DEEP MMT^* TRANSIT SURVEY OF THE OPEN CLUSTER M37 IV: LIMIT ON THE FRACTION OF STARS WITH PLANETS AS SMALL AS 0.3R_J

The raw light curves returned from the image subtraction procedure exhibit substantial scatter due to instrumental artifacts as well as astrophysical variations. Before searching these light curves for low-amplitude transit signals we must take steps to reduce the time-correlated noise. This process may reduce the sensitivity to high signal-to-noise ratio (S/N) planets, so for selection criteria set 3 we only apply a limited postprocessing routine. Our postprocessing pipeline consists of the following steps:

• 1.  
  (Selection criteria sets 1–3) We first clip points from each light curve that are more than five standard deviations from the mean magnitude. We perform two iterations of this procedure.
• 2.  
  (Selection criteria sets 1–3) For each image i, we determine f_i, the fraction of light curves for which image i is at least a 3σ outlier. Images with f_i>f_c, the cutoff fraction, are removed from all the light curves. This process is performed independently for each of the 36 chips. We choose f_c for each chip based on a visual inspection of the histogram of f_i values; we use values that range from 3% to 5%. This process typically removes ∼500 images from the light curves.
• 3.  
  (Selection criteria sets 1 and 2 only) We then remove one sidereal day or 0.9972696 day period signals from the light curves. This is done to remove artifacts due to, for example, rotating diffraction spikes that have a period of exactly one sidereal day. Figure 1 shows an example of a light curve that exhibits brightenings at a period of one sidereal day as diffraction spikes from nearby bright stars sweep over the star. The diffraction spikes rotate in the image due to the need to rotate the camera with respect to the secondary mirror supports to keep stars on the same pixels throughout the night on the alt-az MMT. The number, width, depth, and shape of the brightenings seen in the light curves vary from star to star; moreover, additional artifacts such as color-dependent atmospheric extinction will also give rise to apparent variability with a period of one sidereal day. We remove these signals by binning the light curves in phase, using 200 bins, and then adjusting the points in a bin by an offset so that the average of the bin is equal to the average of the light curve.
• 4.  
  (Selection criteria sets 1 and 2 only) As discussed in Paper III, we found that ∼1/3 of the probable cluster members that we observed show quasi-periodic variations with amplitudes of ∼1% and periods ranging from 0.3 to 15 days. These variations are due to the presence of large spots on the surfaces of these relatively young (550 Myr), rapidly rotating stars. While it may be possible to identify a ∼1% transit on top of such a signal, it would be increasingly difficult to identify shallower transits without taking steps to remove these variations. Using the Lomb–Scargle algorithm (Lomb 1976; Scargle 1982; Press & Rybicki 1989), we identify the period P of the best-fit sine curve to each light curve. We then fit to the poststep-3 light curve a signal of the form

  $$\begin{eqnarray} m &=& A_{0} + A_{1}\sin (2Pt/2\pi + \phi _{1}) + A_{2}\sin (Pt/2\pi + \phi _{2}) \nonumber \\ && \quad + A_{3}\sin (0.5Pt/2\pi + \phi _{3}), \end{eqnarray} \tag{ 1 }$$

  where A_i and ϕ_i are free parameters and we calculate Δχ²_Harm, the reduction in χ² after subtracting the best-fit model from the light curve. Note that we adopt the convention that a more negative value of Δχ² indicates a better fit of the model to the data. This process increases the sensitivity to shallow transits but decreases the sensitivity to deep transits. We therefore fit a box-car transit signal to the poststep-3 light curve, without subtracting the sine curve, phased at period P using the box-fitting least squares (BLS) algorithm (Kovács et al. 2002) and calculate Δχ²_BLS,fix, the reduction in χ² after subtracting the box-car transit model from the light curve. We subtract Equation (1) from light curves with Δχ²_Harm <   Δχ²_BLS,fix, that is, we only apply the correction to light curves for which a harmonic series fits the phased light curve better than a transit signal. To test this technique, we simulate light curves including variability due to spots as well as transits. We use the Dorren (1987) spot model and the Mandel & Agol (2002) transit model. Figure 2 compares the results of a full BLS search on light curves with deep transits relative to spots and vice versa for the two cases of removing and not removing a harmonic signal before running BLS. We see that removing a harmonic series from light curves with Δχ²_Harm < Δχ²_BLS,fix yields BLS results that are consistent with the injected transit signal, while not removing the harmonic signal from light curves with Δχ²_Harm>Δχ²_BLS,fix yields better results than removing the signal.
• 5.  
  (Selection criteria sets 1 and 2 only) Finally, we attempt to remove any remaining instrumental or weather-related trends from the poststep-4 light curves using the Trend Filtering Algorithm (TFA; Kovács et al. 2005). This algorithm linearly decorrelates each light curve against a representative sample of other light curves. The trend list for each light curve consists of other stars on the chip with more than 2500 points, rms <0.1 mag, and that are well outside the photometric radius of the star in question. There are typically ∼250 stars in the trend list for each chip.

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3.1.1. Light-Curve Precision

In Figure 3, we show the rms as a function of magnitude for the stars after stages 2 and 5 in the light-curve processing pipeline. We plot both the point-to-point rms and the rms after binning the light curves by 2 hr in time.

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As discussed by Pont et al. (2006), the limiting factor for detecting low-amplitude transits around bright stars for many transit surveys is the presence of time-correlated systematic variations in the light curves (called red noise; noise that is uncorrelated in time will be referred to as white noise). We can estimate the degree of red noise in our light curves by fitting a noise model to the median rms–r relation. We use a model of the form:

$$\begin{equation} {\rm rms} = \sqrt{\left(\frac{2.5}{\ln (10.0)} \right)^{2}\frac{f + s}{f^{2}} + \sigma _{r}^{2}}, \end{equation} \tag{ 2 }$$

where f = 10^{−0.4(m−z)} is the effective number of photoelectrons per image for a source of magnitude m with zero point z, $s = 10^{-0.4(m_{s} - z)}$ is the effective number of photoelectrons per image due to the sky that contaminate a given source and corresponds to a magnitude m_s, and σ_r is the effective red noise in magnitudes. Note that z depends on the aperture and efficiencies of the optics and detector, the exposure time, the atmospheric extinction, and the photometric procedure. Also note that m_s is the magnitude of a source, which has a flux equal to that of the total sky background through the effective photometric aperture. We bin the light curves on timescales of 5, 10, 30, 60, 120, 180, 240, 1440, 2880, and 7200 minutes and calculate the median rms–r relation for each binning. We then fit Equation (2) simultaneously to all 10 relations using the free parameters z_T, m_s and σ_r,T where z_T and σ_r,T depend on the timescale, while m_s is independent of the timescale. Table 1 gives the parameters from the model fit to the median rms–r relations. We also list in Table 1 the r-magnitude of a source for which the red noise on a given timescale is equal to the Poisson noise, that is,

$$\begin{equation} \frac{2.5}{\ln (10)}\frac{\sqrt{f_{T} + s_{T}}}{f_{T}} = \sigma _{r,T}. \end{equation} \tag{ 3 }$$

As expected, σ_r,T decreases with increasing T. At the timescales relavent for a transit (1–3 hr), the red noise is ∼0.9 mmag. We also note that the values of z_T are consistent with the expected values given the gain of the detector and the measured zero point for the reference image, assuming that the effective exposure time for the "average" image in the light curve is ∼1.0 minute.

Table 1. Parameters from Fitting the Noise Model in Equation (2) to the Median rms–r Relations after Binning the Light Curves on Several Timescales

Binning Timescale (minutes)	z (mag)	σr (mmag)	$m_{\sigma _{r}}$ (mag)a
5.0	32.45	1.35	17.57
10.0	32.81	1.21	17.68
30.0	33.46	1.04	17.90
60.0	33.95	1.00	18.21
120.0	34.44	0.92	18.43
180.0	34.64	0.90	18.53
240.0	34.63	0.93	18.58
1440.0	35.28	0.85	18.88
2880.0	37.38	0.47	19.37
7200.0	37.91	0.33	19.23
ms = 18.63 mag

Note. ^aThe r-magnitude of a source that has equal red noise and white noise when the light curve is binned at the specified timescale.

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An alternative method to determine the degree of red noise in the data is to examine the autocorrelation function of the light curves. We find that stars fainter than r ∼ 18.0 are effectively uncorrelated in time while brighter stars appear to be uncorrelated on timescales longer than ∼200 minutes.

To identify the best-fit transit signal in each light curve, we use the BLS algorithm. We apply two slight modifications to the algorithm.

• 1.  
  Rather than searching over a fixed range of fractional transit width q = τ/P values (where τ is the duration of the transit and P is the orbital period), we allow the q range to vary with the trial frequency. We do this by assuming a stellar radius range of R_min to R_max and taking q_min,max = 0.076(R_min,maxf)^2/3, where f is the trial frequency in days⁻¹, and assuming that M/M_☉ ≈ R/R_☉ for lower main-sequence stars.
• 2.  
  While the postprocessing described above substantially reduces the systematic variations in the light curves, some variations remain (see Figure 3). These variations give rise to increased power at low frequencies in the BLS periodogram of each light curve. As a result, long-period transits should be treated with greater caution than short-period transits. To account for this, we subtract a mean-filtered periodogram from the raw periodogram for each light curve before selecting the peak frequency. To mean filter the periodogram we replace each point in the periodogram by the mean value of the 200 points closest in frequency to the given point after applying an iterative 3σ clipping. Testing this modification on light curves with injected transits shows a very slight improvement (less than 1%) in the fraction of light curves for which the recovered period agrees with the injected period.

We conduct both a high-resolution and a low-resolution BLS search on the light curves. The low-resolution search is needed for the transit recovery simulations (Section 5.1.1) to finish in a reasonable amount of time. For the high-resolution search we examine 20, 000 frequency values over a period range from 0.2 to 5.0 days using 200 phase bins at each trial frequency, using R_min = 0.4 R_☉ and R_max = 1.3 R_☉. For the low resolution search we examine 3072 frequency values over the same period range, using the same number of phase bins. In searching for candidates with the low-resolution search, we use parameters identical to those used in conducting the transit recovery simulations (Section 5.1.1). We use slightly different parameters for possible cluster members and for field stars (see the selections of these stars in Sections 5.1.2 and 5.2). For candidate cluster members, we take R_min = 0.1 R_☉ and R_max = 1.1 × R_phot, where R_phot is the radius of the star estimated from its photometry assuming that it is a cluster member. For the field stars, we take R_min = 0.1 R_☉ and R_max = 1.3 R_☉.

As discussed by B06, a simple selection on the Signal Residue (Equation (5) in Kovács et al. 2002) or the Signal Detection Efficiency (Equation (6) in Kovács et al. 2002) generally yields a large number of false positive detections that must then be removed by eye. However, because it is very difficult to accurately include by eye selections in a Monte Carlo simulation of the transit selection process, it is necessary to devise automatic selection criteria that minimize the number of false positive detections to calculate a robust limit on the frequency of stars with planets. We discuss the three distinct selection criteria that we have devised in turn.

3.2.1. Selection Criteria Set 1

To devise the first set of selection criteria we inject fake transit signals into the light curves of 1366 stars that lie near the cluster main sequence on a color–magnitude diagram (CMD) and choose criteria that maximize the selection of high signal-to-noise-simulated transits, while minimizing the selection of false positives. The simulated transits have radii of 0.35, 0.71 and 1.0 R_J, and inclinations of 90°. For each star/radius, we simulate 10 transits with periods ranging from 0.5 to 5.0 days and random phases. We use the relations between r-magnitude and mass, and r-magnitude and radius for the cluster that were determined in Paper I to estimate the mass and radius of each star. The light curves including injected transits are passed through the pipeline described in the previous subsection before running the BLS algorithm on them.

To calibrate the selection criteria, we first must define the set of injected transits that we consider to be recoverable; we then adopt selection criteria that maximize the selection of these light curves while minimizing the selection of all other light curves. Note that we use information from the injected transits to define the recoverable sample, while we cannot use any of this information when defining the selection criteria. We consider the transit to be recoverable if Δχ²_BLS − Δχ²_BLS,0 < −100, where Δχ²_BLS is the reduction in χ² for the best-fit BLS model to the light curve with the injected transit and Δχ²_BLS,0 is for the light curve prior to injecting the transit. When a light curve satisfies this criterion, the best-fit BLS model is strongly influenced by the transit signal. This can happen even if the identified period does not match the injected period, or a harmonic of the injected period, however, in these cases, detailed follow up may eventually yield the correct period. Note that for transit selection criteria sets 2 and 3, we adopt the more conservative recoverability criterion that the recovered period must match the injected period, or one of its harmonics, to within 10%.

We find that the following selection criteria accurately distinguish between the recoverable and nonrecoverable transits.

• 1.  
  Following B06, we use BLS to identify both the best-fit transit signal and the best-fit inverse transit signal for each light curve. For the best-fit transit signal, we calculate the signal-to-pink-noise (SN) ratio (Pont et al. 2006) via

  $$\begin{equation} SN^{2} = \frac{\delta ^{2}}{\sigma _{w}^{2}\big/n_{t} + \sigma _{r}^{2}\big/N_{t}}, \end{equation} \tag{ 4 }$$

  where δ is the depth of the transit, n_t is the number of points in the transit, N_t is the number of distinct transits sampled, σ_w is the white noise, and σ_r is the red noise at the timescale of the transit. To calculate the white noise for a light curve, we subtract the best-fit BLS model from the light curve and set σ_w equal to the standard deviation of the residual. To calculate the red noise, we bin the residual light curve in time with a bin size equal to the duration of the transit and set σ_r equal to its standard deviation. Note that while this technique provides a convenient method to determine an individual red and white noise estimate for each light curve, it overestimates the noise for an uncorrelated signal by a factor of ${\sim} \sqrt{2}$. We select candidates that have SN>10.0. Figure 4 shows this selection.
• 2.  
  As shown in Figure 4, some of the transit signals that are not expected to be detectable pass the selection from the previous step. We can further reduce these potential false alarms with negligible loss of detectable transits by the following selection. Let Δχ²_BLS refer to the reduction in χ² for the best-fit transit signal and Δχ²_BLS,2 refer to the reduction in χ² for the second noninverse transit peak in the BLS spectrum. The light curves passing selection 1 are then selected if they have P>2.0 days and Δχ²_BLS − Δχ²_BLS,2 < −60 or P ⩽ 2.0 days and Δχ²_BLS − Δχ²_BLS,2> − 10.95P^2.45, as shown in Figure 5.

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When the above selections are applied to the actual data, more than 100 light curves are selected. Many of these are obvious false positives that can be eliminated by applying a few additional selections.

• 3.  
  We reject stars with fewer than 1000 points in their light curves; most of these are near saturation and show strong systematic variations even after applying the TFA.
• 4.  
  Many of the false positives are faint stars located near much brighter stars; these can be rejected by only considering stars with an average instrumental magnitude brighter than 17.0 (corresponding roughly to r < 22.0).
• 5.  
  Other false positives come from very noisy light curves that can be rejected by requiring the standard deviation of the residual light curve after subtracting the best-fit box-car transit model to be less than 0.1 mag.
• 6.  
  Many false positives show an anomalous faint set of points that occur on only one night. Following B06, we require that the fraction of Δχ²_BLS that comes from one night must be less than 0.8.
• 7.  
  We reject light curves for which the best-fit box-car transit has a period between 0.99 and 1.02 days or less than 0.4 days.
• 8.  
  Finally, we require that χ² per degree of freedom (dof) of the light-curve residual after subtracting the best-fit box-car transit model must be less than 2.5 times the expected χ² per dof. The expected χ² per dof is taken to be the median value as a function of magnitude for the full ensemble of light curves (Figure 6). Note that the values of χ² per dof for stars between 18 ≲ r ≲ 23 approach ∼0.5 rather than ∼1.0. This is due to a bug in our differential photometry routine whereby the formal differential flux uncertainties returned are not set to the flux scale of the reference image. We note that a similar bug is present in the differential photometry routine of the ISIS 2.1 package on which our photometry routine is based. The flux scale for the reference image that we use is set to that of a 30 s exposure taken under good seeing conditions. The formal errors are overestimated for longer exposures that were taken under poor seeing conditions. We identified this bug after completing most of the analysis presented in this paper; we do not, however, expect it to significantly affect the statistical results presented here. Note that since the photometry that we are using is dominated by red noise, rather than white noise, the formal photometric errors, which assume Gaussian, uncorrelated noise, do not accurately describe the uncertainties in the data. We have therefore used a cut on the SN, which is determined empirically from the light curve, rather than relying on the formal uncertainties. The formal uncertainties do affect the relative weightings of points; modifying them may thus yield slight changes in the TFA fitting procedure as well as in the periods that are identified by BLS. If the formally correct weighting scheme had been adopted, the sample of selected candidates may have been slightly different; however, this is no different from making minor changes to the rather arbitrary selection criteria. For our statistical conclusions regarding the fraction of stars bearing planets, what is important is that we apply the same selections and weighting scheme to the transit injection simulations as we do to the actual data and that the postselection vetting procedure that we apply to the actual data would not reject any of the injected candidates. Making a uniform change to the weighting scheme will affect the selection of real candidates and injected transits in the same manner, so the transit detection efficiency measured by the injection simulations will accurately reflect the detection efficiency for the survey.

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3.2.2. Selection Criteria Set 2

The above selection criteria were developed by manually inspecting the results of transit recovery simulations and defining cuts that appeared, by eye, to distinguish between successful recoveries and nonrecoveries. While this method has the advantage that the criteria can be enumerated and easily visualized, it has the disadvantage that the cuts are fairly subjective and there is no guarantee that the selections optimally distinguish between recoveries and nonrecoveries.

To complement the above selection criteria, we have devised a set of selections using the Support Vector Machine (SVM) classification algorithm, which has the advantage of being much less subjective, but the disadvantage of being difficult to visualize (Vapnik 1995; a discussion of the algorithm can also be found in Press et al. 2007, which we summarize here). This algorithm takes as input a set of training data, which consists of m points, (x_i, y_i), where x_i is an n-dimensional vector of measurable parameters describing object i (called features) and y_i is either +1 or −1 and is used to divide the data into two classes. The algorithm then searches for a real-valued function f(x) such that f(x_i)>0 for y_i>0 and f(x_i) < 0 for y_i < 0. The function is assumed to take the form

$$\begin{equation} f(\mathbf {x}) = \mathbf {W} \cdot \mathbf {\phi }(\mathbf {x}) + B, \end{equation} \tag{ 5 }$$

where ϕ(x) is a fixed n-to-N-dimensional transformation with N being typically larger than n, and W is an N-dimensional vector. To optimize f, one looks for a vector W that is normal to a hyperplane that separates ϕ(x_i) with y_i = +1 from ϕ(x_i) with y_i = −1 and that has a maximum perpendicular distance from the points nearest to it (called support vectors). In general, it is not always possible to find such a hyperplane, so instead one seeks to optimize f by minimizing

$$\begin{equation} \frac{1}{2} \mathbf {W}\;{\bf \cdot}\;\mathbf {W} + \lambda \sum _{i} \Xi _{i} \end{equation} \tag{ 6 }$$

subject to the constraint

$$\begin{equation} \Xi _{i} \ge 0,\quad y_{i}f(\mathbf {x}_{i}) \ge 1 - \Xi _{i},\quad\ i=1,\ldots,m, \end{equation} \tag{ 7 }$$

where Ξ_i are free parameters that allow for discrepancies between the model and actual classes and λ is a fixed regularization parameter used to control the tradeoff between accurately classifying the training data and maximizing the perpendicular distance between the support vectors and the hyperplane. In practice, the problem is recast in the Lagrangian formalism so that one specifies a kernel matrix with the property K_ij = K(x_i, x_j) = ϕ(x_i) · ϕ(x_j) rather than the transformation ϕ(x).

To apply the SVM algorithm to our problem of devising transit selection criteria we use the SVMlight package⁸ (Joachims 1999, 2002). Before using the algorithm, we apply a set of simple cuts which we found to be necessary to minimize the number of false positives.

• 1.  
  Reject stars with an average instrumental magnitude fainter than 17.0 (corresponding roughly to r>22.0).
• 2.  
  Reject stars that have rms >0.1 mag, where the rms here is the standard deviation of the residual light curve after subtracting the best-fit box-car transit model.
• 3.  
  Reject stars with SN < 10.0, where SN is the SN given by Equation (4). Note for training the algorithm, we use a less restrictive cut of SN < 9.0.
• 4.  
  Reject light curves for which the best-fit box-car transit has a period between 0.99 and 1.02 days or less than 0.4 days.

We train the algorithm on the simulated transit data described in Section 5.1.1. We use only ∼5000 of the ∼250 million simulated transits to allow the algorithm to converge in a reasonable amount of time. We take y_i = +1 for simulated transits that have 0.95 <  P_recover/P_inject <  1.05, where P_recover and P_inject are the recovered and injected transit periods, respectively, and y_i = −1 for all other simulated transits. There are 15 features in the xx_i vectors including: the recovered period, the fractional transit width, the transit depth, SN, the white noise, the red noise, Δχ²_BLS, Δχ²_BLS/Δχ²_BLS,inv, (Δχ²_BLS,2 − Δχ²_BLS)/Δχ²_BLS, the fraction of Δχ²_BLS that comes from one night, the number of points in transit, the number of points observed less than τ minutes prior to transit ingress (τ is the duration of the transit), the number of points observed less than τ minutes after transit egress, the number of distinct transits observed, and the ratio of χ² per dof of the light-curve residual after subtracting the best-fit box-car transit model to the expected χ² per dof. When applying the algorithm, we consider a transit to be recovered if the estimated value of y is greater than 0.1 as we found a number of false positives for which the algorithm returned values between 0.0 and 0.1.

3.2.3. Selection Criteria Set 3

30137. This source is an eclipsing binary as evidenced by the secondary eclipse having an unequal depth to the primary eclipse when phased at twice the period shown in Figure 7. This source is V1187 in the catalog of variables presented in Paper II. We reject this candidate.

60161. The transitlike variation in this light curve is completely removed by the TFA so this source only passes selection criteria set 3. The transit feature is due primarily to two nights. The images on these nights show a gradient in the background counts near the corner of Chip 6 where this star is located. The sense of the gradient is that points are fainter in the corner than near the center of the image. The use of a single master flat-field image for the entire run appears to have failed for these two nights; the result is that many of the stars in the corner of Chip 6 show dips in their raw light curves on the two nights in question. This transit feature therefore appears to be spurious, and we reject it as such. We note that this is the only candidate that was selected by the high-resolution search but not by the low-resolution search.

70127. This source is a promising transit candidate around a field star. Crosscorrelation analysis of the Hectochelle spectrum for the source yields T_eff ∼  5000 K though the uncertainty is very high due to the faintness of the source. Note that there is a nearby source to 70127 that may also contaminate this spectrum. Combining the B, V, and I_C photometry for the source, we estimate that (B − V)₀ ∼  1.0, and E(B − V) ∼  0.16 so that the star has a radius of R_⋆ ∼  0.75R_☉ and a mass of M_⋆ ∼  0.8 M_☉. The light curve has Tingley & Sackett (2005) parameters of η_p = 0.7 and η_⋆ = 0.7, so it appears to be consistent with the transiting planet hypothesis. The source is located less than 2'' from the edge of the chip; this appears to add some scatter to the light curve that is correlated with the seeing and removed by the TFA. Also note that there is a 1.625 mag fainter source, 70181, that is located only 08 from 70127. The fainter source also shows a dip in its light curve; however, the dip in flux appears to be greater for 70127 than 70181, and the centroid of 70181 on the residual image appears to shift slightly in phase with the transit, whereas 70127 does not. Both of these factors suggest that 70127 is the real variable. The "shoulder" just before transit is an artifact of postprocessing the fairly high signal-to-noise light curve and does not appear in the unprocessed light curve. Fitting a Mandel & Agol (2002) transit model to the light curve with quadratic limb darkening coefficients fixed to u₁ = 0.54 and u₂ = 0.2 from Claret (2004), which are appropriate for a ∼5000 K dwarf star in the r filter, and fixing a/R_⋆ = 4.38 assuming the stellar mass, radius, and the orbital period of 0.77353 days, we find R_P ∼  1.0R_J and sin i ∼  0.99. The expected RV amplitude for the star would be K ∼  220 m s^-1(M_P/M_J), where M_P is the mass of the planet. To rule out astrophysical false positives, such as an M-dwarf secondary, would require additional follow-up spectroscopy which would be challenging given the faintness of the source (r ∼ 19.6, V = 19.85). Note that Sahu et al. (2006) achieved a formal RV precision of ∼200 m s^-1 for their candidate SWEEPS-11, which has a comparable magnitude (V = 19.83), so conducting spectroscopic follow up for 70127 is not beyond the realm of possibility.

80009. This source appears to be an eclipsing binary, though there is not enough data to determine the period. There is a significant amount of scatter in the light curve that results from the source being nearly saturated. From the B, V, and I_C photometry, we estimate that the reddening to the source is E(B − V) ∼  0.18 so that the source would have (B − V)₀ ∼  0.65 and M ∼  1.0 M_☉. The primary eclipse depth appears to be deeper than the value returned by BLS (closer to ∼0.1 mag); the secondary source would have to be significantly larger than 2.0R_J to create such a deep eclipse for this star. We reject this candidate.

80014. This shallow transit-like signal is the result of a blend with the r ∼  18.3 mag eclipsing binary, V29, that is located ∼3'' away. We reject this candidate.

90279. This is an eclipsing binary with a period of 2.283 days. Phasing the light curve at twice the period shown in Figure 7 reveals that the secondary and primary eclipses are slightly unequal in depth. It is V1182 in the catalog of variables presented in Paper II. We reject this candidate.

110021. This source is a candidate cluster member with an estimated mass of ∼1.2 M_☉. The secondary object must be significantly larger than 2.0R_J to yield an eclipse deeper than 0.1 mag; the source is thus an eclipsing binary. The source is V827 in the catalog of variables presented in Paper II. The period displayed in Figure 7 and listed in Table 2 is the period returned by BLS. When phased at a period of 6.7667 days the presence of a secondary eclipse becomes apparent; it is also apparent that the system has significant eccentricity as well as an out-of-eclipse variation that phases with the proper orbital period. We reject this candidate.

120050. This is an eclipsing binary; note the dual eclipse depths at phase 0.0. When phased at twice the period shown in Figure 7, it is clear that the light curve shows primary and secondary eclipses of differing depths. This source is V140 in the catalog of variables presented in Paper II. We reject this candidate.

160017. This source is an eclipsing binary. Its position on the CMD makes it a candidate cluster member with a mass of ∼1.2 M_☉. For such a primary, the deep eclipse could not be caused by a planetary-sized companion. There is an out-of-eclipse variation that phases at 4.0989 days; however, the eclipses do not phase at this period. There is not enough data to accurately determine the orbital period. In Figure 7, we show the raw light curve phased at the period returned by BLS for the light curve processed through the TFA. The parameters listed in Table 2 for this system are for the raw light curve, the S/N value is well below 10.0 in this case. The σ-clipping routine clips much of the eclipse yielding a shallow transit that phases roughly at the period shown and yields an S/N value above 10.0. This source is V791 in the catalog of variables presented in Paper II. We reject this candidate.

160311. This is an eclipsing binary; there is a shallow secondary eclipse with a depth of ∼0.05 mag visible at phase 0.5. This source is V716 in the catalog of variables presented in Paper II. We reject this candidate.

170049. The transit feature seen in the noisy unprocessed light curve is completely eliminated by TFA. The source lies within 2'' of the edge of the frame and the variations in the light curve are strongly correlated with the image seeing. The transit feature is removed when the light curve is decorrelated against seeing. We reject this candidate as the variations appear to be spurious.

170100. This source is an eclipsing binary. When phased at twice the period displayed in Figure 7, it is clear that the primary and secondary eclipses are of unequal depth. The out-of-eclipse variations also phase at this period. This source is V1028 in the catalog of variables presented in Paper II. We reject this candidate.

230082. This source, V485, is most likely an F–M eclipsing binary. Note that when phased at the period recovered by BLS (as shown in Figure 7), there appears to be a secondary eclipse at phase −0.5, when phased at a period of 1.6676 days; however, the putative secondary matches to the primary eclipse. Also note that the clipping procedure removes some points from the bottom of the eclipse; the eclipse in the raw light curve is ∼0.01 mag deeper. From the B, V, and I_C photometry, we estimate E(B − V) ∼  0.38, so (B − V)₀ ∼  0.41, which corresponds to a ∼1.5 M_☉, and ∼1.4R_☉, primary star. For a star of this size, a 0.034 mag transit requires that the companion have a radius greater than 2R_J.

270149. This source, V457, is an eclipsing binary. Note the deep, and distinct, primary and secondary eclipses. We plot the raw light curve for this star in Figure 7; the clipping routine removed many of the points in eclipse, which caused BLS to identify the wrong period and underestimate the depth.

280287. This is a grazing eclipsing binary. When the raw light curve is phased at twice the period found by BLS, it is clear that the primary and secondary minima are of unequal depths. This source is V226 in the catalog of variables presented in Paper II. We reject this candidate.

330224. This source, V141 in the catalog of variables, is an eclipsing binary given its depth. The raw light curve reveals a strong out-of-eclipse variation with a peak-to-peak amplitude of ∼0.1 mag that phases at the orbital period.

In Figure 9, we show the selection of stars near the cluster main sequence on g−r and g−i CMDs. We select a total of 2475 stars that have an instrumental magnitude of less than 17, rms <0.1 mag, and that have at least 1000 points in their light curves. Note that we expect ∼1450 of these stars to be cluster members. We use these stars to place a limit on the planet occurrence frequency for the cluster. The total number of planets in the cluster that we expect to detect with our survey is given by

$$\begin{equation} N_{\rm det} = f \sum _{i=1}^{N}P_{{\rm det},i}, \end{equation} \tag{ 8 }$$

where f is the planet occurrence frequency over a specified planet radius and orbital period range, P_det,i is the probability of having detected such a planet for star i, and there are N candidate cluster members in the survey. For a binomial distribution, the detection of no planets is inconsistent at the ∼95% level when N_det ≳ 3. The 95% confidence upper limit on the planet occurrence frequency is then

$$\begin{equation} f \le 3\bigg/\bigg(\sum _{i=1}^{N}P_{{\rm det},i} \bigg). \end{equation} \tag{ 9 }$$

The planet detection probability for star i is given by

$$\begin{equation} P_{{\rm det},i} = \int\!\!\int \frac{d^{2}p}{dR_{p}dP}P_{\epsilon,i}(P,R_{p})P_{T,i}(P,R_{p})P_{{\rm mem},i}dR_{p}dP, \end{equation} \tag{ 10 }$$

where P_,i is the probability of detecting a transit of a planet with the orbital period P and radius R_p for star i if it has an orbital inclination that yields transits, P_T,i is the probability that the planet has an inclination that yields transits, P_mem,i is the probability that star i is a cluster member, and d²p/dR_pdP is the joint probability distribution of R_p and P.

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There are four terms that contribute to Equation (10) that must be determined separately. Assuming random orientations, the term P_T,i is given analytically by P_T = (R_⋆ + R_p)/a, where R_⋆ is the radius of the star and a is the semimajor axis for an orbital period P and stellar mass M_⋆. For the term P_mem,i we take the photometric membership probability of the star, which we calculate by comparing the CMD of the cluster to a CMD of a field off the cluster. The term d²p/dR_pdP must be treated as a prior, and the term P_,i is calculated via Monte Carlo simulation as discussed below.

5.1.1. Calculating P_,i

We follow the procedure described by B06 to calculate P_,i. This involves injecting limb-darkened transits into the light curves of potential cluster members and attempting to recover them. Transits are injected into the raw light curves; we then run each simulated light curve through the postprocessing and transit selection routines described in Section 3. Note that we only conduct simulations on light curves that are not selected as transit candidates. Since there are only a few candidates, not including their efficiencies should not change our results significantly.

To simulate transit light curves, we use the Mandel & Agol (2002) analytic model with r-band quadratic limb-darkening coefficients from Claret (2004). Note that we assume circular orbits for the short-period planets under consideration. We estimate the mass, radius, and surface temperature of each star using the best-fit YREC isochrone (An et al. 2007; also see Paper I), we then estimate the limb-darkening coefficients for each star via linear interpolation within the Claret (2004) grid, assuming [M/H] = 0.045 (Paper I) and v_turb = 2 km s^-1.

To determine the dependence of P_,i on the orbital period, we inject the transits in period bins ranging from 0.4 to 5.0 days with a logarithmic step size of 0.022. For each period bin, we inject N_trial transits with periods distributed uniformly in logarithm over the bin, random phases and inclinations distributed uniformly in cos i over the range 0 ⩽ cos i ⩽ (R_⋆ + R_p)/a. Following B06, we estimate that the error in the recovery fraction f = N_recover/N_trial is given by

$$\begin{equation} \sigma _{f} = \sqrt{f(1 - f)/N_{\rm trial}}. \end{equation} \tag{ 11 }$$

For each period bin, we initially adopt N_trial = 20 and continue simulating transits until σ_f ⩽ 0.05 using selection criteria set 1. We conduct simulations for planetary radii of 0.3, 0.35, 0.4, 0.45, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.5, and 2.0R_J.

5.1.2. Calculating P_mem,i

To calculate the membership probability for each star, we use the luminosity functions of the cluster and Galactic field along a strip in the g−r and g−i CMDs enclosing the cluster main sequence (see Figure 9; the determination of the luminosity functions is described in Paper I). This gives the membership probability as a function of r-magnitude only. In Figure 10, we show the membership probability as a function of r. Since this method ignores color information in assigning a probability to stars, it tends to give too high a probability to stars lying away from the cluster main sequence and too low a probability to stars lying close to the main sequence. If the transit detection probability depends only on the r-magnitude of the star, then this simplification should not bias the result. However, as shown in Paper II, cluster members have a higher probability of exhibiting photometric variability than field stars of the same magnitude, so we caution that it is possible that the transit detection probability may be lower on average for cluster members than for field stars. We consider this possibility in Section 6.3.

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To calculate the detection efficiency for field stars, we follow a procedure that is similar to what we use for cluster members. In this case, however, the mass and radius of a star cannot be determined simply from the magnitude of the star. Instead, we use Galactic models to estimate the mass and radius for each star.

First, we select a sample of observed field stars that includes all stars with g, r, i, B, and V photometry that have more than 1000 points in their light curves, have a light-curve rms that is less than 0.1 mag, an instrumental r-band magnitude less than 17 (corresponding roughly to r ≲ 22), and were not selected as candidate cluster members. The sample of stars is shown on B−V and V − I_C CMDs in Figure 11. A total of 7824 stars are selected in this fashion. We then obtain simulated photometric observations of a 24' × 24' field centered at the Galactic latitude/longitude of the cluster using the Besançon model (Robin et al. 2003). We assume an interstellar extinction of 0.7 mag kpc⁻¹. We caution that this model is known to be unreliable along certain lines of sight (LOSs), in particular for low Galactic latitudes (e.g., see Ibata et al. 2007). For each observed star in our sample, we choose the simulated star that minimizes

$$\begin{equation} (V_{o} - V_{s})^{2} + (B_{o} - B_{s})^{2} + (I_{C,o} - I_{C,s})^{2}, \end{equation} \tag{ 12 }$$

where the o subscripts denote photometric measurements for observed stars and the s subscripts denote photometric measurements for simulated stars. We then assign the mass and radius of the simulated star to the observed star. We reject 10 stars that match to white dwarfs since all these lie in an isolated region of the CMD. The majority of the stars in the sample (97%) match to dwarf stars (log g>4.0, cgs) rather than subgiants or giants. Figure 12 shows the estimated radii of the field stars as a function of magnitude compared with the values for the cluster. Note that at fixed magnitude, the majority of field stars have larger radii than the cluster stars. As a result, we expect the transit probability to be larger, but the overall S/N to be smaller at fixed magnitude for the field stars. Thus for signals for which the S/N is much larger than the threshold, the detection efficiency is larger for field stars, but for planets near the S/N threshold, the detection efficiency will be smaller. Thus, the minimum detectable planet radius will be smaller for cluster stars, all else being equal. For the less common foreground field dwarfs, the opposite is true.

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We determine the detection efficiency by conducting transit injection and recovery simulations as described in Section 5.1.1. We conduct the simulations for 1000 randomly selected field stars. For each star, we calculate P_det,i using Equation (10) where we now take P_mem,i = 1. The 95% upper limit on the occurrence frequency, assuming no detections, can then be calculated using Equation (9) with the sum in the dominator being multiplied by 7814/1000 to scale from the simulated subset to the full sample.

For cluster members, we place a 95% confidence upper limit on the frequency of stars with Jupiter-sized EHJ, VHJ, and HJ planets of 1.1%, 2.7%, and 8.3%, respectively. Note that all of these limits, as well as those we discuss below, come from selection criteria set 2. For smaller planets, the limits rise. For the EHJ period range, we can place a limit of 25% on planets down to 0.35R_J, which is roughly the size of Neptune. For the VHJ period range, we can place a limit of 44% on planets down to 0.45R_J, while for the HJ period range, we can place a limit of 49% on planets down to 0.6R_J.

The sample of field stars provides more stringent upper limits on the planet occurrence frequency than the sample of cluster members. For this sample we can place a 95% confidence upper limit on the frequency of stars with Jupiter-sized EHJ, VHJ, and HJ planets of 0.3%, 0.8%, and 2.7% respectively. This assumes that the candidate 70127 is not a real planet; if 70127 is real, the frequency of ∼1.0R_J planets in the EHJ period range would be 0.002% <  f <  0.5% with 95% confidence. In principle, this frequency is small enough that planets in this period range could have escaped detection in most other RV and transit surveys. Some transit surveys have probed enough stars to be sensitive to planets with these frequencies, as some of them have looked at as many or more stars than we have, but these surveys have generally not been very sensitive to hosts with masses as small as this host (see Gould et al. 2006; Beatty & Gaudi 2008). For the EHJ period range, we can place a limit of 16% on planets down to 0.3R_J; for the VHJ range, a limit of 8.8% on planets down to 0.5R_J; and for the HJ range, a limit of 47% on planets down to 0.5R_J. Extrapolating the curves in Figure 17, we note that for the EHJ period range, we do have some sensitivity even to planets as small as ∼2.5R_⊕.

There are a number of factors that may contribute errors to the upper limits. This includes uncertainties in P from using a finite number of simulations, errors in P_mem from using a finite sample, and systematic errors from blends and binaries, from errors in the assumed Galactic model, and from the possibility that cluster members are more likely to be variable than noncluster members. B06 gave a detailed discussion of how most of these errors can be estimated.

The fractional uncertainties in the 95% upper limits for cluster members ($\sigma _{f_{<95\%}}/f_{<95\%}$) due to using a finite number of simulations to determine P range from ∼1% for 0.3R_J simulations to ∼0.04% for 1.0R_J EHJ simulations. For field stars, the fractional uncertainties are all less than 1%. These errors are negligible compared to other sources of uncertainty.

Following B06, we note that the fractional uncertainty in the 95% upper limit for candidate cluster members due to uncertainties in P_mem is equal to the fractional uncertainty in the effective number of cluster members: $\sigma _{N_{\star,{\rm eff}}}/N_{\star,{\rm eff}}$, where N_⋆,eff = N_⋆〈P_mem〉. We find $\sigma _{N_{\star,{\rm eff}}}/N_{\star,{\rm eff}} = 2\%$.

We expect binarity to be a more significant effect than chance alignments. As we noted in Paper II, we expect ∼1 chance alignment within 01 of two point sources brighter than our detection threshold in our entire field. The effect of binarity on a transit survey is not straightforward. While the blending of light from two stars will dilute the transit signal, the primary star will be slightly smaller than what is assumed when injecting transits. Moreover, if the precision of the light curve is good enough that one could still detect the transit for a given planetary radius and primary star radius if the transit were more than a factor of 2 shallower, then such a planet would be detectable orbiting the primary star for any mass ratio and it may also be detectable orbiting the secondary star. In this case the total number of stars to which one is sensitive to planets is greater than estimated when binarity is neglected (see Gould et al. 2006; Beatty & Gaudi 2008, for further discussion about the effects of binarity). Binarity will only affect the detectability of a transit if the mass ratio is high; B06 estimated that the requirement is q ≳ 0.6, which they argued is true for only ∼11% of dwarf stars when the binary fraction ∼50%. There is an indication that the binary fraction along the main sequence in M37 is closer to 20% (Kalirai & Tosi 2004), which would reduce the number of high-mass ratio binaries to ∼4% assuming that the mass ratio distribution in the cluster is the same as for the Galactic field.

To estimate how uncertainties in the inferred masses and radii of field stars that results from uncertainties in the Galactic model affect the planetary frequency upper limits, we recompute the upper limits using the Trilegal 1.2 Galactic model (Girardi et al. 2005). We match the sample of field stars to a simulated set of photometric observations generated with the Trilegal model following the same procedure that we used in Section 5.2 to match to the Besançon model. The Trilegal model is shown in Figures 11 and 12 and is generated assuming a local extinction law of 0.7 mag kpc⁻¹. With this model, we find that 96.5% of stars in our field sample have log g>4.0, which is comparable to the dwarf fraction found with the Besançon model. In Section 5.2, we computed the transit detection probability for a sample of 1000 field stars by conducting transit injection/recovery simulations with stellar masses and radii adopted from the match to the Besançon model. Here, we avoid the expensive task of conducting additional transit injection/recovery simulations by matching stars in the Trilegal-matched sample to appropriate stars in the Besançon-matched sample. We match each star, i, in the Trilegal-matched sample to the star, j, in the Besançon-matched sample that minimizes

$$\begin{equation} (0.2 \ln (10) (r_{i} - r_{j}))^{2} + (2(R_{i} - R_{j})/R_{i})^{2}, \end{equation} \tag{ 14 }$$

where r is the r-magnitude of the star and R is its radius. We then set the transit detection probability for star i equal to the probability for star j determined from the simulations. This choice of weighting between magnitude and radius minimizes differences between the expected transit S/N of the two stars. The upper limits on the planet occurrence frequency are then computed with Equation (9). The results using selection criteria 2 are listed in Table 4. We find that the fractional difference in the planet frequency upper limits from the two Galactic models is ≲10%. The Trilegal model yields a systematically lower upper limit for the 0.3R_J and 0.5R_J planets but a systematically higher upper limit for larger planets. In a similar manner, we recalculate the upper limits using the Besançon model generated with less extinction (0.5 mag kpc⁻¹). In this case we find that the upper limits for planets smaller than 1.0R_J are systematically smaller while the upper limits for larger planets are systematically larger. The fractional differences in the upper limits for 0.7R_J and smaller planets range from ∼5% to ∼30%, while for larger planets they range from ∼2% to ∼20%. We adopt a fractional uncertainty of ∼10% on the upper limits due to uncertainties in the Galactic model.

To estimate the effect of variability on the upper limits for cluster members, we compute the upper limits under the extreme case where all candidate cluster members that are variable have P_mem = 1 and other stars have P_mem = (N(r)P_mem,0(r) − N_var(r))/(N(r) − N_var(r)), where P_mem,0 is the value of P_mem when variable and nonvariable stars are weighted equally, N(r) is the number of candidate cluster stars in magnitude bin r, and N_var is the number of variable candidate cluster stars in magnitude bin r. The resulting 95% upper limits for selection criteria set 2 are given in Table 5. For small radius planets, the effect is significant, so that the upper limit on 0.35R_J EHJ planets, for instance, increases to 38% from 25%. For larger planets, the effect of variability is less important; above 1.0R_J, the fractional increase in the upper limit is less than 10%.

Assuming that binarity and variability only increase the upper limits, the fractional uncertainty on the upper limits for 1.0R_J planets orbiting candidate cluster members is ∼+11%, − 2%, and for 0.35R_J planets, it is ∼+50%, − 2%. For field stars, we can neglect the uncertainty due to variability and the uncertainty on the membership probability, but we must include the uncertainty on the Galactic model, so that the fractional uncertainty on the upper limit for all planetary radii is ∼+15%, − 10%.

Our results can be compared with both previous results for transit surveys of open clusters and results from transit and RV surveys of the Galactic field.

Our limits of 1.2%, 2.3%, and 6.9% for the cluster on the frequency of EHJ, VHJ, and HJ planets, respectively, with R = 1.5R_J are substantially better than the corresponding limits of 1.5%, 6.4%, and 52% for NGC 1245 by B06. The primary differences between the two surveys are that we observed approximately twice as many cluster members as B06 and we obtained significantly higher precision photometry at fixed stellar radius than B06. Note, however, that B06 used a period range of 3–9 days for the HJ range, whereas we use a range of 3–5 days. Since, by selection, we have zero sensitivity to planets with 5 days < P < 9 days, extending our HJ range to the B06 range results in a limit of ∼15% on the fraction of stars with planets of radius 1.5R_J (assuming an underlying distribution in P that is uniform in logarithm). Miller et al. (2008) have also conducted detailed Monte Carlo simulations of their open cluster transit survey and placed 95% upper limits of 14% and 45% on a VHJ and HJ, respectively, with radii of 1.5R_J.

We can estimate the frequency of planets detected by RV surveys in the EHJ, VHJ, and HJ period ranges using the results from Cumming et al. (2008, see Figure 5 in their paper). Out of their sample of 585 FGKM stars, there are seven planets with M>0.1M_J and 3 days < P < 5 days, one planet with M>0.1M_J and 1 day < P < 3 days, and no planets detected with P < 1 day. This yields 95% confidence intervals for the occurrence frequencies of 0.48% <  f_HJ <  2.5%, 0.0043% <  f_VHJ <  0.95%, and f_EHJ <  0.51%. Our results are not directly comparable with the RV survey results for two reasons: first, our limits are based on planetary radius while the RV survey limits are based on planetary mass; second, the distribution of stellar masses and metallicities is not the same for the two surveys. Nonetheless, if we assume that all extrasolar giant planets have radii of ∼1.0R_J and that the planet occurrence frequency does not depend on stellar properties, then our upper limits of f_EHJ <  0.3%, f_VHJ <  0.8%, and f_HJ <  2.7% for field stars are consistent with the measured RV frequencies.

Gould et al. (2006) used the results of the OGLE transit survey to determine the frequency of VHJ and HJ planets. Assuming that planets follow a uniform distribution in radius between 1.0R_J and 1.25R_J, they found f_VHJ = 0.14^+0.19_−.08% and f_HJ = 0.31^+0.35_−0.17% with 90% confidence error bars. Furthermore, they placed a 95% confidence upper limit of less than 1.7% on planets with periods between one and five days and radii distributed uniformly between 0.78 and 0.97R_J. Similarly, Fressin et al. (2007) found f_VHJ = 0.18% and f_HJ = 0.29% for the OGLE survey, and they found that the fraction of stars bearing planets with periods less than two days is 0.079^+0.066_−0.040% with 90% confidence error bars. Note that the frequency of an HJ is lower for the OGLE survey than for the RV surveys because the RV surveys tend to be biased toward higher metallicity stars whereas the OGLE survey is not. Our upper limits on the VHJ and HJ occurrence frequencies for field stars are consistent with the frequencies determined by both of these groups.

We can also compare our results with the results from the SWEEPS survey. Sahu et al. (2006) found that ∼0.4^+0.4_−0.2% of stars larger than 0.44 M_☉ host a Jupiter-sized planet with P <  4.2 days, assuming that all 16 of their candidates are real, which is consistent with the limits that we set. Focusing specifically on the EHJ period range, we note that Sahu et al. (2006) found five candidate planets in this range and estimated that at least ∼2 of them are likely to be real. Assuming that all 16 candidates are real planets, and that 5 of them are an EHJ, their results suggested that ∼0.13% of stars host an EHJ. If, on the other hand, we estimate that only half of the candidates are real, and that two of them are an EHJ, the frequency of an EHJ would be ∼0.05%. Note that we have assumed that the detection probability is constant over the entire period range. Accounting for the enhanced probability of detecting an EHJ over a VHJ and HJ may lower the frequency estimates of these planets by as much as a factor of ∼5. The SWEEPS frequency of an EHJ is below the 95% upper limit that we set assuming that 70127 is not a real candidate. If 70127 is real, our EHJ frequency of 0.09^+0.2_−0.077% (with 1σ errorbars) would be consistent with the SWEEPS results.

Based on the planet distribution determined by other surveys, we can estimate the expected yield of our survey. Cumming et al. (2008) found that the distribution of planet masses down to ∼0.1M_J is given by

$$\begin{equation} \frac{d^{2}N}{d\ln M d\ln P} = C M^{\alpha } P^{\beta }, \end{equation} \tag{ 15 }$$

where α = −0.31 ± 0.2 and β = 0.26 ± 0.1, M is in Jupiter masses, P is in days, and C is a constant. For notational simplicity, in the following discussion, we do not write factors of M_J and R_J. We will adopt the mass dependence from this relation, but not the period dependence, since this relation does not account for the pile-up of planets at short periods. For the period dependence we adopt a constant $C_{\bar{P}}$ that is appropriate for each period range $\bar{P}$ (i.e., EHJ, VHJ, or HJ). The model distribution is then given by

$$\begin{equation} \frac{dN_{\bar{P}}}{d\ln M} = C_{\bar{P}} M^{\alpha }, \end{equation} \tag{ 16 }$$

where

$$\begin{equation} C_{\bar{P}} = \frac{f_{\bar{P}}\alpha }{M_{\rm max}^{\alpha } - M_{\rm min}^{\alpha }}, \end{equation} \tag{ 17 }$$

and $f_{\bar{P}}$ is the fraction of stars with planets in period range $\bar{P}$ with masses between M_min = 0.1 and M_max = 10.

We assume a simple planetary mass–radius relation of the form

$$\begin{equation} R(M) = \left\lbrace\!\!\!\begin{array}{ll} R_{0}M^{\gamma }, & M {<} 1.0 \\ R_{0}, & M \ge 1.0 \end{array}, \right. \end{equation} \tag{ 18 }$$

with γ = 0.5, R₀ = 1.5, or γ = 0.38, R₀ = 1 (see Figure 19). We consider two different mass-radius relations to determine the sensitivity of our results to the assumed relation. The expected number of planet detections in the EHJ, VHJ, and HJ period ranges is then given by

$$\begin{eqnarray} && N_{\bar{P}} = \sum _{i} \bigg(\int _{R_{\rm min}}^{R_0} \gamma ^{-1} C_{\bar{P}} \left(\frac{R}{R_{0}} \right) ^{\alpha /\gamma } R^{-1} P_{{\rm det},\bar{P},i}(R) dR \nonumber \\ && \qquad + P_{{\rm det},\bar{P},i}(R_{0})C_{\bar{P}}\alpha ^{-1}\big(M_{\rm max}^{\alpha } - 1\big) \bigg), \end{eqnarray} \tag{ 19 }$$

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where the sum is over all stars in the sample, $P_{{\rm det},\bar{P},i}(R)$ is the planet detection probability for star i for planets with radii R restricted to the period range $\bar{P}$, and R_min is a minimum planetary radius below which P_det = 0 for all stars (we take R_min = 0.3). Note that we assume that the planetary mass and period distribution are independent of the stellar mass.

We expect that the sample of stars surveyed by the RV planet searches, which is biased toward high metallicity, provides a better match to the metallicity of the cluster than the stars surveyed by the OGLE transit search, whereas the OGLE sample provides a better match to the metallicity of the field stars than the RV sample. We therefore use the frequencies of the HJ and VHJ planets from Cumming et al. (2008) for the cluster and the frequencies from Gould et al. (2006) for the field. We take the frequency of an EHJ inferred from the SWEEPS survey for both the field and the cluster. Fixing C_HJ, C_VHJ, and C_EHJ for cluster members such that the fraction of stars with an HJ, VHJ, and EHJ larger than 0.1 M_☉ is 0.012, 0.0017, and 0.0005, respectively, we find C_HJ = 2.4 × 10⁻³, C_VHJ = 3.4 × 10⁻⁴ and C_EHJ = 1.0 × 10⁻⁴. For the field stars, we use the frequencies from Gould et al. (2006) to set C_HJ = 6.2 × 10⁻⁴ and C_VHJ = 2.8 × 10⁻⁴.

Using the above model and P_det(R) values from selection criteria set 2, we find that for cluster members we would expect to detect ∼0.10–0.12 EHJ planets, 0.11–0.16 VHJ planets, and 0.23–0.35 HJ planets where the range depends on the assumed mass-radius relation. For field stars, on the other hand, the expected number of detections are ∼0.34–0.51, 0.30–0.54, and 0.18–0.39. We conclude that for the above model, we would have expected to detect ∼1.3–2 stars in our entire survey, and therefore our observations are consistent with the model. Finally, we note that Beatty & Gaudi (2008) predicted that there are ∼6 transiting HJ and VHJ planets per square degree orbiting Sun-like stars with V ⩽ 20 at Galactic latitude b = 31 (see Figure 8 of that paper). For our 0.16 deg² survey, they would predict ∼1 transiting planet detection, which is comparable to the expected planet yield around field stars from our simulations.