Introduction
Extrasolar planets are now routinely discovered orbiting
solar-type stars by radial velocimetry, but the discovery of tran-
siting planets by photometric surveys is just beginning. Although
still marginal, the late success of transit surveys has given an
additional impulse to exoplanetology with the possibility to es-
timate the radius, density and hence composition of extrasolar
planets.
Quantitatively, we know to date 206 extrasolar planets with
masses below 13 MJup (e.g. Udry et al. 2007; Butler et al. 2006).
Among those, a list of 14 currently known transiting planets is
presented in table 1. These planets have been discovered by ra-
dial velocimetry followed by photometry for 3 of them, and by
photometric surveys for the remaining 11.
When considering the score of projects devoted to the detec-
tion of planets by transit photometry, the present harvest appears
meager. The discrepancy between predictions (e.g. Horne 2001)
and reality has been attributed to various factors such as: im-
perfect duty cycle, a reduced number of stars for which tran-
siting planets are detectable (Gould et al. 2006) and the pres-
ence of correlated noises that can greatly limit the detectability
of small planetary transits (Pont et al. 2006b). Several generic
studies have been conducted to understand the yield of different
transit surveys. Pepper & Gaudi (2005) studied the optimization
of transit searches as a function of the observational setup, the
site properties and the planet properties. Gillon et al. (2005) an-
alyzed and compared deep field surveys, considering individual
stellar ranges and observation windows, but did not include the
effects of stellar crowding nor time-correlated noises.
Gould et al. (2006) studied the yield of OGLE survey
(Udalski et al. 2002), the most successful so far in term of num-
ber of transiting planets discovered, with a model populating the
line of sight with stars drawn from the Hipparcos Catalogue.
They estimated with that model the proportion of stars with sen-
sitivity to close-in giant planets to derive from OGLE results
the frequency of planets as a function of their period. They find
that the yield of the OGLE survey is globally consistent with
the detections by radial velocimetry and with planet radii dis-
tributed between 1 and 1.25 jovian radii. The aim of the present
work is to further test these data sets and the underlying physical
model by a forward calculation of transit events with realistic
stellar and planetary populations. In particular, we include up-
to-date models of the evolution and structure of Pegasids (close-
in extrasolar planets) based on models reproducing the observa-
tional constraints from known transiting planets (Guillot 2005;
Guillot et al. 2006). As a consequence, we should be able to de-
termine whether the presently known population of transiting
planets represent the “tip of the iceberg”, i.e. that many more
small, dense extrasolar giant planets exist and await discovery
by the transit method, or whether it is relatively representative
of the global population.
We first describe the model that is used to simulate transit
surveys in general. In Section 3, we describe more particularly
the OGLE surveys and the hypothesis chosen for their mod-
elling. We then discuss the results of the simulation. A summary
of the main conclusions and predictions for future transit surveys
are provided in Section 5.
1. Simulating transit surveys
1.1. General remarks
The search for planets in transit in front of their star naturally
arised with the discovery that a non-negligeable fraction of plan-
ets orbit very close to their stars. If orbital planes are randomly
oriented, the probability that a planet will transit in front of its
star at each orbital revolution is:
Ptransit ≃ R⋆/aplanet, (1)
where R⋆ is the stellar radius, and aplanet the planet’s orbital semi-
major axis. For systems such as 51 Peg b, this probability is close
to 10%. Because the probability for a solar-type star to possess
such a Pegasid (i.e. a 51 Peg b-like planet, planets semi-major
axis up to 0.1 AU) is about 0.5% (e.g. Marcy et al. 2005), 1 in
2000 solar-type star should possess a transiting Pegasid. Using
current results from radial velocity surveys and integrating over
all periods, we estimate that about 1 in 1100 solar-type stars
possesses a transiting giant planet. Of course, depending on the
magnitudes and field considered, giant stars may severly out-
number the dwarfs, so that in a real field, only one in, say, 3000
stars may harbor a transiting giant planet.
Because of the dependence on a, and period distribution,
most of the transit events concerning giant planets occur for or-
bital periods between 1 and 5 days. The transits typically last for
a couple of hours, as this quantity is weakly dependant on the
orbital period P:
τtransit = 1.82
1 day
)1/3 (
)−1/3 (R′⋆
hours, (2)
where R′⋆ is the length of the cord traced on the stellar disk by
the planet’s trajectory. (more precisely: R′⋆ = R⋆ cos b + Rplanet,
where b is the impact parameter of the transit).
The depth of the transits themselves is directly given by the
ratio of the planetary to the stellar disk surfaces:
Rtransit ≃ (Rplanet/R⋆)
. (3)
This value is of order 1% for a Jupiter-size planet orbiting a
Sun-like star. For an F-type star with radius ∼ 1.2 R⊙, the ratio
decreases to 0.7%. Furthermore, transiting giant planets discov-
ered so far have radii between 0.72 and 1.44 RJup (see table 1).
Allowing for stellar radii to vary between 0.8 and 1.3 R⊙ (a typ-
ical range, in magnitude limited surveys), this implies that we
should expect Rtransit to vary between 0.3% and 3%, for giant
planets only. The lower limit is in reality even smaller because
for detection purposes we have to account for the fact that plan-
ets also orbit stars that are in multiple systems (like HAT-P-1),
and hence a dilution factor may apply. Although grazing transits
are ignored in this simple analysis, they are included afterwards
in our simulations.
This altogether implies that in order to detect transiting gi-
ant planets, many thousands of dwarf stars have to be monitored
over periods of weeks for a photometric precision reaching be-
low a fraction of a percent on an equivalent integration time of
about one hour. This is typically done by following a relatively
dense stellar field over a long time with a stable telescope, and
using a camera equiped with a good CCD camera.
1.2. Principle of the simulations
On paper, the simulation of the forward problem is simple: one
has to generate a complete stellar field, or obtain it from observa-
tions, put it on a discrete grid (the CCD), include on probabilistic
arguments the planetary companions, calculate lightcurves in-
cluding the various sources of noise, and determine which events
are detectable. This is the principle of CoRoTlux, a code we first
developed to predict the transit yield of CoRoT space telescope
http://arxiv.org/abs/0704.1919v1
Table 1. Known transiting planets by 2006⋆
# Name Mplanet Rplanet Period a M⋆ R⋆ Teff Metallicity
[MJup] [RJup] [day] [AU] [M⊙] [R⊙] [K] [Fe/H]
OGLE planets
6 OGLE-TR-10 0.63±0.14 1.26+0.07
−0.07 3.10129 0.04162 1.18 ±0.04 1.16±0.06 6075±86 0.28±0.10
2 OGLE-TR-56 1.17±0.04 1.32+0.06
−0.06 1.211909 0.0225 1.17±0.04 1.32±0.06 6119±62 0.19±0.07
5 OGLE-TR-111 0.52±0.13 1.067+0.054
−0.054 4.0144479 0.047 0.81±0.02 0.831±0.031 5044±83 0.19±0.07
3 OGLE-TR-113 1.35±0.22 1.09+0.03
−0.03 1.4324757 0.0229 0.78±0.02 0.77±0.02 4804±106 0.15±0.10
4 OGLE-TR-132 1.14±0.12 1.18+0.07
−0.07 1.689868 0.0299 1.26±0.03 1.34±0.08 6210±59 0.37±0.07
Other transit survey planets
7 TrES-1 0.76±0.05 1.081+0.029
−0.029 3.0300737 0.0393 0.89±0.035 0.811±0.020 5250±75 -0.02±0.06
11 TrES-2 1.28±0.07 1.24+0.09
−0.06 2.47063 0.0367 1.08±0.08 1.00±0.05 5960±100 0.15±0.03
10 XO-1 0.90±0.07 1.184+0.028
−0.018 3.941634 0.0488 1.0±0.03 0.928±0.033 5750±13 0.015±0.03
12 HAT-P-1 0.53±0.04 1.36+0.011
−0.09 4.46529 0.0551 1.12±0.09 1.15±0.09 5975±45 0.13±0.02
13 WASP-1 0.867±0.073 1.443+0.039
−0.039 2.519961 0.0382 1.15±0.09 1.453±0.032 6200±200
14 WASP-2 0.88±0.07 1.038+0.05
−0.05 2.152226 0.0307 0.79±0.08 0.813±0.032 5200±200
Transiting planets discovered with Radial velocities
9 HD189733 1.15±0.04 1.154+0.032
−0.032 2.218573 0.0313 0.82±0.03 0.758±0.016 5050±50 -0.03±0.04
8 HD149026 0.330±0.02 0.726+0.064
−0.064 2.87598 0.042 1.3±0.1 1.45±0.1 6147±50 0.36±0.05
1 HD209458 0.657±0.006 1.320+0.025
−0.025 3.52474859 0.047 1.09±0.09 1.148±0.002 6117±26 0±0.02
MJup = 1.8986112 × 10
30 g is the mass of Jupiter. RJup = 71, 492 km is Jupiter’s equatorial radius.
OGLE-TR-10: Bouchy et al. (2005); Udalski et al. (2002); Konacki et al. (2005); Santos et al. (2006); Pont et al. (2006a)
OGLE-TR-56: Konacki et al. (2003); Udalski et al. (2002); Torres et al. (2003)
Bouchy et al. (2005); Santos et al. (2006); Pont et al. (2006a)
OGLE-TR-111: Pont et al. (2004); Santos et al. (2006); Udalski et al. (2002); Winn et al. (2007); Bouchy et al. (2005)
OGLE-TR-113: Bouchy et al. (2004); Udalski et al. (2002); Konacki et al. (2004); Gillon et al. (2006)
OGLE-TR-132: Bouchy et al. (2004); Udalski (2003); Moutou et al. (2004); Magain et al. (2007)
TRES-1: Alonso et al. (2004); Laughlin et al. (2005); Winn et al. (2007)
TRES-2: O’Donovan et al. (2006)
XO-1: McCullough et al. (2006); Holman et al. (2006); Wilson et al. (2006)
HAT-P-1: Bakos et al. (2006)
WASP-1: Collier Cameron et al. (2006); Shporer et al. (2007); Charbonneau et al. (2006)
WASP-2: Collier Cameron et al. (2006); Charbonneau et al. (2006)
HD-189733: Bouchy et al. (2005); Bakos et al. (2006)
HD-149026: Sato et al. (2005); Charbonneau et al. (2006)
HD209458: Brown et al. (2001); Cody & Sasselov (2002); Wittenmyer et al. (2005); Winn et al. (2005); Knutson et al. (2007)
# is the label of planets in figures ; they are ranked in the order of their discovery.
(Baglin et al. 2002) and quantify the need for follow-up obser-
vations, which is here applied to the case of OGLE.
The interesting point of such a forward simulation is the pos-
sibility to include relatively easily fine details such as the fact
that planets are found more frequently around metal-rich stars,
or, on the basis of planetary evolution models, the fact that young
planets orbiting close to bright stars will be larger than old plan-
ets orbiting smaller stars at larger orbital distances. This requires
however that a relatively large number of physically relevant
parameters (for example, the mass, size, metallicity, age of the
stars) be properly defined.
We further detail the assumptions that we made for these
simulations by describing how we generate the stellar and plane-
tary populations, and how we attempt to include realistic sources
of noise.
1.3. The stellar population
1.3.1. Main targets and background stars
Stellar fields differ enormously in terms of densities and stellar
populations. It is therefore most important to properly account
for this in order to simulate any given transit survey.
It would be very appealing to use direct observations as much
as possible to closely match the observed target fields. But as
we will see hereafter, many different characteristics of the stars
(stellar metallicity, age and subtype ...) are required, and these
are difficult to obtain with generic observations. We therefore
adopt the following procedure:
– The observed stellar densities are obtained from stellar
counts by magnitude, on the real stellar fields (see § 2.1)
– The characteristics of the stars are obtained following a
Monte-Carlo method using the output of the Besançon model
of the galaxy (Robin et al. 2003) obtained for the proper lo-
cation of the survey.
– Where stellar counts are not available, or uncomplete (i.e.
for faint stars), we use both stellar counts and characteristics
from the Besançon model.
Specifically, we keep track of the following parameters ob-
tained directtly from the Besançon model:
– The mass of each star, used to compute orbital parameters of
the transiting object;
– The apparent magnitude of the star in the observed spectral
range (the I filter in the case of the OGLE survey);
– The V magnitude of the star, important to qualify the con-
firmability of a transit event with radial velocimetry;
– The surface temperature of the star
– The luminosity of the star, calculated from its absolute mag-
nitude;
– The radius of the star, calculated from total luminosity and
effective temperature.
Stellar radius [RSun]
Stellar mass [MSun]
Effective temperature [K]
Fig. 1. From top to bottom: Distribution functions for the radii,
masses and effective temperatures for our fiducial stellar popu-
lation corresponding to the simulated OGLE Carina field. The
black line represents the ensemble of stars in the field. The filled
red region is a subset for dwarf stars with stellar type F4 and
later, as these are the only stars for which a transiting planet has
a reasonnable chance of being detected by present-day transit
surveys.
The mass, and effective temperature of the stars are dis-
tributed linearly around values given by the Besançon model
(at ±20%). Figure 1 shows a simulated distribution of stars for
the OGLE Carina field. The ensemble of dwarf stars with types
F4 and later are highlighted as these represent targets for which
planetary transit events are detectable, and, within observational
limits, confirmable by radial velocimetry.
The metallicity distribution is obtained from the model
of Nordström et al. (2004), which is based on the Geneva-
Copenhagen survey of the Solar neighbourhood. These authors
find that the distribution of the metallicities [Fe/H] is well ap-
proximated by a Gaussian function with a mean of −0.14 and
a dispersion of 0.19 dex. We use this gaussian distribution and
choose to ignore possible dependencies between stellar parame-
ters (e.g. masses, ages...) and the metallicities. (The link between
stellar type and metallicity appears to be negligible for F4 and
later types anyway (F. Thévenin, pers. communication 2007)).
1.3.2. Binary and triple systems
Multiple stellar systems are important especially because of the
possibility that stellar eclispes mimic planetary transits (Brown
2003). However, we choose to defer this problem to a later ar-
ticle. Multiple systems are taken into account anyway because
they can yield a dilution of the planetary transit events that
makes them more difficult to detect. Planets may be present in-
differently on the primary, secondary or tertiary components of
a stellar system. (However, we find that only planets around the
primary targets have a non-negligible chance of being discov-
ered by current ground based photometric survey.)
Specifically, following Duquennoy & Mayor (1991), we
consider that 50% of the stars are binaries and 20% of those
are ternaries. Multiple systems are considered as individual stars
at the same position on the CCD. We choose to estimate their
properties more simply than for the other stars, on the basis of
DM91:
– We randomly add companions to the initial draw of primary
stars, without changing their properties. The total mass and
luminosity of each multiple system is thus slightly higher
than initially.
– The mass ratio (primary/secondary) is defined as a gaussian
of median value 0.23 and a full width at half maximum of
0.42. Outside a range of 0.05 and 1, we redraw the mass
ratio.
– The radius is defined as R⋆ = R⊙(M⋆/M⊙) when M⋆ ≤ M⊙
and R⋆ = R⊙(M⋆/M⊙)
1/2 otherwise.
– The luminosity is assumed to be proportional to M2 so that:
lsecondary = lprimary(Msecondary/Mprimary)
– Other stellar parameters are calculated on the basis of these
ones and of those of the primary component (same age, same
distance, same metallicity).
– Triple components are treated with the same method as bi-
naries, and are defined relatively to the primary star.
1.4. The planetary companions
With more than 200 planets known to orbit stars other than our
Sun, we are beginning to have a rather precise view of at least
part of this population. We can expect that biases on the detec-
tions are small in the case of massive planets (the mass of Saturn
and more) and planets that are relatively close to their star (or-
bital distances smaller than ∼ 1 AU). These two conditions hap-
pen to match quite exactly the requirement for detectability by
transit photometry, with one assumption: that only massive giant
planets can have large radii. Although not proven, this assump-
tion seems quite reasonnable.
Hence we choose to focus this study on this well-
characterized population of objects. From the current list of 209
detected exoplanets, we select the ones discovered by radial ve-
locities with mass higher than 0.3 Jupiter masses and known host
star metallicity. Our list of planets includes 153 objects, to which
we may add very-close in planets detected by transit photometry,
as discussed below. We are using this list as representative of an
unbiased sample of giant planets known from radial velocimetry,
even though planetary distribution models may have been made
from slighlty different samples.
1.4.1. Planet incidence
A first important step is the determination of the probability for a
star to harbor a planet. As shown by numerous studies (Gonzalez
1998; Santos et al. 2004; Fischer & Valenti 2005), this probabil-
ity depends mostly on the metallicity of the parent star. Figure 2
shows one such probability function, as well as the result in
terms of planet counts on a simulated stellar field.
In this work, we will use the dependency from Santos et al.
(2004) shown in Fig. 2. Several points are to be considered how-
ever:
1. This probability relation is only valid for solar-like stars, i.e.
F, G, K dwarf stars. Although there are strong indications
that it may change for other stars (e.g. M dwarfs), we will
assume it to hold independently of stellar properties. This
[Fe/H]
[Fe/H]
Fig. 2. Upper panel: Probability for a solar-type star to possess
a giant planet companion as a function of the stellar metallicity
(from Santos et al. 2004). Lower panel: Relative normalised dis-
tributions of stellar metallicities for stars in the field (black line),
and for stars with a giant planet companion (red line).
assumption is sufficient because F, G and K dwarf stars form
by far the majority of stars with detectable planets in photo-
metric surveys.
2. This relation has been calculated independently of the prop-
erties of the planetary companion, in particular orbital dis-
tance. Because in our case we are strongly biased towards
short-period planets, the distribution may be different. This
point will be considered in § 3.4.
3. The possibility of multiple planetary systems is not consid-
ered. This approach is justified because the probability that
several planets belonging to the same system are transiting
planets is small for giant planets.
1.4.2. Planetary masses and orbits
The masses and orbital characteristics of the planet population
are inferred almost entirely from the present radial-velocimetry
surveys. This technique yields an accurate determination of the
orbital period, and less accurately, of the eccentricity of the orbit.
It also yields the value of the mass of the planetary companion
times the sine of the orbital inclination from the knowledge of
the mass of the parent star. With these values, we can then derive
from a random inclination of the orbital planes the planets that
are transiting and those that are not as well as the characteristics
of their orbit.
We test several approaches for the derivations of these quan-
tities:
– An analytical model: In this approach, we consider inde-
pendantly the planet period and its mass. The period of the
planet Π follows the model of Brown (2003), the proba-
bility density P from a piecewise linear fit to the distribu-
tion P(logΠ) = {0.509, 0.165, 0.533} for three period inter-
vals bounded by logΠ = {0.43, 0.65, 2.3, 3.5}. The distribu-
tion in mass is linear in log from 0.3 to 10 Jupiter masses
(Zucker & Mazeh 2001). There is no dependency of these
two parameters linked to metallicity.
– The radial velocity mass-period “carbon-copy” model: A
second approach is to make direct use of the list of planets
discovered by radial velocimetry. This is possible because in
terms of masses and orbital periods the list is almost unbi-
ased for the objects that we consider (massive enough to be
above detection thresholds, and with periods much shorter
than the lifetimes of the surveys). In this case, we select plan-
ets randomly in the RV list, and then allow for a small ran-
dom deviation of their mass and period (a uniform deviation
from −20% to +20%) in order to avoid too much cluster-
ing on the same value. This is particularly important in the
case of the period because of the importance of stroboscopic
effects in planetary transits (e.g. Pont et al. 2005).
– The radial velocity mass-period-metallicity “carbon-copy”
model: As a modification to the previous approach, we also
consider using the metallicity entry in the RV list, because
of correlations between metallicity and orbital period that
are otherwise not taken into account (see discussion in sec-
tion 3). We proceed slightly differently however than for the
mass and orbital period because of the limitations caused
by the finite number of planets in the RV list. In this case,
we choose to split the list into two low- and high-metallicity
lists, and then select the mass and periods in the relevant list.
Our fiducial cutoff value is [Fe/H]=−0.07.
Figure 3 shows a comparison between observations, the car-
bon copy model and the analytical model. It is interesting to no-
tice at this point that the carbon copy and analytical models are
essentially indistinguishable in these diagrams. The differences
with the observations arise only because of our choice to smear
the masses and orbital periods when generating our planet pop-
ulation.
Last but not least, we have to consider the existence of plan-
ets that orbit extremely close to their star, with periods shorter
than 2 days, as discovered by transit surveys (see table 1).
Companions with such short orbital periods have been discov-
ered by radial velocimetry in two occasions: HD 41004 b, and
Gliese 876 d, with respective masses 18.4 and 0.023 Jupiter
masses. These objects are outside the mass range considered for
this study, and therefore, there is no giant planets with periods
shorter than 2 days in the present radial velocimetry list. In or-
der to account for these very close-in planets anyway, we add
the planets with periods smaller than 2 days discovered by tran-
sit photometry to the list, but with a small tunable probability
weight. The fiducial value of this parameter is set so that, on av-
erage, the planet list contains one and a half such planet (added
to the list of 153 RV planets described in § 1.4). Tests on the
effect of this parameter are presented in § 3.3.4.
Our fiducial model is the mass-period-metallicity carbon
copy model, includes addition of very-close in planets and it is
that model which is used in all cases except where specified oth-
erwise. Other approaches are also tested depending on the model
to highlight particular points.
1.4.3. Physical characteristics and the planetary evolution
model
Because we are focussing on planets more massive than Saturn,
we expect most of them to be made of a significant amount of
hydrogen and helium. These giant planets thus undergo a pro-
gressive contraction and cooling that depends on four quanti-
ties: their age, mass, the amount of flux the planet receives from
the central star, and the global amount of heavy elements in the
planet (e.g. Guillot 2005).
Orbital period [days]
Planetary mass [MEarth]
Planetary radius [RJup]
Fig. 3. From top to bottom, distributions of orbital periods,
masses and radii, respectively, of the planets observed by radial
velocimetry (black lines), simulated as part of the mass-period
“carbon copy” model (red lines), and simulated as part of the
analytical model (dotted blue lines) (see text).
Models attempting to reproduce the radii of known tran-
siting giant planets have however had problems in explain-
ing the large radii of some of them (Bodenheimer et al. 2001;
Guillot & Showman 2002; Baraffe et al. 2005; Laughlin et al.
2005). Several possibilities have been proposed to explain the
discrepancy. We can separate them into two categories:
– Mecanisms invoking chance configurations of the plane-
tary orbits in the case of these anomalously large plan-
ets: this includes the tidal circularization of an eccentric
orbit (Bodenheimer et al. 2001), and tidal dissipation for a
planet locked in a Cassini spin-orbit resonnance with the star
(Winn & Holman 2005).
– Effects that would apply to all planets, including problems
with the equations of state or opacities, and the dissipation
by stellar tides of kinetic energy first generated in the atmo-
sphere (Showman & Guillot 2002).
The first mecanisms appear to have a low probability of oc-
curence (Laughlin et al. 2005; Deming et al. 2005; Levrard et al.
2007). The second possibility therefore seems more likely, but
requires in some case the presence of relatively large masses of
heavy elements to reproduce the observed radii.
A model-dependant estimate of the masses of heavy el-
ements present in the currently known transiting Pegasids is
shown in Fig. 4. This model relies on the hypothesis that 0.5%
of the absorbed stellar flux is used to generate kinetic energy
that is subsequently dissipated deep into the planetary inte-
rior (Guillot & Showman 2002). As proposed by Guillot et al.
(2006), there appears to be a correlation between the amount of
heavy elements present in the planet and the metallicity of their
parent star.
This correlation has to be ascertained, but we choose for our
fiducial model to adopt a unique relation between metallicity and
mass of heavy elements (treated as a central core in our calcula-
tions), corresponding to the dotted line in Fig. 4:
MZ = 43.75 × 10
[Fe/H]
− 23.7 M⊕. (4)
Fig. 4. Mass of heavy elements in transiting Pegasids known by
2006 as a function of the metal content of the parent star rela-
tive to the Sun. The mass of heavy elements required to fit the
measured radii is calculated on the basis of evolution models
including an additional heat source slowing the cooling of the
planet. This heat source is assumed equal to 0.5% of the incom-
ing stellar heat flux (Showman & Guillot 2002). Horizontal er-
ror bars correspond to the 1σ errors on the [Fe/H] determina-
tion. Vertical error bars are a consequence of the uncertainties
on the measured planetary radii and ages. The metallicity of re-
cently discovered planets WASP-1 and WASP-2 (right panel) is
not precisely known. The dotted line corresponds to a best fit
model. [Adapted from Guillot et al. (2006)].
We limit the range of possible values of MZ to [0, 100M⊕].
Similarly, we adopt a simple boundary condition for our evo-
lution calculations:
T1bar = 1.25Teq0, (5)
where T1bar is the temperature at the 1 bar pressure level and
Teq0 is the equilibrium temperature for a zero albedo (see Guillot
2005 for a description), calculated as a function of stellar effec-
tive temperature and radius and planet semi-major axis.
For simplicity, and because it yields only minor changes on
the results, we further choose to neglect the time-dependence in
the evolution calculations, and to adopt the equilibrium radius,
or the 10 Gyr solution, whichever occurs first.
Practically, planetary radii are obtained from interpolations
in a table based on three parameters: the planetary mass ranging
from 100 to 3000M⊕, the core mass from 0 to 100M⊕ and the
equilibrium temperature from 100 to 2000 K. Models were not
calculated beyond these values of Teq because of convergence
problems. However we allowed for a slight extrapolation of the
tables to 2600 K to account for rare extremely hot planets. 1
Similarly, because of convergence problems for planets with
small total masses and large core masses, we limited the mass of
the core to 75 M⊕ for planets with masses smaller than 180 M⊕.
More detailed work is required to better simulate this parame-
ter space, including planets less massive than considered in this
study.
1 An electronic version of the table is available at www.obs-
nice.fr/guillot/pegasids/
Figure 5 shows examples of radii obtained for Teq = 1000
and 2000, K, and core masses of 0 and 100Moplus, respectively,
compared to available measurements.
eq =2000K
eq=1000K
no core
e,  n
100 M⊕ core
planets
brown
dwarfs
stars
Fig. 5. Theoretical and observed mass-radius relations. The
black line is applicable to the evolution of solar composition
planets, brown dwarfs and stars, when isolated or nearly isolated
(as Jupiter, Saturn, Uranus and Neptune, defined by diamonds
and their respective symbols), after 5 Ga of evolution. The dotted
line shows the effect of a 15M⊕ core on the mass-radius relation.
Orange and yellow curves represent the mass-radius relations
for heavily irradiated planets with equilibrium temperatures of
1000 and 2000 K, respectively, and assuming that 0.5% of the
incoming stellar luminosity is dissipated at the center (see sec-
tion 1.4.3). For each irradiation level, two cases are considered: a
solar-composition planet with no core (top curve), and one with
a 100M⊕ central core (bottom curve). The transiting extrasolar
giant planets for which a mass and a radius was measured are
shown with points that are color-coded in function of the planet’s
equilibrium temperature. The masses and radii of very low mass
stars are also indicated as blue points with error bars.
1.5. Modeling transit events and their detectability
We now descibe how this population of stars, planets and their
interactions during transits are modelled to reproduce real obser-
vations.
1.5.1. PSFs and CCDs
Each image of a star is spread by the atmosphere and by the
telescope to grow to a specific size and shape when reaching
the CCD in the focal plane of the telescope, the so-called point
spread function (PSF). The CCD being composed of many dis-
crete pixels, these PSFs are then effectively discretized, so that
the signal to be analyzed for any given star is composed of many
different lightcurves corresponding to the many pixels over the
size of its PSF. A further complication arises from the fact that
the stellar fields generally chosen by transit surveys are dense,
so that many PSFs overlap. Recovering individual stellar light
curves from real data is a complex problem. Two popular meth-
ods are aperture photometry (Stetson 1987) and image subtrac-
tion (Alard & Lupton 1998). (An adaptation of the latter was
used to extract the OGLE lightcurves).
A refined simulation could include possible spatial and tem-
poral variations of the PSFs, and a realistic data reduction
pipeline. In our case, we choose to simplify the problem by rely-
ing on a posteriori analyses of real light curves to provide us with
a global noise budget. We however include background stars be-
cause of the important effect of signal dilution.
In order to do so, we assume that the PSFs are gaussian with
a uniform, constant FWHM. (Non-gaussian PSFs are not diffi-
cult to include but we tested in the OGLE case that for a fixed
equivalent FWHM, they have a negligible effect on the resulting
signal-to-noise ratio of simulated transit events). We consider
for each target of the survey the global flux from the main star
and the background stars in its neighborhood up to magnitude
22 in the spectral band of observation. The neighborhood zone
for background stars is defined as a circle of diameter equal to
4 times the PSF’s FWHM around the photocenter of each target
star. Each background star whose photocenter is located in that
zone is taken into account for the calculation of the global flux.
The global flux is calculated as the sum of the pixels located at
less than twice the FWHM of the central star.
We thus simulate aperture photometry when image subtrac-
tion was used for OGLE (Udalski et al. 2002). The choice of the
reduction algorithm indeed affects the sensitivity obtained from
real observations. In our simulations, i.e. a relatively idealized
case, it would have marginal effects since realistic noises are
included a posteriori from the analysis of real lightcurves (see
hereafter).
1.5.2. Noise budget and event detectability
We choose to separate noise sources into two categories:
– ‘White noise’ sources, following gaussian and Poisson laws.
The main source of white noise is the photon noise of target
stars and their background. The level of white noise for a
given target star is obtained from the simulation of the flux
of that star and its background in the photometric aperture.
– ‘Red noise’, or systematic effects on photometry, that un-
dergo temporal correlation. The structure of these systemat-
ics in the OGLE photometry have been explored in details by
Pont et al. (2006b). These noise sources are both instrumen-
tal (jitter and breathing of the CCD, frequency spectrum of
stellar field moves on the camera, change of the PSF shapes
accross the CCD during the night), and environmental (dif-
ferential refraction and extinction, changes of airmass and
sky brightness, temperature changes). Rather than trying to
simulate instrumental and environmental noise sources accu-
rately, which is difficult with the relatively poor knowledge
we have of the time spectrum of their combined effects, we
use the effective global ‘red noise’ measurements of OGLE-
III survey real light curves mentionned in Pont et al. (2006b),
which consider the combined effect of these noise sources.
Pont et al. (2006b) calculated that, in the presence of a mix-
ture of white and red noise (i.e. accounting for photometric sys-
tematics), the detection threshold for a transit survey is well de-
scribed by a limit on the signal-to-noise ratio defined as:
S 2r =
k=1 nk
2(σw2/nk + σr2)
where Ntr is the number of transits sampled, nk the number of
data points in the k-th transit. σw and σr are the standard devia-
tion of measurement points of white and red noises, respectively,
d is the event depth and n the total number of measurement
points during the transit. Specifically, we obtain nk by counting
for each transit the number of observation points between the
middle of ingress and the middle of egress.
Equation 6 makes the disctinction between “white” noise
sources that decrease with n1/2, where n is the number of succes-
sive measurements, and “red” noise sources that are limited by
temporal correlation. Pont et al. (2006b) indeed show that taking
the red noise into account makes a large difference on the detec-
tion threshold – in general as well as in its dependence to the
planet parameters – and that models based on the assumption of
white noise can be poor approximations of the actual detection
threshold.
2. The OGLE survey: input parameters
2.1. Basic parameters and observational procedure
The Optical Gravitational Lensing Experiment (OGLE) has
done 6 observation campaigns looking for transiting planets to-
wards different fields of view from 2001 (Udalski et al. 2002).
It took place at the Las Campanas Observatory, Chile, using the
1.3 m Warsaw telescope and the 8k MOSAIC camera, with a to-
tal field of view of 0.34◦2. All observations were made through
the I filter. We assume for our PSF simulation an average seeing
of 1 arcsec.
We analyze in this work the first three OGLE-III observation
campaigns dedicated to transit search, as their treatment, anal-
ysis and follow-up (with current data processing pipelines) has
been completed:
– OGLE-III-1 (June 12 to July 28, 2001, described in
Udalski et al. (2002); Udalski (2002)). More than 800 im-
ages of three fields in the direction of the galactic bulge were
collected within 32 nights. The exposure time was 120 s, and
each field was observed every 12 min.
– OGLE-III-2 (February 17 to May 22, described in Udalski
(2003)). More than 1100 images of three fields located in the
Carina region of the galactic disk were collected in 76 nights.
The exposure time was 180 s, and the temporal resolution
was about 15 min.
– OGLE-III-3 (February 12 to March 26, described in
Udalski et al. (2004)). The photometric data were collected
during 39 nights spanning the 43 days of the survey. Three
fields of the galactic disk were observed with a time resolu-
tion of about 15 min. The exposure time was 180 s.
In this article, we will refer to these three observation cam-
paigns respectively as ‘Bulge’, ‘Carina’, and ‘Centaurus’ fields.
The simulations include the real observation windows of
each survey, as kindly provided by A. Udalski. For any transit-
ing planet in the simulation, the number of effectively observed
transits is used in eq. 6.
In order to construct a realistic stellar population, we use
the stellar counts per magnitude range obtained by Gould et al.
(2006) based on OGLE-II data, which have calibrated photome-
try. We then randomly select that number of stars per magnitude
from the Besançon model. In order to test the validity of our
approach, we calculated the fraction of “stars for which tran-
sits are detectable” and compared it to the one determined by
Gould. This fraction is defined for a given magnitude range as
the number of stars around which a planet orbiting edge-on with
r = 1.2R jup and a = 7.94R⊙ can be detected, divided by the
total number of stars of that magnitude. As shown by table 2,
there is an excellent agreement between our results and those of
Table 2. Fraction of stars suitable for transit detection
Carina Bulge
Vmax Gould 2006 This work Gould 2006 This work
15.5 0.11 0.16 0.138 0.141
16 0.14 0.16 0.125 0.128
16.5 0.16 0.15 0.098 0.105
17 0.16 0.15 0.068 0.080
17.5 0.16 0.14 0.041 0.052
Gould et al. (2006). Note however that for the global simulation,
the complete star list is used as the above definition for suitable
stars is restricted to planets of a given size and orbital distance.
We calculated the average flux for target stars, companions
and all the background stars near enough to contribute to the
target PSF. We then checked that the average photon noise simu-
lated for target stars at a given magnitude was close to real values
obtained in OGLE light curves at given magnitude presented in
figure 4 of Pont et al. (2006b).
2.2. Modelling the detection threshold
The candidates in the OGLE survey have been identified with
the BLS transit-search algorith of Kovács et al. (2002). A sub-
set of the candidates selected with cuts in the α and SDE pa-
rameters of the BLS were examined by eye, and only the best
were included in the final list. Therefore, the selection thresh-
old is mainly defined by subjective appreciation from an ex-
perienced specialist. Recently, Pont et al. (2006b) have pointed
out that the effective detection threshold of ground-based tran-
sit surveys such as OGLE is importantly affected by correlated
noise (photometric systematics). The subjective selection of can-
didates is in large part necessary because of the presence of this
correlated noise, which produce many spurious detections near
the threshold. Gould et al. (2006) chose to model the OGLE se-
lection threshold with an α > 12 cut (alpha is equivalent to
the signal-to-noise ratio of the transit signal assuming uncor-
related noise and homogeneous distribution of the data points
in phase). Pont et al. (2006b) have included the effect of corre-
lated noise in the signal-to- noise calculation and found that the
OGLE selection could be better described by a threshold of 8
on the signal-to-noise ratio of the transit signal calculated in-
cluding correlated noise (”S r” in their notation, see Sec. 2.5.2),
and without the assumption of homogeneous coverage. While
the two thresholds have similar effects on the global number of
planet detection, they have a very different dependence on some
parameters, such as planet period and host star magnitude. Since
the objective or our study is to examine the detection statistics
in a multi-dimensional parameter space, we use the Pont et al.
(2006b) description of the OGLE detection threshold.
To calculate S r, one needs an assumption on the level of red
noise present in the photometry. Following Pont et al. (2006b),
we use a single-parameter description and assume σr = 3.6
mmag in the Bulge fields, σr= 3.1 mmag in the Carina and
Centaurus fields, and σr= 2.1 mmag in all fields after applica-
tion of decorrelation algorithms.
2.3. Confirmability of transit-like events with follow-up
High-resolution spectra allow the confirmation of the planetary
events if spectral lines are deep enough. Several scenarios make
the follow up of candidates too difficult: early type stars have
lines too weak and too broadened by rotation (type F4 and ear-
lier). Stars with magnitudes V > 17.5 are too faint for present
instruments and telescopes. This is the limit at which observers
estimated not being able to provide low-metallicity stars. Those
stars having weaker lines, could also be difficult to follow cor-
rectly, but as planets are unlikely to be found near this kind of
stars in our model, we did not take that parameter into account.
To simulate the feasability of follow-up, we only considered
in CoRoTlux the stars matching the criteria V < 17.5 and of type
F4 and later.
3. Results of the simulations
We present hereafter runs for the three OGLE-III campaigns for
the fields in the Galactic bulge, in Carina and in Centaurus. In
order to obtain a statistically significant population of detected
planets, the simulations were run multiple times.
We first examine the consistency between the models and ob-
servations for relevant physical variables. In doing so, we choose
to compare our model population to the global population of
transiting planets discovered by OGLE and other surveys. There
is a slight inconsistency in assuming that the parameter compari-
son is almost independant of the type of survey and observational
strategy. In some cases, this is not true, and a clear distinction
between the OGLE planets and the other detections has to be
made.
We then discuss the problem of the detection statistics,
whether observations and models are consistent, and whether a
constraint on the (low) frequency of very close-in planets can be
deduced.
3.1. Deviation of OGLE planets from maximum likelihood of
the simulations
We use a Maximum-Likelihood (ML) technique in order to test
whether model results and observations agree with each other.
We do the tests in two-dimension spaces, in order to qualify pos-
sible correlation and exclusion zones. The ML technique is our
method of choice as it is a powerful tool for fitting a model to a
multi-dimentionnal independant-data distribution (Lyons 1986).
Instead of determining an approximate analytical law fitting
our results, we use the results of a very large Monte-Carlo draw
(1000 times the whole OGLE survey, corresponding to ∼ 9000
planets) to get a map of the density of probability in each 2-
dimension grid. We bin our data on a 20x20 grid as a compro-
mise between resolution of the models and characteristic varia-
tions of the parameters.2 The probability of an event in each bin
is considered equal to the normalized number of draws in that
Figure 6 shows the logarithm of the probability that an event
occurs in each of the 20x20 bins of the mass-radius diagram. The
likelihood of a draw of several independant events is defined as
the sum of the logarithms of the probabilities of these events. In
order to compare our results to any n real discoveries, we first
estimate the standard deviation of any n-planets-random-draw
compared to the maximum likelihood of the model. We ran-
domly select n planets among the simulated detections and cal-
culate the likelihood of this draw. We repeat this selection 1000
times in order to have the maximum likelihood and its standard
deviation σ, then we compare the deviation of the likelihood
of the n real detecions calculated the same way in terms of σ.
2 Tests with different grids yield small variations of the results. As
an example, the mass-radius deviation from maximum likelihood is re-
spectively 0.67, 0.65 and 0.72 σ for 20x20, 30x30 and 40x40 grids.
Henceforth, quantitative comparisons between the simulation re-
sults and the known planets are systematically given in the figure
captions, whilst the text discusses qualitative comparisons and
their implications. For the different figures showing the results
of our simulation, we compare the distribution of planets over
the detection threshold to the 5 OGLE planets. We also compare
our results to the 11 planets discovered by all transit surveys, as
their detection biases are similar to OGLE, and to the 14 planets
which radius is known (11 from transits and 3 from radial ve-
locity surveys) to show how our model can reproduce the whole
known population.
3.2. Depth of the transit events and magnitude of the targets
stars
We first attempt to confirm whether the events detected by
the model are consistent with those found in the OGLE fields.
Figure 7 is a plot showing transit depth as a function of the mag-
nitude of the primary star. Model results are considered detected
when the signal-to-noise ratio is sufficient for a detection (see
§ 2.2). We also show events that are considered photometrically
detectable but very hard or impossible to confirm by radial ve-
locimetry.
The figure evidently shows a good correlation between the
black crosses and the red circles that indicate real detections by
OGLE, with a range of transit depths and V magnitudes that
is very similar between the models and the observations. Our
models overpredict slightly the number of transit events around
faint stars (V ≥ 17), but this may be due to the difficulty of the
follow-up work for these targets. Overall, the agreement between
models and observations is good.
3.3. Compatibility of transit surveys with radial-velocimetry
observations
3.3.1. Compatibility in the mass-period diagram
Figure 8 compares the model and observated mass-period rela-
tion. As it is independant of the planetary evolution model, it is a
direct test of the compatibility between the results of transit sur-
veys and those of radial-velocimetry observations that drive our
model results. Again, the comparison is very good, assuming a
high-enough frequency of very-close in planets (see discussion
in § 3.3.4). One can note especially the absence of planets of rel-
atively large mass (several times that of Jupiter) at short orbital
distances (P < 5 days), and of detectable transit events for peri-
ods longer than ∼ 5 days. This is due especially to the fact that
only events with a relatively large number of observed transits
are detectable, as required by the S r threshold, which, given the
day/night interruptions, imposes a constraint of a short orbital
period. Note that this feature is not well reproduced by mod-
els in which the threshold is computed from white-noise only
(Gould et al. 2006; Gillon et al. 2005).
3.3.2. The OGLE yields with a fixed red noise level
We have tested the efficiency of the fiducial model at estimat-
ing quantitatively the yield of transit surveys. Gillon et al. (2005)
have also simulated OGLE yield in their generic study of multi-
ple transit surveys, but with restrictive assumptions on transit de-
tectability (only complete events matter for detection purposes)
and without considering background stars and red noise, also
not using OGLE-fields specific stellar population. We also in-
cluded in our simulations the recent RV follow-up that has been
Fig. 6. Logarithm of the probability that a simulated detection event occurs in each one of the 20x20 bins of the mass/radius
diagram. The likelihood of a multiple-events draw is the sum of the logarithms of the probabilities of the events of this draw. Bins
without any occuring event in the large Monte-Carlo draw do not have any probability stated. The likelihood of a n-events draw is
the sum of the probabilities of its n events. In this mass-radius diagram, OGLE planets are shown as red circles, planets from other
surveys are in orange, and planets from radial velocity surveys are in blue. The likelihood of the 5 OGLE discoveries as a result of
a Monte-Carlo draw is −8.7, the maximum likelihood is −7 and the standard deviation to maximum likelihood is 2.54. Hence, the
result of the OGLE planets mass-radius distribution is at 0.67σ of the maximum likelihood of the model.
Table 3. OGLE yields with fixed red noise level
Field Mean red RV follow-up Number of planets
of view noise level to Vmag detected simulated with
0 1.5 3
VHJ added (P < 2 days)
Bulge 3.6 17.5 2 0.4 0.6 0.9
Carina original 3.1 17.5 3 3.4 4.1 4.8
updated 2.1 17.5 +(0 − 1) +1.1 +1.1 +1.1
Centaurus 3.1 17.0 0 1.4 1.8 2.2
Total 6 6.3 7.6 9.0
done on Centaurus and Carina. We use unpublished information
from the OGLE/ESO follow-up team, who found one promising
planetary candidate among the Carina fields reprocessed with
the systematics- removal algorithm from Tamuz et al. (2005) and
none in the Centarus fields, with a magnitude limit near V=17
for the radial velocity follow-up. Table 3 compares the average
number of planets detected for 1000 Monte-Carlo draws to real
detections from the OGLE survey.
The total number of simulated discoveries obtained from this
quantitative analysis is in good agreement with the real detec-
tions. The differences in the number of detections between the
Carina and Centaurus surveys are mainly due to the lower duty
cycle of the observations towards Centaurus. A red noise level
fixed at 3.6 mmag in the direction of the galactic bulge bans
most hot Jupiter detections. The agreement between our quan-
titative result and the number of real detections is an indicator
of the global efficiency of our approach (stellar and planetary
distributions, evolution model and noise budget) for estimating
transit survey yield.
3.3.3. The OGLE yields with a variable red noise level
So far, we have considered the level of red noise to depend only
on the field considered. We attempt now to refine this by consid-
ering how the stellar density may affect it. Whereas most ground-
based transit surveys have a global red noise level from 2 to
3.5 mmag (Superwasp: Smith et al. (2006), Monitor: Irwin et al.
(2007), Hatnet: Pont & ISSI team (2007) and OGLE), the causes
of these noise levels seem different, with instruments ranging
from 10-cm wide field reflectors to deep-sky several-meter tele-
scopes. As seen from table 3, the OGLE fields have differ-
ent mean red noise levels (σr = 3.6 mmag for the bulge and
σr = 3.1 mmag for Centaurus and Carina before SYS-REM),
although the instrument and observational strategy were un-
Fig. 7. Depth of the planetary transit events versus magnitude of the parent star in the V band. The five confirmed OGLE detections
are shown as circles. Model results are shown as black plusses for detectable events and orange crosses for events that are considered
undetectable based on the photometric signal (see text). Blue diamonds correspond to events that would be detectable by photometry
alone but that cannot be confirmed by radial velocimetry. Note that the model results correspond to 3 times the full OGLE campaign
for more statistical significance. The OGLE planets depth-magnitude distribution is at 0.69σ from the maximum likelihood of the
model.
Fig. 8. Mass versus period of transiting giant planets. (OGLE planets are red circles, other transit surveys in orange, planets from
radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses). The OGLE planets
mass-period distribution is at 0.62σ from the maximum likelihood of the model (0.72σ considering the 11 planets discovered by
transit surveys and 0.66σ considering the 14 known planets).
Fig. 9. Distribution of the crowding index (see text) of target
stars in Carina (black) and in the bulge (red).
changed. Looking at what distinguishes these fields, it appears
that the most significant difference is the stellar density and
therefore the amount of crowding: The bulge field is about twice
as dense as the Carina and Centaurus fields. Pont & ISSI team
(2007) raise the suspicion that the level of red noise depends
strongly on the presence and characteristics of contaminating
stars, because e.g. of their different colors and differential re-
fraction in the atmosphere. It is hence natural to consider a red
noise that depends on a crowding index.
We define this crowding index as the fraction of the flux
coming from background stars versus that from the target in
the photometric aperture. Importantly, we do not consider stel-
lar companions as contributing to the red noise because they are
generally on the same CCD pixel as the target star and should
affect the noise budget much less.
Figure 9 shows the differences of crowding index for the tar-
get stars with planetary transits (detectable or not) in simulations
of the Carina and Bulge fields of view. The mean crowding in-
dex for target stars of I < 17 is 0.11 in the Carina field and 0.233
in the Bulge field.
We can exclude the fact that all red noise is linked with con-
tamination as many stars in the Carina fields are unblended by
background stars but still show a high noise level.
In order to estimate of the influence of the crowding on the
red noise level, we use the following simple relation between red
noise level and crowding index:
σr = α × Fb + β, (7)
where Fb is the fraction of total flux from background stars,
determined on a star-by-star basis in our simulations, and α
and β are parameters to be determined. This is justified by the
behaviour of the red noise seen for instance in SuperWASP,
showing a linear increase as a function of background flux
(Smith et al. 2006). In order to get the same mean red noise
values as Pont et al. (2006b), we obtain α = 0.4 mmag and
β = 2.65 mmag. This value of β corresponds to the minimum
red noise level obtained for non-contaminated stars in the OGLE
fields.
Table 4 shows the new number of detections when consid-
ering this crowding-dependant red noise level. Compared to ta-
ble 3, the number of detections is found to be essentially un-
changed for the Carina and Centaurus fields, but it increases by a
factor ∼ 3 for the bulge field. This result is more satisfactory be-
cause in the previous case, only ∼ 5% of the simulations would
yield the detection of 2 planets in the bulge, as observed.
3.3.4. Models, observations and the frequency of very
close-in planets
As discussed in § 1.4.1, three OGLE planets have orbital periods
shorter than 2 days and thus belong to a class of objects yet to be
detected by radial velocimetry. So far, we have added one such
planet (on average) to our carbon copy list of nearly 200 radial
velocimetry planets. In Section 3.3, we have shown that with this
assumption, radial-velocity and photometric transit surveys are
compatible. We now test the range of frequencies of very close-
in planets for which this remains true.
In order to do so, we compute the deviation from maximum
likelihood in the mass-radius diagram like in Section 3.3, as a
function of the number of planets which period is less than 2
days added to the RV list. The result is presented in Fig. 10 and
shows that a good match is obtained by adding 1 to 3 short-
period planets. Larger numbers are also possible from the point
of view of the transit surveys, but would conflict with their
non-detection by radial-velocimetry. Adding the other transit-
ing planets discovered thus far yields smaller probabilities of
occurence of these short-period planets, but not by significant
amounts.
All in all, and assuming that the radial velocity planets sam-
ple is unbiased, we constrain the fraction of main-sequence late
stars orbited by very hot giant planets with orbital periods less
than 2 days to be (1/1265)(1+0.33
−0.33) at a 60 % confidence level or
(1/1265)(1+0.83
−0.5 ) at a 90 % confidence level.
The distribution of planets in period between 2 and 5
days is directly obtained from the metallicity-linked distribution
(Santos et al. 2004) and the RV planets sample. Adding the dis-
tribution we found for planets between 1 and 2 days, we obtain a
fraction of (1/215) late main-sequence stars orbited by planets in
the 1 to 5 days period range, in good agreement with the results
obtained in Gould et al. (2006), who obtained (1/220)(1+1.10
−0.45).
Similarly, the distribution we obtain by cutting this sample into
two parts with the cut-off at 3 days is compatible, showing:
– a slightly higher fraction of really short-period planets (1-3
days) of (1/560) instead of (1/710)(1+1.10
−0.54) at a 90 % confi-
dence level in Gould et al. (2006).
– a similar fraction of short-period planets (3-5 days) of
(1/350) instead of (1/320)(1+1.39
−0.59) at a 90 % confidence level
in Gould et al. (2006).
The results presented hereafter use the variable red noise
level approach, and an RV planet list that is complemented with,
on average, 1.5 very-close in planets with periods P < 2 days
taken from the OGLE detections.
3.4. The metallicity of the stars harboring transiting planets
We now compare the metallicity of the parent stars for our ob-
served and modelled populations. A first test using the analyti-
cal scenario for the radial-velocity population (Fig. 11) yields a
clearly different metallicity distribution, with most of the tran-
siting planets observed around low-metallicity stars. We veri-
fied that this problem occurs independantly of the assumed stel-
lar metallicity distribution, for any realistic stellar population. It
arises fundamentally because the global metallicity bias as ob-
tained by Santos et al. (2004) or Fischer & Valenti (2005) is not
Table 4. OGLE yields with variable red noise level
Field RV follow-up Number of planets
of view to Vmag detected simulated with
0 1.5 3
VHJ added (P < 2days)
Bulge 17.5 2 1.2 1.6 2
Carina original 17.5 3 3.6 4.3 4.9
updated 17.5 +(0 − 1) +1.1 +1.1 +1.1
Centaurus 17 0 1.3 1.9 2.3
Total 5-6 7.2 8.9 10.3
Fig. 11. Period of transiting exoplanets versus metallicity of their parent star. The model is based on analytic relations for the mass
and period distributions of planetary companions (see § 1.4.2). (OGLE planets are red circles, other transit surveys in orange, planets
from radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses). The OGLE planets
period-metallicity distribution is at 2.94σ from the maximum likelihood of the model (2.51σ considering the 11 planets discovered
by transit surveys and 2.63σ considering the 14 known planets).
strong enough to compensate for the rarity of very metal-rich
stars in the Galaxy.
As seen in Fig. 12, the problem disappears when one con-
siders the carbon-copy model. Thus, we are led to an important
conclusion, that the metallicity distribution of pegasids (periods
shorter than 10 days) is fundamentally different from the global
exoplanet population. More specifically, there are no Pegasids
orbiting F, G, K stars with metallicities smaller than [Fe/H]=
−0.07. This has strong consequences for planet formation mod-
els (see also Guillot et al. 2006). This work shows that this con-
clusion is robust, and is needed to explain the results of the pho-
tometric surveys.
A finer examination of Fig. 12 shows that while our model
planets reproduce globally the metallicity of the ensemble of
transiting planets, OGLE stars with planets are on average ∼
0.1 dex more metal-rich.
This can tentatively be explained with a metallicity gradient
in the galaxy for OGLE TR-10 ([Fe/H] = 0.28±0.10) and OGLE
TR-56 ([Fe/H] = 0.19± 0.07), the two planets discovered in the
direction of the galactic bulge. The study of galactic cepheids by
Andrievsky et al. (2004) shows a metallicity gradient as a func-
tion of distance to the galactic center. In the [6.6, 10.6] kpc-range
distance from galactic center, this study finds a linear relation be-
tween [Fe/H] and galactocentric distance RG:
[Fe/H] = −0.044(±0.004)RG + 0.363(±0.032) (8)
Following that relation, the two stars with planets discovered in
the direction of the galactic bulge both at a distance around 1500
pc would thus be in a 0.04 dex more metal rich region than the
solar neighborhood.
Concerning the high metallicity of stars with transiting plan-
ets discovered by OGLE in the Carina region, we do not have
any reason to think that the metallicity distribution would be
different from the solar neighborhood. Our only hypothesis is
a low-probability draw for metallicity for the 3 OGLE-Carina
planets.
Fig. 12. Period of transiting exoplanets versus metallicity of their parent star. The figure differs from Fig. 11 in that our fiducial
model, i.e. the mass-period-metallicity “carbon-copy” model is used (see § 1.4.2). (OGLE planets are red circles, other transit
surveys in orange, planets from radial velocitiy surveys in blue. Simulated planets detected: black plusses, under threshold: or-
ange crosses). The OGLE planets period-metallicity distribution is at 0.76σ from the maximum likelihood of the model (0.36σ
considering the 11 planets discovered by transit surveys and 0.39σ considering the 14 known planets) .
3.5. Atmospheric potential energy and orbital distances
Because evaporation may affect the planet population, it is in-
structive to check whether the potential energy of the atmosphere
and the orbital period, two crucial quantities for this process (e.g.
Lecavelier des Etangs et al. 2004), also possess a relatively con-
sistent distribution. We first test the behavior of the analytical
model for the distribution of planets (Fig. 13). This results in
a prediction of many planets with large radii (small values of
the potential energy for atmospheric escape) at small orbital dis-
tances, in patent contradiction with the observations.
The problem mostly disappears with the carbon-copy model:
Fig. 14) shows that in this case, although we do not obtain a
linear correlation between the two variables, we get detections
in the right area of the diagram. This is explained as stemming
from:
– The absence of low-mass planets at small orbital distances,
with a possible limiting relation between these two quantities
(Mazeh et al. 2005);
– The difficulty in detecting planets with larger values of po-
tential energy per unit mass (smaller radii) at large orbital
distances –although we predict that some of these should be
detected by future transit surveys.
Our results strengthen the case for the existence of a rela-
tion between mass and orbital distance for short-period plan-
ets, as advocated by Mazeh et al. (2005): Indeed, the analytic
model which is characterized by the presence of small mass
planets at small distances yields a distribution of detectable plan-
ets that is significantly different from the observations (Fig. 13).
Our carbon-copy model that includes implicitely this correlation
does not (Fig. 14).
3.6. Planetary radii and stellar irradiation
Radius and stellar irradiation (or equivalently equilibrium tem-
perature) should be positively correlated, as a planet with a
higher irradiation dose will tend to cool and contract more
slowly than one that endures less stellar insolation. As Fig. 15
shows, the correlation exist, but is weak, and with a signficant
scatter. This is well reproduced by the model.
However, it can be noted that HD 149026 b lies away from
the cloud of points. In general, we find that our fiducial model
generates few points in this region. This can be easily accounted
for by slightly modifying the metallicity-core mass relation to
allow for larger masses. As planets of small masses and large
core masses are more difficult to model anyway, we chose not to
attempt fine-tuning the model to this level of detail. This should
be postponed for further studies, especially with the discovery of
more Saturn-mass transiting planets.
3.7. The mass-radius relation
We have checked that our fiducial model predicts the detection
of transiting planets with properties that are globally consistent
with the observations. We can now examine in more detail the
mass-radius relation thus obtained, as it is directly tied to as-
sumptions on the compositions and evolutionary models of ex-
oplanets. The predictions also have implications for transit sur-
veys as it is not clear whether they have detected only the “tip of
the iceberg”, ie the few largest giant planets while many smaller
ones would lie undetected or not.
Results with our fiducial model are presented in Fig. 16.
We find that planets with low masses (say, less than Jupiter’s
Fig. 13. Potential energy per unit mass (Ep = GM/R) versus orbital period of transiting planets. (OGLE planets are red circles, other
transit surveys in orange, planets from radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold:
orange crosses). Observations are compared to models based on the analytical relations for the mass and period distribution of
planetary companions (see § 1.4.2). The OGLE planets energy-period distribution is at 2.18σ from the maximum likelihood of the
model (1.86σ considering the 11 planets discovered by transit surveys and 2.47σ considering the 14 known planets).
Fig. 14. Potential energy per unit mass versus orbital period of transiting planets. The figure is similar to Fig. 13, except for the
fact that our fiducial model is used (see § 1.4.2). (OGLE planets are red circles, other transit surveys in orange, planets from radial
velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses). The OGLE planets energy-
period distribution is at 0.55σ from the maximum likelihood of the model. (0.84σ considering the 11 planets discovered by transit
surveys and 0.66σ considering the 14 known planets)
Fig. 15. Radius as a function of equilibrium temperature of transiting exoplanets. (OGLE planets are red circles, other transit surveys
in orange, planets from radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses).
The OGLE planets equilibrium temperature-radius distribution is at 1.22σ from the maximum likelihood of the model (1.05σ
considering the 11 planets discovered by transit surveys and 2.25σ considering the 14 known planets).
mass) can both have very large or very small radii, depending
on whether they contain a significant mass in heavy elements
or not. On the other hand, massive planets have radii which are
comparatively better defined. This is mostly due to the fact that
we assume a maximum mass of heavy elements of 100 M⊕, a
hypothesis that will be tested directly by the discovery of a few
massive transiting planets.
Our model results once again agree well with the detections
made by photometry. Importantly, the yellow crosses in Fig. 16
do not lie significantly below the black ones: we predict that
future surveys will not discover a population of small-sized giant
planets, at least for masses above that of Saturn.
The presence of planets with larger masses of heavy ele-
ments should remain marginal because otherwise they would
have been detected by present-day surveys, Fig. 16 showing
that planets below 1 RJup are already detectable, although in fa-
vorable cases (small radius of the primary and bright targets).
Quantitatively, simulations in the OGLE fields indicate that if
planets had radii uniformly distributed between 0.5 and 1.5 RJup,
18.5% of the planets discovered by the survey would have radii
below 1 RJup. This fraction is not negligible and is (although
marginally) inconsistent with the sample of 0/11 planets with
R < RJup discovered by transit surveys thus far.
Therefore, although we cannot statistically rule out the pres-
ence of a population of small planets, these would require the
formation of extremely metal-rich planets. Our prediction is a
consequence of evolution models and of our assumption that
planets with masses of heavy elements beyond 100 M⊕ should
be rare.
Figure 17 shows the ensemble of planets obtained for an ex-
tremely large number of draws, with our fiducial model. Voids
in the ensemble of crosses correspond to the absence of planets
with these masses in the radial-velocimetry list. They should not
be considered as significant. The contours in the figure indicate
the ensemble of masses and radii expected for planets with dif-
ferent masses of heavy elements, from 0 to 100 M⊕. Importantly,
the location of these contours is linked to our assumption of an
energy source in the planetary interior equal to 0.5% times the
irradiation received by the planet. Independently of the details of
this assumption, this shows that a statistically significant ensem-
ble of known transiting planets would allow a determination of
the presence or lack of heavy elements in these objects.
We have also tested another assumption regarding the plan-
etary evolution model: all planets possess 20 M⊕ mass in heavy
elements, 70% of them have no extra heat source, whereas 30%
have 3 × 1026 erg s−1 dissipated at the center. With this assump-
tion, one can qualitatively explain the observed transiting planets
(i.e. the “normal” planets and the “anomalously large” ones, re-
spectively) with the exception of HD 149026 b, for which one
could argue that the planet comes from a different population. In
this case, Fig. 18 shows a distribution of radii that is relatively
similar to the previous one (Fig. 16), with the exception that no
planet has a radius smaller than 0.8 RJup. In this case, the 2 re-
gions corresponding to the “standard” model, and to the “heat
dissipation” case are clearly different, especially at the low-mass
range of the diagram.
Present observations cannot distinguish between the two
models, showing the need for additional detections of transit-
ing giant planets. Particularly important are planets between the
mass of Saturn and that of Jupiter, as this is a mass regime where
expected compositional differences have the largest impact.
Fig. 16. Mass-radius relation for transiting extrasolar giant planets. (OGLE planets are red circles, other transit surveys in orange,
planets from radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses). The OGLE
planets mass-radius distribution is at 0.67σ from the maximum likelihood of the model (0.72σ considering the 11 planets discovered
by transit surveys and 0.97σ considering the 14 known planets) .
4. Conclusions
We have presented a simulation of photometric transiting sur-
veys based on basic knowledge of the stellar and planetary pop-
ulations in the galactic neighborhood and on a planetary evo-
lution model tuned to the information obtained from transiting
giant planets with masses above that of Saturn. This simulation
was applied to the OGLE survey, and shown to yield a generally
excellent agreement with the transiting planets detected by the
survey.
We have thus shown that radial velocimetry and photometric
surveys are compatible within statistical uncertainties, in agree-
ment with Gould et al. (2006). We have derived a frequency of
very close-in planets with orbital periods shorter than 2 days
around solar-type stars, of (1/1265)(1+0.33
−0.33) at a 60 % confidence
level or (1/1265)(1+0.83
−0.5 ) at a 90 % confidence level.
Using null results by photometric surveys for given ranges of
parameters, we are able to strengthen two results already present
in the radial velocimetry data:
– Stars with low metallicities ([Fe/H]< −0.07) do not, or are
very unlikely to harbour close-in giant planets with orbital
periods P < 10 days. This is unlike stars above that metallic-
ity threshold (see Fig. 12).
– There is a lack of small-mass giant planets below the mass
of Jupiter and above that of Saturn for orbital periods P <
3 days (see Fig. 8).
Further data is required to precisely quantify these empirical re-
sults that bear important consequences for our understanding of
planet formation and migration.
On the basis of our model, and assumptions concerning the
composition of giant planets (i.e. masses of heavy elements be-
tween 0 and 100 M⊕), we find that the present detections of tran-
siting planets have sampled a population that is quite represen-
tative of the main population of giant planets, at least for the
ones that are above about half the mass of Jupiter. We hence pre-
dict that future transit surveys with higher sensitivities will not
discover a significant population of yet undetected Jupiter-mass
planets with small sizes, i.e. radii smaller than that of Jupiter (see
Fig. 16).
Many ground-based transit surveys are in progress, and
with the space missions CoRoT (Baglin et al. 2002) and Kepler
(Borucki et al. 2003), the number of known transiting planets is
expected to rise rapidly over the next few years. This will en-
able us to better test the models and quantify some of the results
presented in this article. We also hope to be able to discriminate
between various models of the evolution and compositions of gi-
ant planets, a matter of great importance for formation models.
We wish to stress however that a continuation of ground-
based transit surveys is desirable even in the presence of simi-
lar programs from space. CoRoT will survey 60,000 dwarf stars
over five 150 days periods and Kepler about 100,000 over 4
years, implying a maximum potential yield of 55 and 90 transit-
ing giant planets, respectively, plus many other smaller planets.
For what concerns giant planets, quantifying the fraction of very
close-in planets with a 10% accuracy at the 3σ level would re-
quire the discovery of ∼ 200 transiting planets. Understanding
the evolution and compositions of giant planets will require an
even larger number of detections. The radius of a giant planet
itself depends mainly on four parameters: the planetary mass,
equilibrium temperature, age, and its composition (note that the
composition can be considered as a simple parameter only in the
case of planets mostly made of hydrogen and helium: smaller
planets will be more difficult to model!). Additional energy
Fig. 17. Mass-radius relation for a very large number of Monte-Carlo trials using the fiducial model. The curves show the ensemble
of planets with masses of heavy elements between 0 and 25, 25 and 50, 50 and 75, 75 and 100 M⊕, respectively. Symbols are as in
Fig. 8.
Fig. 10. Deviations from a maximum likelihood obtained as a
function of NVHJ , the number of very hot jupiter of orbital pe-
riods shorter than 2 days added to the radial velocities carbon-
copy list. Thick line: Deviation from the maximum likelihood
obtained in the mass-radius diagram for the OGLE planets. Thin
line: Same deviation but when compared to the ensemble of
planets. Dashed line: Standard deviation obtained from a com-
parison between the number of simulated planets and the number
of detected ones for the OGLE survey (see table 4). Dotted line:
Standard deviation obtained from the non-detection of these very
close-in planets by radial-velocimetry.
sources may occur (such as in the presence of tidal heat dissipa-
tion), and the initial conditions and formation history may have
their say in the matter as well. Furthermore, the observational
uncertainties are generally large. For example, the planetary ra-
dius is generally only known to∼ 10%, for a global variation that
is relatively small (1 to 1.5 RJup). This implies that to constrain
a given correlation to, say 10%, and with four independant vari-
ables, hundreds of data points are needed, and thousands would
be desirable.
This motivates us to seek programs capable of detecting
thousands of transiting planets in the mid-term future, and ways
to reduce the error bars on the different parameters. One direc-
tion is to test the Dome C plateau in Antarctica for such an am-
bitious program, which is the purpose of A STEP (Fressin et al.
2005). Other directions exist, such as proposals for similar sur-
veys from space. In any case, it is most important that a statisti-
cally significant population of exoplanets be characterized for a
better understanding of planet formation and our origins.
Acknowledgments
The code used for this work, CoRoTlux, has been devel-
opped as part of the CoRoT science program by the au-
thors with major contributions by Aurélien Garnier, Maxime
Marmier, Martin Vannier, Suzanne Aigrain and help from Claire
Moutou, Stéphane Lagarde, Antoine Llebaria, Didier Queloz
and François Bouchy. We want to thank Andrzej Udalski
and Michael Gillon for their communications on OGLE data,
Frédéric Thévenin for his advices on stellar populations sim-
ulation. F.F. has been funded by grants from the French
Ministère de la Recherche and by the Société des Amis des
Sciences. V.M. was funded by a grant from the C.N.R.S.. This
work has used extensively Jean Schneider’s exoplanet database
www.exoplanet.eu, and the Besançon model of the Galaxy at
physique.obs-besancon.fr/modele/. The planetary evolu-
tion models used for this work can be downloaded at www.obs-
nice.fr/guillot/pegasids/.
with heat dissipation
stand
ard m
Fig. 18. Mass-radius relation obtained for an alternative model with 70% of “standard” planets with no extra-energy source, and
30% planets receiving an additional 3× 1026 erg s−1 luminosity dissipated at the center. All planets are assumed to possess 20 M⊕ in
heavy elements. Symbols are as in Fig. 8.
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	Simulating transit surveys
	General remarks
	Principle of the simulations
	The stellar population
	Main targets and background stars
	Binary and triple systems
	The planetary companions
	Planet incidence
	Planetary masses and orbits
	Physical characteristics and the planetary evolution model
	Modeling transit events and their detectability
	PSFs and CCDs
	Noise budget and event detectability
	The OGLE survey: input parameters
	Basic parameters and observational procedure
	Modelling the detection threshold
	Confirmability of transit-like events with follow-up
	Results of the simulations
	Deviation of OGLE planets from maximum likelihood of the simulations
	Depth of the transit events and magnitude of the targets stars
	Compatibility of transit surveys with radial-velocimetry observations
	Compatibility in the mass-period diagram
	The OGLE yields with a fixed red noise level
	The OGLE yields with a variable red noise level
	Models, observations and the frequency of very close-in planets
	The metallicity of the stars harboring transiting planets
	Atmospheric potential energy and orbital distances
	Planetary radii and stellar irradiation
	The mass-radius relation
	Conclusions

  Transiting extrasolar planets are now discovered jointly by photometric
surveys and by radial velocimetry. We want to determine whether the different
data sets are compatible between themselves and with models of the evolution of
extrasolar planets. We simulate directly a population of stars corresponding to
the OGLE transit survey and assign them planetary companions based on radial
velocimetry discoveries. We use a model of the evolution and structure of giant
planets assuming a variable fraction of heavy elements. The output list of
detectable planets of the simulations is compared to the real detections. We
confirm that the radial velocimetry and photometric survey data sets are
compatible within the statistical errors, assuming that planets with periods
between 1 and 2 days are approximately 5 times less frequent than planets with
periods between 2 and 5 days. We show that evolution models fitting present
observational constraints predict a lack of small giant planets with large
masses. We also identify distinct populations of planets: those with short
periods (P < 10d) are only found in orbit around metal-rich stars with [Fe/H] >
-0.07. We further confirm the relative absence of low-mass giant planets at
small orbital distances.

Introduction
Extrasolar planets are now routinely discovered orbiting
solar-type stars by radial velocimetry, but the discovery of tran-
siting planets by photometric surveys is just beginning. Although
still marginal, the late success of transit surveys has given an
additional impulse to exoplanetology with the possibility to es-
timate the radius, density and hence composition of extrasolar
planets.
Quantitatively, we know to date 206 extrasolar planets with
masses below 13 MJup (e.g. Udry et al. 2007; Butler et al. 2006).
Among those, a list of 14 currently known transiting planets is
presented in table 1. These planets have been discovered by ra-
dial velocimetry followed by photometry for 3 of them, and by
photometric surveys for the remaining 11.
When considering the score of projects devoted to the detec-
tion of planets by transit photometry, the present harvest appears
meager. The discrepancy between predictions (e.g. Horne 2001)
and reality has been attributed to various factors such as: im-
perfect duty cycle, a reduced number of stars for which tran-
siting planets are detectable (Gould et al. 2006) and the pres-
ence of correlated noises that can greatly limit the detectability
of small planetary transits (Pont et al. 2006b). Several generic
studies have been conducted to understand the yield of different
transit surveys. Pepper & Gaudi (2005) studied the optimization
of transit searches as a function of the observational setup, the
site properties and the planet properties. Gillon et al. (2005) an-
alyzed and compared deep field surveys, considering individual
stellar ranges and observation windows, but did not include the
effects of stellar crowding nor time-correlated noises.
Gould et al. (2006) studied the yield of OGLE survey
(Udalski et al. 2002), the most successful so far in term of num-
ber of transiting planets discovered, with a model populating the
line of sight with stars drawn from the Hipparcos Catalogue.
They estimated with that model the proportion of stars with sen-
sitivity to close-in giant planets to derive from OGLE results
the frequency of planets as a function of their period. They find
that the yield of the OGLE survey is globally consistent with
the detections by radial velocimetry and with planet radii dis-
tributed between 1 and 1.25 jovian radii. The aim of the present
work is to further test these data sets and the underlying physical
model by a forward calculation of transit events with realistic
stellar and planetary populations. In particular, we include up-
to-date models of the evolution and structure of Pegasids (close-
in extrasolar planets) based on models reproducing the observa-
tional constraints from known transiting planets (Guillot 2005;
Guillot et al. 2006). As a consequence, we should be able to de-
termine whether the presently known population of transiting
planets represent the “tip of the iceberg”, i.e. that many more
small, dense extrasolar giant planets exist and await discovery
by the transit method, or whether it is relatively representative
of the global population.
We first describe the model that is used to simulate transit
surveys in general. In Section 3, we describe more particularly
the OGLE surveys and the hypothesis chosen for their mod-
elling. We then discuss the results of the simulation. A summary
of the main conclusions and predictions for future transit surveys
are provided in Section 5.
1. Simulating transit surveys
1.1. General remarks
The search for planets in transit in front of their star naturally
arised with the discovery that a non-negligeable fraction of plan-
ets orbit very close to their stars. If orbital planes are randomly
oriented, the probability that a planet will transit in front of its
star at each orbital revolution is:
Ptransit ≃ R⋆/aplanet, (1)
where R⋆ is the stellar radius, and aplanet the planet’s orbital semi-
major axis. For systems such as 51 Peg b, this probability is close
to 10%. Because the probability for a solar-type star to possess
such a Pegasid (i.e. a 51 Peg b-like planet, planets semi-major
axis up to 0.1 AU) is about 0.5% (e.g. Marcy et al. 2005), 1 in
2000 solar-type star should possess a transiting Pegasid. Using
current results from radial velocity surveys and integrating over
all periods, we estimate that about 1 in 1100 solar-type stars
possesses a transiting giant planet. Of course, depending on the
magnitudes and field considered, giant stars may severly out-
number the dwarfs, so that in a real field, only one in, say, 3000
stars may harbor a transiting giant planet.
Because of the dependence on a, and period distribution,
most of the transit events concerning giant planets occur for or-
bital periods between 1 and 5 days. The transits typically last for
a couple of hours, as this quantity is weakly dependant on the
orbital period P:
τtransit = 1.82
1 day
)1/3 (
)−1/3 (R′⋆
hours, (2)
where R′⋆ is the length of the cord traced on the stellar disk by
the planet’s trajectory. (more precisely: R′⋆ = R⋆ cos b + Rplanet,
where b is the impact parameter of the transit).
The depth of the transits themselves is directly given by the
ratio of the planetary to the stellar disk surfaces:
Rtransit ≃ (Rplanet/R⋆)
. (3)
This value is of order 1% for a Jupiter-size planet orbiting a
Sun-like star. For an F-type star with radius ∼ 1.2 R⊙, the ratio
decreases to 0.7%. Furthermore, transiting giant planets discov-
ered so far have radii between 0.72 and 1.44 RJup (see table 1).
Allowing for stellar radii to vary between 0.8 and 1.3 R⊙ (a typ-
ical range, in magnitude limited surveys), this implies that we
should expect Rtransit to vary between 0.3% and 3%, for giant
planets only. The lower limit is in reality even smaller because
for detection purposes we have to account for the fact that plan-
ets also orbit stars that are in multiple systems (like HAT-P-1),
and hence a dilution factor may apply. Although grazing transits
are ignored in this simple analysis, they are included afterwards
in our simulations.
This altogether implies that in order to detect transiting gi-
ant planets, many thousands of dwarf stars have to be monitored
over periods of weeks for a photometric precision reaching be-
low a fraction of a percent on an equivalent integration time of
about one hour. This is typically done by following a relatively
dense stellar field over a long time with a stable telescope, and
using a camera equiped with a good CCD camera.
1.2. Principle of the simulations
On paper, the simulation of the forward problem is simple: one
has to generate a complete stellar field, or obtain it from observa-
tions, put it on a discrete grid (the CCD), include on probabilistic
arguments the planetary companions, calculate lightcurves in-
cluding the various sources of noise, and determine which events
are detectable. This is the principle of CoRoTlux, a code we first
developed to predict the transit yield of CoRoT space telescope
http://arxiv.org/abs/0704.1919v1
Table 1. Known transiting planets by 2006⋆
# Name Mplanet Rplanet Period a M⋆ R⋆ Teff Metallicity
[MJup] [RJup] [day] [AU] [M⊙] [R⊙] [K] [Fe/H]
OGLE planets
6 OGLE-TR-10 0.63±0.14 1.26+0.07
−0.07 3.10129 0.04162 1.18 ±0.04 1.16±0.06 6075±86 0.28±0.10
2 OGLE-TR-56 1.17±0.04 1.32+0.06
−0.06 1.211909 0.0225 1.17±0.04 1.32±0.06 6119±62 0.19±0.07
5 OGLE-TR-111 0.52±0.13 1.067+0.054
−0.054 4.0144479 0.047 0.81±0.02 0.831±0.031 5044±83 0.19±0.07
3 OGLE-TR-113 1.35±0.22 1.09+0.03
−0.03 1.4324757 0.0229 0.78±0.02 0.77±0.02 4804±106 0.15±0.10
4 OGLE-TR-132 1.14±0.12 1.18+0.07
−0.07 1.689868 0.0299 1.26±0.03 1.34±0.08 6210±59 0.37±0.07
Other transit survey planets
7 TrES-1 0.76±0.05 1.081+0.029
−0.029 3.0300737 0.0393 0.89±0.035 0.811±0.020 5250±75 -0.02±0.06
11 TrES-2 1.28±0.07 1.24+0.09
−0.06 2.47063 0.0367 1.08±0.08 1.00±0.05 5960±100 0.15±0.03
10 XO-1 0.90±0.07 1.184+0.028
−0.018 3.941634 0.0488 1.0±0.03 0.928±0.033 5750±13 0.015±0.03
12 HAT-P-1 0.53±0.04 1.36+0.011
−0.09 4.46529 0.0551 1.12±0.09 1.15±0.09 5975±45 0.13±0.02
13 WASP-1 0.867±0.073 1.443+0.039
−0.039 2.519961 0.0382 1.15±0.09 1.453±0.032 6200±200
14 WASP-2 0.88±0.07 1.038+0.05
−0.05 2.152226 0.0307 0.79±0.08 0.813±0.032 5200±200
Transiting planets discovered with Radial velocities
9 HD189733 1.15±0.04 1.154+0.032
−0.032 2.218573 0.0313 0.82±0.03 0.758±0.016 5050±50 -0.03±0.04
8 HD149026 0.330±0.02 0.726+0.064
−0.064 2.87598 0.042 1.3±0.1 1.45±0.1 6147±50 0.36±0.05
1 HD209458 0.657±0.006 1.320+0.025
−0.025 3.52474859 0.047 1.09±0.09 1.148±0.002 6117±26 0±0.02
MJup = 1.8986112 × 10
30 g is the mass of Jupiter. RJup = 71, 492 km is Jupiter’s equatorial radius.
OGLE-TR-10: Bouchy et al. (2005); Udalski et al. (2002); Konacki et al. (2005); Santos et al. (2006); Pont et al. (2006a)
OGLE-TR-56: Konacki et al. (2003); Udalski et al. (2002); Torres et al. (2003)
Bouchy et al. (2005); Santos et al. (2006); Pont et al. (2006a)
OGLE-TR-111: Pont et al. (2004); Santos et al. (2006); Udalski et al. (2002); Winn et al. (2007); Bouchy et al. (2005)
OGLE-TR-113: Bouchy et al. (2004); Udalski et al. (2002); Konacki et al. (2004); Gillon et al. (2006)
OGLE-TR-132: Bouchy et al. (2004); Udalski (2003); Moutou et al. (2004); Magain et al. (2007)
TRES-1: Alonso et al. (2004); Laughlin et al. (2005); Winn et al. (2007)
TRES-2: O’Donovan et al. (2006)
XO-1: McCullough et al. (2006); Holman et al. (2006); Wilson et al. (2006)
HAT-P-1: Bakos et al. (2006)
WASP-1: Collier Cameron et al. (2006); Shporer et al. (2007); Charbonneau et al. (2006)
WASP-2: Collier Cameron et al. (2006); Charbonneau et al. (2006)
HD-189733: Bouchy et al. (2005); Bakos et al. (2006)
HD-149026: Sato et al. (2005); Charbonneau et al. (2006)
HD209458: Brown et al. (2001); Cody & Sasselov (2002); Wittenmyer et al. (2005); Winn et al. (2005); Knutson et al. (2007)
# is the label of planets in figures ; they are ranked in the order of their discovery.
(Baglin et al. 2002) and quantify the need for follow-up obser-
vations, which is here applied to the case of OGLE.
The interesting point of such a forward simulation is the pos-
sibility to include relatively easily fine details such as the fact
that planets are found more frequently around metal-rich stars,
or, on the basis of planetary evolution models, the fact that young
planets orbiting close to bright stars will be larger than old plan-
ets orbiting smaller stars at larger orbital distances. This requires
however that a relatively large number of physically relevant
parameters (for example, the mass, size, metallicity, age of the
stars) be properly defined.
We further detail the assumptions that we made for these
simulations by describing how we generate the stellar and plane-
tary populations, and how we attempt to include realistic sources
of noise.
1.3. The stellar population
1.3.1. Main targets and background stars
Stellar fields differ enormously in terms of densities and stellar
populations. It is therefore most important to properly account
for this in order to simulate any given transit survey.
It would be very appealing to use direct observations as much
as possible to closely match the observed target fields. But as
we will see hereafter, many different characteristics of the stars
(stellar metallicity, age and subtype ...) are required, and these
are difficult to obtain with generic observations. We therefore
adopt the following procedure:
– The observed stellar densities are obtained from stellar
counts by magnitude, on the real stellar fields (see § 2.1)
– The characteristics of the stars are obtained following a
Monte-Carlo method using the output of the Besançon model
of the galaxy (Robin et al. 2003) obtained for the proper lo-
cation of the survey.
– Where stellar counts are not available, or uncomplete (i.e.
for faint stars), we use both stellar counts and characteristics
from the Besançon model.
Specifically, we keep track of the following parameters ob-
tained directtly from the Besançon model:
– The mass of each star, used to compute orbital parameters of
the transiting object;
– The apparent magnitude of the star in the observed spectral
range (the I filter in the case of the OGLE survey);
– The V magnitude of the star, important to qualify the con-
firmability of a transit event with radial velocimetry;
– The surface temperature of the star
– The luminosity of the star, calculated from its absolute mag-
nitude;
– The radius of the star, calculated from total luminosity and
effective temperature.
Stellar radius [RSun]
Stellar mass [MSun]
Effective temperature [K]
Fig. 1. From top to bottom: Distribution functions for the radii,
masses and effective temperatures for our fiducial stellar popu-
lation corresponding to the simulated OGLE Carina field. The
black line represents the ensemble of stars in the field. The filled
red region is a subset for dwarf stars with stellar type F4 and
later, as these are the only stars for which a transiting planet has
a reasonnable chance of being detected by present-day transit
surveys.
The mass, and effective temperature of the stars are dis-
tributed linearly around values given by the Besançon model
(at ±20%). Figure 1 shows a simulated distribution of stars for
the OGLE Carina field. The ensemble of dwarf stars with types
F4 and later are highlighted as these represent targets for which
planetary transit events are detectable, and, within observational
limits, confirmable by radial velocimetry.
The metallicity distribution is obtained from the model
of Nordström et al. (2004), which is based on the Geneva-
Copenhagen survey of the Solar neighbourhood. These authors
find that the distribution of the metallicities [Fe/H] is well ap-
proximated by a Gaussian function with a mean of −0.14 and
a dispersion of 0.19 dex. We use this gaussian distribution and
choose to ignore possible dependencies between stellar parame-
ters (e.g. masses, ages...) and the metallicities. (The link between
stellar type and metallicity appears to be negligible for F4 and
later types anyway (F. Thévenin, pers. communication 2007)).
1.3.2. Binary and triple systems
Multiple stellar systems are important especially because of the
possibility that stellar eclispes mimic planetary transits (Brown
2003). However, we choose to defer this problem to a later ar-
ticle. Multiple systems are taken into account anyway because
they can yield a dilution of the planetary transit events that
makes them more difficult to detect. Planets may be present in-
differently on the primary, secondary or tertiary components of
a stellar system. (However, we find that only planets around the
primary targets have a non-negligible chance of being discov-
ered by current ground based photometric survey.)
Specifically, following Duquennoy & Mayor (1991), we
consider that 50% of the stars are binaries and 20% of those
are ternaries. Multiple systems are considered as individual stars
at the same position on the CCD. We choose to estimate their
properties more simply than for the other stars, on the basis of
DM91:
– We randomly add companions to the initial draw of primary
stars, without changing their properties. The total mass and
luminosity of each multiple system is thus slightly higher
than initially.
– The mass ratio (primary/secondary) is defined as a gaussian
of median value 0.23 and a full width at half maximum of
0.42. Outside a range of 0.05 and 1, we redraw the mass
ratio.
– The radius is defined as R⋆ = R⊙(M⋆/M⊙) when M⋆ ≤ M⊙
and R⋆ = R⊙(M⋆/M⊙)
1/2 otherwise.
– The luminosity is assumed to be proportional to M2 so that:
lsecondary = lprimary(Msecondary/Mprimary)
– Other stellar parameters are calculated on the basis of these
ones and of those of the primary component (same age, same
distance, same metallicity).
– Triple components are treated with the same method as bi-
naries, and are defined relatively to the primary star.
1.4. The planetary companions
With more than 200 planets known to orbit stars other than our
Sun, we are beginning to have a rather precise view of at least
part of this population. We can expect that biases on the detec-
tions are small in the case of massive planets (the mass of Saturn
and more) and planets that are relatively close to their star (or-
bital distances smaller than ∼ 1 AU). These two conditions hap-
pen to match quite exactly the requirement for detectability by
transit photometry, with one assumption: that only massive giant
planets can have large radii. Although not proven, this assump-
tion seems quite reasonnable.
Hence we choose to focus this study on this well-
characterized population of objects. From the current list of 209
detected exoplanets, we select the ones discovered by radial ve-
locities with mass higher than 0.3 Jupiter masses and known host
star metallicity. Our list of planets includes 153 objects, to which
we may add very-close in planets detected by transit photometry,
as discussed below. We are using this list as representative of an
unbiased sample of giant planets known from radial velocimetry,
even though planetary distribution models may have been made
from slighlty different samples.
1.4.1. Planet incidence
A first important step is the determination of the probability for a
star to harbor a planet. As shown by numerous studies (Gonzalez
1998; Santos et al. 2004; Fischer & Valenti 2005), this probabil-
ity depends mostly on the metallicity of the parent star. Figure 2
shows one such probability function, as well as the result in
terms of planet counts on a simulated stellar field.
In this work, we will use the dependency from Santos et al.
(2004) shown in Fig. 2. Several points are to be considered how-
ever:
1. This probability relation is only valid for solar-like stars, i.e.
F, G, K dwarf stars. Although there are strong indications
that it may change for other stars (e.g. M dwarfs), we will
assume it to hold independently of stellar properties. This
[Fe/H]
[Fe/H]
Fig. 2. Upper panel: Probability for a solar-type star to possess
a giant planet companion as a function of the stellar metallicity
(from Santos et al. 2004). Lower panel: Relative normalised dis-
tributions of stellar metallicities for stars in the field (black line),
and for stars with a giant planet companion (red line).
assumption is sufficient because F, G and K dwarf stars form
by far the majority of stars with detectable planets in photo-
metric surveys.
2. This relation has been calculated independently of the prop-
erties of the planetary companion, in particular orbital dis-
tance. Because in our case we are strongly biased towards
short-period planets, the distribution may be different. This
point will be considered in § 3.4.
3. The possibility of multiple planetary systems is not consid-
ered. This approach is justified because the probability that
several planets belonging to the same system are transiting
planets is small for giant planets.
1.4.2. Planetary masses and orbits
The masses and orbital characteristics of the planet population
are inferred almost entirely from the present radial-velocimetry
surveys. This technique yields an accurate determination of the
orbital period, and less accurately, of the eccentricity of the orbit.
It also yields the value of the mass of the planetary companion
times the sine of the orbital inclination from the knowledge of
the mass of the parent star. With these values, we can then derive
from a random inclination of the orbital planes the planets that
are transiting and those that are not as well as the characteristics
of their orbit.
We test several approaches for the derivations of these quan-
tities:
– An analytical model: In this approach, we consider inde-
pendantly the planet period and its mass. The period of the
planet Π follows the model of Brown (2003), the proba-
bility density P from a piecewise linear fit to the distribu-
tion P(logΠ) = {0.509, 0.165, 0.533} for three period inter-
vals bounded by logΠ = {0.43, 0.65, 2.3, 3.5}. The distribu-
tion in mass is linear in log from 0.3 to 10 Jupiter masses
(Zucker & Mazeh 2001). There is no dependency of these
two parameters linked to metallicity.
– The radial velocity mass-period “carbon-copy” model: A
second approach is to make direct use of the list of planets
discovered by radial velocimetry. This is possible because in
terms of masses and orbital periods the list is almost unbi-
ased for the objects that we consider (massive enough to be
above detection thresholds, and with periods much shorter
than the lifetimes of the surveys). In this case, we select plan-
ets randomly in the RV list, and then allow for a small ran-
dom deviation of their mass and period (a uniform deviation
from −20% to +20%) in order to avoid too much cluster-
ing on the same value. This is particularly important in the
case of the period because of the importance of stroboscopic
effects in planetary transits (e.g. Pont et al. 2005).
– The radial velocity mass-period-metallicity “carbon-copy”
model: As a modification to the previous approach, we also
consider using the metallicity entry in the RV list, because
of correlations between metallicity and orbital period that
are otherwise not taken into account (see discussion in sec-
tion 3). We proceed slightly differently however than for the
mass and orbital period because of the limitations caused
by the finite number of planets in the RV list. In this case,
we choose to split the list into two low- and high-metallicity
lists, and then select the mass and periods in the relevant list.
Our fiducial cutoff value is [Fe/H]=−0.07.
Figure 3 shows a comparison between observations, the car-
bon copy model and the analytical model. It is interesting to no-
tice at this point that the carbon copy and analytical models are
essentially indistinguishable in these diagrams. The differences
with the observations arise only because of our choice to smear
the masses and orbital periods when generating our planet pop-
ulation.
Last but not least, we have to consider the existence of plan-
ets that orbit extremely close to their star, with periods shorter
than 2 days, as discovered by transit surveys (see table 1).
Companions with such short orbital periods have been discov-
ered by radial velocimetry in two occasions: HD 41004 b, and
Gliese 876 d, with respective masses 18.4 and 0.023 Jupiter
masses. These objects are outside the mass range considered for
this study, and therefore, there is no giant planets with periods
shorter than 2 days in the present radial velocimetry list. In or-
der to account for these very close-in planets anyway, we add
the planets with periods smaller than 2 days discovered by tran-
sit photometry to the list, but with a small tunable probability
weight. The fiducial value of this parameter is set so that, on av-
erage, the planet list contains one and a half such planet (added
to the list of 153 RV planets described in § 1.4). Tests on the
effect of this parameter are presented in § 3.3.4.
Our fiducial model is the mass-period-metallicity carbon
copy model, includes addition of very-close in planets and it is
that model which is used in all cases except where specified oth-
erwise. Other approaches are also tested depending on the model
to highlight particular points.
1.4.3. Physical characteristics and the planetary evolution
model
Because we are focussing on planets more massive than Saturn,
we expect most of them to be made of a significant amount of
hydrogen and helium. These giant planets thus undergo a pro-
gressive contraction and cooling that depends on four quanti-
ties: their age, mass, the amount of flux the planet receives from
the central star, and the global amount of heavy elements in the
planet (e.g. Guillot 2005).
Orbital period [days]
Planetary mass [MEarth]
Planetary radius [RJup]
Fig. 3. From top to bottom, distributions of orbital periods,
masses and radii, respectively, of the planets observed by radial
velocimetry (black lines), simulated as part of the mass-period
“carbon copy” model (red lines), and simulated as part of the
analytical model (dotted blue lines) (see text).
Models attempting to reproduce the radii of known tran-
siting giant planets have however had problems in explain-
ing the large radii of some of them (Bodenheimer et al. 2001;
Guillot & Showman 2002; Baraffe et al. 2005; Laughlin et al.
2005). Several possibilities have been proposed to explain the
discrepancy. We can separate them into two categories:
– Mecanisms invoking chance configurations of the plane-
tary orbits in the case of these anomalously large plan-
ets: this includes the tidal circularization of an eccentric
orbit (Bodenheimer et al. 2001), and tidal dissipation for a
planet locked in a Cassini spin-orbit resonnance with the star
(Winn & Holman 2005).
– Effects that would apply to all planets, including problems
with the equations of state or opacities, and the dissipation
by stellar tides of kinetic energy first generated in the atmo-
sphere (Showman & Guillot 2002).
The first mecanisms appear to have a low probability of oc-
curence (Laughlin et al. 2005; Deming et al. 2005; Levrard et al.
2007). The second possibility therefore seems more likely, but
requires in some case the presence of relatively large masses of
heavy elements to reproduce the observed radii.
A model-dependant estimate of the masses of heavy el-
ements present in the currently known transiting Pegasids is
shown in Fig. 4. This model relies on the hypothesis that 0.5%
of the absorbed stellar flux is used to generate kinetic energy
that is subsequently dissipated deep into the planetary inte-
rior (Guillot & Showman 2002). As proposed by Guillot et al.
(2006), there appears to be a correlation between the amount of
heavy elements present in the planet and the metallicity of their
parent star.
This correlation has to be ascertained, but we choose for our
fiducial model to adopt a unique relation between metallicity and
mass of heavy elements (treated as a central core in our calcula-
tions), corresponding to the dotted line in Fig. 4:
MZ = 43.75 × 10
[Fe/H]
− 23.7 M⊕. (4)
Fig. 4. Mass of heavy elements in transiting Pegasids known by
2006 as a function of the metal content of the parent star rela-
tive to the Sun. The mass of heavy elements required to fit the
measured radii is calculated on the basis of evolution models
including an additional heat source slowing the cooling of the
planet. This heat source is assumed equal to 0.5% of the incom-
ing stellar heat flux (Showman & Guillot 2002). Horizontal er-
ror bars correspond to the 1σ errors on the [Fe/H] determina-
tion. Vertical error bars are a consequence of the uncertainties
on the measured planetary radii and ages. The metallicity of re-
cently discovered planets WASP-1 and WASP-2 (right panel) is
not precisely known. The dotted line corresponds to a best fit
model. [Adapted from Guillot et al. (2006)].
We limit the range of possible values of MZ to [0, 100M⊕].
Similarly, we adopt a simple boundary condition for our evo-
lution calculations:
T1bar = 1.25Teq0, (5)
where T1bar is the temperature at the 1 bar pressure level and
Teq0 is the equilibrium temperature for a zero albedo (see Guillot
2005 for a description), calculated as a function of stellar effec-
tive temperature and radius and planet semi-major axis.
For simplicity, and because it yields only minor changes on
the results, we further choose to neglect the time-dependence in
the evolution calculations, and to adopt the equilibrium radius,
or the 10 Gyr solution, whichever occurs first.
Practically, planetary radii are obtained from interpolations
in a table based on three parameters: the planetary mass ranging
from 100 to 3000M⊕, the core mass from 0 to 100M⊕ and the
equilibrium temperature from 100 to 2000 K. Models were not
calculated beyond these values of Teq because of convergence
problems. However we allowed for a slight extrapolation of the
tables to 2600 K to account for rare extremely hot planets. 1
Similarly, because of convergence problems for planets with
small total masses and large core masses, we limited the mass of
the core to 75 M⊕ for planets with masses smaller than 180 M⊕.
More detailed work is required to better simulate this parame-
ter space, including planets less massive than considered in this
study.
1 An electronic version of the table is available at www.obs-
nice.fr/guillot/pegasids/
Figure 5 shows examples of radii obtained for Teq = 1000
and 2000, K, and core masses of 0 and 100Moplus, respectively,
compared to available measurements.
eq =2000K
eq=1000K
no core
e,  n
100 M⊕ core
planets
brown
dwarfs
stars
Fig. 5. Theoretical and observed mass-radius relations. The
black line is applicable to the evolution of solar composition
planets, brown dwarfs and stars, when isolated or nearly isolated
(as Jupiter, Saturn, Uranus and Neptune, defined by diamonds
and their respective symbols), after 5 Ga of evolution. The dotted
line shows the effect of a 15M⊕ core on the mass-radius relation.
Orange and yellow curves represent the mass-radius relations
for heavily irradiated planets with equilibrium temperatures of
1000 and 2000 K, respectively, and assuming that 0.5% of the
incoming stellar luminosity is dissipated at the center (see sec-
tion 1.4.3). For each irradiation level, two cases are considered: a
solar-composition planet with no core (top curve), and one with
a 100M⊕ central core (bottom curve). The transiting extrasolar
giant planets for which a mass and a radius was measured are
shown with points that are color-coded in function of the planet’s
equilibrium temperature. The masses and radii of very low mass
stars are also indicated as blue points with error bars.
1.5. Modeling transit events and their detectability
We now descibe how this population of stars, planets and their
interactions during transits are modelled to reproduce real obser-
vations.
1.5.1. PSFs and CCDs
Each image of a star is spread by the atmosphere and by the
telescope to grow to a specific size and shape when reaching
the CCD in the focal plane of the telescope, the so-called point
spread function (PSF). The CCD being composed of many dis-
crete pixels, these PSFs are then effectively discretized, so that
the signal to be analyzed for any given star is composed of many
different lightcurves corresponding to the many pixels over the
size of its PSF. A further complication arises from the fact that
the stellar fields generally chosen by transit surveys are dense,
so that many PSFs overlap. Recovering individual stellar light
curves from real data is a complex problem. Two popular meth-
ods are aperture photometry (Stetson 1987) and image subtrac-
tion (Alard & Lupton 1998). (An adaptation of the latter was
used to extract the OGLE lightcurves).
A refined simulation could include possible spatial and tem-
poral variations of the PSFs, and a realistic data reduction
pipeline. In our case, we choose to simplify the problem by rely-
ing on a posteriori analyses of real light curves to provide us with
a global noise budget. We however include background stars be-
cause of the important effect of signal dilution.
In order to do so, we assume that the PSFs are gaussian with
a uniform, constant FWHM. (Non-gaussian PSFs are not diffi-
cult to include but we tested in the OGLE case that for a fixed
equivalent FWHM, they have a negligible effect on the resulting
signal-to-noise ratio of simulated transit events). We consider
for each target of the survey the global flux from the main star
and the background stars in its neighborhood up to magnitude
22 in the spectral band of observation. The neighborhood zone
for background stars is defined as a circle of diameter equal to
4 times the PSF’s FWHM around the photocenter of each target
star. Each background star whose photocenter is located in that
zone is taken into account for the calculation of the global flux.
The global flux is calculated as the sum of the pixels located at
less than twice the FWHM of the central star.
We thus simulate aperture photometry when image subtrac-
tion was used for OGLE (Udalski et al. 2002). The choice of the
reduction algorithm indeed affects the sensitivity obtained from
real observations. In our simulations, i.e. a relatively idealized
case, it would have marginal effects since realistic noises are
included a posteriori from the analysis of real lightcurves (see
hereafter).
1.5.2. Noise budget and event detectability
We choose to separate noise sources into two categories:
– ‘White noise’ sources, following gaussian and Poisson laws.
The main source of white noise is the photon noise of target
stars and their background. The level of white noise for a
given target star is obtained from the simulation of the flux
of that star and its background in the photometric aperture.
– ‘Red noise’, or systematic effects on photometry, that un-
dergo temporal correlation. The structure of these systemat-
ics in the OGLE photometry have been explored in details by
Pont et al. (2006b). These noise sources are both instrumen-
tal (jitter and breathing of the CCD, frequency spectrum of
stellar field moves on the camera, change of the PSF shapes
accross the CCD during the night), and environmental (dif-
ferential refraction and extinction, changes of airmass and
sky brightness, temperature changes). Rather than trying to
simulate instrumental and environmental noise sources accu-
rately, which is difficult with the relatively poor knowledge
we have of the time spectrum of their combined effects, we
use the effective global ‘red noise’ measurements of OGLE-
III survey real light curves mentionned in Pont et al. (2006b),
which consider the combined effect of these noise sources.
Pont et al. (2006b) calculated that, in the presence of a mix-
ture of white and red noise (i.e. accounting for photometric sys-
tematics), the detection threshold for a transit survey is well de-
scribed by a limit on the signal-to-noise ratio defined as:
S 2r =
k=1 nk
2(σw2/nk + σr2)
where Ntr is the number of transits sampled, nk the number of
data points in the k-th transit. σw and σr are the standard devia-
tion of measurement points of white and red noises, respectively,
d is the event depth and n the total number of measurement
points during the transit. Specifically, we obtain nk by counting
for each transit the number of observation points between the
middle of ingress and the middle of egress.
Equation 6 makes the disctinction between “white” noise
sources that decrease with n1/2, where n is the number of succes-
sive measurements, and “red” noise sources that are limited by
temporal correlation. Pont et al. (2006b) indeed show that taking
the red noise into account makes a large difference on the detec-
tion threshold – in general as well as in its dependence to the
planet parameters – and that models based on the assumption of
white noise can be poor approximations of the actual detection
threshold.
2. The OGLE survey: input parameters
2.1. Basic parameters and observational procedure
The Optical Gravitational Lensing Experiment (OGLE) has
done 6 observation campaigns looking for transiting planets to-
wards different fields of view from 2001 (Udalski et al. 2002).
It took place at the Las Campanas Observatory, Chile, using the
1.3 m Warsaw telescope and the 8k MOSAIC camera, with a to-
tal field of view of 0.34◦2. All observations were made through
the I filter. We assume for our PSF simulation an average seeing
of 1 arcsec.
We analyze in this work the first three OGLE-III observation
campaigns dedicated to transit search, as their treatment, anal-
ysis and follow-up (with current data processing pipelines) has
been completed:
– OGLE-III-1 (June 12 to July 28, 2001, described in
Udalski et al. (2002); Udalski (2002)). More than 800 im-
ages of three fields in the direction of the galactic bulge were
collected within 32 nights. The exposure time was 120 s, and
each field was observed every 12 min.
– OGLE-III-2 (February 17 to May 22, described in Udalski
(2003)). More than 1100 images of three fields located in the
Carina region of the galactic disk were collected in 76 nights.
The exposure time was 180 s, and the temporal resolution
was about 15 min.
– OGLE-III-3 (February 12 to March 26, described in
Udalski et al. (2004)). The photometric data were collected
during 39 nights spanning the 43 days of the survey. Three
fields of the galactic disk were observed with a time resolu-
tion of about 15 min. The exposure time was 180 s.
In this article, we will refer to these three observation cam-
paigns respectively as ‘Bulge’, ‘Carina’, and ‘Centaurus’ fields.
The simulations include the real observation windows of
each survey, as kindly provided by A. Udalski. For any transit-
ing planet in the simulation, the number of effectively observed
transits is used in eq. 6.
In order to construct a realistic stellar population, we use
the stellar counts per magnitude range obtained by Gould et al.
(2006) based on OGLE-II data, which have calibrated photome-
try. We then randomly select that number of stars per magnitude
from the Besançon model. In order to test the validity of our
approach, we calculated the fraction of “stars for which tran-
sits are detectable” and compared it to the one determined by
Gould. This fraction is defined for a given magnitude range as
the number of stars around which a planet orbiting edge-on with
r = 1.2R jup and a = 7.94R⊙ can be detected, divided by the
total number of stars of that magnitude. As shown by table 2,
there is an excellent agreement between our results and those of
Table 2. Fraction of stars suitable for transit detection
Carina Bulge
Vmax Gould 2006 This work Gould 2006 This work
15.5 0.11 0.16 0.138 0.141
16 0.14 0.16 0.125 0.128
16.5 0.16 0.15 0.098 0.105
17 0.16 0.15 0.068 0.080
17.5 0.16 0.14 0.041 0.052
Gould et al. (2006). Note however that for the global simulation,
the complete star list is used as the above definition for suitable
stars is restricted to planets of a given size and orbital distance.
We calculated the average flux for target stars, companions
and all the background stars near enough to contribute to the
target PSF. We then checked that the average photon noise simu-
lated for target stars at a given magnitude was close to real values
obtained in OGLE light curves at given magnitude presented in
figure 4 of Pont et al. (2006b).
2.2. Modelling the detection threshold
The candidates in the OGLE survey have been identified with
the BLS transit-search algorith of Kovács et al. (2002). A sub-
set of the candidates selected with cuts in the α and SDE pa-
rameters of the BLS were examined by eye, and only the best
were included in the final list. Therefore, the selection thresh-
old is mainly defined by subjective appreciation from an ex-
perienced specialist. Recently, Pont et al. (2006b) have pointed
out that the effective detection threshold of ground-based tran-
sit surveys such as OGLE is importantly affected by correlated
noise (photometric systematics). The subjective selection of can-
didates is in large part necessary because of the presence of this
correlated noise, which produce many spurious detections near
the threshold. Gould et al. (2006) chose to model the OGLE se-
lection threshold with an α > 12 cut (alpha is equivalent to
the signal-to-noise ratio of the transit signal assuming uncor-
related noise and homogeneous distribution of the data points
in phase). Pont et al. (2006b) have included the effect of corre-
lated noise in the signal-to- noise calculation and found that the
OGLE selection could be better described by a threshold of 8
on the signal-to-noise ratio of the transit signal calculated in-
cluding correlated noise (”S r” in their notation, see Sec. 2.5.2),
and without the assumption of homogeneous coverage. While
the two thresholds have similar effects on the global number of
planet detection, they have a very different dependence on some
parameters, such as planet period and host star magnitude. Since
the objective or our study is to examine the detection statistics
in a multi-dimensional parameter space, we use the Pont et al.
(2006b) description of the OGLE detection threshold.
To calculate S r, one needs an assumption on the level of red
noise present in the photometry. Following Pont et al. (2006b),
we use a single-parameter description and assume σr = 3.6
mmag in the Bulge fields, σr= 3.1 mmag in the Carina and
Centaurus fields, and σr= 2.1 mmag in all fields after applica-
tion of decorrelation algorithms.
2.3. Confirmability of transit-like events with follow-up
High-resolution spectra allow the confirmation of the planetary
events if spectral lines are deep enough. Several scenarios make
the follow up of candidates too difficult: early type stars have
lines too weak and too broadened by rotation (type F4 and ear-
lier). Stars with magnitudes V > 17.5 are too faint for present
instruments and telescopes. This is the limit at which observers
estimated not being able to provide low-metallicity stars. Those
stars having weaker lines, could also be difficult to follow cor-
rectly, but as planets are unlikely to be found near this kind of
stars in our model, we did not take that parameter into account.
To simulate the feasability of follow-up, we only considered
in CoRoTlux the stars matching the criteria V < 17.5 and of type
F4 and later.
3. Results of the simulations
We present hereafter runs for the three OGLE-III campaigns for
the fields in the Galactic bulge, in Carina and in Centaurus. In
order to obtain a statistically significant population of detected
planets, the simulations were run multiple times.
We first examine the consistency between the models and ob-
servations for relevant physical variables. In doing so, we choose
to compare our model population to the global population of
transiting planets discovered by OGLE and other surveys. There
is a slight inconsistency in assuming that the parameter compari-
son is almost independant of the type of survey and observational
strategy. In some cases, this is not true, and a clear distinction
between the OGLE planets and the other detections has to be
made.
We then discuss the problem of the detection statistics,
whether observations and models are consistent, and whether a
constraint on the (low) frequency of very close-in planets can be
deduced.
3.1. Deviation of OGLE planets from maximum likelihood of
the simulations
We use a Maximum-Likelihood (ML) technique in order to test
whether model results and observations agree with each other.
We do the tests in two-dimension spaces, in order to qualify pos-
sible correlation and exclusion zones. The ML technique is our
method of choice as it is a powerful tool for fitting a model to a
multi-dimentionnal independant-data distribution (Lyons 1986).
Instead of determining an approximate analytical law fitting
our results, we use the results of a very large Monte-Carlo draw
(1000 times the whole OGLE survey, corresponding to ∼ 9000
planets) to get a map of the density of probability in each 2-
dimension grid. We bin our data on a 20x20 grid as a compro-
mise between resolution of the models and characteristic varia-
tions of the parameters.2 The probability of an event in each bin
is considered equal to the normalized number of draws in that
Figure 6 shows the logarithm of the probability that an event
occurs in each of the 20x20 bins of the mass-radius diagram. The
likelihood of a draw of several independant events is defined as
the sum of the logarithms of the probabilities of these events. In
order to compare our results to any n real discoveries, we first
estimate the standard deviation of any n-planets-random-draw
compared to the maximum likelihood of the model. We ran-
domly select n planets among the simulated detections and cal-
culate the likelihood of this draw. We repeat this selection 1000
times in order to have the maximum likelihood and its standard
deviation σ, then we compare the deviation of the likelihood
of the n real detecions calculated the same way in terms of σ.
2 Tests with different grids yield small variations of the results. As
an example, the mass-radius deviation from maximum likelihood is re-
spectively 0.67, 0.65 and 0.72 σ for 20x20, 30x30 and 40x40 grids.
Henceforth, quantitative comparisons between the simulation re-
sults and the known planets are systematically given in the figure
captions, whilst the text discusses qualitative comparisons and
their implications. For the different figures showing the results
of our simulation, we compare the distribution of planets over
the detection threshold to the 5 OGLE planets. We also compare
our results to the 11 planets discovered by all transit surveys, as
their detection biases are similar to OGLE, and to the 14 planets
which radius is known (11 from transits and 3 from radial ve-
locity surveys) to show how our model can reproduce the whole
known population.
3.2. Depth of the transit events and magnitude of the targets
stars
We first attempt to confirm whether the events detected by
the model are consistent with those found in the OGLE fields.
Figure 7 is a plot showing transit depth as a function of the mag-
nitude of the primary star. Model results are considered detected
when the signal-to-noise ratio is sufficient for a detection (see
§ 2.2). We also show events that are considered photometrically
detectable but very hard or impossible to confirm by radial ve-
locimetry.
The figure evidently shows a good correlation between the
black crosses and the red circles that indicate real detections by
OGLE, with a range of transit depths and V magnitudes that
is very similar between the models and the observations. Our
models overpredict slightly the number of transit events around
faint stars (V ≥ 17), but this may be due to the difficulty of the
follow-up work for these targets. Overall, the agreement between
models and observations is good.
3.3. Compatibility of transit surveys with radial-velocimetry
observations
3.3.1. Compatibility in the mass-period diagram
Figure 8 compares the model and observated mass-period rela-
tion. As it is independant of the planetary evolution model, it is a
direct test of the compatibility between the results of transit sur-
veys and those of radial-velocimetry observations that drive our
model results. Again, the comparison is very good, assuming a
high-enough frequency of very-close in planets (see discussion
in § 3.3.4). One can note especially the absence of planets of rel-
atively large mass (several times that of Jupiter) at short orbital
distances (P < 5 days), and of detectable transit events for peri-
ods longer than ∼ 5 days. This is due especially to the fact that
only events with a relatively large number of observed transits
are detectable, as required by the S r threshold, which, given the
day/night interruptions, imposes a constraint of a short orbital
period. Note that this feature is not well reproduced by mod-
els in which the threshold is computed from white-noise only
(Gould et al. 2006; Gillon et al. 2005).
3.3.2. The OGLE yields with a fixed red noise level
We have tested the efficiency of the fiducial model at estimat-
ing quantitatively the yield of transit surveys. Gillon et al. (2005)
have also simulated OGLE yield in their generic study of multi-
ple transit surveys, but with restrictive assumptions on transit de-
tectability (only complete events matter for detection purposes)
and without considering background stars and red noise, also
not using OGLE-fields specific stellar population. We also in-
cluded in our simulations the recent RV follow-up that has been
Fig. 6. Logarithm of the probability that a simulated detection event occurs in each one of the 20x20 bins of the mass/radius
diagram. The likelihood of a multiple-events draw is the sum of the logarithms of the probabilities of the events of this draw. Bins
without any occuring event in the large Monte-Carlo draw do not have any probability stated. The likelihood of a n-events draw is
the sum of the probabilities of its n events. In this mass-radius diagram, OGLE planets are shown as red circles, planets from other
surveys are in orange, and planets from radial velocity surveys are in blue. The likelihood of the 5 OGLE discoveries as a result of
a Monte-Carlo draw is −8.7, the maximum likelihood is −7 and the standard deviation to maximum likelihood is 2.54. Hence, the
result of the OGLE planets mass-radius distribution is at 0.67σ of the maximum likelihood of the model.
Table 3. OGLE yields with fixed red noise level
Field Mean red RV follow-up Number of planets
of view noise level to Vmag detected simulated with
0 1.5 3
VHJ added (P < 2 days)
Bulge 3.6 17.5 2 0.4 0.6 0.9
Carina original 3.1 17.5 3 3.4 4.1 4.8
updated 2.1 17.5 +(0 − 1) +1.1 +1.1 +1.1
Centaurus 3.1 17.0 0 1.4 1.8 2.2
Total 6 6.3 7.6 9.0
done on Centaurus and Carina. We use unpublished information
from the OGLE/ESO follow-up team, who found one promising
planetary candidate among the Carina fields reprocessed with
the systematics- removal algorithm from Tamuz et al. (2005) and
none in the Centarus fields, with a magnitude limit near V=17
for the radial velocity follow-up. Table 3 compares the average
number of planets detected for 1000 Monte-Carlo draws to real
detections from the OGLE survey.
The total number of simulated discoveries obtained from this
quantitative analysis is in good agreement with the real detec-
tions. The differences in the number of detections between the
Carina and Centaurus surveys are mainly due to the lower duty
cycle of the observations towards Centaurus. A red noise level
fixed at 3.6 mmag in the direction of the galactic bulge bans
most hot Jupiter detections. The agreement between our quan-
titative result and the number of real detections is an indicator
of the global efficiency of our approach (stellar and planetary
distributions, evolution model and noise budget) for estimating
transit survey yield.
3.3.3. The OGLE yields with a variable red noise level
So far, we have considered the level of red noise to depend only
on the field considered. We attempt now to refine this by consid-
ering how the stellar density may affect it. Whereas most ground-
based transit surveys have a global red noise level from 2 to
3.5 mmag (Superwasp: Smith et al. (2006), Monitor: Irwin et al.
(2007), Hatnet: Pont & ISSI team (2007) and OGLE), the causes
of these noise levels seem different, with instruments ranging
from 10-cm wide field reflectors to deep-sky several-meter tele-
scopes. As seen from table 3, the OGLE fields have differ-
ent mean red noise levels (σr = 3.6 mmag for the bulge and
σr = 3.1 mmag for Centaurus and Carina before SYS-REM),
although the instrument and observational strategy were un-
Fig. 7. Depth of the planetary transit events versus magnitude of the parent star in the V band. The five confirmed OGLE detections
are shown as circles. Model results are shown as black plusses for detectable events and orange crosses for events that are considered
undetectable based on the photometric signal (see text). Blue diamonds correspond to events that would be detectable by photometry
alone but that cannot be confirmed by radial velocimetry. Note that the model results correspond to 3 times the full OGLE campaign
for more statistical significance. The OGLE planets depth-magnitude distribution is at 0.69σ from the maximum likelihood of the
model.
Fig. 8. Mass versus period of transiting giant planets. (OGLE planets are red circles, other transit surveys in orange, planets from
radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses). The OGLE planets
mass-period distribution is at 0.62σ from the maximum likelihood of the model (0.72σ considering the 11 planets discovered by
transit surveys and 0.66σ considering the 14 known planets).
Fig. 9. Distribution of the crowding index (see text) of target
stars in Carina (black) and in the bulge (red).
changed. Looking at what distinguishes these fields, it appears
that the most significant difference is the stellar density and
therefore the amount of crowding: The bulge field is about twice
as dense as the Carina and Centaurus fields. Pont & ISSI team
(2007) raise the suspicion that the level of red noise depends
strongly on the presence and characteristics of contaminating
stars, because e.g. of their different colors and differential re-
fraction in the atmosphere. It is hence natural to consider a red
noise that depends on a crowding index.
We define this crowding index as the fraction of the flux
coming from background stars versus that from the target in
the photometric aperture. Importantly, we do not consider stel-
lar companions as contributing to the red noise because they are
generally on the same CCD pixel as the target star and should
affect the noise budget much less.
Figure 9 shows the differences of crowding index for the tar-
get stars with planetary transits (detectable or not) in simulations
of the Carina and Bulge fields of view. The mean crowding in-
dex for target stars of I < 17 is 0.11 in the Carina field and 0.233
in the Bulge field.
We can exclude the fact that all red noise is linked with con-
tamination as many stars in the Carina fields are unblended by
background stars but still show a high noise level.
In order to estimate of the influence of the crowding on the
red noise level, we use the following simple relation between red
noise level and crowding index:
σr = α × Fb + β, (7)
where Fb is the fraction of total flux from background stars,
determined on a star-by-star basis in our simulations, and α
and β are parameters to be determined. This is justified by the
behaviour of the red noise seen for instance in SuperWASP,
showing a linear increase as a function of background flux
(Smith et al. 2006). In order to get the same mean red noise
values as Pont et al. (2006b), we obtain α = 0.4 mmag and
β = 2.65 mmag. This value of β corresponds to the minimum
red noise level obtained for non-contaminated stars in the OGLE
fields.
Table 4 shows the new number of detections when consid-
ering this crowding-dependant red noise level. Compared to ta-
ble 3, the number of detections is found to be essentially un-
changed for the Carina and Centaurus fields, but it increases by a
factor ∼ 3 for the bulge field. This result is more satisfactory be-
cause in the previous case, only ∼ 5% of the simulations would
yield the detection of 2 planets in the bulge, as observed.
3.3.4. Models, observations and the frequency of very
close-in planets
As discussed in § 1.4.1, three OGLE planets have orbital periods
shorter than 2 days and thus belong to a class of objects yet to be
detected by radial velocimetry. So far, we have added one such
planet (on average) to our carbon copy list of nearly 200 radial
velocimetry planets. In Section 3.3, we have shown that with this
assumption, radial-velocity and photometric transit surveys are
compatible. We now test the range of frequencies of very close-
in planets for which this remains true.
In order to do so, we compute the deviation from maximum
likelihood in the mass-radius diagram like in Section 3.3, as a
function of the number of planets which period is less than 2
days added to the RV list. The result is presented in Fig. 10 and
shows that a good match is obtained by adding 1 to 3 short-
period planets. Larger numbers are also possible from the point
of view of the transit surveys, but would conflict with their
non-detection by radial-velocimetry. Adding the other transit-
ing planets discovered thus far yields smaller probabilities of
occurence of these short-period planets, but not by significant
amounts.
All in all, and assuming that the radial velocity planets sam-
ple is unbiased, we constrain the fraction of main-sequence late
stars orbited by very hot giant planets with orbital periods less
than 2 days to be (1/1265)(1+0.33
−0.33) at a 60 % confidence level or
(1/1265)(1+0.83
−0.5 ) at a 90 % confidence level.
The distribution of planets in period between 2 and 5
days is directly obtained from the metallicity-linked distribution
(Santos et al. 2004) and the RV planets sample. Adding the dis-
tribution we found for planets between 1 and 2 days, we obtain a
fraction of (1/215) late main-sequence stars orbited by planets in
the 1 to 5 days period range, in good agreement with the results
obtained in Gould et al. (2006), who obtained (1/220)(1+1.10
−0.45).
Similarly, the distribution we obtain by cutting this sample into
two parts with the cut-off at 3 days is compatible, showing:
– a slightly higher fraction of really short-period planets (1-3
days) of (1/560) instead of (1/710)(1+1.10
−0.54) at a 90 % confi-
dence level in Gould et al. (2006).
– a similar fraction of short-period planets (3-5 days) of
(1/350) instead of (1/320)(1+1.39
−0.59) at a 90 % confidence level
in Gould et al. (2006).
The results presented hereafter use the variable red noise
level approach, and an RV planet list that is complemented with,
on average, 1.5 very-close in planets with periods P < 2 days
taken from the OGLE detections.
3.4. The metallicity of the stars harboring transiting planets
We now compare the metallicity of the parent stars for our ob-
served and modelled populations. A first test using the analyti-
cal scenario for the radial-velocity population (Fig. 11) yields a
clearly different metallicity distribution, with most of the tran-
siting planets observed around low-metallicity stars. We veri-
fied that this problem occurs independantly of the assumed stel-
lar metallicity distribution, for any realistic stellar population. It
arises fundamentally because the global metallicity bias as ob-
tained by Santos et al. (2004) or Fischer & Valenti (2005) is not
Table 4. OGLE yields with variable red noise level
Field RV follow-up Number of planets
of view to Vmag detected simulated with
0 1.5 3
VHJ added (P < 2days)
Bulge 17.5 2 1.2 1.6 2
Carina original 17.5 3 3.6 4.3 4.9
updated 17.5 +(0 − 1) +1.1 +1.1 +1.1
Centaurus 17 0 1.3 1.9 2.3
Total 5-6 7.2 8.9 10.3
Fig. 11. Period of transiting exoplanets versus metallicity of their parent star. The model is based on analytic relations for the mass
and period distributions of planetary companions (see § 1.4.2). (OGLE planets are red circles, other transit surveys in orange, planets
from radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses). The OGLE planets
period-metallicity distribution is at 2.94σ from the maximum likelihood of the model (2.51σ considering the 11 planets discovered
by transit surveys and 2.63σ considering the 14 known planets).
strong enough to compensate for the rarity of very metal-rich
stars in the Galaxy.
As seen in Fig. 12, the problem disappears when one con-
siders the carbon-copy model. Thus, we are led to an important
conclusion, that the metallicity distribution of pegasids (periods
shorter than 10 days) is fundamentally different from the global
exoplanet population. More specifically, there are no Pegasids
orbiting F, G, K stars with metallicities smaller than [Fe/H]=
−0.07. This has strong consequences for planet formation mod-
els (see also Guillot et al. 2006). This work shows that this con-
clusion is robust, and is needed to explain the results of the pho-
tometric surveys.
A finer examination of Fig. 12 shows that while our model
planets reproduce globally the metallicity of the ensemble of
transiting planets, OGLE stars with planets are on average ∼
0.1 dex more metal-rich.
This can tentatively be explained with a metallicity gradient
in the galaxy for OGLE TR-10 ([Fe/H] = 0.28±0.10) and OGLE
TR-56 ([Fe/H] = 0.19± 0.07), the two planets discovered in the
direction of the galactic bulge. The study of galactic cepheids by
Andrievsky et al. (2004) shows a metallicity gradient as a func-
tion of distance to the galactic center. In the [6.6, 10.6] kpc-range
distance from galactic center, this study finds a linear relation be-
tween [Fe/H] and galactocentric distance RG:
[Fe/H] = −0.044(±0.004)RG + 0.363(±0.032) (8)
Following that relation, the two stars with planets discovered in
the direction of the galactic bulge both at a distance around 1500
pc would thus be in a 0.04 dex more metal rich region than the
solar neighborhood.
Concerning the high metallicity of stars with transiting plan-
ets discovered by OGLE in the Carina region, we do not have
any reason to think that the metallicity distribution would be
different from the solar neighborhood. Our only hypothesis is
a low-probability draw for metallicity for the 3 OGLE-Carina
planets.
Fig. 12. Period of transiting exoplanets versus metallicity of their parent star. The figure differs from Fig. 11 in that our fiducial
model, i.e. the mass-period-metallicity “carbon-copy” model is used (see § 1.4.2). (OGLE planets are red circles, other transit
surveys in orange, planets from radial velocitiy surveys in blue. Simulated planets detected: black plusses, under threshold: or-
ange crosses). The OGLE planets period-metallicity distribution is at 0.76σ from the maximum likelihood of the model (0.36σ
considering the 11 planets discovered by transit surveys and 0.39σ considering the 14 known planets) .
3.5. Atmospheric potential energy and orbital distances
Because evaporation may affect the planet population, it is in-
structive to check whether the potential energy of the atmosphere
and the orbital period, two crucial quantities for this process (e.g.
Lecavelier des Etangs et al. 2004), also possess a relatively con-
sistent distribution. We first test the behavior of the analytical
model for the distribution of planets (Fig. 13). This results in
a prediction of many planets with large radii (small values of
the potential energy for atmospheric escape) at small orbital dis-
tances, in patent contradiction with the observations.
The problem mostly disappears with the carbon-copy model:
Fig. 14) shows that in this case, although we do not obtain a
linear correlation between the two variables, we get detections
in the right area of the diagram. This is explained as stemming
from:
– The absence of low-mass planets at small orbital distances,
with a possible limiting relation between these two quantities
(Mazeh et al. 2005);
– The difficulty in detecting planets with larger values of po-
tential energy per unit mass (smaller radii) at large orbital
distances –although we predict that some of these should be
detected by future transit surveys.
Our results strengthen the case for the existence of a rela-
tion between mass and orbital distance for short-period plan-
ets, as advocated by Mazeh et al. (2005): Indeed, the analytic
model which is characterized by the presence of small mass
planets at small distances yields a distribution of detectable plan-
ets that is significantly different from the observations (Fig. 13).
Our carbon-copy model that includes implicitely this correlation
does not (Fig. 14).
3.6. Planetary radii and stellar irradiation
Radius and stellar irradiation (or equivalently equilibrium tem-
perature) should be positively correlated, as a planet with a
higher irradiation dose will tend to cool and contract more
slowly than one that endures less stellar insolation. As Fig. 15
shows, the correlation exist, but is weak, and with a signficant
scatter. This is well reproduced by the model.
However, it can be noted that HD 149026 b lies away from
the cloud of points. In general, we find that our fiducial model
generates few points in this region. This can be easily accounted
for by slightly modifying the metallicity-core mass relation to
allow for larger masses. As planets of small masses and large
core masses are more difficult to model anyway, we chose not to
attempt fine-tuning the model to this level of detail. This should
be postponed for further studies, especially with the discovery of
more Saturn-mass transiting planets.
3.7. The mass-radius relation
We have checked that our fiducial model predicts the detection
of transiting planets with properties that are globally consistent
with the observations. We can now examine in more detail the
mass-radius relation thus obtained, as it is directly tied to as-
sumptions on the compositions and evolutionary models of ex-
oplanets. The predictions also have implications for transit sur-
veys as it is not clear whether they have detected only the “tip of
the iceberg”, ie the few largest giant planets while many smaller
ones would lie undetected or not.
Results with our fiducial model are presented in Fig. 16.
We find that planets with low masses (say, less than Jupiter’s
Fig. 13. Potential energy per unit mass (Ep = GM/R) versus orbital period of transiting planets. (OGLE planets are red circles, other
transit surveys in orange, planets from radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold:
orange crosses). Observations are compared to models based on the analytical relations for the mass and period distribution of
planetary companions (see § 1.4.2). The OGLE planets energy-period distribution is at 2.18σ from the maximum likelihood of the
model (1.86σ considering the 11 planets discovered by transit surveys and 2.47σ considering the 14 known planets).
Fig. 14. Potential energy per unit mass versus orbital period of transiting planets. The figure is similar to Fig. 13, except for the
fact that our fiducial model is used (see § 1.4.2). (OGLE planets are red circles, other transit surveys in orange, planets from radial
velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses). The OGLE planets energy-
period distribution is at 0.55σ from the maximum likelihood of the model. (0.84σ considering the 11 planets discovered by transit
surveys and 0.66σ considering the 14 known planets)
Fig. 15. Radius as a function of equilibrium temperature of transiting exoplanets. (OGLE planets are red circles, other transit surveys
in orange, planets from radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses).
The OGLE planets equilibrium temperature-radius distribution is at 1.22σ from the maximum likelihood of the model (1.05σ
considering the 11 planets discovered by transit surveys and 2.25σ considering the 14 known planets).
mass) can both have very large or very small radii, depending
on whether they contain a significant mass in heavy elements
or not. On the other hand, massive planets have radii which are
comparatively better defined. This is mostly due to the fact that
we assume a maximum mass of heavy elements of 100 M⊕, a
hypothesis that will be tested directly by the discovery of a few
massive transiting planets.
Our model results once again agree well with the detections
made by photometry. Importantly, the yellow crosses in Fig. 16
do not lie significantly below the black ones: we predict that
future surveys will not discover a population of small-sized giant
planets, at least for masses above that of Saturn.
The presence of planets with larger masses of heavy ele-
ments should remain marginal because otherwise they would
have been detected by present-day surveys, Fig. 16 showing
that planets below 1 RJup are already detectable, although in fa-
vorable cases (small radius of the primary and bright targets).
Quantitatively, simulations in the OGLE fields indicate that if
planets had radii uniformly distributed between 0.5 and 1.5 RJup,
18.5% of the planets discovered by the survey would have radii
below 1 RJup. This fraction is not negligible and is (although
marginally) inconsistent with the sample of 0/11 planets with
R < RJup discovered by transit surveys thus far.
Therefore, although we cannot statistically rule out the pres-
ence of a population of small planets, these would require the
formation of extremely metal-rich planets. Our prediction is a
consequence of evolution models and of our assumption that
planets with masses of heavy elements beyond 100 M⊕ should
be rare.
Figure 17 shows the ensemble of planets obtained for an ex-
tremely large number of draws, with our fiducial model. Voids
in the ensemble of crosses correspond to the absence of planets
with these masses in the radial-velocimetry list. They should not
be considered as significant. The contours in the figure indicate
the ensemble of masses and radii expected for planets with dif-
ferent masses of heavy elements, from 0 to 100 M⊕. Importantly,
the location of these contours is linked to our assumption of an
energy source in the planetary interior equal to 0.5% times the
irradiation received by the planet. Independently of the details of
this assumption, this shows that a statistically significant ensem-
ble of known transiting planets would allow a determination of
the presence or lack of heavy elements in these objects.
We have also tested another assumption regarding the plan-
etary evolution model: all planets possess 20 M⊕ mass in heavy
elements, 70% of them have no extra heat source, whereas 30%
have 3 × 1026 erg s−1 dissipated at the center. With this assump-
tion, one can qualitatively explain the observed transiting planets
(i.e. the “normal” planets and the “anomalously large” ones, re-
spectively) with the exception of HD 149026 b, for which one
could argue that the planet comes from a different population. In
this case, Fig. 18 shows a distribution of radii that is relatively
similar to the previous one (Fig. 16), with the exception that no
planet has a radius smaller than 0.8 RJup. In this case, the 2 re-
gions corresponding to the “standard” model, and to the “heat
dissipation” case are clearly different, especially at the low-mass
range of the diagram.
Present observations cannot distinguish between the two
models, showing the need for additional detections of transit-
ing giant planets. Particularly important are planets between the
mass of Saturn and that of Jupiter, as this is a mass regime where
expected compositional differences have the largest impact.
Fig. 16. Mass-radius relation for transiting extrasolar giant planets. (OGLE planets are red circles, other transit surveys in orange,
planets from radial velocity surveys in blue. Simulated planets detected: black plusses, under threshold: orange crosses). The OGLE
planets mass-radius distribution is at 0.67σ from the maximum likelihood of the model (0.72σ considering the 11 planets discovered
by transit surveys and 0.97σ considering the 14 known planets) .
4. Conclusions
We have presented a simulation of photometric transiting sur-
veys based on basic knowledge of the stellar and planetary pop-
ulations in the galactic neighborhood and on a planetary evo-
lution model tuned to the information obtained from transiting
giant planets with masses above that of Saturn. This simulation
was applied to the OGLE survey, and shown to yield a generally
excellent agreement with the transiting planets detected by the
survey.
We have thus shown that radial velocimetry and photometric
surveys are compatible within statistical uncertainties, in agree-
ment with Gould et al. (2006). We have derived a frequency of
very close-in planets with orbital periods shorter than 2 days
around solar-type stars, of (1/1265)(1+0.33
−0.33) at a 60 % confidence
level or (1/1265)(1+0.83
−0.5 ) at a 90 % confidence level.
Using null results by photometric surveys for given ranges of
parameters, we are able to strengthen two results already present
in the radial velocimetry data:
– Stars with low metallicities ([Fe/H]< −0.07) do not, or are
very unlikely to harbour close-in giant planets with orbital
periods P < 10 days. This is unlike stars above that metallic-
ity threshold (see Fig. 12).
– There is a lack of small-mass giant planets below the mass
of Jupiter and above that of Saturn for orbital periods P <
3 days (see Fig. 8).
Further data is required to precisely quantify these empirical re-
sults that bear important consequences for our understanding of
planet formation and migration.
On the basis of our model, and assumptions concerning the
composition of giant planets (i.e. masses of heavy elements be-
tween 0 and 100 M⊕), we find that the present detections of tran-
siting planets have sampled a population that is quite represen-
tative of the main population of giant planets, at least for the
ones that are above about half the mass of Jupiter. We hence pre-
dict that future transit surveys with higher sensitivities will not
discover a significant population of yet undetected Jupiter-mass
planets with small sizes, i.e. radii smaller than that of Jupiter (see
Fig. 16).
Many ground-based transit surveys are in progress, and
with the space missions CoRoT (Baglin et al. 2002) and Kepler
(Borucki et al. 2003), the number of known transiting planets is
expected to rise rapidly over the next few years. This will en-
able us to better test the models and quantify some of the results
presented in this article. We also hope to be able to discriminate
between various models of the evolution and compositions of gi-
ant planets, a matter of great importance for formation models.
We wish to stress however that a continuation of ground-
based transit surveys is desirable even in the presence of simi-
lar programs from space. CoRoT will survey 60,000 dwarf stars
over five 150 days periods and Kepler about 100,000 over 4
years, implying a maximum potential yield of 55 and 90 transit-
ing giant planets, respectively, plus many other smaller planets.
For what concerns giant planets, quantifying the fraction of very
close-in planets with a 10% accuracy at the 3σ level would re-
quire the discovery of ∼ 200 transiting planets. Understanding
the evolution and compositions of giant planets will require an
even larger number of detections. The radius of a giant planet
itself depends mainly on four parameters: the planetary mass,
equilibrium temperature, age, and its composition (note that the
composition can be considered as a simple parameter only in the
case of planets mostly made of hydrogen and helium: smaller
planets will be more difficult to model!). Additional energy
Fig. 17. Mass-radius relation for a very large number of Monte-Carlo trials using the fiducial model. The curves show the ensemble
of planets with masses of heavy elements between 0 and 25, 25 and 50, 50 and 75, 75 and 100 M⊕, respectively. Symbols are as in
Fig. 8.
Fig. 10. Deviations from a maximum likelihood obtained as a
function of NVHJ , the number of very hot jupiter of orbital pe-
riods shorter than 2 days added to the radial velocities carbon-
copy list. Thick line: Deviation from the maximum likelihood
obtained in the mass-radius diagram for the OGLE planets. Thin
line: Same deviation but when compared to the ensemble of
planets. Dashed line: Standard deviation obtained from a com-
parison between the number of simulated planets and the number
of detected ones for the OGLE survey (see table 4). Dotted line:
Standard deviation obtained from the non-detection of these very
close-in planets by radial-velocimetry.
sources may occur (such as in the presence of tidal heat dissipa-
tion), and the initial conditions and formation history may have
their say in the matter as well. Furthermore, the observational
uncertainties are generally large. For example, the planetary ra-
dius is generally only known to∼ 10%, for a global variation that
is relatively small (1 to 1.5 RJup). This implies that to constrain
a given correlation to, say 10%, and with four independant vari-
ables, hundreds of data points are needed, and thousands would
be desirable.
This motivates us to seek programs capable of detecting
thousands of transiting planets in the mid-term future, and ways
to reduce the error bars on the different parameters. One direc-
tion is to test the Dome C plateau in Antarctica for such an am-
bitious program, which is the purpose of A STEP (Fressin et al.
2005). Other directions exist, such as proposals for similar sur-
veys from space. In any case, it is most important that a statisti-
cally significant population of exoplanets be characterized for a
better understanding of planet formation and our origins.
Acknowledgments
The code used for this work, CoRoTlux, has been devel-
opped as part of the CoRoT science program by the au-
thors with major contributions by Aurélien Garnier, Maxime
Marmier, Martin Vannier, Suzanne Aigrain and help from Claire
Moutou, Stéphane Lagarde, Antoine Llebaria, Didier Queloz
and François Bouchy. We want to thank Andrzej Udalski
and Michael Gillon for their communications on OGLE data,
Frédéric Thévenin for his advices on stellar populations sim-
ulation. F.F. has been funded by grants from the French
Ministère de la Recherche and by the Société des Amis des
Sciences. V.M. was funded by a grant from the C.N.R.S.. This
work has used extensively Jean Schneider’s exoplanet database
www.exoplanet.eu, and the Besançon model of the Galaxy at
physique.obs-besancon.fr/modele/. The planetary evolu-
tion models used for this work can be downloaded at www.obs-
nice.fr/guillot/pegasids/.
with heat dissipation
stand
ard m
Fig. 18. Mass-radius relation obtained for an alternative model with 70% of “standard” planets with no extra-energy source, and
30% planets receiving an additional 3× 1026 erg s−1 luminosity dissipated at the center. All planets are assumed to possess 20 M⊕ in
heavy elements. Symbols are as in Fig. 8.
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	Simulating transit surveys
	General remarks
	Principle of the simulations
	The stellar population
	Main targets and background stars
	Binary and triple systems
	The planetary companions
	Planet incidence
	Planetary masses and orbits
	Physical characteristics and the planetary evolution model
	Modeling transit events and their detectability
	PSFs and CCDs
	Noise budget and event detectability
	The OGLE survey: input parameters
	Basic parameters and observational procedure
	Modelling the detection threshold
	Confirmability of transit-like events with follow-up
	Results of the simulations
	Deviation of OGLE planets from maximum likelihood of the simulations
	Depth of the transit events and magnitude of the targets stars
	Compatibility of transit surveys with radial-velocimetry observations
	Compatibility in the mass-period diagram
	The OGLE yields with a fixed red noise level
	The OGLE yields with a variable red noise level
	Models, observations and the frequency of very close-in planets
	The metallicity of the stars harboring transiting planets
	Atmospheric potential energy and orbital distances
	Planetary radii and stellar irradiation
	The mass-radius relation
	Conclusions
