Find relationship between mass and a pseudo-variable

[June_cosmo]

Homework Statement

https://dept.astro.lsa.umich.edu/~mmateo/Astr404_W16/WebPage/Assignment_Jan21.pdf

Homework Equations

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?

 

[vela]

"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?

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 $\frac{d}{d\xi}$.

Even if we do this, we get a function with a few constants in it. How can we plot this function?

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