Towards a Deterministic Model of Planetary Formation I: a Desert in the Mass and Semi Major Axis Distributions of Extra Solar Planets

We examine the accretion of cores of giant planets from planetesimals, gas accretion onto the cores, and their orbital migration. We adopt a working model for nascent protostellar disks with a wide variety of surface density distributions in order to explore the range of diversity among extra solar planetary systems. If some cores can acquire more mass than a critical value of several Earth masses during the persistence of the disk gas, they would be able to rapidly accrete gas and evolve into gas giant planets. The gas accretion process is initially regulated by the Kelvin-Helmholtz contraction of the planets' gas envelope. Based on the assumption that the exponential decay of the disk-gas mass occurs on the time scales $\sim 10^{6}-10^{7}$ years and that the disk mass distribution is comparable to those inferred from the observations of circumstellar disks of T Tauri stars, we carry out simulations to predict the distributions of masses and semi major axes of extra solar planets. Since planets' masses grow rapidly from $10 M_{\oplus}$ to $100 M_{\oplus}$, the gas giant planets rarely form with asymptotic masses in this intermediate range. Our model predicts a paucity of extra solar planets with mass in the range 10-$100 M_{\oplus}$ and semi major axis less than 3AU. We refer to this deficit as a ``planet desert''. The effect of migration is to sharpen the boundaries and to enhance the contrast of the planet desert. The mass and semi major axis distributions generated in our simulations for the gas giants are consistent with those of the known extra solar planets.