HAT-P-26b: A LOW-DENSITY NEPTUNE-MASS PLANET TRANSITING A K STAR^*

The transits of HAT-P-26b were detected with the HAT-5 and HAT-6 telescopes in Arizona, and with the HAT-8 and HAT-9 telescopes in Hawaii. Two regions around GSC 0320−01027, corresponding to fields internally labeled as 376 and 377, were both observed on a nightly basis between 2009 January and 2009 August, whenever weather conditions permitted. For field 376 we gathered 11,500 exposures of 5 minutes at a 5.5 minute cadence. Approximately 1500 of these exposures were rejected by our photometric pipeline because they yielded poor photometry for a significant number of stars. Each image contained approximately 17,000 stars down to Sloan r ∼ 14.5. For the brightest stars in the field, we achieved a per-image photometric precision of 4 mmag. For field 377 we gathered 5200 exposures with the same exposure time and cadence; we rejected approximately 700 exposures. Each image contained approximately 19,000 stars down to Sloan r ∼ 14.5. We achieved a similar photometric precision for the brightest stars in this field.

The calibration of the HATNet frames was carried out using standard photometric procedures. 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. 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. We detected a significant signal in the light curve of GSC 0320−01027 (also known as 2MASS 14123753+0403359; α = 14^h12^m3744, δ = +04°03'360; J2000; V = 11.744 Droege et al. 2006), with an apparent depth of ∼4.9 mmag (∼5.5 mmag when using TFA in signal-reconstruction mode), and a period of P = 4.2345 days (see Figure 1). The drop in brightness had a first-to-last-contact duration, relative to the total period, of q = 0.0242 ± 0.0002, corresponding to a total duration of Pq = 2.455 ± 0.025 hr.

We removed the transits from the combined 376/377 light curve and searched for additional transiting objects using the BLS method; no significant signals were found in the data. We also searched the transit-cleaned light curve for periodic variations (e.g., due to stellar rotation) using the Discrete Fourier Transform (e.g., Kurtz 1985), and found no coherent variation with an amplitude greater than 0.4 mmag.

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 km s^-1) that are easily recognized.

To carry out this reconnaissance spectroscopy, we made use of the Tillinghast Reflector Echelle Spectrograph (TRES; Fűrész 2008) on the 1.5 m Tillinghast Reflector at FLWO. This instrument provides high-resolution spectra which, with even modest signal-to-noise ratios (S/Ns), are suitable for deriving RVs with moderate precision (≲0.3 km s^-1) for slowly rotating stars. We also use these spectra to estimate effective temperatures, surface gravities, and projected rotational velocities of the target star. Using the medium fiber on TRES, we obtained two spectra of HAT-P-26 on the nights of 2009 December 26 and 2009 December 27. The spectra have a resolution of λ/Δλ≈ 44,000 and a wavelength coverage of ∼3900–8900 Å. The spectra were extracted and analyzed according to the procedure outlined by Buchhave et al. (2010) and Quinn et al. (2010). The individual velocity measurements of 14.62 km s^-1 and 14.81 km s^-1 were consistent with no detectable RV variation within the measurement precision. Both spectra were single-lined, i.e., there is no evidence for additional stars in the system. The atmospheric parameters we infer from these observations are the following: effective temperature T_eff⋆ = 5125 ± 125 K, surface gravity log g_⋆ = 4.5 ± 0.5 (log cgs), and projected rotational velocity vsin i = 1.0 ± 1.0 km s^-1. The effective temperature corresponds to an early K dwarf. The mean heliocentric RV of HAT-P-26 is γ_RV = +14.72 ± 0.10 km s^-1.

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Given the significant transit detection by HATNet, and the encouraging TRES results that rule out obvious false positives, 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 HIRES instrument (Vogt et al. 1994) on the Keck I telescope located on Mauna Kea, Hawaii, between 2009 December and 2010 June. 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 obtained 12 exposures 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). 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. The RV measurements and their uncertainties are listed in Table 1. The period-folded data, along with our best fit described below in Section 3, are displayed in Figure 2.

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In the same figure we show also the S_HK 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 (Figure 3 shows representative Keck spectra including the H and K lines for HAT-P-26). This index was computed and calibrated to the scale of Vaughan et al. (1978) following the procedure described by Isaacson & Fischer (2010). We find a median value of S_HK = 0.182 with a standard deviation of 0.004. Assuming B − V = 0.89 based on the effective temperature measured in Section 3.1, this corresponds to log R'_HK = −4.992 (Noyes et al. 1984). From Isaacson & Fischer (2010), the lower tenth percentile S_HK value among California Planet Search (CPS) targets with B − V = 0.89 is 0.168. The measured S_HK value is only slightly higher than this, implying that HAT-P-26 is a chromospherically quiet star. We do not detect any significant variation of the S_HK 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.

In order to permit a more accurate modeling of the light curve, we conducted additional photometric observations with the KeplerCam CCD camera on the FLWO 1.2 m telescope. We observed five transit events of HAT-P-26 on the nights of 2010 January 5, March 31, April 4, May 8, and May 25 (Figure 4). These observations are summarized in Table 2.

The reduction of these images, including basic calibration, astrometry, and aperture photometry, was performed as described by Bakos et al. (2010). 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 are shown in the top portion of Figure 4, along with our best-fit transit light curve model described below; the individual measurements are reported in Table 3. The combined phase-folded follow-up light curve is displayed in Figure 5.

Fundamental parameters of the host star HAT-P-26 such as 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 spectrum 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). SME yielded the following values and uncertainties: effective temperature T_eff⋆ = 5079 ± 88 K, stellar surface gravity log g_⋆ = 4.53 ± 0.06 (cgs), metallicity [Fe/H] = −0.04 ± 0.08 dex, and projected rotational velocity vsin i = 1.8 ± 0.5 km s^-1, in which the uncertainties for T_eff⋆ and [Fe/H] have been increased by a factor of two over their formal values to include our estimates of the systematic uncertainties.

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In principle the effective temperature and metallicity, along with the surface gravity taken as a luminosity indicator, could 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 quite subtle, and as a result it is typically difficult to determine accurately, so that it is a rather poor luminosity indicator in practice. For planetary transits a stronger constraint is often provided by ρ_⋆ the mean stellar density, which is closely related to the a/R_⋆ normalized semimajor axis. The quantity ρ_⋆ can be derived directly from the combination of the transit light curves (Seager & Mallén-Ornelas 2003; 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. The validity of our assumption, namely that the adequate physical model describing our data is a planetary transit (as opposed to a blend), is shown later in Section 3.2.

Our 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 ρ_⋆ and other fitted variables. For further details we refer the reader to Pál (2009). When combining ρ_⋆ (a luminosity proxy) 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 (YY) series by Yi et al. (2001).

For the case of HAT-P-26b, the eccentricity is poorly constrained by the RV data due to the low semiamplitude of the signal. This in turn leads to a significant uncertainty on ρ_⋆. However, not all combinations of [Fe/H], T_eff⋆, and ρ_⋆ are realized by physical stellar models. In particular, if we conservatively adopt a maximum stellar age of 13.8 Gyr, corresponding approximately to the age of the universe (Komatsu et al. 2010 find 13.75 ± 0.11 Gyr), and a minimum age of 100 Myr, corresponding roughly to the zero-age main sequence,¹³ stars with T_eff⋆ = 5079 and [Fe/H] = −0.04 are not found to have densities in the range 0.24 g cm⁻³ < ρ_⋆ < 2.06 g cm⁻³ or surface gravities in the range 3.915 < log g_⋆ < 4.514 (here 0.24 g cm⁻³ corresponds to an evolved star with M = 0.94 M_☉, while 2.06 g cm⁻³ corresponds to a main-sequence star with M = 0.79 M_☉). Figure 6 shows the inferred location of the star in a diagram of ρ_⋆ versus T_eff⋆, analogous to the classical H-R diagram, for three cases: fixing the eccentricity of the orbit to zero, allowing the eccentricity to vary, and allowing the eccentricity to vary, but only accepting parameter combinations which match to a position in the YY isochrones with 0.1 Gyr < age < 13.8 Gyr. The stellar properties and their approximate 1σ and 2σ confidence boundaries are displayed against the backdrop of Yi et al. (2001) isochrones for the measured metallicity of [Fe/H] = −0.04, and a range of ages. For the zero eccentricity case the comparison against the model isochrones was carried out for each of 30,000 Monte Carlo trial sets (see Section 3.3). We find good overlap between the trials and the model isochrones—in 71% of the trials, the [Fe/H], T_eff⋆, and ρ_⋆ parameter combination matched to a physical location in the H-R diagram that has an age that is within the aforementioned range. However, when the eccentricity is allowed to vary, the model for the light curves and RV data tends toward low values of ρ_⋆ which may only be fit by pre-main-sequence stellar models, or stellar models older than the age of the universe. In this case, only 15% out of 100,000 Monte Carlo trial sets match to physical locations in the H-R diagram with ages within the allowed range. By requiring the star to have an age between 0.1 Gyr and 13.8 Gyr, we effectively impose a tighter constraint on the orbital eccentricity than is possible from the RV data alone (we find an eccentricity of e = 0.124 ± 0.060, as compared with e = 0.24 ± 0.12 when not requiring a match to the stellar models; see also Section 3.3).

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Adopting the parameters which result from allowing the eccentricity to vary while requiring the star to have an age between 0.1 Gyr and 13.8 Gyr yields a stellar surface gravity of log g_⋆ = 4.56 ± 0.06, which is very close to the value from our SME analysis. The values for the atmospheric parameters of the star are collected in Table 4, together with the adopted values for the macroturbulent and microturbulent velocities.

The stellar evolution modeling provides color indices that may be compared against the measured values as a sanity check. The best available measurements are the near-infrared magnitudes from the 2MASS Catalog (Skrutskie et al. 2006), J_2MASS = 10.080 ± 0.022, H_2MASS = 9.685 ± 0.023, and K_2MASS = 9.581 ± 0.023; which we have converted to the photometric system of the models (ESO) using the transformations by Carpenter (2001). The resulting measured color index is J − K = 0.530 ± 0.035. This is within 1σ of the predicted value from the isochrones of J − K = 0.55 ± 0.02. The distance to the object may be computed from the absolute K magnitude from the models (M_K = 3.98 ± 0.19) and the 2MASS K_s magnitude, which has the advantage of being less affected by extinction than optical magnitudes. The result is 134⁺¹⁸₋₈ pc, where the uncertainty excludes possible systematics in the model isochrones that are difficult to quantify.

An additional check on our stellar model can be performed by comparing the isochrone-based age estimate to activity-based age estimates. Using the activity–rotation and activity–age relations from Mamajek & Hillenbrand (2008, Equations (5) and (3)) we convert the value of log R'_HK determined in Section 2.3 into a Rossby number (R_O = P_rot/τ_c, where P_rot is the rotation period and τ_c is the convective turnover timescale) and an age. We find R_O = 2.2 ± 0.2, and log(τ) = 9.80 ± 0.15 or τ = 6.4^+2.7_−1.9 Gyr, where we adopt the estimated uncertainties on R_O and log(τ) from Mamajek & Hillenbrand (2008). The R_O value may be converted to a rotation period using the relation for τ_c given by Noyes et al. (1984). We find P_rot = 48 ± 4 days. The rotation period and color may also be used to obtain a separate age estimate from the gyrochronology relation given by Mamajek & Hillenbrand (2008, Equation (12)). This gives τ = 7.8^+1.4_−1.2 Gyr, where the scatter on this relation is less than the scatter on the age–activity relation because it includes a correction for stellar color. The age inferred from log R'_HK is consistent with the isochrone-based age of 9.0^+3.0_−4.9 Gyr. The equatorial rotation period inferred from the spectroscopic determination of vsin i assuming sin i = 1 is P_rot,vsin i = 22.3^+17.2_−4.6 days, which is shorter than the expected value based on log R'_HK, though the upper limit is poorly constrained.

As discussed below in Section 3.3 we measure a RV jitter of 1.6 m s^-1 for HAT-P-26. From Isaacson & Fischer (2010) the lower tenth percentile jitter among HIRES/Keck observations for CPS stars with 0.7 < B − V < 1.0 and S_HK = 0.182 is 2.17 m s^-1, implying that HAT-P-26 has an exceptionally low jitter value—only a handful of stars in this color range have measured jitter values less than that of HAT-P-26. We note that the jitter may be higher (2.4 m s^-1) if the orbit is circular, though this value is still quite low.

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 sections, we consider and rule out the possibility that such scenarios may have caused the observed photometric and spectroscopic features.

3.2.1. Spectral Line-bisector Analysis

3.2.2. Contamination from a Background Eclipsing Binary

Following our earlier work (Bakos et al. 2010; Hartman et al. 2009) we make use of the high proper motion of HAT-P-26 to rule out the possibility that the observed transits and RV variation may be due to a background eclipsing binary that is aligned, by chance, with the foreground K1 dwarf HAT-P-26. To reproduce the observed ∼0.6% deep transit, the background object cannot be more than 5.6 mag fainter than HAT-P-26 (objects fainter than this would contribute less than 0.6% of the total combined light and so could not cause the transit even if they were to be completely eclipsed by an object that emits no light). Because HAT-P-26 has a high proper motion (148.5 ± 2.7 mas yr^-1; Roeser et al. 2010) it is possible to use the Palomar Observatory Sky Survey plates from 1950 (POSS-I, red and blue plates) to view the sky at the current position of HAT-P-26. Between 1950 and the follow-up observations in 2010, HAT-P-26 has moved ∼89. Figure 7 shows an image stamp from the POSS-I plate compared with a recent observation from the FLWO 1.2 m. We can rule out a background object down to R ∼ 19 mag within ∼3'' of the current position of HAT-P-26. Any background object must be ≳7.5 mag fainter than HAT-P-26 and thus could not be responsible for the observed transit.

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3.2.3. Detailed Blend Modeling of a Hierarchical Triple

Following Bakos et al. (2010), Hartman et al. (2009), and Torres et al. (2005), we attempt to model the observations as a hierarchical triple system. We consider four possibilities.

• 1.  
  One star orbited by a planet.
• 2.  
  Three stars, two fainter stars are eclipsing.
• 3.  
  Two stars, one planet, planet orbits the fainter star.
• 4.  
  Two stars, one planet, planet orbits the brighter star.

Here, case 1 is the fiducial model to which we compare the various blend models. We model the observed follow-up and HATNet light curves (including only points that are within one transit duration of the primary transit or secondary eclipse assuming zero eccentricity) together with the 2MASS and TASS photometry. In each case, we fix the mass of the brightest star to 0.788 M_☉; this ensures that we reproduce the effective temperature, metallicity, and surface gravity determined from the SME analysis when using the Padova isochrones (see below). We have also attempted to perform the fits described below allowing the mass of the brightest star to vary. We find that in this case the mass of the brightest star is still constrained to be close to 0.788 M_☉ by the broadband photometry, even if the spectroscopic parameters are not included. We therefore conclude that fixing the mass of the brightest star is justified. In all cases, we vary the distance to the system and two parameters allowing for dilution in the two HATNet light curves, and we include simultaneous EPD and TFA in fitting the light curves (see Section 3.3). In each case, we draw the stellar radii and magnitudes from the 13.0 Gyr Padova isochrone (Girardi et al. 2000), extended below 0.15 M_☉ with the Baraffe et al. (1998) isochrones. We use these rather than the YY isochrones for this analysis because of the need to allow for stars with M < 0.4 M_☉, which is the lower limit available for the YY models. We use the JKTEBOP program (Southworth et al. 2004a, 2004b) which is based on the Eclipsing Binary Orbit Program (EBOP; Popper & Etzel 1981; Etzel 1981; Nelson & Davis 1972) to generate the model light curves. We optimize the free parameters using the Downhill Simplex Algorithm together with classical linear least squares for the EPD and TFA parameters. We rescale the errors for each light curve such that χ² per degree of freedom is 1.0 for the out of transit portion of the light curve. Note that this is done prior to applying the EPD/TFA corrections, as a result χ² per degree of freedom is less than 1.0 for each of the best-fit models discussed below. If the rescaling is not performed, the difference in χ² between the best-fit models is even more significant than what is given below, and the blend-models may be rejected with higher confidence.

Case 1: one star, one planet. In addition to the parameters mentioned above, in this case we vary the radius of the planet and the impact parameter of the transit. The best-fit model has χ²_Case1 = 3140.1 for 3567 degrees of freedom. The parameters that we obtain for the planet are comparable to those obtained from the global modelling described in Section 3.3.

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Case 2: three stars. For case 2, we vary the masses of the eclipsing components, and the impact parameter of the eclipse. We find no model of three stars which reproduces the observations. The transit depth and duration cannot be fit when the three stars are constrained to fall on the same isochrone, and the brightest star has M = 0.788 M_☉. The best-fit case 2 model consists of equal masses for the brightest two stars, and 0.08 M_☉ (the lowest stellar mass in the Baraffe et al. 1998 isochrones) for the transiting star. Such a model is inconsistent with our spectroscopic observations (it would have been easily identified as a double-lined binary at one of the quadrature phases), and as we will show, can be rejected from the light curves alone. The best-fit case 2 model yields χ²_Case2 = 3310.2 for 3566 degrees of freedom and produces model transits that are too deep compared to the observed transit. The case 1 model achieves a lower χ² with fewer parameters than the case 2 model, so the case 1 model is preferred over the case 2 model. Assuming that the errors are uncorrelated and follow a Gaussian distribution, the case 2 model can be rejected in favor of the case 1 model at the >116σ confidence level. Alternatively, one might suppose that any apparent correlations in the residuals of the best-fit case 2 model are not due to errors in the model but instead are due to uncorrected systematic errors in the measurements; large systematic errors in the measurements would increase the probability that case 1 might give a better fit to the data, by chance, than case 2. To establish the statistical significance with which we may reject case 2 while allowing for possible systematic errors in the measurements, we conduct 1000 Monte Carlo simulations in which we assume the best-fit case 2 model scenario is correct, shuffle the residuals from this fit in a manner that preserves the correlations (this is done by taking the Fourier Transform of the residuals, randomly changing phases of the Transform while preserving the amplitudes, and transforming back to the time domain), and fit both the case 2 and case 1 models to the simulated data. The median value of χ²_Case2 − χ²_Case1 is −17.6 with a standard deviation of 26.3. None of the 1000 trials have χ²_Case2 − χ²_Case1>170.1, the measured value. Based on this analysis we reject case 2, i.e., the hierarchical triple star system scenario, at the 7σ level.

Case 3: two stars, planet orbits the fainter star. In this scenario, HAT-P-26b is a transiting planet, but it would have a radius that is larger than what we infer (it may be a Saturn- or Jupiter-size planet rather than a Neptune-size planet). For this case we vary the mass of the faint planet-hosting star, the radius of the planet, and the impact parameter of the transit. We assume the mass of the planet is negligible relative to the mass of its faint host star. Figure 8 shows χ² as a function of the mass of the planet-hosting star for this scenario. The best-fit case 3 solution has χ² = 3147.6 for 3566 degrees of freedom, and corresponds to a system where the two stars are of equal mass and the planet has a radius of 0.8 R_J. As the mass of the planet host is decreased the value of χ² increases. Repeating the procedure outlined above to establish the statistical significance at which we may reject case 3 we find that the 3σ limit on χ²_Case3 − χ²_Case1 is 9.0, which results in a 3σ lower limit of 0.77 M_☉ on the mass of the planet hosting star. We may thereby place a 3σ lower limit of 0.74 on the V-band luminosity ratio of the two stars. A second set of lines with a luminosity ratio of >0.74 would have easily been detectable in both the Keck and TRES spectra unless the stars had very similar γ velocities (the spectral lines are quite narrow with vsin i = 1.8 ± 0.5 km s^-1). The poor fit for this blend model relative to the fiducial model together with the tight constraints on the relative γ velocities and luminosity ratios of the stars in the blend models that may yet fit the data leads us to reject this blend scenario in favor of the simpler model of a single star hosting a transiting planet.

Case 4: two stars, planet orbits the brighter star. As in the previous case, in this scenario HAT-P-26b is a transiting planet, but the dilution from the blending star means that the true radius is larger than what we infer in case 1. In this case, we vary the radius of the planet, the mass of the faint star, and the impact parameter of the transit. Again we assume the mass of the planet is negligible relative to the mass of its bright host star. Figure 8 shows χ² as a function of the mass of the faint star. The smallest value of χ² is achieved when the faint star contributes negligible light to the system, which is effectively equivalent to the fiducial scenario represented by case 1. Two effects cause χ² to increase with stellar mass. First, the shape of the transit is subtly changed in a manner that gives a poorer fit to the observations. Second, when the mass of the faint star is less than that of the transit host the model broadband photometry for the blended system is redder than for the single star scenario, and is inconsistent with the observed photometry. This gives rise to the peak in χ² at M ∼ 0.65 M_☉. The case 4 model where the faint star has M ≳ 0.77 M_☉ can be rejected as in case 3. For lower masses, we place a 3σ upper limit of 0.55 M_☉ on the mass of the faint star, yielding a 3σ upper limit on the luminosity ratio of 0.1. We conclude that at most the planet radius R_p may be 8% larger than what we find in Section 3.3 if there is an undetected faint secondary star in the system.

Here we summarize the procedure that we followed to model the HATNet photometry, the follow-up photometry, and the RVs simultaneously. This procedure is described in greater detail in Bakos et al. (2010). The follow-up light curves were modeled using analytic formulae based on Mandel & Agol (2002), with quadratic limb darkening coefficients for the Sloan i band interpolated from the tables by Claret (2004) for the spectroscopic parameters of the star as determined from the SME analysis (Section 3.1). We modeled the HATNet data using an approximation to the Mandel & Agol (2002) formulae as described in Bakos et al. (2010). The RVs were fitted with an eccentric Keplerian model.

Our physical model consisted of eight main parameters, including: the time of the first transit center observed with HATNet (taken to be event −74), T_c,−74, and that of the last transit center observed with the FLWO 1.2 m telescope, T_c,+40, the normalized planetary radius p ≡ R_p/R_⋆, the square of the impact parameter b², the reciprocal of the half duration of the transit ζ/R_⋆ as given in Bakos et al. (2010), the RV semiamplitude K, and the Lagrangian elements k ≡ ecos ω and h ≡ esin ω, where ω is the longitude of periastron. Five additional parameters were included that have to do with the instrumental configuration. These are the HATNet blend factors B_inst,376, and B_inst,377, which account for possible dilution of the transit in the HATNet light curves from background stars due to the broad PSF (∼24'' FWHM), the out-of-transit magnitudes for each HATNet field, M_0,HATNet,376 and M_0,HATNet,377, and the relative zero-point γ_rel of the Keck RVs. The physical model was extended with an instrumental model for the follow-up light curves that describes brightness variations caused by systematic errors in the measurements. We adopted a "local" EPD and "global" TFA model (Bakos et al. 2010), using 20 template stars for the TFA procedure and six EPD parameters for each follow-up light curve.¹⁴ In summary, the total number of fitted parameters was 13 (physical model with five configuration-related parameters) + 30 (local EPD) + 20 (global TFA) = 63, i.e., much smaller than the number of data points (1450, counting only RV measurements and follow-up photometry measurements).

As described in Bakos et al. (2010), we use a combination of the downhill simplex method (AMOEBA; see Press et al. 1992), the classical linear least squares algorithm, and the Markov Chain Monte Carlo method (MCMC; see Ford 2006) to obtain a best-fit model together with a posteriori distributions for the fitted parameters. These distributions were then used to obtain a posteriori distributions for other quantities of interest, such as ρ_⋆. As described in Section 3.1, ρ_⋆ was used together with stellar evolution models to obtain a posteriori distributions for stellar parameters, such as M_⋆ and R_⋆, which are needed to determine M_p and R_p.

To check for correlations in the residuals which may indicate systematic errors in the data which are not included in the parameter uncertainties we computed the auto-correlation function of the light curve residuals. We find that the width of the peak near zero time-lag is significantly smaller than the time sampling, indicating that the residuals are effectively uncorrelated. We also performed a "prayer bead" analysis (i.e., the model is fitted to the data after performing cyclic permutations of the residuals; see for example Gillon et al. 2007a) and we find that the resulting parameter uncertainties are comparable to, or less than, the uncertainties determined from the MCMC analysis. This demonstrates that any correlations in the residuals are small enough that they have negligible impact on the parameter uncertainties.

The resulting parameters pertaining to the light curves and velocity curves, together with derived physical parameters of the planet, are listed under the "Adopted Value" column heading of Table 5. Included in this table is the RV "jitter." This is a component of assumed astrophysical noise intrinsic to the star, possibly with a contribution from instrumental errors as well, 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. Auxiliary parameters not listed in the table are: T_c,−74 = 2454860.02709 ± 0.00147 (BJD), T_c,+40 = 2455342.76185 ± 0.00041 (BJD), the blending factors B_instr,376 = 0.92 ± 0.05, and B_instr,377 = 0.84 ± 0.10 for the HATNet field 376 and 377 light curves, respectively, and γ_rel = −2.8 ± 0.8 m s^-1. The latter quantity represents an arbitrary offset for the Keck RVs, but does not correspond to the true center-of-mass velocity of the system, which was listed earlier as γ_RV in Table 4.

Table 5. Orbital and Planetary Parameters

Parameter	Adopted Value	Value	Value
	{ζ/R⋆, b2, p}	{ζ/R⋆, b2, p}, e ≡ 0	{ϒ/R⋆, b, p2}
Light curve parameters
P (days)	4.234516 ± 0.000015	4.234515 ± 0.000015	4.234508+0.000021−0.000022
Tc (BJD)a	2455304.65122 ± 0.00035	2455304.65118 ± 0.00036	2455304.65120+0.00048−0.00049
T14 (days)a	0.1023 ± 0.0010	0.1025 ± 0.0010	0.1023+0.0012−0.0011
T12 = T34 (days)a	0.0077 ± 0.0007	0.0078 ± 0.0007	0.00724+0.00081−0.00027
a/R⋆	13.06 ± 0.83	13.44+0.44−0.59	13.28+0.70−0.76
ζ/R⋆	21.15 ± 0.16	21.14 ± 0.16	21.10+0.20−0.21
Rp/R⋆	0.0737 ± 0.0012	0.0738 ± 0.0012	0.07341+0.00104−0.00093
b2	0.092+0.087−0.053	0.110+0.076−0.059	0.035−0.089−0.032
b ≡ acos i/R⋆	0.303+0.112−0.122	0.332+0.095−0.123	0.00+0.26−0.00
i (deg)	88.6+0.5−0.9	88.6 ± 0.5	89.14+0.59−0.72
Limb-darkening coefficientsb
ai (linear term)	0.3862	0.3862	0.3862
bi (quadratic term)	0.2576	0.2576	0.2576
RV parameters
K (m s-1)	8.5 ± 1.0	8.3 ± 1.0	7.6+1.2−1.2
kRVc	0.099 ± 0.060	0.000 ± 0.000	0.09+0.12−0.11
hRVc	0.027 ± 0.076	0.000 ± 0.000	0.028+0.063−0.060
e	0.124 ± 0.060	0.000 ± 0.000	0.127+0.094−0.068
ω (deg)	54 ± 165	0 ± 0	74+266−59
$\dot{\gamma }$ (m s−1 day−1)	0	0	−0.028+0.014−0.013
ttroj (days)d	0	0	0.01+0.24−0.23
RV jitter (m s-1)	1.6	2.4	...
RV fit rms (m s-1)	2.4	3.0	...
Secondary eclipse parameters
Ts (BJD)	2455307.037 ± 0.162	2455306.768 ± 0.000	2455307.01+0.31−0.29
Ts,14	0.1074 ± 0.0162	0.1025 ± 0.0010	0.108+0.014−0.011
Ts,12	0.0082 ± 0.0067	0.0078 ± 0.0007	0.00782+0.00104−0.00083
Planetary parameters
Mp (MJ)	0.059 ± 0.007	0.057 ± 0.007	0.0522+0.0084−0.0083
Rp (RJ)	0.565+0.072−0.032	0.549+0.034−0.023	0.553+0.037−0.031
C(Mp, Rp)e	0.07	0.08	0.059
ρp (g cm-3)	0.40 ± 0.10	0.42 ± 0.08	0.378+0.099−0.084
log gp (cgs)	2.65+0.08−0.10	2.67 ± 0.07	2.621+0.084−0.092
a (AU)	0.0479 ± 0.0006	0.0478 ± 0.0006	0.04780+0.00064−0.00061
Teq (K)	1001+66−37	981 ± 29	991+42−36
Θf	0.012 ± 0.002	0.012 ± 0.002	0.0110+0.0020−0.0019
Fperi (108 ergs-1cm−2)g	2.91+7.54−0.48	2.10+0.30−0.20	2.87+1.03−0.63
Fap (108 ergs-1cm−2)g	1.81 ± 0.32	2.10+0.30−0.20	1.72+0.28−0.31
〈F〉 (108 ergs-1cm−2)g	2.27+1.08−0.31	2.10+0.30−0.20	2.18+0.40−0.30

Notes. ^aT_c: reference epoch of mid-transit that minimizes the correlation with the orbital period. It corresponds to N_tr = +31. BJD is calculated from UTC. T₁₄: total transit duration, time between first and last contact; T₁₂ = T₃₄: ingress/egress time, time between first and second, or third and fourth contact. ^bValues for a quadratic law, adopted from the tabulations by Claret (2004) according to the spectroscopic (SME) parameters listed in Table 4. ^cLagrangian orbital parameters derived from the global modeling, and primarily determined by the RV data. ^dTime-offset in the radial velocities due to companion planets in Trojan orbits. ^eCorrelation coefficient between the planetary mass M_p and radius R_p. ^fThe Safronov number is given by $\Theta = \frac{1}{2}(V_{\rm esc}/V_{\rm orb})^2 = (a/R_{p})(M_{p}/ M_\star)$ (see Hansen & Barman 2007). ^gIncoming flux per unit surface area, averaged over the orbit.

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We find a mass for the planet of M_p = 0.059 ± 0.007 M_J and a radius of R_p = 0.565^+0.072_−0.032 R_J, leading to a mean density ρ_p = 0.40 ± 0.10 g cm^-3. We also find that the eccentricity of the orbit may be different from zero: e = 0.124 ± 0.060, ω = 54° ± 165°. However, as we show in Section 4.3, this is at best significant at only the 88% confidence level.

We also carried out the analysis described above with the eccentricity fixed to zero. The resulting parameters are given in Table 5 under the column heading "{ζ/R_⋆, b², p}, e ≡ 0." The results are discussed further in Section 4.3.

Finally, we conducted an independent model of the system based on Kipping & Bakos (2010b). The primary differences between this model and the adopted model are differences in the choice of parameters to vary in the fit: we use ϒ/R_⋆ as defined in Kipping & Bakos (2010b) rather than ζ/R_⋆, b rather than b², and p² rather than p. We also allowed for a linear drift in the RVs $\dot{\gamma }$, and a time shift t_troj in the RVs due to possible additional bodies in the system on Trojan orbits with HAT-P-26b. We chose to include both t_troj and $\dot{\gamma }$ rather than fixing them to zero as the value of the Bayesian Information Criterion (BIC; e.g., Kipping et al. 2010) was lower for the best-fit model including these parameters, than for models where one or both of these parameters were fixed to zero. The resulting parameters are given in Table 5 under the column heading "{ϒ/R_⋆, b, p²}." The parameter values from this model are consistent with those from the adopted model, which gives confidence that our results are robust to changes in the choice of fitting parameters.

Figure 9 compares HAT-P-26b to the other known TEPs on a mass–radius diagram. With a density of 0.40 ± 0.10 g cm^-3, HAT-P-26b is significantly less dense than the four other Neptune-size planets with well measured masses and radii (Uranus, Neptune, GJ 436b, HAT-P-11b). For Kepler-4b, Kipping & Bakos (2010a) find a large uncertainty on the radius which results from significant uncertainties on the eccentricity and the transit impact parameter. Kepler-4b may be comparable in size to GJ 436b and HAT-P-11b, or it could be even less dense than HAT-P-26b.

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From the theoretical models of Fortney et al. (2007), HAT-P-26b has a radius that is well above the maximum radius of 0.3 R_J for a 0.06 M_J planet lacking a hydrogen–helium envelope (i.e., a planet with a 100% water–ice composition). The best-fit mass and radius for HAT-P-26b falls just below the 4 Gyr model with a 10 M_⊕ rocky core and 8 M_⊕ gas envelope, implying that a 4 Gyr model with a slightly higher core mass would provide a better match to the mass and radius. We note that the isochrone-based age (9.0^+3.0_−4.9 Gyr) and the activity-based age (7.8^+1.4_−1.2 Gyr) for the HAT-P-26 system are somewhat older than 4 Gyr, so the inferred core mass would therefore be somewhat smaller.

We also compare HAT-P-26b to the theoretical models of Baraffe et al. (2008) which predict more significant inflation due to irradiation for low-mass planets than do the Fortney et al. (2007) models. In this case, the radius of HAT-P-26b is intermediate between the Z = 0.5 and Z = 0.9 heavy-element enrichment models.

Observations of the transiting hot Jupiters HD 209458b and HD 189733b in the H i Lyα line have indicated that both planets are evaporating at a rate of up to ∼10¹⁰ g s⁻¹ (e.g, Vidal-Madjar et al. 2003; Lecavelier des Etangs et al. 2011). Prompted by the observations for HD 209458b, several theoretical studies have indicated that atmospheric evaporation is likely to be important for close-in planets, particularly those with low surface gravities, such as hot Neptunes (see, for example, Lammer et al. 2003 and the review by Yelle et al. 2008). It has even been suggested that some close-in Neptune-mass and smaller planets may be the evaporated cores of planets which initially had masses comparable to Saturn or Jupiter (e.g., Baraffe et al. 2005). In the case of energy-limited escape, the evaporative mass loss is given by (see Erkaev et al. 2007 and Yelle et al. 2008; see also Valencia et al. 2010 and Jackson et al. 2010 for applications to CoRoT-7b)

$$\begin{equation} \dot{M_{p}} = -\frac{\pi R_{p}^{3}\epsilon F_{\rm XUV}}{G M_{p}K_{\rm tide}}, \end{equation} \tag{ 1 }$$

where F_XUV is the incident flux of extreme-ultraviolet (XUV) stellar radiation, is the heating efficiency and is estimated to be ∼0.4 for the case of HD 209458b (Yelle et al. 2008), and K_tide is a factor that accounts for an enhancement of the evaporation rate in the presence of tides, and is given by

$$\begin{equation} K_{\rm tide} = 1 - \frac{3}{2\xi } + \frac{1}{2\xi ^{3}}, \end{equation} \tag{ 2 }$$

where ξ = (M_p/(3M_⋆))^1/3a/R_p is the ratio of the Roche radius to the planet radius. Ribas et al. (2005) find that for solar type stars the XUV flux at 1 AU integrated over the wavelength range 1 Å to 1200 Å is given by

$$\begin{equation} F_{\rm XUV,1\, AU} = 29.7\tau ^{-1.23}\,{\rm erg}\,{\rm s}^{-1}\,{\rm cm}^{-2}, \end{equation} \tag{ 3 }$$

where τ is the age in Gyr. To our knowledge, a similar study has not been completed for K dwarfs, however long term X-ray observations of the 5–6 Gyr α Cen AB system reveal that on average the K1 dwarf star α Cen B has an X-ray luminosity in the 6–60 Å band that is approximately twice that of the Sun, while the G2 dwarf α Cen A has a luminosity that is approximately half that of the Sun (Ayers 2009). For simplicity we therefore assume that the total XUV luminosity of HAT-P-26 is comparable to that of the Sun (4.64 erg s⁻¹ cm⁻² at 1 AU; Ribas et al. 2005), which is likely correct to within an order of magnitude. Assuming = 0.4, we estimate that the expected present-day mass-loss rate for HAT-P-26b is ∼3 × 10¹⁰ g s⁻¹ = 0.17 M_⊕ Gyr⁻¹. To determine the total mass lost by HAT-P-26b over its lifetime, we integrate Equation (1) assuming an age of 4.5 Gyr, F ∝ τ^−1.23 for τ>0.1 Gyr and F ≡ constant for τ < 0.1 Gyr, neglecting tidal evolution of the orbit, and assuming that the radius is constant. We find that HAT-P-26b may have lost a significant fraction its mass (∼30%); the exact value depends strongly on several poorly constrained parameters including F_XUV and its dependence on age for a K1 dwarf, , and the age of the system.

Using the relation given by Adams & Laughlin (2006), the expected tidal circularization timescale for HAT-P-26b is ∼1 Gyr which is much less than the age of the system. This timescale is estimated assuming a large tidal quality factor of Q_P = 10⁶, and that there are no additional bodies in the system exciting the eccentricity. However, because at least two of the three hot Neptunes have significant eccentricities (GJ 436b has e = 0.14 ± 0.01, Demory et al. 2007; and HAT-P-11b has e = 0.198 ± 0.046, Bakos et al. 2010; the eccentricity for Kepler-4b is poorly constrained, Kipping & Bakos 2010a), we cannot conclude that the eccentricity must be zero on physical grounds, and therefore do not adopt a zero-eccentricity model for the parameter determination.

As discussed in Section 3.1 the eccentricity of HAT-P-26b is poorly constrained by the RV observations, and is instead constrained by requiring that the star be younger than the age of the universe (without the age constraint we get e = 0.24 ± 0.12, whereas including the age constraint gives e = 0.124 ± 0.060). To establish the significance of the eccentricity measurement, we also fit a model with the eccentricity fixed to zero. An F-test (e.g., Lupton 1993) allows us to reject the null hypothesis of zero eccentricity with only 79% confidence. Alternatively, the Lucy & Sweeney (1971) test for the significance of an eccentricity measurement gives a false alarm probability of ∼12% for detecting e>0.124 with an error of 0.060, or 88% confidence that the orbit is eccentric. If the eccentricity is fixed to zero, the required jitter to achieve χ²/N_dof = 1 is 2.4 m s^-1, which is closer to the typical Keck/HIRES jitter of other chromospherically quiet early K dwarfs than the jitter of 1.6 m s^-1 that is obtained with an eccentric orbit fit. We therefore are not able to claim a significant eccentricity for HAT-P-26b, and instead may only place a 95% confidence upper limit of e < 0.22. For the {ϒ/R_⋆, b, p²} model discussed at the end of Section 3.3, the 95% confidence upper limit is e < 0.32. Further RV observations, or a photometric detection of the occultation of the planet by its host star, are needed to determine if the eccentricity is nonzero.

The {ϒ/R_⋆, b, p²} model discussed in Section 3.3 with parameters given in Table 5 includes a linear drift in the RVs, $\dot{\gamma }$, as a free parameter. We find $\dot{\gamma } = -0.028_{-0.013}^{+0.014}$ m s⁻¹ day⁻¹. Conducting an odds ratio test (see Kipping et al. 2010), we conclude that the drift is real with 96.6% confidence, making this a 2.1σ detection. While the detection is not significant enough for us to be highly confident that there is at least one additional body in the system, this suggestive result implies that HAT-P-26 warrants long-term RV monitoring. We also searched for a linear time-shift in the RVs due to potential Trojans. We do not detect a significant shift, and may exclude |t_troj| < 0.50 days with 95% confidence. This translates to an upper limit of 84 M_⊕ on the mass of a Trojan companion, which is greater than the mass of the planet. With the present data we are thus not able to place a meaningful limit on the presence of Trojan companions.

HAT-P-26 has a number of features that make it an attractive target for potential follow-up studies. At V = 11.744, it is bright enough that precision spectroscopic and photometric observations are feasible with moderate integration times. The equatorial declination of δ = +04°03'360 also means that HAT-P-26 is accessible to both Northern and Southern ground-based facilities. The exceptionally low jitter will facilitate further RV observations, which might be used to confirm and refine the eccentricity determination, to measure the Rossiter–McLaughlin effect (R-M; discussed in more detail below) and to search for additional planets in the system.

A detection of the occultation of HAT-P-26b by HAT-P-26 with IRAC/Spitzer would provide a strong constraint on k ≡ ecos ω, while the duration of the occultation would provide a constraint on h ≡ esin ω. We note that the median value of the a posteriori distribution for the time of occultation that results from our global fit when the eccentricity is allowed to vary (Section 3.3) is 6.5 hr after the expected time of occultation assuming a circular orbit. The expected depth of the occultation event is a challenging 0.012% and 0.020% at 3.6 μm and 4.5 μm, respectively. Scaling from TrES-4, a somewhat fainter star at these wavelengths, for which Knutson et al. (2009) measured occultations at 3.6 μm and 4.5 μm using IRAC/Spitzer with precisions of 0.011% and 0.016%, respectively, one may hope to achieve a ∼1.4σ and 1.6σ detection for HAT-P-26b for one event at each bandpass. One would need to observe five and four occultations, respectively, to achieve a 3σ detection.

Recent measurements of the R-M effect for TEPs have revealed a substantial population of planets on orbits that are significantly misaligned with the spin axes of their host stars (e.g., Triaud et al. 2010). Winn et al. (2010a) note that misalignment appears to be more prevalent for planets orbiting stars with T_eff>6250 K, and suggest that most close-in planets migrate by planet–planet or planet–star scattering mechanisms, or by the Kozai effect, rather than disk migration, and that tidal dissipation in the convective surfaces of cooler stars realigns the stellar spin axis to the orbital axis of the close-in massive planet. Schlaufman (2010) also finds evidence that planets orbiting stars with M_⋆>1.2 M_☉ are more likely to be misaligned than planets orbiting cooler stars using a method that is independent of the R-M measurements. One prediction of the Winn et al. (2010a) hypothesis is that lower mass planets orbiting cool stars should show a greater degree of misalignment than higher mass planets due to their reduced tidal influence. The detection of misalignment for HAT-P-11b (Winn et al. 2010b; Hirano et al. 2010) is consistent with this hypothesis. Measuring the R-M effect for HAT-P-26b would provide an additional test. Using Equation (40) from Winn (2010), the expected maximum amplitude of the R-M effect for HAT-P-26b is ∼9 m s^-1, which given the low jitter of HAT-P-26, should be detectable at ≳10σ.

By measuring the primary transit depth as a function of wavelength it is possible to obtain a transmission spectrum of an exoplanet's atmosphere. Such observations have been made for a handful of planets (e.g., Charbonneau et al. 2002; see also the review by Seager & Deming 2010). Following Brown (2001), the expected difference in transit depth between two wavelengths is given approximately by

$$\begin{equation} \Delta \delta = \frac{2R_{p}H}{R_{\star }^2}N_{\rm H}, \end{equation} \tag{ 4 }$$

where H = k_BT_eq/g_pμ is the scale height of the atmosphere, g_p is the planet surface gravity, μ is the mean molecular weight of the atmosphere, and N_H = ln(σ₁/σ₂) where σ₁ and σ₂ are the opacities per gram of material at wavelengths in a strong atomic or molecular line and in the nearby continuum, respectively. Assuming a pure H₂ atmosphere, μ = 3.347 × 10⁻²⁷ kg, we find for HAT-P-26b H = 920 km, and Δδ = 0.0246N_H%. If instead we assume that the atmosphere has the same composition as Neptune (e.g., de Pater & Lissauer 2001, p. 80), we have μ = 4.655 × 10⁻²⁷ kg, H = 660 km, and Δδ = 0.0177N_H%. For comparison, assuming a pure H₂ atmosphere, the planet HD 209458b has Δδ = 0.0198N_H%, while GJ 436b has Δδ = 0.0107N_H%, HAT-P-11b has Δδ = 0.0072N_H% and Kepler-4b has Δδ = 0.0038N_H%. Due to its low surface gravity, HAT-P-26b easily has the highest expected transmission spectrum signal among the known transiting Neptune-mass planets. While it is relatively faint compared to the well studied planets HD 209458b and HD 189733b, we note that Sing et al. (2010) used the Gran Telescopio Canarias (GTC) to detect a 0.058% ± 0.016% absorption level at 7582 Å due to Potassium in the atmosphere of XO-2b, which orbits a V = 11.2 early K star. Scaling from this observation, it should be possible to detect components in the atmosphere of HAT-P-26b with N_H ≳ 3 at the ∼3σ level using the GTC.

In summary, HAT-P-26b is a low-density Neptune-mass planet. Its low density relative to the other known Neptune-mass planets means that HAT-P-26b likely has a more significant hydrogen–helium gas envelope than its counterparts. The existence of HAT-P-26b provides empirical evidence that, like hot Jupiters, hot Neptunes also exhibit a wide range of densities. Comparing to the Fortney et al. (2007) models, we find that HAT-P-26b is likely composed of a gas envelope and a heavy-element core that are approximately equal in mass, while the Baraffe et al. (2008) models prefer a higher heavy-element fraction. It is also likely that irradiation-driven mass loss has played a significant role in the evolution of HAT-P-26b —we find that the planet may have lost ∼30% of its present-day mass over the course of its history, though this conclusion depends strongly on a number of very poorly constrained parameters, particularly the XUV flux of HAT-P-26 and its evolution in time. We place a 95% confidence upper limit on the eccentricity of e < 0.22. If further observations detect a nonzero eccentricity, it would mean that at least three of the four known Neptune-mass TEPs have nonzero eccentricities, which may imply that the tidal quality factor is higher than expected for these planets. Observations of the planetary occultation event for HAT-P-26b with IRAC/Spitzer would greatly constrain the eccentricity, however the low expected depth is likely to make this a challenging observation. We find suggestive evidence for a linear drift in the RVs which is significant at the 2.1σ level. If confirmed, this would imply the existence of at least one additional body in the HAT-P-26 system. With an expected R-M amplitude of ∼9 m s^-1 and a low stellar RV jitter, HAT-P-26b is a good target to measure the R-M effect and thereby test the hypothesis that low-mass planets are more likely to be misaligned than high-mass planets. The low surface gravity also makes HAT-P-26b a good target for transmission spectroscopy.