# Red Giant Core Contration: How does it change the star's radius?

By how, I mean what is the equation.

### Context

If the core of a red giant contracts the outer radius will expand; but I cannot find an equation that says this, but I have equations that say parts of this.

The surface radius, $R$, of a red giant is porportial to the core temperature, $T_c$, and stellar mass, $M$, in this way: $$R \propto \frac{M}{T_c}.$$ We can assume that the core radius is much smaller than the stellar radius, $R_c \ll R$.

## Question

A red giant with a non-fusing helium core contracts from $R_{c,0}$ to $R_{c,1}$, what are the stellar radii $R_0, R_1$?

Answers in terms of proportions to stellar mass, $M$, core mass, $M_c$, and core radius $R_c$, are preferred but any light (hehe stars) on the subject is appreciated.