EXO-ZODI MODELING FOR THE LARGE BINOCULAR TELESCOPE INTERFEROMETER

For specific stars, the stellar luminosity L_⋆ (in solar units) is easily derived by fitting stellar atmosphere models to photometry, and the flux density F_{ν, ⋆} at any wavelength can be inferred from these models. Given a distance d, the stellar radius can be estimated from the same model, though in some cases it may have been directly observed. In any case, the contribution of the resolved stellar disk to an LBTI observation will generally be negligible. Here we use the stellar fluxes for HOSTS survey stars derived by Weinberger et al. (2015).

LBTI observations are sensitive to the surface brightness distribution S_disk in the N' band at 11 μm (the bandpass spans 9.81–12.41 μm). Our model can be considered a means of parameterizing this surface brightness distribution using parameters that have some physical relevance. Unless stated otherwise, in what follows our calculations are made at 11 μm.

The surface brightness profile of the disk, if viewed face-on, is modeled as

$$\begin{equation} S_{\rm disk} = 2.35 \times 10^{-11} \Sigma _{\rm m} B_\nu (\lambda,T_{\rm BB}). \end{equation} \tag{ 1 }$$

Here T_BB is the temperature of disk material that behaves like a black body

$$\begin{equation} T_{\rm BB}(r) = 278.3 L_\star ^{0.25} r^{-0.5} {\rm K}, \end{equation} \tag{ 2 }$$

where the distance to the star, r, is in AU. The numerical factor in Equation (1) is a conversion (1 AU²/1 pc²) so that the surface brightness is in units of Jy arcsec⁻². The parameter Σ_m is representative of the disk's face-on surface density of cross-sectional area (in AU²/AU²), so is analogous to optical depth and is assumed to have a power-law distribution

$$\begin{equation} \Sigma _{\rm m}(r) = z \Sigma _{\rm m,0} (r/r_0)^{-\alpha } \end{equation} \tag{ 3 }$$

between radii r_in and r_out (both in AU). The normalization Σ_{m, 0} is to be set at some r₀ (in AU) such that the surface density is in units of zodis z (see Section 2.2.3).

For generality we include an inclination I, which is half the total opening angle of the disk, and represents the maximum inclination of the parent bodies (whose nodes would be assumed to be randomized). Here we assume that our disks have negligible opening angles so I is not used, but that I is always sufficiently large that the disk is radially optically thin.

If all of the disk particles behave like black bodies, the parameter Σ_m can be interpreted as the disk's surface density of cross-sectional area. Its interpretation becomes more complicated for disks with more realistic particle optical properties because the observed dust can be several times hotter than the black body temperature, and not all radiation incident on a particle is absorbed and thermally re-emitted. However, as the surface brightness is typically assumed to exhibit a power-law dependence across the relevant radius range, we have not lost any generality by expressing the disk surface brightness in this way, even if the disk's temperature profile differs from that of a black body. These issues will be discussed in more detail in Sections 2.6 and 2.7.

The final parameters, which specify the direction to the observer relative to the disk, are the disk inclination i and the position angle Ω. The inclination is defined such that i = 0 is face-on and the position angle is measured anticlockwise (i.e., east) from north. The range of inclinations is 0 to 90° and the range of position angles 0 to 180°,⁹ with these ranges set because we cannot distinguish between the near and far side of a disk. That is, the faint disks considered here are optically thin viewed from any direction, and in thermal emission look the same mirrored in the sky plane or for 180° rotations. The model parameters and their meanings are summarized in Table 1.

Table 1. Model Parameters

Symbol	Unit	Parameter	Reference
d	pc	Distance to star+disk system	⋅⋅⋅
L⋆	L☉	Stellar luminosity	1
F⋆	Jy	Stellar flux density at wavelength λ	⋅⋅⋅
R⋆	R☉	Stellar radius	1
rin	AU	Inner disk radius (1500 K)	0.034
rout	AU	Outer disk radius (88 K)	10
r0	AU	Reference radius (${=}\sqrt{L_\star /L_\odot }$ AU)	1
Σm, 0	⋅⋅⋅	Surface density at r0 for a z = 1 disk	7.12 × 10−8
z	⋅⋅⋅	Surface density at r0 in zodis	1
α	⋅⋅⋅	Index for Σm	0.34
I	°	Disk half-opening angle	small
i	°	Disk inclination relative to face-on	⋅⋅⋅
Ω	°	Disk position angle E of N on the sky	⋅⋅⋅

Notes. Model parameters are separated into those related to the star, disk, and observation. The rightmost column gives parameters for the reference disk model.

Download table as:  ASCIITypeset image

2.2.1. Zodipic

Zodipic is an implementation of the Kelsall et al. (1998) zodiacal cloud model.¹⁰ This parametric model has many parameters, whose values were derived via fitting to COBE/DIRBE observations. Zodipic produces two-dimensional images of the solar system's zodiacal dust cloud as might be seen by an external observer, and has been used as a reference model for calculating the limits set by KIN observations (Millan-Gabet et al. 2011). Though it is a complex model and can include dust components such as the Earth's resonant ring and trailing blob, only the main smooth axisymmetric component is normally used. Because this main component is simply a radial power law, with an approximately Gaussian vertical density distribution, it is possible to reproduce the Zodipic surface brightness using our parameterized model.

The vertical dust distribution is the main difference between the Kelsall et al. (1998) model and ours. The Kelsall et al. (1998) model assumes that the radial and vertical components of the cloud are separable

$$\begin{equation} n = n_0 r^{-p} \exp \left(-\beta g^\gamma \right), \end{equation} \tag{ 4 }$$

where

$$\begin{equation} g = \left\lbrace \begin{array}{@{}ll}\xi ^2/2\mu & \xi < \mu \\ \xi - \mu /2 & \xi \ge \mu \end{array} ,\right. \end{equation} \tag{ 5 }$$

ξ = |Z/r|, and Z is the height above the disk midplane in AU. The best-fit values for the COBE data were n₀ = 1.13 × 10⁻⁷, p = 1.34, β = 4.14, γ = 0.942, and μ = 0.189. The units of the normalization are AU²/AU³ (i.e., a volume density of surface area), and the integrated length of the vertical term is not constant with radius, so their exponent p is not the same as our exponent α. Numerically integrating Equation (5) over −∞ < ξ < ∞ yields 0.63, so the surface density at 1 AU is 7.12 × 10⁻⁸ (AU²/AU²). Assuming for mathematical simplicity that γ = 1, the integral of the exponential term in Equation (4) is ∝r for both cases of g, and hence α ≈ p − 1 and α ≈ 0.34. Therefore, the Zodipic model can be reasonably expressed in terms of our model with r₀ = 1 AU Σ_{m, 0} = 7.12 × 10⁻⁸, and α = 0.34.

There remain a few small differences in the surface brightness of our model relative to Zodipic, as shown in the right panel of Figure 1. The main difference is that the Zodipic model is slightly brighter at larger radii because the radial temperature profile is flatter. Setting the Zodipic temperature profile to equal our black body prescription leads to nearly indistinguishable face-on surface brightness profiles. Some minor differences arise because the Kelsall et al. (1998) model defines radius as $r=\sqrt{x^2+y^2+z^2}$, whereas our narrow opening angle means that our model is effectively cylindrical.

Zoom In Zoom Out Reset image size

2.2.2. Reference Model

2.2.3. Normalization (What is a Zodi?)

The original concept for nulling interferometry was put forward by Bracewell (1978), and for a reasonably non-technical description of the instrument we refer the reader to Defrère et al. (2015). Basically speaking, the LBTI combines the beams from the two LBT mirrors, with one beam having a half-wavelength phase shift. Light in the viewing direction, and along lines perpendicular to the mirror baseline vector, is therefore suppressed. Light from off axis sources at odd multiples of the angular distance λ/(2B) is transmitted (B is the baseline length). Similarly, light from sources at even multiples of this angular distance is suppressed. The transmission (or "fringe") pattern is therefore a sin ² function parallel to the baseline vector, and is constant perpendicular to this vector. A model disk, the transmission pattern, and the transmitted disk image are illustrated in Figure 2. This illustration is for an object at transit; at other times the fringe pattern is not aligned with north.

Zoom In Zoom Out Reset image size

The key measurement from an LBTI observation is the ratio of the transmitted flux to the total photometric flux. Because the transmission pattern is designed to suppress light from the star, leaving light from a much fainter disk, this ratio is approximately the ratio of the transmitted disk flux to the stellar flux. Deriving the transmitted disk flux is not straightforward however, as some on-axis (i.e., stellar) flux is always transmitted because the instrument is not perfect and the star has a finite angular size. Removal of these effects is an integral part of the data analysis (see Defrère et al. 2015), so the quantity derived from a given observation is the "calibrated null depth" or "source null depth." In what follows, we generally refer to this measurement and the same quantity derived from our models as simply the "null depth." Because the total flux is almost exactly the stellar flux and dominates over any disk emission, we therefore compute the null depth for our models as the transmission for a starless model divided by the stellar flux density.

2.3.1. Disk Transmission Calculation

Our disk model is radially extended, but to illustrate how surface brightness is transmitted through the LBTI transmission pattern as a function of angular scale we first consider a series of discrete annuli of angular radii ϕ = r/d. We later combine these annuli to model the transmission for our disk models.

The angular distance to the first transmission peak is ϕ_null = λ/(2B); for the LBTI λ = 11 μm and B = 14.4 m, so the distance is 79 mas. The transmission function perpendicular to the fringes T_null is sin ²(πϕ/[2ϕ_null]), and is shown as the gray line in Figure 3, where the x axis is the distance from the annulus center in units of the distance to the first transmission peak (i.e., ϕ/ϕ_null). A common definition for the inner working angle of an instrument is where the sensitivity first reaches half of the peak value, which for LBTI is therefore at λ/(4B) = 39 mas.

Zoom In Zoom Out Reset image size

To calculate the transmission for a point at some azimuth around the annulus requires finding the sky-plane component of this vector point (relative to the star) that is perpendicular to the fringe pattern. This component can be calculated using three rotations, and is

$$\begin{equation} \phi _{\rm proj} = \phi \times \left(\sin \Omega _{\rm LBTI} \cos \theta + \cos \Omega _{\rm LBTI} \sin \theta \cos i \right), \end{equation} \tag{ 6 }$$

where θ is the angle from the sky plane around the annulus to the point of interest, and Ω_LBTI is the position angle of the annulus relative to the LBTI fringe pattern. The angle Ω_LBTI varies with hour angle, and because the LBTI baseline is always perpendicular to the local vertical (i.e., a great circle through the target and zenith), is equal to Ω for an object at transit. The transmission from a point in the annulus is then¹¹

$$\begin{equation} T_{\rm null} = \sin ^2(\pi \phi _{\rm proj} / [2 \phi _{\rm null}]) \,. \end{equation} \tag{ 8 }$$

For a face-on (i = 0) annulus at radius ϕ, the transmission is clearly a function of azimuth around the annulus. Averaging around an annulus yields the total transmission for that annulus, and repeating this calculation for annuli of different angular sizes gives the blue dashed line in Figure 3.

This exercise is finally repeated for a large number of annuli with random orientations, so that cos i is distributed evenly from 0 to 1, and Ω_LBTI is evenly distributed from 0 to 180°. Again, the transmission around each annulus is azimuthally averaged, which results in the dots shown in Figure 3. Averaging these points yields the average transmission as a function of annulus radius for a population of disks with random orientations, shown as the red line. As can be surmised from the decreasing amplitude, the average transmission tends to 0.5 at large separations (i.e., the average of sin ² is 0.5). The minimum separation at which this transmission is achieved is ϕ_null, twice the inner working angle.

The origin of the dot distribution can be understood by considering how annuli of different orientations are transmitted through a given transmission peak. For example, the upper envelope of dots at about 70% transmission are all transmitted through the first transmission peak, and those at higher ϕ/ϕ_null are nearer to edge-on with position angles closer to Ω_LBTI = 0° (i.e., perpendicular to the baseline and parallel to the fringes). This effect is relatively common because the average inclination is about 60° (i.e., biased toward edge-on). By comparing the phase of the gray line with the red line, it is clear that the peak average transmission is actually about a quarter of the way beyond a transmission peak, and that the peak transmission for face-on annuli lies somewhere in between. The phase shift of the face-on transmission can be understood by realizing that an annulus with a radius ϕ that is slightly greater than ϕ_null has more emission in the peak transmission region than an annulus with ϕ = ϕ_null. The average transmission for random orientations is phase shifted slightly further because most disks are inclined, and therefore on average appear somewhat smaller on the sky than they actually are.

Using the reference model of Section 2.2.2, Figure 4 shows the total and transmitted disk flux as a function of radius for a face-on geometry and a distance of 10 pc. That is, the figure is a histogram showing where the total and transmitted flux originates, so the solid line is an azimuthally summed radial profile created from the right panel of Figure 2. Much of the total disk emission originates from the inner regions, and since the disk inner edge is well inside the first transmission peak significant flux from these inner regions is not transmitted. Some flux is also lost at larger radii, and in total only about 30% of the disk flux is transmitted. Overall therefore, the radial distance over which the disk emits strongly in the mid-IR and can be detected is not particularly large; emission is strongly reduced inside the inner working angle, and the faint Wien side of cooler emission means that the surface brightness drops steeply at larger radii. Given the spacing of the transmission peaks and the distance to nearby stars, the transmitted flux detected by the LBTI is constrained to come from near the HZ, even if it cannot be certain that it originates within it.

Zoom In Zoom Out Reset image size

Figure 5 shows how the total transmitted flux is distributed for a random distribution of disk orientations. The concentration of transmitted fluxes near the maximum arises because any disk with a position angle perpendicular to the transmission pattern has the same transmitted flux, regardless of inclination (and any disk near to face-on also has the same transmitted flux). The low transmitted flux tail arises from disks that have a position angle parallel to the transmission pattern and are sufficiently near to edge on such that almost the entire disk lies in the central null transmission region. Such a geometry is relatively unlikely, as can be seen by the lack of dots with low transmission in Figure 3. These relatively rare low transmitted fluxes are allowed because we assume disks with negligible scale height. This distribution therefore represents a pessimistic (but possible) case, and becomes tighter as the disk scale height increases. The width of the distribution also depends on the range of hour angles over which a target is observed, as discussed in Section 2.5.

Zoom In Zoom Out Reset image size

The point here is that the transmitted flux depends on the orientation, and since this is expected to be unconstrained for LBTI targets (unless we use stellar inclination or the orientation of a resolved outer disk as an estimate, which will be possible for some of the HOSTS sample), there will be a corresponding uncertainty in any disk parameters that are derived from the observations.

Since the transmitted flux scales linearly with Σ_{m, 0}, the distribution of Σ_{m, 0}, and hence z, required to reproduce a given transmitted flux is directly related to that in Figure 5. The distribution of possible null depths for our reference model at 10 pc is therefore calculated by dividing the distribution of transmitted fluxes in Figure 5 by the stellar flux. The distribution of zodi levels z implied for a given observed null depth (or upper limit) is found by dividing the observed null depth by the distribution of model null depths for z = 1. This was the method employed by Millan-Gabet et al. (2011) in modeling KIN observations with Zodipic.

Any real LBTI observation takes a finite amount of time, during which any disk will rotate relative to the transmission pattern on the sky. Thus, even a vertically thin edge-on disk, which can be instantaneously invisible to the LBTI, will be visible when the length of the observation is taken into account.

To compute how the disk transmission changes for an observation of a given object requires computing the position angle of the LBTI fringe pattern on the sky at a given hour angle. The common mirror mount for the LBT means that this angle is the difference between the vector in the direction of the local vertical and a vector pointing toward equatorial north along a line of constant right ascension through the object in question (i.e., the hour circle). This angle is commonly called the parallactic angle.

The null depth is then computed as the average null depth over the range of observed hour angles. For modeling purposes here, these are spaced evenly in hour angle, but future models will use the true distribution for each specific observation. Figure 6 shows the difference between the instantaneous sensitivity for a transiting target (left panel) and a 4 hr observation (right panel), for a target at 60° declination over the parameter space of different disk orientations. This calculation is purely related to the changing position angle of the disk, and does not represent the time needed to reach a given S/N. Here, as before, we have assumed a vertically thin disk. The dark region in the upper middle of the left panel shows that an edge on disk is not instantaneously detectable when the disk is aligned with the LBTI fringe pattern. In the right panel, the rotation of the fringe pattern with respect to north over a four hour observation means that the sensitivity relative to a face-on disk is reduced by about 50% for the worst case disk orientation.

Zoom In Zoom Out Reset image size

To understand the origin of this increased sensitivity, Figure 7 again shows the transmission relative to a face-on disk, but now for Ω = 0 at each point in the range of hour angles from −2 to 2 hr. Each curve shows how the transmission for a different disk inclination varies with hour angle (or Ω_LBTI). At HA=0, the curves correspond to a cut at Ω = 0 in the left panel of Figure 6. The average of each line over the hour angle range corresponds to the same cut, but in the right panel. Because the average of each curve is greater than the minimum value at HA=0, the disk sensitivity is greater.

Zoom In Zoom Out Reset image size

Of course, the inclination and position angle dependence in Figures 6 and 7 shows that it would be highly desirable to discern how the transmitted disk flux changes as a function of hour angle in order to learn about the disk geometry and the possibility of non-axisymmetric structure. For axisymmetric disks there is also a potential degeneracy between the disk geometry and the vertical scale height; an edge-on disk with sufficient vertical extent will vary much less than the curves in Figure 7, and may be indistinguishable from a less inclined disk that is oriented such that the transmission variation is small. If a large variation in transmission is seen, the disk may be vertically thin, but could alternatively be non-axisymmetric. The detection of such effects will certainly be sought, but given that we expect most LBTI detections to be near the sensitivity limits and therefore at low S/N we assume for now that LBTI observations can be modeled by simply averaging over the hour angle.

The LBTI is well suited to observing dust levels in the habitable regions around nearby stars, both in terms of baseline length and observing wavelength. LBTI results will therefore broadly address the goal of characterizing warm dust levels without requiring additional information from other observations. However, the specific location and radial extent of the dust will be poorly constrained, and as illustrated by Figure 4, significant disk emission can arise from regions both inside and outside the habitable zone and still be detected with the LBTI. In general, such degeneracy in disk models will be hard to break because the goal of the LBTI is to detect disks that are fainter than current detection limits (hence the need for the modeling framework outlined in this paper).

In some cases, however, LBTI results will be compared to other observations, most likely (spectro)photometric detections or limits for the total disk flux density at the same wavelength, and in some cases limits on disk size from high-resolution imaging or interferometry. Detections and/or upper limits from near-IR interferometry will provide constraints on dust levels inside the LBTI IWA. LBTI will similarly provide very strong constraints on the emission spectrum and location of hot dust detections (Absil et al. 2013; Ertel et al. 2014).¹² Mid-IR photometric data will come from Spitzer InfraRed Spectrograph (IRS; Werner et al. 2004; Houck et al. 2004), WISE observations (Wright et al. 2010), or in the future the spectrometers on the Mid-Infrared Instrument (MIRI) on the JWST, which cover the same wavelength range as the LBTI. Such photometry will provide the most useful constraints on the disk location, so we now briefly outline how the total and LBTI-transmitted disk fluxes vary with model parameters, in particular the disk inner and outer radii.

To illustrate the flux density distribution, Figure 8 shows disk spectra for our reference model with z = 1, with examples for both Sun-like and A-type host stars at 10 pc. Because our standard model has the same temperature range all disk spectra have the same shape, but are brighter or fainter depending on the zodi level, the distance to the host star, and the host star spectral type (which changes the area of the disk and hence total brightness). The A5V star has L_⋆ = 14 L_☉, so the disk around this star is 14 times brighter than for the Sun-like star. However, the A-star is only about 5 times brighter at 11 μm (10 vs. 2 Jy), so the disk/star flux ratio has increased by a factor of about 3.

Zoom In Zoom Out Reset image size

Figure 8 also shows how the spectrum varies when r_in, r_out, and α are changed from their standard values, and what the effect is at the LBTI wavelength of 11 μm. Larger values of α concentrate the disk emission closer to the star, and hence shift the spectrum toward warmer temperatures. Changing r_in and r_out remove emission from the inner and outer disk regions, leading to a spectrum that is closer to a single-temperature black body spectrum. As noted above and shown by the curve for r_out = 5 AU, the outer disk radius is large enough that the exact value matters little for the level of emission at 11 μm.

While our model gives a reasonable idea of the true disk spectrum, and could in principle be used to constrain disk structure based on observations at other wavelengths, it is inaccurate on a detailed level because grains do not emit like black bodies. Three important differences are that (1) the disk will be fainter than our model at long wavelengths because grains emit inefficiently at wavelengths longer than their size, (2) the spectrum may also be shifted to shorter wavelengths due to the presence of small grains with temperatures greater than that of a black body, and (3) the spectrum may show non-continuum spectral features. These differences pose problems for extrapolating our model to other wavelengths, or at least mean that additional parameters such as grain properties would need to be considered.

Such differences are of minor importance here, however, as important comparisons with other observations can be made at the LBTI wavelength, so are largely independent of the disk spectrum. The most important comparison is between the total and transmitted disk fluxes, or equally the total disk to star flux ratio and the null depth. To illustrate why combining such measurements is important, Figure 9 shows how the total and transmitted model disk flux changes at 11 μm as the disk inner and outer radii change. As r_in is increased the total disk flux decreases because hot emission is being removed. However, the LBTI-transmitted flux changes little while r_in remains small, because these changes occur behind the central transmission minimum and are invisible. When r_in is outside the first transmission peak (at about 0.7 AU here) both the total and transmitted fluxes decrease in the same way. As r_out is decreased there is initially little difference in the total and transmitted fluxes because the outer disk is faint at 11 μm, but when r_out moves behind the central transmission minimum the flux drops much more steeply than the total flux. As noted earlier, Figure 9 shows that the LBTI transmission is insensitive to our choice of disk inner and outer radii, as long as the disk is much wider than the habitable zone.

Zoom In Zoom Out Reset image size

Given such relations between the total and transmitted disk fluxes, it is clear that observations of both may constrain the disk location. However, the transmitted disk flux is a function of both the disk size and the orientation, so the best constraints on the disk size require the orientation to be known (or take some assumed value). It is unlikely that the position angle will be inferred from LBTI measurements in many cases, so the disk orientation would probably be assumed based on other system information, such as coplanarity with a resolved outer cool disk component, the known inclination and/or position angle of the host star's rotation axis (Le Bouquin et al. 2009; Greaves et al. 2014), or of planet orbits (Reidemeister et al. 2009), or a combination of all three (Kennedy et al. 2013). See Defrère et al. (2015) for an application of this assumption to η Crv.

Currently, NASA is focused on developing concepts for three optical (0.4–1 μm) exoplanet-imaging missions, Exo-C (a coronagraph), Exo-S (a starshade), as well as an ambitious precursor, the coronagraph on WFIRST-AFTA. Therefore, the thermal dust emission information from the LBTI must be converted to predict the impact on scattered light imaging. We now derive a simple prescription for converting the dust levels in the above model into scattered light surface brightness estimates.

Scattered light predictions are in general difficult, and have largely proven unsuccessful to date (e.g., Krist et al. 2010), with debris disks imaged in scattered light generally seen to be much fainter than predicted based on theoretical grain models that match observed thermal emission. The typical minimum grain size in debris disks is thought to 1–10 μm for Sun-like stars (e.g., Gustafson 1994; Krivov 2010; Pawellek et al. 2014), and the steepness of the size distribution means that these grains dominate the surface area. Such grains are expected to scatter optical light fairly isotropically, and have fairly large albedos of ≳0.5. The scattered light faintness of debris disks, where albedos of 0.05–0.1 are seen (e.g., Kalas et al. 2005; Golimowski et al. 2011; Meyer et al. 2007), likely arise from incorrect assumptions about grain properties and sizes, and how these grains scatter starlight.

The properties of exo-zodi are not sufficiently well known that considering physically motivated possibilities for different grain sizes and properties would make predictions for their scattered light brightness more certain. Therefore, regardless of the physical reason, we will assume that the ∼0.1 effective albedos seen for scattered light disks around nearby stars are representative, and assume isotropic scattering. It may be that the properties of warm dust are different to those inferred from scattered light detections, since for example hot dust detected with near-IR interferometry is generally inferred to originate in much smaller grains (e.g., Defrère et al. 2011; Lebreton et al. 2013). For higher/lower albedos, our scattered light predictions would be higher/lower by the same factor.

While the model surface density Σ_m is connected to the true optical depth and the surface area of grains from which the emission arises, the model surface density is the true optical depth only if the grains behave like black bodies. That is, real grains reflect starlight, so the true total surface area in the disk is always higher than Σ_m. To be more realistic, and to allow for the possibility of scattered light, the grain absorption and scattering properties need to be considered.

Consider a disk with particles with a range of sizes D, and Σ_true(D)dD the true cross-sectional area per unit area of particles in the size range D to D+dD. Using the absorption efficiency as a function of size and wavelength Q_abs(λ, D), the thermal emission is

$$\begin{equation} S_{\rm th} = 2.35 \times 10^{-11} \int \Sigma _{\rm true}(D) Q_{\rm abs}(\lambda,D) B_\nu (\lambda,T[D]) dD. \end{equation} \tag{ 9 }$$

Though the dependence is not included explicitly here for simplicity, all quantities in this equation also vary with location in the disk due for example to the disk structure and changing composition. Using the scattering efficiency Q_sca(λ, D) the scattered light emission can be written in various ways, but a convenient form is

$$\begin{equation} S_{\rm sca} = \frac{F_{\nu,\star }}{4 \pi } \left(\frac{d}{r}\right)^2 \int \Sigma _{\rm true}(D) Q_{\rm sca}(\lambda,D)\, dD, \end{equation} \tag{ 10 }$$

which we could also express in terms of albedo ω = Q_sca/(Q_abs + Q_sca) by substituting Q_sca = Q_absω/(1 − ω). If we assume the albedo ω is the empirical value of 0.1, independent of grain size and wavelength, then because Q_abs = 1 − ω the surface density Σ_m is approximately the true optical depth, but is underestimated by a factor of 1 − ω (i.e., Σ_m = Σ_true[1 − ω]). That is, with these assumptions the thermal surface brightness could be written

$$\begin{equation} S_{\rm th} = 2.35 \times 10^{-11} \Sigma _{\rm true} B_\nu (\lambda,T_{\rm BB}) (1-\omega). \end{equation} \tag{ 11 }$$

The scattered light brightness is therefore calculated using the dust surface density and the empirical effective albedo. With these assumptions, the predicted scattered light emission from the model of Section 2.2 would be

$$\begin{equation} S_{\rm sca} = \frac{F_{\nu,\star }}{4 \pi } \left(\frac{d}{r}\right)^2 \frac{\omega }{1-\omega } \Sigma _{\rm m}. \end{equation} \tag{ 12 }$$

For simplicity we use the value of Σ_m derived from our modeling, and thus include an extra factor of 1/(1 − ω). Therefore, with our model the scattered light surface brightness relative to the stellar flux¹³ in the habitable zone (i.e., r = r₀) of a 1 zodi disk decreases as 1/L_⋆, as the habitable zone is pushed farther from the star by the increased luminosity. This is the approach we use below to estimate the limiting scattered light surface brightnesses for HOSTS, and for the LBTI η Crv detection.

An extension to this approach would be to add a phase function that accounts for forward scattering properly. The albedo could then be derived theoretically and calibrated with observations. While in the above case the scattered light can be calculated simply along the radial direction, and for a given orientation turned into an image, use of a phase function g(θ) requires a three-dimensional calculation that includes the star-particle-observer scattering angle θ at each location in the disk. Given that we expect relatively large uncertainties in disk parameters, even in the case of a detection, use of a detailed three-dimensional calculation including (assumed) grain properties and a phase function will in general be unwarranted.

To convert the simple scattered light prediction of Equation (12) to an observable for an arbitrary disk inclination, a three-dimensional calculation must be made to account for brighter disk ansae. However, the disk inclination will in general be unknown, so a simple approximation for deriving a representative scattered light surface brightness would be to assume an average inclination of 4/π, and for a vertically thin disk an increase of 1/cos (4/π) ≈ 3.

Using η Crv with a null depth of 4.4% ± 0.35% as an example, we first show how the distribution of zodi levels is derived assuming an unknown disk orientation. This procedure is generic in that it can be applied to any LBTI observation, whether a significant detection or an upper limit was found. The LBTI is still in commissioning so the zodi sensitivity derived here is not illustrative of the expected performance.

Figure 10 shows the uv plane coverage of the LBTI observation relative to the disk position angle, and the distribution of zodi levels calculated using our reference model. We include the rotation of the disk relative to the fringe pattern during the observation to compute the average null depth for each disk orientation. As described at the end of Section 2.4, the distribution is essentially the inverse of that shown in Figure 5.

Zoom In Zoom Out Reset image size

The distribution is strongly peaked at about z = 1350, with a tail of larger values due to unfavorable disk orientations. The 1σ zodi uncertainty due purely to the calibrated null depth measurement is given by the width of the "median ± null uncertainty" error bar in the right panel of Figure 10. Similarly, the 16% and 84% levels from the cumulative distribution give a representative 1σ range due to the orientation distribution. The upper uncertainty on the zodi level is therefore set by the orientation distribution, while the lower uncertainty is set by the null depth measurement. The median zodi level for η Crv is z = 1376 ± 102 if the uncertainty is set by the LBTI null depth measurement. Including the 1σ range from the orientation distribution in quadrature, the range covered on either side of the median value is 1236 < z < 1869. In general, we expect that the null depth uncertainty will dominate the lower bound on the zodi level, and the orientation distribution will dominate the upper bound.

For comparison, based on a KIN detection and using the Zodipic model Mennesson et al. (2014) found z = 1813 ± 209 for η Crv at 8.5 μm with no assumptions about the disk orientation. Our derived zodi level is different for two reasons: (1) η Crv is hotter than the Sun (6900 K), so our luminosity-dependent zodi definition will lead to a zodi level about 1.7 times smaller than Zodipic, and (2) the mid-IR spectrum of the η Crv disk increases more steeply with wavelength than a black body (i.e., has a silicate spectral feature), so although our model has nearly the same temperature profile as Zodipic, our derived zodi level will be slightly larger because LBTI observes at 11 μm. Therefore, direct comparisons between our zodi levels and those using Zodipic should not be made.

3.1.1. Generic Scattered Light Surface Brightness

Given a prediction for the sensitivity of the LBTI and a sample of stars that will be observed, we can make predictions for the sensitivity of the HOSTS survey. In what follows we assume a 1σ uncertainty on the LBTI calibrated null depth of 10⁻⁴, and hence the limits presented are also at 1σ. Limits are calculated as above, using the median of the distribution of zodi levels over the random distribution of orientations. Though we do not account for it here, there is some uncertainty in the stellar flux densities and luminosities used, which we estimate to contribute at about the 5% level. Here we calculate the sensitivity assuming an observation at a single hour angle, but when the model is applied to real observations in the future the calculations will account for sky rotation as described in Section 2.5.

These calculations are carried out for both our reference disk model, and a "worst-case" scenario where the dust is restricted to lie only within the habitable zone, which we assume to lie between temperatures of 320 and 210 K. As shown in Figures 4 and 9, the LBTI is sensitive to emission that lies interior and exterior to the habitable zone, so the dust surface density must be higher in the latter case for an LBTI detection at the same sensitivity. As noted above in Section 2.2.3, the "zodi levels" derived with this different radial structure are also different and should really be considered as enhancements over the surface density derived for the solar zodiacal cloud, rather than zodi levels to be compared with other values.

The HOSTS survey sample is described by Weinberger et al. (2015), the key aspect being that targets are chosen such that their habitable zones have larger angular sizes than the first LBTI transmission peak, so observations directly probe the levels of habitable zone dust. The sample is split by B−V color at 0.42 into "Sun-like" and "sensitivity" sub-samples, which simply reflects the levels of dust that can be detected.

Figure 11 shows the predicted disk to star flux ratios, null depths, and sensitivity in zodi units for the HOSTS sample using our reference model (red and blue symbols) and our worst case scenario (gray symbols, which we discuss below). The top and middle panels show the flux ratios and null depths expected for a 1 zodi model around these stars, and the bottom panel shows the sensitivity in zodis for the predicted LBTI sensitivity. There are clear trends with stellar luminosity, which can be understood as follows. With our model the disk flux density at fixed wavelength scales with the angular area, i.e., ∝zΣ_{m, 0}L_⋆/d² (with our zodi definition Σ_{m, 0} is constant, but we include it to consider other zodi definitions below). The stellar flux, on the other hand, scales $\propto L_\star /(T_\star ^3 d^2)$ in the Rayleigh–Jeans regime so the total disk to star flux ratio only depends on the star, and is higher for earlier spectral types

$$\begin{equation} \frac{F_{\rm disk}}{F_{\nu,\star }} \propto z \Sigma _{\rm m,0} T_\star ^3 \,. \end{equation} \tag{ 13 }$$

This dependence in the top panel of Figure 11 arises due to the stronger scaling of stellar luminosity with temperature than flux density at 11 μm, with the variations from a perfect correlation arising due to variation in L_⋆ at fixed T_⋆ (i.e., different stellar radii). At fixed distance, as the stellar temperature and luminosity increase the habitable zone is pushed outward (${\propto}\sqrt{L_\star }$) and its area and brightness increase more rapidly than the stellar flux. Alternatively, for a fixed disk angular size (and hence fixed disk brightness), increasing the stellar temperature and luminosity pushes the system to greater distances and the star becomes fainter due to increasing distance faster than it becomes brighter due to an increased temperature.

Zoom In Zoom Out Reset image size

The null depths for a 1 zodi disk are shown in the middle panel, and the trend has a similar origin as the disk to star flux ratio. At fixed distance, increasing stellar luminosity increases the disk surface brightness at fixed angular radius because r₀ increases (i.e., increases if α is positive). Assuming that most of the transmitted disk flux originates from a constant angular scale (i.e., near the first transmission peak), the transmitted disk flux therefore increases as $\propto (\sqrt{L_\star }/d)^\alpha$ (assuming that the effect of the changing disk temperature in the first transmission peak is small). Combining this expression with the stellar flux yields a null depth

$$\begin{equation} {\rm null} \propto z \Sigma _{\rm m,0} T_\star ^3 \left[ \left(\sqrt{L_\star }/d\right)^\alpha \right] \,. \end{equation} \tag{ 14 }$$

HOSTS stars are chosen to have habitable zones with similar angular sizes and for our reference model α = 0.34, so the term in square parentheses varies relatively little and the null depth almost entirely depends on the stellar temperature.

The absolute null depth level of course also varies linearly with the zodi level, so the zodi limits can be derived by dividing the expected sensitivity of 10⁻⁴ by the null depth values in the middle panel (i.e., by solving Equation (14) for z)

$$\begin{equation} z \propto \Sigma _{\rm m,0}^{-1} T_\star ^{-3} \left[ \left(d/\sqrt{L_\star }\right)^\alpha \right] \,. \end{equation} \tag{ 15 }$$

The resulting sensitivities show that what we define as solar system levels of zodiacal dust are at the predicted 1σ noise level for early-type stars, as are 3–10 zodi disks around Sun-like stars.

This discussion of scalings applies equally to the narrow worst case scenario, shown as gray symbols in Figure 11. As expected for a narrower disk that emits over a smaller total physical area, the disk to star flux ratios are lower than for our reference model, as are the null depths. While the difference between the two models is about a factor of five in disk/star flux ratio, the difference in null depths is only a factor of two to three because most of the flux removed from the disk in the narrower model is hidden behind the central transmission minimum (e.g., Figure 9). Similarly, the difference in zodi levels is a factor two to three higher in this case compared to our reference model. Therefore, the effect of this pessimistic scenario in terms of habitable zone dust levels is relatively minor.

Because our zodi definition is based on constant surface density in the habitable zone, the z dependence on luminosity directly shows that the LBTI can truly detect lower surface densities of dust in the habitable zones of earlier type stars. This conclusion does not depend on our zodi definition because the LBTI is sensitive to Σ_m(r₀). A different definition, for example z∝F_disk/F_⋆, makes the zodi limit approximately constant ($\propto [d/\sqrt{L_\star }]^\alpha$), but also implies $\Sigma _{\rm m,0} \propto T_\star ^{-3}$ (and the sensitivity to habitable zone surface density is the same as with our definition).

3.2.1. Scattered Light for HOSTS

Given the detection limits in the bottom panel of Figure 11, we can also derive the face-on habitable zone ($r=r_0=\sqrt{L_\star /L_\odot }$ AU) scattered light surface brightness of disks at the HOSTS survey detection limits. These limits are shown in Figure 12 for ω = 0.1 as defined in Section 2.7. Red and blue symbols show limits for our standard disk model, which again depend on the stellar luminosity.

Zoom In Zoom Out Reset image size

The origin of the dependence on stellar luminosity can be understood by rewriting Equation (12) using the zodi limit scaling above

$$\begin{equation} S_{\rm sca} \propto \frac{F_{\nu,\star }}{T_\star ^3} \left(\frac{d}{\sqrt{L_\star }}\right)^{2+\alpha } \,. \end{equation} \tag{ 16 }$$

Like the zodi limits, the scattered light levels in the habitable zone at these limits decrease with stellar luminosity. The squared distance and luminosity dependence is simply the geometric effect that accounts for 1/r² dilution of light and that surface brightness is an angular measure. The extra α dependence is because the habitable zone is not at the first transmission peak for all targets. If the LBTI observation is dominated by emission near the first transmission peak, but the habitable zone is slightly farther out (i.e., $d/{\rm 1pc}/\sqrt{L_\star /L_\odot } > 1$), the α dependence represents a model-dependent extrapolation that is relatively unimportant while α is small.

As with the LBTI sensitivity to habitable zone surface density, our zodi definition has no effect on the scattered light predictions and is merely an intermediate step for deriving the true quantity of interest. That is, as described above the LBTI sensitivity to Σ_m in the habitable zone (i.e., zΣ_{m, 0}), which sets the limits on S_sca, will always be the same. For the example of a constant F_disk/F_⋆ zodi definition therefore, z is constant but $\Sigma _{\rm m,0} \propto T_\star ^{-3}$ and Equation (16) is the same.

Because the HOSTS sample is explicitly chosen such that the LBTI is sensitive to thermal emission from dust in the region of interest for Earth-imaging, the scattered light predictions do not strongly depend on the choice of our disk model parameters (but of course depends on albedo). The predictions are not totally independent of our disk model however, since for example changing the steepness of the radial profile makes the dust in the model more or less concentrated relative to the LBTI transmission pattern, thus changing the derived zodi level. Varying α between −1 and 1 yields changes of ≲1 mag in the scattered light predictions, meaning that the model dependent uncertainty is similar to the variation expected from the unknown disk inclination.

For solar-type stars, the scattered light levels in Figure 12 for our face-on reference model at the ∼4 zodi detection limit is 20–21 mag arcsec⁻². For comparison, using a 4 zodi disk the Zodipic model predicts 21.2 mag arcsec⁻² at 1 AU with default parameters (a solar analog with ω = 0.18 and use of a phase function), and 21.6 mag arcsec⁻² for ω = 0.1 and isotropic scattering. Our simple model therefore compares well with the more complex calculation made by Zodipic. For a 60° inclined disk, Zodipic gives 21.1 mag arcsec⁻², and for 90° gives 20.6 mag arcsec⁻², and therefore the increase from face-on to edge-on is about 1 magnitude. Our model has smaller vertical extent than Zodipic, so the increase in surface brightness for inclined disks will be larger, roughly a factor of three for a 60° inclined disk.

The gray symbols in Figure 12 show the scattered light surface brightness for narrower disks that only lie in the habitable zone. These limits are about one magnitude brighter than our reference model, so similar to the variation in brightness with the (unknown) disk orientation. Therefore, the scattered light limits from LBTI observations are fairly robust to disk width.