Astronomy & Celestial Mechanics
the sky, computed
Concept

Close encounter — where it appears

A passage of two bodies near enough that each substantially changes the other's orbit. What counts as near is set by comparing the deflection angle with unity rather than by any distance, so the same separation is a close encounter at low relative speed and a negligible one at high.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

The adaptive step starts 5.7× more accurate and ends 7.5× worse. The envelope of the relative energy error against elapsed revolutions — the worst error within each plotted interval rather than the error at one sample in it — for the same second-order symplectic integrator run two ways on a Kepler orbit at e = 0.5. The flat band is a fixed step of one 200th of a period: its error oscillates once per revolution and the envelope does not grow, because a symplectic method at constant h is the exact solution of a nearby Hamiltonian and is conserving that one. The rising curve is the same method with the step refined where the orbit is fast — h ∝ r^3/2, the free-fall time, varying by a factor of 5 around the orbit — which is the first thing anybody reaches for at a close encounter and which is strictly more accurate step for step: over its first three revolutions it stays a factor of 5.7 below the fixed run. By 1500 revolutions it is a factor of 7.5 above it and still climbing. Changing the step changes which Hamiltonian is being conserved, the errors from successive steps stop cancelling, and what is left is a random walk with no bound at all. The practical consequence is that a solar-system integration cannot adapt its step: it either keeps a step short enough for the closest encounter it will ever meet, or it detects the encounter and hands that piece of the trajectory to an entirely different, non-symplectic method for the duration — which is what every long-term integration of the planets actually does.

A step that must not be adapted

A symplectic integrator's bounded energy error is a property of a fixed step. It is conserving a Hamiltonian a step-size away from the intended one, and changing the step changes which Hamiltonian — so refining the step at a close encounter, which is the first thing anybody does, destroys the only property the method was chosen for.

gravitation · Numerical integration Hard below 8.3 astronomical units, soft above, and nothing settles at the line. The binding energy of a binary of two 0.7 solar-mass stars against its separation, both axes logarithmic, with the mean kinetic energy of a single cluster star at a velocity dispersion of 5 kilometres a second drawn as a level. Where the curve is above the level the binary is bound more tightly than a passing star's motion, and encounters on average take energy out of the field and put it into the pair; where it is below, they do the reverse. The crossing at 8.3 astronomical units is the hard–soft boundary, and it is the entire content of Heggie's law: hard binaries harden and soft binaries soften. The arrows are the direction each side moves, and they point away from the crossing in both directions rather than toward it. That is not a coincidence but a negative heat capacity, the same property that makes a star contract when it radiates: taking energy out of a bound pair moves it closer together and speeds it up, so a hard binary that gives energy to the cluster becomes harder still and gives more. A cluster with binaries in it therefore has a heat source that turns itself up, and the boundary drawn here is a watershed rather than an equilibrium.

A pair that heats what is trying to cool

A star cluster has a negative heat capacity, so it cannot reach equilibrium — its core contracts and gets hotter without limit. What stops it is a binary, and which way a binary exchanges energy with the stars around it is settled by one comparison of two energies.

gravitation · Binary heating 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 An invariant that moves by 8.0e-3 once the planet's orbit is real. The Tisserand parameter of a comet on an orbit of semi-major axis 5 and eccentricity 0.8, followed through a close passage of Jupiter, integrated twice. The lower trace has Jupiter on a perfect circle, which is the problem the parameter is an exact constant of: it survives the encounter having moved by 1.0e-4, which is the integrator's own error and not a physical change, and the spike at the moment of closest approach is the osculating elements being briefly meaningless while the comet is inside Jupiter's sphere of influence rather than the constant failing. The upper trace is the same encounter with Jupiter on its real orbit, eccentricity 0.0489. The parameter comes out changed by 8.0e-3, 79 times as much, because the Jacobi constant exists only when the rotating frame is uniformly rotating and a planet on an ellipse does not provide one. That number is small and it is not negligible: comet families are separated by boundaries in this parameter placed to two decimal places, and a comet that drifts across one over several encounters has changed class without anything having happened to it that a single encounter could account for.

An invariant that is only almost one

The Tisserand parameter survives a close encounter with Jupiter exactly, and comet families are separated by boundaries in it drawn to two decimal places. The exactness holds for a Jupiter on a circle. Jupiter's eccentricity is 0.0489, and integrating the same encounter twice shows what that costs.

orbits · Tisserand parameter A gravity assist with a 68° turn. The velocity triangle of a flyby. In the planet's frame the spacecraft's speed is unchanged and only its direction turns; adding the planet's own velocity converts that turn into a gain in speed measured from the Sun.

The planet pays, and it shows

A gravity assist takes energy from a planet and gives it to a spacecraft, and the planet's loss is exactly the spacecraft's gain. For a two-tonne probe past Jupiter that loss is unmeasurable. Do it with a hundred Earth masses of icy debris and the same bookkeeping moves Neptune outward by several astronomical units.

spaceflight · Gravity assist

Named alongside it

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

Gravity assistHyperbolic orbitRelaxation timeTisserand parameterAdaptive stepAdiabatic invariantAngular momentumBinary heatingBinding energyCircular restricted problemCometCore-collapse

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