Two-body relaxation

Two-body relaxation is the cumulative randomization of stellar velocities by many weak gravitational encounters. Its timescale in a system of $N$ comparable stars is of order $N/\log N$ crossing times.

For a disk with thickness $z_0$, random relative speeds $u=\lambda v$ and small-impact cutoff $b_0\ll z_0\ll R$, the leading three-dimensional logarithmic encounter term has upper cutoff of order $z_0$. At fixed mass and radius, its number volume density exceeds the spherical density by order $R/z_0$. Hence $t_{\mathrm{disk}}/t_{\mathrm{sphere}}\sim {\lambda }^3(z_0/R)\log (R/b_0)/\log (z_0/b_0)$ when the same cutoff is used for comparison. Different speed-dependent cutoffs must be distinguished in numerical applications. Distant planar encounters contribute a convergent, non-logarithmic correction. This scaling ceases to apply if thickness is comparable to the cutoff.

In an unsoftened plane with number surface density $n_2$ and relative speed $u$, an encounter strip has area $2udtdb$. Thus the velocity diffusion rate is $8G^2{m}^2{n}_2/u\int db/b^2$, dominated by the small-impact cutoff. Setting this cutoff to $Gm/u^2$ and requiring a change of order $u^2$ gives $t_{\mathrm{rel}}\sim u/(Gm{n}_2)$. With $u=\lambda v$, $n_2\sim N/R^2$ and $v^2\sim GNm/R$, this is of order $\lambda R/v$, independent of $N$ at fixed total mass. The point-particle, zero-thickness limit is singular; fixed softening or nonzero thickness changes it. Numerical coefficients are not controlled by extending weak kicks down to order-one deflections.

For a mass $m$ passing at relative speed $u$ and impact parameter $b$, integration of $Gmb/(b^2+u^2{t}^2{)}^{3/2}$ gives the transverse kick $2Gm/(bu)$. Weak deflection requires $b\gg Gm/u^2$, up to the equal-mass order-one factor. This scale supplies a small-impact cutoff for the weak-scattering diffusion approximation, not a physical exclusion of stronger encounters.

A collisionless stellar system is one for which discrete gravitational encounters change its distribution negligibly over the time under study, usually $T\ll t_{\mathrm{rel}}$. Stars can orbit and respond collectively to a varying smooth gravitational potential even in this regime; collisionless does not mean force-free or stable against collective gravitational instabilities. Its smooth distribution follows the Collisionless Boltzmann equation.

The stellar relaxation time is the time on which random gravitational encounters change a star's velocity by an amount comparable to its characteristic random velocity. A weak straight-line encounter at relative speed $u$ and impact parameter $b$ gives a kick $2Gm/(bu)$. Three-dimensional encounter counting gives $d⟨|\Delta \mathbf{v}{|}^2⟩/dt=8\pi G^2{m}^{2}n\log (b_{\max }/b_{\min })/u$. Definitions of relaxation, velocity averaging and system structure alter order-one coefficients; the robust spherical scaling is $t_{\mathrm{rel}}\sim v^3/(G^2{m}^{2}n\log \Lambda )\sim (N/\log N)t_{\mathrm{dyn}}$. Collisionless simulation requirements depend on the duration and permitted fractional relaxation, not a universal minimum particle number.

Weak equal-mass small-angle gravitational encounters give a single-star transverse kick $2Gm/(bu)$ and a relative-velocity kick twice as large. With encounter rate $2\pi bdbnu$, the single-star squared-velocity diffusion rate is the displayed $D_v$, while the relative rate is $4D_v$. A chosen reference squared velocity divided by either rate defines a relaxation estimate; specifying which rate and reference are used is essential to a numerical prefactor. If $u=\sqrt{2}\sigma$, the convention $3{\sigma }^2/(4D_v)$ gives $3{\sigma }^3/(16\pi \sqrt{2}n{G}^2{m}^2\ln \Lambda )$. The single-star convention $3{\sigma }^2/D_v$ is four times larger. Velocity-distribution averaging further changes numerical coefficients, while the scaling and Coulomb logarithm in stellar dynamics persist. The factor $n$ must be number density, not a dimensionless population count.

Small-angle gravitational encounters occur at rate $2\pi bdbnu$ and supply kicks $2Gm/(bu)$. Independent kick variances add to $d⟨\Delta v^2⟩/dt=8\pi G^2{m}^{2}n\ln \Lambda /u$. Setting the accumulated variance to $u^2$ and using virial scaling gives a stellar relaxation time of order $0.1N/\ln N$ stellar crossing times, with the coefficient dependent on profile and crossing convention.

The Coulomb logarithm $\log \Lambda =\log (b_{\max }/b_{\min })$ collects the logarithmically equal contributions of gravitational encounters over a wide range of impact parameters. For a self-gravitating system of $N$ comparable stars, $\Lambda$ is of order $N$.

A stellar system is collisional on a timescale of interest when its two-body relaxation time is shorter than that timescale, so discrete encounters materially alter its distribution function.