THE STRUCTURE OF THE β LEONIS DEBRIS DISK

Details about the method of nulling interferometry employed by BLINC can be found in Appendix A. Briefly, the method involves delaying a beam from one aperture by half a wavelength and then overlapping it with a beam from a second aperture. The destructive interference that results can be arranged to suppress the light coming from the central star, allowing faint, extended emission to be observed. Specifically, BLINC is sensitive between radii of ∼012 and 08.

Observations were carried out on 2008 March 27 at the 6.5 m MMT, located on Mount Hopkins, AZ. They consisted of taking ten 1 s frames on targets and calibrators with BLINC set to interfere the apertures destructively. These measurements were followed by nodding the telescope 15'' to take ten 1 s background frames. The process was then repeated with BLINC set to interfere the apertures constructively. To maximize observing efficiency, 40 frame sets of destructive images were occasionally taken for β and o Leo, since these frames had low signal to noise, while constructive images were skipped. All of the observations reported here used an N-band filter, covering roughly 8–13 μm.

Table 1 lists the number of frames taken for each star in both destructive and constructive modes of BLINC, as well as the number of usable frames. We discarded the observed frames when the adaptive optics (AO) system broke lock in the middle of taking a data set (often due to wind gusts), and when BLINC did not properly set the pathlength difference between beams. In addition, some frames taken at the end of observing β and o Leo, as well as all the frames taken on the calibrator star α Boo were adversely affected by weather conditions and were discarded. The results of comparing the weather-affected β and o Leo data with the α Boo data are consistent with the data presented here, but with substantially larger error bars.

Observations of β Leo and o Leo were obtained with KIN during runs on 2008 February 16–18 and 2008 April 14–16. The 85 m baseline on KIN provides the ability to detect circumstellar material in the N band at angular separations of a few milliarcseconds from the primary star, within the KIN field of view defined as an elliptical Gaussian with FWHM of 05 × 044 (Colavita et al. 2009). Thus, the observations taken by KIN are spatially complementary to the shorter baseline of BLINC. Each primary aperture at Keck is further divided into two subapertures, with a baseline of 5 m, which are perpendicular to the long baseline. This results in a second, broader set of interference fringes, perpendicular to the primary fringes, which are modulated to take sky background measurements. A detailed description of this procedure can be found in Koresko et al. (2006).

Each individual observation of a star consists of about 10 minutes of data collection, made up of several hundred null/peak (i.e., destructive/constructive) measurements. These measurements are averaged to produce a final value for the null. Observations of science objects are alternated with observations of calibrator stars. Table 2 lists the observations taken at KIN. Further details of the KIN design, observation procedures, and data reduction can be found in Colavita et al. (2006), Koresko et al. (2006), Colavita et al. (2008), and Colavita et al. (2009).

β Leo was observed using all three instruments on Spitzer: InfraRed Array Camera (IRAC; Fazio et al. 2004), InfraRed Spectrograph (IRS; Houck et al. 2004), and Multiband Imaging Photometer for Spitzer (MIPS; Rieke et al. 2004). Details about the observations are listed in Table 3. The observations at 24 μm were done at two epochs (2006 January 13 and 2006 June 8) in standard small-field photometry mode with four cycles with 3 s integrations at five sub-pixel-offset cluster positions, resulting in a total integration of 902 s on source for each epoch. The 70 μm observations were done in both default- and fine-scale modes on 2004 May 31 with a total on-source integration of 250 s for the default scale and 190 s for the fine scale. The MIPS SED-mode observations were obtained on 2006 June 12 with 10 cycles of 10 s integrations and a 1' chop distance for background subtraction, resulting in a total of 600 s on source. The 160 μm observations were obtained with the original photometry mode with an effective exposure of ∼150 s near the source position. All of the MIPS data were processed using the Data Analysis Tool (Gordon et al. 2005) for basic reduction (e.g., dark subtraction, flat fielding/illumination corrections), with additional processing to minimize instrumental artifacts (Engelbracht et al. 2007; Gordon et al. 2007; Lu et al. 2008; Stansberry et al. 2007). After correcting these artifacts in individual exposures, the final mosaics were combined with pixels half the size of the physical pixel scale for photometry measurements. The calibration factors used to transfer the instrumental units to the physical units (mJy) are adopted from Engelbracht et al. (2007) for 24 μm; Gordon et al. (2007) for 70 μm; Lu et al. (2008) for MIPS-SED mode data, and Stansberry et al. (2007) for 160 μm.

The IRS spectral data were processed starting with the Basic Calibrated Data (BCD) products from the Spitzer Science Center (SSC) IRS pipeline S18.7. To maximize the quality of the final spectrum, we adopted the extraction methods developed by the Formation and Evolution of Protoplanetary Systems (FEPS; Meyer et al. 2006) and Cores to Disks (C2D; Evans et al. 2003) Spitzer legacy science teams, and based in part on the SMART software package (Higdon et al. 2004). Full details of the process are presented in Bouwman et al. (2008) and Swain et al. (2008). Here, we briefly summarize the important elements of the extraction technique.

We begin with the droopres intermediate BCD product. Background subtraction is accomplished by subtracting associated pairs of the two-dimensional imaged spectra from the two nodded positions along the slit. This also eliminates stray light contamination and anomalous dark current signatures. Bad pixels were replaced by interpolating the values in the surrounding 8 pixels. A 6.0 pixel fixed-width aperture was used in the extraction. We optimized the position of the extraction aperture for each order by fitting a sinc profile to the collapsed and normalized source profile in the dispersion direction. The irsfringe package⁷ was used to remove low-level fringing for wavelengths >20 μm.

Our custom extraction relies on relative spectral response functions (RSRFs) derived independently using stars from the FEPS legacy program free of any thermal excess emission (Bouwman et al. 2008; Carpenter et al. 2008). This RSRF assumes that the object is perfectly centered in the slit, but slight order mismatches are evident in the extracted spectra suggesting that the object was not centered. Such an offset can induce a wavelength-dependent curvature in the extracted spectra because of the fixed slit size and diffraction-limited point-spread function (PSF). We employed an algorithm developed by J. Bouwman and F. Lahuis (described in Swain et al. 2008) to correct for this offset. This process dramatically reduced the order offsets. However, a small residual offset remains between the SL1 and LL2 modules.

The final absolute flux density calibration is derived by processing the SSC primary IRS calibrators with the exact same algorithms as used for the FEPS stars (Carpenter et al. 2008) and our program star. The uncertainty in the final absolute calibration is estimated to be ∼10%.

β Leo was observed with IRAC at two epochs: 2004 May 7 (AOR Key 3921152 and 4306176); and 2006 December 30 (AOR Key 18010112). In both the 3921152 and 18010112 data sets, the star was observed in full-frame mode with 30 s frame times and a five-position small-scale dither pattern, for a total of 134 s integration time on source in each band. While the 3921152 data set contains data in all IRAC bands, images only at 4.5 and 8.0 μm were acquired in 18010112. In the 4306176 data set, the star was observed in all bands in full-frame mode with a 12 s frame time, repeated twice and with no dithers. Due to the much shorter exposure and the absence of dithering (limiting the spatial sampling of the star on the IRAC arrays), we have discarded the 4306176 data set. For the other two data sets, we first created individual mosaics in each band starting from the BCD (pipeline version S18.5.0), using our custom post-BCD software IRACproc (Schuster et al. 2006), which is based on the SSC mosaiker MOPEX.

As an initial test for data set quality, a 16 × 20 pixel box centered around the star (which corresponds to 08 × 1''on the sky) is used to calculate the BLINC instrument null. An elongated box is used instead of a circular aperture because the aperture for BLINC is elliptical, making the PSF of a star elliptical as well. The 16 × 20 pixel box size ensures most (though not necessarily all) of the flux from the system is measured. Figure 1 shows the average null for each set. Since each set contains either 10 or 40 frames, error bars are determined by the deviation of the nulls in individual frames from the average null of the whole set. A filled point contains good data and is used for further analysis, while a hollow data point is rejected due to weather. Weather-rejected data occurred consistently for all stars when the telescope rotated into a substantial wind. This wind resulted in problems maintaining the telescope's AO system stability and, consequently, the null.

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For the good data sets, the calibration star β Gem has only small variations both within a set (as indicated by the small error bars) and from set to set. The fact that β Gem has a non-zero null demonstrates the necessity of having calibrator stars: minor movements of the star due to vibrations in the system and sky turbulence result in imperfect suppression (chromatic effects on the null are minimized by a ZnSe corrector in the non-shifted beam path). For the science stars, there is greater variation than is seen for β Gem. This is primarily due to the science stars being much dimmer in the N band, making them more sensitive to changes in background sky brightness. Using weighted means to combine all the good measurements for each star, β Gem has an instrumental null of 3.19% ± 0.09%, o Leo has 3.33% ± 0.28%, and β Leo has 4.93% ± 0.29%. Using the β Gem instrumental null as a baseline, we find source nulls of 0.14% ± 0.30% for o Leo and 1.74% ± 0.30% for β Leo. This indicates that, within errors, o Leo does not have resolved emission, while β Leo has resolved emission at 5σ significance.

Information pertaining to the variation of the null with respect to separation from the star can be gained by measuring the null within strips of increasing vertical separation from the center of the PSF. Due to the vertically changing transmission pattern of BLINC (illustrated in Figure A1(a)), such strips are more natural to use than circular annuli. Each data set has its instrumental null calculated within 16 × 3 pixel strips. For each separation, all good data sets for a star are combined via weighted means, which also provides the errors. The instrumental nulls for β Gem, o Leo, and β Leo are shown in Figure 2(a), while Figure 2(b) shows the source null for o Leo and β Leo. The x-axis is expressed as an angular separation from the center of the system, since 1 pixel has a width of 0054.

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In Figure 2(a), both β Gem and β Leo show an increase in the instrumental null beyond ∼05. This is likely caused by residuals from the AO systems, as well as imperfect beam alignment (both of which are also evident in the fact that the calibrator star has a non-zero null). Data for o Leo only extend to 04 since beyond this separation, slow variations in sky background became comparable to the brightness of o Leo itself when being imaged destructively. As a result, the destructive signal became lost in the noise.

The source null of o Leo (Figure 2(b)) does not show evidence (at a 3σ confidence or greater) of any resolved emission within 04. For β Leo, however, an excess null is present out to a separation of at least 06 (beyond which the error bars become too large to determine whether or not extended emission is present). It must be cautioned that while Figure 2 provides evidence of extended emission, it cannot be straightforwardly interpreted to be evidence for continuous emission from the inner limit of BLINC out to at least 06: the measured excess is a result of the incoming excess multiplied by the interferometer transmission function and convolved with the PSF of the telescope aperture. Modeling is necessary to interpret the figure and is presented in Section 4.3.2.

Table 4 shows the calibrated nulls derived from the KIN observations. The nulls in the table are an average of the observations taken on each night and are analogous to the "source nulls" derived for the BLINC data, with both having been calibrated using a calibrator star. The quoted errors are taken to be the larger of the "external" error for that date, or the average of the "formal" errors for all the scans on that date. The "formal" error is defined to be the scatter in the null within a given scan, while the "external" error is calculated as the standard deviation among all scans on a given night, divided by the square root of the number of scans (additional information on these errors can be found in Colavita et al. 2009).

3.2.1. β Leo

3.2.2. o Leo

The photometry at 24 μm was obtained using aperture photometry on each of the five cluster data sets for both epochs. Two aperture settings were used (small: a radius of 623 with sky annulus between 1992 and 2988 and an aperture correction of 1.699; large: a radius of 1494 with sky annulus between 2988 and 4233 and an aperture correction of 1.142). Such aperture-corrected MIPS photometry generally agrees to within ∼1% for a clean, non-saturated point source. The large aperture gives a total integrated flux density of 1623 ± 33 mJy (2% error assumed), which is ∼2.5% higher than the small one (This behavior is similar to that of the resolved disk of γ Oph at 24 μm (Su et al. 2008)). This result suggests β Leo is slightly more extended than a true point source at 24 μm. Therefore, we adopt the large aperture result as the final photometry in the 24 μm band.

However, the FWHM measurements⁸ of the source at 70 μm are consistent with it being a point source as compared to calibration stars that have a similar brightness at this wavelength. The 70 μm photometry was extracted using PSF fitting, giving a total flux density of 711 ± 50 mJy (7% error assumed) in the 70 μm band. The MIPS SED-mode data were extracted using an aperture of 5 native pixels (∼50'') in the spatial direction. The final MIPS SED-mode spectrum was further smoothed to match the resolution at the long-wavelength portion of the spectrum (R = 15).

The 160 μm observation needs additional attention to eliminate the filter leak contamination. The expected stellar photospheric brightness at 160 μm is ∼26 mJy, resulting in a leak strength of ∼390 mJy (∼15 times) that is brighter than the expected disk brightness assuming its emission is blackbody-like. Visual inspection of the 160 μm data confirms that the β Leo observation was mostly dominated by the ghost image (off position). We used the 160 μm observation (AOR Key 15572992 from PID 52) of Achernar (HD10144, B3Ve) for the leak subtraction as both 160 μm observations were obtained in the same way (dithered cluster position). A constant background value of 7.5 MJy sr⁻¹ for Achernar and of 10 MJy sr⁻¹ for β Leo was taken out in each of the mosaics first. We then scaled the Achernar data by a factor of 0.34 for subtraction, and a faint source appeared in the expected position of β Leo after the subtraction. Because the background (most due to instrument artifacts) is not very uniform, the source is only weakly detected. The pixel-to-pixel variation of the observation suggests a point source 1σ sensitivity of 30 mJy in this observation. We placed a 16'' radius aperture at the position of the source to estimate the source brightness. After proper aperture correction (Stansberry et al. 2007), a flux density of 90 mJy is estimated. Achernar is a nearby (42.75 pc; van Leeuwen 2007) fast rotating Be star with a close (015) companion (Kervella & Domiciano de Souza 2007). Kervella et al. (2008) estimated that the companion has spectral type of A1V-A3V based on the near-infrared colors and contributes 3.3% of the combined photospheric emission. The color-corrected IRAS flux densities are consistent with the expected levels of the photosphere represented by the Kurucz model of T_*∼15,000 K, logg = 3.5, and solar metallicity (Kervella et al. 2009), scaled to match the distance of the star and stellar radius of 8.5 R_☉. Using the same photospheric model, the expected 160 μm value for Achernar is 68 mJy. Part of Achernar's photosphere is subtracted off from the β Leo data for the leak subtraction. After correcting this, the faint source at the expected β Leo position has a flux density of 114 mJy. However, this value is a lower limit due to the uncertainty in scaling the leak subtraction. Varying the scaling by ±10% results in a 30% change in the photometry. Because of the difficulty of leak subtraction and low signal to noise of the data, we do not consider the source to be detected at 160 μm. The source brightness is likely within 114–217 mJy. Furthermore, Herschel has measured β Leo with PACS at 100 and 160 μm with flux densities of 500 ± 50 mJy and 230 ± 46 mJy, respectively (Matthews et al. 2010). We then adopted the Herschel values to constrain the amount of cold dust in the β Leo system in later analysis.

Since β Leo is saturated in the IRAC observations at all bands, we extracted the photometry based on PSF fitting. We have derived its Vega magnitudes by fitting the unsaturated wings and diffraction spikes with a PSF derived from the observations of bright stars (Vega, Sirius, Eridani, Fomalhaut, and Indi). The construction of this PSF and the details of the adopted PSF-fitting technique are described in Marengo et al. (2009). The PSF files are available at the SSC Web site⁹. We estimated the magnitude uncertainty by bracketing the best fit with over and undersubtracted fits in which the PSF-subtraction residuals were comparable to the background and sampling noise. The IRAC photometry for β Leo based on the epoch 1 data is magnitudes (Vega) of 1.905 ± 0.011, 1.909 ± 0.011, 1.875 ± 0.011, and 1.927 ± 0.011 for the IRAC [3.6], [4.5], [5.6], and [8.0] bands, respectively. The data quality in the second epoch is lower than in the first one, so we used it only as a rough confirmation of these results. We converted the PSF-fit magnitudes into fluxes by adopting the zero point (Vega) fluxes listed in the IRAC Data Handbook version 3.0 (2006). The resulting photometry and associated errors are listed in Table 5. The spatial extension of the β Leo disk could not be constrained with the IRAC images because of the PSF core saturation. The measured photometry and spectra along with previous measurements from the literature are shown in the spectral energy distribution (SED) in Figure 3.

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To determine the stellar photospheric emission, we fit all available optical to near-infrared photometry (Johnson UBV, Strömgen uvby photometry, Hipparcos Tycho BV photometry, Two Micron All Sky Survey (2MASS) JHK_s photometry) with the synthetic Kurucz model (Castelli & Kurucz 2003) based on a χ² goodness of fit test. Because of the star's proximity, the 2MASS photometry is saturated and unusable. In addition, β Leo has been found to have a hot K-band (2.14 μm) excess of 2.17 ± 1.4 Jy (Akeson et al. 2009) (∼2.5% above the photosphere). To overcome these difficulties, we tried to include other ground-based near-infrared photometry. β Leo is a primary bright standard in the UKIRT/Mauna Kea System, with m_K = 1.98, H − K = 0.01, and J − K =0.04¹⁰. Combining all the available data and transforming to the 2MASS system resulted in a K_s of 1.93 mag for the photospheric fitting.

A value of T_eff = 8500 K with a total stellar luminosity of 13.44 L_☉ integrated over the Kurucz model spectrum at the given distance gives the best χ² value. This suggests that the stellar radius is 1.7 R_☉ using the Stefan–Boltzmann equation. In comparison, Akeson et al. (2009) estimate the stellar radius to be 1.54 ± 0.021 R_☉ based on the long-baseline interferometric observation from CHARA array. Their radius was derived including a fit with an incoherent flux (hot excess), while the early VLTI interferometric observation by Di Folco et al. (2004) derived a much larger radius (1.728 ± 0.037 R_☉) without including the incoherent flux. Additionally, Decin et al. (2003) use various methods to determine a stellar radius in the range of 1.68–1.82 R_☉ for β Leo, consistent with a spectral type of A3V. A stellar radius of 1.5 R_☉ is the nominal value for a F0V star with an expected effective temperature of 7300 K, making it too cool to fit the observed photometry for β Leo. We do not attempt to solve the mystery of stellar radius for β Leo. However, we use 1.7 R_☉ as the stellar radius in the thermal models for computing dust temperatures in order to be consistent with the stellar radiation field (the difference in stellar radius produces a ∼22% difference in input stellar luminosity). Using a stellar mass of 2.0 M_☉ and the derived stellar luminosity, the blowout size (a_bl in radius) is ∼3 μm for astronomical silicate grains (Laor & Draine 1993).

Based on the best-fit Kurucz model, we then estimate the stellar photospheric flux densities at the wavelengths or bands of interest. For MIPS observations, the expected photospheric flux densities are computed based on the monochromatic wavelength of each band, while the expected photospheric fluxes are integrated over the bandpasses for the IRAC observations. The additional 2% errors of the photosphere determination (mostly limited by the K-band photometry accuracy) have been added to the IRAC measurements. The broadband photometry obtained by this work and their corresponding photospheric flux densities are listed in Table 5.

4.2.1. Near Infrared

4.2.2. Spitzer Mid- and Far-infrared Photometry

The expected photospheric contribution in the MIPS 24 μm band is 1190 mJy, indicating that the flux seen at 24 μm is dominated by the star and that the excess emission is 433 mJy (before color correction). However, the flux seen in the 70 μm band is dominated by the excess emission (583 mJy before color correction) compared to the stellar photosphere. This suggests that the dust emitting at MIPS wavelengths has a color temperature of ∼120 K. Therefore, color correction factors of 1.055 and 1.054 are applied to the excess emission at 24 and 70 μm, respectively. The infrared excess flux densities are then 457 and 615 mJy in the 24 and 70 μm bands with assumed calibration errors of 2% and 7%, respectively.

At 24 μm, the stellar photosphere was subtracted from the data by scaling an observed PSF to the expected photospheric flux density of β Leo and to 1.25 times this value. This oversubtraction technique has been applied to other 24 μm resolved disks like γ Oph (Su et al. 2008) and Fomalhaut (Stapelfeldt et al. 2004) to reveal excess emission lying close to the star. The results are shown in Figure 4. The FWHM¹¹ of the 24 μm image is 577 × 575 before the photospheric subtraction, but is 688 × 661 after the subtraction, which is considerably larger than a typical red PSF (for example, the ζ Lep disk has a FWHM of 561 × 555; Su et al. 2008). The resolved core emission at 24 μm is best shown in the average radial surface brightness profile (Figure 5). It is evident that the first dark Airy ring (between radii of 55 and 75) in the β Leo profile is filled in compared to a true point source, and the profile matches well with that of a point source at larger radii. The position angle of the disk (after photospheric subtraction) is 118°with an error of 3° estimated from two epochs of data. The ratio (1.041) between the semimajor and semiminor radii in the FWHM suggests that the disk is viewed at 20° ± 10°, close to face-on (a ratio of 1.011 is expected for a point source while a ratio of ∼1.115 is derived from the γ Oph disk with inclination of ∼50°). The low inclination angle of the disk is consistent with the star being inclined at 21 5 from pole-on (Akeson et al. 2009).

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The excess flux densities at 24 and 70 μm suggest a color temperature of 119 K. A blackbody emission of 119 K represents the excess emission (from the spectral shape of both the IRS and MIPS-SED data) reasonably well, although the emission at 15–20 μm is a bit low (see Figure 6). For blackbody-like grains, a dust temperature of ∼119 K corresponds to a radial distance of ∼20 AU in the stellar radiation field of β Leo, consistent with the disk being marginally resolved at 24 μm but not at 70 μm. The total dust luminosity is 1.2 × 10³⁰ erg s⁻¹ based on the 119 K blackbody radiation at the given distance of 11 pc, suggesting a dust fractional luminosity (f_d) of 2.4 × 10⁻⁵.

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4.2.3. IRS Spectra

Using the interferometric observations of o Leo, we were able to put constraints on any dust that may be present in the o Leo system and confirm the calibration that was used in the analysis of the β Leo system. Details on the interferometric analysis of o Leo are presented in Appendix B.

4.3.1. KIN

4.3.2. BLINC

To compare physical models of the β Leo disk to the BLINC data (shown in Figure 2), a program was created to take an input disk geometry, interfere it, convolve it with the aperture PSF, and pass it through the same data reduction pipeline used on real images. Because of the limited data points, these calculations assumed an optically and geometrically thin disk composed of blackbody grains. The dust in the disk was given an optical depth of the form

$$\begin{equation} \tau (r) = Cr^{p}, \end{equation} \tag{ 1 }$$

where C is a brightness scaling constant and p is the power-law index. Assuming a sharp inner (R_in) and outer (R_out) cutoff beyond which τ is zero, there were four variables: R_in, R_out, C, and p.

We first compare the possible disk geometries from the interferometry, using the integrated excess from the Spitzer observations as an additional constraint. Figure 7 shows a set of disk geometries compared to the source null of β Leo. In this and similar figures it should be noted that the hump located near 08 does not correspond to a physical excess at that separation from the star; rather, it is a feature relating to the PSF of the aperture. The disk geometries in Figure 7 use different brightness scaling constants, C, and all assume a uniform (p = 0) disk that extends from 0 AU to 10 AU. For a uniform disk, the values of C are equivalent to the optical depth of the disk. Disk geometries with R_out > 10 AU do not differ significantly from these simulations, since the BLINC data are not sensitive outside this radius. These models are thus good approximations of continuous disks that extend into the outer regions imaged by Spitzer. The best fit for such a "continuous disk" simulation is given as a dotted line in Figure 7 and corresponds to an optical depth of 2.25 × 10⁻⁵. This geometry has a reduced χ² of 4.2, making it a relatively poor fit. More importantly, such a disk is required to have a flux density of 0.515 Jy at 10.5 μm (or ∼9% excess emission above the stellar photosphere). This is inconsistent with the SED-measured excess shown in Figure 3 and the IRS excess spectrum in Figure 6. We therefore rule out a continuous disk extending from 0 AU to 10 AU or beyond as being the source of the excesses measured by BLINC.

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Better fits can be achieved by varying the inner and outer radii. The best possible fits for both a uniform and an inverse (p = −1) disk are with a ring extending from 1 to 2 AU. In both models, there is a reduced χ² of around 0.7. However, these models require the disk to be even brighter at 10.5 μm than the best-fit continuous disk models, and thus are also inconsistent with our other observations. In order to bring the models more in line with the IRS data, we consider only those that are at most 0.30 Jy in brightness (∼5% of the photosphere). Doing so, we find the best-fit models to have reduced χ² of around 2. The models are shown in Figure 8. One model assumes a uniform disk and the other assumes an inverse disk. Both of these model disks have an inner radius of 2 AU and a width of less than 1 AU (although the best-fit uniform disk is slightly more extended than the best-fit inverse disk). Both disks have a flux of 0.30 Jy at 10.5 μm  with the uniform disk having a slightly better reduced χ² (χ² = 1.9) than the inverse disk (χ² = 2.0). The uniform disk has an optical depth of 1.04 × 10⁻⁴ while the inverse disk has an optical depth of 1.50 × 10⁻⁴ at R_in.

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Overall, both uniform and inverse models are similar with regards to their quality of fit. The similarity of these two cases shows that the best fit is insensitive to the power-law index when considering thin rings. Also, as noted above, better fits can be found if we allow for higher fluxes; however, such brighter disks would be inconsistent with other data.

While the above models are the best fits to the BLINC data, R_in and R_out have some flexibility: R_out can be extended to 4 AU before the reduced χ² exceeds 3. Similarly, disks that have brightness less than 0.30 Jy and reduced χ² less than 3 exist for models with R_in as small as 1 AU. Reducing R_in to less than this extends the disk to regions where BLINC strongly suppresses emission and is thus insensitive, resulting in a rapidly increasing disk brightness while minimally affecting the quality of fit. However, KIN becomes increasingly sensitive to regions inside of ∼2 AU, which restricts the amount of excess that can be there and still result in a null detection by KIN. Disk widths larger than 1.5 AU result in poor fits (χ² greater than 3). However, disks narrower than 1 AU do not strongly degrade the fit; at widths less than ∼0.5 AU, the resolution of BLINC makes the models degenerate in brightness and width.

With these possible geometries, we look in more detail at the flux constraints from the interferometry. Based on the null analysis in Section 3.1, the excess was found to be 1.74% ± 0.30% over the photosphere. If we take the photosphere to be 6.3 Jy at 10.5 μm, this excess would be 0.11 ± 0.02 Jy. However, for a near face-on disk, the interference pattern will suppress at least half of the light, and possibly more depending on the location of the dust, which sets a lower limit of 0.22 ± 0.04 Jy on the dust emission in this ring. This is consistent with our uniform and inverse model simulations that find that any disk with a flux fainter than 0.2 Jy would result in poor fits. Taking into consideration the IRAC 8 μm and IRS data, the maximum brightness for this BLINC component is 0.3 Jy.

We have used the VMT to model what level of signal, if any, KIN would see from this belt. We find that a 0.3 Jy belt extending from 2 to 3 AU creates a null in KIN of −0.2%. This negative null is likely due to the complex transmission pattern of KIN (see Figure A1(b) in Appendix A) and the fact that the belt would lie in the sidelobes of this pattern. Combined with the partially resolved photosphere of β Leo, the net signal from the system is 0%. For a belt that is less than 0.3 Jy in brightness, the combined null approaches 0.2% (as we would expect). For a belt that has an inner radius as small as 1 AU, the predicted null would still not be large enough to exceed the 0.9% limit that has been established. So, for belts less than 0.3 Jy, the resulting nulls predicted by the VMT are fully consistent with our KIN data. In summary, using the simple models outlined above we then conclude that the excess component detected by BLINC is consistent with a narrow ring of 2–3 AU with a brightness of 0.25 ± 0.05 Jy at 10.5 μm.

We assume a simple, geometrically thin (one-dimensional) debris disk where the central star is the only heating source in the system (optically thin). The dust is distributed radially from the inner radius (R_in) to the outer radius (R_out) and governed by a power law of radius r for the surface number density with an index p (Σ(r) ∼ r^p, where p = 0 is a disk with constant surface density expected from a Poynting–Robertson drag dominated disk while p = −1 is a disk expected from grains ejected out of the system by radiation pressure). We further assume that the grains in the disk have a uniform size distribution at all radii, following a power law with minimum cutoff radius of a_min, maximum cutoff radius of a_max, and a power index of q = −3.5 (n(a) ∼ a^−3.5), consistent with grains generated in theoretical collisional equilibrium. No obvious dust mineralogical features are seen in the IRS spectrum to favor a specific dust composition; therefore, we adopt the grain properties (size-dependent absorption Q_abs and scattering Q_scat) of astronomical silicates (Laor & Draine 1993) with an assumed grain density of 2.5 g cm⁻³. We then compute the dust temperature as a function of the grain size, the incident stellar radiation (best-fit Kurucz model), and the distance r based on balancing the energy between absorption and emission by the dust (scattering is ignored in this simple model). The final SED is then integrated over the grain size and density distribution.

Figure 9 shows the excess SED of the β Leo system. We find that a 119 K blackbody emission matches the excess emission for wavelengths longward of 20 μm. We refer to this excess component as dust from the main planetesimal belt and try to set constraints using our simple SED model. We first start to fit the SED with a_min ∼ a_bl, the smallest grains that are bound to the system against the radiation pressure force, and a_max∼1000 μm, the largest grain size in our opacity function. In the case of β Leo, a_bl∼ 3 μm. Note that grains with sizes larger than 1000 μm contribute insignificantly to the infrared output due to the combination of the grain properties and the steep size distribution. In addition to the MIPS, color-corrected IRAS 60 μm, and Herschel PACS broadband photometry, six fiducial points (12, 18, 20, 26.5, 30, and 35 μm as shown in Figure 6) from the IRS spectrum and four points (55.4, 65.6, 75.8, and 86.0 μm) from the MIPS-SED data are selected to compute χ² in order to determine the best-fit debris disk model (Table 6). We tried both p = 0 and p = −1 cases with various combinations of R_in and R_out. The resultant model emission has a wrong spectral slope in the regions of 15–25 and 55–95 μm. We then tried to relax a_min to be larger or smaller than a_bl in the fit, and found that the a_min ∼ 5 μm with p = 0 case gives the best χ² value (χ²_ν=0.4). Using the same parameters but with a_min= 3 μm gives a χ²_ν value of 2.8. By fixing these parameters (a_min = 5 μm, a_max = 1000 μm, and p = 0), we can then derive the best-fit inner and outer radii to be 3 ± 2 AU and 55 ± 8 AU, respectively, with a total dust mass of (1.9 ± 0.3) × 10⁻⁴ M_⊕. The SED using these best-fit parameters is shown in Figure 9.

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Based on these parameters from the excess SED, we also construct a model image at 24 μm to compare with the observations. We assume the disk mid-plane is aligned with the stellar equator, inclined by 215. The model image was projected to the inclination angle, and then convolved with model STinyTim PSFs. The azimuthally averaged radial profile computed from the model image is also shown in Figure 5 and matches well with the observed profile at 24 μm. This is consistent with the β Leo disk being marginally resolved with a disk radius less than 80 AU.

The next step is to see whether the model we construct for the main planetesimal belt is consistent with the resolved structure detected by BLINC. The model excess flux from the main planetesimal belt is ∼57 mJy at 10.5 μm, which is lower by a factor of five compared to the required level (0.25 ± 0.05 Jy) to fit the null level derived from the BLINC data. Reducing the inner radius of the main planetesimal belt will increase the flux contribution at 10.5 μm (up to ∼130 mJy for R_in = 1 AU); however, it also increases the flux levels in 12–15 μm range, making poor matches with the observed IRS excess (see Figure 9). We also generated high-resolution model images at 10.5 μm given the various inner radii and used them as input to compute the BLINC null levels. The resultant null levels are shown in Figure 10, and they all fail to provide satisfactory fits inside of 06. Therefore, a surface brightness deficit (gap) in the disk structure is required to explain both the BLINC and Spitzer data.

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We then explore the possibility of an inner warm belt separated from the main planetesimal belt. Judging from the amount of excess emission in the range of 5.8–10.5 μm, the typical dust temperature for this emission is ∼600 K. From the constraints derived from the BLINC data, we know this warm component is located at 2–3 AU. Figure 11 shows the thermal equilibrium dust temperature distribution around β Leo. For astronomical silicate grains with sizes smaller than 1 μm, the thermal equilibrium temperatures are generally lower than 500 K at distances of 2–3 AU from β Leo. Only sub-micron-size silicates can reach such a high temperature at a distance of a few AUs. In addition, sub-micron silicate grains have prominent features near 10 and 20 μm. Alternatively, sub-micron carbonaceous grains have equilibrium temperatures of ∼600 K, but produce featureless emission spectrum in the infrared.

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Due to the uncertainty in the exact amount of excess emission for wavelengths shortward of ∼11 μm, we cannot put real constraints on grain compositions and sizes. Our goal is to have the simplest model for the warm belt that can fit the global SED when combined with the emission from the main belt, and with a null level computed from the model image that matches the BLINC data. Given the high dust temperatures in the 2–3 AU belt, we adopted grain properties for amorphous carbon grains (a density of 1.85 g cm⁻³; Zubko et al. 1996) with a radius of 0.5 μm. The surface density in this warm belt is assumed to be constant (uniform distribution at all radii between 2 and 3 AU). With these assumed parameters, we find that we need very little dust (M_d ∼ 2.6 × 10⁻⁸M_⊕ or f_d ∼ 7.9 × 10⁻⁵) to produce the emission shortward of ∼11 μm. Since this warm belt will contribute some fraction of the emission longward of ∼11 μm, we have to adjust the inner radius of the main planetesimal belt to 5 AU so the resultant combined excess SED fits the observed IRS spectrum within the errors. The final two-component excess SED is shown in Figure 12, and its corresponding null level is shown in Figure 10 as well. The final parameters for the disk are summarized in Table 7.

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We use a simple model, created with the VMT provided by the NASA Exoplanet Science Institute. The tool allows one to create a hypothetical system, consisting of any number of point sources, disks, and rings that may be either uniform or Gaussian, and predicts the observational signature if the hypothetical system were to be observed by KIN. We used the model to test whether the variation in null detected can be explained solely by the stellar components of the binary system. Hummel et al. (2001) determine the necessary stellar parameters and orbital data necessary to model the system with two point sources. We adopt a difference in N-band magnitude between the primary and secondary of δm_N = 1.5. Given an orbital period of 14.5 days for the stellar pair, we can use the date of the observations to estimate the relative positions of the primary and secondary to determine a hypothetical observational signature. Values calculated for the expected and observed nulls are shown in Table 8. Comparing the two, we can say that the predicted and actual data are marginally consistent, indicating little evidence for additional extended emission.

Given the behavior of the null with respect to time as compared to the detailed orbital parameters determined by Hummel et al. (2001), we can conclude that the non-zero nulls detected by our KIN observations are likely due to the stellar components being resolved, with no evidence for excess emission from warm dust. A conservative 3σ upper limit for 10 μm emission from hot dust in the system can be taken to be 3%.

Although there was no significant excess emission around o Leo, we can set constraints on what dust might be present there. o Leo is a spectroscopic binary located at a distance of 41.5 pc with a separation of 0.21 AU (making the semimajor axis 5 mas). The orbit has a 576 inclination from face-on (Hummel et al. 2001), and we will make the assumption that any circumbinary debris disk will have a similar inclination to us.

If we assume blackbody grains and the above parameters for the system, we can calculate the radius where we would expect to find 300 K grains. This is the temperature at which the blackbody emission peaks around 10 μm. The physical parameters for the host stars are derived to be 5.9 R_☉ and 6000 K for the first star and 2.2 R_☉ and 7600 K for the second star (Hummel et al. 2001). For spherical blackbody grains,

$$\begin{equation} T_g = \left(\frac{T_1^4 R_1^2 + T_2^4 R_2^2}{2r^2}\right)^{.25}. \end{equation} \tag{ B1 }$$

So, T_g = 300 K when r = 9 AU, which for o Leo is at an angular separation of 022 along the semimajor axis. Assuming the disk is oriented vertically on the detector (and thus perpendicular to the interference pattern), this would put such a ring close to the constructive peak, making BLINC especially sensitive to any dust located in that region.

Figure B1 shows the source null of several ring models plotted along with the o Leo data points. The models assume that the orientation of the disk is perpendicular to the interference pattern of the interferometer. The optical depth of each model is listed in the legend, and the rings extend from 8 to 10 AU, directly through the 300 K region around the star. Plotted as a dotted line is the 2σ limit (based on the error bars of each data point) above which we would have a marginal detection. Thus, if there were a dust ring equivalent to the 3.75 × 10⁻⁴ model in Figure B1, we would not expect to have a positive detection, whereas we would expect to have gotten a marginal detection if the dust ring had an optical depth of 5 × 10⁻⁴. Since models that are brighter than ∼3.75 × 10⁻⁴ begin to lie above the 2σ threshold, this model marks the upper limit we can place on dust in such a ring, which corresponds to a flux of 0.17 Jy at 12 μm.

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If, on the other hand, the disk were oriented parallel to the interference pattern, a brighter disk could be present without being detected. For this sub-optimal orientation, an upper limit of 0.31 Jy can be placed on the disk.

o Leo has previously been identified as having a very hot excess by Trilling et al. (2007) using MIPS on Spitzer. Trilling et al. (2007) reported 24 and 70 μm excess ratios for o Leo of 1.23 and 1.30, respectively, suggesting the presence of circumbinary dust. They derived a dust temperature of 815 K with a minimum radius of 0.85 AU (002). Although too concentrated to be detectable by BLINC, such a disk would be odds with the lack of detection by KIN. However, Trilling et al's analysis appears to be based on saturated 2MASS measurements for o Leo, from which they derive a K-band magnitude of 2.58 ± 0.15. If instead we use Johnson K-band photometry (Johnson et al. 1966) converted to a 2MASS K_S magnitude, we find that o Leo is 2.39 ± <0.07, 0.19 mag brighter than calculated by Trilling et al. Since the 2MASS data are saturated for o Leo, we believe this Johnson K magnitude to be a better measurement. Using this magnitude, our model fitting predicts flux densities of 816 and 89.5 mJy (or ratios of 1.00 and 1.06 to the stellar photosphere) respectively at 24 and 70 μm. The MIPS measurements reported by Trilling et al. (2007) then indicate no excess at 24 μm above the ∼6% (2σ) level (and above 15% at 70 μm). The new ratios indicate that the system harbors little, if any, dust. The lack of excess detections at longer wavelengths is then consistent with the null detections by BLINC and KIN at 10 μm.