HAT-P-44b, HAT-P-45b, AND HAT-P-46b: THREE TRANSITING HOT JUPITERS IN POSSIBLE MULTI-PLANET SYSTEMS^*

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 (BLS; see Kovács et al. 2002) method. Figure 1 shows phase-folded HATNet light curves for HAT-P-44, HAT-P-45, and HAT-P-46 which were selected as showing highly significant transit signals based on their BLS spectra. Cross-identifications, positions, and the available photometry on an absolute scale are provided later in the paper together with other system parameters (Table 10).

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We removed the detected transits from the HATNet light curves for each of these systems and searched the residuals for additional transits using BLS, and for other periodic signals using the Discrete Fourier Transform (DFT). Using DFT we do not find a significant signal in the frequency range 0 d⁻¹ to 50 d⁻¹ in the light curves of any of these systems. For HAT-P-44 we exclude signals with amplitudes above 1.2 mmag, for HAT-P-45 we exclude signals with amplitudes above 1.1 mmag, and for HAT-P-46 we exclude signals with amplitudes above 0.6 mmag. Similarly we do not detect additional transit signals in the light curves of HAT-P-44 or HAT-P-45. For HAT-P-46 we do detect a marginally significant transit signal with a short period of P = 0.388 d, a depth of 2.3 mmag, and a S/N in the BLS spectrum of 8.5. The period is neither a harmonic nor an alias of the primary transit signal. Based on our prior experience following up similar signals detected in HATNet light curves we consider this likely to be a false alarm, but mention it here for full disclosure.

High-resolution, low-S/N "reconnaissance" spectra were obtained for HAT-P-44, HAT-P-45, and HAT-P-46 using the Tillinghast Reflector Echelle Spectrograph (TRES; Fűresz 2008) on the 1.5 m Tillinghast Reflector at FLWO. Medium-resolution reconnaissance spectra were also obtained for HAT-P-45 and HAT-P-46 using the Wide Field Spectrograph (WiFeS) on the ANU 2.3 m telescope at Siding Spring Observatory. The reconnaissance spectroscopic observations and results for each system are summarized in Table 2. The TRES observations were reduced and analyzed following the procedure described by Quinn et al. (2012) and Buchhave et al. (2010), yielding RVs with a precision of ∼50 m s⁻¹, and an absolute velocity zero-point accuracy of ∼100 m s⁻¹. The WiFeS observations were reduced and analyzed as described in Bayliss et al. (2013), providing RVs with a precision of 2.8 km s⁻¹.

Based on the observations summarized in Table 2 we find that all three systems have rms residuals consistent with no significant RV variation within the precision of the measurements (the WiFeS observations of HAT-P-46 have an rms of 3.3 km s⁻¹ which is only slightly above the precision determined from observations of RV stable stars). All spectra were single-lined, i.e., there is no evidence that any of these targets consist of more than one star. The gravities for all of the stars indicate that they are dwarfs.

We obtained high-resolution, high-S/N spectra of each of these objects using HIRES (Vogt et al. 1994) on the Keck-I telescope in Hawaii. The data were reduced to radial velocities in the barycentric frame following the procedure described by Butler et al. (1996). The RV measurements and uncertainties are given in Tables 3–5 for HAT-P-44 through HAT-P-46, respectively. The period-folded data, along with our best fit described below in Section 3, are displayed in Figures 2–4.

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We also show the chromospheric activity S index. The S index for each star was computed following Isaacson & Fischer (2010) and converted to $\log R^{\prime }_{\rm HK}$ following Noyes et al. (1984). We find median values of $\log R^{\prime }_{\rm HK} = -5.247\pm 0.058\pm 0.10$, −5.394 ± 0.072 ± 0.25, and −5.257 ± 0.036 ± 0.21 for HAT-P-44 through HAT-P-46, respectively. The listed uncertainties are the standard errors on the median given the scatter in the individual measurements, followed by the estimated systematic uncertainties assuming an 11% uncertainty in the calibration for S (Isaacson & Fischer 2010). Taken at face value these imply that all three stars are chromospherically quiet. However, since HAT-P-45 is hotter than the T_eff = 6200 K upper limit over which the Noyes et al. (1984) relation is calibrated, its value should be treated with skepticism. Similarly for HAT-P-46, which has a temperature just below this limit. For HAT-P-44 the low activity index is consistent with the slow projected rotation velocity of vsin i = 0.2 ± 0.5 km s⁻¹ (see Section 3.1).

Additionally we show the spectral line bisector spans. The bisector spans were computed as in Torres et al. (2007) and Bakos et al. (2007) and show no detectable variation in phase with the RVs, allowing us to rule out various blend scenarios as possible explanations of the observations (see Section 3.2). For HAT-P-45 the bisector spans may be correlated with the residual RVs which may indicate that the jitter for this object is due to stellar activity (see Section 3).

Additional photometric observations of each of the transiting planet systems were obtained using the following facilities: the KeplerCam CCD camera on the FLWO 1.2 m telescope, the CCD imager on the 0.8 m remotely operated Byrne Observatory at Sedgwick (BOS) reserve in California, and the Spectral Instrument CCD on the 2.0 m Faulkes Telescope North (FTN) at Haleakala Observatory in Hawaii. Both BOS and FTN are operated by the Las Cumbres Observatory Global Telescope (LCOGT; Brown et al. 2013). The observations for each target are summarized in Table 1.

The reduction of the KeplerCam images was performed using the aperture photometry procedure described by Bakos et al. (2010). The BOS and FTN observations were reduced in a similar manner. The resulting differential light curves were further filtered using the External Parameter Decorrelation (EPD) and Trend Filtering Algorithm (TFA)²⁰ methods applied simultaneously with light curve modeling so that uncertainties in the noise filtering process contribute to the uncertainties on the physical parameters (for more details, see Bakos et al. 2010). The light curves, and best-fit models, are shown in Figures 5–7 for HAT-P-44 through HAT-P-46, respectively; the individual measurements are reported in Tables 6–8.

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Stellar atmospheric parameters for each star were measured using the Spectroscopy Made Easy (SME; Valenti & Piskunov 1996) and the Valenti & Fischer (2005) atomic line database. We analyzed the Keck/HIRES template spectra for each star, which yielded the following initial values and uncertainties.

• 1.  
  HAT-P-44—effective temperature T_eff⋆ = 5295 ± 100 K, metallicity [Fe/H] = 0.33  ±  0.1 dex, stellar surface gravity log g_⋆ = 4.42  ±  0.1 (cgs), and projected rotational velocity vsin i = 0.2  ±  0.5 km s⁻¹.
• 2.  
  HAT-P-45—effective temperature T_eff⋆ = 6270  ±  100 K, metallicity [Fe/H] = 0.03  ±  0.1 dex, stellar surface gravity log g_⋆ = 4.26  ±  0.1 (cgs), and projected rotational velocity vsin i = 9.0  ±  0.5 km s⁻¹.
• 3.  
  HAT-P-46—effective temperature T_eff⋆ = 6280  ±  100 K, metallicity [Fe/H] = 0.38  ±  0.1 dex, stellar surface gravity log g_⋆ = 4.38  ±  0.1 (cgs), and projected rotational velocity vsin i = 4.5  ±  0.5 km s⁻¹.

These values were used to determine initial values for the limb-darkening coefficients, which we fix during the light curve modeling (Section 3.4). This modeling, when combined with the Yonsei-Yale (YY) theoretical stellar evolution models (Yi et al. 2001), provides a refined determination of the stellar surface gravity (Sozzetti et al. 2007) which we then fix in a second SME analysis of the spectra yielding our adopted atmospheric parameters. For HAT-P-44 the revised surface gravity is close enough to the initial SME value that we do not conduct a second SME analysis. The final adopted values of T_eff⋆, [Fe/H], and vsin i are listed for each star in Table 10. We compare the measured a/R_⋆ and T_eff⋆ values for each star to the YY model isochrones in Figure 8.

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The values of log g_⋆, as well as of properties inferred from the evolution models (such as the stellar masses and radii) depend on the eccentricity and semi-amplitude of the transiting planet's orbit, which in turn depend on how the RV data are modeled. In modeling these data we varied the number of planets considered for a given system, and whether or not these planets are fixed to circular orbits. Although T_eff⋆, [Fe/H], and vsin i will also depend on the fixed value of log g_⋆ we found generally that log g_⋆ did not change enough between the models that provide a good fit to the data to justify carrying out a separate SME analysis using the log g_⋆ value determined from each model. As we discuss in Section 3.4.2 we tested numerous models; our final adopted values for these model-dependent parameters are presented in that section.

We determine the distance and extinction to each star by comparing the J, H, and K_S magnitudes from the 2MASS Catalogue (Skrutskie et al. 2006), and the V and I_C magnitudes from the TASS Mark IV Catalogue (Droege et al. 2006), to the expected magnitudes from the stellar models. We use the transformations by Carpenter (2001) to convert the 2MASS magnitudes to the photometric system of the models (ESO), and use the Cardelli et al. (1989) extinction law, assuming a total-to-selective extinction ratio of R_V = 3.1, to relate the extinction in each bandpass to the V-band extinction A_V. The resulting A_V and distance measurements are given with the other model-dependent parameters. We find that HAT-P-44 is not significantly affected by extinction, consistent with the Schlegel et al. (1998) dust maps which yield a total extinction of A_V = 0.038 mag along the line of sight to HAT-P-44. HAT-P-45 and HAT-P-46, on the other hand, have low Galactic latitudes (b = 60 and b = 96, respectively), and are significantly affected by extinction. We find A_V = 1.900  ±  0.169 mag and A_V = 0.832  ±  0.145 mag for our preferred models for HAT-P-45 and HAT-P-46, respectively. For comparison, the Schlegel et al. (1998) maps yield a total line of sight extinction of A_V = 5.89 mag and A_V = 3.25 mag for HAT-P-45 and HAT-P-46, respectively, or A_V ∼ 0.8 mag to both sources after applying the distance and excess extinction corrections given by Bonifacio et al. (2000). At these low Galactic latitudes the extinction estimates based on the Schlegel et al. (1998) dust maps are not reliable, so the discrepancy between the dust-map-based and photometry-based A_V estimates for HAT-P-45 is not unexpected. After correcting for extinction the measured and expected photometric color indices are consistent for each star.

To rule out the possibility that any of these objects might be a blended stellar eclipsing binary system we carried out a blend analysis as described in Hartman et al. (2012).

We find that for HAT-P-44 we can exclude most blend models, consisting either of a hierarchical triple star system, or a blend between a background eclipsing binary and a foreground bright star, based on the light curves. Those models that cannot be excluded with at least 5σ confidence would have been detected as obviously double-lined systems, showing many km s⁻¹ RV and BS variations.

For HAT-P-45 and HAT-P-46 the significant reddening (Section 3.1) allows a broader range of blend scenarios to fit the photometric data. For a system like HAT-P-44, where there is no significant reddening and the available calibrated broadband photometry agrees well with the spectroscopically determined temperature, the calibrated photometry places a strong constraint on blend scenarios where the two brightest stars in the blend have different temperatures. For HAT-P-45 and HAT-P-46, on the other hand, such blends can be accommodated by reducing the reddening in the fit. Indeed we find for both HAT-P-45 and HAT-P-46 that the calibrated broadband photometry are fit slightly better by models that incorporate multiple stars (blends) together with reddening, than by a model consisting of only a single reddened star. The difference between these models is small enough, however, that we do not consider this improvement to be significant; such differences may be due to the true extinction law along this line of sight being slightly different from our assumed R_V = 3.1 Cardelli et al. (1989) extinction law.

To better constrain the possible blend scenarios we obtained a partial g band light curve for HAT-P-45 using Keplercam on the night of 2013 May 20. The photometry was reduced as described in Section 2.4 and included in our blend analysis procedure. We show this light curve in Figure 6, though we note that it was not included in the planet parameter determination which was carried out prior to these observations. Even though it is only a partial event, this light curve significantly restricts the range of blends that can explain the photometry for HAT-P-45, excluding scenarios that predict substantially different g and i band transit depths.

Although the broadband photometry permits a wide range of possible blend scenarios, for both HAT-P-45 and HAT-P-46 the nonplanetary blend scenarios which fit the photometric data can be ruled out based on the BS and RV variations. For HAT-P-45 we find that blend scenarios that fit the photometric data (scenarios that cannot be rejected with >5σ confidence) yield several km s⁻¹ BS and RV variations, whereas the actual BS rms is 24 m s⁻¹. Without the g band light curve for HAT-P-45 some of the blend scenarios consistent with the photometry for this system predict BS and RV variations only slightly in excess of what was measured, illustrating the importance of this light curve. For HAT-P-46 the blend scenarios that fit the photometric data would result in BS variations with rms >80 m s⁻¹, much greater than the measured scatter of 14 m s⁻¹.

We conclude that for all three objects the photometric and spectroscopic observations are best explained by transiting planets. We are not, however, able to rule out the possibility that any of these objects is actually a composite stellar system with one component hosting a transiting planet. Given the lack of definite evidence for multiple stars we analyze all of the systems assuming only one star is present in each case. If future observations identify the presence of stellar companions, the planetary masses and radii inferred in this paper will require moderate revision (e.g., Adams et al. 2013).

For each object initial attempts to fit the data as a single planet system following the method described in Section 3.4 yielded an exceptionally high χ² per degree of freedom (80.5, 19.7, and 23.0 for the full RV data of HAT-P-44, HAT-P-45, and HAT-P-46, respectively). Inspection of the RV residuals showed systematic variations (linear or quadratic in time) suggestive of additional components. We therefore continued to collect RV observations with Keck/HIRES for each of the objects. In all three cases the new RVs did not continue to follow the previously identified trends.

Figure 9 shows the harmonic Analysis of Variance (AoV) periodograms (Schwarzenberg-Czerny 1996) of the residual RVs from the best-fit single-planet model for each system.²¹ In each case strong aliasing gives rise to numerous peaks in the periodograms which could potentially phase the residual data; we are thus not able to identify a unique period for the putative outer companions in any of these systems.

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For HAT-P-44 the two highest peaks are at f = 0.0023626 day⁻¹ (P = 423.26 days) and f = 0.0044477 day⁻¹ (P = 224.83 days), with false alarm probabilities of ∼2.5 × 10⁻⁵ and 3.3 × 10⁻⁴, respectively. The periodogram of the residuals of a model consisting of the transiting planet and a planet with P ∼ 220 days (when fitting the data simultaneously for two planets this model provides a slightly better fit than when the outer planet has a period of P = 423 days) yields a peak at P = 17.7 days with a false alarm probability of 0.16. Alias peaks are also seen at P = 17.3 days, P = 18.6 days, P = 11.4 days, P = 11.7 days, P = 13.3 days, and several other values with decreasing significance.

For HAT-P-45 a number of frequencies are detected in the periodogram of the RV residuals from the best-fit single-planet model. These periods are all aliases of each other. The highest peak is at f = 0.065289 day⁻¹ (P = 15.316 days), with a false alarm probability of ∼10⁻². For HAT-P-46 the two highest peaks are at f = 0.012977 day⁻¹ (P = 77.061 days) and f = 0.014708 day⁻¹ (P = 67.988 days), each with false alarm probabilities of ∼10⁻² (or ∼10⁻⁴ if uniform uncertainties are adopted as discussed further below).

The false alarm probabilities given above include a correction for the so-called "bandwidth penalty" (i.e., a correction for the number of independent frequencies that are tested by the periodogram); here we restricted the search to a frequency range of 0.02 day⁻¹ < f < 0.2 day⁻¹ and used the Horne & Baliunas (1986) approximation to estimate the number of independent frequencies tested (the resulting false alarm probability may be inaccurate by as much as a factor of ∼10). Note that adopting a broader frequency range for the periodograms (e.g., up to the Nyquist limit, which for the HAT-P-44 data would be ∼250 day⁻¹) significantly increases the false alarm probabilities. We expect, however, that systems containing multiple Jupiter-mass planets with orbital periods less than 5 days would be dynamically unstable, allowing us to restrict the frequency range to consider on physical grounds.

For HAT-P-44 and HAT-P-45 the false alarm probabilities are approximately the same for high jitter as they are when the jitter is set to 0. For HAT-P-46 the false alarm probabilities are smaller when the errors are dominated by jitter (10⁻⁴ with jitter versus 10⁻² without jitter).

We modeled simultaneously the HATNet photometry, the follow-up photometry, and the high-precision RV measurements using a procedure similar to that described in detail by Bakos et al. (2010) with modifications described by Hartman et al. (2012). For each system we used a Mandel & Agol (2002) transit model, together with the EPD and TFA trend-filters, to describe the follow-up light curves, a Mandel & Agol (2002) transit model for the HATNet light curve(s), and a Keplerian orbit using the formalism of Pál (2009) for the RV curve. A significant change that we have made compared to the analysis conducted in our previous discovery papers was to include the RV jitter as a free parameter in the fit, which we discuss below. We then discuss our methods for distinguishing between competing classes of models used to fit the data, and comment on the orbital stability of potential models.

3.4.1. RV Jitter

It is well known that high-precision RV observations of stars show non-periodic variability in excess of what is expected based on the measurement uncertainties. This "RV jitter" depends on properties of the star including the effective temperature of its photosphere, its chromospheric activity, and the projected equatorial rotation velocity of the star (see Wright 2005; Isaacson & Fischer 2010, who discuss the RV jitter from Keck/HIRES measurements). In most exoplanet studies the typical method for handling this jitter has been to add it in quadrature to the measurement uncertainties, assuming that the jitter is Gaussian white noise. One then either adopts a jitter value that is found to be typical for similar stars, or chooses a jitter such that χ² per degree of freedom is unity for the best-fit model. In our previous discovery papers we adopted the latter approach.

When testing competing models for the RV data the jitter is an important parameter—the greater the jitter the smaller the absolute χ² difference between two models, and the less certain one can be in choosing one over the other. Both of the typical approaches for handling the jitter have shortcomings: the former does not allow for the possibility that a star may have a somewhat higher (or lower) than usual jitter, while the latter ignores any prior information that may be used to disfavor jitter values that would be very unusual. An alternative approach is to treat the jitter as a free parameter in the fit, but use the empirical distribution of jitters as a prior constraint.

The method of allowing the jitter to vary in an MCMC analysis of an RV curve was previously adopted by Gregory (2005). As was noted in that work, when allowing terms which appear in the uncertainties to vary in an MCMC fit, the logarithm of the likelihood is no longer simply ln L = −χ²/2 + C where C is a normalization constant that is independent of the parameters, and can be ignored for most applications. Instead one should use $\ln L = -\chi ^2/2 + \sum _{i=1}^{N}\ln (1/e_{i}) + C$, where e_i is the error for measurement i and in this case is given by $e_{i} = \sqrt{\vphantom{A^A}\smash{{{\sigma _{i}^2 + \sigma _{\rm jitter}^2}}}}$ for formal uncertainty σ_i and jitter σ_jitter. When the uncertainties do not include free parameters, the term $\sum _{i=1}^{N}\ln (1/e_{i})$ is constant, and included in C.

The analysis by Gregory (2005) used an uninformative prior on the jitter, which effectively forces the jitter to the value that results in χ²/dof = 1; here we make use of the empirical jitter distribution found by Wright (2005) to set a prior on the jitter. Wright (2005) provides the distributions for stars in a several bins separated by B − V, activity, and luminosity above the main sequence. The histograms appear to be well-matched by log-normal distributions of the form:

$$\begin{equation} P(\sigma _{\rm jitter})d\sigma _{\rm jitter} = \frac{1}{\sigma _{\rm jitter}\sqrt{2\pi \bar{\sigma }^2}}e^{-\frac{(\ln \sigma _{\rm jitter} - \bar{\mu })^2}{2 \bar{\sigma }^2}}d\sigma _{\rm jitter}. \end{equation} \tag{ 1 }$$

Figure 10 compares this model to the jitter histograms. For HAT-P-44, which falls in the low-activity bin with ΔM_v < 1 and 0.6 < B − V < 1.4, we find $\bar{\sigma } = 0.496$, $\bar{\mu } = 1.251$, with σ_jitter measured in units of m s⁻¹. For HAT-P-45 and HAT-P-46, which fall in the bin of low-activity stars with ΔM_V < 1 and B − V < 0.6, we find $\bar{\sigma } = 0.688$, and $\bar{\mu } = 1.419$.

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The posterior probability density for the parameters θ, given the data D and model M is given by Bayes' relation:

$$\begin{equation} P(\theta | D, M) = \frac{P(\theta | M)P(D | \theta,M)}{P(D | M)} \end{equation} \tag{ 2 }$$

which in our case takes the form:

$$\begin{eqnarray*} P(\theta | D, M) & = & C\exp (\ln L + \ln P(\theta | M)) \\ & = & C\exp (-\chi ^{2}/2 + \sum _{i=1}^{N}\ln (1/e_{i}) + \ln P(\sigma _{\rm jitter})) \end{eqnarray*}$$

where C represents constants that are independent of θ (note we adopt uniform priors on all jump parameters other than σ_jitter). We use a differential evolution MCMC procedure (ter Braak 2006; Eastman et al. 2013) to explore this distribution.

3.4.2. Model Selection

As discussed in the previous subsection, modeling these objects as single-planet systems yields RV residuals with large scatter and evidence of long-term variations. We also attempted fitting the data for HAT-P-44 and HAT-P-46 including linear or quadratic trends in addition to the single-planet systems, but found that these models do not provide good fits to the data. We therefore performed the analysis of each system including additional Keplerian components in their RV models.

We use the Bayes Factor to select between these competing models; here we describe how this is computed. The Bayesian evidence Z is defined by

$$\begin{equation} Z = P(D|M) = \int d\theta P(\theta |M)P(D|\theta,M) \end{equation} \tag{ 3 }$$

where P(D|M) is the probability of observing the data D given the model M, marginalized over the model parameters θ. The Bayes factor K_{1, 2} comparing the posterior probabilities for models M₁ and M₂ given the data D is defined by

$$\begin{equation} K_{1,2} = \frac{P(M_{1}|D)}{P(M_{2}|D)} = \frac{P(M_1)P(D|M_{1})}{P(M_2)P(D|M_{2})} \end{equation} \tag{ 4 }$$

where P(M) is the prior probability for model M. Assuming equal priors for the different models tested, the Bayes factor is then equal to the evidence ratio:

$$\begin{equation} K_{1,2} = \frac{Z_{1}}{Z_{2}}. \end{equation} \tag{ 5 }$$

If K_{1, 2} > 1 then model M₁ is favored over model M₂.

In practice Z is difficult to determine as it requires integrating a complicated function over a high-dimensional space (e.g., Feroz et al. 2009). Recently, however, Weinberg et al. (2013) have suggested a simple and relatively accurate method for estimating Z directly from the results of an MCMC simulation. Their method involves using the MCMC results to identify a small region of parameter space with high posterior probability, numerically integrating over this region, and applying a correction to scale the integral from the subregion to the full parameter space. The correction is determined from the posterior parameter distribution estimated as well from the MCMC. We use this method to estimate Z and K for each model. However, for practical reasons we use the MCMC sample itself to conduct a Monte Carlo integration of the parameter subregion, rather than following the suggested method of using a uniform resampling of the subregion. As shown by Weinberg et al. (2013) the method that we follow provides a somewhat biased estimate of Z, with errors in ln Z ≲ 0.5. For this reason we do not consider ln Z differences between models that are <1 to be significant.

In Table 9 we list the models fit for each system, and provide estimates of the Bayes Factors for each model relative to a fiducial model of a single transiting planet on an eccentric orbit. For reference we also provide the Bayesian Information Criterion (BIC) estimator for each model, which is given by

$$\begin{equation} {\rm BIC} = -2\ln L_{\rm max} + N_{\rm p} \ln N_{\rm d} \end{equation} \tag{ 6 }$$

for a model with N_p free parameters fit to N_d data points yielding a maximum likelihood of L_max. The BIC is determined solely from the highest likelihood value, making it easier to calculate than K. Models with lower BIC values are generally favored. Note, however, that the BIC is a less accurate method for distinguishing between models than is K. We also provide, for reference, the Bayes Factors determined when the jitter of each system is fixed to a typical value throughout the analysis.

Table 9. Bayes Factor and BIC Differences for Models Tested

Modela	Pcb	ecc	Pd	ed	Trend	Npd	ln (K)e	Fixed Jitter	ΔBICg
Number					Order			ln (K)f
HAT-P-44
2	220	...	...	...	...	7	10.4	90.7	46.9
3	438	...	...	...	...	7	10.2	89.7	45.2
4	221	$\checkmark$	...	...	...	9	8.9	90.6	46.3
5	219	...	17	...	...	10	3.1	86.0	57.7
6	219	$\checkmark$	17	...	...	12	−22.5	82.9	54.9
7	216	$\checkmark$	18	$\checkmark$	...	14	2.5	82.0	59.2
8	...	...	...	...	1	6	−2.8	9.5	0.9
9	...	...	...	...	2	7	3.2	76.1	32.3
10	1981	...	...	...	...	7	5.4	94.7	32.2
11	872	$\checkmark$	...	...	...	9	12.4	95.2	54.5
HAT-P-45
2	15.3	...	...	...	...	7	−3.0	−10.1	29.2
3	...	...	...	...	2	6	−10.7	−10.6	−1.0
HAT-P-46
2	78	...	...	...	...	7	2.8	55.3	38.8
3	78	$\checkmark$	...	...	...	9	−4.2	51.3	35.9
4	78	...	8.1	...	...	10	−7.5	48.1	46.2
5	...	...	...	...	1	6	−4.7	−12.7	−2.0
6	...	...	...	...	2	7	−12.8	−15.2	−3.4

Notes. ^aThe number associated with this model in Tables 11–14. Model 1 for each system is the fiducial model of a single planet on an eccentric orbit. By definition this model has ln (K) = 0 and ΔBIC = 0. ^bThe orbital period used for component c or d in days. Models for which the period of a component is listed as "⋅⋅⋅" did not include that planet. We list here the median value from the posterior distribution for this parameter. For HAT-P-44 model 10, this value is substantially different from the value of P = 872 d used to initialize the fit. ^cFlag indicating whether or not the component is allowed to be eccentric (indicated by a $\checkmark$), or if the eccentricity was fixed to 0 (indicated by "⋅⋅⋅"). ^dNumber of varied parameters constrained by the RV observations, including four parameters for the inner transiting planet and one parameter for the jitter. Although the two parameters used to describe the ephemeris of the inner planet are varied in the joint fit of the RV and photometric data, they are almost entirely determined by the photometric data alone, so we do not include them in this accounting. ^eThe natural logarithm of the Bayes Factor between the given model, and a fiducial model of a single planet on an eccentric orbit. Models with higher values of ln K are preferred. In this case the RV jitter is allowed to vary in the fit, subject to a prior constraint from the empirical jitter distribution found by Wright (2005). ^fThe natural logarithm of the Bayes Factor between the given model, and a fiducial model of a single planet on an eccentric orbit. In this case the RV jitter is fixed to a typical value for each star (these were determined such that χ² per degree of freedom was unity for one of the models; we adopted 9.1 m s⁻¹ for HAT-P-44, 12.7 m s⁻¹ for HAT-P-45, and 2.7 m s⁻¹ for HAT-P-46). We provide these to show how the model selection depends on the method for treating the RV jitter. ^gΔBIC = BIC_fiducial − BIC_model, i.e., the difference between the BIC for the fiducial model and for the given model. Models with higher values of ΔBIC are preferred.

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The planet and stellar parameters for each system that are independent of the models that we test are listed in Table 10. Stellar parameters for HAT-P-44, and parameters of the transiting planet HAT-P-44b, that depend on the class of model tested are listed in Table 11, while parameters for the candidate outer component HAT-P-44c are listed in Table 12. In both cases we only show results for the fiducial model of a single transiting planet and for those models that have |ln K − ln K₁₁| < 5, where K₁₁ is the Bayes factor for model 11, which has the highest evidence. Stellar parameters for HAT-P-45 and HAT-P-46, and parameters for the transiting planets HAT-P-45b and HAT-P-46b that depend on the class of model tested are listed in Table 13, while parameters for the candidate outer component HAT-P-45c and HAT-P-46c are listed in Table 14. For both systems we only show parameters for models 1 and 2, which have the highest Bayesian evidences.

Table 10. Model-independent Stellar and Light Curve Parameters for HAT-P-44–HAT-P-46

Parameter	HAT-P-44b	HAT-P-45b	HAT-P-46b	Sourcea
	Value	Value	Value
Stellar astrometric properties
GSC ID	GSC 3465-00123	GSC 5102-00262	GSC 5100-00045
2MASS ID	2MASS 14123457+4700528	2MASS 18172957-0322517	2MASS 18014660-0258154
R.A. (J2000)	14h12m3456	18h17m2940	18h01m4656
Decl. (J2000)	+47°00'529	−03°22'517	−02°58'154
μR.A. (mas yr−1)	−29.0 ± 12.6	9.2 ± 4.1	−7.5 ± 4.3
$\mu _{\rm Dec{\rm l}.}$ (mas yr−1)	12.3 ± 11.4	−3.1 ± 8.2	2.0 ± 7.9
Stellar spectroscopic properties
Teff⋆ (K)	5295 ± 100	6330 ± 100	6120 ± 100	SMEb
[Fe/H]	0.33 ± 0.10	0.07 ± 0.10	0.30 ± 0.10	SME
vsin i (km s−1)	0.2 ± 0.5	9.3 ± 0.5	4.9 ± 0.5	SME
vmac (km s−1)	3.28	4.88	4.55	SME
vmic (km s−1)	0.85	0.85	0.85	SME
γRV (km s−1)	−33.45 ± 0.05	23.903 ± 0.1	−20.911 ± 0.1	TRES
$\log R^{\prime }_{\rm HK}$	−5.247 ± 0.058 ± 0.10	−5.394 ± 0.072 ± 0.25	−5.257 ± 0.036 ± 0.21	Keck/HIRESc
Stellar photometric properties
V (mag)	13.21 ± 0.12	12.79 ± 0.18	11.94 ± 0.12	TASS
V − IC (mag)	0.90 ± 0.24	1.11 ± 0.19	0.87 ± 0.14	TASS
J (mag)	11.729 ± 0.021	10.730 ± 0.027	10.330 ± 0.024	2MASS
H (mag)	11.360 ± 0.019	10.350 ± 0.026	9.972 ± 0.022	2MASS
Ks (mag)	11.275 ± 0.018	10.201 ± 0.023	9.924 ± 0.023	2MASS
Transiting planet light curve parameters
P (days)	4.301219 ± 0.000019	3.128992 ± 0.000021	4.463129 ± 0.000048
Tc (BJD)d	2455696.93695 ± 0.00024	2455729.98612 ± 0.00041	2455701.33646 ± 0.00047
T14 (days)d	0.1302 ± 0.0008	0.1436 ± 0.0013	0.1291 ± 0.0018
T12 = T34 (days)d	0.0158 ± 0.0006	0.0154 ± 0.0011	0.0174 ± 0.0017
ζ/R⋆	17.49 ± 0.07	15.60 ± 0.08	17.83 ± 0.13
Rp/R⋆	0.1343 ± 0.0010	0.1110 ± 0.0021	0.0942 ± 0.0017
b2	$0.030_{-0.018}^{+0.037}$	$0.079_{-0.043}^{+0.058}$	$0.392_{-0.059}^{+0.047}$
b ≡ acos i/R⋆	$0.172_{-0.074}^{+0.079}$	$0.281_{-0.106}^{+0.085}$	$0.626_{-0.051}^{+0.036}$
Assumed limb-darkening coefficientse
c1, i (linear term)	0.3648	0.1935	0.2222	Claret (2004)
c2, i (quadratic term)	0.2817	0.3680	0.3651	Claret (2004)

Notes. ^aWe list the source only for the stellar properties. The listed transiting planet light curve parameters are determined from our joint fit of the RV and light curve data, but are primarily constrained by the light curves. ^bSME = "Spectroscopy Made Easy" package for the analysis of high-resolution spectra (Valenti & Piskunov 1996). These parameters rely primarily on SME, but have a small dependence also on the iterative analysis incorporating the isochrone search and global modeling of the data, as described in the text. ^cMedian values of $\log R^{\prime }_{\rm HK}$ (Noyes et al. 1984) are computed from the Keck/HIRES spectra following the procedure of Isaacson & Fischer (2010). Uncertainties are the standard error on the median given the scatter in the individual measurements, followed by our estimate of the systematic uncertainty assuming an uncertainty of 11% in the calibration of S (Isaacson & Fischer 2010). Note that this index has not been calibrated for stars with T_eff⋆ ≳ 6200 K so the results for HAT-P-45 and HAT-P-46, which are above and just below this threshold, should be treated with caution. ^dT_c: reference epoch of mid transit that minimizes the correlation with the orbital period. T₁₄: total transit duration, time between first to last contact; T₁₂ = T₃₄: ingress/egress time, time between first and second, or third and fourth contact. Barycentric Julian dates (BJD) throughout the paper are calculated from Coordinated Universal Time (UTC). ^eValues for a quadratic law.

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Table 11. Model-dependent System Parameters for HAT-P-44

Parameter	Model 1	Model 2	Model 3	Model 4	Adopted
					Model 11
	Value	Value	Value	Value	Value
Transiting planet (HAT-P-44b) light curve parameters
a/R⋆	9.34 ± 1.70	$11.11_{-1.03}^{+0.70}$	10.13 ± 1.08	$11.37_{-0.97}^{+0.54}$	$11.49_{-0.85}^{+0.46}$
i (deg)	$88.7_{-1.6}^{+0.6}$	89.0 ± 0.5	$88.8_{-0.8}^{+0.5}$	89.1 ± 0.5	89.1 ± 0.4
Transiting planet (HAT-P-44b) RV parameters
K (m s−1)	47.1 ± 10.7	51.5 ± 3.9	52.5 ± 4.7	48.1 ± 4.1	45.9 ± 3.6
$\sqrt{e} \cos \omega$	$-0.206_{-0.189}^{+0.257}$	−0.006 ± 0.121	−0.113 ± 0.113	−0.046 ± 0.125	−0.023 ± 0.121
$\sqrt{e} \sin \omega$	$0.421_{-0.312}^{+0.172}$	$0.219_{-0.236}^{+0.151}$	$0.364_{-0.231}^{+0.122}$	0.147 ± 0.185	0.114 ± 0.175
ecos ω	−0.099 ± 0.129	−0.001 ± 0.037	−0.041 ± 0.047	$-0.008_{-0.042}^{+0.030}$	−0.003 ± 0.032
esin ω	0.214 ± 0.170	$0.055_{-0.059}^{+0.098}$	0.144 ± 0.103	$0.029_{-0.042}^{+0.092}$	$0.019_{-0.036}^{+0.080}$
e	0.272 ± 0.155	0.072 ± 0.071	0.158 ± 0.098	0.054 ± 0.062	0.044 ± 0.052
ω (deg)	117 ± 50	98 ± 84	108 ± 44	114 ± 82	113 ± 90
RV jitter (m s−1)	31.0 ± 4.2	12.7 ± 2.5	13.4 ± 2.3	11.5 ± 2.1	10.7 ± 2.0
Derived transiting planet (HAT-P-44b) parameters
Mp (MJ)	0.347 ± 0.077	0.392 ± 0.031	0.394 ± 0.036	0.368 ± 0.033	0.352 ± 0.029
Rp (RJ)	$1.523_{-0.226}^{+0.442}$	$1.280_{-0.074}^{+0.145}$	$1.403_{-0.130}^{+0.190}$	$1.256_{-0.059}^{+0.126}$	$1.242_{-0.051}^{+0.106}$
C(Mp, Rp)a	0.18	0.10	0.11	0.16	0.06
ρp (g cm−3)	$0.12_{-0.05}^{+0.09}$	0.23 ± 0.05	$0.18_{-0.05}^{+0.07}$	0.23 ± 0.05	0.23 ± 0.04
log gp (cgs)	2.55 ± 0.19	$2.77_{-0.09}^{+0.06}$	2.69 ± 0.10	$2.75_{-0.08}^{+0.06}$	$2.75_{-0.07}^{+0.05}$
a (AU)	$0.0509_{-0.0008}^{+0.0014}$	0.0507 ± 0.0007	0.0507 ± 0.0008	0.0507 ± 0.0007	0.0507 ± 0.0007
Teq (K)	$1238_{-107}^{+173}$	$1126_{-42}^{+67}$	$1181_{-64}^{+84}$	$1114_{-36}^{+59}$	$1108_{-32}^{+51}$
Θb	0.024 ± 0.007	0.033 ± 0.003	0.030 ± 0.004	0.031 ± 0.003	0.030 ± 0.003
〈F〉 c	$5.30_{-1.54}^{+4.25}$	$3.63_{-0.50}^{+1.02}$	$4.40_{-0.85}^{+1.49}$	$3.48_{-0.42}^{+0.86}$	$3.40_{-0.37}^{+0.72}$
Derived stellar properties
M⋆ (M☉)	$0.953_{-0.045}^{+0.083}$	0.939 ± 0.041	0.938 ± 0.042	0.941 ± 0.041	0.942 ± 0.041
R⋆ (R☉)	$1.165_{-0.173}^{+0.334}$	$0.979_{-0.055}^{+0.110}$	$1.072_{-0.099}^{+0.144}$	$0.960_{-0.043}^{+0.096}$	$0.949_{-0.037}^{+0.080}$
log g⋆ (cgs)	4.28 ± 0.16	4.43 ± 0.07	4.35 ± 0.09	4.45 ± 0.06	4.46 ± 0.06
L⋆ (L☉)	$0.96_{-0.27}^{+0.71}$	$0.68_{-0.10}^{+0.19}$	$0.81_{-0.16}^{+0.27}$	$0.66_{-0.09}^{+0.16}$	$0.64_{-0.08}^{+0.14}$
MV (mag)	4.97 ± 0.46	5.34 ± 0.23	5.15 ± 0.28	5.38 ± 0.21	5.41 ± 0.19
MK (mag,ESO)	3.06 ± 0.44	3.44 ± 0.19	3.24 ± 0.24	3.48 ± 0.16	3.50 ± 0.14
Age (Gyr)	$11.5_{-4.4}^{+3.3}$	8.9 ± 3.9	11.5 ± 3.8	8.1 ± 3.8	7.5 ± 3.6
AV (mag)	0.000 ± 0.083	0.000 ± 0.081	0.000 ± 0.083	0.000 ± 0.080	0.000 ± 0.080
Distance (pc)	$445_{-66}^{+127}$	$374_{-23}^{+42}$	$409_{-39}^{+55}$	$367_{-19}^{+37}$	$363_{-17}^{+31}$

Notes. ^aCorrelation coefficient between the planetary mass M_p and radius R_p. ^bThe Safronov number is given by Θ = 1/2(V_esc/V_orb)² = (a/R_p)(M_p/M_⋆) (see Hansen & Barman 2007). ^cIncoming flux per unit surface area, averaged over the orbit, measured in units of 10⁸ erg s⁻¹ cm⁻².

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Table 12. Model-dependent Parameters for Outer Planet in HAT-P-44

Parameter	Model 1	Model 2	Model 3	Model 4	Adopted
					Model 11
	Value	Value	Value	Value	Value
RV and derived parameters for candidate planet HAT-P-44c
Pc (days)	...	219.9 ± 4.5	437.5 ± 17.7	221.4 ± 4.4	872.2 ± 1.7
Tc, c (BJD)a	...	2455686.2 ± 4.1	2455727.0 ± 4.5	2455713.4 ± 4.3	2455711.2 ± 13.4
T14, c (days)a	...	0.505 ± 0.039	0.691 ± 0.067	0.652 ± 0.091	0.503 ± 0.047
Kc (m s−1)	...	56 ± 6	104 ± 12	93 ± 4	99 ± 33
$\sqrt{e} \cos \omega _{c}$	...	0	0	−0.373 ± 0.105	−0.453 ± 0.152
$\sqrt{e} \sin \omega _{c}$	...	0	0	−0.474 ± 0.113	0.510 ± 0.103
ecos ωc	...	0	0	−0.226 ± 0.071	−0.313 ± 0.124
esin ωc	...	0	0	$-0.286_{-0.103}^{+0.079}$	0.349 ± 0.078
ec	...	0	0	0.379 ± 0.075	0.494 ± 0.081
ωc (deg)	...	0	0	232.1 ± 13.3	131.3 ± 14.4
Mpsin ic (MJ)	...	1.6 ± 0.2	3.7 ± 0.5	2.5 ± 0.1	$4.0_{-0.8}^{+1.4}$
ac (AU)	...	0.699 ± 0.014	$1.104_{-0.027}^{+0.039}$	0.702 ± 0.014	1.752 ± 0.025

Note. ^aT_c: reference epoch of mid transit that minimizes the correlation with the orbital period. T₁₄: total transit duration, time between first to last contact; Barycentric Julian dates (BJD) throughout the paper are calculated from Coordinated Universal Time (UTC).

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Table 13. Model-dependent System Parameters for HAT-P-45 and HAT-P-46

Parameter	HAT-P-45		HAT-P-46
			Adopted	Adopted
	Model 1	Model 2	Model 1	Model 2
	Value	Value	Value	Value
Transiting planet (HAT-P-45b and HAT-P-46b) light curve parameters
a/R⋆	$7.36_{-0.62}^{+0.39}$	7.56 ± 0.31	$8.84_{-1.76}^{+1.04}$	$8.86_{-1.24}^{+0.89}$
i (deg)	87.8 ± 0.9	87.9 ± 0.8	$85.4_{-4.1}^{+1.0}$	$85.5_{-2.3}^{+0.8}$
Transiting planet (HAT-P-45b and HAT-P-46b) RV parameters
K (m s−1)	106.6 ± 13.6	106.9 ± 4.5	40.9 ± 19.7	52.2 ± 6.8
$\sqrt{e} \cos \omega$	−0.009 ± 0.160	−0.031 ± 0.086	$0.180_{-0.317}^{+0.239}$	$0.143_{-0.134}^{+0.098}$
$\sqrt{e} \sin \omega$	0.045 ± 0.192	−0.084 ± 0.131	0.242 ± 0.269	$0.305_{-0.258}^{+0.170}$
ecos ω	$-0.001_{-0.045}^{+0.061}$	−0.004 ± 0.017	$0.064_{-0.140}^{+0.183}$	$0.047_{-0.042}^{+0.066}$
esin ω	$0.006_{-0.045}^{+0.089}$	$-0.010_{-0.039}^{+0.024}$	$0.091_{-0.108}^{+0.217}$	$0.105_{-0.089}^{+0.149}$
e	0.049 ± 0.063	0.025 ± 0.026	0.195 ± 0.167	0.123 ± 0.120
ω (deg)	146 ± 98	239 ± 87	82 ± 113	70 ± 87
RV jitter (m s−1)	22.5 ± 6.3	4.7 ± 3.5	28.4 ± 6.7	6.6 ± 3.0
Derived transiting planet (HAT-P-45b and HAT-P-46b) parameters
Mp (MJ)	$0.892_{-0.099}^{+0.137}$	0.890 ± 0.046	$0.383_{-0.127}^{+0.218}$	$0.493_{-0.052}^{+0.082}$
Rp (RJ)	$1.426_{-0.087}^{+0.175}$	1.382 ± 0.076	$1.286_{-0.150}^{+0.426}$	$1.284_{-0.133}^{+0.271}$
C(Mp, Rp)a	0.45	0.40	0.43	0.72
ρp (g cm−3)	0.38 ± 0.09	0.42 ± 0.06	$0.20_{-0.09}^{+0.14}$	0.28 ± 0.10
log gp (cgs)	3.03 ± 0.07	3.06 ± 0.04	$2.72_{-0.28}^{+0.16}$	2.86 ± 0.10
a (AU)	0.0452 ± 0.0007	0.0451 ± 0.0006	$0.0577_{-0.0010}^{+0.0020}$	$0.0577_{-0.0009}^{+0.0014}$
Teq (K)	$1652_{-52}^{+90}$	1627 ± 44	$1465_{-89}^{+220}$	$1458_{-75}^{+140}$
Θb	0.044 ± 0.006	0.046 ± 0.003	0.025 ± 0.009	0.034 ± 0.004
〈F〉 c	$1.68_{-0.20}^{+0.44}$	1.58 ± 0.17	$1.04_{-0.22}^{+0.89}$	$1.02_{-0.19}^{+0.50}$
Derived stellar properties
M⋆ (M☉)	1.259 ± 0.058	1.246 ± 0.050	$1.289_{-0.068}^{+0.144}$	$1.284_{-0.060}^{+0.095}$
R⋆ (R☉)	$1.319_{-0.072}^{+0.155}$	1.281 ± 0.060	$1.398_{-0.156}^{+0.465}$	$1.396_{-0.136}^{+0.293}$
log g⋆ (cgs)	4.30 ± 0.06	4.32 ± 0.03	4.25 ± 0.15	4.25 ± 0.11
L⋆ (L☉)	$2.51_{-0.33}^{+0.71}$	2.35 ± 0.30	$2.48_{-0.57}^{+2.08}$	$2.46_{-0.49}^{+1.25}$
MV (mag)	3.75 ± 0.21	3.83 ± 0.15	3.78 ± 0.47	3.79 ± 0.34
MK (mag,ESO)	2.58 ± 0.18	2.65 ± 0.11	2.47 ± 0.46	2.48 ± 0.32
Age (Gyr)	2.0 ± 0.8	1.7 ± 0.8	$2.4_{-1.0}^{+0.7}$	$2.5_{-1.0}^{+0.7}$
AV (mag)	1.900 ± 0.169	1.895 ± 0.167	0.831 ± 0.145	0.832 ± 0.145
Distance (pc)	$305_{-17}^{+35}$	296 ± 14	$296_{-33}^{+98}$	$296_{-29}^{+61}$

Notes. ^aCorrelation coefficient between the planetary mass M_p and radius R_p. ^bThe Safronov number is given by Θ = 1/2(V_esc/V_orb)² = (a/R_p)(M_p/M_⋆) (see Hansen & Barman 2007). ^cIncoming flux per unit surface area, averaged over the orbit, measured in units of 10⁸ erg s⁻¹ cm⁻².

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Table 14. Model-dependent Parameters for Outer Planets in HAT-P-45 and HAT-P-46

Parameter	HAT-P-45		HAT-P-46
			Adopted	Adopted
	Model 1	Model 2	Model 1	Model 2
	Value	Value	Value	Value
RV and derived parameters for candidate planets HAT-P-45c, HAT-P-46c
Pc (days)	...	15.3 ± 0.1	...	77.7 ± 0.6
Tc, c (BJD)a	...	2455700.0 ± 0.3	...	2455695.6 ± 2.0
T14, c (days)a	...	0.242 ± 0.009	...	0.446 ± 0.058
Kc (m s−1)	...	36 ± 5	...	81 ± 9
$\sqrt{e} \cos \omega _{c}$	...	0	...	0
$\sqrt{e} \sin \omega _{c}$	...	0	...	0
ecos ωc	...	0	...	0
esin ωc	...	0	...	0
ec	...	0	...	0
ωc (deg)	...	0	...	0
Mpsin ic (MJ)	...	0.5 ± 0.1	...	2.0 ± 0.3
ac (AU)	...	0.130 ± 0.002	...	$0.387_{-0.007}^{+0.010}$

Notes. ^aT_c: reference epoch of mid transit that minimizes the correlation with the orbital period. T₁₄: total transit duration, time between first to last contact; Barycentric Julian dates (BJD) throughout the paper are calculated from Coordinated Universal Time (UTC).

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For HAT-P-44 we find that the preferred model, based on the estimated Bayes Factor, consists of two planets, the outer one on an eccentric orbit. This model, labeled number 11 in Tables 9, 11, and 12, includes: the transiting planet HAT-P-44b with a period of P = 4.301219  ±  0.000019 days, a mass of M_p = 0.352  ±  0.029 M_J, and an eccentricity of e = 0.044  ±  0.052; an outer planet HAT-P-44c with a period of P = 872.2  ±  1.7 days, a minimum mass of $M_{p}\sin i = 4.0_{-0.8}^{+1.4}$ M_J, and an eccentricity of e = 0.494  ±  0.081. We find that we are not able to definitively choose this model over alternative models, labeled numbers 2, 3, and 4, which are similar in form to the preferred model, but with different periods for the outer planet (P = 220 days, or P = 438 days). Note that due to the sharpness of the peaks in the likelihood as a function of the period of the outer planet, an MCMC simulation takes an excessively long time to transition between the different periods. For this reason we treat these as independent models. We adopt the model with the long period and high eccentricity for the outer component because it has the highest Bayes factor. This model is favored over the fiducial model of a single planet transiting the host star by a factor of ∼2 × 10⁵, indicating that the data strongly favor the two-planet model over the single-planet model. The preferred model has an associated jitter of 10.7  ±  2.0 m s⁻¹ and a χ² per degree of freedom, including this jitter, of 1.67. Based on Equation (1), one expects only 1.2% of stars like HAT-P-44 to have jitter values ⩾10.7 m s⁻¹ thus the excess scatter in the RV residuals from the best-fit two-planet model suggests that perhaps more than two planets are present in this system, though we cannot conclusively detect any additional planets from the data currently available.

For HAT-P-45 the fiducial model of a single planet on an eccentric orbit is preferred over the other models that we tested. This model, labeled number 1 under the HAT-P-45 headings in Tables 9, 13, and 14, includes only the transiting planet HAT-P-45b with a period of P = 3.128992  ±  0.000021 days, a mass of $M_{p}= 0.892_{-0.099}^{+0.137}$ M_J, and an eccentricity of e = 0.049  ±  0.063. The preferred model has a jitter of 22.5  ±  6.3 m s⁻¹ and χ² per degree of freedom of 2.1. Only ∼0.8% of stars like HAT-P-45 are expected to have a jitter this high. Moreover, the RV residuals from the preferred best-fit model appear to show a variation that is correlated in time (see the third panel down in Figure 3). Both these factors may suggest that a second planet is in the HAT-P-45 system. Nonetheless the data do not at present support such a complicated model. The single-planet model has a Bayes factor of ∼20 relative to the two-planet model, indicating a slight preference for the single-planet model. We also note that the BSs and RV residuals from the single-planet model may be correlated. A Spearman rank-order correlation test (e.g., Press et al. 1992) yields a correlation coefficient of r_s = 0.54 which indicates a weak correlation with a 10% false alarm probability. A correlation between these quantities suggests that at least some of the excess RV scatter in this system may due to stellar activity.

For HAT-P-46 the preferred model consists of a transiting planet together with an outer companion on a circular orbit. This model, labeled number 2 under the HAT-P-46 headings in Tables 9, 13, and 14, includes: the transiting planet HAT-P-46b with a period of P = 4.463129  ±  0.000048 days, a mass of $M_{p}= 0.493_{-0.052}^{+0.082}$ M_J, and an eccentricity of e = 0.123  ±  0.120; and an outer planet HAT-P-46c with a period of P = 77.7  ±  0.6 days, and a minimum mass of M_psin i = 2.0  ±  0.3 M_J. Although the two-planet model is preferred, it has a Bayes factor of only K ∼ 16 relative to the fiducial single-planet model, indicating that the preference is not very strong. The preferred model has a jitter of 6.6  ±  3.0 m s⁻¹ and χ² per degree of freedom of 1.6. The resulting jitter is typical for a star like HAT-P-46 (∼25% of such stars have a jitter higher than 6.6  ±  3.0 m s⁻¹), so there is no compelling reason at present to suspect that there may be more planets in this system beyond HAT-P-46c.

For both HAT-P-44 and HAT-P-46 allowing the jitter to vary in the fit substantially reduces the significance of the multi-planet solutions relative to the single planet solution. If we had not allowed the jitter to vary, we would have concluded that the two-planet model is ∼10³⁹ times more likely than the one-planet model for HAT-P-44, and ∼10²⁴ times more likely for HAT-P-46. For HAT-P-45, it is interesting to note that allowing the jitter to vary actually increases the significance of the two-planet model, perhaps due to the relatively high jitter value that must be adopted to achieve χ²/dof = 1.

3.5.1. Orbital Stability