Property (1).

The first important fact about the solution $\gamma (\xi )$, away from the origin, comes from Equations (14) and (24). From the first of these, since all quantities on the right hand side are non-negative, and the only one that can be zero away from the origin is $\rho(r)$, it follow that $\beta'(r)$ is a non-negative function, that is zero only within a vacuum region. It therefore follows that $\beta(r)$ is a monotonically increasing function, which is a constant if and only if we are within a vacuum region. In addition to this, since according to the asymptotic boundary condition we have that $\beta(\infty)=1$, it also follows that $\beta(r)$ is limited from above by $1$. It then follows from Equation (24) that similar conclusions can be drawn for $\gamma (\xi )$, which is therefore a monotonically increasing function which is limited from above, the limit in this case being the parameter $\xi_{M}$, and which is constant within vacuum regions.

There are two more important properties of the solutions $\gamma (\xi )$ and $\pi (\xi )$ of Equations (37) and (38), away from the origin, that we can establish analytically. Let us therefore consider Equation (37), in which $F(\xi,\pi)$ is given as in Equation (32). Let us also recall that we are working under the assumption that we have $n>1$.