Rarefaction shock in plasma with a bi-Maxwellian electron distribution function

The one-dimensional collisionless expansion into a vacuum of a plasma with a bi-Maxwellian electron distribution function and a single ion species is studied both theoretically and numerically. A shock wave occurs when the ratio of the temperatures between the hot and the cold electrons is larger than 5+√(24)[B. Bezzerides, D. W. Forslund, and E. L. Lindman, Phys. Fluids 21, 2179 (1978)]. The theoretical model presented here gives a coherent and complete description of the rarefaction shock and its effects on the ion acceleration process. Analytical expressions of the characteristics of the shock are given. The analytical findings are compared to the results of a hybrid code describing the plasma expansion, and an excellent agreement is obtained.