Precession and Nutation

The two fundamental celestial reference systems used in heliospheric science are the ecliptic system defined by the mean orbit of the Earth at J2000.0 and the equatorial system defined by the mean orientation of the Earth equator at J2000.0 (see Fig.1).

The intersection of the Earth equatorial plane and the Earth orbital plane (ecliptic) defines the line of the equinoxes (Fig.1). The ascending node of the geocentric ecliptic defines the vernal equinox (first point of Aries). The obliquity of the ecliptic at epoch J2000.0 with respect to the mean equator at epoch J2000.0 is given by [A. K6]

$$\epsilon_{0 J2000}=23^\circ 26^\prime 21^{\prime\prime}.448 \approx 23^\circ.439 291 111$$ (3)

The orientation of both planes changes over time by solar, lunar and planetary gravitational forces on the Earth axis and orbit. The continuous change is called 'general precession', the periodic change 'nutation'. Mean quantities include precessional corrections, true quantities both precessional and nutational corrections.

The mean obliquity of the ecliptic of date with respect to the mean equator of date is given by [S. 3.222-1,A. B18]

$$\epsilon_{0D} = \epsilon_{0 J2000} - 46''.8150 T_0 -0''.00059 T_0^2 + 0''.001813 T_0^3$$ (4)

$$\approx 23^\circ.439 291 111 - 0^\circ.013 004 167 T_0 - 0^\circ.000 000 164 T_0^2 + 0^\circ.000 000 504 T_0^3.$$

The true obliquity of date $\epsilon_D = \epsilon_{0D} + \Delta \epsilon$ includes the effects of nutation which are given to a precision of $2^{\prime\prime}$ for the period 1950-2050 by [S. 3.225-4]:

$$\Delta \epsilon = 0^\circ.0026 \cos(125^\circ.0-0^\circ.05295 d_0)+ 0^\circ.0002 \cos(200^\circ.9+1^\circ.97129 d_0).$$ (5)

For the calculation of true equatorial positions one also needs the longitudinal nutation which is given to first order by [S. 3.225-4]:

$$\Delta\psi = -0^\circ.0048 \sin(125^\circ.0-0^\circ.05295 d_0) - 0^\circ.0004 \sin(200^\circ.9+1^\circ.97129 d_0).$$ (6)

The corresponding rotation matrix from the mean equator of date to the true equator of date is then given by [S. 3.222.3]:

$$N(GEI_D,GEI_T) = E(0^\circ,-\epsilon_D,0^\circ)*E(-\Delta\psi,0^\circ,0^\circ) *E(0^\circ,\epsilon_{0D},0^\circ).$$ (7)

To achieve higher precision one has to add further terms for the series expansion for nutation from [S. Tables .3.222.1-3.224.2]¹⁰.

The orientation of the ecliptic plane of date ($\epsilon_D$) with respect to the the ecliptic plane at another date ($\epsilon_F$) is defined by the inclination $\pi_A$, the ascending node longitude $\Pi_A$ of the plane of date $\epsilon_D$ relative to the plane of date $F$, and the difference in the angular distances $p_A$ of the vernal equinoxes from the ascending node. Values for J2000.0 are given in [S. Table 3.211.1]:

$$\small\pi_A = (47^{\prime\prime}.0029-0^{\prime\prime}.06603T_0+0^{\prime\prime... ...^{\prime\prime}.03302+0^{\prime\prime}.000598T_0)t^2+0^{\prime\prime}.000060t^3$$ (8)

$$\Pi_A = 174^{\circ}52^{\prime}34^{\prime\prime}.982+3289^{\prime\prime}.4... ...-869^{\prime\prime}.8089-0^{\prime\prime}.50491T_0)t +0^{\prime\prime}.03536t^2$$

$$p_A = (5029^{\prime\prime}.0966 + 2^{\prime\prime}.22226T_0 - 0^{\prime... ...\prime\prime}.11113 -0^{\prime\prime}.000042T_0)t^2 -0^{\prime\prime}.000006t^3$$

where $T_0=\epsilon_F-\epsilon_{J2000}$ and $t=\epsilon_D-\epsilon_F$ are the distances in Julian centuries between the fixed epoch $\epsilon_F$ and J2000.0 and between $\epsilon_D$ and $\epsilon_F$ respectively. The corresponding Eulerian rotation matrix is

$$P(HAE_{J2000},HAE_D) = E(\Pi_A,\pi_A,-p_A-\Pi_A).$$ (9)

Coordinates defined on the equator of epoch are transformed to the equator of date by the Eulerian precession matrix

$$P(\epsilon_F,\epsilon_D) = E(90^{\circ}-\zeta_A,\theta_A,-z_A-90^{\circ})$$ (10)

The Eulerian angles are defined in [S. Table 3.211.1]:

$$\footnotesize\theta_A = (2004^{\prime\prime}.3109-0^{\prime\prime}.85330T_0-0^{\prime\pri... ...^{\prime\prime}.42665-0^{\prime\prime}.000217T_0)t^2-0^{\prime\prime}.041833t^3$$

$$\zeta_A = (2306^{\prime\prime}.2181+1^{\prime\prime}.39656T_0-0^{\prime\pri... ...^{\prime\prime}.30188-0^{\prime\prime}.000344T_0)t^2+0^{\prime\prime}.017998t^3$$ (11)

$$z_A = (2306^{\prime\prime}.2181+1^{\prime\prime}.39656T_0-0^{\prime\pri... ...^{\prime\prime}.09468+0^{\prime\prime}.000066T_0)t^2+0^{\prime\prime}.018203t^3$$

where $t$ and $T_0$ are defined as above. These formulae define the precession to the precision used for the Astronomical Almanac but may be easily reduced to lower order.

Hapgood (1997) gives only the first order transformation between epoch of J2000.0 and epoch of date which is a reduction of the above formulae and also given to higher precision in [A. B18]:

$$\theta_A = 0^{\circ}.55675T_0-0^{\circ}.00012T_0^2$$

$$\zeta_A = 0^{\circ}.64062T_0+0^{\circ}.0008T_0^2$$ (12)

$$z_A = 0^{\circ}.64062T_0+0^{\circ}.00030T_0^2$$

For the heliocentric position of the Earth a complete neglect of precession results in an error of $1^\circ.0$ for the period 1950-2050, a neglect of nutation results in an error of $20^{\prime\prime}$. Using first order nutation and precession reduces the error to $2^{\prime\prime}.0$.

Figure: Ecliptic and Equatorial Systems: the ecliptic plane is inclined by the obliquity $\epsilon$ towards the Earth equatorial plane. The vernal equinox $\vernal$ defines the common +X-axis, the +Z-axes are defined by the Northern poles $P$ and $K$. The position of an object $S$ is defined by Right Ascension $\alpha$ and Declination $\delta$ in the equatorial system, by ecliptic longitude $\lambda$ and latitude $\beta$ in the ecliptic system.