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Ayame17
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[SOLVED] Density of a star
For a star of mass M and radius R, the density increases from the centre to the surface as a function of radial distance r, according to
[tex]\rho = \rho_{c}[1-(\frac{r}{R})^2][/tex]
where [tex]\rho_{c}[/tex] is the central density constant.
a) Find M(r).
b) Derive the relation between M and R and show that the average density of the star is [tex]0.4\rho_{c}[/tex].
See above and below.
Right. Firstly, I believe there is a mistake in the question paper - I don't think that the (r/R) is meant to be squared, as I've seen the formula used many times before and it's never had it.
Part a I've done (without the squared bit), and got [tex]M_{r}=\frac{4\pi}{3}\rho_{c}r^3(1-\frac{3r}{4R})[/tex], which I know to be right.
Part b I've done most of, ending up with [tex]\rho_{c}=\frac{3M}{\pi(R^3)}[/tex], which I also know to be right. The bit I'm having trouble with is getting to the [tex]0.4\rho_{c}[/tex]. Can anyone offer any assistance?
Homework Statement
For a star of mass M and radius R, the density increases from the centre to the surface as a function of radial distance r, according to
[tex]\rho = \rho_{c}[1-(\frac{r}{R})^2][/tex]
where [tex]\rho_{c}[/tex] is the central density constant.
a) Find M(r).
b) Derive the relation between M and R and show that the average density of the star is [tex]0.4\rho_{c}[/tex].
Homework Equations
See above and below.
The Attempt at a Solution
Right. Firstly, I believe there is a mistake in the question paper - I don't think that the (r/R) is meant to be squared, as I've seen the formula used many times before and it's never had it.
Part a I've done (without the squared bit), and got [tex]M_{r}=\frac{4\pi}{3}\rho_{c}r^3(1-\frac{3r}{4R})[/tex], which I know to be right.
Part b I've done most of, ending up with [tex]\rho_{c}=\frac{3M}{\pi(R^3)}[/tex], which I also know to be right. The bit I'm having trouble with is getting to the [tex]0.4\rho_{c}[/tex]. Can anyone offer any assistance?