Secular interactions between inclined planets and a gaseous disk

In a planetary system, a secular particle resonance occurs at a location wh ere the precession rate of a test particle (e.g., an asteroid) matches the frequency of one of the precessional modes of the planetary system. We inve stigate the secular interactions of a system of mutually inclined planets w ith a gaseous protostellar disk that may contain a secular nodal particle r esonance. We determine the normal modes of some mutually inclined planet-di sk systems. The planets and disk interact gravitationally, and the disk is internally subject to the effects of gas pressure, self-gravity, and turbul ent viscosity. The behavior of the disk at a secular resonance is radically different from that of a particle, owing mainly to the effects of gas pres sure. The resonance is typically broadened by gas pressure to the extent th at global effects, including large-scale warps, dominate. The standard reso nant torque formula is invalid in this regime. Secular interactions cause a decay of the inclination at a rate that depends on the disk properties, in cluding its mass, turbulent viscosity, and sound speed. For a Jupiter-mass planet embedded within a minimum-mass solar nebula having typical parameter s, dissipation within the disk is sufficient to stabilize the system agains t tilt growth caused by mean-motion resonances.