---
abstract: 'We investigate the nature of the unusual eclipsing star OGLE LMC-ECL-11893 (OGLE J05172127-6900558) in the Large Magellanic Cloud recently reported by Dong [*et al.*]{}  The eclipse period for this star is 468 days, and the eclipses exhibit a minimum of $\sim$1.4 mag, preceded by a plateau of $\sim$0.8 mag.  Spectra and optical/IR photometry are consistent with the eclipsed star being a lightly reddened B9III star of inferred age $\sim$150 Myr and mass $\sim$4 $M_{\odot}$. The disk appears to have an outer radius of $\sim$0.2 AU with predicted temperatures of $\sim$1100-1400K. We model the eclipses as being due to either a transiting geometrically thin dust disk or gaseous accretion disk around a secondary object; the debris disk produces a better fit.  We speculate on the origin of such a dense circumstellar dust disk structure orbiting a relatively old low-mass companion, and on the similarities of this system to the previously discovered EE Cep.' 
author:
- 'Erin L. Scott, Eric E. Mamajek, Mark J. Pecaut, Alice C. Quillen, Fred Moolekamp, and Cameron P. M. Bell'
title: 'Modeling Transiting Circumstellar Disks: Characterizing the Newly Discovered Eclipsing Disk System OGLE LMC-ECL-11893'
---

Introduction
============

In the discovery paper for the eclipsing circumsecondary disk system orbiting 1SWASP J140747.93-394542.6, @Mamajek12 explored the possibility of finding other such systems, calculating the probability to be nonnegligable given continuous monitoring of a large number of young stars.  Examples of eclipsing disks are notably rare, including famous examples of $\epsilon$ Aur, EE Cep, and OGLE-LMC-ECL-17782 [e.g., @Kloppenborg10; @Galan12; @Graczyk11], and recent surveys have yielded few, if any, candidates [e.g., @Meng14].  Modeling of eclipsing disks can provide constraints on the geometry and structure of circumstellar disks–details which are difficult to constrain from spectral energy distributions alone. 

OGLE 11893 (OGLE J05172127-6900558 = LMC SC8 354550; hereafter “OGLE 11893”)[^1] was first cataloged as a LMC member by @Udalski00, flagged as a variable by @Zebrun01, and classified as an eclipsing variable by @Graczyk11.  @Dong14 first reported OGLE 11893 to be a particularly unusual eclipsing binary with a long period, deep eclipse, and asymmetric shape.  @Dong14 proposed that the eclipses could be due to a transiting circumstellar disk.  The properties of OGLE 11893 are compiled in Table \[tab:star\].  The eclipses are a repeating phenomenon, with a period of 468 days and depth of $\sim$1.4 mag [@Graczyk11; @Dong14].  The phase-folded light curve reveals an eclipse structure that is highly consistent on the timescale over which the star has been observed [$\sim$17 yr, $\sim$13 orbital periods; @Dong14], indicating that the disk is not precessing at a detectable rate.  @Dong14 reported preliminary observations from the H$\alpha$ line profile that the star appears to be a Ae/Be star exhibiting H$\alpha$ emission. 

In this contribution, we further characterize OGLE 11893 and model its eclipses using circumstellar disk models.  In Section 2 we discuss observations for the star.  In Section 3 we analyze the properties of OGLE 11893 and its light curve, and model the eclipses as due to various flavors of disks.  In Section 4 we summarize our findings regarding the eclipsing disk. 

Data \[sec:data\]
=================

Photometry \[sec:phot\]
-----------------------

In order to model the light curve, we used data from the photometric observations of the OGLE-II and OGLE-III surveys, spanning 12 yr in total.  The OGLE-II survey took place from 1996 December to 2000 November, using standard $V$ and $I$ photometric bands and generating 300-500 frames per field in $I$ [@Wyrzykowski09].  OGLE-III took place from 2001 July to 2009 May and produced a total of 866 photometric points for OGLE 11893 in $I$ band and 61 points in $V$ over the course of 8 yr [@Graczyk11].  Further details on the star can be found in @Dong14 and in Table \[tab:phot\]. 

Spectroscopy \[sec:spec\]
-------------------------

We analyze the blue optical spectrum of OGLE-LMC-ECL-11893 (Figure \[fig:bluespectrum\]) presented in @Dong14, taken by J. Prieto on UT 2011 December 23 with the IMACS [Inamori-Magellan Areal Camera & Spectrograph; @Dressler06] instrument on the Baade 6.5 m telescope at Las Campanas Observatory.  The blue spectrum was taken with the long 0”.7 slit and 300 L mm$^{-1}$ grism, and yielded resolution $R$ $\sim$ 1800 between $\lambda\lambda$ 3500-6574 Å. The S/N of the IMACS blue spectrum is $\sim$22 ($\sim$5000Å).  Further details are described in Section 2 of @Dong14. 

![ \[fig:bluespectrum\] Comparison of the IMACS spectrum of OGLE 11893 to that of spectral standards HD 141774 (B9 V) and $\gamma$ Lyr (B9 III).  Here the B9 V standard is HD 141774 [@Garrison79] and the B9 III standard is $\gamma$ Lyr [@Johnson53].](f1.eps)

[lcll]{}\[htb!\] $\alpha$(ICRS) & 05:17:21.27 & h:m:s & @Udalski00\
$\delta$(ICRS) & -69:00:55.8 & d:m:s & @Udalski00\
Spec.  Type & B9III: & ... & This paper\
$U$ & 17.854$\pm$0.060 & mag & @Zaritsky04\
$B$ & 17.954$\pm$0.021 & mag & @Udalski00\
$V$ & 17.753$\pm$0.01 & mag & @Graczyk11\
$I_c$ & 17.556$\pm$0.01 & mag & @Graczyk11\
$J$ & 17.61$\pm$0.04 & mag & @Kato07\
$H$ & 17.39$\pm$0.10 & mag & @Kato07\
$K_s$ & 17.43$\pm$0.23 & mag & @Kato07\
$[$3.6$]$ & 16.932$\pm$0.107 & mag & @Meixner06\
$[$4.5$]$ & 16.955$\pm$0.104 & mag & @Meixner06\
Period & 468.045440 & day & @Graczyk11\
Epoch(min. ,HJD) & 2454011.5436 & day & @Graczyk11\
Dist.  Mod.(LMC) & 18.48$\pm$0.05 & mag & @Walker12\
$R_V$(LMC) & 3.41$\pm$0.06 & ... & @Gordon03\
[$T_{\rm eff}$]{} & 12000$\pm$300 & K & This paper\
$E(B-V)$ & 0.28$\pm$0.02 & mag & This paper\
$A_V$ & 0.91$\pm$0.06 & mag & This paper\
$M_V$ & -1.64$\pm$0.08 & mag & This paper\
$M_{\rm bol}$ & -2.32$\pm$0.10 & mag & This paper\
[log($L$/$L_{\odot}$)]{} & 2.83$\pm$0.04 & dex & This paper\
Radius & 6.03$\pm$0.42 & [$R_{\odot}$]{}& This paper\
Mass & $\sim$4.1 & [$M_{\odot}$]{}& This paper\
Age & $\sim$150 & Myr & This paper \[tab:phot\]

Analysis \[sec:anal\]
=====================

Light Curve \[sec:lightcurve\]
------------------------------

Figure \[fig:model\] shows the phase-folded light curve of OGLE 11893.  The light curve shows a dimming of $\sim$0.8 mag over $\sim$8 days, which deepens to $\sim$1.4 mag before returning to full brightness.  The total duration of the eclipse is $\sim$16 days.  As we will show, this pattern is consistent with a disk that is nearly edge-on to the observer’s line of sight, and which has a cleared inner region. 

![ image](f2.eps)

Spectral Classification \[sec:spt\]
-----------------------------------

As shown in Figure \[fig:bluespectrum\], through comparison of the IMACS spectrum of OGLE 11893 to a grid of Galactic MK standards [@Morgan73] observed at a resolution of 4.3 with the RC spectrograph on the SMARTS 1.5 m telescope, we find that the Balmer lines are consistent with that of a B9 giant star.  The luminosity class, however, is still somewhat uncertain due to the low S/N of the spectrum. 

Reddening \[sec:red\]
---------------------

We estimate the reddening by comparing to examples of unreddened Galactic late B-type giants.  @Garrison94 classified 10 stars as B9III or variant.  Of those stars, the revised [*Hipparcos*]{} parallaxes of @vanLeeuwen07 only place two within 100 pc (and hence negligibly reddened): HD 145389 and HD 129174.  Using the @vanLeeuwen07 parallaxes, and $UBV$ photometry from @Mermilliod97, we estimate their mean properties as [$(B - V)_0$]{} $\simeq$ -0.09$\pm$0.03, [($U$-$B$)$_0$]{} $\simeq$ -0.329$\pm$0.10, [$M_V$]{} $\simeq$ 0.20$\pm$0.08.  Based solely on comparison of OGLE 11893’s spectral type and colors in Table \[tab:phot\] to those of these well-studied nearby B9 III templates, we would expect $E(B-V) \simeq$ 0.29$\pm$0.05, $A_V$ $\simeq$0.94$\pm$0.17 [assuming $E(B-V)-A_V$ relation of @Olson75], and distance modulus $(m-M)_o$ $\simeq$ 16.6.  Hence if it were comparable to Galactic B9 III stars, we would predict a distance of $\sim$21 kpc [well short of the LMC’s distance of $\sim$50 kpc; @Walker12].  The reddening is somewhat larger than predicted for the star’s position in the LMC. 

We can also estimate the intrinsic color of the B-type star using the $Q$-method with the observed $UBV$ photometry.  We assume that the star suffers visual reddening similar to that of the Galactic average value [$E(U-B)$/$E(B-V)$ = 0.72; @Johnson53], which is statistically consistent with values derived for hot stars in the LMC not near the 30 Dor complex [@Walker68; @Fitzpatrick86; @Gordon03].  Using the revised calibration of $Q$ versus spectral type for Be stars from @Halbedel93, and the observed colors (see Table \[tab:phot\]; = -0.10$\pm$0.06, $\simeq$ 0.20$\pm$0.04), we estimate the star to have $Q$ = -0.245 and a photometric spectral type of B8.3e.  Unfortunately [@Halbedel93] does not provide the intrinsic colors for the Be stars.  However, if we adopt the intrinsic color sequence for B stars from @Pecaut13, the dereddened photometry corresponds to [$(B - V)_0$]{} = -0.095, [($U$-$B$)$_0$]{}= -0.314, $E(B-V)$ = 0.296.  According to the updated spectral type-effective temperature relations of @Pecaut13, for a Galactic main sequence star, this would correspond to a $\sim$B8 star with [$T_{\rm eff}$]{} $\simeq$ 12100 K.

The foreground Galactic reddening to the LMC varies from $E(B-V)
\simeq$ 0.04-0.09 [@Bessell91].  Typical reddenings for hot stars in the LMC are $E(B-V)\simeq$ 0.15-0.31 mag [@Gordon03], consistent with what we see for OGLE 11893.  However this is at odds with the extinction inferred from analyses of LMC RR Lyraes and red clump giants [@Pejcha09; @Haschke11], which yield $E(V-I_c$) = 0.153 and 0.070, respectively, in the vicinity of OGLE 11893 [corresponding to $A_V$ $\simeq$ 0.38 and 0.17, adopting [$A_V$]{}/[$E$($V - I_C$)]{} = 2.49; @Stanek96].  So while OGLE 11893 appears to have a typical reddening compared to hot stars studied in the LMC, it appears to be anomalously redder compared to that predicted from reddening maps of particular classes of star (e.g. RR Lyraes). 

Using the SED fitting routine described in @Pecaut13, we fit the $UBVIJHK_s$ photometry of OGLE 11893 through comparison to a grid of colors of Galactic B-type dwarfs of varying [$E(B-V)$]{} (assuming a Galactic reddening curve).  The best fit SED was for a star with [$(B - V)_0$]{} = -0.093, [$E(B-V)$]{} = 0.260 mag, [$A_V$]{} = 0.84 mag, and [$T_{\rm eff}$]{}$\simeq$ 11800K. These values are similar to what was previously derived using the $Q$-method. 

HR Diagram Position
-------------------

Based on the spectrum and photometry, we adopt for this star [$T_{\rm eff}$]{} $\simeq$ 12000$\pm$300 K, with reddening $E(B-V)$ = 0.28$\pm$0.02 mag, extinction $A_V$ = 0.91$\pm$0.06 mag.  Adopting the mean LMC distance modulus from @Walker12 of 18.48$\pm$0.04 mag ($\varpi$ = 0.0201$\pm$0.0004 mas), and a [$T_{\rm eff}$]{}-appropriate bolometric correction from @Code76 of BC$_V$ = -0.68$\pm$0.06 mag, we derive [$M_V$]{} = -1.64$\pm$0.08, $M_{\rm bol}$ = -2.32$\pm$0.10, [log($L$/$L_{\odot}$)]{} = 2.83$\pm$0.04, and radius 6.03$\pm$0.42 [$R_{\odot}$]{}. 

Age and Primary Mass
--------------------

In order to estimate an age and mass based on HR diagram position, we use the PARAM 1.2 program,[^2] which uses the @Bressan12 evolutionary tracks.  Adopting [$T_{\rm eff}$]{} = 12000$\pm$300 K, \[Fe/H\] = -0.3$\pm$0.05 dex [typical for LMC; @Luck98], $V_o$ = 16.84$\pm$0.06 mag, and LMC parallax from @Walker12, and assuming Bayesian priors of a @Chabrier01 lognormal IMF and constant star-formation rate, PARAM 1.2 estimates the following a posteriori stellar parameters: age 154$\pm$8 Myr, mass = 4.09$\pm$0.14 [$M_{\odot}$]{}, [log($g$)]{} = 3.50$\pm$0.02 dex, radius 5.76$\pm$0.14 [$R_{\odot}$]{}.  Using the piecewise linear fit from @Ekstrom12, including rotation, we derive a main sequence lifetime of $\sim$185 Myr, hence OGLE 11893 is closer to the end of its lifetime than the beginning. 

Given a period of 468 days and a primary mass of 4.09 [$M_{\odot}$]{}, if we assume that the mass of the secondary is negligible compared to the primary, the orbital separation of the secondary is approximately 1.7 AU.  Using the eclipse duration and the orbital period, we infer that the radius of the disk is $\sim$0.20 AU.  The disk radius should be smaller than the Hill radius of the secondary, placing a lower limit on the mass of the secondary of $M_2 \simeq 3M_1(r_{\rm disk}/a)^3 >$ 0.014 [$M_{\odot}$]{} $\simeq$ 15 [$M_{\rm Jup}$]{}.  Hence the disked secondary is almost certainly a star or brown dwarf rather than a giant planet.  Given the high luminosity of the primary, we find that the equilibrium disk temperature due to the primary alone is probably $T_{\rm eq}$ $\simeq$1250K, not far below the sublimation temperatures of silicate dust. 

Eclipse Modeling
----------------

The disk structure was analyzed using a code[^3] that uses a model to generate synthetic light curves for varying parameters.  The object-oriented model is made up of a star, a secondary object, and a disk.  The star, which is held at a fixed mass, is modeled as a spherical grid of points, each of which contains a fraction of the star’s total flux, using the limb darkening values from @vanHamme93.  The secondary is modeled as a solid sphere.  Our calculations show that the secondary is between 0.15 and 0.38 [$M_{\odot}$]{} (see Equations (\[eqn:secmass\]) and (\[sec:gas\])).  To calculate the secondary radius and temperature, we adopted an isochrone from @Bressan12, which covers mass ranges from 0.1 to 3.8 [$M_{\odot}$]{}, adopting LMC composition $Y$ = 0.2572, $Z$ = 0.0047 for age 200 Myr. 

These objects are put together into an orbital system, the parameters of which are the period and the inclination with respect to the observer’s line of sight.  To avoid degeneracy in the best-fit solution, we assume a perfectly circular orbit.  The program then uses vector addition to determine whether any point on the star is eclipsed by any part of the secondary or disk as seen by the observer.  The observed magnitude is then calculated and recorded. 

To find a model that best fits the observed data, the program utilizes a simplex routine from @NumericalRecipes to minimize $\chi^2$. Due to a systematic variation in the error, which was consistently larger for dimmer points in the light curve, resulting in overfitting in the ingress and egress regions of the eclipse and underfitting in the central regions, all errors were set equal to the median value of 0.02 mag. 

The free parameters are the orientation of the disk ($\theta_x$ and $\theta_y$), the dimensions of the disk ($R_{\rm in}$ and $R_{\rm out}$), and the inclination of the orbit.  If the model is of a flared gas disk rather than a flat dust disk, the inner radius is automatically set to the dust sublimation radius, but additional parameters are also allowed to vary: the density power law 1 $\leq \alpha \leq 3$ (where $\rho \propto r^{-\alpha}$), the secondary mass 0.15 [$M_{\odot}$]{}$\leq$ $M_2$ $\leq$ 0.38 [$M_{\odot}$]{}whose bounds were determined by calculating the Hill radius (see Equations (\[eqn:hill\]) and (\[eqn:secmass\])), and the disk mass $M_{\rm disk}$, which was varied on a logarithmic scale of $-2 \leq \gamma \leq-11$, where $\gamma =$ log$\left(M_{\rm disk}/M_2\right)$. At the present time we are unable to constrain the mass opacity of the disk matter, so we adopt a realistic opacity from @Miyake93 of $\kappa\,=\,0.1$ m$^2$kg$^{-1}$ which would correspond to the $V$-band mass opacity of a population of compact dust particles with composition representative of interstellar dust, grain sizes between 10$^{-8}$ and 10$^{-1}$m following a size$^{-3.5}$ power-law distribution.  Examination of the multi-band light curve from @Dong14 shows that the transiting dust appears to be consistent with $A_B$/$A_V$ $\simeq$ 1.2, which is close to that expected for a $R_V$$\simeq$5 interstellar dust extinction curve [@Mathis90].  The transiting dust is clearly not gray, and likely is reflecting a power-law distribution of dust grains down to submicron size, not too unlike an interstellar dust grains population. 

### Flat Dust Disk

The eclipse was first fit with the simplest model, an infinitely thin annulus.  The best fit for this flat dust disk was found to be an inner radius of $R_{\rm in}$ = 26.2 $R_\odot$, an outer radius of $R_{\rm out}$ = 45.8 $R_\odot$, a disk tilt of $\theta_x = 2.0^{\circ}$, a disk obliquity of $\theta_y = 7.0^{\circ}$ from edge-on, an orbital inclination of $i = 
89.38^{\circ}$, and a flat optical depth of $\tau_0=1.8$. This simple disk produced a model light curve that was a reasonable visual fit to the data, with $\chi^2 / \nu$ of $1186 / 113 = 10.5$.

A slightly more complex model for the debris disk allowed the optical depth to vary according to a power law, which produced a best fit model with $R_{\rm in}$ = 26.1 $R_\odot$, an outer radius of $R_{\rm out}$ = 44.8 $R_\odot$, a disk tilt of $\theta_x = 1.5^\circ$, a disk obliquity of $\theta_y = 5.9^\circ$ from edge-on, an orbital inclination of $i = 
89.46^\circ$, and an optical depth of $\tau_0 = 7.9$ at the inner edge of the disk which varied according to a power law of $p = 0.6$. This model had $\chi^2 / \nu = 1073.8 / 112 = 9.6$, and was a better overall fit to the data. 

The infinitely thin debris disk model captures the overall light curve structure relatively well, but has some trouble matching the data in the central regions of the eclipse (see Figure \[fig:model\]). 

### Flared Gas and Dust Disk \[sec:gas\]

Since the lack of detailed substructure indicates the possibility that the eclipses are due to a disk with non-negligible scale height, we next attempted to fit the light curve with a flared gas and dust disk model.  Assuming vertical hydrostatic equilibrium, the scale height of such a disk is given by @Shakura73:

$$\label{eqn:SH}
h\left(r\right) = \frac{c_s}{\Omega}$$

where $c_s$ is the sound speed

$$\label{eqn:sound}
c_s = \sqrt{\frac{k_bT\left(r\right)}{\mu m_H}}$$

and, assuming circular Keplerian orbits, $\Omega$ is the rotation rate

$$\label{eqn:rotation}
\Omega = \sqrt{\frac{GM_2}{r^3}},$$

which gives a scale height of

$$\label{eqn:SHfinal}
h\left(r\right) = \sqrt{\frac{k_br^3T\left(r\right)}{\mu m_HGM_2}}.$$

The temperature profile can be simplified for modeling purposes.  Given the size of the disk, the outer disk is almost certainly dominated by heating from the primary star, as we discuss in Section 3.5, putting it at a temperature of $T_{\rm out}$ $\simeq$ 1100K. The inner edge of the disk is likely at the silicate dust sublimation temperature $T_{\rm in}$ $\simeq$ 1400K, as commonly seen among T Tauri disks [@Muzerolle03].  Hence the radial variation in temperature throughout the disk is likely less than $\sim$25%.  Since $h \propto T^{1/2}$, the scale height $h$ likely varies by only order $\sim$15% between the inner and outer radii.  To simplify our calculations, we adopt $T$ = 1250K, which should translate to accuracy of scale heights $\sim$10%–15% over the whole disk. 

For a roughly constant temperature, the density structure of a flared accretion disk can be modeled by a Gaussian curve [@Shakura73]: $$\label{eqn:density}
\rho\left(r,z\right) = \rho\left(r,0\right)e^{z^2/2h(r)^2}$$

[where $1\leq\alpha\leq3$. The midplane density $\rho_0$ is set so as to give the correct disk mass[^4] for a given value of $\alpha$. Assuming that the temperature of the disk is constant, the density equation can be integrated:]{}

$$M_{\mathrm{disk}} = \int_{R_{\mathrm{in}}}^{R_{\mathrm{out}}}\int_{-\infty}^\infty\int_0^{2\pi}\rho_0e^{z^2/2h(r)^2}\left(\frac{r}{R_2}\right)^{-\alpha}r\mathrm{d}r\mathrm{d}\phi\mathrm{d}z$$

$$M_{\mathrm{disk}} = 2\pi\rho_0\int_{R_{\mathrm{in}}}^{R_{\mathrm{out}}}r\left(\frac{r}{R_2}\right)^{-\alpha}\mathrm{d}r\int_{-\infty}^\infty e^{z^2/2h(r)^2}\mathrm{d}z$$

$$M_{\mathrm{disk}} = (2\pi)^{\frac{3}{2}}\rho_0R_2^\alpha\int_{R_{\mathrm{in}}}^{R_{\mathrm{out}}}r^{1-\alpha}h(r)\mathrm{d}r$$

$$M_{\mathrm{disk}} = \frac{2(2\pi)^{3/2}}{7-2\alpha}\rho_0R_2^\alpha\left[R_{\mathrm{out}}^{2-\alpha}h\left(R_{\mathrm{out}}\right) - R_{\mathrm{in}}^{2-\alpha}h\left(R_{\mathrm{in}}\right)\right],$$

which can then be solved for $\rho_0$.

In this model, the secondary mass becomes important, as it determines the structure of the disk.  Though we lack radial velocity measurements and therefore do not have a precise mass value for the secondary, it is possible to constrain its mass by using the period and duration of the eclipse to put limits on the Hill radius, the point at which the gravitational influence of the secondary balances that of the primary.  For a circular orbit,

$$\label{eqn:hill}
r_H = a\left(\frac{\mu}{3}\right)^{\frac{1}{3}},$$

where $a$ is the semi-major axis and $\mu$ is the mass ratio $\mu \equiv M_2/\left(M_1 + M_2\right)$. The outer radius of a gas disk extends out to $\xi r_H$, where estimates of $\xi$ are typically between $\xi \sim 0.3$ [@Ayliffe09] and $0.4$ [@Martin11].  The semi-major axis can be calculated using the orbital period and the mass of the primary, giving $a = 367R_\odot = 1.7$ AU.  The outer radius of the disk can be calculated from the ratio of the eclipse duration to the period

$$\label{eqn:angles}
\frac{T_{\mathrm{ecl}}}{P} = \frac{\Theta}{2\pi},$$

where $\Theta$ is the angular distance traversed by the secondary/disk system during the eclipse. 

As seen in Figure \[fig:eclipse\], $\Theta = 2\varphi + 2\beta$. Since the eclipse-to-period ratio is small, we use the small angle approximation $\varphi \simeq$ sin($\varphi$) $\simeq R_1/a$ and $\beta \simeq$ sin($\beta$) $\simeq \xi r_H/a$, giving $\Theta = 
\left(2\xi r_H + 2R_1\right)/a$. Inserting this and Equation (\[eqn:hill\]) into Equation (\[eqn:angles\]), we obtain the cubic equation

$$\label{eqn:secmass}
\left(\xi^3 - 3\pi^3\frac{T^3}{P^3}\right)M_2 + 3\xi^2k^{\frac{1}{3}}M_2^{\frac{2}{3}} + 3\xi k^{\frac{2}{3}}M_2^{\frac{1}{3}} + k - 3\pi^3\frac{T^3}{P^3}M_1 = 0,$$

where $k = 12\pi^2R_1^3 / GP^2$. Solving for $M_2$, we obtain bounds[^5] on the secondary mass of $M_2 = 0.23 M_\odot$ for $\xi = 0.4$ and $M_2 = 0.61 M_\odot$ for $\xi = 0.3$, assuming a circular orbit.  In addition, the lack of a visible secondary eclipse puts an upper limit of 2.1 $M_\odot$ on the mass of the secondary for this system’s age and metallicity. 

The best-fit result for a gas disk was an edge-on disk ($\theta_y =
0^{\circ}$) with an obliquity of $\theta_x = 0.9^{\circ}$, an orbital inclination of $i = 89.7^{\circ}$, an inner radius of $R_{\rm in}$ = 2.4 $R_\odot$, an outer radius of $R_{\rm out}$ = 35.5 $R_\odot$, a secondary mass of $M_2 = 0.32 M_\odot$, a radial density dropoff of $\alpha = 2.8$, and a disk mass of log$\left(M_{\rm disk}/M_2\right)$ = -7.8.  The best fit for a gas and dust rich model produced $\chi^2 
/ \nu = 1322 / 112 = 11.8$. The best-fit parameters for both models are listed in Table \[tab:fits\]. 

### Hybrid Disk \[sec:hybrid\]

While both models show general agreement with the overall eclipse trend, both have problems fitting the finer details.  Therefore, we attempted to see whether the fit could be improved by implementing a hybrid model that combined the flat dust disk and flared gas disk.  The best fit for the hybrid model had an obliquity of $\theta_x = 
1.1^{\circ}$, a tilt of $\theta_y = 3.0^{\circ}$, an orbital inclination of $i = 89.74^{\circ}$, an inner radius of $R_{\rm in}$ = 16.7 $R_\odot$, an outer radius of $R_{\rm out}$ = 41.8 $R_\odot$, a secondary mass of $M_2 = 1 M_\odot$, a radial density dropoff of $\alpha = 1$, a disk mass of log$\left(M_{\rm disk}/M_2\right) = -9$ dex, a debris disk optical depth of $\tau = 6.5$, and a debris disk density dropoff power law of $p = 1.5$, which gave $\chi^2 / \nu = 1419 / 110 = 12.9$.

Setting the dust disk (which produced the better fit out of the two previous models) as the null model and the hybrid disk as the alternative model, we then used a likelihood ratio test to determine whether the hybrid model produced a statistically significant improvement in the fit.  The likelihood function

$$\label{eqn:likelihood}
L = \prod_{i=1}^n\left(\frac{1}{2\pi\sigma_i^2}\right)^{1/2}\mathrm{exp}\left(-\frac{1}{2}\left(\frac{y_i-y(x_i)}{\sigma_i}\right)^2\right)$$

[@Bevington69] becomes

$$\label{eqn:Lfinal}
L = \left(\frac{1}{\pi\sigma^2}\right)^{n/2}\mathrm{exp}\left(-\frac{\chi^2}{2}\right)$$

when the errors of each data point are assumed homoskedastic.  In order to evaluate whether the hybrid model produced a statistically significant improvement in the fit, we made use of the Bayesian Information Criterion (BIC)

$$\label{eqn:BIC}
\mathrm{BIC} = -2\mathrm{ln}\left(L_{\mathrm{max}}\right) + k\mathrm{ln}\left(N\right) = \chi^2 + k\mathrm{ln}\left(N\right)$$

[@Liddle07; @Schwarz78], where $k$ is the number of free parameters, $N$ is the number of data points, and $L_{\rm max}$ is the maximum likelihood.  The dust and hybrid models were fit with 7 and 9 free parameters, respectively, to 120 data points, hence the dust and hybrid models had 112 and 110 free parameters, respectively.  The best-fit debris disk model produces $\chi^2 = 1073.8$; therefore the hybrid model will have produced a statistically significant improvement only if it has a BIC lower than that produced by the dust disk model, or

$$\chi_{\mathrm{hybrid}}^2 + 9\mathrm{ln}\left(120\right) < 1073.4 + 7\mathrm{ln}\left(120\right),$$

which gives $\chi_{\rm hybrid}^2 / \nu < 1064.2 / 110$ for the hybrid disk model.  Since the best-fit hybrid model produced $\chi^2 
/ \nu = 1073.8 / 110$, we conclude that the hybrid model does not produce a statistically significant improvement in the fit. 

Discussion
==========

OGLE 11893 appears to be near the end of its main sequence life, and it is unlikely to have a companion that has retained its primordial protoplanetary disk.  This leaves the possibility that the companion is surrounded by either a dusty debris disk or a second-generation gas-rich disk.  The disk-eclipse system OGLE 11893 bears a strong resemblance to EE Cep, the only other known system discovered to date with a disk eclipsing a Be primary.  @Galan12 speculate that the circumsecondary disk of EE Cep is fed by mass loss from the primary Be star, and given the similarity of the two systems, this is a possibility for OGLE 11893 as well. 

The simplest flat debris disk model produced the best fit to the data (see Figure \[fig:model\]), with a $\chi^2 / \nu$ value of $1073.8 / 
112 = 9.6$. Even the best-fit parameters reproduced the light curve poorly in the central region of the eclipse.  Currently the primary is modeled as a perfect sphere even though it has the spectral signatures of a rapidly rotating Be star [@Dong14].  Modifying the model star to account for the elongation and gravitational darkening at the equator due to its rapid rotation would be an important step in improving the model. 

OGLE 11893 and other eclipsing systems like it can be used to learn more about the composition and geometry of circumstellar disks through continued modeling efforts and photometric and spectroscopic observations.  The discovery of this system in addition to J1407 and EE Cep provides evidence that eclipsing circumsecondary disk systems may not be as rare as previously thought, and that both archival searches and long-term photometric monitoring of large areas of the sky are likely to uncover more of them. 

[lcccl]{}\[htb!\] $\theta_x$ & 1.5 & 0.9 & 1.1 & deg\
$\theta_y$ & 5.9 & 0.0 & 3.0 & deg\
$i$ & 89.46 & 89.70 & 89.74 & deg\
$R_{\rm in}$ & 26.1 & 2.8 & 16.7 & [$R_{\odot}$]{}\
$R_{\rm out}$ & 44.8 & 35.5 & 41.8 & [$R_{\odot}$]{}\
$\alpha$ & – & 2.8 & 1.0 & ...\
$M_2$ & 0.12 & 0.32 & 1.00 & [$M_{\odot}$]{}\
log($M_{\rm disk}/M_2$) & -6.9 & -7.8 & -9.0 & dex\
$\kappa$ & 0.1 & 0.1 & 0.1 & m$^2$kg$^{-1}$\
$\tau_0$ & 7.9 & – & 6.5 & ...\
$p$ & 0.6 & – & 1.5 & ...\
$\chi^2 / \nu$ & 1073.8 / 112 & 1322 / 112 & 1419 / 110 & ...\
\[tab:fits\]

E.L.S., E.E.M., and F.M. acknowledge support from NSF award AST-1313029.  E.E.M. and M.J.P. acknowledge support from NSF AST-1008908.  E.E.M., E.L.S., F.M., M.J.P., and C.P.M.B. acknowledge support from the University of Rochester College of Arts and Sciences.  This research has made use of NASA ADS, Vizier, and SIMBAD, and data from the OGLE survey.  This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.  The authors thank Subo Dong, Jose Prieto, Szymon Kozlowski, and Matt Kenworthy for discussions, and we thank the referee for a thoughtful and timely report. 

[22]{} natexlab\#1[\#1]{}

Ayliffe, B. A. & Bate, M. R. 2009, MNRAS, 397, 657

Bessell, M. S. 1991, , 242, L17

Bevington, P. R. & Robinson, D. K. 1968, Data Reduction and Error Analysis for the Physical Sciences (2nd ed.;  New York, NY: McGraw-Hill, Inc.), 144

Bressan, A., Marigo, P., Girardi, L., et al.  2012, , 427, 127

Chabrier, G. 2001, , 554, 1274

Code, A. D., Bless, R. C., Davis, J., & Brown, R. H. 1976, , 203, 417

Dong, S., Katz, B., Prieto, J. L., et al.,  2014, , 788, 41

Dressler, A., Hare, T., Bigelow, B. C., & Osip, D. J. 2006, , 6269

Ekström, S., Georgy, C., Eggenberger, P., et al.  2012, , 537, 18

Fitzpatrick, E. L. 1986, , 92, 1068

Ga[ł]{}an, C., Miko[ł]{}ajewski, M., Tomov, T., et al.  2012, , 544, A53

Garrison, R. F., & Gray, R. O. 1994, , 107, 1556

Garrison, R. F., & Schild, R. E. 1979, , 84, 1020

Gordon, K. D., Clayton, G. C., Misselt, K. A., et al.  2003, , 594, 279

Graczyk, D., Soszynski, I., Poleski, R., et al.  2011, [Acta Astronomica]{}, 61, 103

Halbedel, E. M. 1993, , 105, 465

Haschke, R., Grebel, E. K., & Duffau, S. 2011, , 141, 158

Johnson, H. L., & Morgan, W. W. 1953, , 117, 313

Kato, D., Nagashima, C., Nagayama, T., et al.  2007, , 59, 615

Kloppenborg, B., Stencel, R., Monnier, J. D., et al.  2010, , 464, 870

Liddle, A. R. 2007, , 377, L74

Luck, R. E., Moffett, T. J., Barnes, T. G. III, & Gieren, W. P. 1998, , 115, 605

Mamajek, E. E., Quillen, A. C., Pecaut, M. J., et al.  2012, , 143, 72

Martin, R. G. & Lubow, S. H. 2011, MNRAS, 413, 1447

Mathis, J. S. 1990, , 28, 37

Meixner, M., Gordon, K. D., Indebetouw, R., et al.  2006, , 132, 2268

Meng, Z., Quillen, A. C., Bell, C., & Scott, E. L. 2014, , 441, 4

Mermilliod, J. C. 1997, yCat, 2168, 0

Miyake, K. & Nakagawa, Y.  1993, Icar 106, 20

Morgan, W. W., & Keenan, P. C. 1973, , 11, 29

Muzerolle, J., Calvet, N., Hartmann, L., & D’Alessio, P. 2003, , 597, L149

Olson, B. I. 1975, , 87, 349

Pecaut, M., & Mamajek, E. E. 2013, , 208, 9

Pejcha, O., & Stanek, K. Z. 2009, , 704, 1730

Press, W. H., Teulolsky, S. A., Vetterling, W. T., & Flannery, B. P.  1992, Numerical Recipes in C: The Art of Scientific Computing (New York, NY: Cambridge Univ.  Press)

Schwarz, G. 1978, Ann.  Statist.,  6, 2

Shakura, N. I. & Sunyaev, R. A. 1973, , 24, 337

Stanek, K. Z. 1996, , 460, L37

Udalski, A., Szymanski, M., Kubiak, M., et al.  2000, [Acta Astronomica]{}, 50, 307

van Hamme, W. 1993, , 106, 5

van Leeuwen, F. 2007, Astrophysics and Space Science Library, 350

Walker, A. R. 2012, , 341, 43

Walker, G. A. H., & Morris, S. C. 1968, , 73, 772

Wyrzykowski, Ł., Kozłowski, S., Skowron, J., et al.  2009, , 397, 3

Zaritsky, D., Harris, J., Thompson, I. B., et al.  2004, , 128, 1606

Zebrun, K., Soszynski, I., Wozniak, P. R., et al.  2001, [Acta Astronomica]{}, 51, 317

[^1]: The Optical Gravitational Lensing experiment, http://ogle.astrouw.edu.pl/ [@Udalski00]. 

[^2]: http://stev.oapd.inaf.it/cgi-bin/param

[^3]: Written by Fred Moolekamp, maintained and updated by Erin Scott (UR). 

[^4]: The density calculation only accounts for the small dust grains that are the primary source of opacity.  Gas and larger grains that contribute most of the mass, but are not a significant source of opacity, are not included in the calculation, resulting in a model disk mass that is orders of magnitude lower than the actual mass. 

[^5]: The upper mass limit is not a hard limit, as a disk can potentially be smaller relative to the Hill radius of the secondary.