Predicting the Detection Rates of Transiting Hot Jupiters and Very Hot Jupiters in Wide-field Photometric Surveys

abstraks:

Seven years after the first sighting of a transiting planet around HD209458, it now appears that early predictions of the detection rate of wide-field photometric surveys substantially overestimated the rate at which extrasolar transiting planets have actually been seen. In this paper, we use a $\chi^2$ test to develop a mathematical formalism that allows for a more accurate prediction of the number of transit detections that a given photometric survey should see. We have taken into account such factors as the frequency of gas giants around main sequence stars, the probability of transits occurring, stellar density changes from galactic structure, and the effects of interstellar extinction. We then apply our method to the Trans-Atlantic Exoplanet Survey (TrES), a currently ongoing ground-based search for transiting planets, and the space-based Kepler mission, due to launch in 2008. For both surveys, we offer a prediction for the total number of hot Jupiters and very hot Jupiters that each can expect to detect, as well as the expected distribution of the detections across a number of parameters.

1. Introduction
Since the first detection of a planetary-mass object in 1989 (Latham et al. 1989) and
the explicit detection six years later of a Jupiter-sized planet in orbit about 51 Peg (Mayor
& Queloz 1995), the number of known extrasolar planets has ballooned to almost 190. By
any standard of measure, these discoveries have dramatically altered our conception of the
universe. Indeed, until the discovery of the planet around 51 Peg, one could plausibly argue
that planet formation was rare, and that our Solar System was one of a few (if not the
only) planetary systems in the galaxy. Instead, we are confronted with the prospect that planetary formation is a relatively routine occurrence during the formation of stars; with
all that this implies about the chances of finding a world similar to Earth elsewhere in the
cosmos.

In addition to confirming the frequency of planetary systems, the properties displayed
by the discovered extrasolar planets have forced a rethinking of modern planet formation
and dynamical interaction theories. Specifically, the presence of large numbers of jovian
planets orbiting very close to their parent stars (so called “hot Jupiters”) have necessitated
the creation of new models of planetary migration that allow jovian-worlds formed outside
the “snow-line” to spiral in closer to their parent star (Lin et al. 1996; Ward 1997).
To date, there are three reliable ways by which extrasolar planets have been detected.
The first is the Radial Velocity (RV) method, which uses the Doppler shift of observed
stellar spectra to look for periodic variations in the target star’s radial velocity. By then
determining the mass of stellar target (or otherwise estimating it from models), the observed
radial velocity curve and velocity semi-amplitude can then be used to directly calculate the
inclination-dependent mass (M sin i) of the companion object. While this gives only the
minimum mass of these objects, the large number of detections of systems whose unseen
companion has a mass on the order of 1MJup sin i statistically ensures that the majority of
these are planetary bodies.

Currently, 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. Capable
of radial velocity measurements precise to under 1 ms?1 for extended periods of time, it is able to detect planets with masses on the order of 3 to 4M© in a wide range of
orbits. Unfortunately, smaller planets will be increasingly difficult to detect using RV
methods. 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 the system
mentioned in the previous footnote) through extensive knowledge of the seismology of the
the target star and long integration times that allow the stellar variability to average out.
Another technique for detecting extrasolar planets is microlensing surveys. Microlensing
of a star occurs when a massive object passes through the line of sight of the observer to the
star. The gravity of the object acts as a lens on the light emitted by the star, which causes
the star to become momentarily brighter as more light is directed towards the observer.
The most well known microlensing survey is the Optical Gravitational Lensing Experiment
(OGLE), which was originally conceived as a narrow-field search for distant microlensing
events caused by dark matter. It was quickly realized, however, that the massive amount
of photometry generated by OGLE was also conducive to searching for planet-induced
microlensing. Indeed, in the last two years, four planets have been discovered by monitoring
microlensing events detected by the OGLE collaboration: OGLE-2003-BLG-235 (Bond et
al. 2004), OGLE-2005-BLG-071 (Udalski et al. 2005), OGLE-2005-BLG-390 (Beaulieu et
al. 2006), and OGLE-2005-BLG-169 (Gould et al. 2006a).

The third method by which extrasolar planets have been discovered is the Transit
Detection technique, which looks for the periodic dimming of a target star that occurs
when an orbiting planet passes in front of the stellar disk. This requires a very specific set
of orbital characteristics to yield a transit visible from Earth (the orbital plane has to be
aligned to within a few degrees of the line of sight), and therefore transiting planets are
expected to be a rarer sight than planets detectable through RV observations. Nevertheless, a transiting extrasolar planet offers the opportunity to not only determine the mass of
that planet (assuming that follow-up RV work is feasible) since the i in M sin i is now
measurable, but also the planetary radius. This allows for a deeper understanding of
not only the composition of extrasolar planets, but also the dynamics of the interiors of
jovian worlds. Additionally, and unlike RV surveys, transiting planets should be readably
detectable down to 1R© and beyond, even for relatively long periods. NASA’s Kepler
mission, which is presently scheduled for launch in the fall of 2008, will consist of a
space-based telescope whose primary mission is to search for transiting planets of just this
size within the habitable zones of main-sequence stars.

The first transiting planet was discovered around HD209458 in late 1999. It was
initially identified as an extremely short-period planetary system by RV measurements in
the spring of that year; it was its short (3.5 day) period and the relative brightness of the
parent star (mV = 7.65) that prompted photometric observations of HD209458 in the hope
that the orbital geometry was sufficient for an observable transit (Charbonneau et al. 2000;
Henry et al. 2000). Since that time, eight more transiting planets have been discovered.
Two of them, HD189733b (Bouchy et al. 2005b) and HD149026b (Sato et al. 2005), were
identified first by spectroscopic RV surveys; similar to how HD209458b was found. Five of
the known transiting extrasolar planets were discovered in the OGLE data sets,2 proving
the efficacy of that project to detect not only microlensing planets, but transits as well.
Unfortunately, because OGLE concentrates its observations on relatively faint (mV ¼ 16)
stars, follow-up spectroscopic observation of the OGLE planets is extremely difficult and
has only been done for a portion of the OGLE planets. In the end, only one transiting planet, TrES-1 (Alonso et al. 2004), has been detected

by a photometric survey specifically designed to find planets around bright stars that allow
for spectroscopic follow-up, despite there being over 20 such searches currently underway.
This dearth of results is in stark contrast to the general expectation following the discovery
of HD209458b that wide-field photometric transit surveys would discover literally thousands
of transiting extrasolar planets (Horne 2003).

The reasons for the low number of transit detections relative to expectations are varied
and complex. Partly, this is because the frequency of planets in close orbits about their
parent stars (the planets most likely to transit) is much lower than was originally expected.
Recent examinations of the results from the OGLE-III field by Gould et al. (2006b) indicate
that the frequency of short-period jovian-worlds is on the order of 1
400 , not 1
100 as is often
assumed from looking at RV surveys. They point out that most spectroscopic planet
searches are usually intentionally biased by the planet-hunting observer towards targeting
metal-rich stars, which are expected to have more planets than the average solar-metallicity
star.

Furthermore, many of the older estimates for the number of expected detections
assumed that all the stars with a photometric precision of less than 1% in a given field
would allow for successful detections, provided the orbital geometry yielded a transit (itself
a roughly 10% chance). However, not all of the stars in a field are on the main-sequence.
Evolved stars constitute a non-negligible population that are capable of allowing precise
photometric measurements, but any planets in orbit around such stars are beyond the
detectable reach of present-day photometric surveys.

Also, further reflection reveals that the photometric precision afforded by a star may
not be the proper metric by which to judge the detection abilities of a transit survey. After
all, one may observe three transits of a star with a precision of 1%, but if one only has three in-transit data points on a 2% deep transit, then the later identification of that event
as a transit will be difficult to say the least. Therefore, instead of photometric precision, a
better detection metric is the signal-to-noise (S/N) ratio of a transit.

In this paper, we use the S/N of transits to statistically calculate the number of
hot-Jupiters (HJs) and very hot-Jupiters (VHJs) that a given transiting planet survey
should be able to detect. We account for factors such as the low frequency of short period
jovian planets, variations in stellar density due to galactic structure, and extinction due to
interstellar dust. We also realistically account for the number of main-sequence stars that
will be in a given field. We restrict ourselves to HJs and VHJs because their short periods
(1-5 days) dramatically increase not only their probability of transiting their parent star,
but also the probability that they will be observed in transit within a reasonable amount
of time by a survey (the so-called “window probability”). It should be noted, however,
that our methods are easily expanded to consider a wider set of possible planets. We first
describe the mathematical formalism with which we have chosen to address this problem,
and then move on to discuss our specific assumptions and our choices for fixed fiducial
parameters. We then offer our own predictions for the detection rate of a ground-based
(TrES) and a space-based photometric survey (Kepler), and compare these to either the
actual detection rates (for TrES) or the detection rates proposed elsewhere (for Kepler).