WASP-12b AND HAT-P-8b ARE MEMBERS OF TRIPLE STAR SYSTEMS

WASP-12b is a highly irradiated transiting hot Jupiter that orbits a G0V star with a 1.09 day period (Hebb et al. 2009). RM measurements yield a sky-projected spin orbit angle of λ = 59$^{+15}_{-20}$ deg (Albrecht et al. 2012). WASP-12b may have a prolate shape and be undergoing Roche-Lobe overflow that results in substantive mass loss (Li et al. 2010; Fossati et al. 2010, 2013). It has been suggested that this planet's dayside emission spectrum is consistent with a super-solar carbon-to-oxygen ratio (Madhusudhan et al. 2011; Moses et al. 2013; see, however, Crossfield et al. 2012). Recent observations of WASP-12b's transmission spectrum indicate that it may also have a high-altitude haze or cloud layer (Swain et al. 2013; Stevenson et al. 2013).

Bergfors et al. (2011) detected a faint source separated by 1.047 ± 0021 from the WASP-12 primary. Using Keck/NIRSPEC archival data, Crossfield et al. (2012) analyzed the near-infrared spectrum of the candidate companion and found that that it is consistent with an M dwarf. Crossfield et al. (2012) also note that the candidate is abnormally bright for an M dwarf if situated at the same distance as the primary. Bergfors et al. (2013) find that the companion's point-spread function (PSF) appears to be elongated in two separate epochs, possibly indicating that it is a marginally resolved triple system.

HAT-P-8b is a transiting hot Jupiter that orbits an F5V star with a period of 3.07 days (Latham et al. 2009). Initially suspected to have an inflated radius, recent observations by Mancini et al. (2013) indicate a higher density than previously reported. Simpson et al. (2011) measure a sky-projected spin-orbit angle of λ = 15$^{+33}_{-43}$ deg and Moutou et al. (2011) find λ = −17$^{+9.2}_{-11.5}$ deg, both consistent with a reasonably well aligned prograde orbit. High spatial resolution imaging by Bergfors et al. (2011, 2013) indicates that HAT-P-8 may be part of a binary star system although Faedi et al. (2013) were unable to confirm the candidate companion, which had a purported angular separation of 1027 ± 0011.

We perform a Bayesian analysis to model the AO observations of WASP-12 and HAT-P-8 at each epoch. Specifically, Markov Chain Monte Carlo (MCMC) numerical methods are employed to compute companion relative brightnesses, astrometric positions, and determine uncertainties. The Metropolis–Hastings algorithm efficiently explores regions of parameter space to find the best-fitting global minimum and calculate posterior distributions for each fit parameter.

We simultaneously model three PSFs to self-consistently account for contamination from a nearby companion star. Free parameters include: rectilinear coordinates for each source; peak brightness of each source; sky background levels (which we model as spatially uniform); and PSF fitting parameters, α, β, γ, r_s, and w. The observations are modeled using a modified Moffat function given by:

$$\begin{equation} I(x,y)= \sum _{i=1}^{i=3}\: \left\lbrace \alpha _i \left[1+\left(\frac{r_i}{r_s}\right)^{2}\right]^{-\beta }+\gamma _i\:e^{-r_i^2/w^2} \right\rbrace, \end{equation} \tag{ 1 }$$

where $r_i= \sqrt{{{{(x_i-{x_0}_i)^2+(y_i-{y_0}_i)^2}}}}$ is a polar coordinate corresponding to the angular separation from each source, i, in the image. The term on the left describes the AO halo and the term on the right characterizes the PSF core. By separating the terms, we effectively account for tip/tilt and focal anisoplanatism in the images, although we do not allow r_s and w to vary individually (w_i = w, $r_{s{_i} }= r$) due to the already large number of degrees of freedom (12 when including the sky background). The posterior distributions found by our MCMC algorithm marginalize over all fitting parameters.

Equation (1) captures on-axis AO features but does not replicate low order aberrations or diffraction from the first Airy ring. We have experimented with other PSF forms such as sinc(...) and sinc²(...) functions. Assuming that uncertainties in each reduced image are described by Poisson statistics at the pixel level, resulting from sky-background subtraction shot noise, we find the results from each AO model are consistent with one another but uncertainties are unrealistically small. For example, angular separation measurement uncertainties are less than 1 mas (1σ). The images used for our analysis have been fully processed prior to MCMC calculations. As such, we have stacked frames acquired from different dither positions. This step is required because the companions are so much fainter than their primary star, particularly in the J band. However, by combining images obtained from different locations on the array, we have introduced PSF spatial smearing from uncorrected optical distortions. We estimate the size of this effect using polynomial fits available for the NIRC2 array provided by Keck Observatory.¹¹ Systematic errors are of order 1–2 pixels and change depending on the size of the dither pattern. Distortion corrections may be applied before image stacking but this can introduce significant numerical noise. Furthermore, the correction coefficients also change slowly with time (Yelda et al. 2010).

Our final adopted astrometric uncertainties were found by adding the effects of optical distortion in quadrature with that from photon noise and pixel crosstalk resulting from PSF fitting errors. We self-consistently account for uncertainty in the plate scale and orientation of the NIRC2 array (Ghez et al. 2008) by randomly drawing values for the plate-scale and orientation from a normal distribution and folding the results into calculations of the angular separation and position angle when converting from pixel separations to arcseconds. The effect from optical distortion dominates the uncertainty for each astrometric epoch as it is much greater than both cross-talk and photon noise. Results for relative astrometry measurements are shown in Table 1. Although our observations from 2012 and 2013 were acquired in different filters (K' and K_s), due to a change in the FHJ default observing strategy, this effect appears to be small since the results are nearly identical.

We perform an astrometric analysis to assess the physical association of the WASP-12 BC and HAT-P-8 BC candidates with their primary star. To do so, we compare our astrometric measurements against the null hypothesis that the off-axis sources are infinitely distant unrelated background objects with zero parallax. WASP-12 has a small proper motion [−0.7, −7.8 mas yr⁻¹] comparable to the size (9.963 ± 0.006 mas) of a NIRC2 pixel (Høg et al. 2000; Ghez et al. 2008). HAT-P-8 has a proper motion that is an order of magnitude larger [75.5, 17.5 mas yr⁻¹] (Høg et al. 2000). Neither star has a Hipparcos parallax measurement, which complicates the analysis. Instead, the distance to WASP-12 is estimated using a photometric distance modulus (Bergfors et al. 2013) and the distance to HAT-P-8 is determined using a spectroscopic distance modulus (Latham et al. 2009). We incorporate parallactic motion by converting estimated distances to a trigonometric parallax (ellipse). The resulting differential motion across the sky between the primary star and candidate secondary/tertiary is given by the vector sum of the proper and parallactic motion (Zimmerman et al. 2010).

Our astrometric measurements are shown in Figure 2. Overplotted are previous measurements taken by Bergfors et al. (2013) in 2009 October that identify the combined light signal of WASP-12 BC and HAT-P-8 BC but do not spatially resolve the sources into individual components. Our Keck AO observations from 2012 and 2013 clearly separate the light from each companion star. The angular separation of WASP-12 BC is 84.3 ± 0.6 mas (21 ± 3 AU) and the angular separation of HAT-P-8 BC is only 65.3 ± 0.5 mas (15 ± 1 AU), comparable to the diffraction limit of a 10 m telescope at near-infrared wavelengths (Figure 1). Optical distortion for such small separations is negligible. To compare data on an equal footing with Bergfors et al. (2013), we plot combined light photo-centers for WASP-12 BC and HAT-P-8 BC in Figure 2.

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The a priori probability of finding three point sources in a hierarchical configuration separated by only 1'' on the sky is very low. Our two astrometric epochs for WASP-12 and HAT-P-8 are separated by 393.9 days and 373.9 days, respectively. The expected motion of a background source relative to the primary star is 8.5 ± 1.0 mas (0.9 ± 0.1 pixels) for WASP-12 and 79.3 ± 2.9 mas (8.0 ± 0.3 pixels) for HAT-P-8 over the same time-frame.

With only two observations, the confirmation that WASP-12 BC are bona-fide companions is marginal. However, combining our measurements with the 2009 October initial detection from Bergfors et al. (2011), we can demonstrate that the three point sources are physically associated (Figure 2). To further reinforce our results, we have determined the photometric distance modulus for WASP-12 BC. The combined light apparent magnitude of the WASP-12 System is 10.19 ± 0.02 (Skrutskie et al. 2006). Backing out the individual apparent magnitudes of WASP-12 BC from our relative photometry measurements, we find the distance to WASP-12 B is 263 ± 13 pc and the distance to WASP-12 C is 267 ± 13 pc. These values overlap with the photometric distance estimated by Bergfors et al. (2013) of 250 ± 30 pc, effectively ruling out the possibility that they are foreground or background objects.

HAT-P-8 BC are confirmed using our observations alone due to the large space motion of the host star. We cannot claim detection of orbital motion for either system because of the aforementioned systematic errors and the fact that the stars were observed with different instruments and filters. Dedicated astrometric measurements are required to determine the total dynamical mass of the secondary and tertiary in each case (Dupuy et al. 2010). We note that in both cases, WASP-12 and HAT-P-8, our measurements are ≈20 mas south and ≈20 mas east of Bergfors, suggesting possible systematics between the AstraLux and Keck data sets.

Bergfors et al. (2013) assign a preliminary spectral type of M0V for WASP-12 "B" (combined light), assuming the identified off-axis source is associated. Crossfield et al. (2012) find that WASP-12 "B" is a hot M dwarf with ΔK = 2.45 ± 0.06 mag. We find that WASP-12 B and WASP-12 C are $\Delta K^{A,B}_s = 3.25\pm 0.04$ and $\Delta K^{A,C}_s = 3.28 \pm 0.04$ mag fainter than the primary, respectively (Table 2). Combining the signal from both components, our measurements show that the expected unresolved brightness difference between the secondary/tertiary and primary star should be $\Delta K^{A,BC}_s = 2.51\pm 0.03$ mag, consistent with the interpretation of Crossfield et al. (2012).

Table 2. Secondary and Tertiary Companion Photometric Properties

Companion	ΔJ	ΔKs	J − Ks	$M_{K_s}$	Mass	Spec. Type
					(M☉)
WASP-12 B	3.81 ± 0.05	3.25 ± 0.04	0.85 ± 0.08	6.47 ± 0.27	0.38 ± 0.05	M3V
WASP-12 C	3.92 ± 0.05	3.28 ± 0.04	0.93 ± 0.08	6.50 ± 0.27	0.37 ± 0.05	M3V
HAT-P-8 B	⋅⋅⋅	5.58 ± 0.07	⋅⋅⋅	7.73 ± 0.16	0.22 ± 0.03	≈M5V
HAT-P-8 C	⋅⋅⋅	6.08 ± 0.10	⋅⋅⋅	8.32 ± 0.17	0.18 ± 0.02	≈M6V

Notes. We estimate spectral types using near-infrared color information (when available) and absolute magnitudes by comparing to Kraus & Hillenbrand 2007. Absolute magnitudes are found using (photometric or spectroscopic) distance modulus estimates: d = 250 ± 30 pc for WASP-12 (Bergfors et al. 2013) and d = 230 ± 15 pc for HAT-P-8 (Latham et al. 2009).

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To further characterize the companions around each star, we calculate absolute magnitudes based on previous distance estimates from Bergfors et al. (2013) and Latham et al. (2009). Our uncertainty in absolute magnitude is dominated by the lack of a trigonometric parallax measurement. We estimate the mass of each companion using Girardi et al. (2002) evolutionary models assuming a system age of 5 Gyr. Comparing our absolute magnitudes to those of Kraus & Hillenbrand (2007), we find that WASP-12 BC are consistent with M3V (Table 2). Additionally, the J − K colors of WASP-12 BC (see Table 2) are also consistent with those of M stars (Kraus & Hillenbrand 2007). Although HAT-P-8 BC are detected during second epoch (UT 2013 July 3) observations, they are spatially unresolved in the J band because the images were obtained at an airmass of 2.19. Performing aperture photometry for the pair, we find a combined difference in magnitude of ΔJ^{A, BC}J = 5.9 ± 0.2. We estimate the spectral types of HAT-P-8 B and C to be ≈ M5V and ≈ M6V, respectively, using K-band photometry alone.

As part of the FHJ program, we obtained additional RV measurements for both systems, which we use to constrain the presence of additional companions at shorter orbital periods. Our best-fit RV slopes are:

$$\begin{eqnarray} {dv/dt}_{{\rm WASP}-12}&=& -4.12 \pm 4.37\ {\rm m\ s^{-1} \ yr^{-1}} \nonumber \\ {dv/dt}_{{\rm HAT}-P-8}&=& -2.72 \pm 2.39\ {\rm m\ s^{-1} \ yr^{-1}}, \end{eqnarray} \tag{ 2 }$$

consistent with the absence of massive, m ⩾ 5 M_J, objects out to a ⩽ 8.3 AU for WASP-12 and a ⩽ 10.9 AU for HAT-P-8. Figure 3 displays joint constraints imposed by the combination of Doppler RV measurements (solid lines) and direct imaging observations (dashed lines). Should any additional companions be present in these systems, their masses must reside below both curves. Continued RV monitoring of the host stars will further eliminate regions of mass–semi-major-axis parameter space.

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