Astronomy & Celestial Mechanics
the sky, computed
Concept

Thermosphere — where it appears

The hot, thin upper atmosphere from about 90 to several hundred kilometres up, heated by solar ultraviolet light. Its density varies tenfold over the solar cycle and changes within hours in geomagnetic storms, which sets the drag on low satellites.

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

The zenith sky after sunset, as single scattering predicts it. The brightness of the zenith sky at 550 nm, in magnitudes per square arcsecond with brighter upward, against the Sun's depression below the horizon, computed by single scattering in a spherical atmosphere: US Standard Atmosphere densities, Rayleigh scattering, a 300 Dobson-unit ozone layer, and every photon scattered exactly once. The Earth's shadow climbs the zenith as R(sec d − 1) — 8.7 km at 3°, 35.1 at 6°, 142 at 12° — through air that thins by a factor of e every eight kilometres or so, so the brightness does not fall steadily: it falls 1.4 magnitudes a degree between 3° and 6°, and 2.7 a degree between 7° and 10°. The model gives 14.3 at the end of civil twilight and 27.0 at the end of nautical. The dashed line is the natural night sky, 21.9 magnitudes per square arcsecond. Single scattering reaches it at 9.25° of depression — 8.8 degrees before the 18° at which astronomical twilight is observed to end. The difference is not an error in the arithmetic; it is the light this model leaves out, scattered more than once.

The shadow that climbs the zenith

After sunset the sky overhead is lit only above the Earth's own shadow, and the shadow climbs as the square of the Sun's depression. Scattering computed once from that geometry predicts a sky that dims faster and faster and is as dark as night by nine degrees. The real sky takes eighteen, and the difference is light that has been scattered more than once.

sky · Twilight Two satellites moved apart by the attitude they hold. The along-track separation of two identical 5 kg satellites at 475 km, starting together, when one flies broadside to the airflow and the other edge-on for 15 days and then they swap, so that their relative drift ends at zero. The separation grows quadratically and then levels off. At solar minimum the manoeuvre moves the pair 2,797 km apart; mid-cycle 8,299; at solar maximum 24,620 — 206° of the orbit's 43,060 km circumference. No propellant is spent; the same command produces a tenfold different result depending on the state of the Sun.

A rendezvous paid for in air

Two satellites in the same low orbit can be moved along it relative to each other with no propellant at all, by turning one broadside to the thin air and the other edge-on. The draggier one sinks and speeds up, the gap grows as the square of the time, and swapping attitudes halfway brings the drift back to zero. The cost moves from fuel to time, to altitude, and to knowing an atmosphere that changes tenfold with the Sun.

spaceflight · Rendezvous

Named alongside it

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

Atmospheric dragBallistic coefficientDepression angleDifferential dragEarths shadowFeedbackMultiple scatteringOzonePhasing orbitRayleigh scatteringRendezvousSatellite constellation

All concepts