Derive and Solve the Lane-Emden Equation for a Polytropic Gas Sphere

[MatinSAR]

Homework Statement

1. Assume a polytropic equation of state for the gas: $P=K \rho ^{\gamma}$
2. Using the mechanical part of the stellar equation (mass conservation and
hydrostatic equilibrium) , drive the below equation. $$ \frac{\gamma K}{r^2} \frac{d}{dr} \left[ r^2 \rho^{\gamma - 2} \frac{d\rho}{dr} \right] = -4 \pi G \rho $$ Using the specified change of variables, derive the Lane-Emden equation and solve it for the given boundary conditions when $𝑛 = 0$. Then, interpret the resulting solution.

Relevant Equations

Mass conservation and hydrostatic equilibrium equation: $$\frac{dM(r)}{dr} = 4\pi r^2 \rho(r)$$ $$\frac{dP(r)}{dr} = -\frac{GM(r)\rho(r)}{r^2} $$

We start with hydrostatic equilibrium: $$ \frac{dP(r)}{dr} = K \gamma \rho ^ {\gamma -1} \dfrac {d \rho}{dr} =-\frac{GM(r)\rho(r)}{r^2}$$ $$ K \gamma \rho ^ {\gamma -1} \dfrac {d \rho}{dr} = - \dfrac {G \rho (r) }{r^2} \int 4 \pi r^2 \rho (r) \, dr$$ $$ r^2 K \gamma \rho ^ {\gamma -2} \dfrac {d \rho}{dr} = -G \int 4 \pi r^2 \rho (r)$$ $$\dfrac { \gamma K}{r^2} \dfrac {d}{dr} \left[ r^2 \rho ^ {\gamma -2} \dfrac {d \rho}{dr} \right] = -4 \pi G \rho$$ I think my answer to part 2 is correct. I'm not sure about following parts of the question:
3. Set $\gamma \equiv \frac{(n+1)}{n}$ and change the variables as $$ \rho(r) \equiv \rho_c [D_n(r)]^n, \quad \text{where} \quad 0 \leq D_n \leq 1 \quad r \equiv \lambda_n \xi \quad \lambda_n \equiv \left[ (n+1) \left( \frac{K \rho_c^{(1-n)/n}}{4 \pi G} \right) \right]^{1/2}$$ 4. Now find the Lane-Emden equation $$\frac{1}{\xi^2} \frac{d}{d\xi} \left[ \xi^2 \frac{d D_n}{d\xi} \right] = -D_n^n$$ 5.Assuming proper boundary conditions solve the Lane-Emden equation analytically for n=0.
7.Can you interpret the results? What the results imply about the size of the star?

I thnik should use the change of variables in part 3 to obtain the equation in part 4, then set $n=0$ and solve it. Then, use the above boundary conditions to find $D(\xi)$. If I were right until now, I can use $r \equiv \lambda_n \xi$ to talk about the size of the star. Am I right?

 

[pasmith]

I thnik should use the change of variables in part 3 to obtain the equation in part 4, then set $n=0$ and solve it. Then, use the above boundary conditions to find $D(\xi)$.

That is exactly what the question is asking you to do. Note that the actual boundary conditions at the origin are $D_n(0) = 1$ and $D_n'(0) = 0$; $\xi_1$ is then the first positive zero of $D_n$.

Without finding $D_0$, what do the definitions $$\rho(r) = \rho_c [D_n(r)]^n$$ and $$\lambda_n \propto (n+1)^{1/2}\rho_c^{(1-n)/(2n)}$$ suggest about the case $n = 0$ for $\rho_c > 0$?

 

[MatinSAR]

That is exactly what the question is asking you to do.

I have reached the part where I need two boundary conditions to determine the constants in the relation: $$ D_0 = -\frac{1}{6} \xi^2 - \frac{C_0}{\xi} + C_1 $$ The boundary conditions are:
1. The pressure is zero at the surface.
2. The gradient of the density at the center is zero.

From the second condition, I understand why $C_0 = 0$. Since $\left. \frac{dD_n}{d\xi} \right|_{\xi=0} = 0$ we have $\left. \frac{C_0}{\xi^2} \right|_{\xi=0} = 0$.

so $C_0$ must be zero. However, I am unsure how to apply the first boundary condition. If I use $D_n(0) = 1$, it follows that the other constant $C_1$ will be 1, but I cannot find the relation between this and our first boundary condition. Why can't I use $D_n(\xi_1) = 0$?

Without finding $D_0$, what do the definitions $$\rho(r) = \rho_c [D_n(r)]^n$$ and $$\lambda_n \propto (n+1)^{1/2}\rho_c^{(1-n)/(2n)}$$ suggest about the case $n = 0$ for $\rho_c > 0$?

$\rho (r) = \rho_c$ which is constant. Does $\lambda_n$ tend to infinity?

 

[pasmith]

$\xi_1$ is an unknown; it must be solved for. The requirements that $D_n(0) = 1$ and that $D_n(\xi) > 0$ for $0 \leq \xi < \xi_1$ together with $D_n(\xi_1) = 0$ determine $\xi_1$. Here, $$0 = D_0(\xi_1) = 1 - \frac{\xi_1^2}{6}$$ is the equation to be solved for $\xi_1$.

Note also from the definition of $n$ that $\gamma$ is infinite for $n = 0$. What can you conclude about the validity of this model in the case $n = 0$?

 

[MatinSAR]

$\xi_1$ is an unknown; it must be solved for. The requirements that $D_n(0) = 1$ and that $D_n(\xi) > 0$ for $0 \leq \xi < \xi_1$ together with $D_n(\xi_1) = 0$ determine $\xi_1$. Here, $$0 = D_0(\xi_1) = 1 - \frac{\xi_1^2}{6}$$ is the equation to be solved for $\xi_1$.

We use the conditions $D(0)=1$ and $D'(0)=0$ to find the values of $C_1$ and $C_2$. After that, we can use $D(\xi_1)=0$ to determine $\xi_1$. Why do we need $\xi_1$, and what does it represent? Is it related to the radius of the star? I need to find the star's size (radius). I think $r_{star}=\lambda_n \xi_1$ because at $\xi = \xi_1$ , $D$ becomes 0.

Note also from the definition of $n$ that $\gamma$ is infinite for $n = 0$. What can you conclude about the validity of this model in the case $n = 0$?

$\gamma= \infty$ shows that the temperature is constant...

 

[MatinSAR]

We use the conditions $D(0)=1$ and $D'(0)=0$ to find the values of $C_1$ and $C_2$. After that, we can use $D(\xi_1)=0$ to determine $\xi_1$. Why do we need $\xi_1$, and what does it represent? Is it related to the radius of the star? I need to find the star's size (radius). I think $r_{star}=\lambda_n \xi_1$ because at $\xi = \xi_1$ , $D$ becomes 0.

I think this is correct, But I don't have any idea to how to use it to find star's radius since $\lambda = \infty$ for $n=0$ ...

 

[MatinSAR]

I did it this way:

Then substitute $\lambda_0 = R/ \xi_1$ in $M$'s equation we derived:

Edit:

$R^3 = 3M/4\pi \rho_c$​