Three‐dimensional Interaction between a Planet and an Isothermal Gaseous Disk. II. Eccentricity Waves and Bending Waves

We perform linear calculations to investigate three-dimensional density waves excited by planets on elliptical and inclined orbits in isothermal protoplanetary disks. We consider small planets that have no disk gap around their orbits. Eccentricities and inclinations of planets are assumed to be smaller than the disk aspect ratio. This is reasonable for planets with no disk gap. The density wave excited by a planet with nonzero small eccentricity e and inclination i is decomposed into three components: the waves by a planet with e = i = 0, the eccentricity waves, and the bending waves. The eccentricity waves are related to the noncircular motion of the planet, while the bending waves are excited by the motion normal to the equatorial plane. In our formulation, these waves are described by the same wave equations, and only the perturbing potentials are different. We numerically solve the wave equations and calculate the force exerted on the planet by the waves. The force is not parallel to the velocity of the epicycle motion. From the force obtained, we also find the evolution rates in the eccentricity, the inclination, and the longitudes of the perihelion and the ascending node. The characteristic evolution time of these orbital elements is about 300(r/1 AU)2 yr for Earth-sized planets in the minimum-mass nebula disk. Eccentricity damping is caused by eccentricity waves, while inclination damping is due to bending waves for planets with small eccentricities and inclinations and with no disk gap. This means that to lowest order there is no coupling between the evolutions of the eccentricity and the inclination.