Understanding Jeans Equations in Stellar Dynamics

The Jeans Equations and Collisionless Boltzmann Equation play a crucial role in describing the distribution of stars in a gravitational potential. By applying assumptions like axial symmetry and spherical symmetry, these equations provide insights into the behavior of large systems of stars. Despite the challenges in obtaining unique solutions, studying the velocity moments and simplifying assumptions can lead to practical applications. These equations are fundamental in stellar dynamics and are extensively used in astrophysics research.

• Jeans Equations
• Stellar Dynamics
• Collisionless Boltzmann Equation
• Astrophysics

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1. Jeans Equations See Cappellari MNRAS 2008

2. Collisionless Boltzman Eq The positions x and velocities v of a large system of stars can be described by the distribution function (DF) f(x, v). When the system is in a steady state under the gravitational influence of a smooth potential , the DF must satisfy the fundamental equation of stellar dynamics, the steady-state collisionless Boltzmann equation (Binney & Tremaine 1987, hereafter BT; equation [4-13b])

3. Collisionless Boltzman Eq Since f is a function of six variables, an infinite family of solutions satisfies equation Need additional assumptions and simplifications for a practical application of the equation. classic way of constraining the problem consists of drastically reducing it, from that of recovering the DF, studying the the velocity moments of the DF. This approach leads to the Jeans equations,

4. Jeans Equations Assume axial symmetry, cylindrical coordinates (R,z, ) and = f/ =0 can re-cast the Boltzmann eq as where vr,vz and v are velocities in the r,z and directions

5. Jeans Equations Then multiplying this equation by v vr and vz and integrating over all velocities one obtains the Jeans eq (BT 4-29) where is the number density Assuming steady state hydrodynamical equilibrium (all time derivatives are zero) and spherical symmetry implies v = v = 0

6. And you Get (MIT Course 8-902 notes)

7. Jeans Equations However these equations are still a function of 4 unknowns and getting a unique solution is difficult \