Astronomy & Celestial Mechanics
the sky, computed
Concept

Axial precession — where it appears

The conical wandering of a spinning body's rotation axis under an external torque, which carries the Earth's pole once round a circle in 25,772 years. It moves the equinox and therefore every coordinate measured from it, which is why a stated position is meaningless without the epoch it refers to.

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 Why the solar day is 3m 55.91s longer than the sidereal one. Two positions of the Earth one day apart, with the direction of a fixed star and the direction of the Sun marked at each. After one sidereal day — 23h 56m 4.0905s — the Earth has turned through exactly 360° and the star is back on the meridian; the Sun is not, because the Earth has moved 0.9856° along its orbit, and the further 0.9856° of turning takes 3m 55.91s. That is the whole of the difference: 366.2422 turns against the stars in the 365.2422 solar days of a year, one more turn than the 365.2422 against the Sun. Nothing here is to scale. The orbital arc is drawn at 40°, which exaggerates the real 0.9856° by 41 times, and the Earth's disc is about 4073 times too large for its orbit. The two star sight lines are drawn parallel because they are: a star's distance cannot be put on the same page as an orbit.

The day that is four minutes short

A star crosses the meridian 3 minutes 55.91 seconds earlier each night, and the same star slips 3 minutes 56.56 seconds a day against a civil clock. Those are two different numbers, and the gap between them is the one extra turn the Earth makes against the stars in every year.

sky · Sidereal time 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 Measured moment-of-inertia factors, from the Sun's 0.07 to the Moon's 0.3931. Eleven bodies whose interiors have never been sampled, arranged by the one interior quantity that has been measured for all of them. C/MR² is 2/5 for a uniform sphere and falls as mass is concentrated toward the centre, and the values here span from 0.07 to 0.3931. That spread is the content. The Moon at 0.3931 is barely differentiated — whatever iron core it has is a few per cent of its radius, which is why the Moon is the one large body in the inner solar system without a magnetic field of its own. Mercury at 0.346 is nearly as low as the Earth despite being an eighth of its mass, and for a body that small the only way to get there is an iron core filling most of the radius. The Sun at 0.07 is off the scale of anything a two-layer model describes; a star is not a planet with a bigger core but a body whose density falls by five orders of magnitude between centre and surface. The faint curves behind are the two-layer relation at a few density contrasts, drawn to show what kind of interior each value is consistent with — and the horizontal placement of each body on them is an illustration rather than a result, since one factor never fixes one core. Every number here was obtained by watching the body turn: a precession rate, a libration amplitude, a gravity field sampled on a flyby, or in the Sun's case the frequencies of its own oscillations.

Whether the heavy material sank

A moment of inertia is 0.4 of MR² for a uniform sphere and less for everything that has differentiated, and it is measured by watching a body wobble. Mercury's 0.346 says most of the planet is iron core; the Moon's 0.393 says almost none of it is.

gravitation · Moment of inertia 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

Named alongside it

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

EquinoxNutationEpochObliquitySidereal daySidereal timeAngular momentumCelestial intermediate originCelestial sphereCircumpolarCore formationDeclination

All concepts