2 Direct photometric detection: Probability

We first compute the probability of detecting the transit of a planetary satellite in front of the parent star. We start by computing the luminosity drop $\Delta F_\ast$ produced by an object, planet or satellite of a planet, on the line of sight to a star of luminosity $F_\ast$. This can be expressed as $\Delta F_\ast = \pi r_0^2 I$, where r₀ is the radius of the object, and I is the surface brightness of the star where the object is producing the occultation. We assume that the star is spherical and that the radial dependence of the surface brightness can be described by the standard limb-darkening law $I(r)=I_0 (1- \mu + \mu \sqrt{1 -(r/r_\ast)^2})$. Here, r is the projected distance from the star center to the object, $r_\ast$ is the radius of the star, and $\mu$ is the limb-darkening parameter that depends on wavelength. For reference, in the visible a solar-type star has $\mu \approx0.5$ (Allen 1973, p. 171). Since $F_\ast$ is the integral of I(r) over the star's area, we can write the relative drop in luminosity imprinted by the object as:

$$\Delta F_\ast/F_\ast = (r_0/r_\ast)^2 \times {{ 1- \mu + \mu \sqrt{1-(r/r_\ast)^2}}\over{1-\mu/3}}.$$ (1)

This expression reduces to $\Delta F_\ast/F_\ast = (r_0/r_\ast)^2$ when $r=0.75r_\ast$.

Numerical evaluations of Eq. (1) show that the drop in luminosity caused by an occulting object depends only weakly on the projected distance of the object to the star center, except when the object is very close to the limb. For example, adopting $\mu=0.5$, Eq. (1) indicates that $\Delta F_\ast/F_\ast$ is only 30% smaller for $r=0.9r_\ast$ than for r=0. In the following, therefore, when computing detection probabilities for planets and satellites of planets transiting in front of a parent star, we consider all projected trajectories with $r\leq r_\ast$. As Eq. (1) shows, $\Delta F_\ast/F_\ast$ is roughly of the order of $(r_0/r_\ast)^2$, and for a Jupiter-like or Earth-like planet occulting a solar-type star this implies $\Delta F_\ast/F_\ast\approx 10^{-2}$ or 10^-4, respectively, at $r\mathrel{\hbox to 0pt{\lower 3.5pt\hbox{$\mathchar''218$}\hss} \raise 1.5pt\hbox{$\mathchar''13C$}} r_\ast$. For reference, COROT is expected to detect planets or satellites with $r_0\mathrel{\hbox to 0pt{\lower 3.5pt\hbox{$\mathchar''218$}\hss} \raise 1.5pt\hbox{$\mathchar''13E$}}2 r_{\rm E}$, where $r_ {\rm E}$ is the Earth radius.

We do not speculate here on the origin of satellites with radii $r_0\mathrel{\hbox to 0pt{\lower 3.5pt\hbox{$\mathchar''218$}\hss} \raise 1.5pt\hbox{$\mathchar''13E$}}2 r_{\rm E}$ that could be detected with COROT. Their formation could be associated to that of giant planets, or they could form or be capured ulteriorly. We note, however, that the lifetimes of such satellites before they collapse on their parent planets are expected to be relatively long because of the high tidal dissipation factor of Jupiter-like planets ($Q_{\rm J}\sim 10^5$; see, e.g., Burns 1986). In fact, based on the calculations by Burns, the expected lifetime of a satellite with $r_0\sim 2\,r_{\rm E}$ at a distance $\mathrel{\hbox to 0pt{\lower 3.5pt\hbox{$\mathchar''218$}\hss} \raise 1.5pt\hbox{$\mathchar''13E$}}\! 10\, r_{\rm J}$ from a Jupiter-like planet with orbital period $\mathrel{\hbox to 0pt{\lower 3.5pt\hbox{$\mathchar''218$}\hss} \raise 1.5pt\hbox{$\mathchar''13E$}}\! 20$ days is $\mathrel{\hbox to 0pt{\lower 3.5pt\hbox{$\mathchar''218$}\hss} \raise 1.5pt\hbox{$\mathchar''13E$}}\!10^9\,{\rm yr}$ (where $r_{\rm J}\sim 11\,r_{\rm E}$ is the Jupiter radius). As mentioned in Sect. 1, the expected occurence of Jupiter-like planets is independent of orbital radius. It is conceivable, therefore, that extended satellites be found around Jupiter-like planets in relatively close orbit (i.e., with short periods) around solar-type stars.

Stellar variability, unless it is much stronger than that exhibited by the Sun, should not prevent the detection of transits of even Earth-like planets. In fact, the Sun variability is $\sim\!\! 10^{-5}$ on a time scale of several hours, as measured by the Solar Maximum Mission satellite, the Upper Atmosphere Research Satellite, and the Solar and Heliospheric Observatory for the Sun (Fröhlich 1987; Willson & Hudson 1991). The photometric variability due to sunspot activity may be more important, but it should not be confused with planetary transits because it depends on wavelength and is not periodic.

We now compute the probability for the transit of a planetary satellite to be detected. In this section the satellite is assumed to be extended enough to produce a detectable minimum in the lightcurve. The detection probability is given by the product of two independent probabilities, which we define as "geometric probability'' and "orbital probability''. In the remainder of this paper, we express all distances in units of $r_\ast=1$ and all masses in units of a solar mass.

2.1 Geometric probability

The geometric probability is the probability for the planet-satellite system to have a favorable orbital inclination angle so that the projected satellite orbit can intersect the star. We first describe the geometric conditions required for the transit of a planet and then extend the results to the case in which the planet has a satellite. We assume throughout this paper that the orbits are keplerian and circular. A transit of a planet is geometrically possible when the projected orbit intersects the star. The projected position of the planet with respect to the star center can be most generally parameterized as:

$$\begin{eqnarray} y_{\rm p} & = & a_{\rm p}\left(\cos \nu \sin \Omega_{\rm p} + \... ...} \right) \\ z_{\rm p} & = & a_{\rm p} \sin \nu \sin i_{\rm p}\,,\end{eqnarray}$$ (2) (3)

where $a_{\rm p}$ is the planet orbital radius, $i_{\rm p}$ is the line-of-sight inclination of the orbital plane, $\Omega_{\rm p}$ is the longitude of the ascending node, and $\nu$ is the planet phase angle on the orbit. A transit then takes place when $\sqrt{y_{\rm p}^2 + z_{\rm p}^2} \leq 1$.

The geometric probability for a transit of the planet can be estimated by noticing that it corresponds to all line-of-sight inclinations of the planetary orbital plane less than the critical angle $\sin i_{\rm {p0}}\approx i_{\rm p0} =1/a_{\rm p}$, where again $a_{\rm p}$ is expressed in units of the stellar radius. Therefore, the total geometric probability of a planetary transit is given by the ratio of the area of detectability, $2\times 2\pi a_{\rm p}$, to the entire possible area covered by planetary orbits, $4\pi a_{\rm p}^2$. This is $a_{\rm p}^{-1}$,or $i_{\rm p0}$. The differential geometric probability that a planet transit occurs with orbital inclination angle between $i_{\rm p}$ and $i_{\rm p}+ {\rm d}i_{\rm p}$ can then be written as:

$$\begin{eqnarray} P_{\rm g}(i_{\rm p})\;{\rm d}i_{\rm p} & = & {\rm d}i_{\rm p} \... ...space{2.3cm} {\rm for}\,\, i_{\rm p0}\leq i_{\rm p}\leq \pi/2\,, \end{eqnarray}$$ (4) (5)

with

$$i_{\rm p0} = a_{\rm p}^{-1}.$$ (6)

We now extend this reasoning to the case in which the planet has a satellite. We denote by $a_{\rm s}$ the satellite orbital radius around the planet and by $i_{\rm ps}$ the angle between the planet and satellite orbital planes. If any value of $i_{\rm ps}$ is allowed, by analogy with the above argument the total geometric probability for the planet or satellite to transit in front of the star will be $a_{\rm p}^{-1}(1+a_{\rm s})$. Satellites of the Solar system, however, have orbital planes inclined by a few degrees at most with respect to that of their parent planet. Hence, we fix $i_{\rm ps}$, which has for effect to limit the total probability for a planetary or satellite transit to $a_{\rm p}^{-1}(1+a_{\rm s}\sin i_{\rm s1})\approx a_{\rm p}^{-1}(1+a_{\rm s} i_{\rm s1})$. In this expression, $i_{\rm s1}= (1+a_{\rm p} i_{\rm ps})(a_{\rm p}-a_{\rm s})^{-1}$ is the maximum allowed line-of-sight inclination of the satellite orbital plane, that is determined from the condition $i_{\rm s1}= i_{\rm ps}+a_{\rm p}^{-1}(1+a_{\rm s} i_{\rm s1})$.Consequently, the maximum line-of-sight inclination of the planetary orbital plane leading to a transit of the satellite is $i_{\rm p1}= i_{\rm s1}-i_{\rm ps}=(1+a_{\rm s} i_{\rm ps})(a_{\rm p}-a_{\rm s})^{-1}$.

We can reexpress $a_{\rm s}$ in terms of the planetary orbital radius, $a_{\rm p}$. The satellite orbital radius must satisfy the condition $a_{\rm R} < a_{\rm s} < a_{\rm H}$, where $a_{\rm R}$ and $a_{\rm H}$ are the planet Roche and Hill radii, respectively. We parameterize $a_{\rm s}$ in terms of the Hill radius as $a_{\rm s}=\xi^{-1}a_{\rm H}= \xi^{-1}a_{\rm p} M_{\rm p}^{1/3}$, where $M_{\rm p}$ is the mass of the planet in units of a solar mass, with $1\le \xi \le a_{\rm H}/a_{\rm R}$. We further adopt the notation $X=\xi^{-1} M_{\rm p}^{1/3}$. With these notations, the maximum line-of-sight inclination of the planetary orbital plane leading to a transit of the satellite, $i_{\rm p1}$, can be rewritten in terms of $a_{\rm p}$, X and $i_{\rm ps}$.The differential geometric probability that a satellite transit occurs when the planetary orbital inclination angle ranges between $i_{\rm p}$ and $i_{\rm p}+ {\rm d}i_{\rm p}$ is thus:

$$\begin{eqnarray} P_{\rm g}(i_{\rm p})\;{\rm d}i_{\rm p} & = & {\rm d}i_{\rm p} \... ...hspace{2.3cm} {\rm for}\,\, i_{\rm p1}\leq i_{\rm p}\leq \pi/2\,.\end{eqnarray}$$ (7) (8)

with

$$i_{\rm p1} = (1+a_{\rm p} X i_{\rm ps})\left[ a_{\rm p}(1-X)\right]^{-1}.$$ (9)

Equations (4) and (7) then show immediately that for line-of-sight inclinations of the planetary orbital plane in the range $[i_{\rm p0},\,i_{\rm p1}]$, only the satellite can transit in front of the star.

In the remainder of this paper, all expressions are computed by adopting the above parameterization of $a_{\rm s}$ in terms of $a_{\rm p}$ via the parameter $\xi$, that allows us to sample all possible satellite orbital radii. The results can be reformulated straightforwardly in the context of any alternative parameterization using $X=\xi^{-1} M_{\rm p}^{1/3}=a_{\rm s}a_{\rm p}^{-1}$.

2.2 Orbital probability

For a satellite transit to occur, the satellite must be in the part of its orbit intersecting the star when the planet is in a favorable location. This can be described by an orbital probability function. We assume that the duration of the observations is at least as long as the planetary orbital period. Therefore, the planet will always sample the part of its orbit that is favorable for a satellite transit if the condition $i_{\rm p}\leq i_{\rm p1}$ is satisfied. To express the orbital probability function of the satellite, it is convenient to introduce the variable

$$\tilde a = a_{\rm s} i_{\rm s } -a_{\rm p} i_{\rm p} +1 = a_{\rm p} i_{\rm p}(X-1)+1+X a_{\rm p} i_{\rm ps}\,,$$ (10)

which measures the projected distance from the stellar limb to the point of intersection of the satellite's orbit with the planet-star axis at the time when the planet projects closest to the star center. For a satellite transit to occur one must have $\tilde a\gt$, which is equivalent to the condition $i_{\rm p} < i_{\rm p1}$.

Three limiting conditions characterize the orbital probability associated to a geometrically possible transit ($\tilde a\gt$). First, the planet itself is transiting, and at a given time the satellite's orbit projects entirely on the stellar area. This corresponds to $\tilde a\gt 2a_{\rm s} i_{\rm s}$, and in this case the orbital probability is unity. Second, at any given time the satellite's orbit projects only partially on the star area. In this case the orbital probability depends on the comparison between $\tilde a$ and the stellar diameter. If $\tilde a<2$, the orbital probability, i.e., the fraction of the satellite's orbit which will intersect the star as the planet orbits is $\tilde a(2 a_{\rm s} i_{\rm s})^{-1}$. If, on the other hand, $\tilde a\gt 2$, then the orbital probability reduces to $2\times(2 a_{\rm s} i_{\rm s})^{-1}=(a_{\rm s} i_{\rm s})^{-1}$.

The above limiting conditions on $\tilde a$ translate via Eq. (10) into conditions on the line-of-sight inclination of the planetary orbital plane, $i_{\rm p}$, for given $a_{\rm p}$, X and $i_{\rm ps}$. The orbital probability of a geometrically possible satellite transit is therefore given as a function of $i_{\rm p}$ by:

$$\begin{eqnarray} P_{\rm o}(i_{\rm p}) & = & 1 \hspace{2.05cm} {\rm for}\,\, i_{... ...hspace{0.2cm} {\rm and}\hspace{0.2cm} i_{\rm p}\leq i_{\rm p3}\,,\end{eqnarray}$$ (11) (12) (13)

with

$$\begin{eqnarray} i_{\rm p2} & = & (1-Xa_{\rm p}i_{\rm ps})\left[a_{\rm p}(1+X)\r... ... & (X a_{\rm p} i_{\rm ps} -1)\left[a_{\rm p} (1-X)\right]^{-1}\,.\end{eqnarray}$$ (14) (15)

Equations (11)-(13) can also be rewritten in terms of $i_{\rm p}$, $a_{\rm p}$, X and $i_{\rm ps}$. If the duration of the observations is $n_{\rm t}$ times longer than the planet period, the orbital probability must be multiplied by $n_{\rm t}$,until eventually one reaches $P_{\rm o}=1$.

2.3 Detection probability

The detection probability of the satellite is the product of the geometric and orbital probabilities. We let the planetary orbital radius $a_{\rm p}$ vary and fix the the angle between the planet and satellite orbital planes, $i_{\rm ps}$,and the parameter X. Using the same notations as above, the probability to detect a satellite transit is therefore

$$P_{\rm det}(a_{\rm p}) = \int_0^{i_{\rm p1}} {\rm d}i_{\rm p}\;P_{\rm o}(i_{\rm p})\,,$$ (16)

while the probability to detect a satellite transit without a transit of the planet is

$$P_{\rm det}^{\rm S}(a_{\rm p}) = \int_{i_{\rm {p0}}}^{i_{\rm p1}} {\rm d}i_{\rm p}\;P_{\rm o}(i_{\rm p})\,.$$ (17)

Also of interest is the probability to detect a transit of the satellite when the planet is itself assumed to transit. This is given by

$$P_{\rm det}^{\rm P}(a_{\rm p}) = a_{\rm p}\int_0^{i_{\rm p0}} {\rm d}i_{\rm p}\;P_{\rm o}(i_{\rm p})\,,$$ (18)

since the probability for a transit of the planet is $a_{\rm p}^{-1}$.

As noted in Sect. 2.2, the form of the integrand in Eqs. (16)-(18) depends on the value of $i_{\rm p}$ relative to $i_{\rm 2}$ and i₃. For fixed $i_{\rm ps}$ and X, this depends on the parameter $a_{\rm p}$. More specifically, if $a_1= (Xi_{\rm ps})^{-1}$ and $a_2=(2-X)(Xi_{\rm ps})^{-1}$, then $P_{\rm det}(a_{\rm p})$, $P_{\rm det}^{\rm S}(a_{\rm p})$ and $P_{\rm det}^{\rm P}(a_{\rm p})$ can be computed by breaking the integrals in Eqs. (16)-(18) according to the rule

$$\begin{eqnarray} i_{\rm p3}<0<i_{\rm p2}<i_{\rm p0}<i_{\rm p1} & & \hspace{1cm} ... ... p3}<i_{\rm p1} & & \hspace{1cm} {\rm for}\,\, a_2 < a_{\rm p} \,.\end{eqnarray}$$ (19) (20) (21)

For the values of X under consideration, a₁ is always smaller than a₂.