Collisionless Dynamics Collisionless Dynamics V:V:

Relaxation and Relaxation and EquilibriumEquilibrium

ReviewReview Tensor virial theoremTensor virial theorem relates structural relates structural

properties (dproperties (d22IIijij/dt/dt22) to kinematic properties ) to kinematic properties (random KE + ordered KE + potential E).(random KE + ordered KE + potential E).

Scalar virial theorem: Scalar virial theorem: 2K + W [+ SP] = 02K + W [+ SP] = 0.. Applications:Applications:

MMvirvir from half-light radius and velocity dispersion. from half-light radius and velocity dispersion. M/LM/L from from (assuming spherical, non-rot) (assuming spherical, non-rot) Flattening of spheroids Flattening of spheroids v/v/..

Jeans theoremJeans theorem: Can express any DF as fcn of : Can express any DF as fcn of integrals of motion (E, Lintegrals of motion (E, Lzz, , LL, I, I33, …)., …).

SummarySummary Relaxation is driven by Relaxation is driven by phase mixingphase mixing and and

chaotic mixingchaotic mixing, with , with violent relaxationviolent relaxation being being dominant in collisionless mergers.dominant in collisionless mergers.

The end state of merging appears to be not The end state of merging appears to be not fully relaxed; the origin of NFW-like profiles fully relaxed; the origin of NFW-like profiles still not fully understood.still not fully understood.

Dynamical frictionDynamical friction (braking due to wakes) can (braking due to wakes) can cause orbital decay in several orbital times.cause orbital decay in several orbital times.

Heat capacity of gravitating systems is Heat capacity of gravitating systems is negative, hence cuspy systems with short negative, hence cuspy systems with short relaxation time undergo relaxation time undergo gravothermal gravothermal collapsecollapse (e.g. globulars). (e.g. globulars).