Definition
Nutation, from the Latin nutare, to nod, is a small periodic oscillation of the Earth's rotation axis superimposed on the slow conical motion of precession. Where precession turns the pole around a circle of 23.4° radius in about 25,800 years, nutation adds a wobble of a few arc seconds with periods from a few days to 18.6 years.
Its cause is the same as precession: the Sun and the Moon pull on the Earth's equatorial bulge. But their pull is not constant. The Moon's orbit is inclined by 5° to the ecliptic and its nodes go round in 18.6 years, so the lunar torque changes over that cycle, and the Sun's and Moon's torques also vary with the half-year and the half-month.
Two components
Nutation is described by two angles:
- Nutation in longitude, Δψ: the shift of the equinox along the ecliptic. It reaches about ±17.2″.
- Nutation in obliquity, Δε: the change in the tilt of the equator to the ecliptic. It reaches about ±9.2″.
Combined, the true pole traces a small ellipse around the mean pole every 18.6 years, with semi-axes of about 9.2″ and 6.9″ (17.2″ × sin 23.4°).
Discovery
James Bradley, who had already discovered the aberration of light, noticed residual shifts in the positions of stars that followed the 18.6-year cycle of the Moon's nodes. He followed them for a full cycle, from 1727 to 1747, before announcing nutation in 1748. Its explanation by gravitation, by d'Alembert in 1749, was one of the early triumphs of Newtonian mechanics.
Meeus's formulas
Astronomical Algorithms gives the full IAU 1980 theory as a table of 63 periodic terms, but also a short version that is accurate to 0.5″ in Δψ and 0.1″ in Δε, enough for most purposes. With Ω the longitude of the Moon's ascending node, L the mean longitude of the Sun and L′ that of the Moon:
Δψ = −17.20″ sin Ω − 1.32″ sin 2L − 0.23″ sin 2L′ + 0.21″ sin 2ΩΔε = +9.20″ cos Ω + 0.57″ cos 2L + 0.10″ cos 2L′ − 0.09″ cos 2Ω| Term | Period | Amplitude in Δψ |
|---|---|---|
| Ω (Moon's node) | 18.6 years | 17.20″ |
| 2L (Sun) | half a year | 1.32″ |
| 2L′ (Moon) | 13.7 days | 0.23″ |
| 2Ω | 9.3 years | 0.21″ |
Modern work uses the IAU 2000A model, with more than 1,300 terms and a precision of a fraction of a milli-arc-second, but at the level of naked-eye or small-telescope positions, Meeus's four terms do the job.
Where nutation enters the computations
- Apparent longitude: the Sun's or a planet's longitude referred to the true equinox of date is the mean value plus Δψ.
- True obliquity: ε = ε0 + Δε, used to convert ecliptic coordinates to right ascension and declination.
- Apparent sidereal time: Greenwich apparent sidereal time equals mean sidereal time plus Δψ cos ε, the equation of the equinoxes, at most about 1.1 seconds of time.
- Star positions: to get the apparent place of a star from a catalogue, Meeus applies proper motion, precession, nutation and aberration, in that order.
On this site, the apparent positions of the planets used in the horizon views include these corrections. They are far too small to be seen at screen scale, but they keep the engine consistent with the almanacs.
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Frequently asked questions
What is the difference between precession and nutation?
Precession is the slow, steady turning of the Earth's axis around the ecliptic pole in about 25,800 years. Nutation is a small periodic wobble superimposed on it, at most about 17″ in longitude and 9″ in obliquity, mainly with an 18.6-year period.
Why does nutation have an 18.6-year period?
Because the Moon's orbital plane turns around the ecliptic with that period. The direction of the lunar pull on the Earth's equatorial bulge changes over the cycle, and the axis nods in response.
Do I need nutation for my calculations?
For positions to better than about 20″, or for apparent sidereal time to better than one second, yes. For drawing the sky or planning observations, the mean positions are enough.