EXOZODIACAL DUST LEVELS FOR NEARBY MAIN-SEQUENCE STARS: A SURVEY WITH THE KECK INTERFEROMETER NULLER

The KIN (Colavita et al. 2009) operates in N band (8.0–13.0 μm) and combines the light from the two Keck telescopes as an interferometer with a physical baseline length B ∼ 85 m. The KIN produces a dark fringe through the phase center ("Nulling"). The adjacent bright fringe (through which flux is transmitted) projects onto the sky at an angular separation λ/2B = 10 mas, or 0.1 AU at the median distance to the stars in our sample (10.5 pc), and for λ = 8.5 μm (the effective wavelength of the KIN bandpass). Thus, the instrument is sensitive to circumstellar dust located as close to the central star as these spatial scales (i.e., "inner working angle"). Blackbody emission peaks at 8.5 μm for T ∼ 432 K. For the median luminosity of the stars in our sample (2.2 L_☉), dust particles in thermal equilibrium at this temperature are located at a stellocentric radius of 0.3 AU. Therefore, the KIN fringe spacing matches well the expected location of relatively warm dust, located in the inner disk regions. The KIN can observe objects as faint as N(flux) =1.7 Jy, as long as they also have Kmag <6 (K-band co-phasing limit) and Jmag <8.5 (angle tracking limit).⁷

The response of any interferometer may be understood by projecting the fringe pattern on the sky: what is measured is the astrophysical flux from the surface brightness transmitted through the fringe pattern. For the work presented here, we measure and calibrate the transmitted flux (expected to be small) due to dust surrounding the central target stars. Thus, we refer to the basic observable as the "flux leakage" or simply the "leak" (the inverse quantity, the null depth, is an equivalent and also frequently used term, i.e., leak =0.01 implies null depth 100:1). The amount of flux leakage not attributable to the finite size of the central stars is referred to as "excess leakage," and by measuring it we can learn about the amounts of circumstellar dust present.

In order to achieve good accuracy of the calibrated leak measurements, the KIN utilizes an architecture in which each Keck pupil is split into two halves, resulting in two Keck–Keck long baselines, and two short baselines (4 m) formed between the two halves of each Keck telescope. In order to accommodate the large dynamic range between a star and any surrounding dust, the star is nulled on the Keck–Keck baselines. In order to detect small leakage signals in the presence of the large mid-infrared background, the nulled outputs are combined on the short baselines in a standard Michelson combiner (a.k.a. the "cross-combiner," or XC) with fast optical path difference modulation (i.e., "interferometric chopping," see also Mennesson et al. 2005). The output of the short baseline combiners (for which any object appears essentially unresolved) when the long baselines are "at peak" also provide the necessary flux normalization of the leak measurement. In essence then, the KIN measurement is the ratio of the amplitudes of the short baseline combiner fringes when the long baseline combiner is set at null, divided by the same quantity when the long baseline combiner is set at peak:

$$\begin{equation} L_{{\rm raw}} = \frac{\mbox{XC fringe amplitude at null}}{\mbox{XC fringe amplitude at peak}}. \end{equation} \tag{ 1 }$$

The many details involved in making this measurement are described in Colavita et al. (2009) and Colavita et al. (2010). As emphasized in those references, in a ground-based background limited environment, achieving a high level of suppression of the central star (i.e., deep nulls) is not the most important consideration. Equally important is being able to calibrate the leaks well; and the KIN four-beam architecture results in a better calibration of the measured leakages (or equivalent visibilities) compared to standard Michelson interferometry at mid-infrared wavelengths.

Due to various instrumental factors (diffraction, material absorption, pinhole mode matching), the KIN spectral responsivity is strongly peaked toward the blue end of the bandpass; the 8–9 μm bin contains most of the signal-to-noise ratio (S/N). Also, the red end of the spectrum is affected by poorer calibration quality (believed to be caused by residual correlation in the short baseline combiners arising from telescope thermal emission). Thus, for the analysis presented here, we only use the 8–9 μm spectral bin, most sensitive to exozodi detection.

As noted above, the KIN response may be understood as the flux from the astrophysical source that is transmitted through its fringe pattern projected on the sky. Due to its four-beam architecture, the KIN beam pattern consists of the following three terms.

• 1.  
  Point-spread function (PSF) of each Keck half-aperture (T_PSF); this is well approximated at 8.5 μm by an elliptical Gaussian of FWHM =490  ×  440 mas. For the observations presented here, the Keck rotator angles are oriented such that this pattern has the major axis along the east–west direction and rotates as the target moves across the sky such that the minor axis always points toward north.
• 2.  
  Fringes of the short baseline combiner (T_XC):

  $$\begin{equation} T_{XC} = \cos (2 \pi (x \cdot u_{xc} + y \cdot v_{xc})), \end{equation} \tag{ 2 }$$

  where (x, y) are coordinate offsets in right ascension and declination, and (u_xc, v_xc) are the corresponding short baseline spatial frequencies. These fringes are perpendicular to the half-pupil PSF major axis, and rotate in the same way. Projected on the sky, this fringe pattern is relatively "broad": at 8.5 μm the fringe spacing for the physical 4 m short baseline is 440 mas.
• 3.  
  Fringes of the long baseline combiner (T_long):

  $$\begin{equation} T_{{\rm long}} = \frac{1}{2} \cdot (1 \mp \cos (2 \pi (x \cdot u + y \cdot v)), \end{equation} \tag{ 3 }$$

  where (u, v) are the long baseline spatial frequencies and the ∓ corresponds to the null/peak configuration, respectively. This fringe pattern can have any orientation with respect to T_PSF and T_XC. As mentioned above, in this "fine" fringe pattern the dark and bright fringes are spaced by an angular separation projected on the sky of 10 mas, at 8.5 μm and for the physical 85 m KI baseline.

The total KIN transmission pattern is thus

$$\begin{equation} T(x,y,u,v,u_{xc},v_{xc},\lambda) = T_{{\rm PSF}} \cdot T_{XC} \cdot T_{{\rm long}}. \end{equation} \tag{ 4 }$$

Through the dependence on the spatial frequencies, the instantaneous KIN pattern depends on wavelength (8.5 μm) and on the hour angle and declination of the target being observed. Figure 1 shows an example of the KIN pattern terms. The KIN is sensitive to circumstellar dust by its ability to measure its flux transmitted through the total KIN fringe pattern (center panel of Figure 1), i.e., dust located from ∼10 mas (0.1 AU at 10 pc) out to the ∼490 × 440 mas field of view (FOV). However, the short baseline fringes (T_XC) also act to limit the KIN's ability to detect outer dust. Indeed, in the example shown in Figure 1 (top right panel), T_XC goes to zero at ∼110 mas along the small baseline direction, so that the "effective FOV" is only ∼110 × 440 mas (∼1.1 × 4.4 AU at 10 pc).

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For an object with a brightness distribution I(x, y, λ), the expected monochromatic leak is thus

$$\begin{eqnarray} &&\hspace{-12pt} L_{{\rm calculated}} (u,v,u_{xc},v_{xc},\lambda)\nonumber\\ && =\, \frac{\int \int I(x,y,\lambda) \cdot T(x,y,u,v,u_{xc},v_{xc},\lambda)^{{\rm null}} dx dy}{\int \int I(x,y,\lambda) \cdot T(x,y,u,v,u_{xc},v_{xc},\lambda)^{{\rm peak}} dx dy}, \end{eqnarray} \tag{ 5 }$$

where the double integral is performed over the KIN FOV, set by the half-pupil PSF, as described above.

In order to provide some physical intuition, we note that for an object of angular size much smaller than the fringe spacing of the long baseline, the nulled signal would ideally be zero, resulting in L = 0. If, on the other hand, the object is large and its brightness distribution spans many long baseline fringes, the flux transmitted at null or at peak are similar and L = 1.0. Detailed descriptions of the theory behind the nuller measurement may also be found in Traub & Oppenheimer (2010) and G. Serabyn et al. (2011, in preparation).

End-to-end data reduction and calibration was performed by the Keck Interferometer Project using their pipeline and external calibration package (nullCalib⁸). Here, we summarize the steps involved and the various sources of measurement error.

During observing, a microsequence is executed during which the short baseline optical path modulation is always active (one fringe measurement every 25 ms), and the nullers alternate between the null and peak states (for 250 ms and 50 ms, respectively). Depending on the object brightness, this sequence is repeated for 10–15 minutes, thus many thousand independent estimates of the leak are formed for each observation. From the scatter of these measurements a "formal" error is estimated for the average leak measurement corresponding to each observation.

Following standard practice, in order to monitor and subtract the instrument's transfer function (i.e., the non-zero leak that is measured when observing a point source located at the phase center), observations of targets of interest were interleaved with observations of calibrator stars of known N-band angular diameters. Thus, the calibrated leak is

$$\begin{equation} L_{{\rm calibrated}} = \big(L_{{\rm raw}}^{{\rm target}} - L^{{\rm system}}\big), \end{equation} \tag{ 6 }$$

where L^system, at the time of the target observations, is obtained by interpolation from the net leak measurements of the bracketing calibrator stars, after subtracting the calibrator leak expected given its angular diameter. This calculation is described in more detail in Appendix C of Colavita et al. (2009), and all the steps are applied by the package nullCalib.

The required calibrator angular diameters were measured using the simultaneous K-band fringe tracker data, and converted to N-band diameters using standard limb-darkening relations. The procedure is described in detail in Appendix A of Colavita et al. (2009); for clarity we repeat here some of the most relevant aspects. The approach was to treat the nulling targets as calibrators, since at K band they are expected to effectively be simple naked stars. Although some of the calibrators are small (<1 mas) compared to the KI resolution, for such small calibrators the 20%–30% precision obtained in those cases results in uncertainties on the calibrated leaks well below the leak measurement errors (best case 0.2%, see below). At the other end of the size range, for a 3 mas calibrator, the largest in our sample, the diameter only needs to be known with 10% precision or better in order to not add significant error to the calibrated leaks. Thus, the calibration procedure is largely insensitive to reasonable errors in the adopted calibrator angular diameters. The accuracy of our measured calibrator angular diameters was evaluated against estimates obtained using surface brightness relations; in all cases the discrepancies also result in a calibrated leakage differences smaller than the best-case measurement error. The uncertainties in the calibrator diameters are taken into account and propagated into the formal error of the calibrated leaks. The relevant parameters for the calibrators used for our sample are listed in Table 2.

We note that we make the assumption that there is no source of calibrator leak other than its angular extent, i.e., the calibrators have no excess N-band emission, which if unaccounted for would directly lead to underestimated exozodi emission levels. However, of 56 calibrators used in this study, one is a main-sequence star and all others are giants (i.e., none are supergiants), and all but three have spectral types K0–M0; both of which minimize the possibility of infrared emission above photospheric levels (e.g., Cohen et al. 1999 and references therein). Furthermore, all the calibrators used have been selected to have IRAS 12 μm/25 μm flux ratios that agree, within the photometric errors, with that of a blackbody of the calibrator effective temperature. These methods, however, do not insure that our calibrators are free from low level dust emission at N band (few percent or less), undetected in the K-band diameter comparisons and IRAS flux ratios. But we can use our own data to place some limits on the possible level of exozodi emission around our calibrators. Indeed, if calibrator exozodi emission were high, and variable from star to star, this would be apparent in correspondingly large leak fluctuations among the different calibrators. Specifically, we have compared the variations in system leak estimates made when same calibrator is used on either side of the target observation (only instrument variations expected) with the same quantity computed when two different calibrators are used (variations due to instrument plus calibrator exozodi level). Within measurement errors, we see no difference between those two cases, implying that our calibrators do not contain large amounts of exozodi dust (in the language of Section 5, ≲ 100 zodi impact on average on the exozodi levels derived for the target stars).

We also include "external" errors in the calibrated leaks, which have been estimated from the night-to-night repeatability of multiple sets of calibrated data taken on the same star (Colavita et al. 2009, 2010). In summary then, the errors on L_calibrated contain two terms: (1) a formal error derived from the scatter of the leak measurements in each observation, and which also contains the uncertainties associated with the estimate of the system leak (mainly due to calibrator diameter uncertainties), and (2) the external error just described. Table 3 shows the measured calibrated leak and errors for each observation. As can be seen, for the stars in our sample the formal error is typically $\sigma _{L_{{\rm calibrated}}}^{{\rm formal}} = 0.001$–0.004. The external errors are wavelength and flux dependent; for the stars in our sample they are in the range: $\sigma _{L_{{\rm calibrated}}}^{{\rm ext.}} = 0.002$–0.0035.

Table 3. Log of Observations and Calibrated Leak Data (Wideband Channel 8–9 μm)

Name	Date	HA	u	v	Lcalibrated	$\sigma _{L_{{\rm calibrated}}}^{{\rm formal}}$	$\sigma _{L_{{\rm calibrated}}}^{{\rm ext.}}$	L⋆
			(m)	(m)
107 Psc	UT 2008 Oct 13	2.54	26.5162	76.8456	0.006772	0.001803	0.003000	0.001605 ± 0.000554
107 Psc	UT 2008 Oct 13	3.27	16.6299	78.2699	−0.000451	0.002041	0.003000	0.001555 ± 0.000537
1 Ori	UT 2008 Feb 17	0.41	48.9782	69.3127	−0.003011	0.001677	0.003000	0.003428 ± 0.000979
1 Ori	UT 2008 Feb 17	1.10	43.0509	72.2259	0.003409	0.001118	0.003000	0.003365 ± 0.000961
1 Ori	UT 2008 Feb 17	1.83	35.3784	74.8085	−0.000761	0.001905	0.003000	0.003259 ± 0.000931
1 Ori	UT 2008 Feb 18	0.10	51.1001	67.9093	0.010442	0.002699	0.003000	0.003438 ± 0.000982
1 Ori	UT 2008 Feb 18	1.22	41.8864	72.6852	0.008744	0.003977	0.003000	0.003349 ± 0.000957
47 UMa	UT 2009 Jan 10	−1.08	56.0337	53.2305	0.002110	0.003131	0.003500	0.000846 ± 0.000201
61 Cyg A	UT 2008 Aug 17	0.46	48.5433	67.6194	0.010119	0.002093	0.002000	0.005730 ± 0.002202
61 Cyg A	UT 2008 Aug 17	1.12	42.8783	72.5475	0.005219	0.001828	0.002000	0.005873 ± 0.002257

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Validation of the KIN response and calibration was evaluated by the project using a test system for which the expected leak could be calculated; in this case a binary system with a well-known orbit (Appendix B of Colavita et al. 2009).

Table 3 also contains the calculated leak expected from the target star itself (L_⋆), which is needed in order to derive the excess leak due to the exozodi cloud, as described below. The calibrated leak data are also shown in Figure 2. We emphasize that at this stage we do not average the multiple leak measurements that are available for a given target, because variations among leaks measured at various times may in principle contain a contribution from the changing projected baseline fringe spacing and orientation. Thus, we first model our measurements for a specific exozodi model and average the results after this conversion, as described in Section 4.4. Figure 3 also summarizes our measurements but only for the wideband (8–9 μm) channel used in the analysis presented here.

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In order to interpret our measurements and compare them to the solar system case, we use the Zodipic code⁹ to create images of zodi clouds around each of our targets. Zodipic synthesizes brightness distributions of exozodiacal clouds based on the empirical fits to the observations of the solar zodiacal cloud made by COBE (Kelsall et al. 1998).

When Zodipic generates a model brightness distribution for a zodi disk analog around a star other than the Sun, the dust has the same optical depth at 1 AU and radial density profile as in the solar system. As a convenient unit, we refer to this model as corresponding to "1-zodi," which we denote z = 1. Zodipic scales the radial temperature profile with stellar luminosity, and the inner dust radius is set by a dust sublimation temperature set at 1500 K (the dust inner radius is thus dependent on stellar spectral type, but z = 1 models around any star have a fractional dust luminosity L_dust/L_⋆  ≃  10⁻⁷, the solar system value). In the Zodipic code the dust density can be treated as a free parameter, allowing to generate brightness distributions for scaled version of the solar system (the total flux due to the circumstellar dust scales linearly with z). In the next section, we describe our procedure for converting the calibrated leaks to an exozodi dust density, parameterized in terms of a number (z) of zodis.

We note that the zodiacal models used here include only the smooth component of interest, i.e., the Earth trailing blob and asteroidal dust bands are not included. We also note that increasing the optical depth of the cloud increases the collision rate, which affects the cloud structure, a physical process which is not taken into account by Zodipic. In a zodiacal cloud, grain–grain collisions become important for grains above a critical size, ∼30–150 μm in the solar zodiacal cloud (Fixsen & Dwek 2002), and which scales inversely as the optical depth of the disk (Kuchner & Stark 2010). Since this critical grain size reaches 10 μm for a disk with about 3–15 zodis worth of dust, we expect that disks with more than roughly a few tens of zodis should begin to show morphological changes at KIN wavelengths because of the collision destruction of grains in the center of the disk. Our models, from Zodipic, are strictly linear in the dust density, and do not take this collisional depletion into account; this level of analysis is left to future studies.

We also note that our procedure assumes that the exozodi density in the inner-most regions of KIN sensitivity (∼10 mas or 0.1 AU at 10 pc, as described above) follows the same radial density profile of the Kelsall et al. (1998) model, which was based on COBE/DIRBE measurements made at larger stellocentric radii, 0.9 AU or larger. However, measurements of the solar zodiacal cloud made by the Helios probes as close as 0.3 AU (Leinert et al. 1981) and measurements in the solar F corona (MacQueen & Greeley 1995) both find radial density profiles with exponents of ∼1.3, in agreement with the Kelsall et al. (1998) model.

The Zodipic images generated are 512 × 512 pixels, with a scale of 2 mas pixel⁻¹, i.e., 10× finer than the long baseline fringe spacing. The image size is thus 1024 × 1024 mas, a good match to the KIN FOV (given by T_PSF). Figure 1 includes an example Zodipic image for a solar system analog at 10 pc. We note that this image size (in pixels) keeps the computation times short, but at this spatial resolution the stellar disk would not be well sampled. Therefore, the Zodipic images generated do not include the central stars, they represent only the exozodi brightness distribution (I_zodi).

If we decompose the total brightness distribution into that of the central star and the exozodi cloud, I = I_⋆ + I_zodi, it follows that

$$\begin{eqnarray} L^{{\rm zodi}}_{{\rm calculated}} & {\simeq} & \frac{\int\!\! \int I_{{\rm zodi}}(x,y,\lambda) {\cdot} T(x,y,u,v,u_{xc},v_{xc},\lambda)^{{\rm null}} dx dy}{\int \!\!\int I_{\star }(x,y,\lambda) {\cdot} T(x,y,u,v,u_{xc},v_{xc},\lambda)^{{\rm peak}} dx dy} \nonumber \\ & {=} & \frac{\int \!\!\int I_{{\rm zodi}}(x,y,\lambda) {\cdot} T(x,y,u,v,u_{xc},v_{xc},\lambda)^{{\rm null}} dx dy}{F_{\star }},\nonumber\\ \end{eqnarray} \tag{ 7 }$$

where the approximation holds when the stellar flux (F_⋆) dominates over the exozodi flux, which is always the case here. The calculation is made monochromatically, at the effective wavelength (8.5 μm) of the 8–9 μm spectral bin used. The error introduced by this approximation is negligible, at the level of a fraction of a zodi, compared with typical hundreds of zodi errors from the formal and external leak errors.

Thus, we also remove the stellar contribution from the measured calibrated leak:

$$\begin{equation} L^{{\rm zodi}}_{{\rm measured}} = (L_{{\rm calibrated}} - L_{\star }), \end{equation} \tag{ 8 }$$

where the leak due to the central star (Table 3) is calculated using an estimate of its angular diameter, following Serabyn (2000):

$$\begin{equation} L_{\star } \simeq \frac{\pi ^2}{16} \cdot \left(\frac{B_p(m) \cdot \theta _\star (\mu {\rm rad})}{\lambda (\mbox{$\mu $m})}\right)^2, \end{equation} \tag{ 9 }$$

where B_p is the length of the projection of the baseline on the sky in the direction to the star and θ_⋆ is the stellar disk angular diameter (Table 1). We note that in the above expression we have neglected limb-darkening terms, unimportant for the spectral type and luminosity class of the stars in our sample.

Equation 8 illustrates an important advantage of the KIN measurements compared to spectrophotometric flux excess measurements which require accurate calibration of the stellar photosphere flux level: the excess leak (L^zodi_measured) estimate depends only on the stellar leak estimate (L_⋆), which is easy to determine given the spectral types of the stars in our sample. Furthermore, the stars are generally small (∼1 mas, Table 1) compared to the KIN resolution at 8.5 μm. Therefore the correction is small and the uncertainty in the stellar angular diameters affects very weakly the uncertainty on L_⋆ and on L^zodi_measured.

For each target and for each individual observation, our procedure consists of the following steps.

• 1.  
  Compute the measured excess leak L^zodi_measured. The uncertainty in the angular diameter of each of our target stars propagates into the uncertainty in the excess leak.
• 2.  
  For a given exozodi disk orientation (inclination i_disk and position angle P.A._disk), compute L^zodi_calculated, for a Zodipic model which has a dust density (number of zodis, z) such that the measured excess leak is matched: L^zodi_calculated = L^zodi_measured. Derive uncertainties in the required number of zodis from the formal and external errors. As noted earlier, this conversion to number of zodis must be made for each observation, because the instantaneous KIN fringe pattern that corresponds to each observation must be applied when using Equation (7).
• 3.  
  Repeat for a range of disk inclinations and position angles which span the extremes in predicted leak, namely, face-on (i_disk = 0°) and edge-on (i_disk = 90°) with position angle parallel and perpendicular to the instantaneous direction of the long baseline fringes. In our averaging scheme, described below, we will take the variation resulting from the uncertainty in the exozodi cloud orientation as an additional source of error in the derived numbers of zodis. We note that (as can be seen in Table 4) in most cases the resulting uncertainty is significantly smaller than that due to the formal and external errors. As described in Section 5, for η Crv and γ Oph, imaging observations have constrained the inclination and position angle of the outer disk, and in those cases we also derive the number of zodis implied by our measurements by assuming that the exozodi disk has the same orientation as the outer disk.

Table 4 shows our results in terms of number of zodis (z_ij) for each observation and for each disk orientation. The notation refers to observation i in "cluster" j. The notion of cluster must be introduced now in order to properly account for the external error in the averaging process described below. KIN observations of a given target begin with an adjustment to the static internal optical delay, and an on-sky alignment procedure which maximizes the mid-infrared flux seen by the nulling camera. We call a cluster a certain number of observations (typically 1–4) of the same target in between such changes to the instrument setup. Clusters of observations of the same target may be made during the same night or on distinct nights. Table 4 also shows the error in the number of zodis that result from the formal and external leak errors (σ^formal_ij and σ^ext_j) as well as the error due to the uncertainty in the stellar leak (σ^⋆). We note that although unphysical, negative zodis are allowed as a result of the error bars on the leak measurements. Finally, in order to describe the spatial extent of the exozodi region that the KIN is most sensitive to, the last column of Table 4 gives the half-light radius (R_half-light) of the azimuthally averaged exozodi brightness transmitted through the KIN fringe pattern. These radii may be compared, for example, to the radii of center of habitable zone given in Table 1.

4.3.1. Altair (α Aql)

4.3.2. Binaries

After converting each leak measurement to a number of zodis we average the multiple observations available for each target in order to reduce the measurement errors. Consistent with the characterization of the external error described in Colavita et al. (2009, 2010), we average observations and clusters as follows.

• 1.  
  Within each cluster: compute the weighted mean of the number of zodis, z_j = ∑(z_ij · w_ij)/∑w_ij, where the formal errors provide the weights, w_ij = 1.0/σ²_{ij, formal}. The error in the weighted mean is taken to be the largest of the statistical error in the mean ($\sigma _{z_j}^a = 1.0/\sqrt{\vphantom{A^A}\smash{\hbox{${\sum w_{ij}}$}}}$) and the weighted average standard deviation in the data ($\sigma _{z_j}^b = \sqrt{\vphantom{A^{A^-}}\smash{\hbox{${\frac{\sum (z_{ij} - z_j)^2 \cdot w_{ij}}{\sum w_{ij}}}$}}}$). To this error we add the external error in quadrature: $\sigma _{z_j} = \sqrt{\vphantom{A^A}\smash{\hbox{${ \sigma _{j,ext}^2 + \max \lbrace \sigma _{z_j}^a,\sigma _{z_j}^b\rbrace ^2}$}}}$.
• 2.  
  To combine the clusters: we again compute the weighted mean ($\overline{z} = \frac{\sum (z_j \cdot w_j)}{\sum w_j}$, with w_j = 1.0/σ²_zj), and the error in the weighted mean given by the statistical error ($\sigma _{\overline{z}} = 1.0/\sqrt{\vphantom{A^{A^-}}\smash{\hbox{${\sum w_j}$}}}$).We note that these first two steps imply that (1) the total error per cluster can never be smaller than the external error and (2) when averaging multiple clusters, the total error can become smaller than the external error. The latter however generally provides only a small improvement, because our data sets for each target contain only 1 or 2 clusters.
• 3.  
  The steps above are done for each exozodi orientation, the last step is to average the results over the three disk orientations computed (i_disk = 0° and i_disk = 90° with P.A._disk parallel or perpendicular to the instantaneous long baseline fringes). Thus, the final error contains a term (added in quadrature) computed as the rms of the number of zodis deduced for the three disk orientations ($\sigma _{[i_{{\rm disk}},{\rm P.A.}_{{\rm disk}}]}$). As mentioned above, the "disk orientation error" is typically relatively small, tens of zodis, compared to that resulting from the formal and external error sources. The final number of exozodis for each target, including all data points and uncertainties, is $z = \sum _1^3 (\overline{z})/3$, $\sigma _z = \sqrt{\vphantom{A^A}\smash{\hbox{${\sigma _{\overline{z}}^2 + \sigma _{[i_{{\rm disk}},{\rm P.A.}_{{\rm disk}}]}^2}$}}}$.

Our only significant detection is η Crv (1250 ± 260 zodis). This object was previously known to have high levels of circumstellar dust, including observations with Spitzer/MIPS at 70 μm (Beichman et al. 2006b), Spitzer/Infrared Spectrograph (IRS)/ at 10–35 μm (Chen et al. 2006), and VLTI/MIDI at 10–13 μm (Smith et al. 2009). Also, as can be seen in Figure 2, the KIN spectrum across the N band has adequate S/N to resolve the 10 μm silicate feature and can be used to infer dust properties. A detailed analysis of these data is left to future work.

The outer disk in this object has been directly imaged (Wyatt et al. 2005; Matthews et al. 2010). Thus, instead of taking the disk inclination and position angle as free parameters, we may assume that the exozodi disk has the same orientation. Using the parameters from the Matthews et al. (2010) Herschel observations (i_disk = 50° with P.A._disk = 103°), we find z = 1300 ± 160.

This star was previously known to have excess starting at 15 μm, and growing much larger at longer wavelengths (Su et al. 2008). It is noteworthy that the infrared excess luminosity is several times larger for γ Oph than for η Crv, while the KIN makes a clear detection for η Crv but only a marginal detection (∼2.6σ) for γ Oph; implying perhaps that the dust distributions are very different in these two objects.

The Spitzer images spatially resolve the outer disk and imply i_disk = 50° and P.A._disk = 55°. Assuming the same orientation for the exozodi disk we derive z = 200 ± 60.

This star was not previously know to have warm circumstellar dust. The KIN results indicate a marginal detection (3.0σ). The KIN measures an excess leak of ∼0.8%. This level of excess would have been undetectable by IRAS at 12 μm, and no 8 μm excess measurements have been made with Spitzer. Therefore, assuming the dust temperature is relatively warm, a KIN detection would not be inconsistent with the lack of excess seen in previous measurements.

Excluding our only clear detection (η Crv) and the two possible detections (γ Oph and α Aql), our sample contains 22 non-detections. Table 5 shows the upper limits on the number of zodis for each star that can be derived from our measurements. The average 3σ upper limit is 570 zodis. We note that the weighted rms data scatter for the non-detections (150 zodis) is similar to the mean data uncertainty (160 zodis), which we take as an indication that the errors in the individual measurements have been fairly assigned.

Spitzer/IRS established exozodi limits based on fractional excess flux measurements (F_dust/F_⋆) made in the 8.5–12 μm band (Beichman et al. 2006a; Lawler et al. 2009). Using a sample of 203 stars, the detection rate was only 1%. Based on their estimated error in the measurement of the fractional excess fluxes of ∼1% (1σ), the sample of non-detection was used to place 3σ limits on the number of zodis in the range 600–2700 zodis, with an average of 1100 zodis. Because the IRS short wavelength band is very similar to that of the KIN, it is of interest to compare the results from the two surveys.

First, we note that the KIN measurement can also be expressed in terms of a fractional flux excess measurement, which enables a direct comparison of expected performances (the much larger FOV of Spitzer/IRS is not expected to enter the comparison, because dust located outside the KIN FOV will be too cold to significantly contribute N-band flux). Assuming perfect stellar cancellation, the numerator in Equation 5 (i.e., at null) equals a fraction f of the zodi flux F_dust, where f is the instantaneous fraction of the zodi flux removed by the KIN fringe pattern at null. Likewise, since F_⋆ ≫ F_dust, the denominator (i.e., at peak) is essentially equal to the flux from the central star, therefore

$$\begin{equation} \frac{F_{{\rm dust}}}{F_{\star }} \simeq \frac{L}{f} \end{equation} \tag{ 10 }$$

and the error on this estimate of the fractional flux excess obtained from the KIN data is

$$\begin{equation} \sigma _{\big(\frac{F_{{\rm dust}}}{F_{\star }} \big)} \simeq \frac{\sigma _L}{f}. \end{equation} \tag{ 11 }$$

The factor f can be easily computed for each observation as the ratio of the exozodi flux transmitted through the KIN pattern at null to the total exozodi flux (for any value of z, say z = 1). On average, f = 0.4, and varies by less than 5% for our sample, including the exozodi cloud orientation effects (inclination and position angle, relative to the KIN fringe pattern). Therefore, in this conversion, the KIN measurement errors, σ_L = 0.002–0.004, are degraded by a factor of 1/f ≃ 2.5, to become $\sigma _{(\frac{F_{{\rm dust}}}{F_{\star }})} =$ 0.005–0.0075. This is to be compared with the best-case IRS errors, $\sigma _{(\frac{F_{{\rm dust}}}{F_{\star }})} = 0.01$. It follows that one expects the current KIN implementation to provide up to 2× tighter exozodi limits than Spitzer/IRS. We note however that the errors quoted for Spitzer/IRS do not include possible and potentially significant systematic errors arising from uncertainties in the absolute calibration of the stellar flux, to which the KIN is immune, as described earlier. In that sense, the above comparison represents a lower bound to the improvement that can be expected from the KIN.

Table 6 summarizes the detailed comparison of exozodi limits for the eight objects in common between the Spitzer/IRS and KIN surveys. It can be seen that on a target-by-target basis, either Spitzer/IRS or KIN can provide tighter limits, depending on the precise measurement errors in each case; but that we recover a common 2 × improvement from KIN, as predicted.

Table 6. Spitzer/IRS—KIN Comparison

Name	HD	Spitzer/IRSa		KIN
		$\frac{F_{\rm dust}}{F_{\star }}$	zb	z
			(3σ limit)	(3σ limit)
47 Uma	95128	−0.022 ± 0.013	1110	628
β Com	114719	0.014 ± 0.01	830	972
γ Lep	38393	0.001 ± 0.01	750	252
ι Psc	2223658	−0.007 ± 0.014	970	318
kx Lib	131977	0.002 ± 0.01	1600	1492
τ Boo	120136	0.011 ± 0.014	970	454
θ Per	16895	0.003 ± 0.01	750	333
υ And	9826	−0.003 ± 0.01	970	498

Notes. ^aFrom Beichman et al. (2006a) and Lawler et al. (2009). ^bComputed as L_dust/L_⋆ × 10⁻⁷.

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Having determined exozodi limits for each star in our sample, we now interpret those observations in terms of a statistically defined exozodi disk model, in order to extract as much information as possible from the KIN measurements. Noting that the measured values in Table 5 tend to scatter around zero, and that the typical uncertainties are large compared to most of the measured values, and certainly large compared to the average of the measured values, we consider in this section the possibility that the true underlying exozodi values might be significantly smaller than the upper limits implied by the individual measurements.

For this analysis, we define the sub-sample of N = 23 stars which were not previously known to contain circumstellar dust (excludes η Crv and γ Oph). We assume that this sub-sample is representative of a single class of stars, with respect to the level of warm exozodi emission. In Appendix A we show that the calibrated net leaks are not correlated with any of the parameters in an exhaustive list describing the instrumental conditions. Furthermore, correlations are not expected among targets. Therefore, we assume that the zodi levels inferred for each star form an uncorrelated set, and that it is thus appropriate to use the 23 measurements to estimate the mean zodi level and its error for the class. We choose the bootstrap method to form a robust estimate of the mean and its error, independent of any assumptions on the underlying statistics (Efron & Tibshirani 1998); using with 10⁶ re-samples of the set of 23 measurements we find $\hat{z} = +2 \pm 50$ (1σ error).¹¹

Further interpretation of this result, in terms of a confidence level for the mean exo zodi emission for the class being below a certain level, would require knowledge, which we do not have, of the underlying probability distribution for the measurement errors (i.e., whether or not they are approximately Gaussian) and of the underlying distribution of exozodi levels for the class of stars represented by our sample. If the measurement errors were Gaussian, the above result would imply that the mean exozodi level for the class is <50 zodis (1σ upper limit) with 84% confidence, or <150 zodis (3σ upper limit) with 99.8% confidence. Moreover, as detailed in Appendix B, we have also used Monte Carlo simulations, and a wide range of assumed exozodi distribution, to show that these conclusions are largely insensitive to the shape and mean level of the true underlying exozodi distribution.

Both our limits on individual stars, and the statistical result just discussed, are very positive for the future of direct imaging of exoplanets in the habitable zones around nearby stars, since they suggest that the exozodi levels might not be as high as was once feared. However, to be able to detect an exoEarth with any technique, the noise in a resolution element stemming from exozodi emission must be no more than a few Earth fluxes; and the limits presented here could still allow actual exozodi levels well above what can be tolerated. Measurements with sensitivity to lower exozodi levels are thus needed. The next step will likely be enabled by the Large Binocular Telescope Interferometer (LBTI), which will also use the nulling technique at mid-infrared wavelengths (Hinz 2009). Beyond that a dedicated space mission, and multi-wavelength characterization, might be required.

In order to validate our noise model, we have examined the dependence of the calibrated net leaks (i.e., stellar contribution subtracted) against a set of 26 variables describing the instrumental and environmental conditions, and the astrophysical properties of the target stars and associated calibrators. For this study, we have selected the (68) independent measurements corresponding to the sub-sample of 23 stars not previously known to contain circumstellar dust (excludes η Crv and γ Oph). The independent variables considered are (1) day of year, (2) local time, (3) hour angle, (4) telescope azimuth, (5) telescope elevation, (6) internal optical delay difference, (7) right ascension, (8) declination, (9) stellar age, (10) stellar effective temperature,¹² (11) detected N-band flux, (12) detected K-band flux (fringe tracker), (13) detected J-band flux (angle tracker), (14) angle tracker centroid rms, (15) percent of leak data accepted by the quality gates, (16) precipitable water vapor (PWV), (17) wind speed, (18) wind direction, (19) air temperature, (20) relative humidity, (21) atmospheric pressure, (22) coherence time due to dry air (as in Colavita 2010), (23) coherence time due to water vapor (as in Colavita 2010), (24) calibrator diameter, (25) target–calibrator angular separation, and (27) target–calibrator flux ratio. The simultaneous PWV measurements are from the Caltech Submillimeter Observatory (CSO) tau-meter archive. All other atmospheric quantities are from the Canada–France–Hawaii Telescope (CFHT) weather archive.

Visual inspection of the correlation plots reveals that a linear relation would be sufficient to describe any candidate correlation. In order to obtain robust estimates of the statistical significance of any correlation candidate, we adopt a bootstrap approach. For each item, we perform a linear fit for each of 10⁶ bootstrap re-samples of the data. For each item, the mean of the 10⁶ fitted slopes is a good estimate of the slope; and the standard deviation is a good estimate of its uncertainty, independent of any assumptions on the underlying statistics (Efron & Tibshirani 1998). The ratio of the mean slope and error thus represents the significance of the slope being different from zero.

The result is that for most items (23 of the 26) the slope deviates from zero at a level less than 1σ and for all items the deviation is less than 2σ. We therefore conclude that there are no statistically significant correlations in our data between the calibrated net leaks and any of the instrument/environment/astrophysical parameters that we have considered.