Astronomy & Celestial Mechanics
the sky, computed
Concept

Nutation — where it appears

The 18.6-year wobble superimposed on the precession of the Earth's axis, driven by the turning of the Moon's own orbital plane. It rides on top of the precession rather than replacing it, and its largest term is nine arcseconds, which is a hundred times any astrometric error.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

The path of the celestial pole over 25,772 years. The circle the Earth's rotation axis traces among the stars, at a radius equal to the obliquity, with the bright stars that fall near it and the years at which each is closest. Polaris is the pole star for a few centuries either side of now, and nothing else on the circle is nearly as close.

The pole star has a shelf life, and the sky has a slow hand

The Earth's axis traces a circle among the stars once every 25,772 years. Polaris is at the pole now, was not four thousand years ago, and will not be in two thousand more.

sky · Precession One star, two coordinate systems, at latitude 52°. The equatorial grid and the horizon grid drawn on the same sphere for an observer at latitude 52°. The star marked has declination 20° and hour angle -40° in the first, and altitude 45.5° and azimuth 239.4° in the second. The two frames differ by a single rotation through the co-latitude 38°, which is why the celestial pole stands 52° above the northern horizon.

Where a star is depends on who is asking

The sky needs two coordinate systems because two different things stay still in it — the observer's horizon and the stars themselves. One rotation converts between them, and the angle of that rotation is the time.

sky · Celestial sphere The wobble inside the wobble. Left: the two components of nutation over 40 years, from the four largest terms of the standard series. The long wave is the regression of the Moon's node in 18.613 years, which is where nearly all of it comes from; the ripple on it is the semi-annual solar term at 1.3″ and the semi-monthly lunar one at 0.2″. Right: the loop the pole actually traces over one node cycle, in arcseconds on the sky, with the mean pole at the centre. The loop is 14.89″ by 19.90″ — taller than it is wide, because the longitude term is foreshortened by sin ε while the obliquity term is not, which is the one thing a schematic of this is always drawn getting wrong. Over the same 18.6 years precession itself carries the pole 936″ along its circle, 47 times the loop's own height, so nutation is a wobble on a path and not a path. It is nonetheless 99,480 times the 0.2 mas astrometry of a modern catalogue, which is why a position has to say whether it is referred to the mean pole or the true one.

The wobble inside the wobble

Precession and nutation are the same torque. The difference is that the Moon's orbital plane turns once in 18.6 years, so part of the pull oscillates instead of accumulating — and a catalogue position is not a direction until it says which pole it is measured from.

sky · Precession A sidereal clock that runs up to 1.15 seconds ahead or behind, with the Moon's node. The equation of the equinoxes from 1990 to 2030: apparent sidereal time, counted from the true equinox, minus mean sidereal time, counted from an equinox that only precesses. It is the nutation in longitude times the cosine of the obliquity, and it is drawn here from the four largest nutation terms, which carry it to about a hundredth of a second. The dominant term follows the Moon's node round its 18.61-year cycle with an amplitude of ±1.052 s; riding on it are a half-yearly term of ±0.081 s from the Sun and a fortnightly one of ±0.014 s from the Moon. Over this span the sum runs from −1.147 to 1.146 s. None of it is the Earth's rotation: it is the zero point of the clock moving, because the zero point is the intersection of the equator with the ecliptic and the equator nods.

A clock whose zero is moving

Sidereal time is counted from the equinox, and the equinox does not stay put. Its steady drift makes the sidereal day eight milliseconds short, its acceleration puts a quadratic term into the formula for sidereal time, and the Moon makes it nod by a second every nineteen years. The Earth rotation angle removes all three by counting from a point defined not to move along the equator.

sky · Sidereal time The path of a spin across the sphere of fixed momentum. A body with principal moments 1, 8, 8.6, started about its axis of least inertia with a 3° wobble and an internal energy sink strong enough that the whole motion fits in 80 turns and each circuit of the path can be seen. The disc is the near hemisphere of the sphere of fixed angular momentum, seen from a direction between all three body axes, with the near end of each axis marked. Each thin curve is a contour of kinetic energy on that sphere — a polhode, one of the paths the angular momentum can follow in the body with no dissipation — and the thick curve is the separatrix through the intermediate axis, which divides motions that circle the axis of least inertia from motions that circle the axis of greatest. The coloured path is what the integration did: solid on the near hemisphere, dashed where it passes behind. It starts at the open dot and ends at the filled one.

A spin that left the axis it was given

The first American satellite was spun about its long axis, like a rifle bullet, and soon after launch it was tumbling end over end. Nothing outside it had pushed. A body that cannot change its angular momentum but can lose energy has exactly one place to end up, and a long body spun about its length is as far from that place as a spin can be.

spaceflight · Attitude control

Named alongside it

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

EquinoxAxial precessionEpochAngular momentumCircumpolarDeclinationObliquitySidereal timeAstrometryAttitude controlCelestial intermediate originCelestial sphere

All concepts