Calculating a star's density profile

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The discussion focuses on solving for a star's density profile using the equation of hydrostatic equilibrium. The user initially integrates the equation but struggles with the right-hand side, which involves a non-uniform density profile expressed as ρ(r) = ρc(1 - r²/R²). They are advised to express the mass of the star as M = 4π∫₀ᴿ ρ(r) r² dr, rather than using the uniform density formula. Additionally, the importance of incorporating an equation of state relating pressure, density, and temperature is highlighted. The user expresses some confusion regarding gravitational potential but ultimately gains clarity on the relevant equations.
Angela G
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Homework Statement
Hello!

I am trying to solve this problem but I'm struggling with it, could someone please help me?

The exercise is



A star with density profile: $$ \rho = \rho_c \left( 1- \frac{r^2}{R^2}\right)$$

a) What is the central pressure ##P_c## and the central temperature ##T_c## in terms of M and the ## \rho_c##?

b) What is the pressure profile ## P(r)## and the temperature profile ## T(r) ## in terms of ##P_c##, ##T_c## in addition to r and R

c) What is the value of ## \alpha## in the expression for the total gravitational energy $$ \Omega = - \frac{\alpha GM^2}{R}$$
Relevant Equations
$$\frac{d P}{d r} = - \frac{\rho Gm}{r^2}$$
$$ P = \frac{\rho k T}{\mu m_H}$$
$$ \Omega = - \int_0^M \frac{G m dm}{r} $$
to solve a) I used The equation of hydrostatic equilibrium $$ \frac{d P}{d r} = - \rho \frac{GM}{r^2} \iff dP = - \rho \frac{GM}{r^2}dr \Longrightarrow \int_{P_c}^0 dP = - \int_0^R \rho \frac{GM}{r^2} dr $$
I replaced M as ## \rho V ## and then
I integrated both the left and right-hand sides and got at the left-hand side ## - P_c##, But I'm stuck on the right-hand side. I calculated it and got $$ - \frac{4 \cdot 8 \pi}{3 \cdot 15} G \rho_c^2 R $$
I was thinking to replace R with the definition of density ## \rho = \frac{ 3M} { 4 \pi R^3} ## But I'm not sure how to proceed, some ideas?
 
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The density profile is given as <br /> \rho(r) = \rho_c\left(1 - \frac{r^2}{R^2}\right). This is not uniform.

You have to start from the equation of hydrostatic equilibrium in the form <br /> \frac{dp}{dr} = -\rho(r) \frac{d\chi}{dr} where the gravitational potential \chi satisfies <br /> \frac{1}{r}\frac{d}{dr}\left(r\frac{d \chi}{dr}\right) = 4 \pi G \rho. Multiplying the first equation by r/\rho and differentiating with respect to r yields <br /> \frac{1}{r}\frac{d}{dr}\left(\frac{r}{\rho}\frac{dp}{dr}\right) = -4\pi G \rho. As the density is not uniform, the mass of the star is given by <br /> M = 4\pi\int_0^R \rho(r) r^2\,dr rather than M= (4/3)\pi R^3, which holds only for a uniform density.

I assume you have some equation of state which relates pressure, density and temperature. It would have been helpful to include that as a relevant equation.
 
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pasmith said:
The density profile is given as <br /> \rho(r) = \rho_c\left(1 - \frac{r^2}{R^2}\right). This is not uniform.

You have to start from the equation of hydrostatic equilibrium in the form <br /> \frac{dp}{dr} = -\rho(r) \frac{d\chi}{dr} where the gravitational potential \chi satisfies <br /> \frac{1}{r}\frac{d}{dr}\left(r\frac{d \chi}{dr}\right) = 4 \pi G \rho. Multiplying the first equation by r/\rho and differentiating with respect to r yields <br /> \frac{1}{r}\frac{d}{dr}\left(\frac{r}{\rho}\frac{dp}{dr}\right) = -4\pi G \rho. As the density is not uniform, the mass of the star is given by <br /> M = 4\pi\int_0^R \rho(r) r^2\,dr rather than M= (4/3)\pi R^3, which holds only for a uniform density.

I assume you have some equation of state which relates pressure, density and temperature. It would have been helpful to include that as a relevant equation.
yes, I'm sorry
For solving a) I have to do it with the hydrostatic equilibrium equation, to determine the pressure and to determine the temperature I have to use the expression I got from the pressure and use the ideal gas law ## P = \frac{\rho k T}{\mu m_H}##, for solving c) I think I should use the $$ \Omega = - \int_0^M \frac{G m dm}{r}$$ and then replace m with ## m = \rho V \Longrightarrow \frac{dm}{dr} = \frac{\rho}{dr} \frac{ dV}{dr} ##. I think this is the way to solve it because I did so in the last exercise, but in that case, the central density was ## \rho_c = \rho ##, so the mass could be replaced by ## \rho V(r) ##. but I will try your way, but I think I need a little more guidance because I do not understand that with the gravitational potential, I'm sorry that was a new sign😅😅

edit: I think I understood, thanks
 
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pasmith said:
the gravitational potential \chi satisfies <br /> \frac{1}{r}\frac{d}{dr}\left(r\frac{d \chi}{dr}\right) = 4 \pi G \rho.
For spherical coordinates, I think the left-hand side should be
<br /> \frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{d \chi}{dr}\right) = 4 \pi G \rho.
 
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