Derivation from Boltzmann equation

The Collisionless Boltzmann equation, also called the Vlasov Equation is a special form of Liouville' equation and is given by:^[3]

${\displaystyle {\partial f \over \partial t}+v{\partial f \over \partial r}-{\partial \Phi \over \partial r}{\partial f \over \partial v}=0}$ Or in vector form: ${\displaystyle {\partial f \over \partial t}+{\vec {v}}\cdot {\vec {\nabla }}f-{\vec {\nabla }}\Phi \cdot {\partial f \over \partial {\vec {v}}}=0}$

Combining the Vlasov equation with the Poisson equation for gravity:

$${\displaystyle \nabla ^{2}\phi =4\pi G\rho .}$$

gives the Jeans equations.

More explicitly, If n=n(x,t) is the density of stars in space, as a function of position x = (x₁,x₂,x₃) and time t, v = (v₁,v₂,v₃) is the velocity, and Φ = Φ(x,t) is the gravitational potential, the Jeans equations may be written as:^[4][5]

${\displaystyle {\frac {\partial n}{\partial t}}+\sum _{i}{\frac {\partial (n\langle {v_{i}}\rangle )}{\partial x_{i}}}=0,}$

${\displaystyle {\frac {\partial (n\langle {v_{j}}\rangle )}{\partial t}}+n{\frac {\partial \Phi }{\partial x_{j}}}+\sum _{i}{\frac {\partial (n\langle {v_{i}v_{j}}\rangle )}{\partial x_{i}}}=0\qquad (j=1,2,3.)}$

Here, the ⟨...⟩ notation means an average at a given point and time (x,t), so that, for example, ${\displaystyle \langle {v_{1}}\rangle }$ is the average of component 1 of the velocity of the stars at a given point and time. The second set of equations may alternately be written as

${\displaystyle n{\frac {\partial \langle {v_{j}}\rangle }{\partial t}}+\sum _{i}n\langle {v_{i}}\rangle {\frac {\partial {\langle {v_{j}}\rangle }}{\partial x_{i}}}=-n{\frac {\partial \Phi }{\partial x_{j}}}-\sum _{i}{\frac {\partial (n\sigma _{ij}^{2})}{\partial x_{i}}}\qquad (j=1,2,3.)}$

where the spatial part of the stress–energy tensor is defined as: ${\displaystyle \sigma _{ij}^{2}=\langle {v_{i}v_{j}}\rangle -\langle {v_{i}}\rangle \langle {v_{j}}\rangle }$ and measures the velocity dispersion in components i and j at a given point.

Some given assumptions regarding these equations include:

• The flow in phase space must conserve mass
• The density around a given star remains the same, or is incompressible

Notice that the Jeans equations contain 9 unknowns (3 average velocities and 6 stress tensor terms), but only 3 equations. This means that Jeans equations are not closed. To solve different systems, various assumptions are made about the stress tensor.^[6]

Spherical Jeans equations

One fundamental usage of Jean's equation is in spherical gravitational bodies. In spherical coordinates, the equations are:^[6]

${\displaystyle {\partial \rho \langle v_{r}\rangle \over \partial t}+{\partial \rho \langle v_{r}^{2}\rangle \over \partial r}+{\rho \over r}[2\langle v_{r}^{2}\rangle -\langle v_{\theta }^{2}\rangle -\langle v_{\phi }^{2}\rangle ]+\rho {\partial \Phi \over \partial r}=0}$

${\displaystyle {\partial \rho \langle v_{\theta }\rangle \over \partial t}+{\partial \rho \langle v_{r}v_{\theta }\rangle \over \partial r}+{\rho \over r}[3\langle v_{r}v_{\theta }\rangle +(\langle v_{\theta }^{2}\rangle -\langle v_{\phi }^{2}\rangle )\cot(\theta )]=0}$

${\displaystyle {\partial \rho \langle v_{\phi }\rangle \over \partial t}+{\partial \rho \langle v_{r}v_{\phi }\rangle \over \partial r}+{\rho \over r}[3\langle v_{r}v_{\phi }\rangle +2\langle v_{\theta }v_{\phi }\rangle \cot(\theta )]=0}$

Using the stress tensor with the assumption that it is diagonal and ${\displaystyle \sigma _{\theta }^{2}=\sigma _{\phi }^{2}}$, can reduce these equations to a single simplified equation:

${\displaystyle {\partial (\rho \sigma _{r}^{2}) \over \partial r}+{2\rho \over r}[\sigma _{r}^{2}-\sigma _{\theta }^{2}]+\rho {\partial \Phi \over \partial r}=0}$

Again, there are two unknown functions (${\displaystyle \sigma _{r}^{2}(r)}$ and ${\displaystyle \sigma _{\theta }^{2}(r)}$) that require assumptions for the equation to be solved.

Milky Way dark matter halo

An example of such an analysis is given by the constraints that can be placed on the dark matter halo within the Milky Way. Using Sloan Digital Sky Survey measurements of our Galaxy, researchers were able to simulate the dark matter halo distribution using Jeans equations.^[8] By comparing measured values with Jeans equation simulation results, they confirmed the need for extra dark matter and placed limits on its ellipsoid size. They estimated the ratio of minor to major axis of this halo to be 0.47 ${\displaystyle \pm }$ 0.14. This method has been applied to many other galactic halos ^[9] and have produced similar results regarding dark matter halo topology.

Simulation limitations

The limiting factor of these simulations however, has been the data required to approximate stress tensor parameter values that dictate the Jeans equations behavior. Additionally, some constraints can be placed on Jeans equation simulations in order to produce reliable results ^[10][11] Some of these limitations include a wavelength resolution requirement, variable gravitational softening, and a minimum vertical structure particle resolution.