A Metric Represents a Sphere of Constant Density Comprising Electrically counterpoised Dust (ECD)

Abstract

The metric which represents a sphere of electrically counterpoised dust (ECD) with constant density π ρ 4 1 = is ( ) ( ) ( ) ( ) ( )  ( ) + Ω < < ∞      − +       + = = − + Ω ≤ ≤ dR R d a R R B A R B A dt ds R dR R d R a R dt ds 2 2 2 2 2 2 2 2 2 2 2 2 2 2 θ 0 θ where θ(R) is the Emden function satisfying the Emden equation with n=3, a is the coordinate radius, R is the radial coordinate , A and B are constants given by ( ) ( ) B a ( ) a A a a a θ θ θ = − = + 2 a is a constant whose value is restricted, 0 < a < b <c where b is the first zero of the function θ(R) + Rθ (R) c is the first zero of θ(R). Emden equation is the non-linear differential equation with initial conditions ( ) ( ) 0 ,1 0 0 2 2 2 + = −y y = y = dx dy dx x d y n For n=0,1,5 the differential equation has analytic solutions ,for n=2,3,4 it has solutions in series , for other positive values of n it can be solved numerically. The solutions which are important are those for which n=1.5 and n=3 which are used in astrophysics while the solution that we need is for n=3.The solution for n=3, we denote by θ(x) and call it the Emden function. It is curious that the solution we have found has connections to stars. The graph of θ(R) and a related graph θ(R) + Rθ (R) are drawn in the same figure and shown here.