An Analytical Solution of Dynamics of Self-gravitating Spherical Gas-Dust Cloud

Mohammed Abobaker

doi:10.13189/ujme.2019.070304

Abstract

In this paper, we present a simple model for the dynamics of one dimensional of a self-gravitating spherical symmetrical gas-dust cloud. We take two analytical approaches to study the dynamics of a gravitating system of a gasdust cloud. The first approach solves a set of non-linear equation of dynamics of a gravitating system. The second approach is a Cole-Hopf transformation, which is used to simplify the equations of dynamics and after that, we applied the method of characteristics to reduce partial differential equations to a system of entirely solvable ordinary differential equations. The results found by the analytical method and the Cole-Hopf method are compared with each other, showing that both lead to the same result. The obtained results in this study are presented in plots. We used the Mathematica software package in performing calculation and plotting graphs.

Highlights

Numerical methods often do not Keywords Hydrodynamics, Non-linear PDE, Cole-Hopf provide an opportunity to understand the internal nature of the Method, Gravitating System solutions obtained
All the physical quantities will depend on two independent variables; radius r and time r
The dynamics of a spherically symmetrical compressible gas-dust cloud is governed by the continuity equation

Summary

Fundamental equations

Assume the spherical cloud has mass M, radius R and uniform density ρ0. All the physical quantities will depend on two independent variables; radius r and time r. Let p(r, t), ρ(r, t), v(r, t), and Φ(r, t) be the pressure, mass density, radial velocity, and gravitational potential respectively. The dynamics of a spherically symmetrical compressible gas-dust cloud is governed by the continuity equation. We shall simplify our model even further and assume that the cloud collapses as pressureless dust, which corresponds to the equation of state p = 0; like many authors have been neglected the pressure to simplify the problem [16, 17].

Initial and boundary conditions

Cole-Hopf transformation method

Derivation of the analytical solution of the fundamental equations

Results and discussion

Conclusion and future work