Astronomy & Celestial Mechanics
the sky, computed
Concept

Argument of perigee — where it appears

The angle within an orbit's plane from the ascending node to the point of closest approach. The flattening of the central body turns it steadily, except at the critical inclinations. It means little for a nearly circular orbit, where the eccentricity vector is used in its place.

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

The figure of eight a geosynchronous orbit draws at 5°, 15°, 30° of inclination. The ground track over one sidereal day of a circular orbit whose period is exactly a sidereal day, at inclinations of 5, 15, 30°, centred on its own mean longitude. It is not a point. The latitude swings to ±i and back twice a day, and the longitude falls behind and then runs ahead of the Earth's rotation, because the rate at which an inclined orbit gains longitude is u̇ cos i / cos²φ: slowest at the nodes, where part of the motion is north–south, and fastest at the extremes of latitude, where all of it is eastward and a degree of longitude is shorter — so the track closes as a figure of eight. Its half-width in longitude is ±0.109° at 5°, ±0.993° at 15°, ±4.117° at 30° — exactly arcsin(tan²(i/2)) — against the small-inclination form i²/4 = ±0.109°, ±0.982°, ±3.927°: quadratic in the inclination, so the eight is tall and very thin. The longitude axis is stretched 8 times relative to the latitude axis, and without that stretch every one of these curves would be drawn as a vertical line. This is why a few degrees of inclination, which a geostationary operator spends most of its propellant preventing, moves the satellite a long way north and south and almost not at all east and west.

A stationary satellite that draws a figure of eight

A satellite with a period of exactly one sidereal day returns over the same ground every day, but only an orbit in the equator with no eccentricity returns over a single point. A few degrees of tilt draw a figure of eight, a little eccentricity a swing in longitude, and the two together draw the figure the Sun draws in the sky over a year.

spaceflight · Ground tracks An eccentricity that goes round a point that is not zero. The eccentricity vector of a near-circular orbit at 800 km and 98.6 degrees, with its component along the line of nodes across and its perpendicular component up, in thousandths. Under J₂ alone it would circle the origin. J₃ moves the centre to 1.03 thousandths, on the axis where the argument of perigee is 90 degrees, and every orbit circles there once in 123 days. A satellite placed at that centre does not move.

An orbit held still by the Earth's pear shape

A satellite's eccentricity does not stay put. The Earth's flattening turns it once every few months, in a circle around zero, as anyone would expect. A coefficient four hundred times smaller, the slight pear shape of the planet, moves the centre of that circle to a value of about one part in a thousand. A satellite placed exactly there goes nowhere at all, and its height above every latitude repeats, for no fuel. Nothing else in orbital mechanics costs so little for so much.

spaceflight · Station-keeping

Named alongside it

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

Critical inclinationAnalemmaEccentricityEccentricity vectorEquation of timeFrozen orbitGeostationary orbitGeosynchronous orbitGround trackInclinationSatellite altimetryStation-keeping

All concepts