# Luminosity L(t) for a homologously contracting star

I'm dealing with a homologously contracting star with Mass M, Radius R and a gravitational binding energy of

$E = -a GM^2 / R$

(a is a constant).

I was looking for a way to find an expression for L(t), the luminosity as a function of time. Unfortunately, my attempts using time derivatives have not taken me far.

Fairly sure this is standard bookwork if you are talking about a PMS star on the Hayashi track.

Differentiate your GPE wrt time, assuming that the mass is constant. Take half of this as the luminosity (via the virial theorem). This gives $L$ in terms of $dR/dt$.

Then if you are on the Hayashi track you can assume surface temperature $T$ is constant and Stefan's law gives you $dR/dt$ in terms of $L$, $T$ and $dL/dt$.

Substitute that in and integrate the resulting differential equation to get $L(t)$.

Using this approach, I managed to get $$L = \left(\frac{\alpha GM^2}{6}\right)^{2/3} \left(4\pi \sigma T^4\right)^{1/3} t^{-2/3},$$ where $T$ is the temperature of the Hayashi track in question.

If you don't want to make the Hayashi track assumption, you can say that $L = AM^B T^C$ defines tracks in the HR diagram and this can be used in addition to Stefan's law to eliminate $dT/dt$ and give your result in terms of constants $A$, $B$, $C$. These can come from polytropic theory or by fitting the $L,M,T$ function to numerical calculations. (E.g. see Jackson & Jeffries 2014 https://arxiv.org/abs/1404.0683 ).

• Hi Rob! Thank you very much for your answer! I didn't have time to redo the calculations on my own, yet, but I will let you know as soon as possible. It looks right to me! Thank you very much in advance! Commented Jun 15, 2016 at 16:23