Two-Body Relaxation and Mass Segregation

Stars in a galactic nucleus do not collide like hard spheres, yet their long-range gravitational deflections slowly change orbital energy and angular momentum. The accumulated random walk is called two-body, non-resonant relaxation. Its central scaling is $t_r\propto\sigma^3/(G^2\rho m\ln\Lambda)$ for equal masses: a dynamically hot system relaxes slowly, whereas high density and a heavy mass spectrum accelerate diffusion. Around a massive black hole, this process can build species-dependent cusps and feed low-angular-momentum orbits, but only when the weak, uncorrelated-encounter assumptions remain valid for long enough.

Required background. Classical Field Theory supplies phase-space distributions, fluxes, and the diffusion approximation used in the derivation.

Helpful background. Black-Hole Geometry and Mechanics identifies the central spacetime and relativistic boundary of the Newtonian treatment; Relativistic Matter and Compact Objects supplies the stellar and remnant mass spectrum; Black-Hole Astrophysics provides the surrounding source context; and Relativistic Two-Body Dynamics distinguishes the weak-scattering calculation here from relativistic compact-binary dynamics.

The weak-deflection random walk

Consider a test star passing a field star of mass $m_i$ at relative speed $v$ far from their encounter and impact parameter $b$. Along an approximately straight encounter, integrating the transverse acceleration gives the impulse

$$\lvert\Delta v_\perp\rvert\simeq \frac{2Gm_i}{bv}.$$

The rate of encounters in the annulus $b$ to $b+db$ is $d\Gamma=2\pi b\,db\,n_i v$. Independent impulses add in variance, not in amplitude, so summing over field-star species gives

$$\frac{d\langle(\Delta v_\perp)^2\rangle}{dt} \simeq \frac{8\pi G^2}{v} \left(\sum_i n_i m_i^2\right) \int_{b_{\min}}^{b_{\max}}\frac{db}{b} =\frac{8\pi G^2}{v} \left(\sum_i n_i m_i^2\right)\ln\Lambda.$$

The logarithm $\ln\Lambda=\ln(b_{\max}/b_{\min})$ records equal contributions per logarithmic impact-parameter interval. A usual lower scale is the impact parameter for an order-unity deflection, $b_{\min}\sim G(m_{\rm test}+m_i)/v^2$; a local density scale or the size of the orbit supplies $b_{\max}$. Neither choice is exact. Encounters near either endpoint are precisely where the straight-line, homogeneous approximation is least secure.

Define relaxation as an accumulated velocity-change variance comparable to the original squared speed. For an approximately isotropic, locally Maxwellian background with one-dimensional velocity dispersion $\sigma$, a standard timescale convention gives

$$t_r\simeq \frac{0.34\,\sigma^3} {G^2\ln\Lambda\sum_i n_i m_i^2}.$$

For equal field masses $m$, the denominator becomes $G^2m\rho\ln\Lambda$. The mass-spectrum version above assumes a common characteristic velocity distribution and Coulomb logarithm; distinct species with different dispersions require their velocity-dependent diffusion coefficients. Dimensions provide an immediate independent check: $G^2n m^2$ has units of velocity cubed per time, so the ratio has units of time. Alexander develops the same non-coherent diffusion hierarchy, including the impact-parameter and mass-spectrum qualifications, in Alexander 2017, §§ 2.1.1 and 2.1.3, author manuscript pp. 11–12.

Energy, angular momentum, and mass exchange

The symbol $t_r$ hides different observables. An order-unity fractional change in orbital energy takes a time of order $t_E\sim t_r$. At the same energy, let $J_c(E)$ be the specific angular momentum of a circular orbit. A highly eccentric orbit has $J\ll J_c$ and needs a much smaller angular-momentum change to alter its periapse. In the local random-walk estimate,

$$t_J(E,J)\sim \left(\frac{J}{J_c(E)}\right)^2t_r(E).$$

This separation is why angular-momentum diffusion can feed capture or disruption while the distribution in orbital energy evolves more slowly. It is also why a single quoted relaxation time is inadequate unless the evolving quantity and location are named.

For unequal masses, diffusion contains the second moment $\sum_i n_i m_i^2$, while dynamical friction produces a systematic drift of a massive test body toward lower orbital energy. In a light-star background, a useful order-of-magnitude segregation time for objects of mass $m_h$ is

$$t_{\rm seg}\sim\frac{\langle m\rangle}{m_h}t_r.$$

Heavy remnants therefore sink faster than the background relaxes. They do not, in general, reach global kinetic-energy equipartition: the black hole imposes a nearly Keplerian speed scale, and a sparse heavy component can develop a steep distribution while exchanging energy with a dominant light component. Weak and strong mass-segregation branches depend on the abundance and mass contrast of the heavy species, not merely on $m_h/\langle m\rangle$ Alexander 2017, §2.1.4, author manuscript pp. 12–14.

The Fokker–Planck formulation makes the distinction precise. Let $\mathcal N_i(\boldsymbol I,t)\,d\boldsymbol I$ count species-$i$ stars in an element of chosen orbital-action space. Drift and diffusion enter the number-conserving equation

$$\frac{\partial\mathcal N_i}{\partial t} =-\frac{\partial}{\partial I_a} \left(D_a^{(i)}\mathcal N_i\right) +\frac12\frac{\partial^2}{\partial I_a\partial I_b} \left(D_{ab}^{(i)}\mathcal N_i\right),$$

where $D_a=\langle\Delta I_a\rangle/\Delta t$ and $D_{ab}=\langle\Delta I_a\Delta I_b\rangle/\Delta t$ are the orbit-averaged diffusion-limit coefficients, with action indices summed. The drift coefficients include dynamical friction; the second-order coefficients carry stochastic diffusion. A phase-space distribution $f_i(E,J)$ is not automatically a number density per $dE\,dJ$: changing coordinates requires the phase-volume Jacobian as well as transformed drift and diffusion coefficients. Boundary conditions at large radius and at capture are part of the problem. A steady solution is not determined by the local timescale alone.

Regimes outside the local estimate

First, nearly Keplerian orbits behave as long-lived mass wires over a coherence time. Their residual torques can drive resonant angular-momentum relaxation much faster than uncorrelated two-body scattering. Mass precession, Schwarzschild precession, self-quenching, and changes in the stellar background end the coherence. The coherent mechanism changes angular momentum efficiently but does not share the local Coulomb-logarithm derivation Alexander 2017, §§2.2–2.3, author manuscript pp. 15–17.

Second, rare strong encounters, binaries, massive molecular clouds, and a time-dependent star-formation history violate the identical weak-kick picture. A small population of massive perturbers can dominate $\sum n_i m_i^2$ even when it contributes little to $\rho$. Conversely, an extended perturber does not behave like a point mass below its own size. These cases require explicit encounter kernels or direct orbit calculations.

Third, the local Maxwellian approximation is strained in an anisotropic Kepler potential. A calculation that inserts the observed one-dimensional dispersion and density into the formula estimates a timescale; it does not prove that a stationary cusp has formed. The nucleus must also be older than the relevant energy-relaxation time, and sources, sinks, mergers, or black-hole binaries must not have reset the distribution. A fuller action-space treatment and its boundary conditions are developed in Merritt 2013, Chapter 5.

Checks and physical interpretation

Three checks catch most misuses. In the equal-mass limit, replace $\sum n_i m_i^2$ by $m\rho$; replacing it by $\rho^2$ is dimensionally wrong. If all masses are equal, there is relaxation but no species-dependent mass segregation. If $J/J_c$ is reduced at fixed energy, the angular-momentum time must decrease quadratically in the diffusive regime.

The estimate applies to collisional evolution of the stellar environment tied to a central massive black hole. Collisionless phase mixing, bar dynamics on galaxy scales, and violent relaxation may reshape a galaxy, but they are not two-body relaxation and are not treated as substitutes for this diffusion calculation.

Exercises

An equal-mass component has $\sigma=100\ {\rm km\,s^{-1}}$, $\rho=10^6\,M_\odot\,{\rm pc^{-3}}$, $m=1\,M_\odot$, and $\ln\Lambda=15$. Estimate $t_r$ using $G=4.3009\times10^{-3}\ {\rm pc}\,M_\odot^{-1}({\rm km\,s^{-1}})^2$. Then estimate the segregation time of a rare $10M_\odot$ component, assuming the light background controls the diffusion coefficients.

Solution

Substitution gives $t_r=0.34(100)^3/[G^2(1)(10^6)(15)]\simeq1225\ {\rm pc}/({\rm km\,s^{-1}})$. Since $1\ {\rm pc}/({\rm km\,s^{-1}})\simeq0.978\ {\rm Myr}$, the relaxation time is about $1.20\ {\rm Gyr}$. The rough drift estimate gives $t_{\rm seg}\sim(1/10)t_r\simeq0.12\ {\rm Gyr}$. This second number would change if the heavy remnants were abundant enough to dominate the mass-squared moment or their own self-relaxation.

References

• Alexander, Tal. 2017. “Stellar Dynamics and Stellar Phenomena Near a Massive Black Hole.” Annual Review of Astronomy and Astrophysics 55: 17–57. DOI.
• Merritt, David. 2013. Dynamics and Evolution of Galactic Nuclei. Princeton University Press. DOI.