EXPANSION-FREE CAVITY EVOLUTION: SOME EXACT ANALYTICAL MODELS

We consider spherically symmetric distributions of anisotropic fluids with a central vacuum cavity, evolving under the condition of vanishing expansion scalar. Some analytical solutions are found satisfying Darmois junction conditions on both delimiting boundary surfaces, while some others require the presence of thin shells on either (or both) boundary surfaces. The solutions here obtained model the evolution of the vacuum cavity and the surrounding fluid distribution, emerging after a central explosion, thereby showing the potential of expansion–free condition for the study of that kind of problems. This study complements a previously published work where modeling of the evolution of such kind of systems was achieved through a different kinematical condition.

Propagation of spherical shock waves in stars

Propagation of spherical shock waves through self-gravitating polytropic gas spheres such as stars, caused by an instantaneous central explosion of finite energy E, is discussed theoretically. The problem is characterized by two lengths R0, L, where $R_0 = \left(\frac {E}{4\pi p_0}\right)^{1|3},\;\;\;\;\; L = \left(\frac {3C^2_0}{2\pi \rho_0G} \right)^{1|2}$p0 and C0 are the values of pressure, density and velocity of sound at the centre of the equilibrium pre-explosion state, and G is the constant of gravitation. R0 and L are scales connected with the power of the explosion and the dimensions of the star respectively, and their ratio A = R0/L has a fundamental significance. A solution especially suitable in the case of A = O(1) is developed in the form of power series in R/R0 (R is the distance between the shock front and the centre) by a method similar to that used in previous papers by the present author (1953, 1954). An approximation to this solution is carried out up to the term in R3. In particular, the velocity of the shock wave U is found to be $\frac {U}{C_0} = 1\cdot30 \left(\frac{R}{R_0}\right)^{-3|2} \{1 +0\cdot41A^2\left(\frac{R}{R_0}\right)^2 + 0\cdot 57\left(\frac{R}{R_0}\right)^3 +\ldot\}$ for the case of λ = 1.4, where λ is the ratio of specific heats.

The propagation and structure of shock waves of varying strength in a self-gravitating gas sphere

This paper gives a general method to describe the motion of a spherically symmetric shock wave of varying strength moving in a gas where the density ahead of the shock front varies with distance from the centre. The method applies only as long as the density does not become zero ahead of the shock front at any instant. The motion is initiated by a central explosion which liberates a given amount of kinetic energy. The density distribution ahead of the shock front is shown to have an important effect on the variation in the shock strength, and a first approximation to the equation of motion of the shock front is deduced. A particular example is worked out in detail, and it is shown that for certain density distributions the blast wave consists of a thin shell of gas while the remainder of the original sphere is left intact.