The Propagation of Shock Waves in a Stellar Model with Continuous Density Distribution.

Abstract

view Abstract Citations (55) References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Propagation of Shock Waves in a Stellar Model with Continuous Density Distribution. Carrus, Pierre A. ; Fox, Phyllis A. ; Haas, Felix ; Kopal, Zdenek Abstract The aim of the present paper has been to investigate the properties of progressing waves which arise if a compressible-gas configuration, in which the density po diminishes with increasing distance r from the center, as Po /3 is disturbed from its state of equilibrium by an instantaneous central explosion. It is shown that, if this explosion has been sufficiently energetic for the outgoing disturbance to possess the characteristics of a shock wave, the central part of our confignration will be effectively evacuated by the explosion up to a certain distance from the center, depending on the amount of energy liberated by the initial explosion. If the release has been instantaneous (and, in consequence, the total energy of the wave motion is independent of the time t), the inner boundary of our flow (inclosing the empty core) becomes, by definition, a surface of contact discontinuity. In order to study the properties of the actual flow of gas trapped between the shock wave and the contact discontinnity, which is of the nature of a progressing wave, the fundamental system of partial differential equations of our problem has been converted to ordinary differential equations with = as the sole independent variable and has been integrated numerically for 8 cases corresponding to different strengths (i.e., the Mach numbers) of the head wave and to the ratio of specific heats = . The physical properties of our flow-such as the velocity, pressure, and density at any point of the disturbed medium remain the same along the lines = Constant. In particular, the radii of both boundaries limiting our expanding regime increase as t4/5 and as for the generalized Roche model, the thickness ( wise) of the shell between them is found to increase with the increasing Mach number of the head wave. Table 1 contains the numerical data describing the details of the individual solutions in terms of nondimensional parameters, which can be easily converted into absolute units by a suitable choice of the initial parameters. Table 2 summarizes the physical properties of the respective shock waves and indicates, in particular, the fractional amount of energy necessary to give rise to the computed phenomena. Finally, the appendix contains a proof of the uniqueness of the solutions of the form of progressing blast waves investigated in Sections Il-V. It demonstrates that, if the dependent variables U, F, and of our fundamental system of partial differential equations are to be expressible in terms of an equivalent system of ordinary differential equations with (r, t) as the sole independent variable, (r, t) must necessarily be of the form (r) (t), where (r) is a power of r alone, while `P(t) may be either a power or an exponential of t. The product of the powers of both r and t represents, moreover, the only possible form of which will render the total energy of the corresponding wave motion independent of the time. If, ultimately, the expanding field of flow is to be headed by a shock front characterized by a constant Mach number (i.e., if both sides of the Rankine-Hugoniot conditions are to be functions of alone), the structure of the undisturbed confignration becomes uniquely specified. Publication: The Astrophysical Journal Pub Date: May 1951 DOI: 10.1086/145420 Bibcode: 1951ApJ...113..496C full text sources ADS |