On detecting terrestrial planets with timing of giant planet transits

The transits of a planet on a Keplerian orbit occur at time intervals exactly equal to the period of the orbit. If a second planet is introduced the orbit is not Keplerian and the transits are no longer exactly periodic. We compute the magnitude of these variations in the timing of the transits, dt. We investigate analytically several limiting cases: (i) interior perturbing planets with much smaller periods; (ii) exterior perturbing planets on eccentric orbits with much larger periods; (iii) both planets on circular orbits with arbitrary period ratio but not in resonance; and (iv) planets on initially circular orbits locked in resonance. Case (iv) is perhaps the most interesting case since some systems are known to be in resonances and the perturbations are the largest. As long as the perturber is more massive than the transiting planet, the timing variations would be of order of the period regardless of the perturber mass! For lighter perturbers, we show that the timing variations are smaller than the period by the perturber to transiting planet mass ratio. An earth mass planet in 2:1 resonance with a 3-day period transiting planet (e.g. HD 209458b) would cause timing variations of order 3 minutes, which would be accumulated over a year. These are easily detectable with current ground-based measurements. For the case of both planets on eccentric orbits, we compute numerically the transit timing variations for several cases of known multiplanet systems assuming they were edge-on. Transit timing measurements may be used to constrain the masses and radii of the planetary system and, when combined with radial velocity measurements, to break the degeneracy between mass and radius of the host star. (abstract truncated)