AN UNDERSTANDING OF THE SHOULDER OF GIANTS: JOVIAN PLANETS AROUND LATE K DWARF STARS AND THE TREND WITH STELLAR MASS

Doppler observations were performed with the High Resolution Echelle Spectrograph (HIRES) on the Keck I telescope (Vogt et al. 1994). Observations obtained R = 55, 000 with the B5 decker and a typical S/N of 200. Wavelength calibration was provided by a molecular iodine cell in the beam line. RV solutions were obtained by a forward modeling process in which an intrinsic stellar spectrum is obtained without the iodine cell, multiplied by an R ∼5 × 10⁵ spectrum of the iodine cell taken with a Fourier transform spectrograph, and convolved by the instrumental profile. The relative shift in wavelength between the model and observed spectra is a free parameter. The median formal measurement error in the M2K survey is 1.25 m s⁻¹. We obtained N = 4 or more RV measurements on 159 stars. The median number of measurements for each star used in our analysis is N = 9, however the distribution of number of measurements is very uneven because of our strategy of follow-up of RV-variable stars (Figure 1).

Zoom In Zoom Out Reset image size

Ideally, we would have defined our sample of late K dwarfs in terms of effective temperature T_eff, a fundamental stellar parameter, but not all our stars have spectroscopically-determined values. Thus, we selected stars based on V − J color as a proxy for T_eff. We restricted the sample to 140 stars with 1.8 < V − J < 2.8, corresponding to 3900 < T_eff < 4750 K or spectral subtypes K3–K7 (Gray & Corbally 2009), based on an empirical color–T_eff relation using stars with measured angular diameters (Boyajian et al. 2012). We corroborated this selection by estimating the T_eff of many of these stars using spectra (Figure 2). Stellar parameters (including T_eff and metallicity [Fe/H]) of stars with 1.8 < V − J < 2.3 were estimated using the Spectroscopy Made Easy (SME) package (Valenti & Piskunov 1996; Table 2). SME performs poorly on dwarfs with V − J > 2.3 and T_eff ⩽ 4300 K so for these stars we estimated T_eff by comparing moderate-resolution (R ∼ 1500) visible-wavelength (3500–8500 Å) spectra obtained with the Supernova Integral Field Spectrograph (Lantz et al. 2004) on the University of Hawaii 2.2 m telescope with synthetic spectra generated by the PHOENIX BT-SETTL program (Allard et al. 2011). The comparison procedure is described in Lépine et al. (2013) and was adjusted (A. W. Mann et al. 2013, in preparation) to maximize agreement with the calibrator stars of Boyajian et al. (2012). For stars without spectra, we estimated T_eff using V − J color and the Boyajian et al. (2012) relation (Table 2).

Zoom In Zoom Out Reset image size

The median offset between SME-derived and V − J-based temperatures is 140 K. For this reason, we included the 13 stars with SME-based T_eff > 4750 K (but acceptable V − J) as their actual effective temperatures are probably within the acceptable range. Unsurprisingly, T_eff estimates based on calibrated comparisons between moderate-resolution spectra and PHOENIX models are consistent with the Boyajian et al. (2012) relation. Nearly all of our stars have measured parallaxes and we estimated masses using the relation with absolute K magnitude in Henry & McCarthy (1993). For those few stars lacking parallaxes we used the empirical relations between T_eff, stellar radius, and stellar mass in Boyajian et al. (2012).

We excluded stars exhibiting relatively high emission in the H and K lines of Ca ii. These stars are chromospherically active and tend to exhibit higher astrophysical Doppler noise or "jitter" (e.g., Isaacson & Fischer 2010). Figure 3 shows values of S_HK, the flux in the Ca ii line cores normalized by the continuum, versus V − J color. The trend of increasing S_HK with redder V − J is due to the lower blue continuum rather than elevated Ca ii emission in redder stars. We calculated a running median (N = 20), fit a linear function $\bar{S}_{{\rm HK}}$ with V − J, and subtracted that from these values. The histogram (inset of Figure 3) suggests a cutoff at $S_{{\rm HK}} - \bar{S}_{{\rm HK}} = 0.44$, which rejects 11 stars as exceptionally active. Among the stars we admitted were five for which it was not possible to estimate S_HK because the continuum was not detected.

Zoom In Zoom Out Reset image size

We next considered the distribution of RV standard deviations (rms) of the remaining 129 stars. The majority (100) of the stars fall in a cluster with rms <15 m s⁻¹ (Figure 4). We inspected the RV data of the 29 stars with rms >15 m s⁻¹. One of these is HIP 57274, which has been described elsewhere (Fischer et al. 2012). Eight others have additional stars within 5 arcsec and were excluded because leakage of light into the spectrograph slit is an established source of RV error.

Zoom In Zoom Out Reset image size

We analyzed each of the remaining 20 sets of RVs with both weighted linear and quadratic regressions and applied an F-test to evaluate the significance of any reduction in variation after subtraction of the fit. Although all stars have ⩾4 RV measurements, and thus the number of degrees of freedom for a quadratic fit is ⩾1, irregular sampling means that overfitting and erroneous reduction in RV variation is possible. For example, four observations in two very closely spaced pairs cannot be reliably regressed: a significant RV offset between the pairs could be the result of a linear trend or any RV variation. To identify such cases, we computed an effective N equal to the sum of normalized Voronoi-type weights w_i = (t_{i + 1} − t_{i − 1})/2 that are often assigned by regularization algorithms (Strohmer 2000). End points have weights w₁ = t₂ − t₁ and w_N = t_N − t_{n − 1}. Thus,

$$\begin{equation} N_{\rm eff} = \frac{3(t_N-t_1) - (t_{N-1}-t_2)}{2\; {\rm max}(w_i)}. \end{equation} \tag{ 1 }$$

In the limit of large N, the effective number of points N_eff and thus the maximum order of the polynomial that should be used in a regression approach the total time interval divided by the maximum interval between points. This is analogous to the Nyquist sampling criterion.

Doppler data sets with rms >15 m s⁻¹ were processed in one of three ways: (1) if regressions did not significantly reduce variance (F-test; p < 0.05), the data were analyzed as is. (2) If a regression did significantly reduce variance and N_eff was >3 (or >4, in the case of a quadratic), the best fit is subtracted before analysis. (3) If a regression is significant but N_eff is not sufficiently large, the star and its data were excluded from the analysis. We excluded 11 stars in this way, leaving 110 stars for analysis, including HIP 57274.

HIP 2247 has a long-period super-Jupiter previously identified by Moutou et al. (2009). We fit the combined HARPS and Keck-HIRES data using the RVLIN code (Wright & Howard 2009), generating errors using 100 Monte Carlo realizations of the data by randomly reshuffling the residuals to the previous fit. We find essentially the same planetary parameters as Moutou et al. (2009), but with significantly reduced uncertainties: M_psin i = 5.14 ± 0.02 M_J, P = 655.90 ± 0.22 days, and e = 0.543 ± 0.0011, with a residual rms of 3.8 m s⁻¹ (Figure 5). No significant trend was found (−0.0024 ± 0.0011 m s⁻¹). The uncertainty in msin i does not include errors in the estimated stellar mass of 0.77 M_☉. Any giant planets with P < 245 days can be ruled out: we include this system in our sample but consider it a definitive non-detection.

Zoom In Zoom Out Reset image size

HIP 38117 exhibits RV variation consistent with the presence of a stellar-mass companion on a 81.28 ± 0.01 day orbit with an eccentricity of 0.478 ± 0.012 (Figure 6). Assuming a primary mass of 0.73 M_☉ based on the system's V − J color, and assuming that the secondary contributes negligible flux, the companion's M_*sin i is 0.45 ± 0.16 M_☉, i.e., this is a very late K or M dwarf. (This calculation assumes an average value of 〈sin i〉 = π/4 to calculate the total system mass.) The residual rms is 3 m s⁻¹ (N = 15). As a planetary orbit with a comparable orbital period is unlikely to be stable, we follow the suit of other studies by excluding this binary system from our analysis.

Zoom In Zoom Out Reset image size

We estimated the fraction of stars f with giant planets having M_P > 0.3M_J (i.e., Saturn mass) and Keplerian orbital periods 1.7 days <P_K < 245 days. The choice of outer cutoff in P_K is based on the temporal baseline of our data—nearly all stars were monitored for at least 245 days—and motivated by the longest bin with good statistics in the analysis of Kepler planet candidates by Fressin et al. (2013). The inner cutoff corresponds to the location of the rollover in the period distribution of giant planets around Kepler stars (Howard et al. 2012). We construct and maximize a likelihood function to find the most probable value of f and its uncertainty. The details of the calculations are given in the Appendix and the method is only summarized here.

A standard procedure to estimate the fraction of stars with planets is to maximize a binomial expression involving the product of detections and non-detections. However, with RV data it can be difficult or impossible to rule out all possible planets, e.g., those on face-on orbits. Thus, we replace detections and non-detections with a Bayesian statistic that is sum of the probabilities $p_i^0$ and $p_i^1$ that there are zero or one giant planets around the ith star, with 1 − f and f as priors for zero or one planets, respectively,

$$\begin{equation} \ln \mathcal {L} = \displaystyle \sum _i \ln \left(p_i^0(1-f) + p_i^1f \right), \end{equation} \tag{ 2 }$$

where the sum is over all stars. This counts multi-giant planet systems once, and thus underestimates the planet occurrence (planets per star). The probabilities $p_i^0$ and $p_i^1$ are Gaussian functions of the difference between the predicted and observed RVs v_ij and $\hat{v_{ij}}$, weighted by priors $\tilde{p_i}$ marginalized over all model parameters

$$\begin{equation} p_i^n = \left\langle \tilde{p_i}\exp \left[-\displaystyle \sum _j \frac{\left(v_{ij}-\hat{v}_{ij}^n\right)^2}{2\sigma _{ij}^2}\right] \right\rangle, \end{equation} \tag{ 3 }$$

and where σ_ij are the formal errors, astrophysical noise or "jitter," as well as systematic error and the contribution of small planets to motion of the star around the system's barycenter (see Section 4.1), added in quadrature. This method is analogous to the approach used in Gaidos et al. (2012) but uses the individual RV measurements, not the rms.

The Gaussian form of Equation (3) means that only the best-fit sets of parameter values (barycenter motion v₀, Keplerian period P_K, Doppler amplitude K, eccentricity e, longitude of periastron ω, and time of periastron t₀) make significant contributions to p⁰ or p¹. We used the linear dependence of the RVs on the barycenter velocity v₀ and Doppler amplitude K to analytically solve for the best-fit values of these two parameters given values for the other parameters. To marginalize over planet mass we used the relation between K, planet mass, and orbital inclination and adopt a power-law distribution of log mass with an index α = −0.31 (Cumming et al. 2008). (The sensitivity of our results to this value is explored in Section 4.2.) We marginalized Equation (3) over the full range of possible values of e, ω, t₀, using a Rayleigh function for the prior on eccentricity (Moorhead et al. 2011) and uniform priors for ω and t₀. We further evaluated the probability over 1.7 < P_K < 245 days at intervals of equal prior probability, assuming a power-law distribution for log period having index β = 0.26 (Cumming et al. 2008). To better sample intervals of P_K corresponding to higher probability we used each P_K value as an initial value in a fit of a Keplerian solution to the data with the RVLIN routine (Wright & Howard 2009). We used the fixed, best-fit values of the other parameters for the RVLIN fit, obtained an adjusted value of P_K, and then re-calculated the other parameters as described above. This procedure was repeated twice, which we found was sufficient for convergence.

We calculated p⁰ and p¹ by summing over final values of all the parameters, and normalizing by p⁰ + p¹. For three stars, p₀ and p₁ were both incalculably small due to large disagreements between v and $\hat{v}$ for either the zero- or one-planet models. This could be due to elevated stellar jitter or the presence of smaller planets (see below). For these stars we assigned p₀ = p₁ = 0.5, i.e., the zero- and one-planet models are accepted or rejected with equal likelihood. We then evaluated Equation (2) over all possible values of f, and found the maximum. We calculated an approximate uncertainty by assuming asymptotic normality, iteratively fitting a parabola to the log-likelihood curve, and assigning $\sigma _f = 1/\sqrt{2C}$, where C is the curvature coefficient of the parabola.

Both the zero- and one-planet models do not account for barycenter motion due to the presence of other, smaller planets, as well as any sources of systematic error. As a result, for some stars both models are strongly rejected, leading to an erroneously high value of f. To account for this effect, we treat this barycenter motion as an additional source of uncorrelated, random RV noise or "jitter" that, along with stellar noise, can be described by a single value of σ₀. A value for σ₀ was chosen by assuming that the pronounced cluster of systems with rms <15 m s⁻¹ (Figure 4) represents stars without giant planets, and fitting that distribution by a Monte Carlo model. We constructed 1000 artificial realizations of the data with the same number of RVs per star but drawn from a random normal distribution. The variance of this distribution was set equal to the formal measurement error and a trial value of σ₀ added in quadrature. We computed the rms values and comparing the distribution to the observed distribution (after subtracting any trends) with a two-sided Kolmogorov–Smirnov (K-S) test. We found that a curve with σ₀ = 6.3 m s⁻¹ (solid curve in Figure 4) maximizes the K-S probability of the Monte Carlo distribution (inset of Figure 4), and we use that value in our estimation of f. The poor agreement between the observed and "best-fit" distributions reflects the inability of a single value of σ₀ to capture the diversity of stellar RV behavior.

Values of p¹, the probability that the RV data are consistent with the presence of a giant planet with P_K < 245 days, are reported for the 110 late K dwarfs in Table 2 (note that p⁰ = 1 − p¹). Figure 7 shows the relative likelihood distribution of f constructed from these values using Equation (2). The most probable value of f is 4.0% and the uncertainty based on the assumption of asymptotic normality is ±2.3%. The elevated range relative to the rate of actual detections (1 of 110 stars) is due to the presence of several stars in our sample with significantly non-zero values of p¹. Eighteen stars have p¹ > 0.2 and four, excluding HIP 57274, have p¹ > 0.9 (Table 2). We are continuing to monitor these stars (T. S. Boyajian et al. 2013, in preparation). If we assume that giant planets with P_K < 245 days are ruled out (p¹ = 0) for all stars other than HIP 57274 (the "HIP 57274 only" case in Figure 7), the most probable value of f becomes 0.92% ± 0.75%.

Zoom In Zoom Out Reset image size

Stars may exhibit high RV variation for reasons other than the presence of giant planets with P < 245 days. Many M2K target stars were monitored for intervals ≫245 days and our RV data are sensitive to the presence of planets on wider orbits. Both Doppler and Kepler surveys find such planets, e.g., HIP 2247 and KOI 1466.01 (see below). Some stars may have a lower-mass (M dwarf) companion like that of HIP 38117 (Figure 6), but on a wider orbit. If the trend in RV produced by such a companion is not resolved because of undersampling, it will manifest itself as a high rms. Despite our precautionary elimination of stars with Ca ii HK emission, some stars in our sample may have high intrinsic "jitter" from spots. Many stars in our sample have only a few Doppler observations (Figure 1), confounding these effects. Ultimately, additional observations are required to discriminate between these possibilities.

The relative likelihood distribution of f for late K dwarfs observed by Kepler is plotted as the dashed line in Figure 7. The most likely value of f is 0.7% ± 0.5%. Based on the two distributions, we calculate a 99% probability that the Kepler value is actually lower than the M2K value. Due to the factors discussed above, the M2K value of 4% may be an overestimate: There is a closer correspondence (85% chance that the Kepler estimate is lower) if we rule out giant planets around all stars other than HIP 57274. Wright et al. (2012) report that the occurrence of "hot" Jupiters around the FGK stars in the CPS Doppler survey is 1.2%, compared to 0.4% for Kepler (Howard et al. 2012; Fressin et al. 2013). Gaidos & Mann (2013) proposed that the difference between the transit and Doppler results may be due to the presence of subgiants in the Kepler target catalog: planets around such stars will be more difficult to detect and more likely to experience destructive orbital decay. This explanation may be less applicable to late K spectral types where the giant and main-sequence branches are more distinguishable, and we consider orbits with P_K ≫ 10 days on which orbital decay will be negligible. Another explanation for at least some of this difference is that the exclusion of spectroscopic and resolved binaries from the M2K sample, but not the Kepler sample, may enrich for giant planets, presuming that such binaries are less likely to host planets for dynamical reasons (e.g., Thébault et al. 2006; Bonavita & Desidera 2007; Kaib et al. 2013).

We compare our M2K and Kepler estimates of f for late K dwarfs with previous studies for different ranges of stellar mass (Figure 8). Our estimates bridge the gap between solar-type stars (Fischer & Valenti 2005; Cumming et al. 2008; Johnson et al. 2010; Howard et al. 2010, 2012; Fressin et al. 2013) and M dwarfs (Naef et al. 2005; Cumming et al. 2008; Johnson et al. 2010; Bonfils et al. 2013; Fressin et al. 2013). We have adjusted values by the factor $\ln (P_{{\rm max}}/{\rm 1.7\, d{\rm ays}})/\ln (245/1.7)$ to account for differences in the maximum orbital period P_max of each survey, assuming a flat distribution with ln P_K. (The adjustment is not sensitive to the exact distribution assumed.) The surveys also differ somewhat in the mass or radius ranges of objects counted as giant planets. For example, although our Kepler-based estimate of 0.7% for late K dwarfs seems much lower than those of Fressin et al. (2013) for either GK dwarfs (6.1% ± 0.9%) or M dwarfs (3.6% ± 1.7%), these statistics are for P_K < 418 days rather than 245 days, and R_p > 6 R_⊕, rather than the 8 R_⊕ convention adopted here. Their overall f falls to 4.1% for P < 245 days, and our f rises to 1.1% ± 0.6% if we include planets with R_p > 6 R_⊕, bringing these two figures closer. Taken together, these estimates suggest an overall trend, perhaps linear, of increasing giant planet occurrence with stellar mass, there is not yet any indication of finer structure. A linear least-squares fit to the adjusted Doppler data yields f(%) = −1.11 + 5.33 M_*/M_☉ (dashed line in Figure 8) with weak significance (F-test probability of 0.12). This compilation also suggests that the deficit of giant planets around Kepler stars relative to the targets of Doppler surveys (Wright et al. 2012) depends on host star mass (Figure 8), although clearly a more homogeneous analysis of the collective data sets is needed.

Zoom In Zoom Out Reset image size

A correlation between giant planets and the metallicity of the host star has been unambiguously established for solar-type stars (e.g., Fischer & Valenti 2005), and is strongly supported by the available evidence for M dwarfs (Neves et al. 2013; Mann et al. 2013). The median metallicity of our sample of late K dwarfs is solar ([Fe/H] = 0.004). The metallicities of HIP 57274 and HIP 2247, the two stars known to host giant planets in our sample, are 0.08 and 0.24 dex, consistent with this trend. The difference in the distributions of metallicities of stars with p¹ < 0.1 (median [Fe/H] of −0.01) and those with p¹ > 0.1 (median [Fe/H] of 0.08) is marginally significant (K-S probability of 0.06), further supporting a giant planet–metallicity relation in late K dwarfs. If SME overestimates the T_eff of these stars (Figure 2), the [Fe/H] is also overestimated by about 0.1 dex per 100 K.

Our estimates of f may be sensitive to the values of any one of several parameters we use in our calculations (see the Appendix). These include the computational resolution n with which $p_i^N$ is evaluated over ranges of the various orbital parameters, the power-law indices α and β for the assumed mass and period distributions, the mean value $\bar{e}$ of the Rayleigh-distributed eccentricities, and the RV jitter σ₀ which is assumed for each star. Due to the computational requirements of such studies, we first considered the effect on two stars, HIP 37798 and HIP 66074, with number of observations equal to the median (N = 9) but with the smallest (0.005) and large (0.82) values of p¹, respectively. Based on the outcome sensitivity of p¹ to varying parameter values, we selectively investigated the effects on our estimates of f.

Varying n from 25 to 50 (at rapidly increasing computational cost) had a negligible effect on p¹ for HIP 37798 but decreased the value for HIP 66074 by about 13%. We found that p¹ varied little for n > 50. Thus, we re-analyzed 17 stars with p¹ values >0.2 (excluding HIP 57274) using n = 50. Not all stars were re-analyzed because of the high computational cost. These n = 50 values are used in the calculations of f in Section 4.1. Without the substitution of high-resolution values, the most likely value of f is 5.1% ± 2.7%.

We varied the power-law index α of the planet mass distribution by ±0.2 from its nominal value of −0.31 based on Cumming et al. (2008; see also Howard et al. 2012). p¹ increased significantly and systematically with more negative values of α, by a factor of 3.5 for HIP 37798 and nearly 1.5 for HIP 66074. We found that the most probable value of f changed from 3% to 6.4% when we varied α from −0.11 to −0.51. Also based on Cumming et al. (2008), we varied the power-law index β of the orbital period distribution by ±0.1 from its nominal value of 0.26. We found that the p¹ for HIP 37798 was essentially unchanged, while that of HIP 66074 changed by only ±15%. Varying $\bar{e}$ by ±0.1 from its nominal value of 0.225 (Moorhead et al. 2011) also had a negligible effect on the p¹ of HIP 37798 and changed that of HIP 66074 only slightly. Thus, our estimate of f is not sensitive to the assumed distributions of orbital period and eccentricity, but does depend on the mass distribution. The last occurs because a steeper mass function (more negative α) includes more Saturn-mass planets that could be hidden on low inclination orbits in our RV data.

Finally, we varied the value of σ₀ assigned to each star to account for astrophysical noise and barycenter motion induced by small planets. We considered a range of 6–6.75 m s⁻¹ based on the K-S probability that the observed and simulated RV rms distributions are above 0.05 times the maximum value at σ₀ = 6.3 m s⁻¹ (i.e., 95% confidence) (see Section 2.2). For both stars, values of p¹ increase significantly if σ₀ is decreased from its nominal value, but increased only slightly for higher σ₀. Correspondingly, f increased by a factor of 1.6 for σ₀ = 6 m s⁻¹, and decreased by 0.82 for σ₀ = 6.8 m s⁻¹. A smaller σ₀ means that the more RV variation must be explained by the presence of giant planets, e.g., Saturn-mass planets with low orbital inclinations.

A correlations between the occurrence of giant planets and stellar metallicity (e.g., Fischer & Valenti 2005; Johnson et al. 2010) has been interpreted as supporting the core accretion scenario of giant planet formation. In that scenario, growth of a sufficiently massive solid core leads to the runaway accretion of gas, but only if it occurs before the gas disk is dissipated in a few million years (Lissauer & Stevenson 2007). In disks of higher metallicity gas, dust grains can grow, collide, and settle to the mid-plane more rapidly, thus initiating planet formation at an earlier epoch Johnson & Li (2012). Simulations of rocky planet mass by Kokubo et al. (2006) produced a linear trend between final planet mass and initial disk mass surface density. Thus, disks around high-metallicity disks should produce larger rocky cores around which gas could accrete more quickly.

However, a trend with stellar mass, supported by our results, may require a more complex explanation. First, the dependence of disk mass on stellar mass appears to be weak (Williams & Cieza 2011) and higher disk mass need not translate into higher mass surface density—and more massive planets (Kokubo et al. 2006)—if the radial extent of the disk is larger. Moreover, Doppler and transit surveys of FGK stars thoroughly probe orbital semimajor axes to ≲1 AU; available RV data suggest a "jump" in the population of giant planets just beyond 1 AU (Wright & Howard 2009) and set generous lower limits on their occurrence on much wider orbits (e.g., Wittenmyer et al. 2011). Microlensing surveys suggest that as many as a third of lensing stars (typically late K and M dwarfs) host giant planets at 1–5 AU (Mann et al. 2010; Cassan et al. 2012). If giant planet formation preferentially occurs on these orbits, the correlation with stellar mass may arise from varying efficiency of inward migration, rather than formation, of giant planets.

The coolest giant planet host stars in our M2K and Kepler samples are HIP 57274 and KOI 1176.01 with temperatures of 4640 K and 4625 K. The only cooler K dwarfs hosting reported giant planets are WASP-43, HIP 70849, and WASP-80 (Table 1), but only the WASP planets are on close-in orbits. The effective temperature of WASP-43, based on the shape of the Balmer Hα line, is 4400 K (Hellier et al. 2011), and this is broadly consistent with the V − J color of 2.4. However, this star is active and chromospheric emission may fill in and weaken the Hα line, making the temperature estimate erroneously low. An analysis of transit light curves coupled with stellar models suggests 4520 ± 120 K instead (Gillon et al. 2012). The T_eff assigned to HIP 70849 was based solely on its luminosity and a theoretical temperature–luminosity relation (Ségransan et al. 2011). WASP-80 shares spectral characteristics with both K7 and M0 dwarfs and analyses of a spectrum and infrared photometry suggests temperatures of 4145 ± 100 and 4020 ± 130 K, respectively (Triaud et al. 2013).

Depending on the properties of WASP-43 and WASP-80, these stars may bracket a T_eff range of 4100–4600 K over which giant planets on close orbits have yet to be found. This could be a hint of structure, i.e., a gap or "shoulder" in the giant planet distribution with stellar mass, but any conclusion requires new surveys. Giant planets appear to orbit at wider separations around such stars (e.g., HIP 70849b and KOI 868.01), and future space-based astrometric searches with the Gaia mission (de Bruijne 2012) and microlensing surveys by Euclid (Penny et al. 2012) or the proposed WFIRST observatory (Barry et al. 2011) should reveal such planet populations in detail.

We have used the M2K and Kepler surveys to place approximate constraints on the fraction of late K dwarfs with giant planets, but the target catalogs are of inadequate size to address the question of any "fine structure" in the distribution of giant planets with stellar mass. The Next Generation Transit Survey (NGTS; www.ngtransits.org) will monitor 40,000 late-G- to early-M-type stars to search for "hot" Neptunes. Based on our inferred occurrence rate, we expect there to be ∼10 Jupiters around these target stars; however, most of these will have orbital periods >10 days where the detection efficiency of a ground-based survey at a single site like NGTS is low. The Transiting Exoplanet Survey Satellite will survey ∼2.5 million stars to V = 13 (Deming et al. 2009) and, according to the TRILEGAL stellar model of the Galaxy (Girardi et al. 2005), approximately 50,000 targets will be late K dwarfs with 4000 < T_eff < 4800 K. Monitoring of these should significantly improve the statistics and allow us to see further.