We investigate the formation of terrestrial planets in the late stage of planetary formation using two-planet model. At that time, the protostar has formed for about 3 Myr and the gas disk has dissipated. In the model, the perturbations from Jupiter
and Saturn are considered. We also consider variations of the mass of outer planet, and the initial eccentricities and inclinations of embryos and planetesimals. Our results show that, terrestrial planets are formed in 50 Myr, and the accretion rate is about $60% - 80%$. In each simulation, 3 - 4 terrestrial planets are formed inside Jupiter with masses of $0.15 - 3.6 M_{oplus}$. In the $0.5 - 4$ AU, when the eccentricities of planetesimals are excited, planetesimals are able to accrete material from wide radial direction. The plenty of water material of the terrestrial planet in the Habitable Zone may be transferred from the farther places by this mechanism. Accretion could also happen a few times between two major planets only if the outer planet has a moderate mass and the small terrestrial planet could survive at some resonances over time scale of $10^8$ yr. In one of our simulations, com-mensurability of the orbital periods of planets is very common. Moreover, a librating-circulating 3:2 configuration of mean motion resonance is found.
We perform numerical simulations to study the Habitable zones (HZs) and dynamical structure for Earth-mass planets in multiple planetary systems. For example, in the HD 69830 system, we extensively explore the planetary configuration of three Neptune
-mass companions with one massive terrestrial planet residing in 0.07 AU $leq a leq$ 1.20 AU, to examine the asteroid structure in this system. We underline that there are stable zones of at least $10^5$ yr for low-mass terrestrial planets locating between 0.3 and 0.5 AU, and 0.8 and 1.2 AU with final eccentricities of $e < 0.20$. Moreover, we also find that the accumulation or depletion of the asteroid belt are also shaped by orbital resonances of the outer planets, for example, the asteroidal gaps at 2:1 and 3:2 mean motion resonances (MMRs) with Planet C, and 5:2 and 1:2 MMRs with Planet D. In a dynamical sense, the proper candidate regions for the existence of the potential terrestrial planets or HZs are 0.35 AU $< a < $ 0.50 AU, and 0.80 AU $< a < $ 1.00 AU for relatively low eccentricities, which makes sense to have the possible asteroidal structure in this system.
We perform numerical simulations to study the secular orbital evolution and dynamical structure in the quintuplet planetary system 55 Cancri with the self-consistent orbital solutions by Fischer and coworkers (2008). In the simulations, we show that
this system can be stable at least for $10^{8}$ yr. In addition, we extensively investigate the planetary configuration of four outer companions with one terrestrial planet in the wide region of 0.790 AU $leq a leq $ 5.900 AU to examine the existence of potential asteroid structure and Habitable Zones (HZs). We show that there are unstable regions for the orbits about 4:1, 3:1 and 5:2 mean motion resonances (MMRs) with the outermost planet in the system, and several stable orbits can remain at 3:2 and 1:1 MMRs, which is resemblance to the asteroidal belt in solar system. In a dynamical point, the proper candidate HZs for the existence of more potential terrestrial planets reside in the wide area between 1.0 AU and 2.3 AU for relatively low eccentricities.
We use a multi-dimensional hydrodynamics code to study the gravitational interaction between an embedded planet and a protoplanetary disk with emphasis on the generation of vortensity (Potential Vorticity or PV) through a Baroclinic Instability. We s
how that the generation of PV is very common and effective in non-barotropic disks through the Baroclinic Instability, especially within the coorbital region. Our results also complement previous work that non-axisymmetric Rossby-Wave Instabilities (RWIs, Lovelace et al. 1999) are likely to develop at local minima of PV distribution that are generated by the interaction between a planet and an inviscid barotropic disk. The development of RWIs results in non-axisymmetric density blobs, which exert stronger torques onto the planet when they move to the vicinity of the planet. Hence, large amplitude oscillations are introduced to the time behavior of the total torque acted on the planet by the disk. In current simulations, RWIs do not change the overall picture of inward orbital migration but cause a non-monotonic behavior to the migration speed. As a side effect, RWIs also introduce interesting structures into the disk. These structures may help the formation of Earth-like planets in the Habitable Zone or Hot Earths interior to a close-in giant planet.
