Astronomy & Celestial Mechanics
the sky, computed
Concept

Ascending node — where it appears

The point at which an orbit crosses the reference plane going northward, and the direction to it from the central body. It is undefined for an orbit in the reference plane, which is the second of the two coordinate singularities the classical element set carries.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

An orbit at i = 42°, Ω = 35°, ω = 55°. The three orientation elements. The orbit is generated in its own plane and rotated by the standard sequence, so the inclination, the node line and the argument of periapsis are the angles that produced the drawn curve rather than labels applied to it.

Six numbers that fix an orbit for all time, and the sixth is the awkward one

Five of the orbital elements describe a curve that never changes. The sixth says where on it the body is, and it is the only one that has to keep being measured.

orbits · Orbital elements Three rotations, applied in order. An orbit of eccentricity 0.45 carried into space by the three orientation angles, one panel per rotation: the argument of periapsis ω = 40°, then the inclination i = 42°, then the longitude of the ascending node Ω = 55°. The last panel applies the same three angles in the reverse order and arrives somewhere else, because rotations about different axes do not commute.

Three rotations that put an orbit in space, and they do not commute

An orbit's orientation takes three angles. Give the same three angles in a different order and the orbit ends up somewhere else — which is why the convention is part of the data.

orbits · Orbital elements Three periods that nearly share a multiple. How far the draconic and anomalistic months are from a whole number, after a whole number of synodic months, in hours. At 223 synodic months — 6585.321 days — both are within an hour of closing, which is what makes an eclipse repeat. The draconic residual is 0.87 hours and the anomalistic 5.19 hours.

The eclipse that repeats a third of a world away

Three lunar periods nearly share a multiple after 6,585 days. The word "nearly" is what makes eclipses predictable, and the leftover third of a day is what moves each repeat a third of the way round the Earth.

sky · Phases and eclipses Perihelia of a young family, run backwards until they meet. The longitude of perihelion of each of 26 members of a synthetic young asteroid family at the Karin family's position, relative to the family's mean at each moment, against time before the present. Today the angles are spread round the whole circle: members only a few thousandths of an astronomical unit apart precess at rates that differ by a fraction of an arcsecond a year, which over millions of years is many full turns. Run backwards — with each member's own drift in semi-major axis included — they converge, and the spread is smallest at 5.75 million years ago, where they were made. The nodes turn the other way, and in the real family, whose rates come from a full theory in which the two frequencies differ, they give a second, independent convergence at the same instant. An age read this way is a measurement of a time in the way a light curve's period is, with an error of a few per cent, rather than an inference from a drift rate.

An age read from angles that meet

A family of asteroids a few million years old has had its members' orbits turned round and round by the planets, each at a slightly different rate, until their orientations look random. Turn them back and they come together at one instant — the collision — with an uncertainty of a few tens of thousands of years, but only if each member's own slow drift in sunlight is put back too.

orbits · Asteroid families The cost of turning an orbit, against how much of each revolution the engine runs. The velocity spent per unit of inclination change, relative to an instantaneous turn at a node, when a small out-of-plane thrust is applied only in arcs centred on the two nodes, against the half-width of each arc (blue). Thrusting all the time costs π/2, 1.571; arcs of ±60° cost 1.209; ±30°, 1.047; ±10°, 1.0051. The price is time: the engine is off for the rest of each orbit, and the turn takes longer by the inverse of the duty times the efficiency (orange, right-hand scale) — relative to a push that was always fully effective, 1.57 times as long for continuous thrust, 3.14 for ±30°, 9.0 for ±10°. Most of Edelbaum's penalty is spent in the quarter-orbits near the points where a push turns the node instead of the plane.

A plane turned only where turning is cheap

A low-thrust engine that turns an orbit's plane continuously pays π/2 times what an impulse at the node would cost, because for half of every revolution it is pushing where a push mostly swings the node instead of tilting the plane. Thrusting only in arcs about the nodes recovers almost all of that: arcs of ±30 degrees cost five per cent more than an impulse, at the price of taking twice as long. It is how geostationary satellites hold their planes, and it stops working for large turns, where a chord is shorter than any arc.

spaceflight · Plane change A thrust that leans out of the plane at the nodes. The yaw of the thrust — its angle out of the orbit's plane — against the argument of latitude round one revolution, at the start, half-way through and near the end of a low-thrust transfer from a 7.726 km/s orbit to geostationary speed (3.075 km/s) that also removes 28.5° of inclination. Edelbaum's law (dashed) holds the yaw at one size through each revolution, switching sign at the antinodes. The optimal law (solid) has tan β proportional to |cos u|: the thrust lies in the plane at the antinodes, where out-of-plane thrust would mostly swing the node, and leans out furthest at the nodes, where it turns the plane best — 44° half-way through the transfer, against Edelbaum's 34° all round. Raising the orbit is equally efficient anywhere, so the thrust never needs to coast: the in-plane share fills exactly the parts of the orbit where turning is poor.

A turn spread where the climb leaves room

An electric spacecraft spiralling from low orbit to geostationary has to remove its launch inclination on the way, and the classic steering law tilts its thrust out of the plane by the same angle all the way round each orbit. But out-of-plane thrust turns the plane only near the nodes, while in-plane thrust raises the orbit equally well everywhere. Letting the tilt follow the nodes — leaning hard where turning works and lying flat where it does not — saves 150 metres a second on a transfer from Cape Canaveral, with no coasting, because the climb fills exactly the parts of each orbit where the turn would be wasted.

spaceflight · Plane change How fast a meteor arrives is its orbit's inclination. The speed at which a meteoroid on a near-parabolic orbit with perihelion at 1 AU meets the Earth, against the orbit's inclination to the ecliptic: relative to the Earth far away (dashed) and at the top of the atmosphere after the Earth's gravity has added its pull (solid). A prograde orbit in the ecliptic overtakes the Earth at 12 km/s and enters at 17; a retrograde one meets it head-on at 72 and enters at 73. The points are four showers computed from their parents' elements at the node nearest 1 AU: Draconids 24 km/s at 31.9°, Geminids 38 km/s at 22.3°, Perseids 61 km/s at 113.5°, Leonids 72 km/s at 162.5°. A meteor's speed is a reading of the angle between its orbit and the Earth's.

A shower that is a reading of an orbit

Every August the Earth runs through the dust of comet Swift–Tuttle, and the meteors arrive at sixty kilometres a second from a point in Perseus. Each of those facts is an orbital element in disguise. The date is the longitude of the stream's node, the speed is its inclination, the radiant is the direction it comes from, and the shower exists at all only because one element happens to put a node within a hair of the Earth's orbit.

orbits · Orbital elements

Named alongside it

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

Orbital inclinationOrbital elementsAnomalyArgument of periapsisEccentricityElectric propulsionEpochGeostationary orbitInclinationLow-thrust transferPeriapsisPlane change

All concepts