Propagation of Shock Waves in the Generalized Roche Model.

Abstract

The aim of the present paper has been to investigate the events which take place if the core of a generalized Roche model is made to expand as the result of an instantaneous central explosion. A sudden expansion of the core sets off a wave of condensation traveling through the envelope; and, if the initial explosion has been sufficiently strong, the outgoing disturbance will possess the characteristics of a shock wave. It is shown that, if the initial explosion has been instantaneous, the velocity, pressure, and density at any point of the disturbed medium can be made to depend on a single parameter, = , where t denotes the time and r the radial distance. The equations of the problem have been rewritten in terms of as the sole independent variable and integrated numerically for 18 cases corresponding to different Mach numbers of the shock waves and different ratios, y, of specific heats of the gas constituting the enve lope. Thee anding regime forms a concentric shell, limited on the outside by the shock front of radius varyingast2 3, while the interface of the core confronts us with a "contact discontinuity." For large values of the Mach number M, the thick- ness of the shell increases with increasing M and decreasing value of the ratio of specific heats; for = (corresponding to polyatomic gas) the radius of the core becomes zero for any strength of the shock. It is shown that, for = 5 (corresponding to monatomic gas), a shock wave characterized by a Mach number M = 2.877... is just sufficiently strong to endow the mass particles immediately behind it with a velocity equal to one of escape from the gravitational field of the configuration, while for M > 5.92. . . the whole envelope will eventually be ejected. Table 1 contains the numerical data describing the individual solutions in terms of nondimensional parameters which can be converted into absolute units by a suitable choice of the initial conditions; and Table 2 summarizes the physical properties of the respective shock waves.