3 Direct photometric detection: Results

Figure 1 shows $P_{\rm det}^{\rm S}(a_{\rm p})$, the probability to detect a satellite transit without a transit of the parent planet, for various assumptions on the planet and satellite parameters. The probability is computed assuming that the duration of the observations is always a planetary period ($n_{\rm t}=1$). In Fig. 1a we show the effect of varying the parameter $\xi$ that controls the satellite orbital radius, for fixed planet mass $M_{\rm p}=M_{\rm J}$ (where $M_{\rm J}$ is the Jupiter mass) and angle between the satellite and planet orbital planes $i_{\rm ps}=10^\circ$. For a given $\xi$, $P_{\rm det}^{\rm S}(a_{\rm p})$ decreases with increasing $a_{\rm p}$. The reason for this is that the range of favorable line-of-sight inclinations of the planet orbital plane, $i_{\rm p1}-i_{\rm p0}$,becomes narrower. The drop is more pronounced when $a_{\rm p}$ is small because then, $i_{\rm p1}-i_{\rm p0}$ is roughly proportional to $a_{\rm p}^{-1}$. For large $a_{\rm p}$,$i_{\rm p1}-i_{\rm p0}$ reaches the asymptotic value $Xi_{\rm ps}(1-X)^{-1}$, and $P_{\rm det}^{\rm S}(a_{\rm p})$ becomes nearly constant. When the planet orbital radius reaches the value $a_{\rm p}=a_2$, however, $P_{\rm det}^{\rm S}(a_{\rm p})$ undergoes a new drop as the orbital probability of catching the satellite decreases (see Eqs. (12), (13) and (21)). This drop occurs at $\log(a_{\rm p})\mathrel{\hbox to 0pt{\lower 3.5pt\hbox{$\mathchar''218$}\hss} \raise 1.5pt\hbox{$\mathchar''13E$}}2.6$ for $\xi=3$ and at larger $a_{\rm p}$ for $\xi=6$ and 43.

  

Figure 1: Probability $P_{\rm det}^{\rm S}(a_{\rm p})$ of detecting a satellite transit during a planetary period, when the parent planet does not itself transit over the star. a) For different values of $\xi=M_{\rm p}^{1/3}a_{\rm p}a_{\rm s}^{-1}$,as indicated, at fixed angle $i_{\rm ps}=10^\circ$ between the planet and satellite orbital planes and fixed planetary mass $M_{\rm p}=M_{\rm J}$ ($a_{\rm p}$ and $a_{\rm s}$ are, respectively, the planet and satellite orbital radii). b) For two values of $M_{\rm p}$, as indicated, at fixed $\xi=6$ and $i_{\rm ps}=10^\circ$.c) For different values of $i_{\rm ps}$, as indicated, at fixed $\xi=6$and $M_{\rm p}=M_{\rm J}$

  

Figure 2: Probability $P_{\rm det}^{\rm S}(a_{\rm p})$ of detecting a satellite transit during a period of observations of 3 years, when the parent planet does not itself transit over the star. Two values of $M_{\rm p}$ are considered, as indicated, at fixed $\xi=6$ and $i_{\rm ps}=10^\circ$ (as in Fig. 1b). The dotted line shows the probability P=a-1 of detecting a planet

Figure 1a shows that the detection probability is higher for satellites with smaller $\xi$, corresponding to larger orbital radii, because the geometric probability that defines the upper limit $i_{\rm p1}$ of the integral in Eq. (17) is consequently larger. For example, for $a_{\rm p}=100$,$P_{\rm det}^{\rm S}(a_{\rm p})$ is a factor of 2 and 10 larger for $\xi=3$ than for $\xi=6$ and $\xi=43$, respectively. We note, for reference, that the Moon has $\xi=6$ and the satellite Io of Jupiter would have $\xi=43$ if the planet were located at 1 AU (i.e., $a_{\rm p}=215$) from the Sun. Figure 1a also shows that $P_{\rm det}^{\rm S}(a_{\rm p})$ is not defined at small $a_{\rm p}$ for large $\xi$because for a satellite to subsist, $a_{\rm p}$ must be larger than $a_{\rm R}\,\xi\, M_{\rm p}^{-1/3}$ (Sect. 2.1).

  

Figure 3: Probability $P_{\rm det}^{\rm P}(a_{\rm p})$ of detecting a satellite transit during a planetary period, when the parent planet also transits over the star. The different curves in the three panels correspond to the same model assumptions as in Fig. 1

Figure 1b shows the effect on $P_{\rm det}^{\rm S}(a_{\rm p})$ of changing the planet mass $M_{\rm p}$ for fixed $\xi=6$ and $i_{\rm ps}=10^\circ$, while Fig. 1c shows the effect of changing $i_{\rm ps}$ for fixed $M_{\rm p}=M_{\rm J}$ and $\xi=6$. The fact that the detection probability appears to increase with the planet mass results from our parameterization of $a_{\rm s}=\xi^{-1}\,M_{\rm p}^{1/3}\,a_{\rm p}$, which for fixed $\xi$ and $a_{\rm p}$ implies that $a_{\rm s}$ increases with $M_{\rm p}$. In fact, massive planets have larger Hill spheres, and hence, they may have more distant satellites. The case $M_{\rm p}=0.15M_{\rm J}$ ($\approx 50M_{\rm E}$)in Fig. 1b corresponds to a planet with $r_{\rm p}\approx 2.5r_{\rm E}$ (where $M_{\rm E}$ and $r_ {\rm E}$ are the Earth mass and radius), which according to Guillot et al. (1996) is the maximum radius for a terrestrial planet. As expected, Fig. 1c shows that $P_{\rm det}^{\rm S}(a_{\rm p})$is higher when the angle between the planet and satellite orbital planes is larger, since again the upper limit $i_{\rm p1}$ of the integral in Eq. (17) is consequently larger. For example, for $a_{\rm p}=100$, $P_{\rm det}^{\rm S}(a_{\rm p})$ is a factor of 2.8 and 50 larger for $i_{\rm ps}=30^\circ$ than for $i_{\rm ps}=10^\circ$and $i_{\rm ps}=0^\circ$, respectively.

The detection probability of a pure satellite transit will be increased with respect to the predictions of Fig. 1 if the parent planet orbits more than once during the period of observations. In Fig. 2 we show $P_{\rm det}^{\rm S}(a_{\rm p})$assuming a period of observations of 3 years, for the same satellite and planet parameters as in Fig. 1b. Since the planet period scales as $a_{\rm p}^{3/2}$, independent of mass, the relative increase in $P_{\rm det}^{\rm S}(a_{\rm p})$at fixed $a_{\rm p}$ with respect to Fig. 1b ($n_{\rm t}=1$) is the same for satellites of $M_{\rm p}=M_{\rm J}$ and $M_{\rm p}=0.15M_{\rm J}$ planets. The increase in $P_{\rm det}^{\rm S}(a_{\rm p})$ is slightly larger at small $a_{\rm p}$, corresponding to short periods, than at large $a_{\rm p}$. For example, $P_{\rm det}^{\rm S}(a_{\rm p})$ is a factor of 3.9 larger at $a_{\rm p}=30$ and a factor of 3.6 larger at $a_{\rm p}=100$ than in the case $n_{\rm t}=1$. This result applies to all the curves in Fig. 1. For comparison, we also show in Fig. 2 the probability $a_{\rm p}^{-1}$ of detecting a planet. When $a_{\rm p}$ is large, therefore, the probability to detect a planet is only slightly larger than that of detecting a satellite without a planet.

We now turn to the case in which the planet is known to transit. Figures 3a, b and c show $P_{\rm det}^{\rm P}(a_{\rm p})$ for the same planet and satellite parameters as in Fig. 1. In all cases the probability to detect a satellite during the transit of the planet is very close to unity. The main features on the curves may be understood from the above discussion of Fig. 1, after operating a multiplication by $a_{\rm p}$. Unlike $P_{\rm det}^{\rm S}(a_{\rm p})$, however, $P_{\rm det}^{\rm P}(a_{\rm p})$ increases for combinations of parameters leading to smaller satellite orbital radii and inclinations, since the planet itself transits.

3.1 Examples of possible lightcurves

We now compute examples of stellar lightcurves expected during the transit of a planet and satellite system along the line of sight. Two types of lightcurves are expected in the case in which both the planet and the satellite transit in front of the star, depending on whether the orbital period of the satellite, $T_{\rm S}$, is shorter or longer than the duration of the planetary transit, $D_{\rm T}$. If $T_{\rm S} \ll D_{\rm T}$, planet-satellite transits may occur during the transit of the planet if the line-of-sight inclination of the satellite orbital plane is such that $i_{\rm s}<r_{\rm p}/a_{\rm s}$, where $r_{\rm p}$ is the planet radius (see Sect. 2.1). By planet-satellite transit we refer to both the transit of the satellite over the planet and the transit (occultation) of the planet over the satellite. Therefore, if $T_{\rm S} \ll D_{\rm T}$ and $i_{\rm s}<r_{\rm p}/a_{\rm s}$, there are at least two planet-satellite transits during a satellite orbit, at the two satellite conjunctions. If, on the other hand, $T_{\rm S}\gg D_{\rm T}$, a planet-satellite transit may or may not occur during a planetary transit for $i_{\rm s}<r_{\rm p}/a_{\rm s}$, depending on whether the satellite happens to pass through one of the two favorable conjunctions on its orbit in the time interval $D_{\rm T}$.

The upmost interest of planet-satellite transits is that they produce a relative increase of the apparent stellar luminosity, by an amount equal to the square of the satellite radius, $r_{\rm s}^2$ (see Eq. (1)), during the transit of the planet and satellite system in front of the star. Hence, planet-satellite transits can help to constrain observationally not only the presence, but also the period and orbital radius of the satellites of extrasolar planets. It should be noted that, since the condition for a planetary transit is $i_{\rm p}<a_{\rm p}^{-1}\ll1$, the above condition on $i_{\rm s}$ is almost equivalent to the condition $i_{\rm ps}= i_{\rm s}-i_{\rm p}\approx i_{\rm s}<r_{\rm p}/a_{\rm s}$.Also, we do not show here examples of lightcurves for the case in which the satellite alone transits in front of the star because these are similar to classical lightcurves of planetary transits (see Fig. 4b).

The condition for observing at least one planet-satellite transit, if $i_{\rm s}<r_{\rm p}/a_{\rm s}$, can be written as

$$T_{\rm R} < T_{\rm S} < 2\times D_{\rm T}\,,$$ (22)

where $T_{\rm R}$ is the period corresponding to the Roche limit. The factor of 2 in the above expression follows from the two conjunctions on a satellite orbit that can lead to a planet-satellite transit during a satellite period. The duration of the planetary transit can be rewritten in terms of the planet orbital radius and period as $D_{\rm T} = T_{\rm P} (\pi a_{\rm p})^{-1}$. Applying the third Kepler law to reexpress $a_{\rm p}$ in terms of $T_{\rm P}$, we obtain, for the condition on $T_{\rm S}$ for at least one planet-satellite transit,

$$T_{\rm R} < T_{\rm S} < 0.15 T_{\rm P}^{1/3}\,,$$ (23)

where units are days. The Roche period depends only on the satellite density and is $T_{\rm R}=0.23\,$day for a terrestrial satellite with $\rho= 5\,$g cm^-3. Thus, the satellite period must be very short, at least an order of magnitude shorter than the planet period, for planet-satellite transits to occur. By virtue of the third Kepler law this also implies that satellite orbital radius, $a_{\rm s}$, must be small, and hence, that the condition $i_{\rm s}<r_{\rm p}/a_{\rm s}$ is also likely to apply. Another implication of Eq. (23) is that the period of the planet iself must be larger than about $300\,(T_{\rm R} /{\rm days})^3$, or roughly 4 days for a terrestrial satellite, in order for a planet-satellite transit to occur.

The actual number of planet-satellite transits that will occur during a planetary transit is $N_{\rm PS}=2D_{\rm T}/T_{\rm S}$. Adopting the same parameterization as before of the satellite orbital radius, $a_{\rm s}\propto \xi^{-1}a_{\rm p}$ with $1\le \xi \le a_{\rm H}/a_{\rm R}$, this yields $N_{\rm PS}=0.15\,\xi^{3/2} \,T_{\rm P}^{-2/3}$. Note that since the satellite period at the Hill radius is equal to the planet period, one also has $1\le \xi \le (T_{\rm P}/T_{\rm R})^{2/3}$.Therefore, $\xi$ must be large, implying again that the satellite orbital radius must be small compared to the planet orbital radius, in order for $N_{\rm PS}$ to be at least unity. For example Jupiter satellite Io has $\xi \approx215$ and would transit Jupiter about twice during a transit of the planet in front of the Sun. On the other hand, if the Jupiter-Io system were located at only 1 AU from the Sun, the satellite would have only $\xi\approx 43$ and the corresponding $N_{\rm PS}$ would be of order unity. In this example, however, Io would be too small by a factor of about 4 in radius to produce a relative increase of the stellar luminosity that could be detected directly with a photometric accuracy of 10^-4 (but see Sect. 3.3).

Figure 4a shows the transit lightcurve of a planet with $r_{\rm p}= r_{\rm J}$ and $T_{\rm P}=50\,$days and a satellite with $r_{\rm s}=2.5r_{\rm E}$ and $T_{\rm S}=0.5 \,$days. Here, $r_{\rm J}$ and $r_ {\rm E}$ are the Jupiter and Earth radii, respectively. The stellar flux shows a marked but moderate decrease at $t=-5\,$hr, as the satellite first transits in front of the star. When the planet also starts to transit, at $t=-3.5\,$hr, the stellar flux drops more abruptly. Soon after, at $t=-1\,$hr, the relative flux maximum produced by the planet-satellite transit appears clearly. Then, the stellar flux sharply increases as the planet leaves the star, and the satellite, which now follows the planet, continues to moderately occult the star from about t=3.5 to 4.5 hr. The curvature of the lightcurve is a consequence of the star limb-darkening, the flux minimum corresponding to the point at which the planet is closest to the stellar center. Also shown by crosses in Fig. 4a are the results of simulated 10 min exposure observations with a poisson noise of 10^-4, typical of the photometric accuracy expected with the satellite COROT for 10$^{\rm th}$ magnitude stars. All feature of the lightcurve are mapped faithfully. In particular, the planet-satellite transit is detected with a signal-to-noise ratio of order 10. For reference, Fig. 4b shows the transit lightcurve of the same planet but without a satellite.

  

Figure 4: Examples of transit lightcurves (see text for interpretation). a) For a planet with $r_{\rm p}= r_{\rm J}$ and $T_{\rm P}=50\,$days and a satellite with $r_{\rm s}=2.5r_{\rm E}$ and $T_{\rm S}=0.5 \,$days. b) For a planet with $r_{\rm p}= r_{\rm J}$and $T_{\rm P}=50\,$days, alone. c) For a planet with $r_{\rm p}= r_{\rm J}$ and $T_{\rm P}=50\,$days and a satellite with $r_{\rm s}=2.5r_{\rm E}$ and $T_{\rm S}= 1.5\,$days. d) For a smaller planet with $r_{\rm p}=2.5r_{\rm E}$ and $T_{\rm P}= 100\,$days and a satellite with $r_{\rm s}=1.5r_{\rm E}$ and $T_{\rm S}=2\,$days. In all panels the solid line is the model lightcurve, and the crosses show the results of simulated 10 min exposure observations with a poisson noise of 10-4

If the satellite period does not satisfy condition (23) for at least one planet-satellite transit, the probability for the satellite to pass through one of the two favorable conjunctions that will produce a transit during the time interval $D_{\rm T}$ is simply given by $N_{\rm PS}$. As shown above, the probability will be highest for satellites with smallest orbital radii. The Moon, for example, has $T_{\rm S}\gg0.15T_{\rm P}^{1/3}$ and $\xi\approx6$, and it would have a probability of only 0.04 to be observed in a transit over or behind the Earth during a transit of the planet over the Sun.

Figure 4c shows the transit lightcurve of the same planet as in Fig. 4a, but with a satellite with $r_{\rm s}=2.5r_{\rm E}$ and $T_{\rm S}= 1.5\,$days, which does not satisfy condition (23). The abrupt drop of the stellar flux at the beginning of the transit is caused by the entry of the planet. The pronounced, but more moderate drop at $t\approx0\,$hr is caused by the subsequent entry of the satellite. Then, the planet first leaves the star causing the flux to increase sharply. The satellite continues to occult the star from about t=3.5 to 4.5 hr until it finally also leaves, and the stellar flux retrieves its original value. Again, the crosses show the results of simulated 10 min exposure observations with a poisson noise of 10^-4. For comparison, we show in Fig. 4d the transit lightcurve for a smaller planet with $r_{\rm p}=2.5r_{\rm E}$ and $T_{\rm P}= 100\,$days and a satellite with $r_{\rm s}=1.5r_{\rm E}$ and $T_{\rm S}=2\,$days. The characteristic signatures of the entry and exit of first the planet and then the satellite are similar to those in Fig. 4c. Although the signal-to-noise level is about 4 times smaller than in Fig. 4c, the satellite and planet transits are still detected unambiguously.