1. Collisionless Dynamics V: Relaxation and Equilibrium Collisionless Dynamics V: Relaxation and Equilibrium

2. Review Tensor virial theorem relates structural properties (d 2 I ij /dt 2 ) to kinematic properties (random KE + ordered KE + potential E). Tensor virial theorem relates structural properties (d 2 I ij /dt 2 ) to kinematic properties (random KE + ordered KE + potential E). Scalar virial theorem: 2K + W [+ SP] = 0. Scalar virial theorem: 2K + W [+ SP] = 0. Applications: Applications: M vir from half-light radius and velocity dispersion. M vir from half-light radius and velocity dispersion. M/L from  (assuming spherical, non-rot) M/L from  (assuming spherical, non-rot) Flattening of spheroids  v/ . Flattening of spheroids  v/ . Jeans theorem: Can express any DF as fcn of integrals of motion (E, L z, L, I 3, …). Jeans theorem: Can express any DF as fcn of integrals of motion (E, L z, L, I 3, …).

36. Summary Relaxation is driven by phase mixing and chaotic mixing, with violent relaxation being dominant in collisionless mergers. Relaxation is driven by phase mixing and chaotic mixing, with violent relaxation being dominant in collisionless mergers. The end state of merging appears to be not fully relaxed; the origin of NFW-like profiles still not fully understood. The end state of merging appears to be not fully relaxed; the origin of NFW-like profiles still not fully understood. Dynamical friction (braking due to wakes) can cause orbital decay in several orbital times. Dynamical friction (braking due to wakes) can cause orbital decay in several orbital times. Heat capacity of gravitating systems is negative, hence cuspy systems with short relaxation time undergo gravothermal collapse (e.g. globulars). Heat capacity of gravitating systems is negative, hence cuspy systems with short relaxation time undergo gravothermal collapse (e.g. globulars).