The Jeans equations are a set of partial differential equations that describe the motion of a collection of stars in a gravitational field. The Jeans equations relate the second-order velocity moments to the density and potential of a stellar system for systems without collision. They are analogous to the Euler equations for fluid flow and may be derived from the collisionless Boltzmann equation. The Jeans equations can come in a variety of different forms, depending on the structure of what is being modelled. Most utilization of these equations has been found in simulations with large number of gravitationally bound objects.
History The Jeans equations were originally derived by James Clerk Maxwell. However, they were first applied to astronomy by James Jeans in 1915 while working on stellar hydrodynamics. Since then, multiple solutions to the equations have been calculated analytically and numerically. Some notable solutions include a spherically symmetric solution, derived by James Binney in 1983 and axisymmetric solutions found in 1995 by Richard Arnold.
Mathematics
Derivation from Boltzmann equation The collisionless Boltzmann equation, also called the Vlasov Equation is a special form of Liouville' equation and is given by:
∂ f ∂ t + v ∂ f ∂ r − ∂ Φ ∂ r ∂ f ∂ v = 0 {\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: ∂ f ∂ t + v → ⋅ ∇ → f − ∇ → Φ ⋅ ∂ f ∂ v → = 0 {\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: ∇ 2 ϕ = 4 π G ρ . {\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:
∂ n ∂ t + ∑ i ∂ ( n ⟨ v i ⟩ ) ∂ x i = 0 , {\displaystyle {\frac {\partial n}{\partial t}}+\sum _{i}{\frac {\partial (n\langle {v_{i}}\rangle )}{\partial x_{i}}}=0,}
∂ ( n ⟨ v j ⟩ ) ∂ t + n ∂ Φ ∂ x j + ∑ i ∂ ( n ⟨ v i v j ⟩ ) ∂ x i = 0 ( j = 1 , 2 , 3. ) {\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.)}
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