, and convolved with the 24 μm PSF to simulate the observations.

We varied the four model parameters to minimize χ² relative to the observed radial profile within 25''. We found that a central ring with a radius of ∼13 au and an outer ring peaked at 64 au with a width of ∼24 au can fit the observed radial profile relatively well (red line in Figure 5). It is reassuring that the peak position of the outer ring is found to be similar to the location found in other studies (∼64 au, Backman et al. 2009; MacGregor et al. 2015; Chavez-Dagostino et al. 2016). However, as in the 35 μm profile analysis, this does not indicate there is only one inner excess region, but only that the inner excess emission region is extended at least to ∼13 au.

Details about the Spitzer/IRS observations were given in Backman et al. (2009). As stated in that paper, linearity and saturation are an issue in the IRS low-resolution data. Therefore, we only discuss the high-resolution data (SH: 9.9–19.5 μm with a slit size of 47 × 113 and LH: 18.7–37.2 μm with a slit of 111 × 223). We retrieved the extracted high-resolution spectrum from the CASSIS website¹⁰ and used the "optimal" product, which simultaneously determines the source position and extraction in the two different nod observations. As described in Lebouteiller et al. (2015), this mode of extraction produces the best results when the source is resolved but only marginally extended. The CASSIS spectrum is shown in Figure 6 in the λ⁴ F_λ versus λ format so a Rayleigh–Jeans spectrum is flat. There is a flux jump between the two SH and LH modules: the SH part of the spectrum is lower than the expected photosphere determined in Section 4.1, and the LH part of the spectrum is slightly lower than the MIPS 24 μm photometry. We joined the two modules by (1) scaling the SH module by 1.09 so that the 10–12 μm region matches the expected photosphere level, and (2) scaling the LH module by 1.02 so it matches the MIPS 24 μm photometry. The scalings are within the uncertainty in the absolute flux calibration between the MIPS and IRS instruments.

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We used the final combined and smoothed (to R = λ/Δλ ∼ 30) spectrum (the black line in Figure 6) to estimate the dust temperature of the excess emission. Although the excess is not exactly blackbody-like, the excess emission can be described by a blackbody emission with temperatures between 170 and 130 K. In Figure 6, we overplotted two (dashed) curves to represent the sum of the photosphere and a blackbody emission of 170 and 130 K (both normalized to the MIPS 24 μm photometry point). This comparison suggests that the excess emission is consistent with a dust temperature of ∼150 ± 20 K.

For blackbody-like emitters (1 mm astronomical silicates), these temperatures correspond a stellocentric distance of 1.5–2.5 au; however, for 1 μm silicate-like grains the corresponding distance is larger (∼2–3.5 au) (see Figure 7). This temperature-radius relation is also composition dependent. In Figure 7 we show the temperature distributions using icy silicates (90% of ice by volume). Icy grains are generally poor absorbers; therefore, at the same temperature they are located at smaller stellocentric distances compared to the same size, bare silicates.

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The stellar wind drag for Eri is found to be 28 times stronger than the P–R drag, based on the measured mass-loss rate 30 times higher than that of the Sun (Wood et al. 2002) and the average solar wind velocity. Therefore, a significant inward flow of dust grains from the cold Kuiper-belt-like region is expected, unless there is a massive, shepherding planet interior of the cold belt. Assuming no such planet, Reidemeister et al. (2011) modeled the marginally resolved Spitzer 24, 70, and 160 μm images and found that the Spitzer data are consistent with this possibility.

Using the no-planet configuration (i.e., no dynamical perturbation in the dragged-in dust flow), we computed model images with parameters derived from Reidemeister et al. (2011) to compare with our new data. The derived radial disk profiles are shown in the upper panel of Figure 8 with the SED shown in the bottom panel. The model SED is a replicate of the bottom panel of Figure 8 in Reidemeister et al. (2011) using the same grain parameters and composition. The model disk surface brightness profile at 24 μm is consistent with the one published in Reidemeister et al. (2011) (i.e., the fit is good for the old, large error-bar profile). However, the 35 μm model disk profile under-predicts the disk flux within 4'' (12 au), and overpredicts it outside 6'' (20 au). The proposal by Reidemeister et al. (2011) that the entire inner warm disk could result from dragged-in grains is not consistent with the SOFIA measurements.

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Resolved images of Eri obtained by Herschel suggest inner excess emission at 70 and 160 μm. Greaves et al. (2014) modeled this inner excess as one single disk with simple geometric parameters: a wedged disk with a radial span of 3–21 au in an r⁻¹ density distribution and an opening angle of ±23°. Note that the opening angle is quite large, i.e., the disk is vertically extended. Due to the significant scale height in this model, we computed the model images with the code dustmap ver. 3.1.2 (Stark 2011), which can take a 3D structure as an input, with the geometric parameters listed above and an inclination angle of 30°. We assumed grain parameters using compact astronomical silicates with a minimum grain size (a_min) of 1 μm, a maximum grain size (a_max) of 1000 μm, and a particle size power-law index (q) of −3.5. Figure 9 shows the results. This model fits the disk profiles very well (the upper panel of Figure 9), and reproduces the mid-infrared SED reasonably well with the normalization set by fitting the disk profiles. For the sake of completeness, we also computed the cold-belt SED using the parameters given by Greaves et al. (2014)—a broad, geometrically thin disk from 36 to 72 au with a constant surface density. For this cold component, we used icy silicates with a_min of 1 μm and a_max of 1000 μm in a q = −3.5 size distribution. We did not include this component when computing the 24 and 35 μm disk profiles since its contribution is insignificant at these wavelengths. The combined SED fits the observed points satisfactorily. However, we note that the inner radius of this cold disk (36 au) is significantly smaller than the one inferred from the millimeter observations (∼53 au from MacGregor et al. 2015, and ∼59 au from Chavez-Dagostino et al. 2016). This might suggest that the broad, cold disk geometry is too simplistic, and a more complex distribution (e.g., a narrow cold belt plus an outer warm belt (see Section 5.3) or a small amount of dragged-in grain component) is needed.

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The choice of a_max has no impact on disk profiles nor the SED at the model wavelengths; however, the value of a_min does make a noticeable difference at 24 μm and in the overall shape of the SED shortward of ∼20 μm. In general, the a_min is usually set to the radiation blowout size (${a}_{\mathrm{bl}}=1.14\ {Q}_{{pr}}\ {L}_{* }\ {M}_{* }^{-1}\ {\rho }_{g}^{-1}$ where ρ_g is the grain density and Q_pr is the radiation pressure efficiency averaged over the stellar spectrum, and is assumed to be 1 for grain sizes comparable or larger than the wavelength where the stellar spectrum peaks, Burns et al. 1979). However, as noted by Reidemeister et al. (2011; Figure 2 in that paper), for the luminosity and mass of Eri, the blowout size does not exist. In the Reidemeister et al. (2011) model, a_min was formally set to be 0.05 μm, but much of the dust cross section in their modeled size distribution comes from larger, μm-sized grains (see their Figure 4). Setting a_min to be 0.05 μm initially, we found that it is difficult to obtain simultaneous good fits to the disk profiles at both wavelengths; i.e., a good fit at 35 μm produces a too bright and broad MIPS 24 μm profile. Setting a_min to larger sizes significantly improves the fit at 24 μm as shown in Figure 9. Setting different a_min for the cold broad disk has no noticeable difference in the resultant SED. We will discuss the physical reason why a large a_min is preferred around Eri in Section 6.1.

The third model we tested is similar to the two-belt model proposed by Backman et al. (2009) based on the marginally resolved Spitzer images. As noted by Backman et al. (2009), the exact locations of the warm belts (∼3 au and ∼20 au) are not well constrained by either the images or the SED. As suggested by the Herschel measurements, the outer warm component is likely to be smaller than was inferred from the Spitzer data. Greaves et al. (2014) suggest that the warm planetesimal belt is located at 14 au with a width of 4 au.¹¹ If the inner warm belt were a direct analog of our own Asteroid belt (near the ice line), the expected location should be 1.5–2 au simply from scaling by stellar luminosity. In fact, such a small size for the inner warm belt would be more consistent with the presence of Eri b (see discussion in Section 6.2).

In light of these results, we construct a revised two-belt model with the following parameters. Both belts are assumed to be geometrically thin (no scale height) and with a constant surface density, and to contain grains from a_min = 1 μm to a_max = 1000 μm in a power-law size distribution with q set to −3.65 (e.g., Gáspár et al. 2012). The inner warm belt is assumed to range from 1.5–2 au, and the outer warm belt ranges from 8 to 20 au. Note that the outer warm belt appears to be broad, but most of the emission at the model wavelengths comes from the inner 8–12 au region due to radial dependence of the dust temperatures since we used a constant surface density. Furthermore, in order to produce overall good fit to the system's SED, we have to use icy silicates for the outer warm belt to suppress the total flux in the mid-infrared while providing enough flux in the range of MIPS-SED spectrum (also see Figure 6 in Reidemeister et al. 2011). Similarly, we also include a SED fit for the cold planetesimal disk ranging from 55 to 80 au (the radial span derived by MacGregor et al. 2015) using icy silicates with the same grain size distribution as in the warm belts.

The resultant model profiles are shown in the upper panel of Figure 10 with the combined SED in the bottom panel. This revised two-belt model fits the 35 μm profile reasonably well (within 1σ), but is slightly too broad/bright outside 5'' at 24 μm (but still within 3σ). The largest issue with models of this class is that they fall slightly short of the measured SED between 20 and 30 μm, which might indicate that the assumed geometry is too simple, or might be revealing a problem for this type of model. We also produced the model fits for a larger (3–4 au) asteroid belt as originally suggested by Backman et al. (2009) with the same outer warm belt (3–20 au) and SED parameters. The results are very similar in the disk profiles with a slightly better SED fit in the 20–30 μm region. Limited by the uncertainty in the exact shape of the mid-infrared excess, both asteroid-belt models produce similar results.

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We tested three proposed debris distributions in the inner 25 au of the Eri system using the newly obtained mid-infrared disk profiles. We found that the 24 and 35 μm emission is consistent with the in situ dust distribution produced either by one planetesimal belt at 3–21 au (e.g., Greaves et al. 2014) or by two planetesimal belts at 1.5–2 au (or 3–4 au) and 8–20 au (e.g., a slightly modified form of the proposal in Backman et al. 2009). The observed profiles are not consistent with the case dominated by dragged-in grains (uninterrupted dust flow from the cold Kuiper-belt-analog region) as proposed by Reidemeister et al. (2011). This might suggest the need of a planet interior to the 64 au cold belt to maintain the inner dust-free zone, or a very dense cold belt where the intense collisions destroy the dust grains before they have enough time to be dragged in. In either case, some amount of dragged-in grains from the cold belt can still contribute a fraction of the emission inside 25 au; the exact amount remains to be determined by future high spatial resolution data and improved collisional models for the cold belt.

The model derived dust fractional luminosity (f_d) is 3 × 10⁻⁵ and 7 × 10⁻⁵ for the inner and outer warm belts, respectively, in the revised two-belt model. For the broad disk model, the dust fractional luminosity is 6 × 10⁻⁵. We adopt the simple analytical model proposed by Wyatt et al. (2007) to test whether the dust in the inner Eri system is produced by transient events. Assuming the typical parameters for disks around solar-like stars and an age of 800 Myr, the maximum dust fractional luminosity (f_max) is 1 × 10⁻⁶ and 4 × 10⁻⁵ for a belt at 2 and 10 au. Therefore, the observed dust levels in these belts are close to the expected maximum value for dust being generated through collisional grinding, and do not require to invoking transient events.¹²

We note that the model parameters (especially the grain parameters) are not unique in our test cases, due to degeneracy between grain properties and dust location. This is particularly true for the cold belt component since we do not include the image fits to the far-infrared and mm wavelength data (beyond the scope of the paper). For simplicity, we adopted the astronomical silicates as the grain composition in modeling the component inside 10 au. The mismatch in the 20–30 μm SED region might partially be due to this choice, in addition to the simple geometry assumed for the inner component. Furthermore, a larger (3–4 au) inner warm belt in the two-belt model produces similar results. Nevertheless, the existence of an inner (within 25 au) separate dust source (i.e., planetesimal belt(s) different from the outer cold belt) is a robust conclusion inferred from the newly obtained SOFIA data.

In Section 5, we showed that a_min ∼ 1 μm gives a much better fit to the mid-infrared disk profiles compared to the models using a smaller size cutoff in the particle size distribution. Observationally a wide range of a_min has been inferred from resolved imaging and mid-infrared spectroscopy. Since silicate-like small grains have solid-state features in the 8–25 μm region, a featureless emission spectrum is usually interpreted to indicate a lack of small grains. However, the definition of "small" depends on the dust composition. For example, for amorphous silicates (like astronomical silicates), the 10 and 20 μm features have very similar shapes for grain sizes of 0.05–1 μm (with a slightly sharper 10 μm feature toward smaller sizes). As a result, at the same dust temperature the resultant emission spectrum is very similar, and the only difference lies in the amount of emission. Therefore, the IRS spectrum in the Eri inner region provides no constraint on the minimum grain size, as also has been suggested by Backman et al. (2009; who found that the spectrum only requires a ≲ 3 μm).

As discussed in Reidemeister et al. (2011), the nominal blowout size does not exist in Eri, due to its low mass and luminosity. However, a possible blowout size might exist if pressure exerted by stellar winds is invoked. In a case where the contribution of the stellar wind is strong enough, a sum of the radiation pressure force and stellar wind pressure force could reinforce a_bl, as has been suggested in the AU Mic disk (Figure 1 in Schüppler et al. 2015). However, for a K-star like Eri, the effect is too small, resulting again in no blowout limit.

Theoretically, we expect some depletion in the inner region around Eri due to enhanced drag forces (P–R and stellar wind drags). The P–R timescale depends on the mass of the star, the location of the dust (R) and the ratio between radiation force and gravity (β ∝ a⁻¹), is given as

$$\begin{eqnarray}&&{\tau }_{\mathrm{PR}}=400\ \mathrm{year}\ \displaystyle \frac{{M}_{\odot }}{{M}_{* }}{\left(\displaystyle \frac{R}{\mathrm{au}}\right)}^{2}\displaystyle \frac{1}{\beta }.\end{eqnarray} \tag{ 1 }$$

Therefore, τ_PR ∼ 4 × 10³ yr for a belt at 2 au around Eri (assuming β = 0.5). Such a dust belt with a f_d of 3 × 10⁻⁵ has a collisional timescale of ∼4 × 10⁴ yr, roughly 10 times longer than the P–R timescale. With the additional aid of stellar wind drag from the active star, the drag timescales will be even shorter. Since smaller particles are dragged in faster than larger ones, we expect a flatter size distribution at small sizes, setting an effective a_min larger than the typical size in a collision-dominated system.

One can estimate a_min, or the dominant/critical grain size (a_c), by balancing the two source and sink timescales as suggested by Kuchner & Stark (2010) and Wyatt et al. (2011). In the inner region of Eri, the source timescale is the collisional timescale, and the sink timescale is the transported timescale by stellar wind since the stellar wind drag dominates the P–R drag. We then re-derived Equation (6) of Kuchner & Stark (2010) as follows:

$$\begin{eqnarray}{a}_{c} & = & 1000\ \mu {\rm{m}}\ {Q}_{\mathrm{sw}}{\left(\displaystyle \frac{{\rho }_{g}}{1{\rm{g}}{\mathrm{cm}}^{-3}}\right)}^{-1}\left(\displaystyle \frac{{\dot{M}}_{\mathrm{sw}}}{30{\dot{M}}_{\odot }}\right)\\ & & \times {\left(\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\right)}^{-1/2}{\left(\displaystyle \frac{R}{1\mathrm{au}}\right)}^{-1/2}{\left(\displaystyle \frac{{f}_{d}}{{10}^{-7}}\right)}^{-1},\end{eqnarray} \tag{ 2 }$$

where Q_sw is the stellar wind pressure efficiency, and ${\dot{M}}_{\mathrm{sw}}$ is the stellar wind mass-loss rate. Assuming Q_sw = 1, ρ_g = 2.5 g cm⁻³, a mass-loss rate of 30 times the solar value, M_* = 0.82 M_☉, R = 2 au, and f_d of 3 × 10⁻⁵, the critical grain size is about 1 μm.

Using the energy-conservation criterion, Krijt & Kama (2014) analytically derived the lower boundary of the particle sizes produced in debris disks. Under plausible parameters, they found that a_min could be much larger than a_bl depending on the collision velocity, material parameters, and the size of the largest fragment. Thebault (2016) numerically investigated such an effect, and concluded that the surface energy constraint generally has a weak effect for early-type stars and wide (50–100 au) debris disks, but might be more pronounced for Sun-like stars and narrow belts. The depletion of the small grains is hard to estimate since we lack detailed information on the planetesimal belt(s) in the inner region of Eri. As an optimal case (since some parameters of their model are quite uncertain), the ratio between a_min and a_bl is ∼3 for a 2 au belt around Eri (0.34 L_☉ and 0.82 M_☉) (Equation (7) in Krijt & Kama 2014). Our choice of a_min ∼ 1 μm is consistent with this limit and the critical size estimated in the previous paragraph.

The existence of Eri b and its orbital parameters have been heavily debated since it was reported in 2000 (Hatzes et al. 2000; Benedict et al. 2006; Butler et al. 2006). The Exoplanet Encyclopedia lists the planet at 3.5 au radius as confirmed, but Zechmeister et al. (2013) combined 15 yr of radial velocity data and found that it did not support this status for the case of a highly eccentric orbit (e ∼ 0.6). A similar result was also been found by Anglada-Escudé & Butler (2012). Howard & Fulton (2016) analyzed all available radial velocity measurements from Lick and Keck Observatories by the California Planet Survey, and found that the stellar CaII H& K emission does not correlate with the radial velocity. Thus the radial velocity modulation is likely caused by an external source, i.e., the planet. Combining available ground-based high-contrast imaging, Mizuki et al. (2016) presented an updated contrast curve around Eri, and could marginally rule out the 1.6 M_J, e = 0.7 case (Benedict et al. 2006) if the age of the system is as young as 200 Myr.

As noted by Backman et al. (2009), the possibility that Eri b is on a highly eccentric orbit is inconsistent with the existence of an inner warm debris belt at ∼3 au. If the planet co-exists with an asteroid-belt analog (i.e., one within 2 au), its orbit must have low eccentricity (e ≤ 0.2), based on the dynamical stability study by Brogi et al. (2009). We estimated the width of the planet's chaotic zone by assuming Eri b is 1.6–1.7 M_J with e = 0–0.2. The inner boundary of the chaotic zone is then at 2–2.6 au, with 3.5–5 au for the outer boundary using the formulae from Mustill & Wyatt (2012) and Morrison & Malhotra (2015). Any planetesimal belt in the inner region of the Eri system must be located inside 2 au and/or outside 5 au to be dynamically stable with the assumed Eri b. For this reason, we constructed the asteroid-belt analog at 1.5–2 au in Section 5.3.

If there is no Eri b, the location of the asteroid belt could be at 3 au as proposed by Backman et al. (2009) using the dust temperature argument. Since current data put no constraints on the exact number of planetesimal belts (one or two) in the inner Eri region, the inner warm component could be a dragged-in component from the outer warm planetesimal belt (∼8–20 au). This remains a possibility because the P–R timescale is similar to the collisional timescale in a 10 au belt around Eri (and would be shorter with the aid of stellar wind drag).

Although the new SOFIA data rule out the drag-dominated transported model for the inner debris, another version of the transported dust scenario, involving disintegration of icy comets scattered inward by the planets interior of a cold planetesimal region (Morales et al. 2011; Bonsor & Wyatt 2012), remains plausible for the source of inner debris. As the perturbed icy planetesimals get closer to the ice line, sublimation of volatile material likely causes them to become active comets, populating the inner region with dust. This mechanism is found to be the primary source of the warm dust in the inner region of the solar system (Nesvorný et al. 2010; Ueda et al. 2017). However, there is circumstantial evidence that the warm excesses in some other systems are aligned with their primordial ice lines, not the current-day ones, and thus the dust arises from in situ planetesimal belts (Ballering et al. 2017, submitted). Comets release material primarily in the form of coarse millimeter- to centimeter-sized dust grains, and the radial span of the big grains from active comets should be broad, as is the radial distribution of comets themselves. For an in situ planetesimal belt, the distribution of large dust grains should be narrower, because the eccentricities of the parent bodies are expected to be lower than those of comets. As a result, the millimeter emission from the in situ planetesimal belt should be confined in a narrower distribution, and thus be more easily detectable, than the one from the cometary grains. Therefore, the ultimate test to differentiate the in situ and transported origins of the warm dust is to detect the submillimeter/millimeter emission from large grains in the inner region.

Eri has been observed by many radio single-dish and interferometric facilities, which provide some constraints on the amount of submillimeter/millimeter emission in the inner region. Eri is a young and active star, and is known to possess excess free–free emission at 7 mm (MacGregor et al. 2015), making it difficult to evaluate the possible excess emission at millimeter wavelengths using integrated photometry. No confirmed excess emission is reported by Lestrade & Thilliez (2015) and MacGregor et al. (2015) at 1.2/1.3 mm, while Chavez-Dagostino et al. (2016) report an (∼5σ) excess of 1.3 mJy at 1.1 mm in the inner 18 au region. However, the high-resolution ALMA 1.3 mm image of the system rules out any narrow belt with a total flux density of >0.8 mJy in the inner region (M. Booth 2017, private communication). The ALMA observation was one single pointing offset from the star designed to image the northern part of the cold ring, providing poor coverage of the inner region. The millimeter detection of the inner debris remains controversial.

We predict the flux density of the inner debris at 1.3 mm using the three tested models in Section 5. For the dragged-in model, the millimeter flux is much less than 10 μJy due to lack of large grains. For the puffed-up disk model and the inner 1.5–2 au belt model (asteroid-belt analog), the total flux density is ≲16 μJy at 1.3 mm, making the millimeter detection very challenging. The outer warm belt model (8–20 au) gives a total flux density of 0.7 mJy at 1.3 mm, comparable to the expected stellar emission. Given the difficulty of predicting the stellar output in the millimeter wavelengths, the best way to confirm such a belt is to resolve it from the star. If the outer warm belt is narrow in the millimeter wavelengths, a belt at 13 au (∼4'' radius) would have a surface brightness of 27 μJy/beam at 1.3 mm assuming a beam of 1'' and an unresolved width. The ALMA observation obtained by M. Booth (2017, private communication) reaches an rms of 14 μJy/beam. Under nominal conditions (e.g., if the source region was centered at the primary beam), the proposed outer warm belt could be detected at ∼2σ, making a confirmed detection (>3σ) difficult. If the outer warm belt is slightly larger or broader than the beam size, the expected surface brightness would be less as a result.