INTERFEROMETRY OF AURIGAE: CHARACTERIZATION OF THE ASYMMETRIC ECLIPSING DISK

Our first interferometric data set on Aurigae was acquired using the PTI (Colavita et al. 1999) located on Mount Palomar in California. The facility consisted of three 40 cm siderostats, each located at the termination of one of the interferometer's three arms. Pairwise combination provided baselines between 85 and 110 m. The beam combiner at PTI operated in several low-resolution spectral modes providing up to 11 spectral channels across the K band ($2.2\;\mu {\rm{m}}$).

Much of the wide-band visibility data has been discussed previously in Stencel et al. (2008), therefore, we will not consider it further. Here we have re-reduced the spectrally dispersed (i.e., narrow-band) data subject to the calibrator diameters in Table 2. We have used the narrow- and wide-band calibration routines nbCalib and wbCalib, respectively, from the V2calib software package. These programs are available as a web service, webCalib, on the NASA Exoplanet Science Institute (formerly Michelson Science Center) website.¹⁰ We have selected the PTI defaults with the following exceptions: (1) the calibration window was extended to four hours; (2) no ratio correction was applied, and (3) no minimum uncertainty was enforced for reasons discussed in Stencel et al. (2008). Output from this pipeline were saved in OIFITS format.

The NPOI (Armstrong et al. 1998) is a six telescope optical interferometer that started operation in 1994. The array may be configured in either an astrometric or imaging mode. Data from the imaging subarray comes from six movable 50 cm siderostats with baselines between 16 and 79 m. The NPOI beam combiner operates at visible wavelengths (0.5–0.85 μm) in 16 spectral channels. The NPOI observational setup and data recording procedure can be found in Hummel et al. (2003) and Benson et al. (2003). Post-processing and data reduction were performed using C. Hummel's OYSTER software package.

The NPOI observed Aurigae on a total of 29 nights between 2006 February and 2010 April as shown in Table 1. The data were initially calibrated with respect to HD 32630 assuming a uniform disk diameter of 0.507 ± 0.025 milliarcseconds (mas) and saved as OIFITS files. Prior to modeling these data, we recalibrated the OIFITS files by multiplying the visibilities and closure amplitudes by the ratio of the uniform disk function of the former and new diameter as listed in Table 2.

Georgia State University's CHARA (ten Brummelaar et al. 2005) is an interferometric array located on Mount Wilson, CA. The array consists of six 1-m telescopes that can be combined to form up to 15 baselines ranging in length from 34 to 331 m. Using the longest baselines, a resolution of down to 0.5 mas in the H band (0.7 mas in the K band) can be realized.

Initial calibration observations of Aurigae were taken far in advance of the eclipse, in 2008 October/November/December. Semi-regular observations were scheduled around the photometric eclipse, beginning in 2009 October/November and ending in 2011 November. In total, Aurigae was observed on 38 nights, yielding 19 individual epochs after consecutive nights were merged. The first two eclipse ingress epochs were previously discussed in Kloppenborg et al. (2010) in which reconstructed images and a preliminary model (comprised of an infinitely thin, but optically thick disk seen in projection) for the eclipsing disk were presented.

2.3.1. Michigan InfraRed Combiner (MIRC)

2.3.2. CLassic Interferometry with Multiple Baselines (CLIMB)

In this work, we use Bayesian statistics to assess the relative goodness of fit between our proposed models rather than traditional chi-squared methods that are ill-adapted to make such inferences (Marshall et al. 2006). Bayesian statistics provides a consistent approach to estimate a set of parameters, Θ, in a hypothesis (e.g., an image or model), H, given some observed data, D. Bayes’ theorem states that

$$\begin{eqnarray}&&P({\rm{\Theta }}| D,H)=\displaystyle \frac{P(D| {\rm{\Theta }},H)P({\rm{\Theta }}| H)}{P(D| H)},\end{eqnarray} \tag{ 1 }$$

where $P({\rm{\Theta }}| D,H)$ $\equiv$ $P({\rm{\Theta }})$ is the posterior probability distribution of the parameters, $P(D| {\rm{\Theta }},H)\equiv L({\rm{\Theta }})$ is the likelihood, $P({\rm{\Theta }}| H)=\pi ({\rm{\Theta }})$ is the prior, and $P(D| H)\equiv Z$ is the Bayesian Evidence. In parameter estimation problems, where the model remains the same, the normalization factor, Z, is often ignored as it is independent of the parameters Θ. When selecting between various models, the evidence plays a central role through the Bayes Factor:

$$\begin{eqnarray}&&R=\displaystyle \frac{P({H}_{1}| D)}{P({H}_{0}| D)}=\displaystyle \frac{P(D| {H}_{1})P({H}_{1})}{P(D| {H}_{0})P({H}_{0})}=\displaystyle \frac{{Z}_{1}}{{Z}_{0}}\displaystyle \frac{P({H}_{1})}{P({H}_{0})}.\end{eqnarray} \tag{ 2 }$$

The evidence is the average of the likelihood over the prior. Thus a simple model with greater likelihood over the parameter range will be favored over a more complex model with lower likelihood over the parameter range, unless the latter is significantly better at explaining the data. Therefore the Bayes factor automatically implements the Occam razor principle.

A principal difficulty in using Bayesian evidence for model selection is that the multidimensional integral,

$$\begin{eqnarray}&&P(D| H)\equiv Z=\int L({\rm{\Theta }})\pi ({\rm{\Theta }}){d}^{D}{\rm{\Theta }},\end{eqnarray} \tag{ 3 }$$

must be evaluated. For this work, we decided to use the MultiNest library (Feroz & Hobson 2008; Feroz et al. 2009, 2013) that numerically estimates this integral by intelligently exploring the parameter range using Markov chain methods and ellipsoidal bounding conditions. In the model fitting process we have assumed non-informative (flat) priors and used the standard likelihood function:

$$\begin{eqnarray}&&L({\rm{\Theta }})=\displaystyle \prod _{i}\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_{i}^{2}}}\mathrm{exp}\left[\displaystyle \frac{{(M{({\rm{\Theta }})}_{i}-{D}_{i})}^{2}}{2{\sigma }_{i}^{2}}\right],\end{eqnarray} \tag{ 4 }$$

where $M{(\theta )}_{i}$ is the model prediction, D_i is the data, and ${\sigma }_{i}$ is the uncertainty associated for the ith data point.

Within the SIMTOI framework, the F-star is modeled as a single uniformly illuminated sphere, located at the origin, to which GLSL shaders (simulating limb darkening) were applied. The PTI observations are very high up on the visibility curve, therefore, they provide no reliable measurement of limb darkening. Consequently, we fit all PTI data with a uniform disk model. The CHARA and NPOI observations frequently resolve the F-star near or beyond the first visibility null. These data show clear departures from uniform disk behavior. To account for this, we implemented several limb darkening laws in GLSL. These include linear, logarithmic, square root, power law (via. Hestroffer 1997), and a few multi-parameter laws (e.g., Claret 2000; Claret & Hauschildt 2003; Fields et al. 2003).

The previous models for the Aurigae disk were purely geometric representations with hard edges (e.g., Kloppenborg et al. 2010; Kloppenborg 2012, and references therein). In this work, we have created geometric and astrophysical density distribution models with position-dependent optical properties. Because of the edge-on nature of the eclipse, we elected to represent all disks as a series of concentric rings of infinitesimal thickness and uniform total height, h, that are equally spaced between an inner, ${r}_{\mathrm{in}}$, and outer, ${r}_{\mathrm{out}}$, radius. The rings are connected at the midplane by another surface.

Inspired by the appearance of proplyds in the Orion Nebula (e.g., Ricci et al. 2008), the geometrical models have opacity (via. OpenGL source transparency, ${\mathrm{src}}_{\alpha }$) that is controlled by a double power law that is a function of radius and height:

$$\begin{eqnarray}&&{\mathrm{src}}_{\alpha }(r,z)={\left(\displaystyle \frac{r}{{r}_{\mathrm{out}}}\right)}^{-\alpha }{\left(\displaystyle \frac{z}{h/2}\right)}^{-\beta }.\end{eqnarray} \tag{ 5 }$$

For the two astrophysical disk models, we keep the concentric ring representation of the disk, but modify the opacity of each ring according to a real density distribution. The first model from Pascucci et al. (2004) consists of a power law in the radial direction and a scale height exponential taper in the vertical direction:

$$\begin{eqnarray}&&\rho (r,z)={\rho }_{0}{\left(\displaystyle \frac{r}{{r}_{c}}\right)}^{-\alpha }{e}^{-\displaystyle \frac{1}{2}{\left((,\displaystyle \frac{z}{h},)\right)}^{2}},\end{eqnarray} \tag{ 6 }$$

where ${\rho }_{0}$ is the density, r_c is the scale radius, $h={h}_{c}{\left(\frac{r}{{r}_{c}}\right)}^{-\beta }$, h_c is the scale height, and units are in angular quantities (e.g., mas or mas⁻³ when appropriate). The second density distribution is characterized by a power-law in the inner disk and an exponential taper at large radii (cf. Andrews et al. 2009):

$$\begin{eqnarray}&&\rho (r,z)={\rho }_{0}{\left(\displaystyle \frac{r}{{r}_{c}}\right)}^{-\gamma }{e}^{-\displaystyle \frac{1}{2}{\left((,\displaystyle \frac{z}{h},)\right)}^{2}}{e}^{-{\left((,\displaystyle \frac{r}{{r}_{c}},)\right)}^{2-\gamma }}.\end{eqnarray} \tag{ 7 }$$

For these models, the OpenGL transparency is calculated as ${\mathrm{src}}_{\alpha }=1-{e}^{-\kappa (\lambda )\rho (r,z)}$ to ensure the pixel intensity is computed following radiative transfer conventions. Because the composition and precise optical properties of the disk have yet to be ascertained, we have treated the product ${\kappa }_{0}{\rho }_{0}$ as a single quantity in our minimization.

In this work we also introduce time-dependence by positioning the disk according to a Keplerian orbit. The spectroscopic orbital solutions from Stefanik et al. (2010) and Chadima et al. (2010) are in excellent agreement with each other; however, the astrometric orbital parameters are not well constrained. Therefore, our model adopts the spectroscopic orbital parameters (longitude of periastron, ω, eccentricity, e, period, P, and time of periastron, T) from Stefanik et al. (2010) and we derive the astrometric parameters (inclination, i, longitude of the ascending node, Ω, and total orbital semimajor axis, ${\alpha }_{T}$) via. minimization to our data.

Within the framework of the aforementioned models, the number of parameters needed ranges from 6 in the case of a uniformly illuminated F-star and cylindrical disk (e.g., i, Ω, ${\alpha }_{T}$, ${\theta }_{{\rm{UDD}}}$, ${r}_{\mathrm{disk}}$, and ${h}_{\mathrm{disk}}$) to 10 with a limb darkened disk and more complex opacity model for the disk (see Table 4 for a summary of all parameters used and their permitted ranges). To reduce the number of degrees of freedom, we elected to establish bounds by solving a series of subproblems first, then lift restrictions to generalize the results.

• 1.  
  First, use Bayesian evidence and the post-eclipse CHARA-MIRC data to establish bounds for the diameter and limb-darkening of the F-star.
• 2.  
  Once determined, the diameter was used to approximate the total orbital semimajor axis, ${\alpha }_{T}$, for the system using two methods. To first order, we may assume the orbit is circular and the eclipse transects the equator of the F-star. Then ${\alpha }_{T}$ may be found, to first order, by equating the fraction of the orbit spent in ingress, with the equivalent sector of the orbit via,

  $$\begin{eqnarray}&&\displaystyle \frac{T}{{t}_{\mathrm{ingress}}}=\displaystyle \frac{p}{s}\approx \displaystyle \frac{2\pi {\alpha }_{T}}{{\theta }_{\mathrm{star}}},\end{eqnarray} \tag{ 8 }$$

  where p is the perimeter of the orbit, s is the orbital sector, ${\alpha }_{T}$ is the separation of the components, and ${\theta }_{\mathrm{star}}$ is the diameter of the F-star in radians. If $s\ll p$, we may perform a second-order approximation for the sector using the Ramanujan approximation for the perimeter of an ellipse. In this case, we find:

  $$\begin{eqnarray}\displaystyle \frac{T}{{t}_{\mathrm{ingress}}} & \approx & \pi {\alpha }_{T}\left[3(1+\sqrt{1-{e}^{2}})\right.\\ & & -\left.\sqrt{10\sqrt{1-{e}^{2}}+3(2-{e}^{2})}\right]\displaystyle \frac{\mathrm{cos}\omega }{{\theta }_{\mathrm{star}}},\end{eqnarray} \tag{ 9 }$$

  where ω and e are the aforementioned orbital quantities.
• 3.  
  Approximate the position angle of the ascending node, Ω, by computing the average of the position angles determined from single-epoch in-eclipse minimizations.
• 4.  
  Determine the best-fit disk model by performing a simultaneous fit to the photometric and interferometric data. The F-star's diameter and limb darkening are held constant. ${\alpha }_{T}$ and i are free, whereas Ω, ω, e, P, and T are constant.
• 5.  
  Derive the best-fit F-star's diameter, limb darkening coefficient, disk height, and disk transparency at each epoch while holding the remaining parameters constant.
• 6.  
  Finally, lift the constraints on our models insofar as possible to derive statistical information of the aforementioned parameters via bootstrapping individual epochs.

We have used the four post-eclipse CHARA-MIRC observations to compare seven different limb darkening laws for the F-star. As seen in Table 3, the uniform disk model is always a poor fit to the data compared to the limb darkening models. In all but one epoch, the quadratic limb darkening law was found to be a better fit to the interferometric data than other models. On 2011 October 10, the two-parameter law described in Fields et al. (2003) was a better fit.

Table 3.  Posterior Odds Ratios (${\rm{\Delta }}\mathrm{log}R$) Relative to the Uniform Disk Model

	Posterior Odds Ratio (${\rm{\Delta }}\mathrm{log}R$, See Section 3.1)
Epoch	Claret (2000)	Fields et al. (2003)	Logarithmic	Power Lawa	Quadratic	Square Root
2011-09-18	9504	9518	9515	9506	9545	9506
2011-09-24	1954	1949	1959	1956	1971	1954
2011-10-10	2568	2615	2560	2561	2560	2558
2011-11-03	6151	5900	6165	6166	6227	6134

Note.

^aAlthough there is slight evidence in favor of the quadratic limb darkening law, visual inspection of the visibility data shows no significant difference between it and the power-law limb darkening that we pragmatically adopted in this work.

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Table 4.  A Summary of all Parameters Used in the Modeling Process

Parameter	Range	Units	Description
F-star parameters
${\theta }_{{\rm{UDD}}}$	1–3	mas	Uniform disk diameter
${\theta }_{{\rm{LDD}}}$	1–3	mas	Limb darkened disk diameter
${\alpha }_{\mathrm{LDD}}$	0–1	⋯	Power-law limb darkening coefficient
Geometric disk models
${r}_{\mathrm{in}}$	0	mas	Inner radius
${r}_{\mathrm{out}}$	1–30	mas	Outer radius
${h}_{\mathrm{disk}}$	0–5	mas	Height
α	5–20	⋯	Radial exponent
β	0.1–5	⋯	Height exponent
Astrophysical density disk models
${\kappa }_{0}{\rho }_{0}$	3000–10,000	⋯	Characteristic opacity
α	5–20	⋯	Radial exponent (Pascucci model)
β	0.1–5	⋯	Height exponent (Pascucci model)
γ	0.001–20	⋯	Radial exponent (Andrews model)
hc	0.001–20	mas	Disk scale height
rc	1–4	mas	Disk scale radius
${i}_{\mathrm{disk}}$	±10	deg	Disk inclination
${{\rm{\Omega }}}_{\mathrm{disk}}$	±10	deg	Disk position angle
Orbital parametersa
e	0.227 ± 0.01	⋯	Eccentricy
i	70–110	deg	Inclination
ω	39.2 ± 3.4	deg	Longitude of periastron
P	9,896 ± 1.6	day	Period
Ω	90–145 and 270–325	deg	Position Angle
${\alpha }_{T}$, ${\alpha }_{1}$, ${\alpha }_{2}$	$13\lt {\alpha }_{T}\lt 38$	mas	Semi-major axis (total, F-star, disk)
T	2,434,723 ± 80	days	Time of periastron

Note.

^ae, ω, P, and T from Stefanik et al. (2010).

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It would appear that the quadratic limb darkening law is most appropriate; however, an inspection of the fits reveals that all models predict essentially the same visibility function. Interestingly, all models also predict significantly lower flux at the limb than is implied by either plane parallel or spherical stellar atmosphere codes. We verified that the SIMTOI results matched LitPro's analytical models, therefore, it is unlikely that this effect is fictitious. We have noticed small, few degree, non-zero closure phases and two epochs where the location of the first visibility null differs between baselines. Thus it is possible that the F-star may be slightly oblate or have surface features. We will explore these possibilities in greater detail in a future publication. Because it is simple and can reproduce the data, we pragmatically adopted the power-law limb darkening law to represent the F-star in this work. The mean out-of-eclipse diameter and limb darkening coefficient are 2.22 ± 0.09 mas and 0.50 ± 0.26, respectively. This range of values compares favorably with published diameters from the NPOI (${\theta }_{{\rm{UDD}}}$ 2.18 ± 0.05 mas at $0.5-0.85\;\mu$m, Nordgren et al. 2001), the Mark III (limb darkened diameter, ${\theta }_{{\rm{LDD}}}$, $1.888-2.136$ mas at $0.4-0.8\;\mu {\rm{m}},$ Mozurkewich et al. 2003), and PTI (${\theta }_{{\rm{UDD}}}$ 2.27 ± 0.11 mas at 2.2 μm, Stencel et al. 2008) interferometers.

Using the equations in Section 3.4, we have estimated the orbital semimajor axis. Assuming $T=9896\pm 1.6$ days (Stefanik et al. 2010), the H-band ingress time 145 ± 15 days (Hopkins 2012), and a circular orbit, we estimate ${\alpha }_{T}={\alpha }_{1}+{\alpha }_{2}\sim 24\pm 2.6$ mas where ${\alpha }_{1}$ and ${\alpha }_{2}$ are the semimajor axes of the F-star and disk with respect to the system's center of mass. In the elliptical case, the spectroscopic elements from Stefanik et al. (2010) and Chadima et al. (2010) orbital elements yield nearly identical results of ${\alpha }_{T}\sim 31\pm 3.7$ and $33\pm 4.5$ mas, respectively. The dominant source of uncertainty in these values comes from the ∼10% errors in ω and ${t}_{\mathrm{ingress}}$. Recognizing estimates for the F-star's contribution to ${\alpha }_{T}$ are some 13–24 mas (Strand 1959; van de Kamp 1978; Heintz & Cantor 1994), we establish bounds of $13\lt {\alpha }_{T}\lt 38$ mas for our minimization. A non-equatorial intersection, as seen in our previous work (Kloppenborg et al. 2010), will strictly decrease the upper bound on ${\alpha }_{T}$.

Next, we estimated Ω by fitting the in-eclipse CHARA-MIRC data with a model consisting of a circular F-star with power-law limb darkening and cylindrical disk. We set α_T = 31 mas and ${r}_{\mathrm{disk}}=10$ mas while permitting the stellar and orbital inclination parameters to remain free. An average of Ω = 296 ± 3 deg was obtained from the six totality epochs.

From the aforementioned disk models, we created eight variants. The geometric models are (1) a hard-edged cylinder, and three variants with power-law transparency in (2) both radius and height, (3) height only, and (4) radius only. The models from astrophysical density distributions are (5) a Pascucci et al. disk and (6) an Andrews et al. disk. Although not supported by the interferometric images, we decided to test for the central clearing hypothesis (c.f. Ferluga 1990, and references therein) and created (7) a Pascucci et al. disk with a variable inner radius. Finally we test whether the Kemp et al. (1986) polarization model by creating (8) a Pascucci et al. disk that may be tilted out of the orbital plane.

Using the MultiNest minimizer, we derived $\mathrm{log}Z$ estimates by simultaneously fitting each model to a subset of the H-band photometric data (consisting of 200 observations spaced at approximately equal intervals throughout the eclipse) and six interferometric epochs from CHARA-MIRC (2009-11, 2009-12, 2010-08, 2010-11, 2011-01, and 2011-09-18). The best-fit parameters, posterior odds ratio (${\rm{\Delta }}\mathrm{log}R$), and average ${\chi }^{2}$ values for each model are shown in Table 5. The ${\rm{\Delta }}\mathrm{log}R$ values indicate that model 8 (the tilted Pascucci disk) provides the best simultaneous fit to the data and was therefore adopted for the remainder of our work.

Table 5.  Bayes Factors Relative to the Cylinder Model and Average Reduced ${\chi }^{2}$ for the Six Disk Models Described in the Section 3.3

		Orbit		Disk										Fit Information
Name	Modela	${\alpha }_{T}$	i	${r}_{\mathrm{in}}$	${r}_{\mathrm{out}}$	h	α	β	rc	hc	$\kappa \rho$ ${}^{{\rm{b}}}$	${{\rm{\Omega }}}_{\mathrm{disk}}$	${i}_{\mathrm{disk}}$	${\rm{\Delta }}\mathrm{log}R$	${\chi }_{r}^{2}({\rm{H}})$	$\bar{{\chi }_{r}^{2}}({V}^{2})$	$\bar{{\chi }_{r}^{2}}({T}_{3{\rm{A}}})$	$\bar{{\chi }_{r}^{2}}({T}_{3\phi })$
		(mas)	(deg)	(mas)	(mas)	(mas)			(mas)	(mas)		(deg)	(deg)
Cylinder	1	21.7	88.4	0.0	4.85	0.54	…	…	…	…	…	…	…	0	260	10	12	48
Ringed Disk	2	27.0	88.7	0.0	7.20	0.71	0.13	1.64	…	…	…	…	…	63940	69	4.1	4.3	31
RingedDisk (only β)	3	27.5	88.8	0.0	6.39	0.79	…	0.02	…	…	…	…	…	59811	93	4.4	4.1	32
RingedDisk (only α)	4	27.5	88.8	0.0	7.40	0.70	0.11	…	…	…	…	…	…	63792	67	4.1	4.4	31
Pascucci Disk	5	33.0	89.0	0.0	…	…	11.11	1.75	2.39	0.029	6496	…	…	67058	50	3.8	4.1	30
Andrews Disk	6	33.0	89.0	0.0	…	…	11.14	1.74	2.48	0.032	5161	…	…	67110	49	3.8	4.1	31
Pascucci Disk w/ clearing	7	32.6	88.9	3.8	…	…	10.94	0.75	2.32	0.079	6667	…	…	67254	51	3.8	4.0	31
Tilted Pascucci Diskc	8	31.2	88.9	0.0	…	…	13.33	3.69	2.77	0.007	6287	−0.02	2.98	68806	50	3.7	3.7	28

Notes.

^aSee Section 5.3 for model descriptions. ^bThe product κρ was treated as a single quantity in our minimization process. see Section 5.3 for further details. ^cModel 8 obtains the highest Bayes factor and is therefore adopted in this work.

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We have rendered the best-fit version of each disk model in Figure 2. With the exception of the cylindrical disk, all disk models have a characteristic size of ∼6 mas radius and ∼0.75 mas height before becoming optically thin. Plots of the corresponding H-band photometry in Figure 3 show that although the disk models appear physically different, they all do a reasonable job reproducing the global properties of the light curve. This clearly demonstrates that reproducing the photometry is a necessary, but not sufficient condition to prove the validity of any particular disk model. We note that none of these symmetric disk models are capable of reproducing all of the photometric features seen during the eclipse.

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Using model 7 (a Pascucci disk with a variable inner radius), we test the notion that the disk's central clearing is responsible for the alleged mid-eclipse brightening. We find that the disk remains edge-on, but has an inner radius of some 3.8 mas. This implies that the inner 60% of the disk could be devoid of any opaque material. Despite this fact, the impact on the light curve is minimal. Given the geometry of the eclipse, light penetrating the central clearing cannot be responsible for any mid-eclipse brightening. We will discuss this result in greater detail in Section 6.

The photometry predicted by the symmetric disk models creates an interesting corollary: if one assumes the disk is symmetric, one may immediately conclude, to the contrary, that the disk must be asymmetric because the residuals between the observed and predicted photometry far exceed the F-star's ${\rm{\Delta }}H\sim 0.05\;\mathrm{mag}$ variations seen outside of eclipse. We tested this asymmetric conjecture by performing several single-epoch MultiNest minimizations to the in-eclipse MIRC data. We created three additional disk models which (a) forced the disk to reside in the orbital plane, (b) tilted the disk out of the orbital plane at a fixed angle, and (c) permitted both the position and inclination of the disk with respect to the orbital plane to vary. We find that the disk is tilted out of the orbital plane by less than 4° (1.33 ± 0.67 deg with rejection of one outlier) and has significant variations in structure (see Table 9).

To derive statistical uncertainties and simulated photometry we used the best-fit values from the aforementioned minimizations as starting points and bootstrapped each interferometric epoch 10,000 times. During each bootstrap we dynamically recalibrated the data using the uncertainty distribution of the calibrator. Then we created a new realization of the data using the measured uncertainties, taking into account any known correlations (i.e., as seen in spectrally dispersed visibilities) in the data when required. The results of this effort are shown in Table 6 with a subset of the results plotted in Figure 4.

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Table 6.  Bootstrapped Nominal Values and Uncertainties for Model 8 With a Fixed Tilt Evaluated on a Per-epoch Basis Subject to Only the Interferometric Data

					F-star			Disk					Statistical Information
Data set	$N({V}^{2})$	$N({T}_{3})$	$N({UV})$	Effective JD	${\theta }_{{\rm{UDD}}}$	${\theta }_{{\rm{LDD}}}$	${\alpha }_{\mathrm{LDD}}$	$\kappa \rho$	rc	hc	α	βa	${\chi }_{r}^{2}$	${\chi }_{r}^{2}({V}^{2})$	${\chi }_{r}^{2}({T}_{3A})$	${\chi }_{r}^{2}({T}_{3\phi })$	Notes
					(mas)	(mas)			(mas)	(mas)
							$(\times {10}^{-2})$	$(\times {10}^{-2})$
1997-10-22-PTI	14	…	16	2450744.0200	2.43 ± 0.29	…	…	…	…	…	…	…	0.38	0.38	…	…	⋯
1998-11-07-PTI	66	…	69	2450761.9500	2.13 ± 0.28	…	…	…	…	…	…	…	0.27	0.27	…	…	Distribution not well constrained
1997-11-09-PTI	20	…	21	2451124.9500	2.78 ± 0.15	…	…	…	…	…	…	…	2.21	2.21	…	…	⋯
1998-11-25-PTI	10	…	11	2451142.9400	1.93 ± 0.44	…	…	…	…	…	…	…	0.78	0.78	…	…	Distribution well constrained. HD 30823 sole calibrator this night. Perhaps calibrator diameter over-estimated?
1998-11-26-PTI	5	…	5	2451143.9300	2.06 ± 0.53	…	…	…	…	…	…	…	0.53	0.53	…	…	Distriubtion poorly constrained. Nominal value matches best-fit MultiNest estimate.
2006-02-NPOI	540	135	544	2453791.7434	2.09 ± 0.06	…	…	…	…	…	…	…	2.35	2.65	2.00	1.49	⋯
2007-03-NPOI	660	330	966	2454173.6316	…	${2.28}_{-0.02}^{+0.07}$	${0.42}_{-0.05}^{+0.23}$	…	…	…	…	…	2.28	0.90	1.67	5.67	⋯
2007-10-19-PTI	100	…	104	2454392.9900	2.13 ± 0.13	…	…	…	…	…	…	…	0.59	0.59	…	…	⋯
2007-10-20-PTI	81	…	84	2454393.9700	2.08 ± 0.14	…	…	…	…	…	…	…	0.18	0.18	…	…	⋯
2007-10-21-PTI	40	…	43	2454394.9800	2.16 ± 0.24	…	…	…	…	…	…	…	1.35	1.35	…	…	⋯
2007-11-27-PTI	20	…	22	2454431.8500	2.55 ± 0.32	…	…	…	…	…	…	…	2.16	2.16	…	…	Distribution not constrained, highly skewed toward higher values.
2007-12-23-PTI	5	…	5	2454457.7800	2.09 ± 0.43	…	…	…	…	…	…	…	0.68	0.68	…	…	⋯
2007-12-24-PTI	35	…	39	2454458.7500	2.45 ± 0.26	…	…	…	…	…	…	…	1.75	1.75	…	…	⋯
2008-02-17-PTI	10	…	10	2454513.6400	1.75 ± 0.33	…	…	…	…	…	…	…	3.22	3.22	…	…	Distribution is well constrained. Bad calibration?
2008-02-18-PTI	25	…	27	2454514.6500	2.17 ± 0.34	…	…	…	…	…	…	…	1.77	1.77	…	…	⋯
2008-09-CHARA-MIRC	23	8	50	2454729.0153	…	${2.28}_{-0.08}^{+0.08}$	${0.69}_{-0.20}^{+0.20}$	…	…	…	…	…	1.91	2.46	1.94	0.29	Errors limited by calibrator uncertainty
2008-10-17-PTI	65	…	69	2454757.0000	2.37 ± 0.20	…	…	…	…	…	…	…	5.71	5.71	…	…	⋯
2008-10-26-PTI	80	…	84	2454765.9800	2.01 ± 0.15	…	…	…	…	…	…	…	0.88	0.88	…	…	⋯
2008-11-CHARA-MIRC	138	76	268	2454778.5967	…	${2.22}_{-0.06}^{+0.06}$	${0.39}_{-0.12}^{+0.12}$	…	…	…	…	…	3.81	4.59	3.24	2.96	Errors limited by calibrator uncertainty
2008-11-08-PTI	10	…	11	2454778.8700	2.17 ± 0.40	…	…	…	…	…	…	…	0.08	0.08	…	…	⋯
2008-11-09-PTI	20	…	21	2454779.8700	2.34 ± 0.49	…	…	…	…	…	…	…	2.18	2.18	…	…	Distribution not constrained and highly skewed toward higher values. Bad calibration?
2008-11-16-PTI	10	…	10	2454786.8700	…	…	…	…	…	…	…	…	…	…	…	…	⋯
2008-11-22-PTI	50	…	52	2454792.8200	2.30 ± 0.23	…	…	…	…	…	…	…	1.32	1.32	…	…	⋯
2008-12-CHARA-MIRC	38	8	65	2454810.8249	…	${2.36}_{-0.06}^{+0.06}$	${0.80}_{-0.17}^{+0.17}$	…	…	…	…	…	3.67	4.03	5.42	0.23	Errors limited by calibrator uncertainty
2009-03-NPOI	840	420	1225	2454898.6326	…	${2.16}_{-0.02}^{+0.07}$	${0.37}_{-0.04}^{+0.25}$	…	…	…	…	…	1.85	1.65	1.29	2.80	⋯
2009-11-CHARA-MIRC	1091	672	2575	2455138.9326	…	${2.29}_{-0.03}^{+0.02}$	${0.62}_{-0.10}^{+0.08}$	${6329}_{-2038}^{+2038}$	${1.79}_{-0.02}^{+0.02}$	${2.57}_{-0.40}^{+0.40}$	${9.19}_{-0.06}^{+0.04}$	${1.56}_{-0.10}^{+0.12}$	4.01	2.64	0.00	8.57	⋯
2009-12-CHARA-MIRC	730	392	1662	2455169.0375	…	${2.28}_{-0.04}^{+0.04}$	${1.00}_{-0.12}^{+0.11}$	${6842}_{-2009}^{+2009}$	${3.72}_{-0.07}^{+0.07}$	${12.18}_{-1.00}^{+0.50}$	${19.94}_{-0.03}^{+0.01}$	${0.10}_{-0.02}^{+0.28}$	9.26	4.08	3.77	24.39	⋯
2009-12-NPOI	290	29	293	2455185.6210	…	${2.05}_{-0.04}^{+0.21}$	${0.72}_{-0.20}^{+0.28}$	${7007}_{-1847}^{+1847}$	${3.23}_{-0.03}^{+0.22}$	${6.70}_{-4.10}^{+8.50}$	${16.72}_{-0.06}^{+0.04}$	${0.92}_{-0.16}^{+0.21}$	2.14	2.20	1.89	1.73	Visibities at short baselines are much higher than CHARA model would predict.
2010-01_AB-NPOI	3324	1376	4444	2455205.5082	…	${2.24}_{-0.02}^{+0.03}$	${0.76}_{-0.07}^{+0.14}$	${6211}_{-1690}^{+1690}$	${2.97}_{-0.06}^{+0.08}$	${4.10}_{-0.30}^{+0.30}$	${17.84}_{-0.02}^{+0.02}$	${2.43}_{-0.02}^{+0.07}$	1.23	1.61	1.21	0.33	Disk rc is clearly bimodal
2010-02-NPOI	810	265	814	2455243.1816	…	${2.37}_{-0.10}^{+0.10}$	${0.62}_{-0.24}^{+0.24}$	${6219}_{-2000}^{+2000}$	…	${10.70}_{-1.30}^{+1.30}$	…	${0.70}_{-0.23}^{+0.23}$	1.42	1.47	1.81	0.87	⋯
2010-02-CHARA-MIRC	96	64	236	2455245.7444	…	${2.01}_{-0.04}^{+0.04}$	${0.21}_{-0.13}^{+0.13}$	${3075}_{-1600}^{+1600}$	…	${13.14}_{-0.50}^{+0.50}$	…	${0.55}_{-0.11}^{+0.11}$	6.88	1.64	1.18	20.45	Interferometry + Photometry. Statistics from MultiNest distribution.
2010-04-NPOI	15	0	15	2455289.6026	…	${2.33}_{-0.28}^{+0.28}$	${0.56}_{-0.24}^{+0.24}$	${6500}_{-2000}^{+2000}$	…	${11.50}_{-7.70}^{+7.70}$	…	${0.88}_{-1.05}^{+1.05}$	0.53	0.53	0.00	0.00	⋯
2010-08-CHARA-MIRC	960	640	2164	2455430.5170	…	${2.33}_{-0.04}^{+0.04}$	${0.74}_{-0.11}^{+0.09}$	${9901}_{-1334}^{+1334}$	…	${1.27}_{-0.10}^{+0.10}$	…	${3.75}_{-0.12}^{+0.07}$	10.74	3.97	3.90	27.74	⋯
2010-09-CHARA-MIRC	1176	728	3020	2455464.4883	…	${2.37}_{-0.02}^{+0.03}$	${0.73}_{-0.04}^{+0.06}$	${5944}_{-1888}^{+1888}$	…	${2.35}_{-0.10}^{+0.20}$	…	${3.11}_{-0.08}^{+0.10}$	3.38	1.96	1.98	7.07	⋯
2010-10-CHARA-MIRC	288	152	732	2455496.4319	…	${2.26}_{-0.03}^{+0.03}$	${0.62}_{-0.08}^{+0.08}$	${6629}_{-1973}^{+1973}$	…	${1.47}_{-0.10}^{+0.80}$	…	${3.56}_{-0.62}^{+0.18}$	1.77	1.31	0.77	3.67	⋯
2010-11-CHARA-MIRC	288	192	763	2455505.4193	…	${2.18}_{-0.01}^{+0.01}$	${0.11}_{-0.03}^{+0.05}$	${3720}_{-1817}^{+1817}$	…	${9.66}_{-0.20}^{+0.20}$	…	${0.40}_{-0.07}^{+0.11}$	4.64	2.93	1.16	10.71	⋯
2010-12-CHARA-MIRC	191	112	475	2455543.7059	…	${2.22}_{-0.03}^{+0.04}$	${0.31}_{-0.09}^{+0.10}$	${5548}_{-1985}^{+1985}$	…	${0.72}_{-0.10}^{+0.10}$	…	${4.71}_{-0.28}^{+0.28}$	5.72	2.29	1.36	15.92	⋯
2011-01-CHARA-MIRC	310	182	860	2455580.2465	…	${2.18}_{-0.02}^{+0.02}$	${0.25}_{-0.04}^{+0.04}$	${3013}_{-1838}^{+1838}$	…	${4.94}_{-0.20}^{+1.10}$	…	${1.84}_{-0.52}^{+0.16}$	5.01	2.57	1.36	12.83	⋯
2011-04-CHARA-CLIMB	41	14	45	2455655.0673	…	${2.33}_{-0.06}^{+0.06}$	${0.33}_{-0.48}^{+0.48}$	${6100}_{-2000}^{+2000}$	${3.48}_{-0.30}^{+0.30}$	${13.20}_{-3.80}^{+3.80}$	${18.55}_{-1.96}^{+1.96}$	${0.64}_{-0.50}^{+0.50}$	3.23	4.06	1.04	2.97	⋯
2011-09-18-CHARA-MIRC	201	240	756	2455823.0305	…	${2.25}_{-0.04}^{+0.02}$	${0.62}_{-0.10}^{+0.06}$	…	…	…	…	…	5.06	3.55	3.02	8.37	⋯
2011-09-24-CHARA-MIRC	120	160	394	2455829.0277	…	${2.17}_{-0.03}^{+0.03}$	${0.36}_{-0.07}^{+0.06}$	…	…	…	…	…	5.40	3.32	1.67	10.69	⋯
2011-10-10-CHARA-MIRC	400	480	1412	2455844.9422	…	${2.12}_{-0.05}^{+0.04}$	${0.34}_{-0.12}^{+0.11}$	…	…	…	…	…	16.97	5.49	6.82	36.70	⋯
2011-11-03-CHARA-MIRC	831	1119	2677	2455868.8509	…	${2.25}_{-0.07}^{+0.03}$	${0.57}_{-0.17}^{+0.08}$	…	…	…	…	…	12.14	5.64	7.21	21.90	⋯

Note.

^aThe changes seen in height power, β, hint that there may be some asymmetric structure in the disk.

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By inspection of Table 6, one can see that the angular diameter and limb darkening profile of the F-star are largely consistent within 1σ. This implies that there are no egregious systematic calibration errors between our data sets. Furthermore, these results show that there has been no secular change to the F-star's diameter over the last 14 years. However, our observations are not sufficiently precise to definitively exclude the 0.6% year⁻¹ contraction rate suggested by Saitō & Kitamura (1986).

In the top panel of Figure 4 we plot the H-band photometry as observed, simulated from the symmetric model, and predicted from the bootstrap process described above. Not all of the photometric values are in perfect agreement, but this is expected as the photometry was not used as a constraint in the bootstrapping process. We note that a small change in disk structure can have a substantial impact on the observed photometry (e.g., a 0.1 mas difference in thickness, five times smaller than our resolution limit, results in ${\rm{\Delta }}H\sim 0.1$ mag), thus the predicted photometry is in reasonable agreement.

In the bottom three panels of Figure 4, we plot the angular diameter, limb darkening coefficient, disk scale height, h_c, and height power-law exponent, β, as a function of time. The 1σ estimates for h_c and β from model 8 do not overlap well with the single-epoch bootstraps. This is likely due to the ∼1.6° difference in disk inclinations between the models. The changes in h_c and β (greater h_c, smaller β) indicate that the disk is more spatially extended in height before mid-eclipse than it is after mid-eclipse. Combined with the asymmetric evolution of several neutral absorption lines during eclipse (c.f. Lambert & Sawyer 1986; Leadbeater et al. 2012), the evidence suggests that the disk is not purely symmetric. We suspect that these features could be explained by asymmetric heating (c.f. Takeuti 2011) and sublimation of the disk on the side facing the F-star. Verifying this claim by hydrodynamical radiative transfer simulations is beyond the scope of this work.

Because of the limited UV coverage, the egress CHARA-CLIMB data was fit in conjunction with an interpolated photometric point. The result reveals that the F-star is similar in size, but the disk is more extended in the vertical direction than the CHARA-MIRC observations three months earlier would imply. This interpretation is supported by the appearance of the model-independent images (discussed below); however, we caution the reader that this may simply be an artifact of the limited UV coverage of this data set.

In Table 7, we show the aggregate statistics derived from single-epoch bootstrapping for each beam combiner. We believe these values represent the general characteristics of the system. In Table 8, we summarize these quantities for a variety of distance estimates for the system. Due to the large scatter of possible distances (0.5–4 kpc), we use only the nominal values and do not propagate any uncertainties from the distance measurements. We caution the reader that the aggregation of data to create Table 8 was performed without regard to either the wavelength of observation or any asymmetries we advocate exist. Hence, the outliers have biased and skewed the resulting value. We provide this last table to assist with the creation of a full radiative transfer model of the system including dust physics rather than provide a definitive measurement of the properties of the system.

Table 7.  Aggregate Statistics for all Interferometric Data with Uncertainties Determined from the Maximum of the Upper/Lower Averaged Bootstrapped Uncertainties or the Standard Deviation of the Nominal Values

Quantity	Units	NPOI (V)	MIRC (H)	CLIMB (K)	PTI (K)
Quantity	⋯	NPOI (V)	MIRC (H)	CLIMB (K)	PTI (K)
${\theta }_{{\rm{UDD}}}$	(mas)	2.09 ± 0.06	2.10 ± 0.15	⋯	2.22 ± 0.53
${\theta }_{{\rm{LDD}}}$	(mas)	2.21 ± 0.28	2.22 ± 0.09	2.33 ± 0.06	…
${\alpha }_{\mathrm{LDD}}$	⋯	0.47 ± 0.28	0.50 ± 0.26	0.33 ± 0.48	…
${{\rm{\Omega }}}_{\mathrm{disk}}$	(deg)	⋯	1.30 ± 0.67	⋯	…
${i}_{\mathrm{disk}}$	(deg)	⋯	89.49 ± 1.03	⋯	…
κρa	⋯	6676 ± 2000	5667 ± 2188	6100 ± 2000	…
rc	(mas)	3.10 ± 0.22	2.76 ± 1.36	3.48 ± 0.30	…
hc	(mas)	0.07 ± 0.09	0.05 ± 0.05	0.13 ± 0.04	…
α	⋯	17.28 ± 0.79	14.56 ± 7.60	18.55 ± 1.96	…
β	⋯	1.23 ± 1.05	2.18 ± 1.67	0.64 ± 0.50	…

Notes. Orbital values of ${\rm{\Omega }}=297.60\pm 0.06$ (deg), $i=88.89\pm 0.03$ (deg), and ${\alpha }_{T}=31.2\pm 0.9$ (mas) were used in these models.

^aNot well constrained.

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Table 8.  Representative^a Linear Equivalent of our Results if the System were at Various Distances in Literature

			Nominal distance estimates (pc)
Quantity	This work		600b,c	653d	737e	1000f	1500g	Linear units
Ω	297 ± 3	(deg)	⋯	⋯	⋯	⋯	⋯	⋯
i	89 ± 1	(deg)	⋯	⋯	⋯	⋯	⋯	⋯
${\alpha }_{T}={\alpha }_{1}+{\alpha }_{2}$	31 ± 3	(mas)	18.72 ± 1.80	20.37 ± 1.96	22.99 ± 2.21	31.20 ± 3.00	46.80 ± 4.50	(AU)
F-star Radius	1.11 ± 0.05	(mas)	143.25 ± 5.81	155.90 ± 6.32	175.96 ± 7.13	238.75 ± 9.68	358.13 ± 14.52	(${R}_{\odot }$)
F-star LDD coeff	0.50 ± 0.26	⋯	⋯	⋯	⋯	⋯	⋯	⋯
Disk scale Height (hc)	1.038 ± 0.139	(mas)	0.03 ± 0.03	0.03 ± 0.03	0.04 ± 0.04	0.05 ± 0.05	0.07 ± 0.07	(AU)
Disk scale radius (rc)	7.416 ± 0.276	(mas)	1.66 ± 0.82	1.80 ± 0.89	2.03 ± 1.00	2.76 ± 1.36	4.14 ± 2.04	(AU)

Notes. These values average over all interferometric epochs. Therefore, these estimates are biased and are skewed by the outliers in Table 6.

^aWe caution the reader that this aggregation of data is performed in a wavelength and model-agnostic fashion. Thus, any asymmetries in the system, which we argue exist, have biased the values quoted here. These values are supplied to ease the creation of a radiative transfer model including dust physics. We do not advocate that these values be quoted elsewhere. ^bvan de Kamp (1978). ^cHeintz & Cantor (1994). ^dvan Leeuwen (2008). ^eKloppenborg (2012). ^fStrand (1959). ^gGuinan et al. (2012).

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Table 9.  Best-fit Values and Statistical Information for Single-epoch MultiNest Minimizations Involving Three Variants^a of the Tilted Pascucci Disk Model (Model 8) The Posterior Odds Ratio (${\rm{\Delta }}\mathrm{log}R$) is Relative to the Pascucci Zero tilt Disk Model

	F-star		Disk							Statistical Information
	${\theta }_{{\rm{LDD}}}$(mas)	${\alpha }_{\mathrm{LDD}}$	${i}_{\mathrm{disk}}$(deg)	${{\rm{\Omega }}}_{\mathrm{disk}}$(deg)	α	rc(mas)	β	hc(mas)	$\kappa \rho$	${\rm{\Delta }}\mathrm{log}R$	${\chi }_{r}^{2}$	${\chi }_{r}^{2}({V}^{2})$	${\chi }_{r}^{2}({T}_{3{\rm{A}}})$	${\chi }_{r}^{2}({T}_{3\phi })$
Model: Pascucci zero tilt
2009-11	2.30	0.63	…	…	9.19	1.87	1.67	0.023	4708	…	4.92	4.23	2.62	8.34
2009-12	2.28	1.00	…	…	19.95	3.71	0.10	0.118	7046	…	9.39	4.00	3.78	25.02
2010-02	2.43	0.98	…	…	…	…	0.75	0.173	3046	…	3.47	2.83	3.26	4.66
2010-08	2.33	0.74	…	…	…	…	3.78	0.011	4349	…	10.25	4.00	3.69	26.18
2010-09	2.37	0.72	…	…	…	…	3.00	0.021	3336	…	3.24	1.98	2.05	6.48
2010-10	2.29	0.66	…	…	…	…	4.20	0.007	5569	…	1.78	1.33	0.91	3.49
2010-11	2.18	0.10	…	…	…	…	0.36	0.093	3106	…	4.80	2.91	1.14	11.29
2010-12	2.29	0.41	…	…	…	…	4.83	0.005	3726	…	4.94	2.92	1.49	11.84
2011-01	2.13	0.18	…	…	…	…	3.05	0.015	3030	…	5.73	3.31	1.86	13.71
Average	2.29 ± 0.09	0.60 ± 0.32	0	0	14.57 ± 7.61	2.79 ± 1.30	2.41 ± 1.75	0.052 ± 0.061	4215 ± 1373	…	5.39	3.06	2.31	12.34
Model: Pascucci fixed tilt
2009-11	2.29	0.62	…	…	9.19	1.79	1.56	0.026	6329	−37	4.91	3.98	2.64	8.69
2009-12	2.28	1.00	…	…	19.94	3.72	0.10	0.122	6842	119	9.22	3.93	3.63	24.66
2010-02	2.43	0.99	…	…	…	…	0.72	0.179	−5	296	3.53	2.92	3.39	4.59
2010-08	2.33	0.74	…	…	…	…	3.35	0.014	−493	−2999	10.65	3.93	3.82	27.57
2010-09	2.39	0.76	…	…	…	…	2.81	0.028	−129	5802	3.37	2.02	1.95	6.98
2010-10	2.29	0.66	…	…	…	…	3.66	0.011	−8	1235	1.70	1.34	0.88	3.18
2010-11	2.18	0.10	…	…	…	…	0.37	0.097	29	849	4.69	2.91	1.15	10.91
2010-12	2.29	0.39	…	…	…	…	4.80	0.006	−56	394	5.22	2.97	1.56	12.73
2011-01	2.13	0.17	…	…	…	…	2.27	0.030	81	661	5.50	3.02	1.68	13.53
Average	2.29 ± 0.09	0.60 ± 0.32	1.3	−0.02	14.56 ± 7.60	2.76 ± 1.36	2.18 ± 1.62	0.057 ± 0.061	4948 ± 2245	−55	5.42	3.00	2.30	12.54
Model: Pascucci free tilt
2009-11	2.30	0.62	1.102	0.16	8.93	1.74	1.55	0.025	6018	349	4.65	4.37	2.52	7.25
2009-12	2.27	1.00	1.148	−0.15	19.96	3.73	0.10	0.120	6417	164	9.14	3.65	3.29	25.22
2010-02	2.43	0.95	1.088	−2.72	…	…	1.10	0.176	46	347	3.06	2.37	2.63	4.52
2010-08	2.33	0.78	2.330	0.70	…	…	4.75	0.005	1298	−1208	16.17	7.69	7.96	37.09
2010-09	2.39	0.78	1.153	0.51	…	…	2.70	0.027	422	6353	2.92	1.71	1.67	6.11
2010-10	2.32	0.71	0.353	−0.29	…	…	4.92	0.003	13	1256	4.43	2.11	1.19	12.06
2010-11	2.24	0.29	6.879	−0.95	…	…	0.24	0.000	691	1511	5.22	2.93	1.37	12.52
2010-12	2.23	0.38	2.324	−0.96	…	…	4.97	0.000	330	780	114.24	22.38	14.86	370.25
2011-01	2.20	0.23	1.108	−0.87	…	…	2.08	0.037	1391	1971	1.62	1.25	0.87	3.01
Average	2.30 ± 0.08	0.64 ± 0.28	1.943 ± 1.955b	−0.51 ± 1.03	14.45 ± 7.80	2.73 ± 1.40	2.49 ± 1.97	0.044 ± 0.062	4223 $\pm$ 1319	522	17.94	5.39	4.04	53.12

Notes.

^aThe variations either assume the disk has zero tilt, a fixed tilt, or a per-epoch tilt with respect to the orbital plane. The averaged inclination of $1\buildrel{\circ}\over{.} 33\pm 0\buildrel{\circ}\over{.} 67$ agrees well with the multi-epoch minimizations. The variations in scale height appear to be real. ^bExcluding the 2010-11 result, this becomes $1.33\pm 0.67$ in agreement with our fixed-tilt model.

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In Figure 5, we present the best-fit model and model-independent SQUEEZE reconstructions using the total variation regularizer. By following the qualitative comparison method discussed in Section 4, we may distinguish whether the features seen in these images are real signals or artifacts of the reconstruction process. A detailed discussion of this process, as well as BSMEM and other SQUEEZE reconstructions, can be found in the Appendix and extended Figures A46–A52. Summarizing this account, the true features are as follows: (1) the dark lane in the F-star's southern hemisphere, interpreted to be the disk; (2) flux that appears on the far south–west (or south–east) edge of the F-star during ingress (egress); and (3) the presence or absence of the F-star's southern pole. Although the post-eclipse interferometric observations do feature small non-zero closure phases and photometry of the Aurigae system does show an intrinsic ${\rm{\Delta }}V\sim 0.1\;\mathrm{mag}$ photometric variation outside of eclipse, we presently regard any flux variations on the surface F-star as artifacts.

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Tarball of figure set images

Figure A46 shows the resulting model and image from this set of two four-telescope MIRC observations that were taken at nearly the same hour angle. The image reconstruction in this epoch is quite poor when compared with later epochs. By inspection of the SED (Hoard et al. 2012, 2010) we know that the F-star is the dominant source of flux in the H band, hence we interpret this to be an image of the F-star. All of the images show that the flux is mostly constrained within the bounds of the best-fit model. The model-independent image shows the star as approximately round with some surface features. These features are mostly replicated in the model-independent reconstruction from the synthetic data, hence the non-circular structure and small photometric variations seen on the F-star are artifacts of the reconstruction process. Therefore the F-star before the eclipse does not appear to have any egregious asymmetries which might interfere with later observations.

In Figure A47, we show the best-fit model and model-independent images reconstructed from this eclipse ingress phase epoch. For reasons discussed above (and in greater detail in Kloppenborg et al. 2010), we again interpret the bright source to be the F-star and the dark region occuring in the southern half of the image as the disk intruding into the line of sight. We are not aware of any evidence which suggests the F-stars rotation is misaligned to the binary's orbit, hence we shall call the un-eclipsed portion of the F-star the “northern hemisphere.” Likewise, we will refer to the (mostly) eclipsed portion as the “southern hemisphere.” The North pole of the F-star would be located at a position angle of ∼26°.

By comparing the best-fit model from SIMTOI to the sampled model reconstruction, we may qualitatively access the presence of artifacts in the image. The dark spot in the northern hemisphere and two bright spots in the east/west near the limb appear to be artifacts. The straight edges along the perimeter of the F-star are a common artifact caused by the UV coverage of the data set.

Despite the large number of artifacts, several real features may be discerned. For example, the southern pole is seen in the real image, model, and synthetic reconstruction; hence we feel this feature is real. Likewise, the small amount of flux seen on the western edge of the disk intrusion is also real.

The best-fit model and reconstructed images of this second ingress epcoh are shown in Figure A48. This data has excellent UV coverage and appears similar, in many regards, to the 2009-11 epoch. Like previous observations, the straight-edge appearance of the F-star is an artifact of the UV coverage. It is probable that the spots seen in the northern hemisphere of the F-star are also artifacts.

The southern pole again appears quite strong in the real image, model, and synthetic reconstruction, implying this feature is likely real. For the same reason, we regard the small quantity of flux at the west edge of the disk intrusion to be a real feature rather than an artifact of the reconstruction process.

The UV coverage at this epoch is extremely poor, consisting of two four-telescope observation with MIRC. Hence the model-independent images shown in Figure A49 are difficult to interpret without information garnered from the model and H-band photometry. Both the model and images imply that the entire southern hemisphere and a small fraction of the northern hemisphere are covered. This conclusion is supported, at least circumstantially, by the H-band photometry being at its faintest at this time. The appearance of bright spots in the northern hemisphere is most likely caused by limited UV coverage or the reconstruction process, rather than any real surface flux variations on the F-star. The model independent and synthetic images agree quite well about the over-all appearance of the F-star during this epoch.

The qualitative appearance of these epochs is quite similar (see Figure A50), hence they will be discussed in aggregate. The obscuration by the disk remains remarkably consistent across five months of observations. The occasional spot in the F-stars northern hemisphere, scalloped edge of the disk along the F-stars equator, and flux variations along the F-stars equator are frequently seen in the real data and synthetic reconstructions, hence these are likely artifacts. The southern pole has re-appeared. It appears in the real image, model, and synthetic image therefore we regard this as a true feature in the image.

It is important to note that the mid-eclipse observation (2010-08) shows the disk as entirely opaque. Hence mid-eclipse brightening hypotheses, which rely on a large opening in the disk, are unlikely. Likewise, although photometric variations were seen during this phase of the eclipse, no significant flux variations were seen within the disk plane, hence the photometric variations are not likely a result of flux penetrating semi-transparent gaps in the disk's midplane.

In Figure A51 we present the only interferometric images of the Aurigae during the egress phase and the first image created with data taken by the CLIMB beam combiner. Unlike all other data sets, the L0-norm images also required the use of the uniform disk regularizer in SQUEEZE. These model-independent K-band images show that a portion of the F-star's south–east edge is no longer obscured by the disk. This notion is in excellent agreement with the SIMTOI model and observed photometry (e.g., see Figure 4). A bulk comparison of the reconstructed model versus real data implies that the large concentration of flux in the F-star's northern hemisphere is probably an artifact, whereas the small crescent of the F-star seen in the east is real.

All of the post-eclipse images are displayed in Figure A52. Much like the ingress phase images, these reconstructions show that the F-star has no egregious asymmetries that cannot be explained by UV coverage or reconstruction artifacts. We note that two of the post-eclipse epochs (2011-10-10 and 2011-11-03) do show non-zero closure phase and different locations of the first visibility null, indicating that the F-star may harbor spots or be slightly oblate. Both of these effects, if real, are insignificant compared to the variations that the disk imparts upon the interferometric data during the eclipse. We will attempt to quantify the presence of spots or asymmetry in greater detail in a future publication.