Astronomy & Celestial Mechanics
the sky, computed
Concept

Two-body encounter — where it appears

A single gravitational interaction between two members of a stellar system, treated in isolation from the rest. Summing the squared velocity changes over all impact parameters gives the relaxation time, and the sum is equal per decade of impact parameter.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The wake, computed from the streamlines that make it. Left: 26 streamlines past a point mass, in the mass's own frame, each integrated from far upstream with the same speed and a different impact parameter, and mirrored about the axis. Nothing is drawn to converge — every track is the hyperbola its own impact parameter gives it, and they cross downstream because an attraction focuses. Right: the density that focusing produces at 3.2 focusing radii behind the mass, as (b/y)(db/dy) — the Jacobian of the map from starting radius to arrival radius — which peaks at 15.53 times the background at 0.03 radii off the axis. The overdensity is behind the mass, and that is the entire mechanism: the wake pulls backwards on the body that made it. What the figure cannot show is the steady state, because it has no time in it: a real wake is continuously replenished, and the drag is the sum over an infinite train of these encounters, which is where Chandrasekhar's logarithm comes from.

A drag with nothing to drag against

A massive body moving through a sea of light ones raises an overdensity behind itself and is pulled back by it. The force does not depend on the masses of the background bodies at all — and it is strongest at one particular speed.

gravitation · Dynamical friction The whole kick, delivered in about two encounter times. The transverse force a body feels while a mass sweeps past it on a straight line, and the velocity that force has delivered so far, both against time in units of the impact parameter divided by the relative speed. The force is the component of the inverse-square attraction perpendicular to the path, which is the impact parameter over the cube of the distance, and it is drawn at its peak value of one at closest approach. The rising curve is its running integral, scaled by twice the gravitational constant times the mass over the impact parameter and the speed. Two things are visible and both are the point. The integral of the force over all time is exactly two in these units, so the kick is exactly 2GM/bv with no free constant anywhere — an answer to a three-body-shaped question obtained without solving anything. And it arrives quickly: 71 per cent of it within a single encounter time of closest approach and 98.6 per cent within the 6 drawn, which is what licenses calling the whole thing an impulse. On the scale of anything slower, the velocity changes discontinuously.

An answer obtained along a path that was not taken

Integrate the force of a passing mass along the straight line the body would have followed if the encounter had not happened, and out comes an exact deflection with no free constant in it. The approximation is circular, it is wrong in a known direction by a known amount, and it is the reason stellar dynamics has closed forms at all.

gravitation · Impulse approximation

Named alongside it

The objects these essays reach for when they reach for this one.

Coulomb logarithmDynamical frictionImpulse approximationAdiabatic invariantClose encounterCollisionless systemCross-sectionGalaxy mergerGravitational wakeHyperbolic orbitImpulseMass segregation

All concepts