HAT-P-20b–HAT-P-23b: FOUR MASSIVE TRANSITING EXTRASOLAR PLANETS^*

Table 1 summarizes the HATNet discovery observations of each new planetary system. The calibration of the HATNet frames was carried out using standard procedures correcting for the CCD bias, dark current, and flat-field structure. The calibrated images were then 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. For certain data sets (HAT-P-20, HAT-P-22, HAT-P-23) we also carried out an image subtraction (Alard 2000) based photometric reduction using discrete kernels (Bramich 2008), as described in Pál (2009b). The resulting light curves were decorrelated (cleaned of trends) using the External Parameter Decorrelation (EPD; see Bakos et al. 2010) technique in "constant" mode and the Trend Filtering Algorithm (TFA; see Kovács et al. 2005). The light curves were searched for periodic box-shaped signals using the Box Least-squares (BLS; see Kovács et al. 2002) method. Candidates were passed through a number of automatic filters to reject false alarms and obvious eclipsing binaries or other variable stars (two examples include checking for differences between the depths of even and odd transits when the period is doubled, and checking for evidence of a secondary eclipse at the period of the transit), and then passed a by-eye selection procedure which primarily rejects obvious blends with nearby eclipsing binaries or cases where the candidate selection is due to an artifact which is not captured by one of the automated filters. We detected significant signals in the light curves of the stars as summarized below.

• 1.  
  HAT-P-20—GSC 1910-00239 (also known as 2MASS 07273995+2420118; $\alpha = 07^{\mathrm h}27^{\mathrm m}39\mbox{$.\!\!^{\mathrm s}$}96$, δ = +24°20'119; J2000; V = 11.339). A signal was detected for this star with an apparent depth of ∼10.8 mmag, and a period of P = 2.8753 days (see Figure 1). Note that the apparent HATNet depth stated here is attenuated by the presence of the fainter neighbor star that is not resolved on the coarse resolution (95 pixel⁻¹) HATNet pixels. Also, the depth by fitting a trapezoid instead of the correct Mandel & Agol (2002) model is somewhat shallower than the maximum depth in the Mandel & Agol (2002) model fit (which was 19.6 mmag; see later in Section 3.3). Note that the dilution in the HATNet light curve is accounted for by the dilution factor parameter included in the global fitting procedure described in Section 3.3. The follow-up light curves discussed in Section 2.4 are obtained with instruments that have much higher spatial resolution, and when necessary are reduced using the image subtraction technique, so that they are not affected by dilution from any known neighbors. The drop in brightness had a first-to-last-contact duration, relative to the total period, of q = 0.0268  ±  0.0003, corresponding to a total duration of Pq = 1.848  ±  0.019 hr. HAT-P-20 has a red companion (2MASS 07273963+2420171, J − K = 0.92) at 686 separation that is fainter than HAT-P-20 by ΔR = 1.36 mag.
• 2.  
  HAT-P-21—GSC 3013-01229 (also known as 2MASS 11250598+4101406; $\alpha = 11^{\mathrm h}25^{\mathrm m}05\mbox{$.\!\!^{\mathrm s}$}88$, δ = +41°01'406; J2000; V = 11.685). A signal was detected for this star with an apparent depth of ∼8.0 mmag, and a period of P = 4.1245 days (see Figure 2). The drop in brightness had a first-to-last-contact duration, relative to the total period, of q = 0.0371  ±  0.0007, corresponding to a total duration of Pq = 3.672  ±  0.065 hr.
• 3.  
  HAT-P-22—HD 233731 (also known as GSC 03441-00925 and 2MASS 10224361+5007420; $\alpha = 10^{\mathrm h}22^{\mathrm m}43\mbox{$.\!\!^{\mathrm s}$}68$, δ = +50°07'420; J2000; V = 9.732). A signal was detected for this star with an apparent depth of ∼9.7 mmag, and a period of P = 3.2122 days (see Figure 3). The drop in brightness had a first-to-last-contact duration, relative to the total period, of q = 0.0372  ±  0.0004, corresponding to a total duration of Pq = 2.869  ±  0.033 hr. HAT-P-22 has a close red companion star (2MASS 10224397+5007504, J − K = 0.86) at 91 separation and Δi = 2.58 mag fainter.
• 4.  
  HAT-P-23—GSC 1632-01396 (also known as 2MASS 20242972+1645437; $\alpha = 20^{\mathrm h}24^{\mathrm m}29\mbox{$.\!\!^{\mathrm s}$}88$, δ = +16°45'437; J2000; V = 12.432). A signal was detected for this star with an apparent depth of ∼11.5 mmag, and a period of P = 1.2129 days (see Figure 4). Similarly to HAT-P-20, the depth was attenuated by close-by faint neighbors. The drop in brightness had a first-to-last-contact duration, relative to the total period, of q = 0.0748  ±  0.0006, corresponding to a total duration of Pq = 2.178  ±  0.017 hr.

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As is routine in the HATNet project, all candidates are subjected to careful scrutiny before investing valuable time on large telescopes. This includes spectroscopic observations at relatively modest facilities to establish whether the transit-like feature in the light curve of a candidate might be due to astrophysical phenomena other than a planet transiting a star. Many of these false positives are associated with large RV variations in the star (tens of kilometers per second) that are easily recognized. The reconnaissance spectroscopic observations and results for each system are summarized in Table 2; below we provide a brief description of the instruments used, the data reduction, and the analysis procedure.

One of the tools we have used for this purpose is the Harvard-Smithsonian Center for Astrophysics (CfA) Digital Speedometer (DS; Latham 1992), an echelle spectrograph mounted on the FLWO 1.5 m telescope. This instrument delivers high-resolution spectra (λ/Δλ ≈ 35, 000) over a single order centered on the Mg i b triplet (∼5187 Å), with typically low S/Ns that are nevertheless sufficient to derive RVs with moderate precisions of 0.5–1.0 km s⁻¹ for slowly rotating stars. The same spectra can be used to estimate the effective temperature, surface gravity, and projected rotational velocity of the host star, as described by Torres et al. (2002). With this facility we are able to reject many types of false positives, such as F dwarfs orbited by M dwarfs, grazing eclipsing binaries, and many triple or quadruple star systems. Additional tests are performed with other spectroscopic material described in the next section.

Another of the tools we have used for this purpose is the Fiber-fed Échelle Spectrograph (FIES) at the 2.5 m Nordic Optical Telescope (NOT) at La Palma, Spain (Djupvik & Andersen 2010). We used the medium-resolution fiber which produces spectra at a resolution of λ/Δλ ≈ 46, 000 and a wavelength coverage of ∼3600–7400 Å to observe HAT-P-20. The spectrum was extracted and analyzed to measure the RV, effective temperature, surface gravity, and projected rotation velocity of the host star, following the procedures described by Buchhave et al. (2010).

Based on the observations summarized in Table 2, we find that HAT-P-21, HAT-P-22, and HAT-P-23 have rms residuals consistent with no detectable RV variation within the precision of the measurements. The reconnaissance observations of HAT-P-20 hinted at a K ∼ 1 km s⁻¹ orbital variation roughly in phase with the photometric ephemeris (these velocities are consistent with the K ∼ 1.2 km s⁻¹ orbit of HAT-P-20 measured subsequently with Keck/HIRES given the ∼0.5 km s⁻¹ precision of the DS). All spectra were single-lined, i.e., there is no evidence that any of these targets consist of more than one star. The gravities for all of the stars indicate that they are dwarfs.

We proceeded with the follow-up of each candidate by obtaining high-resolution, high-S/N spectra to characterize the RV variations, and to refine the determination of the stellar parameters. These observations are summarized in Table 3. The RV measurements and uncertainties are given in Tables 4–7 for HAT-P-20–HAT-P-23, respectively. The period-folded data, along with our best fits described below in Section 3, are displayed in Figures 6–9 for HAT-P-20–HAT-P-23. Below we briefly describe the instruments used, the data reduction, and the analysis procedure.

Observations were made of all four planet host stars with the HIRES instrument (Vogt et al. 1994) on the Keck I telescope located on Mauna Kea, Hawaii. The width of the spectrometer slit was 086, resulting in a resolving power of λ/Δλ ≈ 55, 000, with a wavelength coverage of ∼3800–8000 Å. We typically used the B5 decker yielding a 35(H) × 0861(W) slit, and for the last few observations on each target we used the C2 decker that enables a better sky subtraction due to the longer slit 140(H) × 0861(W). The slit height was oriented with altitude (vertical), except for rare cases, when the slit would have run through the faint companion to HAT-P-20 or HAT-P-22. A Keck/HIRES snapshot for each planet host star is shown in Figure 5. Spectra were obtained through an iodine gas absorption cell, which was used to superimpose a dense forest of I₂ lines on the stellar spectrum and establish an accurate wavelength fiducial (see Marcy & Butler 1992). For each target an additional exposure was taken without the iodine cell, for use as a template in the reductions. Relative RVs in the solar system barycentric frame were derived as described by Butler et al. (1996), incorporating full modeling of the spatial and temporal variations of the instrumental profile.

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In each of Figures 6–9 we show also the relative S index, which is a measure of the chromospheric activity of the star derived from the flux in the cores of the Ca ii H and K lines. This index was computed following the prescription given by Vaughan et al. (1978), and as described in Hartman et al. (2009). Note that our relative S index has not been calibrated to the scale of Vaughan et al. (1978). We do not detect any significant variation of the index correlated with orbital phase; such a correlation might have indicated that the RV variations could be due to stellar activity, casting doubt on the planetary nature of the candidate.

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In order to permit a more accurate modeling of the light curves, we conducted additional photometric observations with the KeplerCam CCD camera on the FLWO 1.2 m telescope for each star, and with the Faulkes North Telescope (FTN) of the Las Cumbres Observatory Global Network (LCOGT) at Hawaii for HAT-P-21 only. The observations for each target are summarized in Table 1.

The reduction of these images, including basic calibration, astrometry, and aperture photometry, was performed as described by Bakos et al. (2010). In each case we obtained light curves for all stars in the 2MASS catalog within the field of view of the observations and confirm that the primary target is the source of the transit rather than one of the known neighboring stars. We found that the aperture photometry for HAT-P-20 was significantly affected by the close-by neighbor star 2MASS 07273995+2420118 with Δi = 1.1 mag difference at 686 separation (Figure 5). Thus, we performed image subtraction on the FLWO 1.2 m images with the same toolset used for the HATNet reductions, but applied a discrete kernel with half-size of 5 pixels and no spatial variations. Indeed, for this stellar configuration, the image subtraction results proved to be superior to the aperture photometry. For all of the follow-up light curves, we performed EPD and TFA to remove trends simultaneously with the light curve modeling (for more details, see Section 3, and 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 10–13; the individual measurements are reported in Tables 8–11, for HAT-P-20 through HAT-P-23, respectively.

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Fundamental parameters for each of the host stars, including the mass (M_⋆) and radius (R_⋆), which are needed to infer the planetary properties, depend strongly on other stellar quantities that can be derived spectroscopically. For this we have relied on 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 (which we have conservatively doubled for the temperature and metallicity to include our estimates of the systematic errors).

• 1.  
  HAT-P-20—effective temperature T_eff⋆ = 4626  ±  104 K, stellar surface gravity log g_⋆ = 4.80  ±  0.1 (cgs), metallicity [Fe/H] = +0.31  ±  0.08 dex, and projected rotational velocity vsin i = 3.1  ±  0.5 km s⁻¹.
• 2.  
  HAT-P-21—effective temperature T_eff⋆ = 5701  ±  144 K, stellar surface gravity log g_⋆ = 4.48  ±  0.14 (cgs), metallicity [Fe/H] = +0.04  ±  0.1 dex, and projected rotational velocity vsin i = 4.1  ±  0.5 km s⁻¹.
• 3.  
  HAT-P-22—effective temperature T_eff⋆ = 5338  ±  88 K, stellar surface gravity log g_⋆ = 4.52  ±  0.14 (cgs), metallicity [Fe/H] = +0.26  ±  0.08 dex, and projected rotational velocity vsin i = 1.5  ±  0.5 km s⁻¹.
• 4.  
  HAT-P-23—effective temperature T_eff⋆ = 5987  ±  120 K, stellar surface gravity log g_⋆ = 4.48  ±  0.12 (cgs), metallicity [Fe/H] = +0.18  ±  0.1 dex, and projected rotational velocity vsin i = 8.0  ±  0.5 km s⁻¹.

The effective temperature and metallicity, along with the surface gravity taken as a luminosity indicator, can be used as constraints to infer the stellar mass and radius by comparison with stellar evolution models. However, the effect of log g_⋆ on the spectral line shapes is rather subtle, and as a result it is typically difficult to determine accurately, so that it is a rather poor luminosity indicator in practice. Unfortunately, a trigonometric parallax is not available for any of the host stars, since they were not included among the targets of the Hipparcos mission (Perryman et al. 1997). For planetary transits, another constraint is provided by the a/R_⋆ normalized semi-major axis, which is closely related to ρ_⋆, the mean stellar density. The quantity a/R_⋆ can be derived directly from the combination of the transit light curves (Sozzetti et al. 2007) and the RV data (required for eccentric cases; see Section 3.3). This, in turn, allows us to improve on the determination of the spectroscopic parameters by supplying an indirect constraint on the weakly determined spectroscopic value of log g_⋆ that removes degeneracies. We take this approach here, as described below.

For each system, our initial values of T_eff⋆, log g_⋆, and [Fe/H] were used to determine auxiliary quantities needed in the global modeling of the follow-up photometry and RVs (specifically, the limb-darkening coefficients). This modeling, the details of which are described in Section 3.3, uses a Monte Carlo approach to deliver the numerical probability distribution of a/R_⋆ and other fitted variables. When combining a/R_⋆ (used as a proxy for luminosity) with assumed Gaussian distributions for T_eff⋆ and [Fe/H] based on the SME determinations, a comparison with stellar evolution models allows the probability distributions of other stellar properties to be inferred, including log g_⋆. Here we use the stellar evolution calculations from the Yonsei–Yale group (YY; Yi et al. 2001) for all planets presented in this work. The comparison against the model isochrones was carried out for each of 10,000 Monte Carlo trial sets for HAT-P-21, HAT-P-22, and HAT-P-23, and for 20,000 Monte Carlo trial sets for HAT-P-20 (see Section 3.3). Parameter combinations corresponding to unphysical locations in the H-R diagram (26% of the trials for HAT-P-20, and less than 1% of the trials for the other objects) were ignored, and replaced with another randomly drawn parameter set. For each system we carried out a second SME iteration in which we adopted the value of log g_⋆ so determined and held it fixed in a new SME analysis (coupled with a new global modeling of the RV and light curves), adjusting only T_eff⋆, [Fe/H], and vsin i. This gave the following:

• 1.  
  HAT-P-20: log g_⋆ = 4.64  ±  0.06, T_eff⋆ = 4595  ±  80 K, [Fe/H] = +0.35  ±  0.08, and vsin i = 2.1  ±  0.5 km s⁻¹.
• 2.  
  HAT-P-21: log g_⋆ = 4.31  ±  0.06, T_eff⋆ = 5588  ±  80 K, [Fe/H] = +0.01  ±  0.08, and vsin i = 3.5  ±  0.5 km s⁻¹.
• 3.  
  HAT-P-22: log g_⋆ = 4.37  ±  0.06, T_eff⋆ = 5302  ±  80 K, [Fe/H] = +0.24  ±  0.08, and vsin i = 0.5  ±  0.5 km s⁻¹.
• 4.  
  HAT-P-23: log g_⋆ = 4.33  ±  0.06, T_eff⋆ = 5905  ±  80 K, [Fe/H] = +0.15  ±  0.04, and vsin i = 8.1  ±  0.5 km s⁻¹.

In each case the conservative uncertainties for T_eff⋆ and [Fe/H] have been increased by a factor of two over their formal values, as before. For each system, a further iteration did not change log g_⋆ significantly, so we adopted the values stated above as the final atmospheric properties of the stars. They are collected in Table 12.

With the adopted spectroscopic parameters the model isochrones yield the stellar mass and radius, and other properties. These are listed for each of the systems in Table 12. According to these models HAT-P-20 is a dwarf star with an estimated age of 6.7^+5.7_{− 3.8} Gyr, HAT-P-21 is a slightly evolved star with an estimated age of 10.2  ±  2.5 Gyr, HAT-P-22 is a slightly evolved star with an estimated age of 12.4  ±  2.6 Gyr, and HAT-P-23 is a slightly evolved star with an estimated age of 4.0  ±  1.0 Gyr. The inferred location of each star in a diagram of a/R_⋆ versus T_eff⋆, analogous to the classical H-R diagram, is shown in Figure 14. In all cases the stellar properties and their 1σ and 2σ confidence ellipsoides 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 are also shown (in each case with a triangle).

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The stellar evolution modeling provides color indices (see Table 12) that may be compared against the measured values as a sanity check. For each star, the best available measurements are the near-infrared magnitudes from the 2MASS Catalog (Skrutskie et al. 2006), which are given in Table 12. These are converted to the photometric system of the models (ESO system) using the transformations by Carpenter (2001). The resulting color indices are also shown in Table 12 for HAT-P-20 through HAT-P-23, respectively. Indeed, the colors from the stellar evolution models and from the observations agree for all of the host stars within 2σ. The distance to each object was computed from the absolute K magnitude from the models and the 2MASS K_s magnitudes, which has the advantage of being less affected by extinction than optical magnitudes. The results are given in Table 12, where in all cases the uncertainty excludes possible systematics in the model isochrones that are difficult to quantify.

Our initial spectroscopic analyses discussed in Sections 2.2 and 2.3 rule out the most obvious astrophysical false positive scenarios. However, more subtle phenomena such as blends (contamination by an unresolved eclipsing binary, whether in the background or associated with the target) can still mimic both the photometric and spectroscopic signatures we see. In the following section, we investigate whether such scenarios may have caused the observed photometric and spectroscopic features.

3.2.1. Spectral Line-bisector Analysis

Following Torres et al. (2007), we explored the possibility that the measured RVs are not real, but are instead caused by distortions in the spectral line profiles due to contamination from a nearby unresolved eclipsing binary. A bisector span (BS) analysis for each system based on the Keck spectra was done as described in Section 5 of Bakos et al. (2007). In general, none of the Keck/HIRES spectra suffer significant sky contamination. Nevertheless, we calculated the sky contamination factors (SCF) as described in Hartman et al. (2009), and corrected for the minor correlation between SCF and BS. The results are exhibited in Figure 15, where we show the SCF–BS and RV–BS_SCF (BS after SCF correction) plots for each planetary system. We also calculated the Spearman rank-order correlation coefficients (denoted as R_s for the RV versus BS quantities and the false alarm probabilities (see Table 13). There is no correlation for HAT-P-21, HAT-P-22, and HAT-P-23, and thus the interpretation of these systems as transiting planets is clear. There is an anti-correlation present for HAT-P-20, which is strengthened when the SCF correction is applied. A plausible explanation for this is that the neighboring star at 6'' separation (see Figure 5) is bleeding into the slit, even though we were careful during the observations to keep the slit centered on the main target, and adjusted the slit orientation to be perpendicular to the direction to the neighbor. We simulated this scenario, and calculated the expected BS as a function of RV due to the neighbor, assuming that the two stars have the same systemic velocity and the seeing is 1''. Indeed, we get a slight anti-correlation from this simulation, and the range of magnitude in the BS variation is consistent with the observations. It is also possible that some of the anti-correlation is due to the fact that the slit was not in vertical angle for many of the observations. The non-vertical slit mode may result in wavelength-dependent slit losses due to atmospheric dispersion, and this could bring in a correlation with the sky background, and change the shape of the spectral lines.

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Table 13. Summary of RV versus BS Correlations

Name	Rs1a	FAP1b	Rs2c	FAP2d
HAT-P-20	−0.73	2.46%	−0.92	0.05%
HAT-P-21	−0.24	39%	−0.20	47%
HAT-P-22	0.33	30%	0.27	37%
HAT-P-23	0.27	37%	0.27	37%

Notes. ^aThe Spearman correlation coefficient between the bisector (BS) variations and the radial velocities (RV). ^bFalse alarm probability for R_s1. ^cThe Spearman correlation coefficient between BS corrected for the sky contamination factor (SCF) and the RVs. ^dFalse alarm probability for R_s2.

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This section describes the procedure we followed for each system to model the HATNet photometry, the follow-up photometry, and the RVs simultaneously. Our model for the follow-up light curves used analytic formulae based on Mandel & Agol (2002) for the eclipse of a star by a planet, with limb darkening being prescribed by a quadratic law. The limb-darkening coefficients for the Sloan g-band, Sloan i-band, and Sloan z-band were interpolated from the tables by Claret (2004) for the spectroscopic parameters of each star as determined from the SME analysis (Section 3.1). The transit shape was parameterized by the normalized planetary radius p ≡ R_p/R_⋆, the square of the impact parameter b², and the reciprocal of the half-duration of the transit ζ/R_⋆. We chose these parameters because of their simple geometric meanings and the fact that these show negligible correlations (see Bakos et al. 2010). The relation between ζ/R_⋆ and the quantity a/R_⋆, used in Section 3.1, is given by

$$\begin{equation} a/R_\star = P/2\pi (\zeta /R_\star) \sqrt{1-b^2} \sqrt{1-e^2}/(1+e \sin \omega) \end{equation} \tag{ 1 }$$

(see, e.g., Tingley & Sackett 2005; Kipping 2010). Note the subtle dependency of a/R_⋆ on the k ≡ ecos ω and h ≡ esin ω Lagrangian orbital parameters that are typically derived from the RV data (ω is the longitude of periastron). This dependency is often ignored in the literature, and a/R_⋆ is quoted as a "pure" light curve parameter. Of course, if high-quality secondary eclipse observations are available that determine both the location and duration of the occultation, then k and h can be determined without RV data. Our model for the HATNet data was the simplified "P1P3" version of the Mandel & Agol (2002) analytic functions (an expansion in terms of Legendre polynomials), for the reasons described in Bakos et al. (2010). Following the formalism presented by Pál (2009a), the RVs were fitted with an eccentric Keplerian model parameterized by the semi-amplitude K and Lagrangian elements k and h. Note that we allowed for an eccentric orbit for all planets, even if the results were consistent with a circular orbit. There are several reasons for this: (1) many of the close-in hot Jupiters show eccentric orbits, thus the assumption of fixing e = 0 has no physical justification (while this has been customary in early discoveries relying on very few data points), (2) the error bars on various other derived quantities (including a/R_⋆) are more realistic with the inclusion of eccentricity, and (3) non-zero eccentricities can be very important in proper interpretation of these systems.

We assumed that there is a strict periodicity in the individual transit times. For each system we assigned the transit number N_tr = 0 to a complete follow-up light curve. For HAT-P-20b this was the light curve gathered on 2009 October 21, for HAT-P-21b: 2010 February 19, HAT-P-22b: 2009 February 28, and HAT-P-23b: 2008 September 13. The adjustable parameters in the fit that determine the ephemeris were chosen to be the time of the first transit center observed with HATNet (T_{c, −252}, T_{c, −286}, T_{c, −135}, and T_{c, −312} for HAT-P-20b through HAT-P-23b, respectively) and that of the last transit center observed with the FLWO 1.2 m telescope (T_{c, 0}, T_{c, 1}, T_{c, 19}, and T_{c, 250} for HAT-P-20b through HAT-P-23b, respectively). We used these as opposed to period and reference epoch in order to minimize correlations between parameters (see Pál et al. 2008). Times of mid-transit for intermediate events were interpolated using these two epochs and the corresponding transit number of each event, N_tr. The eight main parameters describing the physical model for each system were thus the first and last transit center times, R_p/R_⋆, b², ζ/R_⋆, K, k ≡ ecos ω, and h ≡ esin ω. For HAT-P-20b, HAT-P-22b, and HAT-P-23b, three additional parameters were included (for each system) that have to do with the instrumental configuration (dilution factor, out-of-transit magnitudes, gamma velocities; see later). For HAT-P-21b seven additional parameters were included, because it was observed in three different HATNet fields. These are the HATNet dilution factor B_inst (one for each HATNet field), which accounts for dilution of the transit in the HATNet light curve from background stars due to the broad point spread function (PSF; 25'' FWHM) and for reduction of the HATNet transit depth due to the TFA procedure, the HATNet out-of-transit magnitude M_{0, HATNet} (one for each HATNet field), and the relative zero-point γ_rel of the Keck RVs.

The physical model was extended with an instrumental model that describes brightness variations caused by systematic errors in the measurements. This was done in a similar fashion to the analysis presented by Bakos et al. (2010), i.e., using the "ELTG" (EPD local, TFA global) method for correcting the follow-up light curves. In all cases the total number of fitted parameters (43, 36, 33, and 67 for HAT-P-20b through HAT-P-23b) was much smaller than the number of data points (755, 1172, 892, and 953, counting only RV measurements and follow-up photometry measurements). The joint fit was performed as described in Bakos et al. (2010) using a Markov Chain Monte Carlo approach (MCMC; see Ford 2006).

The a priori distributions of the parameters for these chains were chosen to be Gaussian, with eigenvalues and eigenvectors derived from the Fisher covariance matrix for the best-fit solution. The MCMC analysis provided the full a posteriori probability distributions of all adjusted variables.

Following this procedure we obtained the a posteriori distributions for all other quantities of interest such as a/R_⋆. As described in Section 3.1, a/R_⋆ was used together with stellar evolution models to infer a value for log g_⋆ that is significantly more accurate than the spectroscopic value. The improved estimate was in turn applied to a second iteration of the SME analysis, as explained previously, in order to obtain better estimates of T_eff⋆ and [Fe/H]. The global modeling was then repeated with updated limb-darkening coefficients based on those new spectroscopic determinations. The resulting geometric parameters pertaining to the light curves and velocity curves for each system are listed in Table 14.

Included in each table is the RV "jitter," which is a noise term that we added in quadrature to the internal errors for the RVs in order to achieve χ²/dof = 1 from the RV data for the global fit. The jitter is a combination of assumed astrophysical noise intrinsic to the star, plus instrumental noise rising from uncorrected instrumental effects (such as a template spectrum taken under suboptimal conditions).

The planetary parameters and their uncertainties were derived by combining the a posteriori distributions for the stellar, light curve, and RV parameters. In this way we find masses and radii for each planet. These and other planetary parameters are listed at the bottom of Table 14, and further discussed in Section 4.

HAT-P-20b is a very massive (M_p = 7.246  ±  0.187 M_J = 2302.9  ±  59.5 M_⊕) and very compact (R_p = 0.867  ±  0.033 R_J) hot Jupiter orbiting a K3 (Skiff 2009) star. With ρ_p = 13.78  ±  1.50 g cm⁻³, HAT-P-20b is among the most massive and most dense transiting planets known (see Figure 16). Modeling HAT-P-20b may be a challenge, as the oldest (4 Gyr, i.e., yielding the most compact planets) Fortney et al. (2007) models with M_p = 2154 M_⊕ total mass and 100 M_⊕ core mass predict a much bigger radius (1.04 R_J). The observed radius of 0.87 M_J would require a very high metal content. We note that the host star is one of the most metal-rich stars that have a transiting planet ([Fe/H] = +0.35  ±  0.08). Curiously, HAT-P-20b orbits a fairly late-type star (K3), as compared to most of the massive hot Jupiters that orbit ∼F5 dwarfs. The irradiation HAT-P-20b receives is one of the smallest, clearly making it a pL class exoplanet (Fortney et al. 2008): 〈F〉 = (2.00  ±  0.19) × 10⁸ erg s⁻¹ cm⁻², comparable to the mean flux per orbit for another "heavy" planet HD 17156b on a 21 day period orbit. HAT-P-20 is an outlier in the M_p–M_⋆ plane; it is a relatively small mass star harboring a very massive planet. Another outlier (albeit to a much lesser extent) with similar planetary radius and stellar mass is WASP-10b (Johnson et al. 2009; Christian et al. 2009), but this planet has less than half of the mass of HAT-P-20b (3.09 M_J). We also calculated the maximum mass of a stable moon for both the prograde and retrograde cases, and derived 0.128 M_⊕ and 8.31 M_⊕, respectively, i.e., HAT-P-20b can harbor a fairly massive moon. An 8.31 M_⊕ retrograde moon would cause ∼10 s variations in the transit times, which is marginally detectable from the ground.

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HAT-P-20 has a close-by faint and red companion star at ∼686 separation. Based on the Palomar sky survey archival plates, we confirm that they form a close common proper motion pair, thus it is very likely that the two stars are physically associated. The binary has appeared in the Washington Double Star compilation (WDS) as POU2795, and was discovered by Pourteau (1933). Furthermore, based on the summary of observations in the WDS, there is already a hint of orbital motion of the companion to HAT-P-20 over the last century. The position angle of the companion changed from P.A. = 323° to P.A. = 320° over the course of 89 years (between 1909 and 1998), and it seems to be retrograde on the sky (clockwise). Thus, HAT-P-20 is yet another example of a massive planet in a binary system (Udry et al. 2002). The binary companion makes this system ideal for high precision ground or space-based studies, as it provides a natural comparison source, even though it has a later spectral type.

HAT-P-21b has a mass of M_p = 4.063  ±  0.161 M_J, radius of R_p = 1.024  ±  0.092 R_J, mean density of ρ_p = 4.68^+1.59_{− 0.99} g cm⁻³, and orbits on a moderately eccentric orbit with e = 0.228  ±  0.016, ω = 309°  ±  3°. The transits occur near apastron. As noted by Buchhave et al. (2010), 4 M_J mass planets are quite rare in the sample of currently known transiting exoplanets, and the only siblings of HAT-P-21b are WASP-32b (3.6 M_J; Maxted et al. 2010), HD 80606b (4.08 M_J; Naef et al. 2001), and HAT-P-16b (4.19 M_J; Buchhave et al. 2010). WASP-32b and HAT-P-16b are similar to each other with P ∼ 2.7–2.8 day orbital periods and large radii (R_p > 1.15 R_J in both cases), but dissimilar from the longer period, smaller radius, and more eccentric planet HAT-P-21b. HD 80606b, on the other hand, has a similar radius than HAT-P-21b, and orbits on an extremely eccentric (e = 0.93) orbit with a period of 111 days. It appears that HAT-P-21b is thus an unusual, short-period, eccentric, massive, and compact planet.

The only models from Fortney et al. (2007) consistent with the observed radius are 4 Gyr models with 100 M_⊕ core mass, yielding 1.05 R_J radius. HAT-P-21b has a very high mean density; similar to HD 80606b and WASP-14b.

The flux received by the planet varies between (10.0  ±  1.5) × 10⁸ erg s⁻¹ cm⁻² and (3.96  ±  0.66) × 10⁸ erg s⁻¹ cm⁻². Interestingly, this puts HAT-P-21b on the bordlerline between pL (low irradiation) and pM (high irradiation) planets. At the time of occultation, HAT-P-21b is just approaching its periastron, thus entering the irradiation level quoted for pM-type planets.

HAT-P-22b has a mass of M_p  =  2.147  ±  0.061 M_J, radius of R_p  =  1.080  ±  0.058 R_J, and mean density of ρ_p = 2.11^+0.40_{− 0.29} g cm⁻³. HAT-P-22b orbits a fairly metal-rich ([Fe/H]  =  + 0.24  ±  0.08), bright (V = 9.732), and close-by (82  ±  3 pc) star. Similar to HAT-P-20, the host star has a faint and red neighbor at 9'' separation that is co-moving with HAT-P-22 (based on the POSS plates and recent Keck/HIRES snapshots), thus they are likely to form a physical pair.

HAT-P-22b belongs to the moderately massive (∼2 M_J) and compact (R_p ≈ 1 R_J) hot Jupiters, such as HAT-P-15b (M_p = 1.95M_J, R_p = 1.07 R_J; Kovács et al. 2010), HAT-P-14b (M_p = 2.23M_J, R_p = 1.15 R_J; Torres et al. 2010), and WASP-8b (M_p = 2.25M_J, R_p = 1.05 R_J; Queloz et al. 2010).

HAT-P-22b is broadly consistent with the models of Fortney et al. (2008). For 300 Myr, 1 Gyr, and 4 Gyr models, it requires a 100 M_⊕, 50 M_⊕, and 25 M_⊕ core, respectively, to have a radius of ∼1.08 R_J. The low incoming flux (see Table 14) means that HAT-P-22b is a pL class planet. HAT-P-22b can harbor a 0.96 M_⊕ mass retrograde moon, which would cause transit timing variations (TTVs) of ∼2 s.

HAT-P-23b has a mass of M_p = 2.090  ±  0.111 M_J, radius of R_p = 1.368  ±  0.090 R_J, and mean density of ρ_p = 1.01  ±  0.18 g cm⁻³. It belongs to the inflated group of 2 M_J planets. A few other planets which are similar to it include TrES-3b (M_p = 1.91M_J, R_p = 1.37 R_J; Sozzetti et al. 2009), Kepler-5b (M_p = 2.10M_J, R_p = 1.31 R_J; Kipping & Bakos 2011; Koch et al. 2010), WASP-46b (M_p = 2.10M_J, R_p = 1.33 R_J; Anderson et al. 2011), and CoRoT-11b (M_p = 2.33M_J, R_p = 1.43 R_J; Gandolfi et al. 2010). The orbit is nearly circular, with the eccentricity being marginally significant. The reason for the somewhat higher than usual errors in the RV parameters is the high jitter of the star (34.7 m s⁻¹), which may be related to the moderately high vsin i = 8.1  ±  0.5 km s⁻¹ and the very close-in orbit of HAT-P-23b. The Fortney et al. (2008) models cannot reproduce the observed radius of HAT-P-23b; even for the youngest, (300 Myr) coreless models, the theoretical radius for its mass is 1.25 R_J. HAT-P-23b orbits its host star on a very close-in orbit. The orbital period is only 1.2129 days; almost identical to that of OGLE-TR-56b (1.21192 days). The nominal planetary radius of the two objects is also the same within 1%, but OGLE-TR-56b is much less massive (1.39 M_J). The flux falling on HAT-P-23b from its host star is one of the highest (i.e., belongs to the pM class objects), and is similar to that of HAT-P-7b and OGLE-TR-56b. We also calculated the spiral infall timescale for each new discovery based on Levrard et al. (2009) and Dobbs-Dixon et al. (2004). By assuming that the stellar dissipation factor is Q_⋆ = 10⁶, the infall time for HAT-P-23b is τ_infall = 7.5^+2.9_{− 1.8} Myr, one of the shortest among exoplanets.

The Rossiter–McLaughlin (R-M) effect for HAT-P-23b should be quite significant, given the moderately high vsin i = 8.1  ±  0.5 km s⁻¹ of the host star and the Δi = 17 mmag deep transit. The impact parameter is also "ideal" (b = 0.324^+0.070_{− 0.101}), i.e., it is not equatorial (b = 0), where there would be a strong degeneracy between the stellar rotational velocity vsin i and the sky-projected angle of the stellar spin axis and the orbital normal, λ, and is also far from grazing, where the transit is short, and other system parameters have lower accuracy. The effective temperature of the star (T_eff⋆ = 5905  ±  80 K) is close to the critical temperature of 6250 K noted recently by Winn et al. (2010), which may be a borderline between systems where the stellar spin axes and planetary orbital normals are preferentially aligned (T_eff⋆ < 6250 K) and those that are misaligned (T_eff⋆ > 6250 K). An alternative hypothesis has been brought up by Schlaufman (2010), where misaligned stellar spin axes and orbital normals are related to the mass of the host star. The mass of HAT-P-23 (1.13  ±  0.04) is sufficiently close to the suggested dividing line of M_⋆ = 1.2 M_☉, and thus it will provide an excellent additional test for these ideas.¹⁷

We presented the discovery of four new massive transiting planets, and provided accurate characterization of the host star and planetary parameters. These four new systems are very diverse, and significantly expand the sample of massive (M_p ≳ 2 M_J) planets. Two of the new discoveries orbit stars that have fainter companions, which are probably physically associated. The new discoveries do not tend to enhance the mass–eccentricity correlation, since only one (HAT-P-21b) is significantly eccentric. Also, the tentative mass–vsin i correlation noted in Section 1 is weakened by the new discoveries. The heavier mass planets (HAT-P-20b and HAT-P-21b) are inconsistent with current theoretical models in that they are extremely dense, and would require a substantial core (or high metal content) to have such small radii. One planet (HAT-P-23b) is also inconsistent with the models (unless we assume that the planet is very young), but in the sense that it has a density that is lower than can be explained by the models. It has been noted by Winn et al. (2010) and Schlaufman (2010) that systems exhibiting stellar spin axis–planetary orbital normal misalignment are preferentially eccentric and heavy mass planets (in addition to the dependence noted by these authors of the spin–orbit angle on the effective temperature or mass of the host star). The four new planets presented in this work will provide additional important tests for checking these conjectures. The host stars are all bright (9.7 < V < 12.4), and thus enable in-depth future characterization of these systems.