Occultations from an Active Accretion Disk in a 72-day Detached Post-Algol System Detected by K2

Approximately 70% of intermediate-mass stars reside in multistellar systems (e.g., Duchêne & Kraus 2013; Moe & Di Stefano 2017), and perhaps only half of them can go through life without experiencing the influence of their partners. Binary stars are important astrophysical laboratories, where the primordial stars share a common age, and these are excellent test beds of binary stellar evolution, mass transfer, and accretion processes. The final states of many binaries are shaped by mass-transfer events during their lifetimes.

Algols are binary stars that experience mass transfer when the primary star in the primordial binary evolves. In these binaries (Batten 1989a, 1989b; Peters 2001), the more massive companion evolves off the main sequence first and expands to fill its Roche lobe. Overflowing matter from the more massive donor star accretes onto the mass-gaining star (hereafter "accretor"), leading to an inverted mass ratio for the resultant system. The actual pathway for each Algol depends on the initial mass ratio of the binary, the initial orbital period, how conservative the mass transfer is, and how much specific angular momentum is carried away with the lost mater (e.g., Nelson & Eggleton 2001; Eggleton & Kiseleva-Eggleton 2002; De Loore & van Rensbergen 2005). Some Algols are stable mass-transferring systems and leave remnant thermally bloated white dwarfs (e.g., Rappaport et al. 2015) or subdwarf B and O stars (e.g., Han et al. 2002). However, shortly after the mass-transfer phase ends, the tenuous residual atmosphere of the mass-losing giant can remain bloated for up to 1 Myr until the remaining hydrogen is consumed, while the envelope shrinks and gets hotter, to the point where the underlying white dwarf or subdwarf is revealed. This might be called a "transitional phase."

It may be possible for the accretion disks to persist in systems that have ended active mass transfer. Occultations induced by persistent disks have been detected in a handful of long-period binaries. Photometric occultations by disks are often distinct from other events, as they exhibit long-duration, deep eclipse-like signals that can be identified in photometric surveys. Aurigae is the classical example of such a system, with a post-AGB F0 supergiant with a B-star companion embedded in a dusty disk with an orbital period of 27 yr (e.g., Kuiper et al. 1937; Huang 1965; Kopal 1971). The occultation is observed in photometry (e.g., Gyldenkerne 1970; Kemp et al. 1986; Carroll et al. 1991), interferometry (Kloppenborg et al. 2010, 2015), and spectroscopy (e.g., Lambert & Sawyer 1986; Chadima et al. 2011; Leadbeater et al. 2012; Griffin & Stencel 2013; Muthumariappan et al. 2014; Strassmeier et al. 2014), and the system's orbit and masses are constrained by the radial velocities (Stefanik et al. 2010). The 2 yr long eclipse is inferred to be from a ∼4 au diameter disk consisting of both dust and gas (e.g., Lissauer et al. 1996). The dusty component is required to explain the strong IR excess in the system (Hoard et al. 2010), while the gaseous component in the vertically extended disk shell induces a series of spectroscopic features that map out the Keplerian disk during occultation (e.g., Chadima et al. 2011; Strassmeier et al. 2014). A central brightening is seen during eclipse, leading to a flared disk geometry interpretation (e.g., Budaj 2011). Similar long-period, long-duration disk occultations in other potential mass-transfer systems have also been identified: the 96-day-period V383 Sco (Zola et al. 1994), the 5.6 yr period EE Cep (Mikolajewski & Graczyk 1999), the 468-day periodiocally recurring eclipses of OGLE-LMC-ECL-11893 (Dong et al. 2014; Scott et al. 2014), and the 1277-day periodic eclipses of OGLE-BLG182.1.162852 (Rattenbury et al. 2015). The recent advent of wide-field photometric surveys is now also enabling numerous new discoveries, including the 69 yr eclipsing disk system TYC 2505-672-1 (Rodriguez et al. 2016).

MWC 882 (V_mag = 10.8) was originally identified in the Mount Wilson Catalog (MWC) of A and B stars owing to its Balmer line emissions (Merrill & Burwell 1949). Here we report that MWC 882 is a 72-day-period binary involving a B7 post-red giant branch (RGB) star and an A0 accretor, exhibiting disk occultations on each orbit. Occultations of the donor by the accretion disk around the accretor were detected by the K2 mission (during Campaign 11) and subsequently identified in pre-discovery observations by ground-based photometric surveys. The basic geometry of the system is illustrated in Figure 1. As we detail below, the disk signatures of MWC 882 are eerily reminiscent of Aurigae. The eclipse light curve exhibits a central brightening that we also argue to be from a flared disk geometry. Spectroscopic signatures of the disk were detected, via a similar set of absorption lines to that from the recent Aurigae eclipse. Unlike Aurigae, both members of the system are directly detected, and their dynamical masses can be measured. Our interpretation of MWC 882, as a post-Algol with a donor star that relatively recently ended its mass-transfer phase, will aid the understanding of similar long-period disk occultation systems.

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The photometric and astrometric properties of MWC 882 are listed in Table 1. A total of 38 photometric occultations were recorded in multiple data sets, including the K2 discovery light curves, pre-discovery data sets from ground-based wide-field photometric surveys, and our multiwavelength follow-up observations. They are summarized below, listed in Table 2 for clarity, and plotted in Figure 2 in full.

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2.1. Identification from K2 Photometry

2.2. Pre-discovery with All-Sky Automated Survey (ASAS) and All Sky Automated Survey for Supernovae (ASAS-SN)

2.3. Ground-based Photometric Follow-up

We obtained spectroscopic observations of MWC 882 with a series of facilities over 2017 August and September to measure the masses of the system, estimate spectral classifications, and monitor for temporal changes in the spectroscopic features. These observations are summarized in Table 4.

A total of 11 observations were obtained using the Tillinghast Reflector Echelle Spectrograph (TRES) on the 1.5 m telescope at Fred Lawrence Whipple Observatory, Mt. Hopkins, Arizona, USA. TRES is a fiber-fed spectrograph with a spectral resolution of $R\equiv \lambda /{\rm{\Delta }}\lambda =44000$ over the wavelength range 3900–9100 Å via 51 echelle orders. Each observation is made of three sequential exposures, combined to minimize the impact of cosmic rays and reduced as per Buchhave et al. (2010). Our observations spanned the period of 2017 July 31–2017 September 25, over orbital phases between 0.25 and 0.75. Of the 11 observations, 5 were obtained during the occultation at phase 0.5.

We observed MWC 882 six times between 2017 August 28 and 2017 September 15 with the 2.7 m Harlan J. Smith Telescope and its Robert G. Tull Coudé Spectrograph (Tull et al. 1995) at McDonald Observatory, Mt. Locke, Texas, USA. We used the TS23 spectrograph configuration, providing R = 60,000 over 58 echelle orders. The wavelength coverage is 3570–10200 Å and is complete below 5691 Å with increasingly large interorder gaps redward of this point; notably, Hα falls into one of these gaps and is not captured. We obtained one spectrum per epoch. The data were reduced and the spectrum extracted and wavelength-calibrated using standard IRAF tasks.

We also obtained an observation of MWC 882 using the 2.4 m Automated Planet Finder (APF) and the Levy spectrograph, located at Lick observatory, Mt Hamilton, California, USA. The APF is coupled with a high-resolution, slit-fed spectrograph that works at a typical resolution of R ≈ 110,000. The telescope is operated by a dynamic scheduler and has a peak overall system throughput of 15%, and its data reduction pipeline extracts spectra from 3750 to 7600 Å (Vogt et al. 2014; Burt et al. 2015). The APF spectrum used in this work was extracted from a single, 20-minute exposure taken with the 1'' × 3'' slit on 2017 July 26.

3.1. Radial Velocities of the Binary

The spectra of MWC 882 show obvious signs of blending by the two stellar components in the system. Luckily, only the donor star is hot enough to exhibit strong He i absorption lines. As such, we make use of the unblended He i lines at 5876 and 6678 Å to measure the radial velocity of the donor star (Figure 3). To measure the velocity of the accretor, we perform a least-squares deconvolution of the spectrum to derive the broadening kernel of the spectral lines. The broadening kernel contains the radial velocity information of the system, since the observed spectrum is the convolution of the kernel of the two stars and a synthetic nonrotating single-star spectral template. Following the technique laid out in Donati et al. (1997), a deconvolution is performed between the observed spectrum and a single-star nonrotating synthetic spectral template. The template is generated with the SPECTRUM code²² (Gray & Corbally 1994) with the ATLAS9 model atmospheres (Castelli & Kurucz 2004), over the wavelength range of 4000–6100 Å, and has the atmospheric properties of a T_eff = 12,000 K, $\mathrm{log}g=4.5$ main-sequence star. We find that broadening profiles from least-squares deconvolutions often resolve blended lines better than classical cross-correlation functions, due to the sharp-edge nature of rotational broadening kernels.

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Temporal variations of the broadening profile, as a function of the orbital phase, are shown in Figure 4. These temporal variations capture the radial velocity orbit of the system and have the potential of revealing line width and asymmetry variations if they are present. We fit the broadening profiles with a two-star model, and each kernel of each star is modeled with the convolution of a rotational kernel and a macroturbulence kernel, along with the parameters for velocity centroid and height ratio. The velocity of the broadening kernel of the donor is fixed to that measured from the He i line centroids, while their broadening values are fitted for simultaneously with the fitting of the donor's profile. The fitting is performed via a Markov chain Monte Carlo (MCMC) exercise, using the emcee affine invariant ensemble sampler (Foreman-Mackey et al. 2013). The derived radial velocities are presented in Table 5 and plotted in Figure 5.

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Table 5.  Relative Radial Velocities

BJD	RV1a	σ RV1	RV2b	σ RV2c	Inst.
(TDB)	(km s−1)	(km s−1)	(km s−1)	(km s−1)
2,457,960.7368	−24.62	0.21	−61.67	0.50	APF
2,457,965.6830d	⋯	⋯	−26.84	1.79	TRES
2,457,990.6635d	⋯	⋯	53.73	0.55	TRES
2,457,993.6303	−31.83	0.15	38.44	0.50	McDonald
2,457,994.6378	−34.05	0.31	34.32	0.50	McDonald
2,457,994.6673	−33.38	0.23	34.94	0.50	TRES
2,458,002.6414	−21.70	0.33	−21.58	2.00	TRES
2,458,006.6275e	⋯	⋯	−47.81	1.31	TRES
2,458,007.6279e	⋯	⋯	−56.62	1.02	McDonald
2,458,008.6041e	⋯	⋯	−63.44	0.50	McDonald
2,458,008.6290e	⋯	⋯	−62.36	0.50	TRES
2,458,009.5981e	⋯	⋯	−71.15	0.50	McDonald
2,458,009.6244e	⋯	⋯	−69.11	0.50	TRES
2,458,011.6228	−14.25	0.16	−79.53	0.50	McDonald
2,458,019.6026	−7.77	0.16	−106.26	0.50	TRES
2,458,020.6046	−8.17	0.30	−107.93	0.96	TRES
2,458,021.6009	−8.74	0.10	−106.84	0.50	TRES

Notes.

^aVelocity of the accretor measured from the line-broadening kernels. ^bVelocity of the donor measured from the He i lines, which are present only in the donor spectrum. ^cA minimum uncertainty of 0.50 $\mathrm{km}\,{{\rm{s}}}^{-1}$ is adopted to account for systematic uncertainties. ^dOnly velocities of the donor were measured; could not disentangle the velocities of the accretor and donor from the line-broadening kernels. ^eOnly velocities of the donor were measured; the photospheric lines of the accretor were contaminated by the disk absorption lines.

A machine-readable version of the table is available.

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Keplerian fits to the radial velocities yield the dynamical masses of the system. The Keplerian orbit parameters and associated uncertainties are determined via an MCMC analysis. To yield realistic uncertainty estimates and to account for underestimation of per-point velocity measurement errors, the per-point velocity uncertainties were inflated such that the reduced χ² for the best-fit solution after the MCMC burn-in chain is at unity. We did not attempt to correct the velocities from different facilities for systematic offsets. Instead, we chose to increase the formal uncertainties to account for this effect. The per-point uncertainties were assigned to be at least 0.5 $\mathrm{km}\,{{\rm{s}}}^{-1}$ to account for these possible systematic uncertainties. We find, for assumed circular orbits, that the radial velocity amplitudes are ${K}_{1}={14.3}_{-1.3}^{+1.4}$ $\mathrm{km}\,{{\rm{s}}}^{-1}$ and ${K}_{2}={85.5}_{-2.2}^{+1.9}$ $\mathrm{km}\,{{\rm{s}}}^{-1}$, resulting in minimum mass measurements of ${M}_{\mathrm{acc}}=3.24\pm 0.29\,{M}_{\odot }$ and ${M}_{\mathrm{don}}=0.542\,\pm 0.053\,{M}_{\odot }$. The residuals from the best-fit model have scatters of 3.0 and 1.4 $\mathrm{km}\,{{\rm{s}}}^{-1}$ for the accretor and donor velocities, respectively. To test the effect of combining velocities from multiple instruments in our analysis, we recomputed the masses from the TRES velocities alone, finding M_acc = 3.14 ± 0.14 M_⊙ and M_don = 0.54 ± 0.02 M_⊙, consistent with the combined analysis. We also performed an independent radial velocity analysis of the TRES spectra via the two-dimensional cross-correlation technique TODCOR (Zucker & Mazeh 1994), finding M_acc = 3.01 ± 0.17 M_⊙ and M_don = 0.508 ± 0.017 M_⊙, consistent with the masses quoted above to within 1σ.

We checked that the orbit is indeed consistent with being nearly circular, with eccentricity formally constrained to be e = 0.021 ± 0.010. While we cannot rule out a small but nonzero eccentricity for the system, significantly more radial velocities are required to avoid biases at near-zero eccentricities (e.g., Lucy & Sweeney 1971; Bassett 1978). We note, however, that nonzero eccentricities have been detected for other Algols (e.g., TT Hydrae Miller et al. 2007), so such follow-up observations are worthwhile.

3.2. Spectral Classification

Stellar classification was particularly difficult given the complexity of the system: the spectra are blended, both stars exhibit signatures of chemical peculiarity (see Section 3.3), active accretion prevents the use of Balmer line strengths for temperature estimates, and both stars are severely reddened by interstellar and circumstellar material. As such, no single set of synthetic template spectra and fluxes fit the observed spectra well.

Since the stars exhibit chemical peculiarity and accretion signatures, we did not perform a global spectral matching of observed and synthetic spectra. Instead, we make use of temperature-sensitive spectral features. For example, strong and broad near-UV/optical He i absorption lines are prominent features in B-star spectra but decrease in strength toward cooler temperatures. As a result, He i lines can be useful temperature indicators. Figure 3 shows the radial velocity orbital phase variations of the donor star by its He i absorption feature. Note that we see He i only at the spectrum of the donor star, not the accretor, suggesting that the donor star is hotter than ≈13,000 K. The Ca ii H and K lines grow weaker and narrower for earlier-type A and B stars, while C ii lines grow in strength for B stars. Figure 6 shows the observed spectra of MWC 882 for the Ca ii K and C ii lines, compared to synthetic spectra from ATLAS12 models (Kurucz 2005) taken from the POLLUX spectral database (Palacios et al. 2010). Synthetic spectra with metallicity of [Fe/H] = +0.5 were chosen, since the stellar surface of both stars appears metal enriched and chemically peculiar (Section 3.3). We also adopt models with a surface gravity of $\mathrm{log}g=4.5$ for the accretor, corresponding to its expected properties after mass transfer (see Section 5), and a gravity of $\mathrm{log}g=3.5$ for the donor, the lowest-gravity template available in the ATLAS12 grid. We take the "observed spectrum" to be the average of spectra taken over 2017 September 23–25, at orbital phase 0.75–0.77, where the radial velocity separation between the accretor and donor was at its greatest. We find the accretor star to be consistent with an A0 spectral type with an effective temperature of 12,000 ± 1000 K, while the donor is likely a B7 star, with an effective temperature of 15,000 ± 1000 K. The similar temperatures between the accretor and donor star are consistent with the gray eclipse observed during our follow-up observations (Section 2.3).

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Broadening of the spectral lines informs us of the projected rotational velocity of the stars. We fit for rotation profiles (defined in Gray 2005) to the least-squares deconvolution kernels taken at phases 0.25 and 0.75, where the velocities of the two stars are well separated. We find that both stars exhibit similar line broadenings, with the accretor rotating at v sin i = 25.6 ± 4.5 $\mathrm{km}\,{{\rm{s}}}^{-1}$, and the donor star with v sin i = 31 ± 11 $\mathrm{km}\,{{\rm{s}}}^{-1}$. We note, however, that our quoted rotational broadening velocities do not account for the effects of macroturbulence. The broadening profiles of both stars in the MWC 882 system deviate significantly from the standard rotational profiles, and the fits that incorporate both $v\sin {I}_{\star }$ and macroturbulence are highly degenerate. A similar issue was encountered when analyzing the spectra of Aurigae, with multiple $v\sin {I}_{\star }$ values quoted, ranging from $5\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (Chadima et al. 2011) to 41 $\mathrm{km}\,{{\rm{s}}}^{-1}$ (Stefanik et al. 2010). Regardless of the assumptions on nonrotational line broadenings, the stars in both MWC 882 and Aurigae are rotating significantly slower than the average A and B stars (e.g., Royer et al. 2007), and certainly not near breakup velocity as expected for actively accreting systems. Dervisoǧlu et al. (2010) suggest a strong stellar wind as a mechanism for angular momentum dumping. They expect that strong differential rotation, induced by accretion, can generate strong magnetic fields that enhance the stellar wind and mass loss, spinning down the stars.

The relative line strengths of the accretor and donor spectra also inform us of the flux ratio of the system. To measure the flux ratio, we fit for the contributions of the donor and accretor stars over the entire TRES spectrum, order by order, using the ATLAS12 models. Only a few orders have high signal-to-noise ratio and enough lines from both stars to constrain the flux ratio. We find that the spectral region over 48 orders between 4000 and 6000 Å satisfies this requirement well, yielding F_acc/(F_don+F_acc) = 0.52 ± 0.18, with the uncertainty being the scatter in the flux ratios measured. An illustrative section of the spectral fitting is shown in Figure 7.

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Reddening, UV, and IR excess are constrained by the spectral energy distribution (SED; Figure 8). We model the SED with 12,000 and 15,000 K synthetic templates from Castelli & Kurucz (2004) and fit for a blackbody component to constrain any IR excess and an extinction coefficient to account for interstellar and circumstellar reddening. Since both stellar components have very similar temperatures, the flux ratio is impossible to constrain using the SED only, so we fix the flux ratio to 0.52 as determined from spectral line fitting. We recover significant reddening of E(B – V) = 1.07, and it is better fit with an extinction law of R(V) = A(V)/E(B–V) = 2.5. Minimal IR excess is detected in the SED and can be rectified by a blackbody of 1200 K. The IR excess may be due to contaminating M dwarfs along the line of sight, or low levels of a dusty envelope around either of the stars.

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3.3. Chemical Peculiarity

Some A and B stars exhibit chemical and magnetic peculiarity in their atmospheres. The spectra of both stars in the MWC 882 system are consistent with classification as chemically peculiar stars. In particular, Figure 9 shows that both the donor and accretor star exhibit signatures of strong Si enhancement. Enhancement in the Cr lines is also seen in the accretor star, but not the donor star. We did not see enhancement in Sr, which is common among stars that are enriched in Si and Cr. We also searched for enhancements in Y, Hg, and Mn (Figure 10), seen in the HgMn class of Bp/Ap stars, but did not detect their presence. The slow rotational ($\lesssim 30$ km s⁻¹) velocities of both stars in the MWC 882 system are consistent with the presence of strong magnetic fields, and both stars reside within the roughly B6–F4 spectral range of known Bp/Ap stars (e.g., Wolff 1981; Smith 1996). Therefore, we have classified the donor and accretor stars in MWC 882 as BpSi and ApSiCr stars, respectively.

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3.4. Spectroscopic Signatures of Accretion

Evidence for active accretion is apparent in the spectra of MWC 882. Hα is seen strongly in emission during all observations, with an equivalent width of 12 ± 2 Å measured from the out-of-occultation spectra. The feature is double peaked and is representative of those seen in longer-period (P > 6 days) Algols (e.g., Richards & Albright 1999; Budaj et al. 2005). The wings of the Hα feature extend to ≈400 km s⁻¹, corresponding to an inferred inner disk radius of ≈3.7 R_⊙.²³ In comparison, the isochrone-fitted radius of the 3.24M_⊙ accreting star is $3.09\pm 0.59\,{R}_{\odot }$ (via fitting of Geneva isochrones; Ekström et al. 2012; as described in Section 4), and, as such, the inner edge of the accretion disk extends close to the stellar surface. The peak emissions of both Hα and Hβ are located at ≈150 km s⁻¹, corresponding to a disk radius of ≈28 R_⊙. The Ca ii triplet (CaT) lines exhibit inverted P- Cygni profiles, again indicative of active accretion. The line profiles and phase variations of the accretion features are shown in Figure 11. We see no He features in the accreting star, nor any signatures of wind and outflows in the He i or O i lines.

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The mass-transfer rate onto the accreting star can be estimated via the Hα luminosity. By comparing the Hα flux against the bolometric luminosity from our SED fit, we measure L_Hα/L_bol,acc = 0.0023, implying ${L}_{{\rm{H}}\alpha }={0.32}_{-0.07}^{+0.15}\,{L}_{\odot }$. Assuming complete conversion of potential energy, the mass-transfer rate estimated from the Hα flux is (1.3 ± 0.3) × 10⁻⁸ M_⊙ yr⁻¹. For comparison, typical accretion rates measured for Algols in orbital periods of 4–13 days range from 10⁻⁹ to 10⁻⁵ M_⊙ yr⁻¹ (Van Rensbergen & De Greve 2016, and references therein). We note that an accurate estimate of the accretion rate should model the Hα profile to extract the accretion luminosity and account for flux from other accretion signatures. Future observations in the UV and more sophisticated modeling of the line profiles will yield more accurate accretion rates for this system.

3.5. Gas Disk Absorption during Occultation

Occultations like those exhibited by MWC 882 present opportunities to study the gas material within the accretion disk. During occultation, light from the donor star passes through the accretion disk, and spectroscopic absorption features from the optically thin parts of the gaseous disk are imprinted on the observed spectrum. These absorption lines trace the velocities of disk gas along the line of sight between the donor star and the observer.

Such occultation disk spectroscopy was obtained during eclipses of Aurigae in 1982–1984 (Lambert & Sawyer 1986) and the more recent 2009–2011 event (Leadbeater et al. 2012; Griffin & Stencel 2013; Muthumariappan et al. 2014; Strassmeier et al. 2014). The thin dusty and gaseous disk is seen to transit in front of an F0 supergiant, which we view almost edge-on (near 90° inclination). Disk lines from the optically thin disk "shell" around Aurigae were observed at varying widths, depths, and velocities through the 2 yr eclipse. In particular, lines of low-excitation potential species were strongly detected, such as Na i, K i, Mg i, Hα, and Hβ. These lines traced the Keplerian velocity of the disk and helped to constrain the morphology, homogeneity, and potential eccentricity of the disk.

Our spectra obtained during the occultation of MWC 882 revealed a series of broad absorption features that are offset in velocity from both the donor and accretor. These features are obvious in the Balmer lines (Figure 11), as well as many neutral and ionized metal lines, such as the Ca ii triplet (Figure 11), Si ii (Figure 12), Fe i, ii, Na i D, K i, and Mg i. The disk features are also present in the ensemble line profiles derived from the least-squares deconvolution analysis, shown in Figure 4. The velocities derived from selected unblended lines are shown in Figure 13.

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Assuming that the greatest contribution in flux to the disk lines comes from the innermost regions of the disk that actually occult the donor star, we can use these disk line velocities to constrain the occultation geometry and the system architecture (Section 4).

Algols undergoing active accretion are rarely detected at periods as long as 72 days (e.g., Richards & Albright 1999). Unraveling the true system architecture is crucial to understanding its evolutionary state.

From the light curves and radial velocities, we know that the disk around the accreting star is occulting the donor star, the two orbiting each other with a period of 72 days. The dynamical masses indicate that the accretor is now $3.24\pm 0.29$ M_⊙ and the donor is $0.542\pm 0.053\,{M}_{\odot }$, separated by a distance of 114.0 ± 3.1 R_⊙. Examinations of the spectroscopic lines find that the donor is slightly hotter than the accretor, at 15,000 and 12,000 K respectively, both contributing approximately equally to the flux of the system. Properties of the accretor can be further estimated by fitting its mass and effective temperature to standard stellar isochrones. Using the Geneva isochrones for high-mass stars (Ekström et al. 2012), at solar metallicity, and no rotation, the radius and luminosity of the accretor are estimated to be $3.09\pm 0.59\,{R}_{\odot }$ and 142₋₃₃⁺⁶⁷ L_⊙. Assuming Stefan–Boltzmann's law and a flux ratio of F_don/(F_don+F_acc) = 0.52 ± 0.18, we calculate the radius of the donor to be 1.79 ± 0.72 R_⊙. We note that the Geneva isochrones are single-star evolution tracks and begin at zero-age main sequence. The accretor star, however, is a lower-mass star that is "rejuvenated" by the mass-transfer episode, with different interior structures. Section 5 discusses binary evolution tracks that account for the co-evolution of the two stars. However, the Geneva grid presents a reasonable estimate of the properties of the accretor star, and an interpolation of the grid allows us to estimate uncertainties on the properties of both stars, which is more difficult with the scenario-specific binary evolution tracks.

Adopting these masses and radii, neither star is close to filling its Roche lobe (Figure 14). The estimated sizes of the Roche lobes are 61.1 ± 2.5 R_⊙ for the accretor and 27.3 ± 0.9 R_⊙ for the donor (following the approximation in Eggleton 1983). In fact, a stable disk can only be as large as 80% of the Roche lobe (Paczynski 1977), with a maximum radius of 49.4 ± 3.2 R_⊙.

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We create a toy disk model to simultaneously fit the eclipse light curve and spectroscopic disk velocities. In this model, the eclipse duration constrains the diameter of the disk. The ingress and egress timescales constrain the radius of the donor star, if we assume that the disk has a sharp edge and uniform optical depth. The central brightening is modeled by including a flaring to the disk, such that a smaller area of the donor star is covered during mid-eclipse. The velocities of the disk absorption lines seen during occultation are also fitted for simultaneously. At each phase, we take the disk line velocity to be described by the Keplerian orbital velocity of the innermost disk annulus that is occulting the limb of the donor star.

We fit for a disk of radius R_disk using the following prior. The probability is not penalized if R_disk < 49.4 but follows a Gaussian distribution with σ = 3.2 R_⊙ (see above) if the disk is larger. We also fit for the donor star radius, R_don, constrained by a Gaussian prior about 1.79 ± 0.72 R_⊙, its inferred radius being taken from spectroscopic analyses. The system is inclined to our line of sight by angle i, and the disk exhibits a flare angle of β, with thickness h at the inner edge, and has uniform optical depth of τ. We make use of all available light curves and fit for a reference eclipse time T_c and period P. We estimated the best-fit values and uncertainties for model parameters with an MCMC exercise. Since the star exhibits photometric jitter only at the 7 mmag level, the per-point uncertainties are inflated to be the scatter of the out-of-occultation light curve. We find a best-fit radius of the donor star of $2.01\pm 0.52\,{R}_{\odot }$, being occulted by a disk of ${R}_{\mathrm{disk}}=59.9\pm 6.2\,{R}_{\odot }$, with a flaring angle of 163 ± 047 and inclined to our line of sight by 187 ± 058. The best-fit light curve model is illustrated in Figure 15, and the best-fit parameters are summarized in Table 6.

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Table 6.  System Parameters

Parameter	Value	Source
Light-curve parameters
Period P (days)	72.416 ± 0.016	Light-curve modeling
Occultation centroid Tc (BJD)	2457713.18 ± 0.18	Light-curve modeling
RV parameters
Krv1 ($\mathrm{km}\,{{\rm{s}}}^{-1}$)	${14.3}_{-1.3}^{+1.4}$	Radial velocity modeling
Krv2 ($\mathrm{km}\,{{\rm{s}}}^{-1}$)	${85.5}_{-2.2}^{+1.9}$	Radial velocity modeling
e cos ω	−0.0127 ± 0.0097	Radial velocity modeling
e sin ω	−0.003 ± 0.017	Radial velocity modeling
Systemic RV ($\mathrm{km}\,{{\rm{s}}}^{-1}$)	−22.7 ± 1.9	Radial velocity modeling
Derived parameters
Macc (M⊙)	$3.24\pm 0.29$	Radial velocity modeling
Racc (R⊙)	$3.09\pm 0.59$	Spectral + Geneva isochrone fitting (Ekström et al. 2012)
Lacc (L⊙)	${142}_{-33}^{+67}$	Spectral + Geneva isochrone fitting (Ekström et al. 2012)
Mdon (M⊙)	$0.542\pm 0.053$	Radial velocity modeling
Rdon (R⊙)	1.79 ± 0.72	Spectral + Geneva isochrone fitting (Ekström et al. 2012)
Rdon (R⊙)	2.01 ± 0.52	Disk modeling + Gaussian prior from isochrone fitting
Ldon/(Ldon+Lacc)	0.52 ± 0.18	Spectral fitting
Distance (pc)	770 ± 140	Isochrone fitting
a (R⊙)	$114.0\pm 3.1$	Radial velocity modeling
e	0.021 ± 0.010	Radial velocity modeling
Derived disk parameters
Rdisk (R⊙)	$59.9\pm 6.2$	Light-curve modeling and Roche lobe constraint
Rdisk (R⊙)	49.2 ± 3.2	From Roche lobe constraint only
Line-of-sight inclination inc (deg)	1.43 ± 0.56	Light-curve modeling
Disk flare angle β (deg)	0.70 ± 0.25	Light-curve modeling
Disk thickness h (R⊙)	0.63 ± 0.21	Light-curve modeling
Disk optical depth τ	0.19 ± 0.13	Light-curve modeling

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Some of the parameters in our model are highly degenerate, and the model is overly simplistic for this system. From the MCMC analysis, the flare angle is highly correlated with the donor star radius and line-of-sight inclination. The true disk will also not have a sharp edge, and the ingress and egress timescales will be degenerate with the optical depth of the disk and the donor star radius. For future reference, we also note that our adopted priors on disk and donor star radii significantly influence the goodness of fit of the light curve. We found that an unconstrained model fits the light curve much better but tends to adopt a significantly larger disk radius of ≈65 R_⊙ and larger donor star of ≈15 R_⊙; neither are physical solutions given our known constraints on the system. Even with the Gaussian prior on the disk radius, our derived value of $59.9\pm 6.2\,{R}_{\odot }$ is in tension with that expected from the Roche lobe constraint at the 1.5σ level. The disk size in our model depends on the eclipse impact factor through a perfectly elliptical projected disk. However, we know from the ingress and egress light curve that the disk is irregular. In this simplistic modeling, we are forcing the disk to be well aligned and symmetric. Asymmetries in the disk can be better explored by future observations that provide better spectroscopic coverage of the disk absorption lines during occultation (Section 3.5).

It is also possible that the disk contains significant amounts of dust. MWC 882 exhibits levels of infrared excess consistent with a 1500 K dusty disk. Similarly, dust $\gtrsim 5$ μm is thought to be the source of infrared excess in Aurigae (Hoard et al. 2010). The dust in the disk of Aurigae may be partially responsible for the mid-eclipse brightening via forward scattering (Budaj 2011).

Forward scattering can also effectively reproduce the mid-eclipse brightening seen in the occultations of MWC 882. During the eclipse, light from the donor star is passing through, or skimming along, the surface of the disk. There are three components to the forward-scattering pattern: (1) the angular size of the light source as seen by a dust grain, (2) the scattering pattern (i.e., "phase function") versus scattering angle for individual dust grains, and (3) the angular size of the dust region as seen from the donor star. These three angular components would be convolved together. The first of these is just a few degrees owing to the small size of the donor. The second component is given approximately from Mie scattering theory by ∼20°/s, where s is the grain size in μm. For 5–10 μm particles, this angle is only 2°–4°. Finally, the size of the whole dust cloud is essentially 60° across, which dominates the system scattering pattern. As such, at orbital phase ϕ, the magnitude of the forward-scattering effect is roughly proportional to the length of the chord through the disk at each angle ϕ:

$$\begin{eqnarray}&&B(\phi )\propto \sqrt{{({R}_{\mathrm{disk}}/a)}^{2}-{\sin }^{2}\phi }.\end{eqnarray} \tag{ 1 }$$

Figure 16 demonstrates the effect of forward scattering on a nonflared disk occultation model. The disk is chosen to be 60 R_⊙ wide, inclined to our line of sight at an angle of 15 with an intrinsic width of 0.5 R_⊙. In this demonstration, the magnitude of the scattering effect is arbitrarily normalized to replicate the central brightening seen in the photometry.

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We propose that MWC 882 is a post-Algol binary system. Although there are a number of different configurations for the progenitor binary that could have evolved to match the currently observed system, we believe that the progenitor binary likely consisted of a 3.6 M_⊙ primary (the donor star) and a 2.1 M_⊙ secondary star (accretor) with an orbital period of about 7.05 days. Some of the difficulties in determining the properties of the progenitor binary are related to (i) uncertainties in the physics describing the amount of mass lost from the binary as it evolves and (ii) the magnitude of orbital angular momentum loss due to systemic mass loss, stellar winds, and tidal friction (Eggleton 2000). It is very difficult to quantify these phenomena, but previous theoretical calculations based on reasonable physical assumptions have matched the properties of many observed post-Algol systems (e.g., Batten 1989a, and references therein). Moreover, the uncertainties can be parameterized and constrained within certain physical limits (e.g., Eggleton 2000).

According to our favored scenario, a 3.6 M_⊙ primary evolves off of the main sequence and burns sufficient hydrogen to form a 0.35 M_⊙ helium ash core that is surrounded by a thin hydrogen-burning shell just as it first begins to overflow its Roche lobe. At this point (see Figure 17) the luminosity of the donor is about 230 L_⊙ and its radius is ∼10 R_⊙. Mass transfer from the donor star to the 2.1 M_⊙ accreting companion proceeds stably as the donor evolves up the RGB.²⁴ Mass-loss rates from the donor can approach 10^−5.5 M_⊙ yr⁻¹ because of its short nuclear timescale and because of angular momentum carried away by the (systemic) mass loss from the binary. After the donor has lost about 85% of its mass, it is largely composed of a compact, degenerate helium core with a mass of approximately 0.5 M_⊙ and a very tenuous envelope composed mostly of hydrogen that has a mass of approximately 0.05 M_⊙ and extends over a cross-sectional radius of 25 R_⊙. Continued mass loss cannot be sustained, as this causes this envelope to collapse once a certain threshold in pressure is reached owing to the ever-decreasing gas densities in the envelope. Thus, the donor contracts within its Roche lobe on its thermal (Kelvin–Helmholtz) timescale with the concomitant cessation of mass transfer (see Nelson et al. 2004, and references therein for a detailed discussion).

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Once mass transfer stops, nuclear burning near the surface of the helium core persists for several million years. This causes the surface luminosity to remain high (more than 100 L_⊙) and approximately constant during this phase. As a result, the effective temperature continues to rise as the donor star evolves thermally. We find that the donor star evolves through this post-RGB "horizontal branch" phase for approximately 0.35 Myr before attaining a surface temperature and luminosity close to what are inferred for MWC 882. At this juncture the donor has a mass of 0.495 M_⊙ and a luminosity of approximately 300 L_⊙. The orbital period of the binary is 72.4 days, and the mass of the companion is 3.03 M_⊙.

In order to reproduce the observed properties of MWC 882, a grid of models was computed using the MESA stellar evolution code (Paxton et al. 2011, 2015). The evolution of the donor star was followed according to the Roche lobe overflow model. The evolution of the accretor was calculated simultaneously, and the computation was stopped after the accretor evolved to become a giant, thereby filling its Roche lobe and leading to a "mass-transfer reversal." The chemical composition of the progenitor binary was assumed to be solar (Z = 0.02), and magnetic braking was assumed to operate in low-mass stars (≲1.5 M_⊙) with convective envelopes and radiative cores.

Although highly uncertain, we set the systemic mass loss and angular momentum loss parameters such that α = 0.4, β = 0.3, γ = 0, and δ = 0 (for details see Tauris & van den Heuvel 2006). Here, α and β are fractions of the mass lost by the donor star that get ejected from the binary and carry away the specific angular momentum of the donor star and accreting star, respectively. γ and δ are parameters that are associated with a circumbinary disk and are not used here. Note that these values imply that (i) mass transfer is quite nonconservative, with an accretion efficiency of 30% (i.e., with 70% of the mass being lost via a fast Jeans's ejection), and (ii) none of the mass lost from the binary forms a circumbinary torus that can extract additional orbital angular momentum.

It is important to note that the track presented in Figure 17 is only one of several possible tracks that could reproduce the observed properties of the system. This implies that the properties of the progenitor binary (e.g., the initial masses and separation) and the physics of systemic mass and angular momentum loss (e.g., α and β) do not have to be fine-tuned in order to match the observations. This increases the probability that the post-Algol hypothesis is correct.

The donor star in MWC 882 is currently evolving through the slowly pulsating B-star ("SPB") instability strip²⁵ and will contract further until substantial helium burning is ignited in its core after ≃3 Myr has elapsed. During the next ≃50 Myr, the donor star resembles a subwarf B (sdB) star and begins the process of alpha-capture to produce oxygen in the core. This corresponds to the loop that is seen in Figure 17 for a temperature of 29,000 K and a luminosity of 25 L_⊙. After an additional ≃35 Myr, burning at the center of the donor will be complete, and the chemical composition at the center is approximately 68% oxygen and 30% carbon. After passing through the sdO (subdwarf O star) region of the H-R diagram, the donor reaches its maximum effective temperature (nuclear burning has been quenched) and subsequently enters the degenerate dwarf cooling phase. During this final phase, its radius is approximately constant as it evolves to a completely electron-degenerate configuration.

We also predict that the accretor will evolve off the main sequence approximately 300 Myr from now and fill its Roche lobe—resulting in a "mass-transfer reversal" episode. The orange track in Figure 17 illustrates the future evolution of the accretor. At that time, the 3 M_⊙ accretor will have evolved up the AGB and have a 0.54 M_⊙ CO core with a thin helium-burning layer surrounding it. Once mass transfer onto the 0.495 M_⊙ "donor" is initiated, it will be dynamically unstable, with the "donor" spiraling in toward the center of the accretor (Webbink 1984). This common envelope (CE) phase of evolution will last for ∼100–1000 yr, after which the envelope of the accretor will be ejected into the interstellar medium, leaving a nascent double-degenerate CO–CO white dwarf binary of nearly equal mass in a tight orbit. In order to determine the likely orbital period of the double-degenerate binary, we follow the approach used by Rappaport et al. (2017, see their Equation (10)) and set their CE efficiency parameter to 0.5. This implies that the remnant binary will have an orbital period of about 1 hr and will be composed of a 0.54 M_⊙ CO white dwarf and a 0.50 M_⊙ CO white dwarf in a nearly circular orbit. The merger of these two white dwarfs as a result of gravitational radiation losses, which will occur within 40 Myr of the CE phase, may even result in a Type Ia supernova if sub-Chandrasekhar mergers are admissible (van Kerkwijk et al. 2010). Because of the fine-tuning required, it is more likely that the merger product will be a very massive single white dwarf such as inferred for J0317–853 and GD362as (see Ji et al. 2013, for a discussion of mergers). If the 0.50 M_⊙ white dwarf has a slightly lower mass than our models predict, it is also possible that the merger could lead to the creation of an R CrB star (see, e.g., Pandey et al. 2006).

Finally, it is important to note that the current-epoch donor star will be a member of the predicted population of subdwarf B stars in wide binaries around A star accretors while it is in the helium-burning phase of evolution. Our model produces a very natural evolutionary pathway that reproduces the properties of members of this population. Binary population synthesis by Han et al. (2003) predicts a population of donor stars with masses sharply peaked at about 0.5 M_⊙ and with orbital periods extending up to $\lesssim 1000$ days, from the first stable Roche lobe overflow channel. The resultant accretors in these systems can have a wide range of spectral types based on their masses and degree of nuclear evolution. We could reasonably expect that most unevolved accretors would fall within the F to B V spectral classes; the evolved systems could have much cooler temperatures but are less likely to be observed because of their relatively short lifetimes. They note that most of these systems are probably beyond the detection thresholds of typical radial velocity surveys (e.g., Maxted et al. 2001), due to the small orbital semiamplitudes of the luminous accretor stars.

5.1. Why Is the Disk Still Around?

In the scenario outlined above, active mass transfer ended ∼0.3 Myr ago, when the donor star evolved off the RGB and underfilled its Roche lobe. The occultation we observe, however, indicates that a substantial disk of size $59.9\pm 6.2\,{R}_{\odot }$ still remains around the accretor. We explore a few simple arguments for the lifetimes of gas and dusty disks below, to show that without the disk being actively fed, it is difficult for the disk to remain in its current state.

To estimate the lifetime of a gaseous disk, we adopt the Shakura & Sunyaev (1973) α-disk model for an approximation of the disk filling timescale τ_fill:

$$\begin{eqnarray}{\tau }_{\mathrm{fill}} & = & \displaystyle \frac{{M}_{\mathrm{disk}}}{\dot{M}}\\ & \simeq & 240{\left(\displaystyle \frac{\alpha }{0.01}\right)}^{-4/5}{\left(\displaystyle \frac{\dot{M}}{{10}^{-8}{M}_{\odot }{\mathrm{yr}}^{-1}}\right)}^{-3/10}{\left(\displaystyle \frac{{r}_{\max }}{60{R}_{\odot }}\right)}^{5/4}\,\mathrm{yr},\end{eqnarray} \tag{ 2 }$$

where $\dot{M}$ is the accretion rate, r_max is the outer radius of the disk, and α specifies the efficiency of viscous angular momentum transport in the disk. Adopting our Hα-derived accretion rate of 10⁻⁸ M_⊙ yr⁻¹ and an α parameter of 0.01, the disk should have dissipated in just 200 yr. In fact, while our accretion rate may be somewhat underestimated, it needs to be smaller by 10 orders of magnitude to achieve a disk lifetime of 0.2 Myr. Alternatively, α would have to be 2 × 10⁻⁶ for the disk to last for 0.25 Myr. We also note that for the disk to maintain its current accretion rate of 1.3 × 10⁻⁸ M_⊙ yr⁻¹ over 0.2 Myr, the disk needs to be more massive than 0.003 M_⊙. However, given that the currently inferred accretion rate is similar in magnitude to the theoretically expected mass-transfer rate during the Roche lobe filling phase, it would be difficult to build up such a massive disk.

We can also explore the possibility that the disk contains significant dust. Dust particles in the disk are subjected to the Poynting–Robertson drag, by which dust grains lose orbital angular momentum. This results from the fact that in the rest frame of the dust there is a small component of the momentum flux of the photons that is in the direction opposed to the orbital motion. Figure 18 shows the expected orbital decay timescales from Poynting–Robertson drag on dust particles in the disk at 49 R_⊙, for dust having different imaginary indices of refraction k. As can be seen from Figure 18, such dust particles have orbital lifetimes of no more than ∼1000 yr, insufficient to sustain the disk in its current state. Furthermore, most dust particles smaller than ∼10 μm have ratios of radiation pressure forces to gravity $\gtrsim 0.5$, and therefore they become unbound from the system on a dynamical timescale.

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MWC 882 is a post-Algol binary with a B7 post-RGB donor star that, until ∼0.3 Myr ago, was transferring mass to its A0 companion. The donor star, now pushed outward to an orbital separation of $114.0\pm 3.1\,{R}_{\odot }$, is occulted by the remnant accretion disk around the accretor once every 72 days. The occultations were observed during Campaign 11 of the K2 mission and subsequently identified in pre-discovery light curves from the ASAS and ASAS-SN surveys. The dynamical masses of the system were measured to be $3.24\pm 0.29\,{M}_{\odot }$ and $0.542\pm 0.053\,{M}_{\odot }$. The strengths of temperature-sensitive lines yielded a spectral type estimate of A0 for the accretor and B7 for the donor star, with the two stars exhibiting approximately equal luminosities. We estimate, via isochrone fitting and light-curve modeling, that the radii of the two stars are $3.09\pm 0.59\,{R}_{\odot }$ and $2.01\pm 0.52\,{R}_{\odot }$.

We coordinated a campaign of multiband photometric and spectroscopic observations over the occultation event centered on 2017 September. Our eclipse light curves agreed well with those observed by K2 and pre-discovery surveys. The eclipse was found to be colorless to within detection limits. We obtained a series of spectroscopic observations over the second half of the eclipse and detected a series of absorption lines from the disk around the accretor, reminiscent of those seen during the eclipse of Aurigae (e.g., Chadima et al. 2011; Strassmeier et al. 2014). We used a toy disk model to simultaneously fit for the photometric and spectroscopic eclipses, finding a disk $59.9\pm 6.2\,{R}_{\odot }$ in radius. The central brightening seen during eclipse is explained by including a flaring geometry for the disk, such that a smaller area of the donor star is covered during mid-eclipse, or by a dusty disk inducing significant forward scattering along the line of sight (similar to the disk model of Aurigae from Budaj 2011).

Our interpretation of the system cannot account for the persistence of the accretion disk. We expect the donor star to have begun contraction ∼0.3 Myr ago, terminating mass transfer, while the disk is nominally expected to survive for only hundreds of years without active feeding, regardless of its dust-to-gas composition. We suggest that future observations could search for signatures of active mass transfer in the system. Full spectroscopic coverage of the orbital phase can reconstruct a Doppler tomographic image of the inner disk (e.g., Marsh & Horne 1988) and help map out the ongoing accretion mechanics. Sporadic hot spots in the disk are also signs of mass transfer and may be identified by frequent monitoring of the system at all phases. In particular, higher signal-to-noise ratio spectra during the eclipse, covering ingress and egress, can help us constrain the optical depths at the edges of the disk and allow us to resolve degeneracies that plague the light-curve modeling. The disk of Aurigae is thought to consist of large dust grains (e.g., Hoard et al. 2010), which may help extend the disk lifetime. Similar mid- and far-infrared observations of MWC 882 may constrain the disk gas-to-dust ratio and help us better understand why the disk is still present. Continuous spectroscopic and spectropolarimetric monitoring of the system over the rotation period of the inner disk and the stars may reveal connections between accretion and the chemical peculiarity of the stars. Ap/Bp stars are known to exhibit strong magnetic fields, spots, and inhomogeneities in the abundances across their surfaces (e.g., Kochukhov et al. 2004; Kochukhov & Wade 2010). We can search for links between the inner disk structure and the potential spot distribution in the stars of MWC 882.

Work by G.Z. is provided by NASA through Hubble Fellowship grant HST-HF2-51402.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. L.N. thanks the Natural Sciences and Engineering Research Council (Canada) for financial support provided through a Discovery grant. C.I.J. gratefully acknowledges support from the Clay Fellowship, administered by the Smithsonian Astrophysical Observatory. M.H.K. would like to acknowledge Allan Schmitt for his LcTools software. Work by C.H. is supported by the Juan Carlos Torres Fellowship. Work by A.V. is performed in part under contract with the California Institute of Technology/Jet Propulsion Laboratory funded by NASA through the Sagan Fellowship Program executed by the NASA Exoplanet Science Institute. This work makes use of the Smithsonian Institution High Performance Cluster (SI/HPC). We also thank Calcul Québec, the Canada Foundation for Innovation (CFI), NanoQuébec, RMGA, and the Fonds de recherche du Québec—Nature et technologies (FRQNT) for computational facilities. We would also like to thank J. Cannizzo for insightful discussions. This paper includes data taken at the McDonald Observatory of the University of Texas at Austin.

Facilities: ASAS - The All Sky Automated Survey, Kepler - , FLWO:1.5 m (TRES) - , Smith (Tull) - , APF - , HAO - .