Astronomy & Celestial Mechanics
the sky, computed
Concept

Formation flying — where it appears

Keeping two or more spacecraft at a controlled separation and orientation relative to each other while each orbits on its own. It lets separate craft act as one instrument many metres or kilometres across, such as an occulter and its telescope.

Named by 2 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 A shadow 77 mm across at 144.3 m: an occulter 1.42 m wide and a 50 mm aperture inside its umbra. Side view of an eclipse made by two spacecraft, distances along the axis in metres and widths across it in metres, the vertical scale magnified about fifty times. Sunlight arrives from the left. The occulting disc, 1.42 m across, casts an umbra — the cone from which none of the Sun's disc is visible — which narrows because the Sun is not a point: its edges converge at a rate set by the Sun's angular radius, 959.6 arcseconds, and would meet 153 m behind the disc. At 144.3 m, where the coronagraph sits, the umbra is 77 mm across and the aperture 50 mm; the penumbra, from which part of the Sun is visible, extends to 2.76 m. From the aperture the occulter's edge hides the Sun out to 1.058 solar radii, so the corona is visible from just beyond the limb.

An eclipse held in place to a millimetre

A total eclipse of the Sun works because the Moon's disc happens to match the Sun's almost exactly, and it lasts a few minutes. Two spacecraft flying 144 metres apart can make one that lasts six hours: a disc 1.4 metres across, with a coronagraph sitting in its shadow. The geometry that sets the shadow's size also sets the price. At that separation the umbra is 77 millimetres across around a 50-millimetre aperture, so the formation has to hold itself in line with the Sun to a centimetre — and to two millimetres in January, when the Sun is at its largest.

sky · Phases and eclipses

Named alongside it

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

Angular diameterApproach ellipseThe Clohessy–Wiltshire equationsCoronagraphDrift rateFresnel diffractionImpulseThe local-vertical local-horizontal framePenumbraProximity operationsR-bar approachRelative motion

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