  In this paper, we study the Riemann problem of relativistic Euler system for
rarefied monatomic and diatomic gases when the constitutive equation for the
energy is the Synge equation that is the only one compatible with the
relativistic kinetic theory. The Synge equation is involved with modified
Bessel functions of the second kind and this makes the relativistic Euler
system quite complex. Based on delicate estimates of the modified Bessel
functions of the second kind, we provide a detailed investigation of basic
hyperbolic properties and the structure of elementary waves, especially for the
structure of shock waves and in this way, the mathematical theory of the
Riemann problem for these relativistic Euler system, which is analogous to the
corresponding theory of the classical ones, is rigorously provided.
