HATP44b, HATP45b, and HATP46b:
Three Transiting Hot Jupiters in Possible MultiPlanet Systems^{1}
Abstract
We report the discovery by the HATNet survey of three new transiting extrasolar planets orbiting moderately bright (V=, and ) stars. The planets have orbital periods of , , and days, masses of , , and , and radii of , , and . The stellar hosts have masses of , , and . Each system shows significant systematic variations in its residual radial velocities indicating the possible presence of additional components. Based on its Bayesian evidence, the preferred model for HATP44 consists of two planets, including the transiting component, with the outer planet having a period of d and a minimum mass of . Due to aliasing we cannot rule out an alternative solution for the outer planet having a period of d and a minimum mass of . For HATP45 at present there is not enough data to justify the additional free parameters included in a multiplanet model, in this case a singleplanet solution is preferred, but the required jitter of is relatively high for a star of this type. For HATP46 the preferred solution includes a second planet having a period of d and a minimum mass of , however the preference for this model over a singleplanet model is not very strong. While substantial uncertainties remain as to the presence and/or properties of the outer planetary companions in these systems, the inner transiting planets are well characterized with measured properties that are fairly robust against changes in the assumed models for the outer planets. Continued RV monitoring is necessary to fully characterize these three planetary systems, the properties of which may have important implications for understanding the formation of hot Jupiters.
Subject headings:
planetary systems — stars: individual (HATP44, GSC 346500123, HATP45, GSC 510200262, HATP46, GSC 510000045) — techniques: spectroscopic, photometric1. Introduction
There is mounting evidence that systems containing closein, gasgiant
planets (hot Jupiters) are fundamentally different from systems that
do not contain such a planet. These differences are seen in the
occurrence rate of multiple planets between systems with and without
hot Jupiters and in the distribution of projected orbital
obliquities
Out of the 187 systems listed in the exoplanets orbit
database
Observations of the RossiterMcLaughlin effect have revealed that hot Jupiters exhibit a broad range of projected obliquities (e.g. Albrecht et al., 2012). In contrast, the multiplanet systems, not containing a hot Jupiter, for which the projected obliquity of at least one of the planets has been determined, are all aligned (Albrecht et al., 2013). Differences in the obliquities have been interpreted as indicating different migration mechanisms between the two populations (SanchisOjeda et al., 2012; Albrecht et al., 2013).
There are, however, selection effects which complicate this picture. While most multiplanet systems have been discovered by RV surveys or by the NASA Kepler space mission, the great majority of hot Jupiters have been discovered by groundbased transiting planet searches. For these latter surveys access to highprecision RV resources may be scarce, and the candidates are usually several magnitudes fainter than those targeted by RV surveys. To deal with these factors, groundbased transit surveys leverage the known ephemerides of their candidates so as to minimize the number of RV observations needed to detect the orbital variation. In practice this means that many published hot Jupiters do not have the longterm RV monitoring that would be necessary to detect other planetary companions, if present. Moreover, groundbased surveys produce light curves with much shorter time coverage and poorer precision than Kepler, so whereas Kepler has identified numerous multitransitingplanet systems, groundbased surveys have not yet discovered any such systems.
In this paper we report the discovery of three new transiting planet systems by the HATNet survey (Bakos et al., 2004). The transiting planets are all classical hot Jupiters, confirmed through a combination of groundbased photometry and spectroscopy, including highprecision radial velocity (RV) measurements made with KeckI/HIRES which reveal the orbital motion of the star about the planet–star centerofmass. In addition to the orbital motion due to the transiting planets, the RV measurements for all three systems show systematic variations indicating the possible presence of additional planetarymass components. As we will show, for two of these systems (HATP44 and HATP46) we find that the observations are best explained by multiplanet models, while for the third system (HATP45) additional RV observations would be necessary to claim an additional planet.
2. Observations
The observational procedure employed by HATNet to discover Transiting Extrasolar Planets (TEPs) has been described in several previous discovery papers (e.g. Bakos et al., 2010; Latham et al., 2009). In the following subsections we highlight specific details of this procedure that are relevant to the discoveries presented in this paper.
2.1. Photometric detection
Table 1 summarizes the HATNet discovery observations of each new planetary system. The HATNet images were processed and reduced to trendfiltered light curves following the procedure described by Bakos et al. (2010). The light curves were searched for periodic boxshaped signals using the Box LeastSquares (BLS; see Kovács et al., 2002) method. Figure 1 shows phasefolded HATNet light curves for HATP44, HATP45, and HATP46 which were selected as showing highly significant transit signals based on their BLS spectra. Crossidentifications, positions, and the available photometry on an absolute scale are provided later in the paper together with other system parameters (Table 10).
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 to 50 d in the light curves of any of these systems. For HATP44 we exclude signals with amplitudes above 1.2 mmag, for HATP45 we exclude signals with amplitudes above 1.1 mmag, and for HATP46 we exclude signals with amplitudes above 0.6 mmag. Similarly we do not detect additional transit signals in the light curves of HATP44 or HATP45. For HATP46 we do detect a marginally significant transit signal with a short period of d, a depth of mmag, and a S/N in the BLS spectrum of . 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.
Instrument/Field  Date(s)  Number of Images  Cadence (sec)  Filter 

HATP44  
HAT5/G145  2006 Jan–2006 Jul  2880  330  band 
HAT6/G146  2010 Apr–2010 Jul  6668  210  band 
KeplerCam  2011 Mar 19  112  134  band 
BOS  2011 Apr 14  176  131  band 
KeplerCam  2011 Apr 14  85  134  band 
KeplerCam  2011 May 27  176  134  band 
HATP45  
HAT5/G432  2010 Sep–2010 Oct  272  330  band 
HAT8/G432  2010 Apr–2010 Oct  7309  210  band 
KeplerCam  2011 Apr 02  133  73  band 
KeplerCam  2011 Apr 05  44  103  band 
FTN  2011 Apr 30  197  50  band 
KeplerCam  2011 May 22  174  64  band 
KeplerCam  2011 Jun 10  146  64  band 
KeplerCam  2011 Jul 05  99  103  band 
KeplerCam 
2013 May 20  229  50  band 
HATP46  
HAT5/G432  2010 Sep–2010 Oct  300  330  band 
HAT8/G432  2010 Apr–2010 Oct  7633  210  band 
KeplerCam  2011 May 05  392  44  band 
KeplerCam  2011 May 14  368  49  band 
KeplerCam  2011 May 23  247  39  band 
2.2. Reconnaissance Spectroscopy
Highresolution, lowS/N “reconnaissance” spectra were obtained for HATP44, HATP45, and HATP46 using the Tillinghast Reflector Echelle Spectrograph (TRES; Fűresz, 2008) on the 1.5 m Tillinghast Reflector at FLWO. Mediumresolution reconnaissance spectra were also obtained for HATP45 and HATP46 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); Buchhave et al. (2010), yielding RVs with a precision of , and an absolute velocity zeropoint accuracy of . The WiFeS observations were reduced and analyzed as described in Bayliss et al. (2013), providing RVs with a precision of 2.8 .
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 HATP46 have an RMS of 3.3 which is only slightly above the precision determined from observations of RV stable stars). All spectra were singlelined, 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.
Instrument 

RV  

(K)  (cgs)  ()  ()  
HATP44  
TRES  55557.01323  5250  4.5  2  34.042 
TRES  55583.91926  5250  4.5  2  34.047 
HATP45  
WiFeS  55646.25535  18.9  
WiFeS  55648.19634  16.6  
WiFeS  55649.24624  18.5  
WiFeS  55666.31876  20.1  
TRES  55691.96193  6500  4.5  10  23.162 
HATP46  
WiFeS  55644.28771  21.1  
WiFeS  55646.25316  29.6  
WiFeS  55647.21574  21.3  
WiFeS  55647.21882  21.7  
WiFeS  55648.17221  23.9  
WiFeS  55649.21348  25.0  
TRES  55659.92299  6000  4.0  6  21.314 
TRES  55728.82463  6000  4.0  6  21.385 
2.3. High resolution, high S/N spectroscopy
We obtained highresolution, highS/N spectra of each of these objects using HIRES (Vogt et al., 1994) on the KeckI 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 35 for HATP44 through HATP46, respectively. The periodfolded data, along with our best fit described below in Section 3 are displayed in Figures 24.
We also show the chromospheric activity index and the spectral line bisector spans. The index for each star was computed following Isaacson & Fischer (2010) and converted to following Noyes et al. (1984). We find median values of , , and for HATP44 through HATP46, respectively. These values imply that all three stars are chromospherically quiet. 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).
BJD  RV 

BS  S 
Phase  

(2,454,000)  ()  ()  ()  ()  
Note. – Note that for the iodinefree template exposures we do not measure the RV but do measure the BS and S index. Such template exposures can be distinguished by the missing RV value.
BJD  RV 

BS  S 
Phase  

(2,454,000)  ()  ()  ()  ()  
Note. – Note that for the iodinefree template exposures we do not measure the RV but do measure the BS and S index. Such template exposures can be distinguished by the missing RV value.
BJD  RV 

BS  S 
Phase  

(2,454,000)  ()  ()  ()  ()  
Note. – Note that for the iodinefree template exposures we do not measure the RV but do measure the BS and S index. Such template exposures can be distinguished by the missing RV value.
2.4. Photometric followup observations
In order to permit a more accurate modeling of the light curves, we conducted additional photometric observations of each of the transiting planet systems. For this purpose we made use of 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 as 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)
BJD  Mag 
Mag(orig) 
Filter  

(2,400,000)  
Note. – This table is available in a machinereadable form in the online journal. A portion is shown here for guidance regarding its form and content.
BJD  Mag 
Mag(orig) 
Filter  

(2,400,000)  
Note. – This table is available in a machinereadable form in the online journal. A portion is shown here for guidance regarding its form and content.
BJD  Mag 
Mag(orig) 
Filter  

(2,400,000)  
Note. – This table is available in a machinereadable form in the online journal. A portion is shown here for guidance regarding its form and content.
3. Analysis
3.1. Properties of the parent star
Stellar atmospheric parameters for each star were measured using our template spectra obtained with the Keck/HIRES instrument, and the analysis package known as Spectroscopy Made Easy (SME; Valenti & Piskunov, 1996), along with the atomic line database of Valenti & Fischer (2005). For each star, SME yielded the following initial values and uncertainties:

HATP44 – effective temperature K, metallicity dex, stellar surface gravity (cgs), and projected rotational velocity .

HATP45 – effective temperature K, metallicity dex, stellar surface gravity (cgs), and projected rotational velocity .

HATP46 – effective temperature K, metallicity dex, stellar surface gravity (cgs), and projected rotational velocity .
These values were used to determine initial values for the limbdarkening coefficients, which we fix during the light curve modeling (Section 3.4). This modeling, when combined with theoretical stellar evolution models taken from the YonseiYale (YY) series by 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 HATP44 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 , and are listed for each star in Table 10. The values of , as well as of properties inferred from the evolution models (such as the stellar masses and radii) depend on the eccentricity and semiamplitude 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 , , and will also depend on the fixed value of we found generally that 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 value determined from each model. As we discuss in Section 3.4.2 we tested numerous models; our final adopted values for these modeldependent parameters are presented in that section.
The inferred location of each star in a diagram of versus , analogous to the classical HR diagram, is shown in Figure 8. In each case the stellar properties and their 1 and 2 confidence ellipsoids are displayed against the backdrop of model isochrones for a range of ages, and the appropriate stellar metallicity. For comparison, the locations implied by the initial SME results for HATP45 and HATP46 are also shown (in each case with a triangle).
We determine the distance and extinction to each star by comparing the , and magnitudes from the 2MASS Catalogue (Skrutskie et al., 2006), and the and 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 totaltoselective extinction ratio of , to relate the extinction in each bandpass to the band extinction . The resulting and distance measurements are given with the other modeldependent parameters. We find that HATP44 is not significantly affected by extinction, consistent with the Schlegel et al. (1998) dust maps which yield a total extinction of mag along the line of sight to HATP44. HATP45 and HATP46, on the other hand, have low Galactic latitudes ( and , respectively), and are significantly affected by extinction. We find mag and mag for our preferred models for HATP45 and HATP46, respectively. For comparison, the Schlegel et al. (1998) maps yield a total line of sight extinction of mag and mag for HATP45 and HATP46, respectively, or 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 dustmapbased and photometrybased estimates for HATP45 is not unexpected. After correcting for extinction the measured and expected photometric color indices are consistent for each star.
3.2. Excluding Blend Scenarios
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 HATP44 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 confidence would have been detected as obviously doublelined systems, showing many RV and BS variations.
For HATP45 and HATP46 the significant reddening (Section 3.1) allows a broader range of blend scenarios to fit the photometric data. For a system like HATP44, 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 HATP45 and HATP46, on the other hand, such blends can be accommodated by reducing the reddening in the fit. Indeed we find for both HATP45 and HATP46 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 Cardelli et al. (1989) extinction law.
To better constrain the possible blend scenarios we obtained a partial band light curve for HATP45 using Keplercam on the night of 20 May 2013. 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 HATP45, excluding scenarios that predict substantially different  and band transit depths.
Although the broadband photometry permits a wide range of possible blend scenarios, for both HATP45 and HATP46 the nonplanetary blend scenarios which fit the photometric data can be ruled out based on the BS and RV variations. For HATP45 we find that blend scenarios that fit the photometric data (scenarios that cannot be rejected with confidence) yield several BS and RV variations, whereas the actual BS RMS is . Without the band light curve for HATP45 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 HATP46 the blend scenarios that fit the photometric data would result in BS variations with RMS , much greater than the measured scatter of .
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).
3.3. Periodogram Analysis of the RV Data
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 per degree of freedom (, and for the full RV data of HATP44, HATP45, and HATP46, 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, indicating that if additional bodies are responsible for the excess scatter, they must have orbital periods shorter than the timespans of the RV data sets.
Figure 9 shows the harmonic Analysis of Variance (AoV)
periodograms (SchwarzenbergCzerny, 1996) of the residual RVs from
the bestfit singleplanet model for each system
For HATP44 the two highest peaks are at d ( d) and d ( d), with false alarm probabilities of and , respectively. The periodogram of the residuals of a model consisting of the transiting planet and a planet with d (when fitting the data simultaneously for two planets this model provides a slightly better fit than when the outer planet has a period of d) yields a peak at d with a false alarm probability of . Alias peaks are also seen at d, d, d, d, d, and several other values with decreasing significance.
For HATP45 a number of frequencies are detected in the periodogram of the RV residuals from the bestfit singleplanet model. These periods are all aliases of each other. The highest peak is at d ( d), with a false alarm probability of . For HATP46 the two highest peaks are at d ( d) and d ( d), each with false alarm probabilities of (or if uniform uncertainties are adopted as discussed further below).
The false alarm probabilities given above include a correction for the socalled “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 d d 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 ). Note that adopting a broader frequency range for the periodograms (e.g. up to the Nyquist limit, which for the HATP44 data would be d) significantly increases the false alarm probabilities. We expect, however, that systems containing multiple Jupitermass 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 HATP44 and HATP45 the false alarm probabilities are approximately the same for high jitter as they are when the jitter is set to 0. For HATP46 the false alarm probabilities are smaller when the errors are dominated by jitter ( with jitter vs. without jitter).
3.4. Global model of the data
We modeled simultaneously the HATNet photometry, the followup photometry, and the highprecision 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 trendfilters, to describe the followup 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.
RV Jitter
It is well known that highprecision RV observations of stars show nonperiodic 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 whitenoise. One then either adopts a jitter value that is found to be typical for similar stars, or chooses a jitter such that per degree of freedom is unity for the bestfit 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 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 where is a normalization constant that is independent of the parameters, and can be ignored for most applications. Instead one should use , where is the error for measurement and in this case is given by for formal uncertainty and jitter . When the uncertainties do not include free parameters, the term is constant, and included in .
The analysis by Gregory (2005) used an uninformative prior on the jitter, which effectively forces the jitter to the value that results in ; 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 , activity, and luminosity above the main sequence. The histograms appear to be wellmatched by lognormal distributions of the form:
(1) 
Figure 10 compares this model to the jitter histograms. For HATP44, which falls in the lowactivity bin with and , we find , , with measured in units of . For HATP45 and HATP46, which fall in the bin of lowactivity stars with and , we find , and .
The posterior probability density for the parameters , given the data and model is given by Bayes’ relation:
(2) 
which in our case takes the form:
where represents constants that are independent of (note we adopt uniform priors on all jump parameters other than ). We use a differential evolution MCMC procedure (ter Braak, 2006; Eastman et al., 2013) to explore this distribution.
Model Selection
As discussed in the previous subsection, modeling these objects as singleplanet systems yields RV residuals with large scatter and evidence of longterm variations. 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 is defined by
(3) 
where is the probability of observing the data given the model , marginalized over the model parameters . The Bayes factor comparing the posterior probabilities for models and given the data is defined by
(4) 
where is the prior probability for model . Assuming equal priors for the different models tested, the Bayes factor is then equal to the evidence ratio:
(5) 
If then model is favored over model .
In practice is difficult to determine as it requires integrating a complicated function over a highdimensional space (e.g. Feroz et al., 2009). Recently, however, Weinberg et al. (2013) have suggested a simple and relatively accurate method for estimating 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 and 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 , with errors in . For this reason we do not consider differences between models that are 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
(6) 
for a model with free parameters fit to data points yielding a maximum likelihood of . The BIC is determined solely from the highest likelihood value, making it easier to calculate than . Models with lower BIC values are generally favored. Note, however, that the BIC is a less accurate method for distinguishing between models than is . We also provide, for reference, the Bayes Factors determined when the jitter of each system is fixed to a typical value throughout the analysis.
3.5. Resulting Parameters
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 HATP44, and parameters of the transiting planet HATP44b, that depend on the class of model tested are listed in Table 11, while parameters for the candidate outer components HATP44c and HATP44d are listed in Table 12. Stellar parameters for HATP45 and HATP46, and parameters for the transiting planets HATP45b and HATP46b, that depend on the class of model tested are listed in Table 13, while parameters for the candidate outer components HATP45c, HATP46c and HATP46d are listed in Table 14.
For HATP44 we find that the preferred model, based on the estimated Bayes Factor, consists of 2 planets, the outer one on a circular orbit. This model, labeled number 2 in Tables 9, 11, and 12, includes: the transiting planet HATP44b with a period of d, a mass of , and an eccentricity of ; an outer planet HATP44c with a period of d, and minimum mass of . We find that an alternative model, labelled number 3, which has the same form as the preferred model, except the outer planet has a period of d, and a minimum mass of is equally acceptable based on the Bayes Factor. 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 two periods. For this reason we treat these as independent models. We adopt the model with the shorter period for the outer component because it gives a slightly higher maximum likelihood. This model is favored over the fiducial model of a single planet transiting the host star by a factor of indicating that the data strongly favor the twoplanet model over the singleplanet model. The preferred model has an associated jitter of and a per degree of freedom, including this jitter, of . Based on equation 1, one expects only 0.5% of stars like HATP44 to have jitter values thus the excess scatter in the RV residuals from the bestfit 2planet model suggests that perhaps more than 2 planets are present in this system, though we cannot conclusively detect any additional planets from the data currently available.
For HATP45 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 HATP45 headings in Tables 9, 13, and 14, includes only the transiting planet HATP45b with a period of d, a mass of , and an eccentricity of . The preferred model has a jitter of and per degree of freedom of . Only of stars like HATP45 are expected to have a jitter this high. Moreover, the RV residuals from the preferred bestfit model appear to show a variation that is correlated in time (see the third panel down in Fig. 3). Both these factors suggest that a second planet may be present in the HATP45 system. Nonetheless the data do not at present support such a complicated model. The singleplanet model has a Bayes factor of relative to the twoplanet model, indicating a slight preference for the singleplanet model.
For HATP46 the preferred model consists of a transiting planet together with an outer companion on a circular orbit. This model, labeled number 2 under the HATP46 headings in Tables 9, 13 and 14, includes: the transiting planet HATP46b with a period of d, a mass of , and an eccentricity of ; and an outer planet HATP46c with a period of d, and a minimum mass of . Although the twoplanet model is preferred, it has a Bayes factor of only relative to the fiducial singleplanet model, indicating that the preference is not very strong. The preferred model has a jitter of and per degree of freedom of . The resulting jitter is typical for a star like HATP46 (% of such stars have a jitter higher than ), so there is no compelling reason at present to suspect that there may be more planets in this system beyond HATP46c.
For both HATP44 and HATP46 allowing the jitter to vary in the fit substantially reduces the significance of the multiplanet solutions relative to the single planet solution. If we had not allowed the jitter to vary, we would have concluded that the twoplanet model is times more likely than the oneplanet model for HATP44, and times more likely for HATP46. For HATP45, it is interesting to note that allowing the jitter to vary actually increases the significance of the twoplanet model, perhaps due to the relatively high jitter value that must be adopted to achieve .
Model 
Trend  Fixed Jitter  

Number 


Order 





HATP44  
HATP45  
HATP46  
Orbital Stability
To check the orbital stability of the multiplanet solutions that we have found, we integrated each orbital configuration forward in time for a duration of Myr using the Mercury symplectic integrator (Chambers, 1999). We find that the adopted solutions for HATP44 and HATP46 (model 2 in each case) are stable over at least this time period, and should be stable for much longer given the large, and nonresonant, period ratio between the components in each case. For HATP44 the threeplanet models that we tested quickly evolved in less than years to a different orbital configuration. In particular, when we start HATP44b on a d period, HATP44c on a d period, and HATP44d on a d period, HATP44d migrates to a 15.1 d period orbit, while HATP44b migrates to a 4.6928 d period. While this final configuration appears to be stable for at least yr, it is inconsistent with the RV and photometric data. We did not carry out a full exploration of the parameter space allowed by our uncertainties, but the fact that the bestfit 3planet model for HATP44 shows rapid planetary migration indicates that this model may very well be unstable. If additional RV observations support a 3planet solution for HATP44, it will also be important to test the stability of this solution.
HATP44b  HATP45b  HATP46b  

Parameter  Value  Value  Value  Source 
Stellar Astrometric properties  
GSC ID  GSC 346500123  GSC 510200262  GSC 510000045  
2MASS ID  2MASS 14123457+4700528  2MASS 181729570322517  2MASS 180146600258154  
R.A. (J2000)  
Dec. (J2000)  
(mas yr)  
(mas yr)  
Stellar Spectroscopic properties  
(K)  SME 

SME  
()  SME  
()  SME  
()  SME  
()  TRES  
Keck/HIRES 

Stellar Photometric properties  
(mag)  13.212  12.794  11.936  TASS 
(mag)  TASS  
(mag)  2MASS  
(mag)  2MASS  
(mag)  2MASS  
Transiting Planet Light curve parameters  
(days)  
()


(days)


(days)


Assumed Limbdarkening coefficients 

(linear term)  Claret,2004  
(quadratic term)  Claret,2004 
Adopted  

Model 1  Model 2  Model 3  Model 4  Model 5  Model 6  Model 7  
Parameter  Value  Value  Value  Value  Value  Value  Value 
Transiting planet (HATP44b) light curve parameters  
(deg)  
Transiting planet (HATP44b) RV parameters  
()  
(deg)  
RV jitter ()  
Derived transiting planet (HATP44b) parameters  
()  
()  


()  
(cgs)  
(AU)  
(K)  




Derived stellar properties  
()  
()  
(cgs)  
()  
(mag)  
(mag,ESO)  
Age (Gyr) 