# Find relationship between mass and a pseudo-variable

1. Feb 2, 2016

### June_cosmo

1. The problem statement, all variables and given/known data
https://dept.astro.lsa.umich.edu/~mmateo/Astr404_W16/WebPage/Assignment_Jan21.pdf [Broken]

2. Relevant equations

3. The attempt at a solution
Apologize for the long question. I was able to solve problem a and b. But for problem c, I was confused. I asked my professor and he gave me this explanation:

"I give you theta (the temperature) as a function of xi; as I recall, that solution is theta = sin(xi) / xi. And you know too that rho(xi) = rho_c * theta since n=1 for this case.

So now you calculate the interior mass by replacing rho(r) with with this expression for rho(xi) and r^2 with xi^2 and dr with dxi. Then integrate from 0 (the center) to values between 0 and pi (3.14159 since the surface of the star corresponds to xi=pi). For example, the integral is from 0 to 1 to get m(xi=1) and so forth. As you will see the integral is analytic, so this is just evaluating a simple function for values of xi between 0 and pi. Then you plot this, normalizing the y axis to go from 0 to 1.0 and you can either leave xi to go from 0 to pi, or define a new varlable xi-prime = xi/pi and plot from 0 to 1 using xi-prime. "

so I am confused because xi=alpha*r,why can we substitute r^2 with xi^2? Even if we do this, we get a function with a few constants in it. How can we plot this function?

Last edited by a moderator: May 7, 2017
2. Feb 7, 2016

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Feb 8, 2016

### vela

Staff Emeritus
You can't. Also, you have it slightly wrong; it's $r = \alpha \xi$. You have to replace $r^2$ with $(\alpha \xi)^2$, and use the chain rule to relate $\frac{d}{dr}$ to $\frac{d}{d\xi}$.

That's why your professor said to plot the normalized function. You want $m_\xi(\pi) = 1$.