HAT-P-32b AND HAT-P-33b: TWO HIGHLY INFLATED HOT JUPITERS TRANSITING HIGH-JITTER STARS^*

Table 1 summarizes the HATNet discovery observations of each new planetary system. The HATNet images were processed and reduced to trend-filtered light curves following the procedure described by Bakos et al. (2010). The light curves were searched for periodic box-shaped signals using the Box Least-Squares (see Kovács et al. 2002) method. For HAT-P-33, supporting observations of the discovery were obtained with the Wise-HAT (WHAT; Shporer et al. 2009) telescope at Wise Observatory in Israel; these were analyzed in the same manner as the HATNet images. We detected significant signals in the light curves of the stars summarized below.

• 1.  
  HAT-P-32. GSC 3281-00800 (also known as 2MASS 02041028+4641162; $\alpha = 02^{\mathrm h}04^{\mathrm m}10\mbox{$.\!\!^{\mathrm s}$}27$, δ = +46°41'162; J2000; V = 11.289; Droege et al. 2006). A signal was detected for this star with an apparent depth of ∼20.3 mmag and a period of P = 2.1500 d (see Figure 1, left).
• 2.  
  HAT-P-33. GSC 2461-00988 (also known as 2MASS 07324421+3350061; $\alpha = 07^{\mathrm h}32^{\mathrm m}44\mbox{$.\!\!^{\mathrm s}$}20$, δ = +33°50'062; J2000; V = 11.188; Droege et al. 2006). A signal was detected for this star with an apparent depth of ∼8.5 mmag and a period of P = 3.4745 d (see Figure 1, right).

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High-resolution, low signal-to-noise ratio (S/N) "reconnaissance" spectra were obtained for both HAT-P-32 and HAT-P-33 using the Harvard-Smithsonian Center for Astrophysics (CfA) Digital Speedometer (DS; Latham 1992) on the FLWO 1.5 m telescope. These observations, which are summarized in Table 2, were reduced and analyzed following the procedure described by Torres et al. (2002) (see also Latham et al. 2009). We find that both stars show no velocity variation at the 1 km s⁻¹ precision of the observations, and all spectra are consistent with single, moderately rotating dwarf stars. For HAT-P-32, the stellar atmospheric parameters that we find, assuming solar composition, are T_eff⋆ = 6500 ± 100 K, log g_⋆ = 4.5 ± 0.25, vsin i = 21.9 ± 1.0 km s⁻¹, and γ_RV = −23.21 ± 0.26 km s⁻¹. For HAT-P-33, we find T_eff⋆ = 6500 ± 100 K, log g_⋆ = 4.0 ± 0.25, vsin i = 15.6 ± 1.0 km s⁻¹, and γ_RV = 23.03 ± 0.28 km s⁻¹.

In addition to the DS observations for HAT-P-33, we also obtained several initial reconnaissance observations of this target with the SOPHIE spectrograph on the Observatoire de Haute-Provence 1.93 m telescope. These observations showed ∼100 m s⁻¹ scatter, with only a very faint hint of phasing with the photometric ephemeris, hinting at the possibility that the system is a blend. Based on these observations we postponed further follow-up of the target for several years.

We obtained high-resolution, high S/N spectra for both stars using the High Resolution Echelle Spectrometer (HIRES; Vogt et al. 1994) on the Keck-I telescope in Hawaii. For HAT-P-32, we gathered 28 spectra with the I₂ absorption cell (Marcy & Butler 1992) between 2007 August and 2010 December, together with a single I₂-free template spectrum. We rejected one low S/N spectrum for which we were unable to obtain a high-precision RV measurement, and exclude from the analysis two spectra which were obtained during transit and thus may be affected by the Rossiter–McLaughlin effect (e.g., Queloz et al. 2000). For HAT-P-33, we gathered 22 spectra with the I₂ cell between 2008 September and 2010 December, and two template spectra. The HIRES spectra were reduced to barycentric RVs following Butler et al. (1996). The resulting measurements are given in Tables 3 and 4 for HAT-P-32 and HAT-P-33, respectively. The phased data, along with our best-fit models for both circular and eccentric orbits, are displayed in Figures 2 and 3.

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For both candidates, the RV residuals from the best-fit model show significant scatter greatly in excess of what is expected based on the measurement uncertainties. For HAT-P-32, the residual rms is 80.3 m s⁻¹ or 75.0 m s⁻¹ for circular and eccentric models, respectively, while the rms expected from instrumental variations plus photon noise is only 15.7 m s⁻¹. For HAT-P-33, the residual rms is 55.7 m s⁻¹ or 54.1 m s⁻¹, again for circular and eccentric models, respectively, while the expected rms is 7.9 m s⁻¹. Because of this high "jitter," we gathered substantially more high S/N spectra for these targets than we do for typical HATNet candidates. The reason for this is twofold: the high jitter could have been due to additional Keplerian motion due to the presence of additional planets in the systems which would be revealed by further observations, and in the presence of high jitter more observations are needed to precisely determine the orbital parameters of the system. We consider further the origin of the jitter for each object in Section 3.2, concluding that it is most likely not due to additional planets, but rather due to stellar activity.

For each spectrum, we also calculated the spectral line bisector span (BS) following the method described in Section 5 of Bakos et al. (2007) and the activity index S calibrated to the scale of Vaughan et al. (1978) following the procedure of Isaacson & Fischer (2010). Note that for the BS we follow the convention BS = v₁ − v₂, where v₁ is the velocity of a point on the bisector near the continuum level and v₂ is the velocity of a point on the bisector near the line core. The BS and S values are plotted in Figures 2 and 3. For both stars, the BS and S values do not phase with the photometric ephemeris; however the scatters in the BS values are comparable to the RV semiamplitudes, so we are not able to rule out blend scenarios for either object based on the bisectors. Instead we rely on detailed blend modeling of the observations to rule out these scenarios as described in Section 3.3. For HAT-P-33, we found that the RV residuals are correlated with the BS (see also Section 3.2), and were thus able to use the BS to correct the RVs, reducing the effective jitter to ∼35 m s⁻¹.

We conducted high-precision photometric observations of HAT-P-32 and HAT-P-33 using the KeplerCam CCD camera on the FLWO 1.2 m telescope. The observations for each target are summarized in Table 1. For HAT-P-32, in addition to observations taken during transit, we also obtained three sets of observations taken out of transit at the predicted times of secondary eclipse (assuming a circular orbit). These data are excluded from the global analysis described in Section 3.4, but are used in Section 3.3 in ruling out blend scenarios.

The reduction of the KeplerCam images was performed as described by Bakos et al. (2010). We performed External Parameter Decorrelation (EPD) and used the Trend Filtering Algorithm (TFA; Kovács et al. 2005) to remove trends simultaneously with the light curve modeling (for more details, see Bakos et al. 2010). The final time series, together with our best-fit transit light curve models, are shown in Figure 4; the individual measurements are reported in Tables 5 and 6 for HAT-P-32 and HAT-P-33, respectively.

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Planetary parameters, such as the mass and radius, depend strongly on the stellar mass and radius, which in turn are constrained by the observed stellar spectra as well as the light curves and RV curves. We followed an iterative procedure, described by Bakos et al. (2010), to determine the relevant stellar parameters. The procedure involves iterating between inferring stellar atmospheric parameters (including the effective temperature T_eff⋆, surface gravity log g_⋆, metallicity [Fe/H], and the projected rotation velocity vsin i) from the Keck/HIRES template spectrum using the Spectroscopy Made Easy package (SME; Valenti & Piskunov 1996) and the Valenti & Fischer (2005) atomic line database, and modeling the light curves and RV curves (see Section 3.4) to determine the stellar density ρ_⋆. At a given cycle in the iteration we combine our estimates of T_eff⋆, [Fe/H], and ρ_⋆ with the Yonsei–Yale (YY; Yi et al. 2001) series of stellar evolution models to determine the stellar mass, radius, and surface gravity among other parameters. If the resulting surface gravity differs significantly from the value determined from the spectrum with SME, we repeat the analysis fixing the surface gravity in SME to the new value.

For each star, the initial SME analysis, in which the surface gravity was allowed to vary, yielded the following values and uncertainties.

• 1.  
  HAT-P-32: T_eff⋆ = 6001 ± 88 K, [Fe/H] = -0.16 ± 0.08 dex, log g_⋆ = 4.02 ± 0.07 (cgs), and vsin i = 21 ± 0.5 km s⁻¹.
• 2.  
  HAT-P-33: T_eff⋆ = 6234 ± 114 K, [Fe/H] = -0.04 ± 0.08 dex, log g_⋆ = 3.86 ± 0.10 (cgs), and vsin i = 13.9 ± 0.5 km s⁻¹.

As described in Section 3.4 the inferred stellar densities, and hence radii, for HAT-P-32 and HAT-P-33 depend strongly on the orbital eccentricities, which due to the high RV jitters are poorly constrained. For each system we conducted separate analyses, first assuming a circular orbit and then allowing the eccentricity to vary. In each case we obtained a new value of log g_⋆ which we held fixed during a second SME iteration. The final stellar parameters for each star, assuming both circular and eccentric models, are listed in Table 7. As discussed in Section 3.4, for each system the assumed circular orbit model is preferred as the simplest model capable of fitting the data. We therefore provisionally suggest adopting the circular orbit models for applications which require a single set of parameters for each system; however we note that this may underestimate the true parameter uncertainties.

The inferred location of each star for both the circular and eccentric models, in diagrams of a/R_⋆ (which is related to ρ_⋆; e.g., see Seager & Mallén-Ornelas 2003) versus T_eff⋆, analogous to the classical H-R diagram, is shown in Figures 5 and 6. In each case the stellar properties and their 1σ and 2σ confidence ellipsoids are displayed against the backdrop of model isochrones for a range of ages and the appropriate stellar metallicity.

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As a check on our stellar parameter determinations we compare the measured photometric colors of the two stars to the predicted values based on the models. HAT-P-32 has J − K = 0.281 ± 0.008¹¹ and V − I_C = 0.62 ± 0.10 (from two-armed spiral shock (TASS); Droege et al. 2006), whereas the predicted values are J − K = 0.32 ± 0.02 and V − I_C = 0.574 ± 0.023 for the circular model, and J − K = 0.34 ± 0.02 and V − I_C = 0.592 ± 0.024 for the eccentric model. In both cases the model V − I_C colors are consistent with the measurements, but the model J−K colors are slightly (∼2σ) redder than the measurements. HAT-P-33 has J − K = 0.280 ± 0.030 and V − I_C = 0.577 ± 0.087 (also from 2MASS and TASS), and has predicted values of J − K = 0.27  ±  0.02 and V − I_C = 0.511  ±  0.022 for the circular model and J − K = 0.28  ±  0.02 and V − I_C = 0.522  ±  0.022 for the eccentric model. The measured photometric colors of HAT-P-33 are within 1σ of the predicted values for both models.

Neither HAT-P-32 nor HAT-P-33 shows significant chromospheric emission in their Ca ii H and K line cores. Following the procedure of Isaacson & Fischer (2010), we find median log R'_HK (Noyes et al. 1984) values of −4.62 and −4.88 for HAT-P-32 and HAT-P-33, respectively.

Both HAT-P-32 and HAT-P-33 exhibit notably high scatter in their velocity residuals. Stellar jitter values of 78.7 m s⁻¹ and 55.1 m s⁻¹ for HAT-P-32 and HAT-P-33, respectively, must be added in quadrature to their velocity errors to achieve reduced χ² values of unity for the best-fit circular orbits (see Section 3.4).

A possible cause of the jitter is the presence of one or more additional planets in either system, though this would not explain the high scatter seen in the BS measurements. Figure 7 shows the Lomb–Scargle frequency spectra (L-S; Lomb 1976; Scargle 1982; Press & Rybicki 1989) of the RV observations for both stars computed up to a maximum frequency of 2.0 d⁻¹. In both cases there is a peak in the spectrum at the transit frequency; in the case of HAT-P-32 this is the highest peak in the spectrum, while in the case of HAT-P-33 the highest peak is at a high-frequency alias of the transit frequency (if we restrict the search to frequencies shorter than 0.49 d⁻¹, the transit frequency is the highest peak). We note that in neither case would the planet be detectable from the RV data alone. For HAT-P-32, the false-alarm probability of detecting a peak in the L-S periodogram with a height greater than or equal to the measured peak height is ∼8% (this is determined by applying L-S to simulated Gaussian white noise RV curves with the same time sampling as the observations). For HAT-P-33, the false-alarm probability of the transit peak is 79% (the false-alarm probability for the highest peak in the spectrum is 30%). We also show the L-S spectra of the RV residuals from the best-fit circular orbit models, in order to see if there is evidence for additional short-period planets in the systems. The highest peaks in these spectra are at periods of 18.104 d and 0.8507 d for HAT-P-32 and HAT-P-33, respectively. If we assume that each system has an additional planet at the periods stated above, the rms of the RV residuals decreases to ∼40 m s⁻¹ and ∼30 m s⁻¹ for HAT-P-32 and HAT-P-33, respectively. However, these peaks have corresponding false-alarm probabilities of 8% and 8.9%, so they are not statistically significant detections. We conclude that while either system may host additional short-period planets, there are not at present enough RV observations to make a statistically significant detection.

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For each system there is some ambiguity in the peak that depends on the method used to scan for additional signals. As an alternative analysis we fit the RV observations of each system using a function of the form

$$\begin{equation} {\rm RV}_{\rm obs}(t) = \gamma + \dot{\gamma }t + {\rm RV}_{\rm orb}(t) + A\sin {2\pi f t} + B\cos {2\pi f t}, \end{equation} \tag{ 1 }$$

where $\dot{\gamma }$ is a constant acceleration (assumed to be zero for HAT-P-33), RV_orb(t) is an eccentric orbit with fixed period and epoch of transit, and A and B are free parameters. We scan through f optimizing all free parameters at each trial frequency. When scanning over the frequency range 0.001 d⁻¹ < f < 2.49 d⁻¹, we find f = 0.0208 d⁻¹ or P = 47.962 d for HAT-P-32 and f = 2.178 d⁻¹ or P = 0.459 d for HAT-P-33. If we limit the frequency scan to f < 0.49 d⁻¹, we find f = 0.09545 d⁻¹ or P = 10.476 d for HAT-P-33.

Another potential source of RV and BS jitter is varying scattered moonlight contaminating the spectra. Following Kovács et al. (2010), we investigated the possibility that the BS and RV may be affected by varying sky contamination, and found no evidence that this is the case. Alternatively, RV and BS jitter might be caused by variable contamination from a nearby star. Based on our KeplerCam observations of HAT-P-32 and HAT-P-33, we can rule out contaminating neighbors with Δi ≲ 5 mag at a separation greater than 3'', or neighbors with Δi ≲ 2 mag at a separation greater than 1'' of either star. We cannot rule out fainter close neighbors which could be responsible for at least some of the jitter. Constraints on potential neighbors based on modeling the photometric light curves are further considered in Section 3.3.

Figure 8 compares the RV residuals to the BSs for HAT-P-32 and HAT-P-33. For HAT-P-33, there is a clear anti-correlation between these quantities (the Spearman rank-order correlation test (e.g., Press et al. 1992) yields a correlation coefficient of r_s = −0.77 with a false-alarm probability of 0.04% assuming a circular orbit, while for the best-fit eccentric orbit r_s = −0.72 with a 0.1% false-alarm probability). For HAT-P-32, there is a hint of an anti-correlation, but it is not statistically significant (r_s = −0.30 with a false-alarm probability of 14% assuming a circular orbit, and r_s = −0.39 with 4.9% false-alarm probability for the best-fit eccentric orbit). The anti-correlation for HAT-P-33 is an indication that the measured jitter for this star is dominated by intrinsic stellar variations, rather than being due to another planet in the system, or to the instrumental errors being underestimated. We make use of this correlation in Section 3.4 to reduce the scatter in the RV residuals for HAT-P-33 and to improve the precision of the fitted parameters. We note that other investigators have previously made use of correlations between RV residuals and BS measurements to improve the precision of RV measurements (e.g., Melo et al. 2007).

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Saar et al. (1998) argue that the RV jitter of F stars is due primarily to convective inhomogeneities (due, for example, to regions of strong magnetic fields which locally suppress convection) which vary in time, rather than temperature inhomogeneities (due, for example, to cool star spots) which vary in time. The latter may be more important for G and K stars. Both sources of jitter would result in significant BS variations that may be correlated with the RV variations, but the lack of detectable out-of-transit photometric variations in the HATNet light curves of HAT-P-32 and HAT-P-33, and the fact that neither star shows particularly high chromospheric emission in its Ca ii H and K line cores, is consistent with the expectation that temporally changing convective inhomogeneities, rather than temporally changing temperature inhomogeneities, are responsible for the jitter of these two stars.

Jitter values as high as those found for HAT-P-32 and HAT-P-33 are typical of stars with similar spectral types and rotation velocities. From the vsin i–jitter correlation measured by Saar et al. (2003), the expected jitter for an F dwarf with vsin i = 20 km s⁻¹ is ∼50 m s⁻¹, while for an F dwarf with vsin i = 14 km s⁻¹ the expected jitter is ∼30 m s⁻¹. The sample used by Saar et al. (2003) to determine this correlation includes only a handful of stars with vsin i > 10 km s⁻¹, and the scatter about the relation is fairly significant—one star with vsin i ∼ 15 km s⁻¹ was found to have a jitter in excess of 100 m s⁻¹. The planet-hosting star HAT-P-2, which has a similar temperature and rotation velocity to HAT-P-32 and HAT-P-33 (T_eff⋆ = 6290 ± 60 K, vsin i = 20.8 ± 0.3 km s⁻¹; Pál et al. 2010), has been reported to have a high jitter of ∼60 m s⁻¹ based on data from Keck and Lick (Bakos et al. 2007). A subsequent analysis by Winn et al. (2007), using only Keck/HIRES data, found a somewhat lower jitter of ∼30 m s⁻¹, though this is still quite a bit higher than most planet-hosting stars discovered to date (e.g., the median jitter of the previously published HATNet planets is ∼7 m s⁻¹). The primary difference between HAT-P-2 and the two systems presented here is that the planet HAT-P-2b is significantly more massive than either HAT-P-32b or HAT-P-33b (HAT-P-2b has M_p = 9.09 ± 0.24 M_J; Pál et al. 2010); as a result, the RV semiamplitude of HAT-P-2b (K = 984 ± 17 m s⁻¹) greatly exceeds the jitter, making this planet more straightforward to confirm than either HAT-P-32b or HAT-P-33b.

Both HAT-P-32 and HAT-P-33 exhibit significant spectral line BS variations (Figures 2 and 3; the rms of the BS is 53 m s⁻¹ and 51 m s⁻¹ for HAT-P-32 and HAT-P-33, respectively). In neither case are the variations in phase with the transit ephemeris, as one might expect if the observed RV variation were due to a blend between an eclipsing binary system and another star. As discussed in Section 3.2, these variations are likely due to temporally changing convective inhomogeneities, perhaps created by variable photospheric magnetic faculae as in the Sun, which we suspect are responsible for the significant RV jitter seen in both of these stars. Nonetheless the large BS variations, which are comparable to the semiamplitudes of the RV signals, prevent us from using the BSs to rule out the possibility that either of these systems is a blend.

To rule out blend scenarios we made use of the blendanal program (Hartman et al. 2011b; see also Hartman et al. 2011a) which models the observed light curves, stellar atmospheric parameters, and calibrated photometric magnitudes using various blended eclipsing binary scenarios as well as scenarios involving a transiting planet system potentially blended with light from another star. The program relies on a combination of the Padova (Girardi et al. 2000) and Baraffe et al. (1998) stellar evolution models, the Eclipsing Binary Orbit Program (EPOP; Popper & Etzel 1981; Etzel 1981; Nelson & Davis 1972) as modified by Southworth et al. (2004a, 2004b), and stellar limb-darkening parameters from Claret (2004). It is similar to the blender program (Torres et al. 2005) which has been used to confirm Kepler planets (e.g., Torres et al. 2011), but with a number of technical differences which are described by Hartman et al. (2011b).

For each object we fit four classes of models.

• 1.  
  A single star with a transiting planet.
• 2.  
  A planet transiting one component of a binary star system.
• 3.  
  A hierarchical triple stellar system.
• 4.  
  A blend between a bright stationary star and a fainter, physically unrelated eclipsing binary.

Initially, we assume that the eclipsing components have a circular orbit. For both HAT-P-32 and HAT-P-33, we find that the class 1 model (a single star with a transiting planet) or the class 2 model with a planet-hosting star that is much brighter than its binary star companion provide better fits (lower χ²) than the class 3 and class 4 models. To evaluate the statistical significance with which the class 3 and class 4 models may be rejected, we follow the Monte Carlo procedure described in Hartman et al. (2011a). We find that for HAT-P-32 we may reject both the class 3 and 4 models with ∼13σ confidence (the best-fit class 4 model consists of a group of stars with similar parameters to the best-fit class 3 model), while for HAT-P-33 we may reject the class 3 and 4 models with ∼6σ confidence (e.g., Figure 9).

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For both HAT-P-32 and HAT-P-33, the eclipsing binary star blend scenarios (classes 3 and 4) are excluded in part due to the lack of an apparent secondary eclipse or out-of-transit variation that are predicted by models capable of fitting the observed primary transits. For HAT-P-33 the HATNet light curve provides these constraints, while for HAT-P-32 three sets of KeplerCam observations collected during predicted secondary eclipses (assuming a circular orbit) augment the constraints provided by the HATNet light curve. Figure 10 shows a few example model light curves for the best-fit class 4 model for HAT-P-32 which illustrate this.

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Because the time of secondary eclipse depends on the eccentricity, which is poorly constrained by the RV observations, we repeat the blend analysis for both systems fixing the eccentricities to the best-fit values as determined in Section 3.4. While for HAT-P-32 this scenario causes the secondary eclipses to not occur during the out-of-transit KeplerCam observations, the eccentric eclipsing binary results in stronger out-of-transit variations which are ruled out by the HATNet data. In this case for HAT-P-32, we may reject the class 3 and 4 models with >11σ confidence. For HAT-P-33, we may reject the class 3 and 4 models with >7σ confidence.

For the class 2 models we consider two cases, one in which the transiting planet orbits the brighter binary star component, and the other in which the planet orbits the fainter binary star component. In both cases the spectroscopic temperature and the photometric colors constrain the brighter star to have a mass of ≳ 1 M_☉ for both systems. We find that the case of the planet orbiting the fainter component can be rejected outright with >4σ confidence for both HAT-P-32 and HAT-P-33. The case of a planet orbiting the brighter binary star component amounts to including third light in the fit. When the secondary star contributes negligible light to the system, the model becomes indistinguishable from the best-fit case 1 model; such a model cannot be ruled out with the available data. Instead we may place an upper limit on the mass of any binary star companion. For HAT-P-32 we find that a putative companion must have M < 0.5 M_☉ with 5σ confidence, while for HAT-P-33 a companion must have M < 0.55 M_☉ with 5σ confidence. These translate into upper limits on the secondary-to-primary V-band luminosity ratio of ∼1% for both HAT-P-32 and HAT-P-33.

Based on the above discussion we conclude that the signals detected in the HAT-P-32 and HAT-P-33 light curves and RV curves are planetary in nature.

We modeled the HATNet photometry, the follow-up photometry, and the high-precision RV measurements using the procedure described in detail by Pál et al. (2008) and Bakos et al. (2010). One significant difference from our previous planet discoveries is that we use a Mandel & Agol (2002) transit model to describe the HATNet photometry, rather than a simplified no-limb-darkening model. This was necessary due to the high S/N HATNet detections, especially for HAT-P-32. To describe the follow-up light curves, we use a Mandel & Agol (2002) transit model together with the simultaneous EPD and TFA model of instrumental variations (Bakos et al. 2010), and we use a Keplerian orbit to describe the RV curves. For HAT-P-33, we include a term α_BS × BS in the RV model, where α_BS is a free parameter describing the residual RV–BS correlation, as we found that this significantly reduces the RV residuals. For HAT-P-32, we do not include this correction because the residual RV–BS correlation is not statistically significant. For each planetary system we fit two models, one in which the eccentricity is fixed to zero, and another in which the eccentricity varies. The parameters for each system are listed in Tables 8 and 9.

As described in Bakos et al. (2010), we use a combination of the downhill simplex method (e.g., Press et al. 1992) and the classical linear least-squares algorithm to initially optimize the free parameters, and then use the Markov Chain Monte Carlo method (MCMC; Ford 2006) to determine the parameter uncertainties (we use the "Hyperplane-CLLS" variant described in Bakos et al. 2010). Here, we provide a few additional details not given in Bakos et al. (2010). For each system/model we generate a single chain of 20,000 links using the Metropolis–Hastings algorithm (e.g., Press et al. 2007). We start the chain from the optimized position, assuming a uniform prior distribution for each of the jump parameters (listed in Bakos et al. 2010), and drawing them from a Gaussian proposal distribution with eigenvalues and eigenvectors derived from the Fisher covariance matrix analytically evaluated at the optimal position. We inspected the resulting parameter chains to verify that with this procedure it is not necessary to include a "burn-in" phase.

Neither system has a clearly non-zero eccentricity. Using the Lucy & Sweeney (1971) test, we find that there is a non-negligible ∼3% probability that the eccentricity found for HAT-P-32 arises by chance from a circular orbit, while for HAT-P-33 the probability is ∼20%. As an alternative approach we calculate the Bayesian Information Criterion (BIC; Schwarz 1978; Liddle 2007) for the circular and eccentric models, taking BIC = χ² + dlog n, where d is 2 or 4 for the circular and eccentric models, respectively, and n is the number of RV data points (e.g., Kipping et al. 2010). We find that for both HAT-P-32 and HAT-P-33 the circular models have lower BIC values than the models in which the eccentricity is allowed to vary (BIC = 30.5 and 33.8 for the circular and eccentric models for HAT-P-32, and BIC = 28.3 and 34.0 for the circular and eccentric models for HAT-P-33), indicating that for both systems a circular orbit model is preferred over a model in which the eccentricity is allowed to vary.

Because the resulting planets have large radii (particularly when the eccentricity is allowed to vary), the assumption that each planet has a spherical surface which lies well within its Roche lobe may no longer be valid. In particular the fact that a planet cannot exceed its Roche lobe places an upper limit on the transit-inferred radius of the planet for a given semimajor axis, eccentricity, and mass ratio. Following the procedure described in the Appendix, we impose the constraint that the planet cannot exceed its Roche lobe in determining the parameters and errors for each system. Figure 11 shows an approximation of the constraints on R_p/a and e for each system. In both cases the best-fit solutions are below this limit; however the uncertainties on the parameters (particularly the upper uncertainties on the eccentricities and radii) are reduced by imposing the Roche lobe constraint.

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