This paper presents the characterization of the transonic stationary solutions for the hydrodynamic escape problem (HEP), which is an important issue in the study of the evolution of planetary atmospheres. The transonic stationary solutions of the HEP involve effects of gravity, heat, and conduction, which are based in reality in this study and the sonic points are singular points in the time independent model. The characterization is established by the geometric singular perturbation method on the adiabatic wind solution of the HEP. The existence and nonexistence of the adiabatic wind solution with or without heat effect has been explored. The smooth transonic stationary solution under the effect of conduction is verified by analyzing the adiabatic wind solution. The singularity at the sonic points can be moved to the thermal critical points under the effect of conduction. By introducing the artificial viscosity, the dynamics at the thermal critical points and sonic points is studied in detail. Such a smooth transonic solution is proved to be a regular perturbation of the adiabatic wind solution. Discontinuous standing shock solutions are constructed and their stationary layer profiles have also been studied. Simulations of the adiabatic wind solution with or without heat and conduction effects are presented to illustrate the relevant theoretical results.
- Adiabatic wind solution
- Conservation law
- Geometric singular perturbation
- Hydrodynamic escape problem