HAT-P-44, HAT-P-45 and HAT-P-46

HAT-P-44b, HAT-P-45b, and HAT-P-46b: Three Transiting Hot Jupiters in Possible Multi-Planet Systems1


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 HAT-P-44 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 HAT-P-45 at present there is not enough data to justify the additional free parameters included in a multi-planet model, in this case a single-planet solution is preferred, but the required jitter of  is relatively high for a star of this type. For HAT-P-46 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 single-planet 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 (HAT-P-44, GSC 3465-00123, HAT-P-45, GSC 5102-00262, HAT-P-46, GSC 5100-00045) — techniques: spectroscopic, photometric

1. Introduction

There is mounting evidence that systems containing close-in, gas-giant 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 obliquities36 of hot Jupiters compared to that of other planets.

Out of the 187 systems listed in the exoplanets orbit database37 (Wright et al., 2011) containing a planet with  d and , only 5 (2.7%) include confirmed, and well-characterized outer planets (these are  And, Butler et al., 1997, 1999; HD 217107, Fischer et al., 1999, Vogt et al., 2005; HD 187123, Butler et al., 1998, Wright et al., 2007; HIP 14810, Wright et al., 2007; and HAT-P-13, Bakos et al., 2009). By contrast there are 87 multi-planet systems among the 395 systems (22%) in the database that do not have a hot Jupiter. In addition to the 5 confirmed multi-planet hot Jupiter systems, there are a number of other hot-Jupiter-bearing systems for which long term trends in their RVs have been reported. These trends could be due to long-period planetary companions, but their periods are significantly longer than the time spanned by the observations, and one cannot generally rule out stellar mass companions (a few examples from the Hungarian Automated Telescope Network, or HATNet, survey include HAT-P-7, Pál et al., 2008; HAT-P-17, Howard et al., 2012, Fulton et al., 2013; HAT-P-19, Hartman et al., 2011; and HAT-P-34, Bakos et al., 2012). Differences in the occurrence rate of multiple planets between hot-Jupiter-hosting systems and other systems are also apparent from the sample of Kepler transiting planet candidates (Latham et al., 2011).

Observations of the Rossiter-McLaughlin effect have revealed that hot Jupiters exhibit a broad range of projected obliquities (e.g. Albrecht et al., 2012). In contrast, the multi-planet 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 (Sanchis-Ojeda et al., 2012; Albrecht et al., 2013).

There are, however, selection effects which complicate this picture. While most multi-planet systems have been discovered by RV surveys or by the NASA Kepler space mission, the great majority of hot Jupiters have been discovered by ground-based transiting planet searches. For these latter surveys access to high-precision RV resources may be scarce, and the candidates are usually several magnitudes fainter than those targeted by RV surveys. To deal with these factors, ground-based 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 long-term RV monitoring that would be necessary to detect other planetary companions, if present. Moreover, ground-based surveys produce light curves with much shorter time coverage and poorer precision than Kepler, so whereas Kepler has identified numerous multi-transiting-planet systems, ground-based 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 ground-based photometry and spectroscopy, including high-precision radial velocity (RV) measurements made with Keck-I/HIRES which reveal the orbital motion of the star about the planet–star center-of-mass. 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 planetary-mass components. As we will show, for two of these systems (HAT-P-44 and HAT-P-46) we find that the observations are best explained by multi-planet models, while for the third system (HAT-P-45) additional RV observations would be necessary to claim an additional planet.

In Section 2 we summarize the detection of the photometric transit signal and the subsequent spectroscopic and photometric observations of each star to confirm the planets. In Section 3 we analyze the data to determine the stellar and planetary parameters. Our findings are discussed in Section 4.

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 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).

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 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  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.

Figure 1.— HATNet light curves of HAT-P-44 (top), HAT-P-45 (middle) and HAT-P-46 (bottom). See Table 1 for a summary of the observations. For each planet we show two panels. The top panel shows the unbinned light curve folded with the period resulting from the global fit described in Section 3. The solid line shows the model fit to the light curve (Section 3.4). The bottom panel shows the region zoomed-in on the transit. The dark filled circles show the light curve binned in phase with a bin size of 0.002. These are plotted with uncertainties.
        Instrument/Field Date(s) Number of Images Cadence (sec) Filter
    HAT-5/G145 2006 Jan–2006 Jul 2880 330  band
    HAT-6/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
    HAT-5/G432 2010 Sep–2010 Oct 272 330  band
    HAT-8/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
    KeplerCam38 2013 May 20 229 50  band
    HAT-5/G432 2010 Sep–2010 Oct 300 330  band
    HAT-8/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
Table 1 Summary of photometric observations

2.2. Reconnaissance Spectroscopy

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); 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 HAT-P-46 have an RMS of 3.3  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.

Instrument 39 RV
(K) (cgs) () ()
    TRES 55557.01323 5250 4.5 2 -34.042
    TRES 55583.91926 5250 4.5 2 -34.047
    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
    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
Table 2 Summary of reconnaissance spectroscopy observations

2.3. High resolution, high S/N spectroscopy

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.

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 HAT-P-44 through HAT-P-46, 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).

Figure 2.— Top left: Keck/HIRES RV measurements for HAT-P-44 shown as a function of orbital phase for the outer planet HAT-P-44c assuming a period of  d, after subtracting off the model orbit for the inner transiting planet HAT-P-44b. Overplotted is our best-fit model (see Table 11). Zero phase corresponds to the time of mid-transit (assuming an edge on orbit; transits of this planet have not been detected). The center-of-mass velocity has been subtracted. Observations shown twice are plotted with open circles. Second left: Same as the top left panel, here we show the observations phased using the ephemeris of the transiting planet HAT-P-44b, after subtracting off Keplerian orbit due to HAT-P-44c. Third left: Bisector spans (BS), with the mean value subtracted, phased using the ephemeris for the transiting planet. The measurement from the template spectrum is included (see Section 3.2). Bottom left: Chromospheric activity index measured from the Keck spectra, phased using the ephemeris for the transiting planet. Top right: RV curve of HAT-P-44 as a function of time, together with our best-fit two planet model. Second right: RV residuals from the best-fit two-planet model as a function of time. Third right: RV curve of HAT-P-44 as a function of time after subtracting the orbit due to the inner planet HAT-P-44b. For reference, the dashed line in this plot shows the model for the  d alias signal. Bottom right: RV curve of HAT-P-44 as a function of time after subtracting the orbit due to the outer planet HAT-P-44c. Note the difference in vertical scales for all panels. RV uncertainties in this figure include the jitter which is estimated during the fitting procedure (see Section 3.4.1) and added in quadrature to the formal uncertainties.
Figure 3.— Top panel: Keck/HIRES RV measurements for HAT-P-45 shown as a function of orbital phase, along with our best-fit single-planet model (see Table 13). Zero phase corresponds to the time of mid-transit. The center-of-mass velocity has been subtracted. Second panel: Velocity residuals from the best fit shown as a function of time. Third panel: Same as the second panel, here we zoom-in on the residuals for the first 8 observations. Fourth panel: Velocity residuals shown as a function of orbital phase. Fifth panel: Bisector spans (BS), with the mean value subtracted. The measurement from the template spectrum is included (see Section 3.2). Bottom panel: Chromospheric activity index measured from the Keck spectra. Note the different vertical scales of the panels. Observations shown twice are represented with open symbols.

Figure 4.— Keck/HIRES observations of HAT-P-46. The panels are as in Figure 2. The parameters used in the best-fit model are given in Table 13.
BJD RV40 41 BS S42 Phase
(2,454,000) () () () ()

Note. – Note that for the iodine-free 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.

Table 3 Relative radial velocities, bisector spans, and activity index measurements of HAT-P-44.
BJD RV43 44 BS S45 Phase
(2,454,000) () () () ()

Note. – Note that for the iodine-free 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.

Table 4 Relative radial velocities, bisector spans, and activity index measurements of HAT-P-45.
BJD RV46 47 BS S48 Phase
(2,454,000) () () () ()

Note. – Note that for the iodine-free 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.

Table 5 Relative radial velocities, bisector spans, and activity index measurements of HAT-P-46.

2.4. Photometric follow-up observations

Figure 5.— Unbinned transit light curves for HAT-P-44. The light curves have been EPD and TFA processed, as described in § 3.4. The dates of the events, the facilities used to collect the light curves, and the filters used are indicated. Curves after the first are displaced vertically for clarity. Our best fit from the global modeling described in Section 3.4 is shown by the solid lines. Residuals from the fits are displayed at the bottom, in the same order as the top curves. The error bars represent the photon and background shot noise, plus the readout noise.
Figure 6.— Similar to Figure 5; here we show the follow-up light curves for HAT-P-45. The -band light curve obtained on 2013 May 20 was used for rejecting blend scenarios but was not used in determining the system parameters as the analysis was performed prior to carrying out this observation.
Figure 7.— Similar to Figure 5; here we show the follow-up light curves for HAT-P-46.

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)49 methods applied simultaneously with the 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 final time series, together with our best-fit transit light curve model, are shown in the top portion of Figures 5-7 for HAT-P-44 through HAT-P-46, respectively; the individual measurements are reported in Tables 6-8.

BJD Mag50 Mag(orig)51 Filter

Note. – This table is available in a machine-readable form in the online journal. A portion is shown here for guidance regarding its form and content.

Table 6 High-precision differential photometry of HAT-P-44.
BJD Mag52 Mag(orig)53 Filter

Note. – This table is available in a machine-readable form in the online journal. A portion is shown here for guidance regarding its form and content.

Table 7 High-precision differential photometry of HAT-P-45.
BJD Mag54 Mag(orig)55 Filter

Note. – This table is available in a machine-readable form in the online journal. A portion is shown here for guidance regarding its form and content.

Table 8 High-precision differential photometry of HAT-P-46.

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:

  • HAT-P-44 – effective temperature  K, metallicity  dex, stellar surface gravity  (cgs), and projected rotational velocity .

  • HAT-P-45 – effective temperature  K, metallicity  dex, stellar surface gravity  (cgs), and projected rotational velocity .

  • HAT-P-46 – effective temperature  K, metallicity  dex, stellar surface gravity  (cgs), and projected rotational velocity .

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 theoretical stellar evolution models taken from the Yonsei-Yale (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 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 ,  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 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 , , 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 model-dependent parameters are presented in that section.

The inferred location of each star in a diagram of  versus , analogous to the classical H-R 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 HAT-P-45 and HAT-P-46 are also shown (in each case with a triangle).

Figure 8.— Model isochrones from Yi et al. (2001) for the metallicities of HAT-P-44 (top), HAT-P-45 (center) and HAT-P-46 (bottom). For HAT-P-44 the isochrones are shown for ages of 0.2 Gyr, 0.6 Gyr, and 1.0 Gyr to 13.0 Gyr in steps of 1.0 Gyr (left to right), while for HAT-P-44 and HAT-P-45 the isochrones are shown for ages of 0.2 Gyr, and 0.5 Gyr to 3.0 Gyr in steps of 0.5 Gyr (left to right). The adopted values of and  are shown together with their 1 and 2 confidence ellipsoids. For HAT-P-45 and HAT-P-46 the initial values of  and  from the first SME and light curve analyses are represented 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 total-to-selective extinction ratio of , to relate the extinction in each band-pass to the -band extinction . The resulting 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  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 ( and , respectively), and are significantly affected by extinction. We find  mag and  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  mag and  mag for HAT-P-45 and HAT-P-46, 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 dust-map-based and photometry-based estimates for HAT-P-45 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 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 confidence would have been detected as obviously double-lined systems, showing many  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 broad-band 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 broad-band 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 HAT-P-45 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 HAT-P-45, excluding scenarios that predict substantially different - and  band transit depths.

Although the broad-band 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 confidence) yield several  BS and RV variations, whereas the actual BS RMS is . Without the  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, 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 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, indicating that if additional bodies are responsible for the excess scatter, they must have orbital periods shorter than the time-spans of the RV data sets.

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 system56. 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.

For HAT-P-44 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 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  d ( d), with a false alarm probability of . For HAT-P-46 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 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  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 HAT-P-44 data would be  d) 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 ( with jitter vs.  without jitter).

Figure 9.— Harmonic AoV periodograms of the RV residuals from the best-fit single planet models for HAT-P-44, HAT-P-45 and HAT-P-46.

3.4. Global model of the data

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.

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 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 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 well-matched by log-normal distributions of the form:


Figure 10 compares this model to the jitter histograms. For HAT-P-44, which falls in the low-activity bin with and , we find , , with measured in units of . For HAT-P-45 and HAT-P-46, which fall in the bin of low-activity stars with and , we find , and .

Figure 10.— Comparison of the distribution of Keck/HIRES jitter values for low activity main sequence stars taken from Wright (2005) to our model distributions. The histograms show the empirical distributions with Poisson error-bars, the open squares show equation 1 integrated over each bin, and the solid curves are included to show the form of the distribution. The top panel is for stars with , appropriate for HAT-P-45 and HAT-P-46, the bottom panel is for stars with , appropriate for HAT-P-44.

The posterior probability density for the parameters , given the data and model is given by Bayes’ relation:


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 single-planet systems yields RV residuals with large scatter and evidence of long-term 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


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


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:


If then model is favored over model .

In practice 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 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


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 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 components HAT-P-44c and HAT-P-44d are listed in Table 12. 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 components HAT-P-45c, HAT-P-46c and HAT-P-46d are listed in Table 14.

For HAT-P-44 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 911, and 12, includes: the transiting planet HAT-P-44b with a period of  d, a mass of , and an eccentricity of ; an outer planet HAT-P-44c 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 two-planet model over the single-planet 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 HAT-P-44 to have jitter values   thus the excess scatter in the RV residuals from the best-fit 2-planet 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 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 913, and 14, includes only the transiting planet HAT-P-45b 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 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 Fig. 3). Both these factors suggest that a second planet may be present 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 relative to the two-planet model, indicating a slight preference for the single-planet model.

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 913 and 14, includes: the transiting planet HAT-P-46b with a period of  d, a mass of , and an eccentricity of ; and an outer planet HAT-P-46c with a period of  d, and a minimum mass of . Although the two-planet model is preferred, it has a Bayes factor of only relative to the fiducial single-planet 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 HAT-P-46 (% 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 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 times more likely than the one-planet model for HAT-P-44, and 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 .

Model57 Trend Fixed Jitter
Number 58 59 Order 60 61 62 63
Table 9 Bayes Factor and BIC differences for models tested

Orbital Stability

To check the orbital stability of the multi-planet 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 HAT-P-44 and HAT-P-46 (model 2 in each case) are stable over at least this time period, and should be stable for much longer given the large, and non-resonant, period ratio between the components in each case. For HAT-P-44 the three-planet models that we tested quickly evolved in less than years to a different orbital configuration. In particular, when we start HAT-P-44b on a  d period, HAT-P-44c on a  d period, and HAT-P-44d on a  d period, HAT-P-44d migrates to a 15.1 d period orbit, while HAT-P-44b 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 best-fit 3-planet model for HAT-P-44 shows rapid planetary migration indicates that this model may very well be unstable. If additional RV observations support a 3-planet solution for HAT-P-44, it will also be important to test the stability of this solution.

HAT-P-44b HAT-P-45b HAT-P-46b
        Parameter Value Value Value Source64

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)
    Dec. (J2000)
     (mas yr)
     (mas yr)
Stellar Spectroscopic properties
     (K) SME65
     () SME
     () SME
     () SME
     () TRES
Stellar Photometric properties
     (mag) 13.212 12.794 11.936 TASS
     (mag) TASS
     (mag) 2MASS
     (mag) 2MASS
     (mag) 2MASS
Transiting Planet Light curve parameters
    () 67
    (days) 68
    (days) 69
Assumed Limb-darkening coefficients 70
    (linear term) Claret,2004
    (quadratic term) Claret,2004
Table 10Model-independent stellar and light curve parameters for HAT-P-44–HAT-P-46
Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7
Parameter Value Value Value Value Value Value Value

Transiting planet (HAT-P-44b) light curve parameters
Transiting planet (HAT-P-44b) RV parameters
   RV jitter ()
Derived transiting planet (HAT-P-44b) parameters
Derived stellar properties
    Age (Gyr)