4 Indirect detection by timing of the planet transits

If a satellite is not extended enough to produce a detectable signal in the stellar lightcurve, it may still be detected indirectly through the associated rotation of the planet around the barycenter of the planet-satellite system. This requires that a planetary transit be observed at least 3 times, as the effect of the rotation will be a periodical time shift of the lightcurve minima induced by the planet transits. We now estimate the expected time shift in the simple case where the satellite and planet orbital planes are aligned along the line of sight ($i_{\rm ps}\approx{i_{\rm p}}\approx0$). In this case, the projected diameter of the planet orbit around the barycenter of the planet-satellite system is $2 a_{\rm s} M_{\rm s}M_{\rm p}^{-1}$, where $M_{\rm s}$ is the satellite mass. Therefore, the expected time shift between transits will be

$$\Delta t\sim 2 a_{\rm s}M_{\rm s}M_{\rm p}^{-1} \times T_{\rm p}(2\pi a_{\rm p})^{-1}\,.$$ (24)

Measurements of $\Delta t$, in addition to revealing the presence of a satellite, provide also an estimate of the product of its mass and orbital radius, $a_{\rm s}M_{\rm s}$. The minimum detectable $a_{\rm s}M_{\rm s}$is determined by the the minimum measurable time shift, and hence, by the accuracy of the timing of lightcurve minima. If $\delta t_{\rm obs}$denotes the sampling time, i.e., the duration of each consecutive exposure, then we can expect to be sensitive to the presence of satellites with

$$M_{\rm s} a_{\rm s} \sim M_{\rm p} a_{\rm p} \pi \delta t_{\rm obs} / T_{\rm p}\,.$$ (25)

We have estimated the minimum sampling time required to determine the position of a lightcurve minimum with a timing accuracy of $\delta t_{\rm obs}$.We simulated observed lightcurves using different values of $\delta t_{\rm obs}$and considering poisson noise only, which we cross-correlated with the corresponding input model lightcurves. We find that the minimum $\delta t_{\rm obs}$ required corresponds to the exposure time needed to detect the depth of the planetary transit minimum at twice the photon noise level, i.e., $\left\vert\Delta F_\ast\right\vert \gt 2\sigma$ (see Eq. (1)). As an example, the minimum $\delta t_{\rm obs}$ would be about $0.6\,$s to time the transit of a Jupiter-like planet over a solar-type star of magnitude 10 with a telescope like the one onboard COROT, while it would be $\delta t_{\rm obs}\approx4\,$min to time the transit of a planet with radius $2r_{\rm E}$ over the same star. According to Eq. (25), one could then infer the presence of satellites with the mass of Io or with a mass a tenth that of the Earth, respectively, in these two cases.

As a final note, we mention some examples of the possible implications of our results for future observations with COROT. Transits of planets and satellites larger than about $2r_{\rm E}$ are expected to be detectable with COROT (see Sect. 2). The results of Sect. 3 and Sect. 4 then indicate that, if present, planets photometrically detectable and with orbital radii $a_{\rm p}=0.1$,0.2, and 0.3 AU should have probabilities 4.6%, 2.3%, and 1.5%, respectively, to produce observable transits. Also, if these planets have satellites that are themselves photometrically detectable, the probability to detect the satellites is found to be $\sim\!100\%$. The main reason for this is that since the planets under consideration have small orbital radii, the inclination of the orbital plane of a satellite is expected to be close to that of the parent planet. A satellite will also induce time shifts between successive planet transits because of the rotation of the planet around the barycenter of the planet-satellite system. Via this effect, with COROT it will be possible to detect or exclude the presence of satellites much smaller than those photometrically detectable (i.e., with $r_{\rm s} \sim 0.3 r_{\rm E}$ around a Jupiter-like planet for a sampling time $\delta t_{\rm obs}\approx 0.6\,$s). COROT, therefore, will set important constraints on the existence of satellites around detectable planets.

Acknowledgements

P.S. is an ESA fellow. We thank the referee for helpful comments.