Demonstration of the jeans equation

[fab13]

Hello,

I have two problems on an article about the demonstration of the jeans equation. I have 2 problems :

1*) Firstly, I begin with the collisionless Boltzmann equation :

$$\dfrac{\partial\,f}{\partial t}+{\bf v}\,\cdot\,\nabla\,f-\nabla\,\Phi\,\cdot\,\dfrac{\partial\,f}{\partial {\bf v}}=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$

By integrating over all possible velocities, we can write :

$$\int\,\dfrac{\partial\,f}{\partial t}\,d^{3}{\bf v}+\int\,v_{i}\,\dfrac{\partial\,f}{\partial x_{i}}\,d^{3}{\bf v}-\dfrac{\partial\,\Phi}{\partial x_{i}}\,\int\,\dfrac{\partial\,f}{\partial v_{i}}\,d^{3}{\bf v}=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)$$

By defining the density and the mean stellar velocity :

$$\nu=\int\,f\,d^{3}{\bf v}\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\bar{v_{i}}=\dfrac{1}{\nu}\,\int\,f\,v_{i}\,d^{3}{\bf v}$$

I don't understand how we can get from eq(2) the continuity equation :

$$\dfrac{\partial\,\nu}{\partial t}+\dfrac{\partial\,(\nu\,\bar{v_{i}})}{\partial x_{i}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(3)$$

How can we make vanish the third term of eq(2) in order to get eq(3) ?

2*) My second problem : get the jeans equation.

If I multiply the collisionless Boltzmann equation (1) by $$v_{j}$$ and integrate over all velocities, we get :

$$\dfrac{\partial}{\partial t}\int\,f\,v_{j}\,d^{3}{\bf v}+\int\,v_{i}\,v_{j}\,\dfrac{\partial\,f} {\partial x_{i}}\,d^{3}{\bf v}-\dfrac{\partial\,\Phi}{\partial x_{i}}\,\int\,v_{j}\,\dfrac{\partial\,f}{\partial v_{i}}\,d^{3}{\bf v}=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(4)$$

In the article, they say that : "using the fact that $$f\rightarrow 0$$ for large $$v$$ and applying the divergence theorem, eq(4) can be written as :

$$\dfrac{\partial\,\nu\,\bar{v_{j}}}{\partial t}+ \dfrac{\partial\,(\nu\,\overline{v_{i}\,v_{j}})}{\partial x_{i}}+\nu\dfrac{\partial\,\Phi}{\partial x_{j}}=0$$

How can I simplify the eq(4), especially the third term of eq(4) with the two above assumptions ?

From the divergence theorem, can I write : ?

$$\dfrac{\partial\,\Phi}{\partial x_{i}}\,\int\,v_{j}\,\dfrac{\partial\,f}{\partial v_{i}}\,d^{3} {\bf v}=\dfrac{\partial\,\Phi}{\partial x_{i}}\,\int\,v_{j}\,f\,d^{2}{\bf v}$$

Any help would be appreciated

 

[AndyW]

.

Hello,

Thank you for your questions. I am happy to provide some clarification on the demonstration of the Jeans equation.

1) In order to get from equation (2) to the continuity equation (3), we need to use the definition of density and mean velocity that you provided:

\nu=\int\,f\,d^{3}{\bf v}\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\bar{v_{i}}=\dfrac{1}{\nu}\,\int\,f\,v_{i}\,d^{3}{\bf v}

Using these definitions, we can rewrite the third term of equation (2) as:

-\dfrac{\partial\,\Phi}{\partial x_{i}}\,\int\,\dfrac{\partial\,f}{\partial v_{i}}\,d^{3}{\bf v} = -\dfrac{\partial\,\Phi}{\partial x_{i}}\,\dfrac{1}{\nu}\,\int\,f\,v_{i}\,d^{3}{\bf v} = -\dfrac{\partial\,\Phi}{\partial x_{i}}\,\bar{v_{i}}\,\nu

Substituting this into equation (2) and rearranging, we get the continuity equation (3). The key is to recognize that the third term of equation (2) can be rewritten using the definition of mean velocity.

2) To simplify equation (4), we need to use the assumptions that f approaches 0 for large velocities and apply the divergence theorem. The third term can be written as:

-\dfrac{\partial\,\Phi}{\partial x_{i}}\,\int\,v_{j}\,\dfrac{\partial\,f}{\partial v_{i}}\,d^{3}{\bf v} = -\dfrac{\partial\,\Phi}{\partial x_{i}}\,\int\,v_{j}\,f\,d^{3}{\bf v}

Using the fact that f approaches 0 for large velocities, we can assume that the integral over all velocities is dominated by small velocities. This allows us to apply the divergence theorem, which states that the integral of a divergence over a closed surface is equal to the integral of the function over the volume enclosed by that surface. In