Editing Stellar dynamics (section)

=== Weak encounters ===
Weak encounters have a more profound effect on the evolution of a stellar system over the course of many passages. The effects of gravitational encounters can be studied with the concept of [[Relaxation (physics)|relaxation]] time.  A simple example illustrating relaxation is two-body relaxation, where a star's orbit is altered due to the gravitational interaction with another star.

Initially, the subject star travels along an orbit with initial velocity, <math>\mathbf{v}</math>, that is perpendicular to the [[impact parameter]], the distance of closest approach, to the field star whose gravitational field will affect the original orbit. Using Newton's laws, the change in the subject star's velocity, <math>\delta \mathbf{v}</math>, is approximately equal to the acceleration at the impact parameter, multiplied by the time duration of the acceleration.

The relaxation time can be thought as the time it takes for <math>\delta \mathbf{v}</math> to equal <math>\mathbf{v}</math>, or the time it takes for the small deviations in velocity to equal the star's initial velocity. The number of "half-diameter" crossings for an average star to relax in a stellar system of <math>N</math> objects is approximately 
<math display="block">{t_\text{relax} \over t_\text{ς}} = N^{\text{relax}} 
\backsimeq \frac{0.123(N-1)}{\ln (N-1)} \gg 1</math>
from a more rigorous calculation than the above mean free time estimates for strong deflection.

The answer makes sense because there is no relaxation for a single body or 2-body system. A better approximation of the ratio of timescales is <math> \left.\frac{N'}{\ln \sqrt{1+ N'^2}}\right|_{N'=0.123(N-2)}</math>, hence the relaxation time for 3-body, 4-body, 5-body, 7-body, 10-body, ..., 42-body, 72-body, 140-body, 210-body, 550-body are about 16, 8, 6, 4, 3, ..., 3, 4, 6, 8, 16 crossings.  There is no relaxation for an isolated binary, and the relaxation is the fastest for a 16-body system; it takes about 2.5 crossings for orbits to scatter each other.  A system with <math> N \sim  10^2 - 10^{10} </math> have much smoother potential, typically takes <math> \sim \ln N' \approx (2-20) </math> weak encounters to build a strong deflection to change orbital energy significantly.