In this paper, we investigate the conditions required for the 3 and 17 Earth mass solid planets in the Kepler-10 system to have formed through collisions and mergers within an initial population of embryos. By performing a large number of N-body simu
lations, we show that the total mass of the initial population had to be significantly larger than the masses of the two planets, and that the two planets must have built-up farther away than their present location, at a distance of at least a few au from the central star. The planets had to grow fast enough so that they would detach themselves from the population of remaining, less massive, cores and migrate in to their present location. By the time the other cores migrated in, the discs inner edge would have moved out so that these cores cannot be detected today. We also compute the critical core mass beyond which a massive gaseous envelope would be accreted and show that it is larger than 17 Earth masses if the planetesimal accretion rate onto the core is larger than 10^{-6} Earth mass per year. For a planetesimal accretion rate between 10^{-6} and 10^{-5} Earth mass per year, the 17 Earth mass core would not be expected to have accreted more than about 1 Earth mass of gas. The results presented in this paper suggest that a planetary system like Kepler-10 may not be unusual, although it has probably formed in a rather massive disc.
The interaction between a planet located in the inner region of a disc and the warped outer region is studied. We consider the stage of evolution after the planet has cleared-out a gap, so that the planetary orbit evolves only under the gravitational
potential from the disc. We develop a secular analysis and compute the evolution of the orbital elements by solving Lagranges equations valid to second order in the eccentricity. We also perform numerical simulations with the full disc potential. In general, the interaction between the disc and the planet leads to the precession of the orbit. The orbital plane therefore becomes tilted relative to the discs inner parts, with no change in the eccentricity. When the inclination approaches 90 degrees, there is an instability and the eccentricity increases. In this case, both the inclination and the eccentricity develop large variations, with the orbit becoming retrograde. As the eccentricity reaches high values, we would expect tidal capture on a short orbit of the planet by the star to occur. This instability happens when the disc is severely warped, or if there is a significant amount of mass in a ring inclined by at least 45 degrees relative to the initial orbital plane. The inclination of the orbit does not depend on the semimajor axis nor on the planets mass. However, for a significant inclination to be generated on a timescale of at most a few Myr, the planet should be beyond the snow line. The process described here would therefore produce two distinct populations of inclined planets: one with objects beyond the snow line with at most moderate eccentricities, and another with objects on short circularized orbits.
