ABSTRACT
We report the discovery of HAT-P-31b, a transiting exoplanet orbiting the V = 11.660 dwarf star GSC 2099-00908. HAT-P-31b is the first planet discovered with the Hungarian-made Automated Telescope (HAT) without any follow-up photometry, demonstrating the feasibility of a new mode of operation for the HATNet project. The 2.17 MJ, 1.1 RJ planet has a period of Pb = 5.0054 days and maintains an unusually high eccentricity of eb = 0.2450 ± 0.0045, determined through Keck, FIbr-fed Échelle Spectrograph, and Subaru high-precision radial velocities (RVs). Detailed modeling of the RVs indicates an additional quadratic residual trend in the data detected to very high confidence. We interpret this trend as a long-period outer companion, HAT-P-31c, of minimum mass 3.4 MJ and period ⩾2.8 years. Since current RVs span less than half an orbital period, we are unable to determine the properties of HAT-P-31c to high confidence. However, dynamical simulations of two possible configurations show that orbital stability is to be expected. Further, if HAT-P-31c has non-zero eccentricity, our simulations show that the eccentricity of HAT-P-31b is actively driven by the presence of c, making HAT-P-31 a potentially intriguing dynamical laboratory.
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1. INTRODUCTION
Transiting extrasolar planets provide invaluable insight into the nature of planetary systems. The opportunities for follow-up include spectroscopic inference of an exoplanet's atmosphere (Tinetti et al. 2007), searches for dynamical variations (Agol et al. 2005; Kipping 2009a, 2009b), and characterization of the orbital elements (Winn et al. 2011). Multi-planet systems in particular offer rich dynamical interactions and their frequency is key to understanding planet formation.
The Hungarian-made Automated Telescope Network (HATNet; Bakos et al. 2004) survey for transiting exoplanets (TEPs) around bright stars operates six wide-field instruments: four at the Fred Lawrence Whipple Observatory (FLWO) in Arizona and two on the roof of the hangar servicing the Smithsonian Astrophysical Observatory's Submillimeter Array, in Hawaii. Since 2006, HATNet has announced and published 30 TEPs (e.g., Johnson et al. 2011). In this work, we report our thirty-first discovery, around the relatively bright star GSC 2099-00908. In addition, a long-period companion is detected through detailed modeling of the radial velocities (RVs), although no transits of this object have been detected or are necessarily expected.
In Section 2, we summarize the detection of the photometric transit signal and the subsequent spectroscopic observations of HAT-P-31 to confirm the planet. In Section 3, we analyze the data to determine the stellar and planetary parameters. Our findings are discussed in Section 4.
2. OBSERVATIONS
As described in detail in several previous papers (e.g., Bakos et al. 2010; Latham et al. 2009), HATNet employs the following methods to discover transiting planets: (1) identification of candidate transiting planets based on HATNet photometric observations, (2) high-resolution, low signal-to-noise (S/N) "reconnaissance" spectra to efficiently reject many false positives, (3) higher-precision photometric observations during transit to refine transit parameters and obtain the light curve derived stellar density, and (4) high-resolution, high-S/N "confirmation" spectroscopy to detect the orbital motion of the star due to the planet, characterize the host star, and rule out blend scenarios.
In this work, step (3) is omitted and this is the biggest difference from the usual HATNet analysis. The detection, and thus verification, of an exoplanet can be made using step (4) alone. Indeed, the majority of exoplanets have been found in this way. Step (1) clearly allows us to intelligently select the most favorable targets for this resource-intensive activity though. Step (2) follows the same logic. Step (3) is predominantly for the purposes of characterizing the system, and therefore its omission does not impinge on the planet detection. In some cases, follow-up photometry is used to confirm marginal HATNet candidate detections as well, but this is not the case for the discovery presented in this work. An additional check on step (4) is that the derived ephemeris is consistent with that determined photometrically.
We did not obtain high-precision photometry for this target as the transits were not observable from our usual site of choice, FLWO, until at least 2012 May. This is because the transiting planet has a near-integer period and the time of transit minimum has now phased into the day-light hours in Arizona (i.e., unobservable). Rather than wait until this time, we have decided to release this confirmed planet detection to the community so that follow-up photometry may be conducted at other sites.
The principal consequence of not having any follow-up photometry is that the obtainable precision of the transit parameters is reduced. In the case of HAT-P-31b, this means that the light curve derived density was less precise than that determined spectroscopically. In practice then, we reverse the usual logic and instead of applying a prior on the stellar density from the light curve, we apply a prior on the light curve from the stellar density. One can see that the decision on this will vary from case to case depending upon the transit depth and target brightness.
Another issue is that in the past HAT analyses have used the HATNet photometry for measuring the planetary ephemeris, P and
In this paper, we will consequently show that HATNet photometry alone is sufficient to constrain the system properties and that future work may not always require step (3) (i.e., follow-up photometry).
In the following subsections we highlight specific details of this procedure that are pertinent to the discovery of HAT-P-31b.
2.1. Photometric Detection
The transits of HAT-P-31b were photometrically detected with a combined confidence of 6.2
Standard photometric procedures were used to calibrate the HATNet frames and then these calibrated images were subjected to star detection and astrometry, as described in Pál & Bakos (2006). Aperture photometry was performed on each image at the stellar centroids derived from the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006) catalog and the individual astrometric solutions. The resulting light curves were decorrelated (cleaned of trends) using the EPD (see Bakos et al. 2010) technique in "constant" mode and TFA (see Kovács et al. 2005).
The light curves were searched for transits using the Box-fitting Least-Squares (BLS; Kovács et al. 2002) method. We detected a significant signal in the light curve of GSC 2099-00908 (also known as 2MASS 18060904+2625359; ,
0; J2000; V = 11.660; Droege et al. 2006), with an apparent depth of ∼5.1 mmag, and a period of P = 5.0050 days. The BLS periodogram is shown in Figure 1. The drop in brightness had a first-to-last-contact duration, relative to the total period, of q = 0.0432 ± 0.0038, corresponding to a total duration of 5.187 ± 0.462 hr. Due to the lack of follow-up photometry, the HATNet photometry was re-processed with more computationally expensive reconstructive TFA, as discussed in Section 2 and shown in Figure 2. The EPD and reconstructive TFA corrected photometry is provided in Table 1.
Figure 1. BLS (Kovács et al. 2002) periodogram of HATNet photometry. The arrow marks the true transit signal and the other peaks are interpreted to be aliases. The y-axis denotes signal residue (SR); see Kovács et al. (2002) for details of the definition.
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Standard image High-resolution imageFigure 2. Unbinned light curve of HAT-P-31 including all 8436 instrumental IC band 5.5 minute cadence measurements obtained with the HAT-5 and HAT-8 telescopes of HATNet (see the text for details), and folded with the period P = 5.005425 days resulting from the global fit described in Section 3. The solid line shows our transit model fit to the light curve (Section 3.2). The bottom panel shows a zoomed-in view of the transit, the filled circles show the light curve binned in phase with a bin size of 50 points.
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Standard image High-resolution imageTable 1. HATNet Differential Photometry of HAT-P-31
BJD | Mag (EPD)a | Mag (TFA)b | |
---|---|---|---|
(2,400,000+) | |||
54178.0007000 | 11.4164 | 11.4203 | 0.0075 |
54178.0045362 | 11.4267 | 11.4273 | 0.0073 |
54178.0083829 | 11.4250 | 11.4317 | 0.0075 |
54178.0122220 | 11.4177 | 11.4275 | 0.0069 |
54178.0160619 | 11.4244 | 11.4220 | 0.0072 |
⋮ | ⋮ | ⋮ | ⋮ |
Notes. aThese magnitudes have been subjected to the EPD procedure. bThese magnitudes have been subjected to the EPD and TFA procedures.
Only a portion of this table is shown here to demonstrate its form and content. Machine-readable and Virtual Observatory (VO) versions of the full table are available.
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2.2. Reconnaissance Spectroscopy
High-resolution, low-S/N reconnaissance spectra were obtained for HAT-P-31 using the Tillinghast Reflector Echelle Spectrograph (TRES; Fűrész et al. 2008) on the 1.5 m Tillinghast Reflector at FLWO, and the echelle spectrograph on the Australian National University (ANU) 2.3 m telescope at Siding Spring Observatory in Australia. The two TRES spectra of HAT-P-31 were obtained, reduced, and analyzed to measure the stellar effective temperature, surface gravity, projected rotation velocity, and RV via cross-correlation against a library of synthetic template spectra. The reduction and analysis procedure has been described by Quinn et al. (2010) and Buchhave et al. (2010). A total of 14 spectra of HAT-P-31 were obtained with the ANU 2.3 m telescope. These data were collected, reduced, and analyzed to measure the RV via cross-correlation against the spectrum of an RV standard star HD 223311 following the procedure described by Béky et al. (2011). The resulting measurements from TRES and the ANU 2.3 m telescope are given in Table 2.
Table 2. Summary of Reconnaissance Spectroscopy Observations of HAT-P-31
Instrument | Date | Number of | Teff⋆ | log (g*(cgs)) | vsin i | |
---|---|---|---|---|---|---|
Spectra | (K) | (km s−1) | (km s−1) | |||
TRES | 2009 Jul 5 | 1 | 6000 | 4.0 | 4 | −2.342 |
TRES | 2009 Jul 7 | 1 | 6000 | 4.0 | 4 | −2.300 |
ANU 2.3 m | 2009 Jul 14 | 5 | ... | ... | ... | −8.07 ± 0.35 |
ANU 2.3 m | 2009 Jul 17 | 5 | ... | ... | ... | −7.13 ± 0.40 |
ANU 2.3 m | 2009 Jul 18 | 2 | ... | ... | ... | −7.64 ± 0.44 |
ANU 2.3 m | 2009 Jul 19 | 2 | ... | ... | ... | −7.54 ± 0.49 |
Notes. aThe mean heliocentric RV of the target. Systematic differences between the velocities from the two instruments are consistent with the velocity zero-point uncertainties. For the ANU 2.3 m observations we give the weighted mean of the observations and the uncertainty on the mean for each night. Note that the systematic difference of −5.3 km s−1 between the ANU 2.3 m and TRES observations is similar to the difference of −5.1 km s−1 found between these same two instruments by Béky et al. (2011) for HAT-P-27.
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These observations revealed no detectable RV variation at the 1 km s−1 precision of the observations. Additionally the spectra are consistent with a single, slowly rotating, dwarf star.
2.3. High-resolution, High-S/N Spectroscopy
We proceeded with the follow-up of this candidate by obtaining high-resolution, high-S/N spectra to characterize the RV variations and to refine the determination of the stellar parameters. For this we used the High Resolution Echelle Spectrometer (HIRES) instrument (Vogt et al. 1994) with the iodine-cell (Marcy & Butler 1992) on the Keck I telescope, the High-Dispersion Spectrograph (HDS; Noguchi et al. 2002) with the iodine-cell (Sato et al. 2002) on the Subaru telescope, and the FIbr-fed Échelle Spectrograph (FIES) on the 2.5 m Nordic Optical Telescope (Djupvik & Andersen 2010). Table 3 summarizes the observations. The table also provides references for the methods used to reduce the data to relative RVs in the solar system barycentric frame. The resulting RV measurements and their uncertainties are listed in Table 4. The different instrumental uncertainties arise from different slit widths, exposure times, and seeing conditions. The period-folded data, along with our best fit described below in Section 3, are displayed in Figure 3.
Figure 3. First row: Keck/HIRES (squares), Subaru (circles), and FIES (triangles) RV measurements for HAT-P-31, along with our best-fit two-planet model (see Table 7). The center-of-mass velocity has been subtracted. Second row: same as the top panel except the RV model of the inner planet has been subtracted from the data and the model, revealing the orbit of the outer planet. The
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Standard image High-resolution imageTable 3. Summary of High-resolution, High-S/N Spectroscopy Observations of HAT-P-31
Instrument | Date | Number of | Reduction |
---|---|---|---|
Range | RV Obs. | Reference | |
HDS | 2009 Aug 8–2010 May 24 | 25 | 1 |
HIRES | 2010 Feb 24–2010 Jul 3 | 9 | 2 |
FIES | 2009 Oct 6–2009 Oct 11 | 6 | 3 |
Notes. aThe mean heliocentric RV of the target. Systematic differences between the velocities from the two instruments are consistent with the velocity zero-point uncertainties. References. (1) Sato et al. 2005; (2) Butler et al. 1996; (3) Buchhave et al. 2010.
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Table 4. Relative Radial Velocities, Bisector Spans, and Activity Index Measurements of HAT-P-31
BJDa | RVb | BS | Sd | Instrument | ||
---|---|---|---|---|---|---|
[2,454,000+] | (m s−1) | (m s−1) | (m s−1) | (m s−1) | ||
1051.87826 | −30.72 | 6.99 | ... | ... | ... | Subaru |
1051.88946 | −27.97 | 6.90 | ... | ... | ... | Subaru |
1051.90067 | −44.27 | 6.56 | ... | ... | ... | Subaru |
1052.81471 | −163.76 | 9.25 | ... | ... | ... | Subaru |
1052.93238 | −200.89 | 6.63 | ... | ... | ... | Subaru |
1052.94706 | −202.52 | 7.29 | ... | ... | ... | Subaru |
1053.80433 | −159.83 | 7.09 | ... | ... | ... | Subaru |
1053.81555 | −166.62 | 7.47 | ... | ... | ... | Subaru |
1053.82676 | −163.36 | 7.95 | ... | ... | ... | Subaru |
1053.89544 | −145.48 | 7.08 | ... | ... | ... | Subaru |
1053.90665 | −140.21 | 6.46 | ... | ... | ... | Subaru |
1053.91786 | −136.05 | 6.39 | ... | ... | ... | Subaru |
1111.33488 | 71.48 | 9.40 | ... | ... | ... | FIES |
1112.33770 | −115.00 | 10.70 | ... | ... | ... | FIES |
1113.34016 | −225.79 | 10.50 | ... | ... | ... | FIES |
1114.34798 | 12.92 | 10.70 | ... | ... | ... | FIES |
1115.38654 | 234.94 | 10.80 | ... | ... | ... | FIES |
1116.33401 | 101.33 | 24.70 | ... | ... | ... | FIES |
1252.10465 | ... | ... | −2.81 | 1.78 | 0.1280 | Keck |
1252.11375 | −34.70 | 2.65 | 5.07 | 1.72 | 0.1190 | Keck |
1286.09877 | 153.49 | 2.56 | 2.47 | 1.48 | 0.1270 | Keck |
1338.90673 | −199.81 | 11.70 | ... | ... | ... | Subaru |
1338.91100 | −199.43 | 11.83 | ... | ... | ... | Subaru |
1338.91526 | −196.43 | 10.73 | ... | ... | ... | Subaru |
1338.91953 | −188.53 | 11.15 | ... | ... | ... | Subaru |
1339.85945 | 129.95 | 16.85 | ... | ... | ... | Subaru |
1339.87200 | 140.14 | 11.90 | ... | ... | ... | Subaru |
1339.88669 | 151.20 | 9.63 | ... | ... | ... | Subaru |
1339.95891 | 180.56 | 7.45 | ... | ... | ... | Subaru |
1339.97360 | 179.56 | 7.50 | ... | ... | ... | Subaru |
1339.98489 | 186.81 | 8.74 | ... | ... | ... | Subaru |
1341.08414 | 167.01 | 13.93 | ... | ... | ... | Subaru |
1341.09189 | 158.61 | 13.28 | ... | ... | ... | Subaru |
1341.10303 | 151.27 | 9.77 | ... | ... | ... | Subaru |
1343.00801 | −167.85 | 2.30 | 10.12 | 1.17 | 0.1350 | Keck |
1372.86477 | −139.73 | 2.11 | −3.69 | 1.30 | 0.1230 | Keck |
1374.99015 | 177.54 | 1.84 | 0.74 | 1.16 | 0.1240 | Keck |
1375.86758 | 207.17 | 1.90 | −3.49 | 1.11 | 0.1240 | Keck |
1378.79885 | −228.26 | 1.90 | −4.23 | 1.29 | 0.1230 | Keck |
1380.79961 | 212.57 | 1.93 | −4.20 | 1.12 | 0.1220 | Keck |
Notes. For the iodine-free template exposures there is no RV measurement, but the BS and S index can still be determined.
aBarycentric Julian dates throughout the paper are calculated from Coordinated Universal Time.
bThe zero point of these velocities is arbitrary. An overall offset
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One false-alarm possibility is that the observed RVs are not induced by a planetary companion, but are instead caused by distortions in the spectral line profiles due to contamination from a nearby unresolved eclipsing binary. This hypothesis may be tested by examining the spectral line profiles for contamination from a nearby unresolved eclipsing binary (Queloz et al. 2001; Torres et al. 2007). A bisector analysis based on the Keck spectra was performed as described in Section 5 of Bakos et al. (2007). The resulting bisector spans, plotted in Figure 3, show no significant variation, and are not correlated with the RVs, indicating that this is a real TEP system.
In the same figure, one can also see the S index (Vaughan et al. 1978), which is a quantitative measure of the chromospheric activity of the star derived from the flux in the cores of the Ca ii H and K lines (Isaacson & Fischer 2010). Following Noyes et al. (1984) we find that HAT-P-31 has an activity index log R'HK = −5.312, implying that this is a very inactive star.
3. ANALYSIS
The analysis of the HAT-P-31 system, including determinations of the properties of the host star and planet, was carried out in a similar fashion to previous HATNet discoveries (e.g., Bakos et al. 2010). Below, we briefly summarize the procedure and the results for the HAT-P-31b system.
3.1. Properties of the Parent Star
Stellar atmospheric parameters were measured from our template Keck/HIRES spectrum using the Spectroscopy Made Easy (SME; Valenti & Piskunov 1996) analysis package and the atomic line database of Valenti & Fischer (2005). SME yielded the following values and uncertainties: effective temperature Teff⋆ = 6065 ± 100 K, metallicity [Fe/H] = +0.15 ± 0.08 dex, and stellar surface gravity log g⋆ = 4.26+0.11−0.13 (cgs), projected rotational velocity vsin i = 0.5 ± 0.6 km s−1.
The above atmospheric parameters are then combined with the Yonsei–Yale (YY; Yi et al. 2001) series of stellar evolution models to determine other parameters such as the stellar mass, radius, and age. The results are listed in Table 5. We find that the star has a mass and radius of M⋆ = 1.218+0.089−0.063 M☉ and R⋆ = 1.36+0.27−0.18 R☉, and an estimated age of 3.17+0.70−1.11 Gyr.
Table 5. Stellar Parameters for HAT-P-31
Parameter | Value | Source |
---|---|---|
Spectroscopic properties | ||
Teff⋆ (K)....... | 6065 ± 100 | SMEa |
[Fe/H]........ | 0.15 ± 0.08 | SME |
vsin i (km s−1).. | 0.5 ± 0.6 | SME |
vmac (km s−1)b.. | 4.47 | SME |
vmic (km s−1)b.. | 0.85 | SME |
−2.40 ± 0.03 | TRES | |
Photometric properties | ||
V (mag)....... | 11.660 | TASS |
V − IC (mag)... | 0.67 ± 0.17 | TASS |
J (mag)........ | 10.423 ± 0.023 | 2MASS |
H (mag)....... | 10.128 ± 0.027 | 2MASS |
Ks (mag)....... | 10.083 ± 0.021 | 2MASS |
Derived properties | ||
M⋆ (M☉)...... | 1.218+0.089−0.063 | YY+SME c |
R⋆ (R☉)....... | 1.36+0.27−0.18 | YY+SME |
log (g* (cgs)).... | 4.26+0.11−0.13 | YY+SME |
L⋆ (L☉)....... | 2.23+1.01−0.58 | YY+SME |
MV (mag)...... | 3.91+0.34−0.41 | YY+SME |
MK (mag,ESO).. | 2.64 ± 0.30 | YY+SME |
Age (Gyr)...... | 3.17+0.70−1.11 | YY+SME |
Distance (pc)... | 354+74−51 | YY+SME |
Notes. aSME: "Spectroscopy Made Easy" package for the analysis of high-resolution spectra (Valenti & Piskunov 1996). bAssumed quantity based upon derived spectral type. cYY+SME: based on the YY isochrones (Yi et al. 2001) and the SME results.
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For previous HATNet planets (e.g., Bakos et al. 2010) we used the normalized semimajor axis, a/R⋆ which is closely related to
Figure 4. Model isochrones from Yi et al. (2001) for the measured metallicity of HAT-P-31, [Fe/H] = 0.15, and ages from 1 to 14 Gyr in steps of 1 Gyr (gray-scale lines, left to right). The adopted values of Teff⋆ and log g⋆ determined from the SME analysis are shown together with their 1
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Standard image High-resolution imageAs an additional check on the stellar evolution modeling, we note that HAT-P-31 has a measured near-infrared color of J − K = 0.364 ± 0.034, which we have taken from 2MASS (Skrutskie et al. 2006) using the Carpenter (2001) transformation to the ESO photometric system. This is within 2
3.2. Global Modeling of the Data
3.2.1. Photometry
In previous HATNet papers, we have used a simplified model for the transit light curve of the HATNet data. For HAT-P-31, no precise photometry exists and thus we fit the HATNet data using a more sophisticated quadratic limb-darkening Mandel & Agol (2002) algorithm with limb-darkening coefficients interpolated from the tables by Claret (2004). One caveat is that the instrumental blending factor, Binst, is unknown as discussed earlier in Section 2. We point out that experience with previous HATNet planets suggests Binst is within 2
Due to the low-precision photometry, the stellar density cannot be determined to high precision using the method of Seager & Mallén-Ornelas (2003) and in fact spectroscopic estimates were found to be more precise. However, we can reverse this well-known trick by implementing a Bayesian prior in our fitting process for the stellar density.
We use the spectroscopically determined stellar density from Section 3.1 as a prior in our fits. Since the period of the transiting planet is well constrained for even low signal-to-noise photometry, reasonably precise estimates for Pb and
The photometry which is fitted in the global modeling is corrected for instrumental systematics through the EPD and reconstructive TFA correction procedures prior to performing the fit (see Section 2.1 and Bakos et al. 2010 for details).
3.2.2. Radial Velocities
For the RV fits, we found a single planet fit gave a very poor fit to the observations (
Table 6. Comparison of RV Models Attempted for HAT-P-31
Model | BICa | |
---|---|---|
Circular planet................ | 3714.9 | 3729.6 |
Eccentric planet............... | 204.2 | 226.3 |
Eccentric planet + drift.......... | 159.6 | 185.5 |
Eccentric planet + trojan......... | 191.1 | 216.9 |
Eccentric planet + drift + trojan ... | 113.2 | 142.7 |
Eccentric planet + drift + quadratic | 33.9 | 63.4 |
Eccentric planet + circular planet.. | 33.6 | 66.8 |
Eccentric planet + eccentric planet | 33.2 | 73.8 |
Note. aBIC: Bayesian Information Criterion (Schwarz 1978; Liddle 2007).
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The most likely physical explanation for a quadratic trend is a third body in the system, described by a Keplerian model. Indeed, the Keplerian model provides an improved
Figure 5. Top row: marginalized posterior distributions for Kc and Pc when we attempted to fit for a second Keplerian signal, instead of a quadratic trend. The multi-MODAL nature of these histograms reflect the unconverged nature of the fits. Bottom row: history of the MCMC trials for the same parameters and the same fit. Each continuous line represents one of the 10 independent MCMC chains. The lines clearly illustrate the inability of the current data to converge upon a solution for HAT-P-31c.
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Standard image High-resolution imageTable 7. Global Fit Results for HAT-P-31
Parametera | Value |
---|---|
Fitted parameters | |
Pb (days) | 5.005425+0.000091−0.000092 |
4320.8866+0.0051−0.0053 | |
![]() |
16300+1000−1000 |
p2b (%) | 0.65+0.18−0.12 |
bb | 0.57+0.23−0.31 |
OOT | 1.00061+0.00016−0.00016 |
Kb (ms−1) | 232.5+1.1−1.1 |
ebsin |
−0.2442+0.0043−0.0043 |
ebcos |
0.0185+0.0080−0.0079 |
−29.0+1.4−1.4 | |
18.8+3.1−3.1 | |
92.7+5.6−5.6 | |
![]() |
0.141+0.025−0.025 |
![]() |
0.00226+0.00021−0.00021 |
SME derived quantities | |
u1b,e | 0.2078* |
u2b![]() |
0.3550* |
![]() |
0.69+0.34−0.26 |
HAT-P-31b derived properties | |
0.4737+0.0065−0.0064 | |
eb | 0.2450+0.0045−0.0045 |
274.3+1.8−1.8 | |
log (gb (cgs)) | 3.61+0.15−0.32 |
(ab/R*) | 8.9+1.4−2.3 |
ib (°) | 87.1+1.8−2.7 |
[T1, 4]b (s) | 18500+1700−1200 |
[T2, 3]b (s) | 14200+1300−2400 |
pb | 0.080+0.022−0.015 |
Mb (MJ) | 2.171+0.105−0.077 |
Rb (RJ) | 1.07+0.24−0.16 |
Corr(Rb,Mb) | 0.795 |
2.18+1.24−0.93 | |
ab ( |
0.055+0.015−0.015 |
[Teq]b (K) | 1450+230−110 |
0.190+0.036−0.056 | |
Fperi (109 erg s−1 cm−2) | 1.69+1.39−0.44 |
Fap (109 erg s−1 cm−2) | 0.62+0.51−0.16 |
〈F〉f (109 erg s−1 cm−2) | 0.99+0.82−0.26 |
HAT-P-31c derived quantities | |
5254.9+7.4−6.8 | |
KcP−2c (ms−1 day−2) | 7.64+0.71−0.70 |
Pc (years) | ⩾2.8 |
Kc (ms−1) | ⩾60 |
Mc (MJ) | ⩾3.4 |
Notes.
a
Quoted values are medians of MCMC trials with errors given by 1 (see Hansen & Barman 2007).
fIncoming flux per unit surface area, averaged over the orbit.
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The quadratic model may be used to infer some physical parameters of the third planet. To make some meaningful progress, we will assume the outer planet is on a circular orbit. One may compare the quadratic and Keplerian model descriptions via
![Equation (1)](https://content.cld.iop.org/journals/1538-3881/142/3/95/revision1/aj400226fd1.gif)
![Equation (2)](https://content.cld.iop.org/journals/1538-3881/142/3/95/revision1/aj400226fd2.gif)
By differentiating both expressions and solving for the time when the signals are minimized, one may write
![Equation (3)](https://content.cld.iop.org/journals/1538-3881/142/3/95/revision1/aj400226fd3.gif)
Differentiating both RV models with respect to time twice and evaluating at the moment when both signals are minimized yields
![Equation (4)](https://content.cld.iop.org/journals/1538-3881/142/3/95/revision1/aj400226fd4.gif)
Equations (3) and (4) may therefore be used to determine some information about HAT-P-31c.
To evaluate the statistical significance of HAT-P-31c, we performed an F-test between the one-planet and two-planet models. In both cases, HAT-P-31b is assumed to maintain non-zero orbital eccentricity. Assuming HAT-P-31c is on a circular orbit, the false-alarm probability (FAP) from an F-test is 3.0 × 10−12, or 7.0
3.2.3. Fitting Algorithm
We utilize a Metropolis–Hastings Markov Chain Monte Carlo (MCMC) algorithm to globally fit the data, including the stellar density prior (our routine is described in Kipping & Bakos (2011)). To ensure the parameter space is fully explored, we used five independent MCMC fits which stop once 1.25 × 105 trials have been accepted and burn out the first 20%. This leaves us with a total of 5 × 105 points for the posterior distributions. At the end of the fit, a more aggressive downhill simplex
There were 14 free parameters in the global fit: {, bb, Pb, ebsin
,
, OOT}, which we elaborate on here.
is the "one-term" approximate equation (Kipping 2010) for the transit duration between the instant when the center of the planet crosses the stellar limb to exiting under the same condition. bb is the impact parameter, defined as the sky-projected planet–star separation in units of the stellar radius at the instant of inferior conjunction. eb is the orbital eccentricity and
(drift) and
(curl) are the first and second time derivatives of
Final quoted results are the median of the marginalized posterior for each fitted parameter with 34.15% quantiles either side for the 1
Figure 6. Posterior distributions of the fitted parameters used in the global fits (described in Section 3.2) from our global fits. Histograms computed from 5 × 105 MCMC trials. Bottom-right panel shows joint-posterior of the radial velocity drift and curl, with white denoting the 2
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Standard image High-resolution imageTable 7 provides estimates for some minimum limits on various parameters of interest relating to HAT-P-31c. These limits are determined by the known constraints on the minimum Pc. We determined this value by forcing a circular orbit Keplerian fit for planet c through the data, stepping through a range of periods from 1 year to 5 years in 1 day steps. The minimum limit on Pc is defined as when
4. DISCUSSION
4.1. Physical Properties of HAT-P-31b and c
HAT-P-31b is a 2.17 MJ hot Jupiter transiting the host star once every 5.005 days. Due to the lack of follow-up photometry obtained for this object (as a consequence of the nearly integer orbital period), we have only HATNet photometry, which is of lower S/N than dedicated follow-up. This fact, combined with our choice to double all uncertainties on depth related transit terms, leads to a large uncertainty on the planetary radius of Rb = 1.07+0.48−0.32 RJ, consistent with many other hot-Jupiter objects (see http://exoplanet.eu).
High-precision RVs also indicate the presence of an outer body, HAT-P-31c, found through an induced quadratic trend in the RV residuals. Keplerian fits are unable to convincingly distinguish between a circular or eccentric orbit for this object. HAT-P-31c has a minimum mass of Mc ⩾ 3.4 MJ and eccentric orbit solutions significantly increase this figure. It is unclear whether HAT-P-31c is a brown dwarf or a "planet," and future work will be needed to determine this.
4.2. Orbital Stability
4.2.1. Circular Fit for HAT-P-31c
Here we discuss our procedure to test the dynamical stability of two possible orbital configurations. It should be noted that the eccentricity of HAT-P-31b is highly secure but the eccentricity of HAT-P-31c remains unclear. For this reason, we repeat our simulations assuming both a circular and eccentric orbit for HAT-P-31c, beginning with the former. We utilize the Systemic Console (Meschiari et al. 2009) for this purpose assuming a coplanar configuration. Employing the Gragg–Bulirsch–Stoer integrator, orbital evolution was computed for 250,000 years for the HAT-P-31 system (see Figure 7).
Figure 7. Two possible realizations for the orbital evolution of planets HAT-P-31b and c. The solid lines show the evolution starting from an eccentric orbit solution for HAT-P-31c. The dashed lines show the evolution starting from a circular orbit solution for HAT-P-31c.
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Standard image High-resolution imageWe first consider the circular case. The orbital period and mass of HAT-P-31c are non-convergent parameters and so we can only provide an orbital solution which gives a good fit to the data, but is not necessarily unique. To this end, we proceeded to input HAT-P-31c with Pc = 4.86 years and Mc = 13.0 MJ, corresponding to the solution presented in Table 6. This test revealed minor evolution over the course of our simulations, indicating a stable and essentially static configuration.
4.2.2. Eccentric Fit for HAT-P-31c
To test the eccentric fit, we again used the lowest 027) using the expressions of Kipping (2010).
The orbital period and semimajor axes of both bodies were stable over the 250,000 years of integration considered here.
4.2.3. Habitable-zone Bodies
We tried adding a habitable-zone Earth-mass planet on a circular orbit into the system and testing stability. One may argue that the probable history of this system involved the inward migration of HAT-P-31b and that this migration through the inner protoplanetary disk would essentially eliminate the possibility of an Earth-like planet forming in the habitable-zone. However, Fogg & Nelson (2007) have shown that this not necessarily true. In their simulations, it is found that >60% of the solid disk survives, including planetesimals and protoplanets, by being scattered by the giant planet into external orbits where dynamical friction is strong and terrestrial planet formation is able to resume. In one simulation, a planet of 2 M⊕ formed in the habitable-zone after a hot-Jupiter passed through and its orbit stabilized at 0.1
For a planet to receive the same insolation as the Earth, we estimate P = 604 days. For our circular orbit solution of HAT-P-31c, the habitable-zone Earth-mass planet is stable for over 100,000 years. For our eccentric orbit solution, the Earth-like planet is summarily ejected in less than 1000 years.
4.3. Circularization Timescale
Due to the poor constraints on the planetary radius, there is a great deal of uncertainty in the circularization timescale (
HATNet operations have been funded by NASA grants NNG04GN74G, NNX08AF23G, and SAO IR&D grants. D.M.K. has been supported by Smithsonian Institution Restricted Endowment Funds. Work of G.Á.B. and J. Johnson were supported by the Postdoctoral Fellowship of the NSF Astronomy and Astrophysics Program (AST-0702843 and AST-0702821, respectively). G.T. acknowledges partial support from NASA grant NNX09AF59G. We acknowledge partial support also from the Kepler Mission under NASA Cooperative Agreement NCC2-1390 (D.W.L., PI). G.K. thanks the Hungarian Scientific Research Foundation (OTKA) for support through grant K-60750. L.L.K. is supported by the "Lendulet" Young Researchers Program of the Hungarian Academy of Sciences and the Hungarian OTKA grants K76816, K83790, and MB08C 81013. Tamás Szalai (University of Szeged) is acknowledged for his assistance during the ANU 2.3 m observations. This research has made use of Keck telescope time granted through NASA (N167Hr). This work is based in part on data collected at Subaru Telescope, which is operated by the National Astronomical Observatory of Japan, and in part on observations made with the Nordic Optical Telescope, operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway, and Sweden, in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias. Thanks to G. Laughlin for useful advise on the Systemic Console, Dan Fabrycky and René Heller for useful comments, and special thanks to the anonymous referee for their helpful suggestions.
Footnotes
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Based in part on observations obtained at the W. M. Keck Observatory, which is operated by the University of California and the California Institute of Technology. Keck time has been granted by NASA (N167Hr). Based in part on data collected at Subaru Telescope, which is operated by the National Astronomical Observatory of Japan. Based in part on observations made with the Nordic Optical Telescope, operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway, and Sweden, in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias.