# Temp of a Blackhole

1. Oct 13, 2015

### Crush1986

1. The problem statement, all variables and given/known data

The problem is to calculate the temperature of a one solar mass black hole
2. Relevant equations

$$S = \frac{8\pi^2GM^2k}{hc}$$
$$E = Mc^2$$
$$\frac{1}{T} = \frac{\partial S}{\partial U}$$
3. The attempt at a solution

My first solution I pulled out an $$Mc^2$$ Which left my equation looking like $$S = \frac{k8\pi^2GM} {hc}*U$$ and did the partial derivative with respect to U of the entropy equation. I found that I was off by a factor of two (I think, I don't for sure know the right answer but some friends got answers 1/2 as much as my answer).

I think I know why and I just want to check out my reasoning. By only factoring out one U instead of U^2 I left an M in the equation. But M and U are intricately related right? So I must take out a U^2 in order to take out all of the M's in the original equation. It is only then that I get the correct result (That being 6.14 *10^-8 K, which I believe to be right but I'm also not entirely sure.)

Does this sound like a reasonable conclusion as to why I'm probably wrong?

Thank you for any help you can offer!

2. Oct 14, 2015

### JorisL

You forgot that through $E=Mc^2$ you have to use that $\frac{\partial M}{\partial U}$ also contributes to the derivative.
Then you get the right result.

Another approach you can use is using the chain rule.
Then you can write $\frac{1}{T} = \frac{\partial S}{\partial U} = \frac{\partial S}{\partial M}\frac{\partial M}{\partial U}$.
Usually I would go for this.
In this case the relation between E and M is simple but for harder problems (in possibly other domains) the algebraic manipulations can get ugly real quick increasing the probabilities of mistakes.

3. Oct 20, 2015

### Crush1986

This is so late I know, but, THANK YOU!!

I didn't quite get it at the time... I mean I knew of the chain rule, I just never really recognized when to use it until JUST now.

I've been shredding through a lot of problems tonight remembering to keep this little mathematical tool in my pocket.

4. Oct 21, 2015

### JorisL

Your welcome. It's useful to look back at old problems whenever you learn something new.
This helps you selecting suitable tools further on.

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