---
abstract: 'We develop a method for predicting the yield of transiting planets from a photometric survey given the parameters of the survey (nights observed, bandpass, exposure time, telescope aperture, locations of the target fields, observational conditions, and detector characteristics), as well as the underlying planet properties (frequency, period and radius distributions). Using our updated understanding of transit surveys provided by the experiences of the survey teams, we account for those factors that have proven to have the greatest effect on the survey yields. Specifically, we include the effects of the surveys’ window functions, adopt revised estimates of the giant planet frequency, account for the number and distribution of main-sequence stars in the survey fields, and include the effects of Galactic structure and interstellar extinction. We approximate the detectability of a planetary transit using a signal-to-noise ratio (S/N) formulation. We argue that our choice of detection criterion is the most uncertain input to our predictions, and has the largest effect on the resulting planet yield. Thus drawing robust inferences about the frequency of planets from transit surveys will require that the survey teams impose and report objective, systematic, and quantifiable detection criteria. Nevertheless, with reasonable choices for the minimum S/N, we calculate yields that are generally lower, more accurate, and more realistic than previous predictions. As examples, we apply our method to the Trans-Atlantic Exoplanet Survey, the XO survey, and the [*Kepler*]{} mission. We discuss red noise and its possible effects on planetary detections. We conclude with estimates of the expected detection rates for future wide-angle synoptic surveys.'
author:
- 'Thomas G. Beatty and B. Scott Gaudi'
title: PREDICTING THE YIELDS OF PHOTOMETRIC SURVEYS FOR TRANSITING EXTRASOLAR PLANETS
---

Introduction
============

There are four ways by which extrasolar planets have been detected. The first method to unambiguously detect an extrasolar planet was pulsar timing, which relies on detecting periodic variations in the timing of the received radio signal that occur as the pulsar orbits about the system’s barycenter. The first system of three planets was found around PSR B1257+12 in 1992 [@wolszczan1992], followed by a single planet around PSR B1620-26 [@backer1993]. Although rare, the pulsar planets are some of the lowest mass extrasolar planets known: PSR B1257+12a is about twice the mass of the Moon.

The second way to find extrasolar planets is through radial velocities (RV), which uses the Doppler shift of observed stellar spectra to look for periodic variations in the target star’s radial velocity. By estimating the mass of primary star, the observed radial velocity curve and velocity semi-amplitude can then be used to directly calculate the inclination-dependent mass ($M_p\sin i$) of the companion object. To date, RV surveys have detected more than 230 planets around other stars, making it the most successful method of extrasolar planet detection. While the large number of detected systems having an unseen companion with a mass on the order of $1
M_{Jup} \sin i$ statistically ensures that the majority of these are planetary bodies, the RV surveys are unable to provide more information than the minimum masses, periods, eccentricities, and the semi-major axes of the planets.

RV surveys also have a limited ability to detect planets much smaller than a few Earth masses. The state of the art in RV surveys is the High Accuracy Radial Velocity Planet Searcher (HARPS) spectrometer at the La Silla Observatory in Chile, which is capable of radial velocity measurements with precisions better than $1 \ \mathrm{ms^{-1}}$ for extended periods of time [@lovis2006]. HARPS is therefore able to detect planets with masses on the order of $3$ to $4 M_{\oplus}$ in relatively short period orbits. Unfortunately, planets closer to an Earth mass will be increasingly difficult to detect since intrinsic stellar variability, in the form of acoustic oscillation modes and granulations on the photosphere, makes more precise spectroscopic radial velocity measurements harder to acquire. However, it may be possible to surmount this obstacle (as in the case of @lovis2006) through the selection of stars with “quiet” photospheres and long integration times which serve to average out the stellar variability.

Gravitational microlensing is another technique for detecting extrasolar planets. Microlensing of a star occurs when a a star passes near the line of sight of the observer to another background star. The gravity of the foreground star acts as a lens on the light emitted by the background star, which causes the star in the background to become momentarily brighter as more light is directed towards the observer. Planetary companions to the lens star can further magnify the background star, and create short-term perturbations to the microlensing light curve. To date, six planets have been detected using microlensing [@bond2004; @udalski2005; @beaulieu2006; @gould2006a; @gaudi2008]. Unfortunately, the one-shot nature of microlensing observations means that information about systems discovered this way is generally sparser than that available for RV systems. Therefore, microlensing is most useful in determining the general statistical properties of extrasolar planets, such as their frequency and distribution, and not the detailed properties of the planetary systems.

Planetary transits are a fourth method by which extrasolar planets have been discovered, and the one that provides the most complete set of information about the planetary system. Only planets with very specific orbital characteristics have a transit visible from Earth, because the orbital plane has to be aligned to within a few degrees of the line of sight. Therefore transiting planets are rare. Nevertheless, a transiting extrasolar planet offers the opportunity to determine the mass of the planet (when combined with RV measurements) since the inclination is now measurable, as well as the planetary radius, the density, the composition of the planetary atmosphere, the thermal emission from the planet, and many other properties (see [@charbonneau2007] for a review). Additionally, and unlike RV surveys, transiting planets should be readily detectable down to $1 R_\oplus$ and beyond, even for relatively long periods.

Having accurate predictions of the number of detectable transiting planets is immediately important for the evaluation and design of current and future transit surveys. For the current surveys, predictions allow the operators to judge how efficient are their data-reduction and transit detection algorithms. Future surveys can use the general prediction method that we describe here to optimize their observing set-ups and strategies. More generally, such predictions allow us to test different statistical models of extrasolar planet distributions. Specifically, as more transiting planets are discovered and characterized, predictions relying on incorrect statistics of extrasolar planet properties will increasingly diverge from the observed set of distributions.

Using straightforward estimates it appears that observing a planetary transit should not be too difficult, presuming that one observes a sufficient number of stars with the requisite precision during a given photometric survey. Specifically, if we assume that the probability of a short-period giant planet (as an example) transiting the disk of its parent star is 10%, and take the results of RV surveys which indicate the frequency of such planets is about 1% [@cumming08], together with the assumption that typical transit depths are also about 1%, the number of detections should be $\approx 10^{-3}N_{\leq
1\%}$, where $N_{\leq 1\%}$ is the number of surveyed stars with a photometric precision better than 1%.

Unfortunately, this simple and appealing calculation fails. Using this estimate, we would expect that the TrES survey, which has examined approximately 35,000 stars with better than 1% precision, to have discovered 35 transiting short period planets. But, at the date of this writing, they have found four. Indeed, overall only 51 transiting planets have been found at this time by photometric surveys specifically designed to find planets around bright stars[^1]. This is almost one hundred times less than what was originally predicted by somewhat more sophisticated estimates [@horne2003].

Clearly then, there is something amiss with this method of estimating transiting planet detections. Several other authors have developed more complex models to predict the expected yields of transit surveys. [@pepper2003] examined the potential of all-sky surveys, which was expanded upon and generalized for photometric searches in clusters [@pepper2005]. [@gould2006b] and [@fressin2007] tested whether the OGLE planet detections are statistically consistent with radial velocity planet distributions. [@brown2003] was the first to make published estimates of the rate of false positives in transit surveys, and [@gillon2005] model transit detections to estimate and compare the potential of several ground- and space-based surveys.

As has been recognized by these and other authors, there are four primary reasons why the simple way outlined above of estimating surveys yields fails.

First, the frequency of planets in close orbits about their parent stars (the planets most likely to show transits) is likely lower than RV surveys would indicate. Recent examinations of the results from the OGLE-III field by [@gould2006b] and [@fressin2007] indicate that the frequency of short-period Jovian-worlds is on the order of $0.45\%$, not $1.2\%$ as is often assumed by extrapolating from RV surveys [@marcy2005a]. [@gould2006b] point out that most spectroscopic planet searches are usually magnitude limited, which biases the surveys toward more metal-rich stars, which are brighter at fixed color. These high metallicity stars are expected to have more planets than solar-metallicity stars [@santos2004; @fischer2005].

Second, a substantial fraction of the stars within a survey field that show better than 1% photometric precision are either giants or early main-sequence stars that are too large to enable detectable transit dips from a Jupiter-sized planet [@gould2003; @brown2003].

Third, robust transit detections usually require more than one transit in the data. This fact, coupled with the small fraction of the orbit a planet actually spends in transit, and the typical observing losses at single-site locations due to factors such as weather, create low window probabilities for the typical transit survey in the majority of orbital period ranges of interest [@vonbraun2007].

Lastly, requiring better than 1% photometric precision in the data is not a sufficient condition for the successful detection of transits: identifiable transits need to surpass some kind of a detection threshold, such as a signal-to-noise ratio (S/N) threshold. The S/N of the transit signal depends on several factors in addition to the photometric precision of the data, such as the depth of the transit and the number of data points taken during the transit event. Additionally, ground-based photometry typically exhibits substantial auto-correlation in the time series data points, on the timescales of the transits themselves. This additional red noise, which can come from a number of environmental and instrumental sources, substantially reduces the statistical power of the data [@pont2006].

In this paper we describe a method to statistically calculate the number of planets that a given transiting planet survey should be able to detect. We have done so in a way that maximizes the flexibility of our approach, so that it may be applied in as many circumstances as possible. Therefore, we have allowed for survey-specific quantities such as the observation bandpass to be defined arbitrarily, and have also allowed for our astrophysical assumptions to be easily altered. We account for factors such as the low frequency of Jovian planets, the increased difficulty of detecting transits at high impact parameters, as well as variations in stellar density due to Galactic structure and extinction due to interstellar dust. We also use the present-day mass function to realistically account for the number of main-sequence stars that will be in a given field of view. We do not treat, or simulate, the distribution and effects of giant stars. We approximate the detectability of a planetary transit using a S/N formulation.

This differs from the previously published work in the field by virtue of the generalized nature of our approach. Other authors, such as [@pepper2003], [@gillon2005], [@gould2006b], and [@fressin2007] have restricted themselves to modeling specific transit surveys or types of transit surveys. Our approach, which allows for variations such as arbitrary telescope parameters, observing bandpasses, cadences, and fields of view, is not restricted to any one type of transit survey. Indeed, we offer examples of how our method may be applied to the four most frequently used modes for transit searches. Furthermore, we have conducted an extensive literature review of basic astrophysical relations, such as the Present Day Mass-Function (PDMF), Galactic structure, and the main sequence mass-luminosity relation, and included these in our predictions. We do not use more complicated (but presumably more accurate) Galactic models like the Besancon model [@robin1986] because these models are less flexible and considerably more time consuming to utilize.

We first describe the mathematical formalism with which we have chosen to address this problem, and then move on to discuss our specific assumptions. As examples, we then offer our predictions for several different types of photometric surveys: point-and-stare, drift-scanning, space-based, and all-sky. For each, we choose a representative survey for that particular mode, and compare our predictions to actual survey results, or to predictions elsewhere in the literature. We also examine the possible effects of red noise on our predictions, and conclude by discussing what our predictions imply for the general state of photometric transiting planet surveys.

General Formalism
=================

In the most general sense, we can describe the average number of planets that a transit survey should detect as the probability of detecting a transit multiplied by the local stellar mass function, integrated over mass, distance, and the size of the observed field (described by the Galactic coordinates $(l,b)$: [ $$\begin{aligned}
\label{eq:10}
\frac{d^6 N_{det}}{dR_p\ dp\ dM\ dr\ dl\ db} &=& \rho_*(r,l,b)\ r^2 \cos b\ \frac{dn}{dM}\ \frac{df(R_p,p)}{dR_p\ dp} \nonumber \\ 
&&\times\ P_{det}(M,r,R_p,p),  \end{aligned}$$ ]{} where $P_{det}(M,r,R_p,p)$ is the probability that a given star of mass $M$ and distance $r$ orbited by a planet with radius $R_p$ and period $p$ will present a detectable transit to the observing set-up. $\frac{df(R_p,p)}{dR_p\ dp}$ is the probability that a star will possess a planet of radius $R_p$ and period $p$. $dn/dM$ is the present day mass function in the local solar neighborhood, and $\rho_*$ is the local stellar density for the three-dimensional position defined by ($r,l,b$). We use $r^2 \cos b$ instead of the usual volume element for spherical coordinates, $r^2 \sin \phi$, because $b$ is defined opposite to $\phi$: $b=90^\circ$ occurs at the pole.

First, we will integrate over both planetary radius and orbital period [ $$\begin{aligned}
\label{eq:15}
\frac{d^4 N_{det}}{dM\ dr\ dl\ db} &=& \rho_*(r,l,b)\ r^2 \cos b\ \frac{dn}{dM}\nonumber \\ 
&\times&\ \int_{R_{p,min}}^{R_{p,max}} \int_{p_{min}}^{p_{max}} \frac{df(R_p,p)}{dR_p\ dp}\nonumber \\
&\times&\ P_{det}(M,r,R_p,p) \,dR_p \,dp.\end{aligned}$$ ]{} To do this, we must first specify the detection probability $P_{det}$, as well as the distribution of planet radius and periods $df(R_p,p)/dR_p d_p$, which we treat in Section 2.2.

The Detection Probability $P_{det}$
-----------------------------------

The probability that a planet orbiting a given star shows a detectable transit may be broken up into two separate probabilities: the chance that a planet with radius $R_p$ and period $p$ around a star of mass $M$ at a distance $r$ will transit with a sufficient signal-to-noise ratio to be detected, and the window probability that a transit will be visible for the particular observing set-up: $$\label{eq:22}
P_{det}(M, r, R_p, p) = P_{S/N}(M, r, R_p, p)\ P_{win}(p)$$

### $P_{S/N}$

To calculate the probability that a planetary system will show detectable transits, we first select the appropriate statistical test to determine whether or not our notional data shows a discernible transit. In the world of transiting planet searches, many different methods are used either individually or together to test the statistical significance of a possible transit and alert the researcher to its existence. For example, the Hungarian-made Automated Telescope Network (HATNet) uses a combination of the Dip Significance Parameter (DSP) and a Box-Least Squares (BLS) power spectrum (amongst others) to test its photometric data. In general though, whatever test one uses, it is generally the S/N of a transit shape in the observed data is what determines whether or not that transit will be identified as an event.

As the square of the signal-to-noise can be approximated by the $\chi^2$ test, we therefore chose to use the $\chi^2$ value to determine if a given transit will be detectable for a set of given observation parameters. For a specific transit, the $\chi^2$ statistic, assuming uncorrelated white noise,[^2] is given by $$\label{eq:30}
\chi^2 = N_{tr} \left(\frac{\delta}{\sigma}\right)^2.$$ Where $N_{tr}$ is the number of data points observed in a transit of fractional depth $\delta$ and with fractional standard deviation $\sigma$, assuming a boxcar transit shape with no ingress or egress. To be detected, the transit must have a $\chi^2$ higher than a certain minimum $\chi_{min}^2$ $$\label{eq:40}
\chi^2 \geq \chi_{min}^2.$$ The fractional depth of the transit is given by the square of the planet-to-star radius ratio $$\label{eq:50}
\delta = \left(\frac{R_p}{R_*}\right)^2.$$ While this formulation ignores the contributions of stellar limb-darkening to the depth of the transit, [@gould2006b] demonstrate that, to a first order approximation, transit detection rates are not affected by limb-darkening. Briefly, they point out that the effects of limb-darkening on the transit lightcurve will be offset by the effects of the planet’s ingress and egress from the stellar disk. Therefore, we have not included stellar limb-darkening terms in our calculations.

To determine $\sigma$, we assume Poisson statistics, such that for a given exposure with an observed number of $N_S$ source photons from the target star and $N_B$ background photons, $$\label{eq:60}
\sigma = \sqrt{\frac{N_S + N_B}{N_S^2} + \sigma_{scint}^2},$$ where $\sigma_{scint}$ accounts for scintillation as per [@young1967], and $N_S$ is defined as $$\label{eq:70}
N_S = e_{\lambda} F_{S,\lambda} t_{exp} A.$$ Here $e_{\lambda}$ is the efficiency of the overall observing set-up, and can take values from 0 to 1. It accounts for photon losses in places such as the filter, the mirrors of the telescope, the CCDs, and passage through the atmosphere. $F_{S,\lambda}$ is the source photon flux in the observed bandpass in units of $\gamma\ m^{-2}\
s^{-1}$. $t_{exp}$ is the exposure time of each observation, and $A$ is simply the light collecting area of the telescope.

$F_{S,\lambda}$, the observed photon flux from the source in a certain bandpass, can be defined as a function of the photon luminosity of the source, the distance to the source, and the effects of extinction due to interstellar dust: $$\label{eq:80}
F_{S,\lambda} = \frac{\Phi_{\lambda}(M)}{4\pi r^2} 10^{-A_\lambda / 2.5},$$ where $10^{-A_\lambda / 2.5}$ accounts for extinction, and will be treated later. It has the effect of decreasing the number of observed photons from more distant stars, particularly in the bluer bands.

$\Phi_{\lambda}(M)$, the photon luminosity of the source star, we set as a function of the stellar mass $M$ and the observational bandpass $\lambda$. We derive this quantity in Appendix A, and only note here that we approximate the source stars as blackbodies. While this is a simplification, we show in Appendix A that it is sufficient for our purposes. Using $\Phi_{\lambda}(M)$ in equation (\[eq:80\]) yields the observed photon flux at a distance $r$ from that same star.

We thus now have an expression for $N_S$ as a function of the stellar mass, distance, and bandpass choice, that we can use in our calculation of $\sigma$. We now only need $N_B$, the number of background photons observed. Similar to equation (\[eq:70\]), $N_B$ is defined as $$\label{eq:140}
N_B = S_{sky,\lambda} \Omega t_{exp} A,$$ where $S_{sky,\lambda}$ is the photon surface brightness of the sky at the particular wavelength $\lambda$ (or, in our case, for a specific band), and $\Omega$ is the effective area of the seeing disk. Assuming that the point-spread function (PSF) is a Gaussian with a full-width half-max (FWHM) of $\Theta_{FWHM}$ arcseconds, $\Omega$ is defined by $$\label{eq:150}
\Omega = \frac{\pi}{\ln 4} \Theta_{FWHM}^2.$$ We can now place both of the expressions for $N_S$ and $N_B$ back into equation (\[eq:60\]) to arrive at a determination of $\sigma$, the photometric uncertainty in the data points.

In certain cases, $N_S$ and $N_B$ will be so great that the CCD detectors of the telescope will be saturated by the number of photons received from a star during the exposure time. In this case, further measurements become useless. Mathematically, we may describe saturation as occurring when the number of photons collected by a single CCD pixel during an exposure exceeds the Full-Well Depth (FWD) of that pixel: $$\label{eq:75}
N_{pixel} \geq N_{FWD}.$$ $N_{pixel}$ is related to $N_S$ and $N_B$ as [ $$\begin{aligned}
\label{eq:76}
N_{pixel} &=& N_S \left(1-\mathrm{exp}\left[-\ln 2 \left(\frac{\Theta_{pix}}{\Theta_{FWHM}}\right)^2\right]\right)\nonumber \\
 &&+ \frac{N_B \Theta_{pix}^2}{\Omega}, \end{aligned}$$ ]{} where $\Theta_{pix}$ is the angular size of an individual pixel.

The only part of equation (\[eq:30\]) that remains to be defined is the number of data points observed while the system is in transit, $N_{tr}$. Assuming a circular orbit, we can say that the number of points seen in transit is directly proportional to the fraction of time that the system spends in eclipse, and given a total number of observed data points $N_{tot}$, $N_{tr}$ is thus $$\label{eq:160}
N_{tr} = N_{tot} \left(\frac{R_*}{\pi a} \right)\sqrt{1-b^2},$$ where $b$ is the impact parameter of the transit in units of the stellar radius, $b\equiv(a/R_*)\cos i$, $i$ is the orbital inclination, and we have assumed that $R_*<<a$. We note that this does not hold for correlated noise, see [@aigrain2007].

Taking equation \[eq:160\] and putting it back into our original formulation for $\chi^2$ gives $$\label{eq:170}
N_{tot} \left(\frac{R_*}{\pi a} \right)\sqrt{1-b^2} \left(\frac{\delta}{\sigma}\right)^2 \geq \chi_{min}^2.$$ We define a new variable, $\chi_{eq}^2$, as the value of equation (\[eq:170\]) when $b=0$, corresponding to the $\chi^2$ of an equatorial transit about a given star. This gives us a new form that makes it easier to determine when a system will have a transit geometry sufficient to be detected by the given observing set-up: $$\label{eq:180}
\chi_{eq}^2 \sqrt{1-b^2} \geq \chi_{min}^2.$$ Rearranging, we find that we are able to detect transits up to a $b_{max}$ of $$\label{eq:190}
b_{max}(M, r, p) = \sqrt{1-\left(\frac{\chi_{min}^2}{\chi_{eq}^2}\right)^2}.$$

Therefore, $b_{max}$ is a function of $M$, $r$, and $p$ as a result of the relations contained within the $\chi_{eq}^2$ term.

Finally, we can now deduce an explicit expression for $P_{S/N}$. Given that detectable transits can only occur between $0 \leq b \leq
b_{max}$, we may simply integrate the normalized probability density function of allowable impact parameters from 0 to $b_{max}$ to find a value for $P_{S/N}$.

Given that $b\equiv(a/R_*) \cos i$, the maximum value of the impact parameter for a circular orbit (different from the maximum at which we will detect transits) is $b=a/R_*$. The normalization for the impact parameter’s probability density function is therefore $$\label{eq:200}
1 = \int_0^{\frac{a}{R_*}} P_b(b) \,db.$$ Since impact parameters are uniformly distributed[^3], this gives the solution $P_b = R_*/a$. It should be noted that this form of $P_b$ is only valid in the case of circular orbits. For the period ranges probed so far by transit surveys, this is not a very restrictive assumption, since only $\sim 10\%$ of the known transiting planets have eccentricities inconsistent with zero. At longer periods, however, the effects of eccentricity must be considered [@barnes07; @burke08].

If we now integrate this over the range of impact parameters that yield detectable transits, we find [ $$\begin{aligned}
\label{eq:210}
P_{S/N}(M,r,R_p,p)&=&\int_0^{b_{max}} \frac{R_*}{a} \,db \nonumber \\
                      &=&\frac{R_*}{a} \sqrt{1-\left(\frac{\chi_{min}^2}{\chi_{eq}^2}\right)^2}.\end{aligned}$$ ]{}

### $P_{win}$

The window probability (or window function) for a transit survey describes the probability that a transiting planet with a certain period will be visible to the survey, given the observational cadence of the survey (i.e. - X hours of observing per night for Y nights). While many numerical routines exist that can exactly calculate the window probability for a given observing set-up, for our purposes calculating an exact numerical answer is generally too time intensive to be practical.

-0.0in -0.0in

We therefore developed an analytical framework that allows for the approximate calculation of a window probability. This has the virtue of requiring much less computing time. This is gained at the cost of a small loss of accuracy.

We give a full description of our analytic window probability in Appendix B. Briefly, we assume randomly sampled observations, which leads to simple expressions for the average window probability. This implicitly ignores the effects of aliasing at integer and half-integer day periods. Thus we are not able to address issues related to non-uniform or biased *a posteriori* distributions of planet parameters resulting from the effect of aliasing (e.g. [@pont2006]). Nevertheless, the analytic solution does follow the “outline” of the exact calculations, and closely tracks the moving average value of the exact calculations sufficiently accurately for our purposes, as is shown in Figure 1.

The Frequency of Planets $\frac{df(R_p,p)}{dR_p\ dp}$
-----------------------------------------------------

$df(R_p,p)/(dR_p\ dp)$ denotes the probability that a given star has a planet with radius $R_p$ and orbital period $p$. In the example surveys that we consider later, we chose to focus on the detection of short period Very Hot- and Hot-Jupiters (VHJs and HJs) with orbital periods of 1 to 5 days, as this is the regime being probed by these surveys. The exception of this focus is the Kepler survey, where we also examine the detectability of transiting planets over a range of radii and orbital periods. In both cases, we consider a fixed planet radius, so that $$\label{eq:230}
\frac{df(R_p,p)}{dR_p\ dp} = k(p) f(p)\delta(R_p - R_p^\prime),$$ where $k(p)$ is the correct normalization for a given period range, $f(p)$ is the distribution within this range, and $R_p^\prime$ is the planet radius. For VHJs and HJs we use $R_p^\prime = 1.1 R_{Jup}$, the average radius of the transiting planets found to date[^4]. In general, we have left the exact distribution of $f(p)$ to be assigned as is deemed appropriate. For the VHJ and HJ period distributions described above, we adopt a distribution that is uniform in period, $f(p)=1$.

The frequency of Hot Jupiters ($P\le 5~{\rm days}$, $M_p=0.1-10\ M_{Jup}$) is commonly taken to be $\sim 1\%$ from the results of the RV planet surveys (e.g., @marcy2005a). However, adopting frequencies from RV surveys can lead to biased estimates of the expected yields from transit surveys, because RV surveys constrain the frequency of planets as a function of mass, rather than radius. Furthermore, as argued by @gould2006b, most RV surveys are more metallicity biased than transit surveys, and therefore the planet frequency found by RV surveys will be correspondingly higher, since planet occurrence is an increasing function of metallicity [@santos2004; @fischer2005]. RV surveys are metallicity biased because the target samples for these surveys are magnitude-limited. Since metal rich stars are brighter at fixed color, they are overrepresented in magnitude-limited samples with respect to a volume-limited sample [@fischer2005]. Indeed, [@marcy2005b] note that because they use a magnitude-limited target list, their target stars have a definite metallicity bias. Transit surveys, on the other hand, are typically signal-to-noise ratio limited. At fixed color, metal-rich stars are both brighter and bigger. These two effects on the signal-to-noise ratio roughly cancel, and as a result transit surveys are very weakly biased with respect to metallicity (see Section 8.1 of @gould2006b for a quantitative discussion.)

Indeed, there is evidence for the effects of this metallicity bias in the comparison of various surveys. From the California and Carnegie Keck magnitude-limited and metallicity-biased [@fischer2005; @marcy2005b] survey sample, @cumming08 find a frequency of $\sim 1.3\%$ for planets with $P\le
5~{\rm days}$ and $M\ge 0.2 M_{Jup}$. On the other hand, for the volume-limited [@udry2000b] CORALIE sample, @udry2007 report a frequency of $\sim 0.8\%$ for $M\ge 0.2 M_{Jup}$ and $P\la
11.5~{\rm days}$ $(a< 0.1~{\rm AU})$. Assuming that $\sim 75\%$ of these planets have $P\le 5~{\rm days}$, this corresponds to a frequency of 0.6% for $P\le 5~{\rm days}$ and $M\ge 0.2 M_{Jup}$, a factor of $\sim 2$ lower than that found for the Keck RV survey. @gould2006b examined the OGLE-III transit survey, and found a frequency of $\sim 0.5\%$ for $P\le 5~{\rm days}$, which was subsequently confirmed and strengthened by [@fressin2007]. Thus, as would be expected based on the arguments above, the frequencies inferred from transit surveys and volume-limited RV surveys are similar (and entirely consistent considering Poisson uncertainties and uncertainties in the mass-radius relation of giant planets), whereas they are substantially lower than the frequencies inferred from magnitude-limited (and metallicity-biased) RV surveys.

For the VHJ and HJ predictions that we later make using TrES, XO, and Kepler, we will adopt the normalizations from @gould2006b of $k(p) = 1/690$ for VHJs with orbital periods uniformly distributed between 1-3 days, and $k(p) = 1/310$ for HJs uniformly distributed between 3-5 day periods. By using statistics from OGLE-III, instead of the RV statistics, we are avoiding the difficulties discussed above, and furthermore any unaccounted-for biases inherent in photometric surveys will be preserved, and therefore any systematic effects unknown to us will still be accurately captured in the final predictions. In other words, it is an “apples-to-apples” comparison, instead of “apples-to-oranges” using RV data.

The Present Day Mass Function
-----------------------------

The next step is to integrate equation (\[eq:10\]) over mass: $$\label{eq:240}
\frac{d^3N_{det}}{dr\ dl\ db} = \rho_*(r,l,b) r^2 \cos b \int_{M_{min}}^{M_{max}} P_{det}(M,r) \frac{dn}{dM} \,dM$$

Having previously derived an expression for $P_{det}$, we need only determine the appropriate formulation for the PDMF, $\frac{dn}{dM}$. Using the results of [@reid2002], who used the Hipparcos data sets along with their own data from the Palomar/Michigan State University survey to create a volume-limited survey out to 25 pc, we adopt $$\begin{aligned}
\label{eq:250}
\frac{dn}{dM} &=& k_{norm} \left(\frac{M_*}{M_\odot}\right)^{-1.35}\ \mbox{ for } 0.1 \leq M_*/M_\odot \leq 1,\\
\frac{dn}{dM} &=& k_{norm} \left(\frac{M_*}{M_\odot}\right)^{-5.2}\ \ \mbox{ for } 1 < M_*/M_\odot, \nonumber\end{aligned}$$ where $k_{norm}$ is the appropriate normalization constant. A fuller discussion for our decision to use this particular form for the PDMF is in Appendix C.

Adopting a stellar mass density in the Solar Neighborhood of $0.032 M_\odot \ \mathrm{pc}^{-3}$ [@reid2002], we find a normalization of $k_{norm} = 0.02124\ \mathrm{pc}^{-3}$.

Galactic Structure
------------------

While our normalization of the mass function properly describes the density of stars in the Solar Neighborhood, beyond about 100 pc we must account for variations in stellar density due to Galactic structure. Additionally, as mentioned previously, we must also take into consideration the effects of extinction due to interstellar dust. This is the final stage of integration that needs to be done on equation (\[eq:10\]): $$\label{eq:260}
N_{det} = \int_{0}^{r_{max}} \int_{l_{min}}^{l_{max}} \int_{b_{min}}^{b_{max}} P_{det}(r) \rho_*(r,l,b) r^2 \cos b \,dr \,dl \,db.$$

### Galactic Density Model

We adopt the Bahcall & Soneira model for the Galactic thin disk [@bahcall1980]. The model treats stellar density in the thin disk as a double exponential function involving both the distance from the Galactic center and height above the Galactic plane. Specifically, the mass density relative to the local density is given by $$\label{eq:280}
\rho_*(r,l,b) = exp\left[-\frac{d - d_{gc}}{h_{d,*}} - \frac{|z|}{h_{z,*}}\right],$$ where $d_{gc}=8$ kpc is the distance from the Galactic Center. $h_{d,*}$ is the scale length, and $h_{z,*}$ is the scale height.

We adopt a value of $h_{d,*} = 2.5\ \mathrm{kpc}$ for the scale length of the disk. For the vertical scale height, we use the following relation, which is a slightly modified form of that originally proposed by [@bahcall1980]: $$\begin{aligned}
\label{eq:285}
h_{z,*} &=& 90\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } M_V \leq 2,\\
        &=& 90 + 200 \left(\frac{M_V-2}{3}\right)\ \mbox{ for } 2 < M_V < 5, \nonumber \\
        &=& 290\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } 5 \leq M_V. \nonumber\end{aligned}$$ We calculate the absolute V-band magnitudes of stars of a given mass using the blackbody assumptions, and the mass-radius and mass-luminosity relations described in Appendix A.

We defer the complete discussion of our reasoning behind our choices for $h_{d,*}$ and $h_{z,*}$ to Appendix D. The exact values for these parameters are not well constrained in the literature, and we show the effects of using different values in the Appendix. We note that within the range of values found in the literature, the final prediction results may differ by as much as 20%.

We only treat the thin disk in our modeling because we decided that an expanded analysis that also looked at stars in the thick disk, halo, and bulge would not add a significant number of detections to our final result. Both the thick disk and the halo have much lower volume densities than the thin disk at distances typically surveyed by transit searches. In the same vein, the bulge is too far away. These populations are also typically older than the thin disk, and so have a fainter turn-off point, meaning that these stars are more difficult to detect. That being said, for deeper transit surveys, such as some of the all-sky surveys, these populations may play an interesting role. In that case, it would be fairly simple to add them into our model.

We chose the Bahcall model instead of other more detailed models due to speed and portability considerations. While others, such as the Besancon models [@robin1986], take into account features such as density variations from the spiral structure of the Galaxy and the Galactic Bulge, they typically require over 30 minutes to simulate a given star field. Additionally, many are only accessible through webpage interfaces, and therefore difficult to include in a self-contained prediction algorithm.

We do note that comparisons between our Galactic model and the Besancon model revealed star count differences of only 10% in the Kepler and TrES fields over a range of magnitude limits.

### Interstellar Extinction Modeling

We also model the density of interstellar dust as a double exponential $$\label{eq:290}
\rho_{dust}(r,l,b) = exp\left[-\frac{d - d_{gc}}{h_{d,dust}} - \frac{|z|}{h_{z,dust}}\right].$$ Which gives the dust density at a distance $r$ along a particular line of sight ($l,b$) relative to the density in the Solar Neighborhood (i.e. - $z=0$, $d=d_{gc}$). To calculate the extinction in a given bandpass for a star at a given distance, we then integrate equation (\[eq:290\]) over $r$ to arrive at the effective path length through an equivalent Solar Neighborhood dust density, $$\label{eq:310}
\ell_{dust} = \int_0^r \rho_{dust}(r,l,b) \, dr.$$ The V-band extinction in magnitudes, $A_V$, is given by this path length multiplied by the appropriate normalization constant: $A_V = k_{dust,v} \ell_{dust}$. We adopt $k_{dust,v} = 1$ mag kpc$^{-1}$ [@struve1847]. We determine the extinction in other bands by using the extinction ratios ($A_\lambda/A_V$) from [@cox2000], assuming that $R_V = 3.1$.

Assumptions
===========

It is important to keep in mind that in the preceding derivations, we made several assumptions about the nature of extrasolar planetary systems, as well as some simplifications to keep the math and computing time more manageable.

1\. [*Detectability is Best Measured Through $\chi^2$:*]{} While S/N is a better metric than photometric precision to determine if a transit will be detectable, and the $\chi^2$ parameter is a good proxy for S/N, actual transit surveys use a wide variety of tests to detect transits. Many tests, such as the visual evaluation of the data by a human being, are difficult to simulate numerically. Thus, using just a $\chi^2$ test is a simplification, as human inspection may impose a much higher effective S/N threshold. As we show for typical surveys, the number of detections is a strong function of the S/N threshold. Thus a proper determination of the effective S/N threshold imposed by all cuts is necessary before any robust conclusions can be drawn about the rate of detections relative to expectations. This is a fundamental - and unavoidable - limitation of our study.

2\. [*No Stellar Limb-Darkening:*]{} We have neglected to include terms for stellar limb-darkening because, as stated above, to first-order its effects are canceled by the the effects of ingress and egress on the transit lightcurve [@gould2006b].

3\. [*Stars are Blackbodies:*]{} We assume that the radiated energy of a star can be described as a blackbody. While in general this is a fair approximation (to within 0.5 mag, see Figures 11 and 12), it does not hold true for the bluer bandpasses we consider (U-band in particular) and for very low-mass stars.

4\. [*Simplified Galactic Structure:*]{} While we do treat variations in stellar density and interstellar dust densities in a general sense, our formalism is a simplified double exponential model that does not account for the spiral structure of the Galaxy or patchiness in the dust distribution. Therefore detailed, accurate predictions for an individual field may be better served by incorporating more sophisticated Galactic models (e.g. [@fressin2007]).

5\. [*No Binary Systems:*]{} While we only consider main sequence stars, we do not model the statistics of binary systems; in our formulation all stars are treated as single. Though this is clearly not the case, it was outside the scope of this project to go deeply into binary frequencies, the ability of planets to form in such systems, and whether or not such planets would be detectable.

However, we can make a rough estimate of the expected magnitude of the effect of binaries. First, we assume that 1/3 of our stars have no binary companion [@duquennoy91]. We assume the remaining 2/3 of the stars are in binary systems. A typical wide-field survey has a PSF size of $\sim 20''$, which corresponds to a separation of $\sim 4000~{\rm AU}$, and a period of $\log (P/{\rm day})\sim 7.8$ for an equal-mass solar-mass binary. Adopting the period distribution for binary stars from @duquennoy91, we estimate that $\sim 90\%$ of the binary systems are unresolved. Of these unresolved binaries, we conservatively assume that those with separations $\la 5~{\rm AU}$, or $\log (P/{\rm day})\sim 3.5$, do not host hot Jupiters because the binary components are too close.

Thus we estimate that $\sim 62\%$ of unresolved binaries can host hot Jupiters. To estimate the detectability of transiting planets in these, consider a binary where $F_1$, $R_1$, $M_1$ and $F_2$, $R_2$, $M_2$ are the flux, radius, and mass of the primary (“1”) and secondary (“2”), respectively. Additionally, define $\ell=F_2/F_1$, ${\cal R}=R_2/R_1$, and $q=M_2/M_1\le 1$. Then, for fixed planet radius and semimajor axis, assuming photon-limited uncorrelated uncertainties, the signal-to-noise ratios if the stars are resolved are $({\rm S/N})_1 \propto R_1^{-3/2} F_1^{1/2}$ and $({\rm S/N})_2 \propto R_2^{-3/2} F_2^{1/2}$, while if the stars are unresolved the signal-to-noise ratios for transits across each component are $({\rm S/N})_{B,1} \propto R_1^{-3/2} F_1^{1/2} (1+\ell)^{-1/2}$ and $({\rm S/N})_{B,2} \propto R_2^{-3/2} F_2^{1/2} (1+\ell)^{-1/2}$. Assuming a constant volume density of stars and no interstellar extinction, the number of detected planets scales as $({\rm S/N})_{\rm min}^{-3}$ [@gould2003b]. Thus the number of planet detections for a population of binaries in the unblended case can be written as $N_0 \propto ({\rm S/N})_1^3 + f(q) ({\rm S/N})_2^3$, where $f(q)$ is the mass ratio distribution. Similarly, for the blended case, this is $N_B \propto ({\rm S/N})_{B,1}^3 + f(q) ({\rm S/N})_{B,2}^3$. Using the scalings for the signal-to-noise ratios just derived, we evaluate the ratio of the number of detections in the blended case, relative to the number of detections one would expect if they were unblended, $$\frac{N_B}{N_0} = (1+\ell)^{-3/2} \frac{1+f(q)\ell^3{\cal R}^{-9/2}}{1+f(q)\ell^{3/2}{\cal R}^{-9/2}}.
\label{eq:291}$$ We adopt $\ell = q^5$, which is roughly appropriate for the $R$-band (see Fig. 11), and ${\cal R} = q^{0.8}$ (Eq. \[eq:121\]). We consider two different distributions for $f(q)$. First, we adopt $f(q)$ from @duquennoy91, which is weighted toward low mass ratio systems. Second, we consider a distribution with a substantial population of ‘twins’ with $q=1$. Specifically, we choose $f(q)\propto 1 +3.5\exp[-0.5(q-1)^2/\sigma_q^2]$, with $\sigma_q=0.2$, which approximates the distribution found by @halbwachs03. When we then average Equation (\[eq:291\]) over $q$, weighting by $f(q)$, we find $\langle N_B/N_0\rangle \simeq 0.89$ for the @duquennoy91 mass ratio distribution, and $\langle N_B/N_0\rangle \simeq 0.68$ for the @halbwachs03 mass ratio distribution. Thus, we estimate that roughly 70-90% of planets in unresolved binaries will still be detectable. The basic point is that, because the mass-luminosity relation is so steep, only those binaries that have $q\sim 1$ result in a substantial reduction in the signal-to-noise ratio. Low mass ratio companions have little affect on the ability to detect transit signals. For the mass ratio distribution of @duquennoy91, these provide the bulk of the companion population, and so in this case the effects of binarity are suppressed even further.

Finally, we can summarize these various components to provide a rough estimate of the fraction of planets that will be detected when we consider binary systems, relative to our fiducial assumption that all stars are single. We find $1/3 + (0.1+ 0.62 \langle N_B/N_0\rangle)\times (2/3) \simeq 0.68-0.77$. Thus we expect that our estimated yields are systematically high by at most $30\%$. This is generally of order or smaller than other sources of uncertainty in our predicted yields.

We note that a corollary of our assertion that binaries have a small effect on the yields of transit surveys is that a substantial fraction of the planets that have been found in wide-field transit surveys should be in binary systems. Only two planets in confirmed common proper-motion binary systems have been reported [@burke2007; @bakos2007], however systematic surveys for stellar companions to the host stars of the known transiting planets have not been performed, and therefore the true incidence of binary systems amongst the sample of transiting planets is not known.

Results
=======

Using the preceding formalism, we simulated four different surveys modes used by transit searches: point-and-stare, drift-scanning, space-based, and all-sky surveys. For the point-and-stare and drift-scanning cases, we use the TrES and XO transit surveys as our examples, respectively. Both surveys have been successful in their hunt for planets, and so we are able to directly compare the results of our calculations with the actual number of detections. For VHJs and HJs, we simulate the detection of planets with $R_p=1.1R_{Jup}$, and with the period distribution from [@gould2006b] that we describe in Section 2.2.

[c|c]{} Band & 550-700 nm\
$m_R$ Limit & $\approx 13$\
Observation Time & Varies\
Exposure Time & 90 seconds\
CCD Read Time & 25 seconds\
Telescope Diameter & 0.1 m\
Throughput & 0.6\
PSF (FWHM) & 25”\
$\theta_{pix}$ & 10”\
CCD Full-Well Depth & $1.2\times10^5\ e^-$\
Field of View & $36\ \mathrm{deg}^2$\
Galactic Longitude & Varies\
Galactic Latitude & Varies

[c|cc]{} Field & Observation Time & Galactic Coordinates\
And0 & 765 hrs & \[126.11, -015.52\]\
Cyg1 & 609 hrs & \[084.49, +010.28\]\
Cas0 & 628 hrs & \[120.88, -013.47\]\
Per1 & 643 hrs & \[156.37, -014.04\]\
UMa0 & 497 hrs & \[168.87, +047.70\]\
Crb0 & 537 hrs & \[053.49, +048.92\]\
Lyr1 & 582 hrs & \[077.15, +017.86\]\
And1 & 435 hrs & \[109.03, -017.62\]\
And2 & 332 hrs & \[115.52, -016.21\]\
Tau0 & 406 hrs & \[169.83, -015.94\]\
UMa1 & 500 hrs & \[156.32, +054.01\]\
Her1 & 592 hrs & \[063.26, +026.42\]\
Lac0 & 686 hrs & \[097.79, -008.86\]

We demonstrate our method for space-based surveys by simulating the upcoming Kepler mission, and comparing our predictions with other predictions for Kepler in the literature. Similarly for the all-sky mode, we examine the future planned observing programs of LSST, SDSS-II, and Pan-STARRS and make predictions for each.

For each mode, we have tried to approximate the effect of red noise on detectability by increasing our S/N cut [@pont2006]. To show the effects of a more accurate red noise calculation, in Section 4.5 we evaluate its effects on the TrES survey.

Point-and-stare - TrES
----------------------

Point-and-stare observing is the traditional observing mode of tracking one field of sky continuously for all, or part, of the night. There are several operational photometric transit surveys that use this particular method of observing, including HATnet [@bakos2004], SuperWasp [@pollacco2006], and KELT [@pepper2007], but we chose to model is the Trans-Atlantic Exoplanet Survey (TrES). TrES is composed of a network of three telescopes: STARE on the Canary Islands, PSST at Lowell Observatory in Arizona, and Sleuth on Mt. Palomar, California. All are small, 10 cm aperture, wide-field ($6^\circ$), CCD cameras that operate in unison to observe the same field of sky nearly continuously over one to two month periods. The survey specifics are described in more detail by [@dunham2004] and by [@brown1999]. The relevant quantities for our modeling are listed in Table 1.

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We fit to the RMS plots posted on the Sleuth observing website[^5] to determine the background sky brightness in the TrES bandpass (550-700nm), using the TrES team’s stated photon throughput of 0.6, and we estimate that the TrES bandpass has a sky background of $19.6\
\mathrm{mag\ arcsec}^{-2}$.

[c|ccccccccccccccc|c]{} & And0 & Cyg1 & Cas0 & Per1 & UMa0 & CrB0 & Lyr1 & And1 & And2 & Tau0 & UMa1 & Her1 & Lac0 & Total\
VHJs & 0.38 & 0.37 & 0.34 & 0.34 & 0.18 & 0.21 & 0.32 & 0.24 & 0.19 & 0.22 & 0.17 & 0.21 & 0.41 & 3.59\
HJs & 0.34 & 0.30 & 0.29 & 0.28 & 0.16 & 0.18 & 0.27 & 0.18 & 0.09 & 0.15 & 0.16 & 0.19 & 0.34 & 2.91\
Both & 0.72 & 0.67 & 0.63 & 0.62 & 0.34 & 0.39 & 0.59 & 0.42 & 0.28 & 0.37 & 0.33 & 0.40 & 0.75 & 6.50

In our simulations, we treat the three TrES telescopes as one single instrument. This is clearly an oversimplification. While these telescopes are quite similar, there are differences between the data quality and quantity obtained from each site. For example, Sleuth must contend with a higher sky background due to its proximity to urban areas, whereas the fraction of clear nights for PSST is typically lower than for the other two sites. However, accounting for these differences is difficult, and furthermore their effect on our predicted detections is small compared to the other uncertainties in our model. We therefore ignore these effects. In dealing with the window probability of the TrES network, we were greatly aided by the plots posted on the Sleuth website which showed the window probability for each field, as well as the total network observing time each field had received.

We simulated thirteen different TrES fields using this methodology. The network observation times and the location of each field are shown in Table 2. The data for this table was collected off from the Sleuth observing website. The expected detections for each of the fields (and the total number of detections over all simulated fields) are shown in Table 3. The effects of our Galactic structure model can clearly be seen in distribution of expected detections. Both the UMa0 and Crb0 fields are substantially above the plane of the Galaxy ($b \approx 48^\circ$), and thus show a much lower detection rate than those fields directed towards denser regions of the sky.

Figure 2 shows the distributions of detections over various parameters in the Lyr1 field. For Figure 2, we show two distributions, one assumes white noise in the photometry, the other assumes 3 mmag of red noise. For the white noise estimates, we used a signal-to-noise limit of S/N $\geq 30$ and a magnitude limit of $m_{R}\leq13$. Lyr1 is the field in which TrES-2 was discovered, and is adjacent to Lyr0, which contains TrES-1. We therefore have plotted the location of these two stars on the distributions shown in Figure 2. Both stars reside close to the peaks of our parent star mass and radius distributions. The third and fourth planets discovered by TrES, TrES-3 and TrES-4, are not plotted in Figure 2, as they reside in the Hercules fields, which are at a higher Galactic latitude than Lyra. Also of note is that with our assumptions half of the expected detections the F-dwarf range. Among the false positives encountered by the TrES team, the most frequent are eclipsing F/M-dwarf binaries. Figure 2 offers a partial explanation for this, showing that roughly half of TrES’ planet-like signals come from F-dwarfs.

[|c|ccc|]{} & VHJ & HJ & Both\
S/N$\geq20$ & 6.06 & 5.72 & 11.78\
S/N$\geq25$ & 4.68 & 4.05 & 8.73\
S/N$\geq30$ & 3.59 & 2.91 & 6.50\
S/N$\geq35$ & 2.77 & 2.13 & 4.90\
S/N$\geq40$ & 2.16 & 1.59 & 3.75

[|c|ccc|]{} & VHJ & HJ & Both\
$m_R\leq12.0$ & 2.13 & 1.95 & 4.08\
$m_R\leq12.5$ & 2.95 & 2.58 & 5.53\
$m_R\leq13.0$ & 3.59 & 2.91 & 6.50\
$m_R\leq13.5$ & 3.86 & 3.01 & 6.87\
$m_R\leq14.0$ & 3.94 & 3.03 & 6.97\
$m_R\leq14.5$ & 3.96 & 3.05 & 7.01

[|c|ccc|]{} & VHJ & HJ & Both\
$R_p=0.9R_{Jup}$ & 1.66 & 1.17 & 2.83\
$R_p=1.0R_{Jup}$ & 2.52 & 1.90 & 4.42\
$R_p=1.1R_{Jup}$ & 3.59 & 2.91 & 6.50\
$R_p=1.2R_{Jup}$ & 4.53 & 3.88 & 8.41\
$R_p=1.3R_{Jup}$ & 5.52 & 5.04 & 10.56

Our predicted number of detections for the selected TrES fields of $N_{det} = 6.50$ covers only half of the fields observed by TrES, so we may roughly extrapolate this result to $13.00$ detections over all of the fields. This is roughly three times the four planets that TrES has actually detected, though we hesitate to draw strong conclusions from this comparison. While we have the luxury in our predictions of imposing draconian signal-to-noise ratio and magnitude cuts in the data, in real life the TrES detections are identified in data using more flexible selection effects. A better comparison between our predictions and the actual results of TrES (or any survey) would therefore require a better understanding of the selection effects in the observed data set.

[c|c]{} Band & 400-700 nm\
$m_V$ Limit & $12$\
Observation Time & $\approx 400$ hrs\
Exposure Time & 54 seconds\
CCD Read Time & n/a\
Telescope Diameter & 0.11 m (two)\
Throughput & 0.16\
PSF (FWHM) & 45”\
$\theta_{pix}$ & 25.4”\
CCD Full-Well Depth & $1.5\times10^5\ e^-$\
Field Area & $7.2\times63\ \mathrm{deg}^2$\
Galactic Longitude & Varies\
Galactic Latitude & Varies

To see how different selection criteria change our predictions for TrES, Tables 4-6 show the effects of changing the S/N limit, the magnitude limit, and the radius of the targeted planets on the detection numbers.

A changing magnitude limit is particularly interesting, since all four of the TrES detections have been around stars with $m_R\leq12$. This might indicate that the effective magnitude limit of the TrES survey is brighter than what is described in the published literature. If this is the case, then using a limit of $m_R\leq12$ lowers the number of expected detections in our simulated fields to $N_{det} = 4.08$, and total detections to $8.16$.

Drift-scanning - XO
-------------------

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The XO survey is a wide-angle ground-based photometric survey that uses drift-scanning for its observations. The XO telescope is a pair of 11 cm cameras in place at the Haleakala Observatory in Hawaii. It monitors thousands of stars with $m_V\leq12$ in six fields spaced evenly around the sky in right ascension[^6]. The cameras scan within these fields, sweeping in declination from $0^\circ$ to $+63^\circ$ every ten minutes. Typically, this provides over 3000 observations per star in each of the XO fields. More on the specifics of the XO system can be found in [@mccullough2005]. The relevant parameters for our predictions are listed in Table 7.

To determine the throughput of XO and the background sky magnitude in the XO band (400-700 nm), we fit to the RMS diagram shown in Figure 8 of [@mccullough2005]. With the addition of their stated calibration and scintillation error, we were able to exactly reproduce the relations shown in that figure using a background sky magnitude of 19.4 mag $\mathrm{arcsec}^{-1}$ and a system throughput of 0.16.

[cccccc|c]{} 00 hrs & 04 hrs & 08 hrs & 12 hrs & 16 hrs & 20 hrs & Total\
0.34 & 0.35 & 0.34 & 0.30 & 0.32 & 0.37 & 2.02

Additionally, we checked to ensure that the values we used for the observation time and the number of nights observed generated an analytic window probability similar to that shown in Figure 7 of [@mccullough2005]. Our ability to do this was greatly aided by the XO team’s description of how many nights they lost to weather and mechanical trouble.

We simulated each of the six XO fields for our predictions. The results for each field using the base parameters of S/N$\geq30$, $m_V\leq12$ and $R_p=1.1R_{Jup}$ are shown in Table 8. The first planet discovered by the XO survey, XO-1b, [@mccullough2006] is located in in the field at 16 hrs, and so we have also plotted the distribution of detections within this field over a variety of parameters in Figure 3. XO-2b [@burke2007] and XO-3b [@johnskrull2007] are in noticeably different fields (08 hrs and 04 hrs, respectively), so were not included on the plot.

[|c|ccc|]{} & VHJ & HJ & Both\
S/N$\geq20$ & 3.45 & 2.41 & 5.86\
S/N$\geq25$ & 2.20 & 1.27 & 3.47\
S/N$\geq30$ & 1.37 & 0.65 & 2.02\
S/N$\geq35$ & 0.83 & 0.33 & 1.16\
S/N$\geq40$ & 0.50 & 0.17 & 0.67

[|c|ccc|]{} & VHJ & HJ & Both\
$m_V\leq11.5$ & 1.09 & 0.55 & 1.64\
$m_V\leq12.0$ & 1.37 & 0.65 & 2.02\
$m_V\leq12.5$ & 1.50 & 0.69 & 2.19\
$m_V\leq13.0$ & 1.54 & 0.70 & 2.24\
$m_V\leq13.5$ & 1.55 & 0.71 & 2.46

[|c|ccc|]{} & VHJ & HJ & Both\
$R_p=0.9R_{Jup}$ & 0.29 & 0.09 & 0.38\
$R_p=1.0R_{Jup}$ & 0.69 & 0.25 & 0.94\
$R_p=1.1R_{Jup}$ & 1.37 & 0.65 & 2.02\
$R_p=1.2R_{Jup}$ & 2.07 & 1.18 & 3.25\
$R_p=1.3R_{Jup}$ & 2.93 & 1.92 & 4.85

Tables 9-11 show the effects of changing the S/N limit, the magnitude limit, and the radii of the targeted planets on XO’s predicted detection numbers. For our base case of $m_V\leq12$ and S/N$\geq30$, we expect that XO will detect 2.02 planets. If we consider the three actual detections from XO, it seems that, similar to TrES, the actual S/N and magnitude limit of the XO survey may differ from what is in the literature. For example, all three XO parent stars are at $m_V\leq11.5$ (XO-1, the dimmest, is at $m_V=11.3$). To see how this affected our calculated distributions, we also simulated the field at 16 hrs assuming S/N$\geq20$, $m_V\leq12$, and S/N$\geq20$, $m_V\leq11.5$. The results are included on Figure 3. In these cases, the total number of detections increases to 5.86 and 4.21, respectively.

Space-based - Kepler
--------------------

To make predictions for a space-based mission, we chose NASA’s Kepler mission, which is primarily designed to look for Earth-sized extrasolar planets. It consists of a single spacecraft that is currently scheduled to be launched into an Earth-trailing heliocentric orbit in early 2009. Onboard, Kepler will have 42 CCDs that will measure the light from 100,000 main-sequence stars within the telescope’s field of view. Ideally, Kepler will identify planets orbiting within the “habitable zone” of their parent stars which would be capable of possessing liquid water on their surfaces.

[c|c]{} Band & 400-850 nm\
$A_{Kepler}$ & 0.86\
$m_V$ Limit & $\approx 14$\
Observation Time & 4 years\
Exposure Time & 3 seconds\
CCD Read Time & Negligible\
Telescope Diameter & 0.95 m\
Throughput & .3\
PSF (FWHM) & 10”\
$\theta_{pix}$ & 3.98”\
CCD Full-Well Depth & $1.2\times10^6\ e^-$\
Field of View & $106\ \mathrm{deg}^2$\
Galactic Longitude (new) & $76.32^\circ$\
Galactic Longitude (old) & $70^\circ$\
Galactic Latitude (new) & $13.5^\circ$\
Galactic Latitude (old) & $5^\circ$

Beyond looking for extrasolar Earths, the immense amount of photometry that Kepler is expected to produce over its four year operational lifetime will also allow for the detection of larger jovian worlds transiting other stars. Previous estimates of the number of VHJs and HJs expected to be found by Kepler have predicted the discovery of about 180 HJs (no VHJs had been detected at the time, and so they were not treated) [@jenkins2003]. Clearly, given that there are only 46 currently known transiting planets, this would be a significant increase in the number of known planets

To simulate Kepler with our methodology, we used the values in Table 12 to simulate Kepler observations out to a distance of $8\ \mathrm{kpc}$, using a signal-to-noise ratio of SN$\geq7$. We gathered the information in Table 12 from both the Kepler mission website[^7] as well as published descriptions of the satellite [@borucki2004]. Three items are of note. First, our value for the photon detection efficiency of Kepler is based upon information from the Kepler website that a 12th magnitude G2 dwarf will show $7.8\times10^8\ \gamma\  \mathrm{hr}^{-1}$. Secondly, the value we use for the extinction ratio in the Kepler bandpass, $A_{Kepler}/A_V = 0.861$, is an average that we calculated by integrating the extinction curve over the Kepler bandpass (400-850nm) [@fitzpatrick1999]. Finally, the Kepler website and much of the literature states that Kepler will use a dim magnitude cutoff of $m_V\leq14$. In reality, there is no “hard” magnitude cutoff at which stars will be excluded from observations. Indeed, Kepler is expected to observe promising stars that are much dimmer than $m_V=14$ (D. W. Latham, private communication). Therefore, for our more explicit calculations of the expected number of HJ and VHJ detections for Kepler, we adjusted the magnitude limit such that the number of stars in the field agreed with the expected number of Kepler targets (100,000 in the case of the new field). For other, more general, simulations, we used several different magnitude limits to determine what effects these had on Kepler’s expected detections.

While the location of the Kepler field is currently centered on Galactic coordinates $(l=76.32^\circ, b=13.5^\circ)$, up until 2004 it was instead situated closer to the Galactic plane at $(l=70^\circ, b=5^\circ)$. The switch in the position of the field to higher Galactic latitudes was done in order to avoid sorting through the larger number of giant stars that are found in fields directed along the plane of the Galaxy. Much of the previously published literature examining the Kepler mission have used the coordinates of the old field. For the purpose of comparison, we therefore simulated both the old and new Kepler fields.

### General Characterisation

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Looking in the new Kepler field with a magnitude limit of $m_V\leq14$, we simulated the number of detections as a function of orbital period for Jupiter-, Earth-, and Mars-sized worlds, assuming that every star possessed a planet with the given period and radius (Figure 4), and that red noise was negligible. Interestingly, the difference between the number Jupiter- and Earth-sized planets detected is minimal for $p \leq 10$ days, which indicates that Kepler is able to detect all the planets within this radius range, assuming that they transit. Therefore, for $m_V \leq 14$ and $p \leq 10$ days, the S/N of transiting Jupiter- and Earth-sized planets is large enough that Kepler’s S/N limit is irrelevant, and the $m_V\leq14$ magnitude cutoff is the limiting factor in the number of detections. For smaller planets the size of Mars, the S/N of their transits are in the neighborhood of the Kepler S/N limit for the brightest stars and shorter periods, and so Mars-sized worlds are detected at a much lower rate.

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To more accurately characterize the magnitude- and S/N-limited detection regimes, we plotted the number of detections as a function of planetary radius for orbital periods of 1 day and 365 days (Figure 5). In addition in a magnitude limit of $m_V=14$, we also plot the same relations for a limit of $m_V=20$. One can clearly see the flattening of these curves as the radius of the target planets increase, indicating the increasing effect of the magnitude limit on the detection numbers. As is expected in the S/N limited regime, detections go as $R_p^6$ [@pepper2003; @gaudi2005; @pepper2005], though at the lower end saturation of the Kepler CCDs serves to cut out some of these detections.

### Habitable Planets

One of the major goals of the Kepler mission is the detection of planets in a star’s habitable zone that would be capable of supporting life. The habitable zone is defined as the distance from the parent star at which liquid water water could exist on a planet’s surface. We estimate the detection rate of habitable zone planets by assuming that all stars have a planet with a semi-major axis equal to $$\label{eq:415}
\frac{a}{{\rm AU}} = \sqrt{\frac{L_{bol,*}}{L_{bol,\odot}}}.$$

This assumption is simple and likely optimistic. In reality, the habitable zone has a finite width, and habitable planets will have a range of radii. Thus, a better estimate would be to adopt a distribution of planet masses and radii, and integrate over the range of radii and periods considered habitable. Furthermore, the assumption of one habitable planet per star is probably optimistic, and so the actual number of detected habitable Earths will be smaller than we estimate, depending on the real frequency of Earth-like planets. For the sake of simplicity, however, we will adopt the assumptions above.

With this assumption, we simulated the detection of habitable Earths in both the old and new Kepler fields (Figure 6). Surprisingly, despite there being nearly twice as many main sequence stars in the old field, the number of detected habitable Earths remained nearly the same in the two fields. This occurred because at the magnitude relevant for Kepler, the increased star count in the old field was mainly due to the larger number of A and F main-sequence stars found closer to the Galactic plane; habitable planets are doubly hard to detect around these stars due to the lessened depth of transit, and the longer periods required for a planet to be in the habitable zone. The exact distribution of stars by spectral type for the old and new fields is given in Table 13, and is discussed further below. For the 106,000 stars in the new Kepler field, we find that Kepler will be able to detect 79 habitable Earth radius planets, assuming that every star posses one.

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In general, Figure 6 provides an excellent example of the benefit of repositioning the Kepler field away from the Galactic plane. Namely, the number of large dwarfs and giant stars (which we have not treated) found in the Galactic plane can be avoided, reducing the false-positive rate, while the number of habitable Earths detected remains nearly the same [@gould2003b].

It is also possible that life may arise on non-Earth-sized worlds. A world the size of Mars might develop quite differently than it has in our own Solar System if it were closer to its parent star, and Jovian worlds in the habitable zone could possess large moons capable of supporting life. Figure 7 shows the number of detections of Jupiter-, Earth-, and Mars-sized habitable worlds, assuming that every star has a planet, against a variety of parameters in the new Kepler field. For completeness, the magnitude limit was set to $m_V\geq 20$. One can see in the mass and radius plots an explicit demonstration of why changing the number of bright, massive main-sequence stars in the field does not effect the number of habitable planet detections. As was mentioned earlier all detections drop to zero beyond $1.4\ M_\odot$. In the plot showing the number of detections binned against the absolute V-magnitude of the parent star we have additionally shown the effect of changing the apparent V-magnitude limit on the distribution of Earth-sized planets.

### Hot and Very Hot Jupiters

Using the HJ and VHJ frequencies provided by [@gould2006b], we were able to make explicit predictions about the number of these worlds that Kepler should find. As mentioned previously, [@jenkins2003] (hereafter JD03) have made their own estimates of Kepler’s HJ detection ability (as noted previously, no VHJs were known at the time, and so were not considered), thus providing a useful point of comparison. As before, we used a limit of S/N$\geq7$, though for Jupiter-sized planets Kepler is magnitude, and not signal-to-noise, limited.

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[ccccccc|c]{} & B & A & F & G & K & M & All\
Jenkins & Doyle: & 11014& 36708& 81085& 61347& 15573& 831& 206558\
Calculated (old field): & 4278& 31370& 76630& 62891& 18243& 1264& 194678\
Calculated (new field): & 338& 5066& 41094& 45568& 13351& 942& 106342

[ccccccc|c]{} & B & A & F & G & K & M & All (HJs)\
Jenkins & Doyle: & 10& 32& 71& 54& 14& 1& 181 (181)\
Calculated (old field): & 4.2& 27.3& 55.5& 33& 8& 0.4& 127 (72.8)\
Calculated (new field): & 0.3& 4.4& 28.6& 23.5& 5.8& 0.33& 62.9 (36)

At the time of the JD03 estimate the Kepler field was centered on its old coordinates. For the purposes of comparison, we therefore examined the old Kepler field in addition to the new field.

Tables 13 and 14 show the distribution of our predictions by spectral type. To facilitate comparison with JD03, we have included not only the final prediction numbers for both old and new Kepler fields, but also the distribution of the calculated total number of stars in each field. As stated above, in both fields we set the magnitude limit of the simulation such that the total number of stars in the fields agreed with the number of expected Kepler targets. In the case of the new field, this is 100,000 stars, while for the old field we matched the number of stars predicted by JD03. This corresponded to a magnitude limit of $m_V=15.7$ in the new field and $m_V=15.9$ in the old field.

Looking at the detections, one can see in Table 14 that our final prediction of $N_{det}=127$ for the old Kepler field is lower than the JD03 prediction. This is primarily a result of the fact that the [@gould2006b] results that we use for $k(p)$ (see Section 2.2) posit a HJ and VHJ frequency approximately $1/3$ of the frequency used in JD03. An important difference, however, between our work and JD03 is our consideration of VHJs, which were not treated by JD03. This has the effect of “doubling” the number of planets we expect Kepler to detect (as demonstrated by the final values in parentheses in Table 14). We therefore arrive at detection rates for VHJs and HJs that are 2/3 that of JD03 for the old Kepler field, and 1/3 that of JD03 in the new Kepler field. For the old field, we predict 54.2 VHJs and 72.8 HJs. For the new field, we predict 26.9 VHJs, and 36.0 HJs.

In the new Kepler field, we see that looking away from the plane of the Galaxy reduces the number of HJs and VHJs detected by half to $N_{det}=62.9$, which matches the overall drop in the star count in the new field. The distribution of detections changes substantially, shifting more towards F-, G-, and K-dwarfs. This is the caused by our model of the Galaxy’s vertical structure, which uses reduced scale heights for earlier stars. Thus, most of the difference in star counts between the old and new fields is caused by fewer B-, A-, and F-dwarfs, leading to proportionally smaller detection numbers around these types of stars.

While the prediction number of detections in the new Kepler field is lower than the corresponding JD03 prediction would be for the same star count, this is accounted for in the same way as in the old field. Indeed, we find that our predictions are consistent with those of JD03, except for the difference in our assumption about the frequency of HJs and VHJs. The eventual results from Kepler therefore offer an excellent opportunity to more robustly determine the actual prevalence of short period Jovian worlds.

All-sky Surveys
---------------

[c|cccc]{} Survey & LSST & SDSS-II & Pan-STARRS & Pan-STARRS Wide\
$m_V$ Limit (Sun-like) & 18.51 & 15.56 & 14.99 & 12.49\
$m_V$ Limit (M-dwarfs) & 23.13 & 20.18 & 19.61 & 17.11\
Observation Time & 10 years & 37.5 days & 5 months & 5 months\
Telescope Diameter & 6.5 m & 2.5 m & 1.8 m & 1.8 m\
Throughput & 0.5 & 0.5 & 0.5 & 0.5\
Field of View & $9.6\ \mathrm{deg}^2$ & $6.25\ \mathrm{deg}^2$ & $7\ \mathrm{deg}^2$ & $7\ \mathrm{deg}^2$\
Area Surveyed & $20,000\ \mathrm{deg}^2$ & $300\ \mathrm{deg}^2$ & $1200\ \mathrm{deg}^2$ & $12,000\ \mathrm{deg}^2$

In addition to wide-field photometric surveys, several current and planned projects aim to repeatedly gather photometry over gigantic swathes of the sky. While not all of them have extrasolar planet detection as one of their goals, the large areas covered by these surveys, together with their long durations (from 3 to 10 years), offer the possibility of detecting hundreds of transiting VHJs and HJs.

We examined three all-sky surveys, LSST, SDSS-II, and Pan-STARRS. The Large Synoptic Survey Telescope (LSST) will be a large 8.4 m telescope, with an effective aperture of 6.4 m, situated in northern Chile. It is scheduled to finish construction sometime in 2014. Using a 9.6 deg$^2$ field of view, the LSST team intends to image 20,000 deg$^2$ of the sky repeatedly over the course of 10 years.[^8]

SDSS-II is the second phase of the Sloan Digital Sky Survey (SDSS) which began in June, 2005 and is scheduled to last for three years until June of 2008. It is composed of three distinct surveys, the Sloan Legacy Survey, SEGUE, and the Sloan Supernova Survey [@sako2005]. The Supernova Survey is the only one of these three that will collect photometry capable of finding transiting planets; the other two survey modes are focused on non-variable objects within and outside the Galaxy. To execute the Supernova Survey, SDSS uses a 2.5 m telescope on Apache Point in New Mexico with a 6.25 deg$^2$ field of view. Over the three year run of SDSS-II, the team expects the Supernova Survey to image about 300 deg$^2$ of the sky near the Galactic poles every two nights for the three months a year the poles are visible. Given the observing cadence the survey expects to use, this translates into a total observing time of 37.5 days.

[|c|ccc|]{} Mag. Limit & Gal. Plane & Gal. Poles & All-Sky Average\
$m_V \leq 20$ & 4.052 & 0.027 & 0.800\
LSST (18.51) & 1.609 & 0.027 & 0.387\
$m_V \leq 18$ & 1.125 & 0.026 & 0.293\
$m_V \leq 16$ & 0.219 & 0.025 & 0.087\
SDSS-II (15.56) & 0.146 & 0.020 & 0.063\
Pan-STARRS (14.99) & 0.083 & 0.016 & 0.041\
$m_V \leq 14$ & 0.029 & 0.009 & 0.017\
Pan-STARRS Wide (12.49) & 0.005 & 0.003 & 0.004\
$m_V \leq 12$ & 0.003 & 0.002 & 0.002

[|c|ccc|]{} Mag. Limit & Gal. Plane & Gal. Poles & All-Sky Average\
LSST (23.13) & 2.7558 & 0.0888 & 0.7764\
SDSS-II (20.18) & 0.2473 & 0.0398 & 0.1136\
$m_V \leq 20$ & 0.2081 & 0.0368 & 0.0989\
Pan-STARRS (19.61) & 0.1422 & 0.0304 & 0.0725\
$m_V \leq 18$ & 0.0257 & 0.0105 & 0.0169\
Pan-STARRS Wide (17.11) & 0.0092 & 0.0048 & 0.0068\
$m_V \leq 16$ & 0.0047 & 0.0015 & 0.0028\
$m_V \leq 14$ & 0.00017 & 0.00015 & 0.00016\
$m_V \leq 12$ & 0.00001 & 0.00001 & 0.00001

Finally, Pan-STARRS - the Panoramic Survey Telescope and Rapid Response System - will be a group of four 1.8 m telescopes atop the summit of Haleakala, Hawaii [@kaiser2004]. The first telescope, PS-1, is nearing completion. When fully operational, Pan-STARRS will embark on a 3 year-long “Medium Deep” survey that will observe 1200 square degrees of the sky near the Galactic poles for a total of five months.[^9] We also looked at a notional “wide” Pan-STARRS survey that looked at 10x the amount of sky as the Medium Deep survey, but used the same amount of observing time.

To characterise each of these surveys and find how many transiting VHJs and HJs they should discover, we first calculated the V-magnitude limits at which each of the surveys will be able to detect transiting Jovian worlds. The complete derivation of this limit is given in Appendix E. Table 15 shows the results of these calculations, as well as the other relevant parameters for each of the surveys. We calculated magnitude limits for detecting giant planets around both Sun-like ($0.8 \leq M_*/M_\oplus \leq 1.2$) stars and M-dwarfs ($0.1 \leq M_*/M_\oplus \leq 0.5$) for S/N$\geq20$.

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To translate these magnitude limits into the number of transiting planets each of the surveys should detect, we used our methodology to calculate the density of VHJs and HJs for various limiting magnitudes, in numbers per square degree, along the Galactic plane, at the Galactic poles, and averaged around the sky. Table 16 shows the resultant values for a series of fiducial magnitude limits, as well as the planet densities at the magnitude limits of our modeled surveys for Sun-like stars; Table 17 shows transiting planet densities around M-dwarfs. Additionally, in Figure 8 we show the transiting planet density around Sun-like stars and M-dwarfs as a function of Galactic latitude. It is interesting to note the dip near the Galactic plane, which occurs as a result of extinction from interstellar dust.

Knowing the angular size of each survey, we use these densities to predict the number of planets that will be seen. For SDSS-II and Pan-STARRS, which are both pointed towards the Galactic poles, we used the polar densities, while for LSST and the wide Pan-STARRS, with their much larger and wide-ranging survey area, we used the all-sky average density. The final predictions for both Sun-like stars and M-dwarfs are shown in Table 18.

We note that these predictions are based on a simpler model than what we have used in other portions of this paper. For instance, we do not include the effects of CCD saturation or sky-background. Nevertheless, our calculations should provide a good expectation for the range of the detection numbers that these all-sky surveys can expect.

The large difference between the number of planets expected to be detected by LSST and the other surveys is caused by two effects. First, SDSS-II and Pan-STARRS cover much smaller areas of the sky to shallower depths than LSST, and second, LSST is not primarily directed towards the Galactic poles, greatly increasing the density of stars for which planet detection is possible. Pan-STARRS Wide, meanwhile, has a substantially brighter magnitude limit than any of the other surveys, which prevents it from detecting as many planets as LSST, despite the similarities in the size of their survey areas. Pan-STARRS Wide does detect more planets than the planned Narrow survey, primarily because the latter “runs up” against the scale height of the Galaxy.

[c|cc]{} & Sun-like (G2) & M-dwarfs (M5)\
LSST & 7740 & 15530\
SDSS-II & 6.0 & 11.9\
Pan-STARRS & 19.2 & 36.5\
Pan-STARRS Wide & 48.0 & 81.6

The real difficulty, however, in managing the 7,740 or 15,530 detections that we expect LSST to make will be properly identifying and separating the false-positives from the actual planetary transit events. Indeed, to date the wide-angle surveys have generated approximately 15 false positive candidates for every actual transiting planet they have seen (D. W. Latham, private communication). If we extend this ratio to LSST, SDSS-II, Pan-STARRS and Pan-STARRS Wide, we would expect them to create 116,000, 90, 290, and 720 false positives, respectively around Sun-like stars, and 233,000, 180, 550, and 1,200 around M-dwarfs, respectively. Working through this many transit candidates (especially in the case of LSST) is an intensive, time-consuming task. We would therefore expect the all-sky surveys, despite the high absolute number of predicted detections, to discover transiting planets at about the same rate as the operational wide-angle surveys, as the identification rate is predominantly limited by the ability to make follow-up observations of candidates to determine whether or not it is a false positive.

Effects of Red Noise
--------------------

As pointed out by [@pont2006], the general mathematical description we have used for the S/N of a transit, equation (\[eq:30\]), assumes uncorrelated errors on the collected photometric data points (white noise). In practice this may not be the case, as phenomenon such as changes in the seeing, airmass, telescope tracking, etc. introduce systematic trends into the observed data that are correlated on the timescales of transits (red noise). This has the effect of substantially decreasing the effective S/N displayed by a transit, thus making detections more difficult. Specifically, [@pont2006] show that to account for red noise, equation (\[eq:30\]) must be modified by adding a term to include a covariance coefficient $C_{ij}$ between the $i^{th}$ and $j^{th}$ measurements, $$\label{eq:600}
\chi^2 = \delta^2 \left( \frac{\sigma^2}{N_{tr}} + \frac{1}{N_{tr}^2} \displaystyle\sum_{i\neq j} C_{ij}\right)^{-1}.$$ This is equation (4) in [@pont2006], using the variables we previously defined for our equation (\[eq:30\]). The authors give a more manageable version on page 8 of their work, which relates the S/N with red noise to the S/N without red noise: $$\label{eq:610}
\chi_r^2 = \chi_w^2\ \left[ 1 + N_k \left(\frac{\sigma_r}{\sigma}\right)^2\right],$$ where $\chi_w^2$ is the white noise $\chi^2$ statistic we calculate in equation (\[eq:30\]), $N_k$ is the number of data points in the single $k^{th}$ transit (which on average is just $N_{tr}/$number of transits), and $\sigma_r$ is the red component of the noise in the data.

[c|ccccccccccccc|c]{} & And0 & Cyg1 & Cas0 & Per1 & UMa0 & CrB0 & Lyr1 & And1 & And2 & Tau0 & UMa1 & Her1 & Lac0 & Total\
VHJs & 0.34 & 0.33 & 0.31 & 0.30 & 0.17 & 0.20 & 0.30 & 0.22 & 0.17 & 0.20 & 0.16 & 0.20 & 0.36 & 3.26\
HJs & 0.25 & 0.22 & 0.21 & 0.21 & 0.11 & 0.13 & 0.19 & 0.12 & 0.07 & 0.11 & 0.11 & 0.14 & 0.25 & 2.15\
Both & 0.59 & 0.55 & 0.52 & 0.51 & 0.28 & 0.33 & 0.49 & 0.34 & 0.24 & 0.31 & 0.27 & 0.34 & 0.61 & 5.41

  --
  --

To properly use this formulation, we must therefore have some idea of what value $\sigma_r$ takes. [@pont2006; @pont2007] infer values of 3 mmag of red noise for typical ground-based surveys, which can be reduced to approximately 2 mmag after applying de-trending algorithms to the data. We therefore took this tentative value of $\sigma_r = 0.002$ and used it to re-simulate the TrES fields to see what effect red noise would have on detection numbers. As noted in [@pont2007], we set a signal-to-noise ratio limit of $S/N_r \geq 12$, which is the $S/N_r$ value of the detection of TrES-1b; TrES-2b has a higher value of $S/N_r = 14$. The results are shown in Table 19. Figure 2 shows (in addition to TrES with only white noise) the distribution of detections using 2 mmag of red noise and $S/N_r \geq 12$.

[c|c]{} $\sigma_r$ & Hab. Earths\
$10^{-4}$ & 14.37\
$5\cdot 10^{-5}$ & 32.53\
$10^{-5}$ & 72.88\
0 & 78.89

Compared to a white S/N limit of $S/N_w \geq 30$, using $\sigma_r = 2$ mmag and $S/N_r \geq 12$ drops the total number of expected detections by about 20%. Looking at the split between HJs and VHJs, the HJ population is disproportionately effected, which is a result of the relatively fewer transits HJs show, lowering the overall S/N of their transit signal. Similarly, Figure 2 shows that red noise also cuts out the detections around higher mass stars, as these have a lower S/N overall.

Without de-trending the data, a red noise of 3 mmag would cause an even further reduction in detections, down 60% as compared to white noise.

We also examined the possible effect that red noise would have on the yield of habitable Earth-radius planets from Kepler mission. Given that Kepler is a space-based mission designed to detect transit signals with fractional depths at the level of $\sim 10^{-4}$, we would expect that the amount of red noise due to instrumental and environmental sources would be small, if it exists at all. However, this is generally difficult to predict before launch. Furthermore, there may exist significant astrophysical red noise due to intrinsic variability.

We therefore estimated the expected Kepler yield for a set of three values for $\sigma_r$, assuming that every star has an Earth-radius planet with $a={\rm AU}(L/L_\odot)^{1/2}$ (Table 20). For a relatively large amount of red noise at $\sigma_r =
10^{-4}$, we find the number of habitable Earths-radius is drastically reduced from our baseline (uncorrelated noise) prediction, from 78.89 down to 14.37. This is not surprising, since Earth passing in front of a Sun-like star has a transit depth of only $8.5\cdot
10^{-5}$. For this amount of red noise, Kepler is restricted to detecting Earth-like planets only around K- and M-dwarfs. This shifts the expected distribution of host stars towards dimmer magnitudes (Figure 6). For smaller values of $\sigma_r = 5\cdot10^{-5}$ and $\sigma_r =
10^{-5}$, the number of habitable Earths detected is reduced to 32.53 and 72.88, respectively. Both of these estimates also assume that every target star has an Earth-sized planet in the habitable zone, and therefore must be viewed as upper limits.

Thus we conclude that Kepler’s yield of habitable zones of habitable planets will be not be strongly affected by red noise, provided that the red noise is $\la 10^{-5}$ on the timescales of the transits, which is $\sim 13~{\rm hrs}(M/M_\odot)^{1.75}$ for solar-type stars. If the red noise is found to be larger than this, it may compromise the primary Kepler science by significantly reducing the number of expected habitable planet detections. For example, if the red noise is $\sigma_r = 10^{-4}$, and the frequency of habitable planets with $r=R_\oplus$ is $\sim 20\%$, there is a $\sim 5\%$ chance that Kepler will not detect any. The effects of red noise on the habitable planet yield can be mitigated by including a larger number of K and M dwarfs in the target sample. Of course, it will also be possible to extrapolate the frequency of shorter-period or more massive planets, where the effects of red noise are mitigated, into the habitable zone.

Conclusion
==========

In this paper we have developed a method to more accurately predict the number of transiting planets detectable by wide-field ground- and space-based photometric surveys. We have taken into account such factors as the frequency of gas giants around main sequence stars, the probability of transits, stellar density changes from Galactic structure, and the effects of interstellar extinction. We adopt a S/N detection threshold criterion to approximate the detectability of planetary transits.

To test our model, we compared our predictions to two operating photometric surveys, TrES and XO. For both, we calculated the number of statistically expected detections, as well as the population characteristics of the host stars. For TrES, we overestimate the number of detections (thirteen versus the four actual detections), while for XO our simulations yield approximately the actual number of detections (two predicted versus three actual). The exact reason for the discrepancy between our predictions and the actual TrES results could be one of many. As we discuss in the TrES section, from the properties of the known TrES planets, it may be that the survey has a brighter magnitude cut off than the one we use in our fiducial case. If this were the case, and we changed the magnitude limit from $m_R\leq13$ to $m_R\leq12$, the number of expected detections would drop to 8.16 from 13. Red noise also has the potential to substantially affect TrES; a specific estimate of the magnitude of red-noise in the TrES survey would be worthwhile.

Indeed, it is generally hard to draw specific statistical conclusions from the comparison of our results and those of the transit surveys because they typically do not adopt the strict detection criteria we have used. Promising planet candidates are often followed up even if they are beyond the stated S/N or magnitude limits of the survey. Understanding and quantifying how the survey teams select candidates is vital to appropriately deriving the statistical properties of extrasolar planets. Indeed, Tables 4-6 and 9-11 demonstrate that the actual predictions depend crucially on the specific magnitude and S/N threshold used.

As more transiting planets are discovered, these statistical properties are increasingly becoming the frontier of research, shifting the focus of the field away from individual detections. Given that transiting planets are so far one of the only ways that we may infer the radius of extrasolar planets and their exact orbital properties, the statistical characteristics of this group of objects is one of the only foreseeable ways that the areas of planetary interiors, system dynamics, migration, and formation will acquire more data.

In the future, besides using the preceding formalism to design more efficient transit surveys, we would hope that a more systematic approach towards transit surveys will allow this model to be used to make more specific statistical comparisons. As the number of known transiting planets grows, we would also expect that this formalism will be used to test different distributions of planet frequencies, periods, and radii against those observational results. This will allow us to better understand the statistics of extrasolar planetary systems, improve our ability to find new planets, and help to understand the implications of the ones that have already been detected.

We would like to thank the referee for a helpful report. We would also like to thank Dave Charbonneau and Francis O’Donovan for providing useful discussions and information about the TrES survey, Peter McCullough for his correspondence about the XO survey, and Scott Schnee for discussions about interstellar dust. TGB would like to thank Dave Latham and Josh Winn for helpful discussions and encouragement.

Photon Luminosity of a Source Star
==================================

The photon luminosity in a bandpass of a given star of a given mass is calculated exactly by dividing the Planck radiation law for wavelength by the energy of a photon at that wavelength, multiplying the result by the surface area of the star, and then integrating over the wavelength range of the relevant bandpass: $$\label{eq:91}
{\Phi_{\lambda}(M) = \int_{\lambda_{min}}^{\lambda_{max}} \frac{B_\lambda (T_*)}{E_\lambda}} 4\pi^2 R_*^2 \, d\lambda,$$ Where $B_\lambda (T_*)$ is the Planck Law, and $E_\lambda = \frac{hc}{\lambda}$ is the energy of a photon. Note that this assumes that stellar radiation can be approximated as a blackbody, a crude assumption that still suffices for our purposes, as we show below.

Because the widths of the bandpasses in the visible and near-infrared that we are considering are generally narrow with respect to $\frac{dB_{\lambda}}{d\lambda}$, we can safely approximate this integral by the product of the central wavelength of each bandpass $\lambda_c$ and a bandpass width $\Delta \lambda$: $$\label{eq:101}
{\Phi_{\lambda}(M) = \frac{B_\lambda (T_*,\lambda_c) \Delta \lambda}{E_{\lambda_c}}} 4\pi^2 R_*^2 = \frac{8\pi^2 c R_*^2 \lambda_c^{-4} \Delta \lambda}{\mathrm{exp}[hc / \lambda_c k T_*(M)]-1},$$ which, once we select a bandpass, only leaves the temperature of the source star $T_*(M)$ to be determined.

Using the Stefan-Boltzmann Law, we may relate the temperature of the source star to its bolometric luminosity and radius: $$\label{eq:111}
T_* = \left(\frac{L_{bol,*}}{4\pi \sigma R_*^2}\right)^{\frac{1}{4}},$$ where $R_*$ and $L_{bol,*}$ are the radius and the bolometric luminosity of the source star, respectively. We chose to treat both as functions of the mass of the source star, as this is an independent variable that allows the use of the present day mass function to describe the general stellar population.

Using stellar data from [@harmanec1988] and [@popper1980], we fit broken power laws for both the $R_*(M)$ and $L_{bol,*}(M)$ relations. For the mass-radius relation, we fit for two breaks in the power law, while the fitting for the mass-luminosity relation was accomplished using only one break. The relations that we thus found are:

$$\begin{aligned}
\label{eq:121}
R_*(M) = R_\odot \left(\frac{M}{M_\odot} \right)^\alpha\ \ \ \alpha&=&0.8\ \mbox{ for } M_* \leq 1,\\
\alpha&=&1.5\ \mbox{ for } 1 < M_* \leq 1.5, \nonumber \\
\alpha&=&.54\ \mbox{ for } 1.5 < M_*, \nonumber\end{aligned}$$

$$\begin{aligned}
\label{eq:131}
L_{bol,*}(M) = L_{bol,\odot} \left(\frac{M}{M_\odot}\right)^\beta\ \ \ \beta&=&2.5\ \mbox{ for } M_* \leq 0.4, \\
\beta&=&3.8\ \mbox{ for } 0.4 < M_*. \nonumber\end{aligned}$$

Equations (\[eq:121\]) and (\[eq:131\]) are plotted against the [@harmanec1988] and [@popper1980] data in Figure 9.

The real test, however, is how well these relations allow for the calculation of the effective temperature of a star at a given mass using (\[eq:111\]), as this feeds directly into the determination of the photon luminosity in a given band. Figure 10 shows a plot of our mass vs. temperature relation along with data points from [@harmanec1988] and [@popper1980]. Additionally, we also show the analytical relation provided by [@harmanec1988]. One can see that our formulation of (\[eq:121\]) and (\[eq:131\]) affords a close approximation to both the data and the relation provided by [@harmanec1988].

Substituting back into (\[eq:101\]) allows us to calculate the photon luminosity of the source star, as well as the absolute magnitude of the source star in various bandpasses. If we compare these calculated absolute magnitudes in different bands for various stellar masses against empirical magnitudes for those same masses (Figures 11 and 12) we can see that the two relations are similar, to within half a magnitude, especially in longer wavelengths. Shorter wavelength bands such as U, B, and to a certain extent V, have calculated magnitudes that diverge from the empirical relation below about a solar mass, which is an unfortunate effect of using blackbody relations to determine the photon luminosity, as in these higher energy bands stellar spectra increasingly do not look like black bodies. Nevertheless, the blackbody assumption is sufficiently accurate for our purposes.

Analytic Window Probability
===========================

Roughly, if we ignore the effects of aliasing, we may say that the window probability of seeing a minimum of $N_{tr,min}$ transits of a planet with orbital period $p$ over the course of $N_n$ nights may be broken up into four different regimes.

First, if the average number of transits during the total time during which the target star is observed exceeds $N_{tr,min}$, then it is statistically “certain” that the transit will occur during the times of observation .

Second, it may be that the entire length of the survey is long enough that there are at least $N_{tr,min}$ transits, but that during the total time spent observing the average number of transits is less than that required for detection. Then one is not certain to detect $N_{tr,min}$ transits, and there is a decreasing window probability (for increasing planet period).

Third, one may have $N_{tr,min}$ transits during an observing run, but only if the first transit occurs sufficiently early during that particular run. The probability of detecting the system then decreases at an even greater rate as the orbital period increases.

Finally, the orbital period of system may be so great that there is no possible way in which one could see the minimum number of transits required for a detection, and $P_{window}=0$. Mathematically then, $$\begin{aligned}
\label{eq:510}
P_{window} &=& 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } 1 \leq \frac{<N_{tr}>}{N_{tr,min}},\\
&=& \left(\frac{<N_{tr}>}{N_{tr,min}}\right)^2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } \frac{<N_{tr}>}{N_{tr,min}} < 1 \leq \frac{(N_n-1)\Lambda+T_{n}}{N_{tr,min}P}, \nonumber \\
&=& \left(\frac{<N_{tr}>}{N_{tr,min}}\right)^2 \left(\frac{(N_n-1)\Lambda+T_{n}-(N_{tr,min}-1)P}{P}\right)^\frac{1}{2} \nonumber \\
&\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } \frac{(N_n-1)\Lambda+T_{n}}{N_{tr,min}P} < 1 \leq \frac{(N_n-1)\Lambda+T_{n}}{(N_{tr,min}-1)P}, \nonumber \\
&=& 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } \frac{(N_n-1)\Lambda+T_{n}}{(N_{tr,min}-1)P} < 1, \nonumber\end{aligned}$$ where $\Lambda = 1$ day. $<N_{tr}>$ is the average number of transits that will be seen during the time spent observing, and is given simply by: $$\label{eq:520}
<N_{tr}> = \frac{N_g T_n}{P}.$$ $N_g$ is defined as the total number of “good” observing nights in a particular run, and may mathematically be described as $N_g=N_n-N_{lost}$, where $N_{lost}$ is the number of nights lost due to weather, technical problems, or other issues. $T_n$ is the length of each night’s observing time. Alternatively, one may simply view the quantity $N_g T_n$ as the total amount of good observing time in a given run.

To test the accuracy of our formulation, we integrated and found the average window probability for both the analytic and exact formulations linearly averaged over several period ranges (1-3, 3-5, and 5-10 days). These are shown as a function of total observing nights under different viewing scenarios in Figure 13.

One can see that as a result of the analytic solution not taking into account aliasing, there is divergence between the two solutions as the analytic form approaches unity, as well as in the cases requiring a minimum of three transits.

The Present Day Mass Function
=============================

A review of the literature shows that while there is agreement that the PDMF beyond $1M_\odot$ can be treated as a power law with a slope of $\alpha \approx5.3$, below a solar mass this agreement breaks down. This is because the low-mass end of the mass-function is unfortunately not as well constrained as it is for higher mass stars, reflecting the difficulty of observing dim M-dwarfs, as well as uncertainties in the mass-luminosity function [@henry1999].

For instance, [@reid2002], who used the Hipparcos data sets along with their own data from the Palomar/Michigan State University (PMSU) survey to create a volume-limted sample out to 25 pc, propose a traditional power law formulation of the PDMF: $$\label{eq:350}
\Psi(M)=\frac{dn}{dM}= k_{PDMF} M^{-\alpha},$$ with $\alpha = 5.2\pm0.4$ for $M>M_\odot$, and $\alpha = 1.35\pm0.2$ for $M\leq M_\odot$, and $k_{PDMF}$ as the correct normalization.

In a recent review, meanwhile, [@chabrier2003] argues for a log-normal function below a solar mass: $$\label{eq:360}
\Psi(M)=\frac{dn}{dM}= \frac{1}{M(\ln10)} A\ exp\left[\frac{-(\log M-\log M_c)^2}{2\sigma^2}\right],$$ with $A = 0.158_{-0.046}^{+0.051}$, $M_c = 0.079_{-0.016}^{+0.021}$, and $\sigma = 0.69_{-0.01}^{+0.05}$.

For comparison, if we normalize both to the [@scalo1986] normalization for 5 Gyr of $(dn/dM)_{1M_\odot} = k_{PDMF} = 0.019 M_\odot^{-1} \mathrm{pc}^{-3}$ and plot the two functions against each other (Figure 14), we can clearly see the divergence both numerically and functionally of these two forms below a solar mass. Interestingly however, for most surveys this divergence makes little difference in the final number of predicted detections: typically Reid’s relation gives only 3% fewer detections than Chabrier’s does, if both are normalized at a solar mass. That this difference is so small comes from the fact that most wide-angle surveys are not sensitive to detections around stars with $\log(M)<-0.15$. Hence, the region of greatest uncertainty in the PDMF is left unseen by these surveys.

Since the difference between the two relations is small, and because the data set used by [@reid2002] is much more recent than that used by [@chabrier2003], we therefore chose to use the [@reid2002] form of the PDMF.

For the purpose of comparing the Reid and Chabrier PDMFs in Figure 14, we normalized both to the value given by [@scalo1986]. Going forward, however, we change the normalization to the integrated mass density found observationally by [@reid2002] of $0.032 M_\odot \ \mathrm{pc}^{-3}$.

Galactic Structure
==================

The values for the scale length and scale height of the Galactic disk that we use in our modeling are $h_{d,*} = 2.5\ \mathrm{kpc}$ and $h_{z,*}$ as a function of absolute magnitude. Specifically, we calculate the vertical scale height of the disk as: $$\begin{aligned}
\label{eq:370}
h_{z,*} &=& 90\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } M_V \leq 2,\\
        &=& 90 + 200 \left(\frac{M_V-2}{3}\right)\ \mbox{ for } 2 < M_V < 5, \nonumber \\
        &=& 290\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } 5 \leq M_V, \nonumber\end{aligned}$$ which is a modified form of the relation found in [@bahcall1980], who instead used a value of 320 pc for the scale height of stars with $5 \leq M_V$.

The large uncertainties in the value of the scale length of the disk reflects the uncertainty in the relevant literature. In their initial paper, [@bahcall1980] use $h_{d,*} = 2.5 - 3.0\ \mathrm{kpc}$. Since then various sources have proposed an even $h_{d,*} = 3.0\ \mathrm{kpc}$ [@kent1991], have covered all the bases by using several values between $h_{d,*} = 2.0 - 3.0\ \mathrm{kpc}$ [@olling2001], and have given the very exact result of $h_{d,*} = 2.264\pm.09\ \mathrm{kpc}$ [@drimmel2001].

To see what effect different scale lengths have on our final detection numbers, we simulated observations of a $6^\circ\times6^\circ$ field along the Galactic plane at intervals of $10^\circ$ in Galactic longitude, as well as fixing the Galactic longitude at certain points ($0^\circ$, $90^\circ$, and $180^\circ$) and varying the latitude of the fields. Figure 15 shows these results for scale lengths of 2, 2.5, and 3 kpc. At the extremes (directly towards or directly away from the Galactic core) the 2 and 3 kpc scale lengths differ by only 0.15 detections per field. We chose 2.5 kpc not only to split this difference, but also because more recent results place the scale length much lower than earlier 3 kpc estimates \[e.g. - [@drimmel2001]\].

In the case of the scale height of the Galactic disk, the literature divides between those who derive the scale height from optical star counts and those who rely on near-infrared observations. In the optical case, the scale height is presented as a function of absolute visual magnitude that varies from 90 pc to values between 250-320 pc. The near-infrared observations, on the other hand, lead to one single value of the scale height, generally around 290 pc. This happens because measurements in the infrared are dominated by older M-dwarfs that have dispersed in an even fashion away from the Galactic plane.

To compare these different estimates, we have plotted the number of detections in a $6^\circ\times6^\circ$ field along $l=90^\circ$ as a function of Galactic latitude in Figure 16. We used two of the optically-derived scale height relations, as well as a constant scale height of 290 pc. One of the optical relations, the Bahcall model, is given by $$\begin{aligned}
\label{eq:375}
h_{z,*} &=& 90\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } M_V \leq 2,\\
        &=& 90 + 230 \left(\frac{M_V-2}{3}\right)\ \mbox{ for } 2 < M_V < 5, \nonumber \\
        &=& 320\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } 5 \leq M_V.\nonumber\end{aligned}$$

The other optical model is taken from Allen’s Astrophysical Quantities (AAQ) [@cox2000] and is: $$\begin{aligned}
\label{eq:380}
h_{z,*} &=& 90\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } M_V, \leq 2\\
        &=& 90 + 160 \left(\frac{M_V-2}{3}\right)\ \mbox{ for } 2 < M_V < 5, \nonumber \\
        &=& 250\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{ for } 5 \leq M_V. \nonumber\end{aligned}$$

Also plotted in Figure 16 are the number of detections above and below $M_V=5$ for the three different scale height relations, as well as the residuals of the constant 290 pc and the AAQ relation against the Bahcall formulation. The exact choice of how the Galactic scale height is treated leads to rather different final predictions. Specifically, while the optical relations are similar, the constant 290 pc scale height from near-infrared observations leads to substantially more detections around stars with $M_V<5$.

For our predictions, we have adopted the form of the [@bahcall1980] and Allen’s Astrophysical Quantities scale height relations. We find the gradated scale height in these models appealing, because it accounts for the fact that more luminous stars with shorter lifetimes should not be dispersed far away from star-forming regions in the plane of the Galaxy. The constant 290 pc relation, because it is based on near-infrared data, reflects the sensitivity of those observations to less massive stars that are brighter in longer wavelengths. Nevertheless, we have used the 290 pc result as our value for the scale height of late-type stars with $5<M_V$, in contrast to the values used in the AAQ and Bahcall models.

All-Sky Surveys’ Limiting Magnitudes
====================================

As per equation (\[eq:30\]), we have $$\label{eq:440}
\sqrt{\chi^2} = S/N = (N_{tr})^{1/2}\ \frac{\delta}{\sigma}.$$

For the all-sky surveys that we are considering, the number of in-transit data points observed is simply the number of data points acquired in each field multiplied by the fraction of time a planet will spend in transit during its orbit: $$\label{eq:450}
S/N = \left(\frac{t_{obs} \Theta_{FOV}}{t_{exp} \Omega_{survey}} \frac{R_*}{\pi a} \right)^{1/2} \frac{\delta}{\sigma},$$ where $t_{obs}$ is the total observation time of the survey, $\Theta_{FOV}$ is solid angle of the survey telescope’s field of view, and $\Omega_{survey}$ is the total solid angle of the sky covered by the survey.

Rearranging: $$\label{eq:460}
\sigma = \left(\frac{t_{obs} \Theta_{FOV}}{t_{exp} \Omega_{survey}} \frac{R_*}{\pi a} \right)^{1/2} \frac{\delta}{S/N}$$

For Poisson, source-noise dominated errors, we have $$\label{eq:470}
\sigma = N_{source}^{-1/2}$$ where $N_{source}$ is the total number of photons from the source collected during $t_{exp}$. We can then write, $$\label{eq:480}
\sigma = \left(\frac{t_{exp}\ e_\lambda}{t_{exp,0}}\right)^{-1/2} \left(\frac{D_{scope}}{D_{scope,0}}\right)^{-1} \sigma_0\ 10^{0.2(m_V-m_{V,0})}$$ where $\sigma_0$ is the fiducial fractional photometric uncertainty achieved for a star with apparent magnitude $m_{V,0}$ using a telescope with diameter $D_{scope,0}$ and an exposure time of $t_{exp,0}$. We have defined $e_\lambda$ as the efficiency of the observing setup relative to the fiducial case. If we rearrange equation (\[eq:480\]) to $$m_V = 5\log{\left[ \left(\frac{t_{exp}\ e_\lambda}{t_{exp,0}}\right)^{1/2}\left(\frac{D_{scope}}{D_{scope,0}}\right) \frac{\sigma}{\sigma_0} \right]} + m_{V,0}$$ we may now substitute in equation (\[eq:470\]) for $\sigma$. Doing so, and simplifying, we find that $$m_V = 5\log{\left[ \left(\frac{t_{obs}\ e_\lambda}{t_{exp,0}} \frac{\Theta_{FOV}}{\Omega_{survey}} \frac{R_*}{\pi a}  \right)^{1/2} \frac{D_{scope}\ \delta}{D_{scope,0}\ \sigma_0\ S/N} \right]} + m_{V,0}$$ Which is the limiting magnitude around which an all-sky survey will detect planets.

Aigrain, S., & Pont, F. 2007, , 378, 741

Backer, D. C., Foster, R. S., & Sallmen, S. 1993, , 365, 817

Bahcall, J. N., & Soneira, R. M. 1980, , 44, 73

Bakos, G., Noyes, R. W., Kov[á]{}cs, G., Stanek, K. Z., Sasselov, D. D., & Domsa, I. 2004, , 116, 266

Bakos, G. [Á]{}., et al.  2007, , 656, 552

Barnes, J. W. 2007, , 119, 986

Beaulieu, J.-P., et al. 2006, , 439, 437

Bond, I. A., et al. 2004, , 606, L155

Borucki, W., et al. 2004, ESA SP-538: Stellar Structure and Habitable Planet Finding, 177

Brown, T. M., & Charbonneau, D. 1999, Bulletin of the American Astronomical Society, 31, 1534

Brown, T. M. 2003, , 593, L125

Burke, C. J., et al.  2007, , 671, 2115

Burke, C. J. 2008, , 679, 1566

Chabrier, G. 2003, , 115, 763

Charbonneau, D., Brown, T. M., Noyes, R. W., & Gilliland, R. L. 2002, , 568, 377

Charbonneau, D., Brown, T. M., Burrows, A., & Laughlin, G. 2007, Protostars and Planets V, 701

Cox, A. N. 2000, Allen’s Astrophysical Quantities, 4th ed. Publisher: New York: AIP Press; Springer, 2000. Editedy by Arthur N. Cox.  ISBN: 0387987460

Cumming, A., Butler, R. P., Marcy, G. W., Vogt, S. S., Wright, J. T., & Fischer, D. A. 2008, , 120, 531

Drimmel, R., & Spergel, D. N. 2001, , 556, 181

Dunham, E. W., Mandushev, G. I., Taylor, B. W., & Oetiker, B. 2004, , 116, 1072 Duquennoy, A., & Mayor, M. 1991, , 248, 485 Fischer, D. A., & Valenti, J. 2005, , 622, 1102

Fitzpatrick, E. L.  1999, , 111, 63

Fressin, F., Guillot, T., Morello, V., & Pont, F. 2007, , 475, 729

Gaudi, B. S. 2005, , 628, L73

Gaudi, B. S., et al.  2008, Science, 319, 927

Gillon, M., Courbin, F., Magain, P., & Borguet, B. 2005, , 442, 731

Gould, A., & Morgan, C. W. 2003, , 585, 1056

Gould, A., Pepper, J., & DePoy, D. L. 2003, , 594, 533

Gould, A., et al. 2006a, , 644, L37

Gould, A., Dorsher, S., Gaudi, B. S., & Udalski, A. 2006b, Acta Astronomica, 56, 1

Halbwachs, J. L., Mayor, M., Udry, S., & Arenou, F. 2003, , 397, 159

Harmanec, P. 1988, Bulletin of the Astronomical Institutes of Czechoslovakia, 39, 329

Henry, T. J., Franz, O. G., Wasserman, L. H., Benedict, G. F., Shelus, P. J., Ianna, P. A., Kirkpatrick, J. D., & McCarthy, D. W., Jr. 1999, , 512, 864

Horne, K. 2003, ASP Conf. Ser. 294: Scientific Frontiers in Research on Extrasolar Planets, 294, 361

Jenkins, J. M., & Doyle, L. R. 2003, , 595, 429

Johns-Krull, C. M., et al. 2007, ArXiv e-prints, 712, arXiv:0712.4283

Kaiser, N. 2004, , 5489, 11

Kent, S. M., Dame, T. M., & Fazio, G. 1991, , 378, 131

Latham, D. W., Stefanik, R. P., Torres, G., Davis, R. J., Mazeh, T., Carney, B. W., Laird, J. B., & Morse, J. A. 2002, , 124, 1144

Lovis, C., et al. 2006, , 441, 305

Marcy, G., Butler, R. P., Fischer, D., Vogt, S., Wright, J. T., Tinney, C. G., & Jones, H. R. A.  2005a, Progress of Theoretical Physics Supplement, 158, 24

Marcy, G. W., Butler, R. P., Vogt, S. S., Fischer, D. A., Henry, G. W., Laughlin, G., Wright, J. T., & Johnson, J. A. 2005b, , 619, 570

McCullough, P. R., Stys, J. E., Valenti, J. A., Fleming, S. W., Janes, K. A., & Heasley, J. N. 2005, , 117, 783

McCullough, P. R., et al. 2006, , 648, 1228

Olling, R. P., & Merrifield, M. R. 2001, , 326, 164

Pepper, J., Gould, A., & Depoy, D. L. 2003, Acta Astronomica, 53, 213

Pepper, J., & Gaudi, B. S. 2005, , 631, 581

Pepper, J., et al.  2007, , 119, 923

Pollacco, D. L., et al. 2006, , 118, 1407

Pont, F., Zucker, S., & Queloz, D. 2006, , 1146

Pont, F., et al. 2007, Transiting Extrapolar Planets Workshop, 366, 3

Popper, D. M. 1980, , 18, 115

Reid, I. N., Gizis, J. E., & Hawley, S. L. 2002, , 124, 2721

Robin, A., & Creze, M. 1986, , 157, 71

Sako, M., et al. 2005, ArXiv Astrophysics e-prints, arXiv:astro-ph/0504455

Santos, N. C., Israelian, G., & Mayor, M. 2004, , 415, 1153

Scalo, J. M. 1986, Fundamentals of Cosmic Physics, 11, 1

Struve, F. G. W. 1847, St. Petersburg : Tip. Acad. Imper., 1847; IV, 165 p. ; in 8.; DCCC.4.211

Udalski, A., et al.  2005, , 628, L109

Udry, S., Mayor, M., Queloz, D., Naef, D., & Santos, N. 2000a, From Extrasolar Planets to Cosmology: The VLT Opening Symposium, 571

Udry, S., et al. 2000b, , 356, 590

Udry, S., & Santos, N. C. 2007, , 45, 397

von Braun, K., & Ciardi, D. R. 2007, ArXiv e-prints, 711, arXiv:0711.4256

Wolszczan, A., & Frail, D. A. 1992, , 355, 145

Young, A. T. 1967, , 72, 747

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[^1]: As of June 2008. See the Extrasolar Planets Encyclopedia at http://exoplanet.eu for an up-to-date list.

[^2]: We consider red noise later, in Section 4.5.

[^3]: The distribution of $\cos i$ is uniform, which follows from the assumption that planetary systems are randomly oriented. The distribution of $\cos i$ may be derived by considering the normalized angular momentum vector of the planetary orbit, which is free to point in any direction, and thus has a parameter space that forms an imaginary sphere around the parent star. Particular values of $\cos i$ are “hoops” along the surface of this sphere, and the ratio of their area to that of the total sphere is the probability that $\cos i$ will take that value.

[^4]: http://www.inscience.ch/transits/

[^5]: http://www.astro.caltech.edu/ ftod/tres/sleuthObs.html

[^6]: 00 hr, 04 hr, 08 hr, 12 hr, 16 hr, and 20 hr.

[^7]: http://kepler.nasa.gov/

[^8]: http://www.lsst.org

[^9]: The remaining balance of the 3 years will be taken up by the other Pan-STARRS science modes; see\
    http://pan-starrs.ifa.hawaii.edu/project/reviews/PreCoDR/...\
    documents/PSCoDD\_1\_4\_design\_reference\_mission.pdf
