Astronomy & Celestial Mechanics
the sky, computed
Concept

Impulse — where it appears

The velocity change delivered by integrating a force along the path a body would have taken without it. For a gravitational flyby the transverse impulse is exactly twice the gravitational constant times the mass over the impact parameter and the speed, with no free constant.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

A burn along the track moves the chaser 8330 m backwards. Three 0.5 m/s impulses from rest alongside a target in a 400 km circular orbit, followed for 2 revolutions in the frame riding on the target. Along-track distance runs across the page with the direction of travel to the left, and radial distance up. The prograde burn ends 8330 metres behind after one revolution — exactly 6πΔv/n, and it is behind rather than ahead because the burn raised the orbit and a higher orbit takes longer. The retrograde burn ends 8330 metres ahead by the same arithmetic with the sign reversed. The radial burn is the third case and the strange one: it opens a closed loop and returns exactly to where it started after a revolution, having gone nowhere at a cost of 0.5 m/s. That is not a curiosity but the basis of the R-bar approach, in which a vehicle closes on a station from below along a path that costs nothing to abandon.

A burn that moves the wrong way

In the frame riding on an orbiting target, a thrust along the direction of travel leaves a chaser eight kilometres behind after one lap, a radial thrust returns it exactly to where it started, and every free relative orbit is the same ellipse — twice as long along the track as it is across.

spaceflight · Rendezvous 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 Two ranges that look identical until something is spent. The bearing to a target, as a camera measures it, for two relative orbits that differ only in size — one 2.5 times further away than the other in every coordinate at every instant. Before the manoeuvre the two histories are identical to machine precision, and that is not a numerical accident but a property of the equations: the linearised relative dynamics are homogeneous, so scaling a whole trajectory gives another trajectory, and a direction is unchanged by scaling. An angles-only tracker is therefore exactly blind to range, and no amount of observing improves it. At 42 per cent of the way through, both vehicles fire the same 0.05 m/s impulse — and the histories separate, by up to 2.3 degrees, because an impulse measured in metres per second is the one quantity in the problem that does not scale with the trajectory. The manoeuvre is what makes range observable, which is why an autonomous vehicle navigating on a camera alone flies a deliberate zig-zag rather than a straight approach, and why the manoeuvre is chosen for what it reveals as much as for where it goes.

A camera that cannot see how far

The equations that make a close approach tractable are linear and homogeneous, so scaling an entire relative trajectory gives another valid one with every bearing identical. A vehicle navigating on angles alone is therefore not merely imprecise about range — it is exactly blind to it, and the only thing that restores the scale is spending propellant.

spaceflight · Rendezvous

Named alongside it

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

The Clohessy–Wiltshire equationsProximity operationsAdiabatic invariantAngles only navigationApproach ellipseClose encounterCoulomb logarithmCross-sectionDegeneracyDrift rateDynamical frictionFormation flying

All concepts