Hydrodynamic instability of the solar nebula in the presence of a planetary core

When a planetary core composed of condensed matter is accumulated in the primitive solar nebula, the gas of the nebula becomes gravitationally concentrated as an envelope surrounding the planetary core. Models of such gaseous envelopes have been constructed subject to the assumption that the gas everywhere is on the same adiabat as that in the surrounding nebula. The gaseous envelope extends from the surface of the core to the distance at which the gravitational attraction of core plus envelope becomes equal to the gradient of the gravitational potential in the solar nebula; at this point the pressure and temperature of the gas in the envelope are required to attain the background values characteristics of the solar nebula. In general, as the mass of the condensed core increases, increasing amounts of gas became concentrated in the envelope, and these envelopes are stable against hydrodynamic instabilities. However, the core mass then goes through a maximum and starts to decrease. In most of the models tested, the envelopes were hydrodynamically unstable beyond the peak in the core mass. An unstable situation was always created if it was insisted that the core mass contain a larger amount of matter than given by these solutions. For an initial adiabat characterized by a temperature of 450°K and a pressure of 5 × 10^?6 atm, the maximum core mass at which instability occurs is approximately 115 earth masses; this value is rather insensitive to the position in the solar nebula or to the background pressure of the solar nebula. However, if the adiabat is lowered, then the core mass corresponding to instability is decreased. Since the core masses found by Podolak and Cameron for the giant planets are significantly less than the critical core mass corresponding to the initial solar nebula adiabat, we conclude that the giant planets obtained their large amounts of hydrogen and helium by a hydrodynamic collapse process in the solar nebula only after the nebula had been subjected to a considerable period of cooling.