# Behaviour of a star as it approaches Ultrarelativistc limit

By considering the potential energy of a degenerate star of mass $$M$$ and radius $$R$$, I can use dimensional arguments to show that the radius of the star depends on its mass as:

$$R ∝ M^\frac{2-n}{4-3n}$$

Where in the non-relativistic regime $$n = 5/3$$ whilst in ultra-relativitic $$n = 4/3$$

By taking $$n = 5/3$$ I can see that non-relativistically

$$R ∝ M^\frac{-1}{3}$$

Could someone explain to me the physical nature of what happens when $$n = 4/3$$ and it becomes relativistic as I am unsure what

$$R ∝ M^{\infty}$$ means physically.

• Have you seen en.wikipedia.org/wiki/… ? In particular, see the section titled Mass–radius relationship and mass limit. – PM 2Ring Mar 27 '19 at 13:16

It means, that $$M \propto R^0$$ so that if you did have a star governed by a relativistic polytrope, there is only one mass the star can have. This is the "classic" Chandrasekhar mass, at which the radius shrinks to nothing and the density increases to infinity.