A STELLAR-MASS-DEPENDENT DROP IN PLANET OCCURRENCE RATES

We adopt the stellar effective temperature T_eff, radius R_*, mass M_*, and surface gravity g from Huber et al. (2014), which addressed some of the issues reported in the initial Kepler Input Catalogue. Huber et al. (2014) also reports in which quarters each star was observed. From this information and the observing time for each quarter (tabulated in the Kepler data release notes³) we calculate the total observing time per star t_obs. We remove giant stars from the sample following the prescription of Ciardi et al. (2011), which is based on effective temperature and surface gravity. The stellar noise is characterized by the Combined Differential Photometric Precision or CDPP (Christiansen et al. 2012), of which we downloaded the 2013 December 12 version from the MAST⁴ archive.

The CDPP is given for 3, 6, and 12 hr periods for each quarter. We describe the stellar noise σ_* for any transit duration t by fitting a power law with index cdpp_index to the median-combined noise per quarter

$$\begin{equation} \sigma _{*} (t) = \sigma _{\rm LC} \left(\frac{t}{t_{\rm LC}}\right)^{\rm cdpp_{index}}, \end{equation} \tag{ 1 }$$

where we normalize the noise (σ_LC) at the shortest period possible period in the long cadence, 1765.5 s (t_LC). Additional details of this approach, in particular the non-Gaussianity of the noise (cdpp_index ≠ −0.5), are described in Appendix B.2). After removing stars without a measurement of the CDPP during the first eight quarters (see below), we are left with a sample of 162,270 stars.

We use the list of planetary candidates in Burke et al. (2014), which presents a set of candidates detected in the first eight quarters, as well as a re-evaluation of the Borucki et al. (2011) candidates. Transit parameters of the new KOIs were derived using the first 10 quarters of Kepler photometry, while KOIs detected in the first six quarters have transit parameters determined from eight quarters (Batalha et al. 2013). Though this sample is not meant to be statistically complete, it does currently present the largest uniform sample, since one can assume the KOIs from the first six quarters could also have been detected in eight quarters. Hence, we use only the first eight quarters for calculating the detection efficiency of each star.

We note that KOIs are available up to Q12 and Q16, but are only partially released to the community. They do not form a uniform sample at the time of submission, which is why we choose to not include them, as it is not clear what the current biases in the Q12 and Q16 lists are. After removing single transits and KOIs not matching any stars in our sample, we identify 3731 planet candidates. This is the list of KOIs we will be using in this paper.

Whether a planet of given radius R_p and orbital period P can be detected around a star depends on the achieved S/N for that star, which is a function of transit depth δ = (R_p/R_*)², noise level σ_* during the transit duration t_dur, and the number of transits n.

$$\begin{equation} {\rm S/N} = \frac{\delta n^{0.5} }{\sigma _{*} (t_{\rm dur})} \end{equation} \tag{ 2 }$$

Note that, even though the reported noise on 3–12 hr timescales does not appear to be Poissonian for all stars (Appendix B.2), we assume the S/N still increases with the square root of the number of transits. We demonstrate in Appendix B.3 that this assumption provides the smallest deviations in the signal-to-noise ratios between the calculated values (S/N) and the measured values tabulated in Burke et al. (2014), S/N_tab. We account for a systematic offset between these two values (see also Morton & Swift 2014) by multiplying the calculated values from Eq. 2 by a factor of 1.33 (see Appendix B.2 for details).

The transit duration time t_dur is given by:

$$\begin{equation} t_{\rm dur} = \frac{PM_* \sqrt{1-e^2}}{\pi a}, \end{equation} \tag{ 3 }$$

where e is the orbital eccentricity and a is the semi-major axis given by:

$$\begin{equation} a= \root 3 \of {\frac{G M_*P^2}{4 \pi ^2}} \end{equation} \tag{ 4 }$$

Using this approach, we have corrected for increased probability of shorter transits in eccentric orbits (Burke 2008), but not for impact parameter b to account for the transit duration anomaly (Plavchan et al. 2012), which we motivate in Appendix B.1. Unfortunately orbital eccentricities are not available for most planets, but for small eccentricities (e < 0.3), the deviations in transit duration are less then 5%. Hence, our results are not influenced by the choice of eccentricity and we assume an average eccentricity of e = 0.1. This value falls in the range of eccentricities (0.1–0.25) derived by Moorhead et al. (2011).

The number of transits n for a given period P is calculated for each individual star from the total observing time: n = t_obs/P. Since a potential planetary candidate requires three transits to be detected, the detection efficiency f_n increases from zero to one between 2 and 3 potential transits. Following Batalha et al. (2013), we use:

$$\begin{eqnarray} t_{\rm obs} \le 2 P:\, f_n&=& 0 \nonumber\\ 2 P < t_{\rm obs} < 3 P:\, f_n&= &(t_{\rm obs}/P -2) \\ t_{\rm obs} \ge 3 P:\, f_n&=& 1 \nonumber \end{eqnarray} \tag{ 5 }$$

The Kepler pipeline considers transits with an S/N of at least 7.1 a detection. However, the vetting procedure to rule out false positives also removes some bona fide transits close to the detection limit (Seader et al. 2013), leading to a modified detection efficiency f_eff given by a linear ramp between an S/N of 6 and 12 (Fressin et al. 2013; Morton & Swift 2014):

$$\begin{eqnarray} {\rm S/N} \le 6:\, f_{\rm eff} &=&0 \nonumber\\ 6 < {\rm S/N} \le 12:\, f_{\rm eff} &=& \frac{ {\rm S/N} - 6}{6} \\ {\rm S/N} > 12:\, f_{\rm eff} &=& 1 \nonumber \end{eqnarray} \tag{ 6 }$$

The number of stars N_* in a given spectral type bin {T_eff} as defined in Section 2, around which a planet with given radius and orbital period {R_p, P} can be detected based on its S/N, is then

$$\begin{equation} N_*(\lbrace T_{\rm eff}\rbrace,R_p,P)= \Sigma _{i=0}^{N_*(\lbrace T_{\rm eff}\rbrace)} (f_{{\rm eff},i} \cdot f_{n,i}), \end{equation} \tag{ 7 }$$

where we round N_* to the nearest integer.

When calculating occurrence rates one also needs to take into account the geometric transit probability. This probability is given by:

$$\begin{equation} f_{\rm geo} = \frac{R_p+R_*}{a(1-e^2)}, \end{equation} \tag{ 8 }$$

where (1 − e²) is a correction factor to take into account the increased transit probability of a planet on an eccentric orbit (Burke 2008).

The occurrence rate for a planet of given radius and orbital period {R_p, P} is

$$\begin{equation} f_{\rm occ} (\lbrace T_{\rm eff}\rbrace,R_p,P) = \frac{1}{f_{\rm geo} N_*(\lbrace T_{\rm eff}\rbrace,R_p,P)}. \end{equation} \tag{ 9 }$$

Figure 2 shows the occurrence rate of the entire sample in the style of Howard et al. (2012)—but on an extended grid. Occurrence rates per grid cell are calculated by adding the occurrence rate contributions from individual KOIs in that cell, and scaled to a logarithmic area unit. The 1σ errors per grid cell are given by the 16th and 84th percentile of the binomial probability distribution for drawing the detected number of planets out of N_*, where the detection efficiency and probabilities are calculated for a planet at the logarithmic center of the cell, analogous to Dressing & Charbonneau (2013). For bins without planets, 1σ upper limits are given by the occurrence rate of a (hypothetical) planet at the logarithmic center of the cell.

The occurrence rates we derive per grid cell are—within errors—consistent with those in Howard et al. (2012) for planet radii larger than 2.8 R_⊕. On average, our rates are slightly higher, reflecting the increased detection efficiency of KOIs in Burke et al. (2014). The deviation at small planet radii arises mainly because Howard et al. (2012) assume their sample was complete at S/N > 10, while in reality incompleteness is an issue up to much higher S/N (Fressin et al. 2013). We are able to reproduce their occurrence rates down to 2 R_⊕ if we use a step function rather than the linear ramp in detection efficiency. We have also benchmarked our occurrence rate calculation against the results in Dressing & Charbonneau (2013). We are able to reproduce the numbers in their Figure 15 when using their list of stars and KOIs, and by making the same assumptions on detection efficiency.

When calculating occurrence rates as a function of a parameter other than orbital period, such as semi-major axis, one has to take into account that the orbital period used in calculating the occurrence rate from Equations (1) to (9) is different for each star. When calculating the occurrence rate as a function of semi-major-axis a, the period carries a dependence on stellar mass:

$$\begin{equation} P= \sqrt{\frac{4\pi ^2 a^3}{GM_*}} \end{equation} \tag{ 10 }$$

replacing Equation (4), and P has to be replaced by a in Equations (7) and (9).

To identify which mechanism might be responsible for setting the steep decrease in planet occurrence rate toward the star, we estimate the location where a cutoff would occur as a function of stellar mass for a range of processes. These mechanisms, described in the introduction, include: inhibiting planet formation inside a disk inner edge, trapping migrating planets at that edge, or removing planets by tidal interactions.

In Figure 5, we show how these processes match the observed turnover in planet occurrence rate, each colored cross representing the calculated turnover location for each KOI based on the stellar mass, radius and effective temperature in Huber et al. (2014). To guide the eye, we have indicated the location where the occurrence rates are reduced by 5%, 10%, 20%, and 50% with respect to the plateau for different spectral type bins. The median truncation location per bin is shown by a thick black cross. These mechanisms include additional free parameters that can be tuned to match the exact location of the cutoff, and the purpose of this plot is mainly to show how the cutoff radius scales with the median mass per spectral type bin. The power-law index of each scaling law with stellar mass is also given in Table 2.

Zoom In Zoom Out Reset image size

Table 2. Power-law Index of Turnover Radius as Function of Stellar Mass for the Different Truncation Mechanisms Discussed in Section 4.1

Mechanism	Location	Assumption	Equation
Co-rotation	$M_*^{1/3}$		11
Sublimation	$M_*^{0.7}$	$L_{\rm PMS} \propto M_*^{1.4}$	12
	$M_*^{11/9}$	q = 2	13
	$M_*^{7/9}$	q = 1	13
Tides (stellar)	$M_*^{9/13}$	R*∝M*	15
Tides (planetary)	$M_*^{3/13}$		16

Notes. The third column lists any additional assumption required for turning the listed equation in a pure stellar-mass dependency. The pre-main-sequence luminosity scaling is based on the Baraffe et al. (1998) evolutionary tracks at 1 Myr.

Download table as:  ASCIITypeset image

Though in principle the index of the power law inside of a_cut contains information on the truncation process, we did not attempt a full forward modeling to explain the shape of this curve, but simply note that it may reflect either a spread in initial disk or observed stellar parameters that define the location of the turnover, or may be a result of long-term dynamical evolution of the planet population such as planet–planet scattering. Our approach is very similar to that of Plavchan et al. (2012), though it is independent of the functional form chosen to describe the radial planet occurrence rates, leading to a different stellar-mass scaling for a_cut, which is discussed in detail in Section 5.2.

Protoplanetary disks are truncated at or around the co-rotation radius, from where material is funneled onto the star. This radius is defined as the location where the stars angular velocity Ω equals the Keplerian frequency:

$$\begin{equation} a_{\rm corot}= \root 3 \of {\frac{G M_*}{\Omega _{\rm PMS}^2}}, \end{equation} \tag{ 11 }$$

where Ω_PMS is the angular velocity of the star. The angular velocity is independent of stellar mass during the protoplanetary disk lifetime (Ω_PMS ∼ 10 Ω_☉), albeit with large scatter (Bouvier 2007). This scatter results in a range of co-rotation radii for any given stellar mass, but we will use the average value for the purpose of showing how the cutoff location is different in each spectral type bin, as illustrated in the top right panel of Figure 5. We note that the co-rotation radius from the minimum and maximum angular velocity (Ω_PMS ∼ 2–30 Ω_☉) bracket the factor 2 and 20 drop in occurrence rate, respectively.

The dust sublimation radius for a passive, illuminated disk is located at:

$$\begin{equation} a_{\rm sub}= a_0 \sqrt{L_{\rm PMS}/L_\odot } \end{equation} \tag{ 12 }$$

where the pre-main-sequence luminosity L_PMS is calculated from the Baraffe et al. (1998) evolutionary tracks at an age of 1 Myr, and a₀ is a normalization factor containing all uncertainties in determining the dust sublimation radius (including age). More sophisticated formulas for calculating the dust sublimation radius are available (e.g., Kama et al. 2009), though these do not significantly alter the stellar-mass-dependence. Picking a later age (up to 10 Myr) results in smaller hole sizes, but does not affect the luminosity–mass dependence in the given mass-range, which is of order $L_{\rm PMS} \propto M_*^{1.5}$. The exact value of the normalization factor is only instrumental in determining the stellar-mass dependency, and we take it to be a₀ = 0.04 au, to be consistent with the dust hole inferred from interferometric observations of disks around low-mass stars (Pinte et al. 2008).

We also explore dust sublimation by viscous heating within the disk, as typical mid-plane temperatures of a low-mass pre-main sequence star within 1 au are not dominated by stellar heating (D'Alessio et al. 1998). Using the analytic expression based on detailed radiative transfer models from Min et al. (2011), the dust⁶ sublimation radius scales as:

$$\begin{equation} a_{\rm sub}= a_1 \left(\frac{M_*}{M_\odot }\right)^{ (3+4q) / 9} \end{equation} \tag{ 13 }$$

where q is the power-law index of the scaling between mass-accretion rate and stellar mass ($\dot{M} \propto M_*^q$), which is observed to be q ∼ 2 (Alcalá et al. 2014), and a₁ is a normalization factor that we take be a₁ = a₀. This dependence is shown in Figure 5 second row right panel.

A drop in the number of close-in exoplanets has been noted before Kepler and attributed to tides raised on a planet's host star by Jackson et al. (2009). The authors assume constant planet formation efficiency and subsequent removal of close-in planets by dissipation of tidal energy over gigayear time scales, which results in a planet distribution as a function of semi-major axis (Figure 4 in Jackson et al. 2009) closely resembling our occurrence rate plots in Figure 4. The location of the turnover is given by Equation (1) in Jackson et al. (2009):

$$\begin{eqnarray} \frac{1}{a} \frac{{\rm d} a}{{\rm d} t} &= & -\left[ \frac{63}{2} \sqrt{G M_*^3} \frac{R_p^5}{Q_p M_p} e^2 \right.\nonumber \\ &&+ \left. \frac{9}{2} \sqrt{G / M_*} \frac{R_*^5 M_p}{Q_*} \left(1+\frac{57}{4} e^2 \right) \vphantom{\frac{57}{4}} \right] a^{-13/2} \\ &= & \, {\rm const} = - a_2^{-13/2} \nonumber \end{eqnarray} \tag{ 14 }$$

where Q_* and Q_p are the modified tidal dissipation parameters for the star and planet, respectively, which we take, as in previous work of Jackson et al. (2009), to be Q_* = 10⁶ and Q_p = 10³. For tidally circularized orbits (e ∼ 0), stellar tides dominate, and Equation (14) simplifies to:

$$\begin{eqnarray} a_{{\rm tides},*} &= & a_2 \left(\frac{9}{2} \sqrt{G / M_*} \frac{R_*^5 M_p}{Q_*} \right)^{2/13}, \end{eqnarray} \tag{ 15 }$$

where a₂ is a constant that we take to be a₂ = 8 × 10⁻¹¹s^2/13, such that truncation occurs at 0.05 AU for a G-type star, as in Jackson et al. (2009).

If eccentricities can be maintained, for example through secular interactions in a multi-planet system, planetary tides—the first term in Equation (14)—dominate even for small eccentricities (e ∼ 0.01), and the tidal truncation radius lies further in, at

$$\begin{equation} a_{{\rm tides},p} = a_2 \left(\frac{63}{2} \sqrt{G M_*^3} \frac{R_p^5}{Q_p M_p} \right)^{2/13} \end{equation} \tag{ 16 }$$

There are significant uncertainties in the location of the tidal truncation radius, due to uncertainties in the tidal parameter Q and orbital eccentricities, though these can be mitigated through a different choice of the normalization parameter a₂. Hence, the tidal truncation radius shown in Figure 5 can move significantly inward or outward to match the observed truncation radius by a different choice of parameters, but these uncertainties do not affect the stellar-mass dependency that is of interest here.

We do not consider photo-evaporation (Owen & Wu 2013), as this process does not reduce the number of planets as long as the planet detection limit is above the core mass, which is always the case for the periods of interest here.

The pre-main-sequence co-rotation radius best matches the location of the stellar-mass dependent turnover, closely followed by planetary tides and the dust sublimation radius of a passive disk. Stellar tides, as well as dust sublimation in a viscous disk, provide a poor fit to the observations. Although this list of processes is not comprehensive, it should be kept in mind that any model invoked to explain the population of Kepler planets must be able to reproduce this trend, which is close to $a_{\rm cut} \propto M_*^{1/3} \propto P(M_*)$. The power-law index of each scaling law with stellar mass is also given in Table 2.

Using updated stellar parameters and additional KOIs, we confirm the trend identified by Howard et al. (2012) that planet occurrence rates increase toward cooler stars. Thanks to the larger sample of KOIs we can also show that this trend is present up to 150-day periods and extend it down to Earth-sized planets (Figure 3). The scaling factor we derive follows the same linear dependence on effective temperature described by Howard et al. (2012) Equation (9), with f₀ = 0.17 and k_T = −0.07, consistent within their errors. Surprisingly, there is no simple linear or power-law correlation with stellar mass.

One possibility is that the higher occurrence rate for cooler stars is due to observational biases. Kepler is a magnitude-limited survey, and different spectral types probe different populations of stars. On the main sequence, earlier (later) spectral types are brighter (fainter), and hence are located farther away (closer by) for the same magnitude. Hence, the earlier spectral types in the Kepler sample probe a population that is on average farther away. As Kepler looks out of the galactic plane, these more distant stars are also located higher above the galactic plane, where stars are on average older, have a lower metallicity, and have a higher probability of belonging to the galactic thick disk which is also enriched in α-elements. In the next two paragraphs we will discuss if the differences in stellar composition between the different spectral types in the Kepler sample can explain the increasing planet occurrence toward later spectral types.

The occurrence of giant planets correlates with metallicity (e.g., Santos et al. 2000; Fischer & Valenti 2005), though lower-mass planets are not preferentially found around metal-rich stars (e.g., Sousa et al. 2008; Buchhave et al. 2014). Metallicities for stars in the Kepler field are not determined with high enough precision to test a possible metallicity-occurrence relation. Howard et al. (2012) explored the metallicity gradient as a function of spectral type in the Kepler sample using the Besancon galactic model (Robin et al. 2003). The authors find that the metallicity gradient is not large enough to explain the increased occurrence toward later spectral types even in the optimistic case when planet occurrence scales with metallicity as it does for giant planets. In addition, they find an upturn in metallicity for F stars due to an age cutoff, which contradicts the persistently lower occurrence toward earlier spectral types.

In the low-metallicity regime, a high abundance of α elements correlates with planet occurrence (Adibekyan et al. 2012). These elements, such as Mg, O, and Si, make up the bulk of the mass of the planet-forming solids. The abundances of α elements for the Kepler stars have not been measured, yet inferences for the entire population can be made based on a similar distance argument as for metallicity. Earlier spectral types are located higher above the galactic plane, and hence contain a larger fraction of stars belonging to the galactic thick disk—which is enriched in α elements—than to the galactic thin disk. Hence they contain on average a higher abundance of α elements, predicting an increased occurrence of planets toward earlier spectral types, contrary to what is observed.

Hence, it is unlikely that the trend of increased planet occurrence toward later spectral types in the Kepler sample is caused by differences in stellar populations. It may be an imprint of a different formation efficiency of planets around low-mass stars, for example, the formation of more numerous, but smaller planets (e.g., Kokubo et al. 2006; Raymond et al. 2007). We will leave this analysis for future work, as it requires a thorough investigation of star-disk-scaling laws, planet mass–radius relations and a comparison of different regions in the disk.

Another potential explanation for this trend is the presence of binary companions that may inhibit planet formation. The binary fraction is a strong function of spectral type, rising from ∼35% in M stars to ∼60% in F stars (e.g., Raghavan et al. 2010). However, only close binaries are able to dynamically disrupt planets within 1 AU. Parker & Quanz (2013) estimate that 65%–90% of G type stars can host a planet within 1 AU, and this number is likely higher for the shorter separations considered in this work. Hence, we consider it unlikely that binaries are the sole explanation for the decreased occurrence by a factor of ∼2 between M and G stars, unless they inhibit planet formation in a way other than through dynamical instability.

We derive a smaller power-law index for the planet occurrence-stellar mass scaling law (0.33) than Plavchan & Bilinski (2013) for the Kepler KOIs (0.38–0.9). We attribute this difference mainly to the functional form chosen by the authors to compare with the observed planet distribution within 0.1 AU. This functional form (Gaussian) is a good approximation for hot Jupiters, with an observed pile-up at 0.03 AU. However, we see no evidence for such a pile-up in the smaller Kepler planets (1 R_⊕ to 4 R_⊕) considered in this work, as the occurrence rate does not show a steep drop outward of the cutoff (Figure 4). A different functional form, for example a double power law, will likely result in a scaling law more consistent with our result.

The different distributions for hot Jupiters (pile-up) versus sub-Neptunes and super-Earths (plateau with cutoff) likely indicate a different mechanism is at work. Hence, the equations for tidal interactions presented here are different from those in Plavchan & Bilinski (2013), as they describe different processes. The equations in the latter paper are mainly based on a single planet scattered on an extremely eccentric orbit and being tidally circularized. However, it is not clear how such a model could account for the flat occurrence rates well outside the cutoff, where tides do not play a role in circularizing the orbits. Instead, we rely on removal of planets interior to a certain cutoff radius, either by in-spiral of planets on a circular orbit due to tides raised on the star, or by tides raised on planets with (low-)eccentricity orbits that are maintained due to secular interactions. However, we note that the power-law index for tidal circularization after Kozai interactions with a binary companion (3/13, Wu et al. 2007), is the same as that for secular interaction in multi-planet systems (Table 2).

In this section we explore the implication that the stellar mass scaling of the cutoff location occurs early on and reflects the disk co-rotation radius. The occurrence rates in Figure 4 show a planet formation rate or stalling mechanism that is independent of distance from the star until it is truncated at the co-rotation radius.

In an in situ formation context, this cutoff would lie at the dust sublimation radius, inward of which no solid building blocks can condense from the gas phase. The dust sublimation radius in protoplanetary disks is typically observed to be outside that of the gas disk (e.g., Eisner et al. 2009, 2010; Salyk et al. 2011), and hence inconsistent with the derived stellar-mass scaling of the cutoff. Only a hybrid scenario (e.g., Hansen & Murray 2012; Cossou et al. 2014), in which building blocks (that are sufficiently large to resist sublimation) migrate inward and get trapped at the co-rotation and farther out, can explain the truncation at the co-rotation radius.

In a planet-migration context, planets would form farther out, migrate inward through the gas disk, and get trapped at the co-rotation radius instead of falling onto the star (e.g., Swift et al. 2013). The lack of a pronounced peak in occurrence rate at the cutoff requires additional planets to get trapped behind the first planet to create the plateau in occurrence rate outward of a 10 day period. Trapping migrating planets in mean-motion resonances and subsequent destabilization has also been proposed to explain the presence of Kepler multi-planet systems (Baruteau & Papaloizou 2013; Goldreich & Schlichting 2014). Typical planet formation times inward of the snow line and for a minimum-mass solar nebula disk are much longer than gas disk lifetimes Hence, these planets need to form outside the snow line (e.g., Kennedy & Kenyon 2008), or form in a disk with a higher surface density inside the snow line (Chiang & Laughlin 2013).

The transit duration of a hypothetical planet with period P, semi-major axis a, eccentricity e, pericenter orientation ω, and impact parameter b is

$$\begin{equation} t_{\rm dur} = \frac{PM_*}{\pi a}\sqrt{1-b^2} f(e,\omega), \end{equation} \tag{ B1 }$$

where f(e, ω) is a pre-factor to correct for the orbital eccentricity, given by (Burke 2008, Equation (7)):

$$\begin{equation} f(e,\omega)= \frac{\sqrt{1-e^2}}{1+e \cos (\omega)}. \end{equation} \tag{ B2 }$$

Marginalizing over all possible pericenter orientations ω, taking into account increased probability of transit near pericenter, but assuming an edge-on orbit (b = 0), gives (Burke 2008, Equation (18)):

$$\begin{equation} f(e,\omega)= \sqrt{1-e^2} \end{equation} \tag{ B3 }$$

Taking all this into account, the transit duration simplifies to:

$$\begin{equation} t_{\rm dur} = \frac{PM_* \sqrt{1-e^2}}{\pi a} \sqrt{1-b^2}. \end{equation} \tag{ B4 }$$

Marginalizing over a uniform impact parameter distribution between b = 0 and b = 1 gives:

$$\begin{equation} \int _0^1 \sqrt{1-b^2} db= \frac{\pi }{4}. \end{equation} \tag{ B5 }$$

However, ignoring the impact parameter (b = 0) gives a better fit to the observed transit duration (Figure 6)

$$\begin{equation} t_{\rm dur} = \frac{PM_* \sqrt{1-e^2}}{\pi a}, \end{equation} \tag{ B6 }$$

which probably indicates stellar radii are systematically underestimated in the Kepler catalog (e.g., Plavchan et al. 2012).

Zoom In Zoom Out Reset image size

The CDPP is reported for three different time scales, which allows for a first-order characterization of the noise profiles as a function of time. We have fitted the noise using a simple power law in time (Equation (1)). The distribution of power-law indices is shown in Figure 7. A significant fraction of the stars deviates from a pure Poissonian noise profile, which corresponds to a power-law index of −0.5. There is a distinct tail toward shallower noise profiles, with an additional peak around cdpp_index = 0 corresponding to noise that does not decrease with time, at least on three to twelve hour time scales. The choice of noise profile impacts how the S/N increases as a function of both transit time and number of transits, and does so differently for each star. Different studies have used different assumptions. We will test them in the next subsection.

Zoom In Zoom Out Reset image size

As reported by Morton & Swift (2014), calculating the S/N of a transit using the CDPP from the MAST database does not always reproduce the S/Ns of KOIs reported by the Kepler team, which are measured from the light-curve. In addition, a significant fraction of the Kepler stars exhibit non-Poissonian noise profiles as a function of time (Appendix B.2). In this section, we explore which assumptions for calculating the total S/N best reproduces the ratios measured by the Kepler team (S/N_tab, Burke et al. 2014) We will use those assumptions for the calculation of S/N for all stars in the Kepler sample in the main text.

When comparing calculated to tabulated S/Ns two things should be kept in mind. First, the tabulated value S/N_tab is based on a transit with given impact parameter b and eccentricity e and is measured from the transit light curve, while we use average eccentricities and impact parameters to calculate the S/N for each star (Appendix B.1). Hence, we expect a significant scatter when comparing calculated to measured values. Second, S/N_tab is measured from the light curve fitting—which is done on 10 quarters of data for detection from the first eight quarters, and eight quarters of data for detection in the first six quarters. The calculated S/N, however, depends—among other things—on how accurately we know the stellar parameters, in which systematic errors are still present (e.g., Plavchan et al. 2012). Hence, the purpose of this comparison is not to exactly reproduce S/N_tab, but to find the best way of correcting for known and unknown biases in the data. We will assume the best solution minimizes the scatter in S/N/S/N_tab, and apply an overall scaling factor akin to Morton & Swift (2014) to correctly calculate the average detection efficiency in the entire Kepler sample of stars.

Figure 8 shows the comparison between observed (S/N_tab) and calculated (S/N) signal-to-noise ratios for all planetary candidates. The left panel shows the assumption made by Morton & Swift (2014), i.e., noise with a Poissonian time-behavior. In the right panel the signal-to-noise of a single transit is calculated by interpolating the CDPP to the transit duration, as in Dressing & Charbonneau (2013), while the total signal to noise is still scaled with the square root of the number of transits. The latter provides a smaller spread, but needs to be adjusted by a scaling factor of 1.33 to match the 1:1 slope. A signal-to-noise calculation where the total noise also scales with $n^{\rm cdpp_{index}}$ performs significantly worse, and is not shown here. We conclude that interpolating the noise from the CDPP during the transit, and assuming Poissonian noise between transits results in the smallest scatter around the tabulated signal-to-noise values, but requires a correction factor of 1.33 to match the median signal-to-noise values.

Zoom In Zoom Out Reset image size