Obliquity of the ecliptic is a name used by astronomers for the inclination of Earth's equator to the ecliptic, or of Earth's rotation axis to a perpendicular to the ecliptic. Currently about 23.4°, it varies slightly due to changes in the plane of the Earth's orbit with planetary precession.
The angular value of the obliquity is found by observation of the motions of the Earth and planets over many years. Astronomers produce new fundamental ephemerides as the accuracy of observation improves and as the understanding of the dynamics increases, and from these ephemerides various astronomical values, including the obliquity, are derived.
Until 1983, the angular value of the obliquity for any date was calculated based on the work of Newcomb, who analyzed positions of the planets until about 1895:
ε = 23° 27′ 08″.26 − 46″.845 T − 0″.0059 T2 + 0″.00181 T3
where ε is the obliquity and T is tropical centuries from B1900.0 to the date in question.
From 1984, the Jet Propulsion Laboratory's DE series of computer-generated ephemerides took over as the fundamental ephemeris of the Astronomical Almanac. Obliquity based on DE200, which analyzed observations from 1911 to 1979, was calculated:
ε = 23° 26′ 21″.45 − 46″.815 T − 0″.0006 T2 + 0″.00181 T3
where hereafter T is Julian centuries from J2000.0.
JPL's fundamental ephemerides have been continually updated. For instance, the Astronomical Almanac for 2010 specifies:
ε = 23° 26′ 21″.406 − 46″.836769 T − 0″.0001831 T2 + 0″.00200340 T3 − 0″.576×10−6 T4 − 4″.34×10−8 T5
These expressions for the obliquity are intended for high precision over a relatively short time span, perhaps ± several centuries. J. Laskar computed an expression to order T10 good to 0″.04/1000 years over 10,000 years.
All of these expressions are for the mean obliquity, that is, without the nutation of the equator included. The true or instantaneous obliquity includes the nutation.
Read more about this topic: Ecliptic