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

<p>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 $dR/dt$.</p>

<p>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 $dL/dt$.</p>

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

<p>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.</p>

<p>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 &amp; Jeffries 2014 <a href="https://arxiv.org/abs/1404.0683" rel="nofollow">https://arxiv.org/abs/1404.0683</a> ).</p>

CC BY-SA 3.0
2016-06-15T16:23:25.463Z
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!
DeltaCentauri
CC BY-SA 3.0
2016-06-13T00:16:40.203Z
2016-06-13T11:31:35.723Z
2016-06-13T11:31:35.723Z
ProfRob
<p>I'm dealing with a homologously contracting star with Mass M, Radius R and a gravitational binding energy of </p>

<p>$E = -a GM^2 / R $ </p>

<p>(a is a constant). </p>

<p>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. </p>

<p>Thank you for your help! </p>

<star-formation><stellar-astrophysics>
Luminosity L(t) for a homologously contracting star
DeltaCentauri
# Luminosity L(t) for a homologously contracting star

(Asked by: username_0 on 2016-06-12)

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.

Thank you for your help!

**Tags:** star-formation, stellar-astrophysics

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## 1 answers

### Answer by username_1 on 2016-06-13. Score: 3

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> ).

> **Comments:**
> **username_0**: 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!  

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