6 Computing true celestial coordinates

The coordinates $\alpha$ and $\delta$ in the intermediate or true system described in the previous section have the same definition as true coordinates in the equinox-based system. They are positions with respect to the true, or intermediate, equator and either the CEO or the equinox, depending upon which system is used. Thus, precession-nutation has been applied to ICRF positions. This is an extension of the use of the intermediary frame to astrometry and it is legitimate to call the resulting $\alpha$ and $\delta$, the true geocentric coordinates and to call the frame, true celestial frame. Their determination from raw observations corrected for instrumental parameters, stellar aberration, and refraction, and reduced, if significant, to the geocenter, is analogous to the following procedure, described by Lieske et al. (1977).

Figure 1: The precession angles $\zeta _{\rm A} , z_{\rm A}$, and $\theta _{\rm A}$. The Ox axis points towards $\gamma$ in the old system, towards $\sigma$ in the new.

For the old system, let us consider a set of three-dimensional reference Cartesian coordinate axes R₀ (0 x y z ) at epoch t₀, centered at the center of the celestial sphere. The X-axis points to the fixed equinox $\gamma_0$. The Y-axis is $90^\circ$ away in an easterly direction along the equator, and the Z-axis points toward the mean pole P₀ (Fig. 1). Let P be the pole of the instantaneous (true geocentric) reference frame R(t) (O x y z ) at time t and, on its equator, the mean equinox of date, $\gamma$, is the origin of right ascensions. The transformation from R₀ to R(t) consists of three rotations.

1.

A rotation of $-\zeta_{\rm A}$ about the Z-axis makes $P_0 \gamma$ pass through P. This great circle meets the mean equator of epoch at right angles at point Q and the mean equator of date at right angles at R. The new x-axis is now in the direction Q and the y-axis points toward the node S, where the two equators cross;

2.

a rotation of $+\theta _{\rm A}$ equal to the angular separation of P from P₀ about the new y-axis brings the mean equator of epoch to the mean equator of date. The new z-axis points to the pole of date P, the y-axis to the node S, and the x-axis now points towards R in the plane of the equator of date;

3.

a rotation of $-z_{\rm A}$ equal to the angle $(\gamma - H)$ about the new z-axis brings H to $\gamma$, so that the x-axis points toward $\gamma$, the equinox of date, and still lies in the plane of the mean equator of date.

In the new system, the situation is technically similar. Now, in Fig. 1, the geocentric ICRF axes replace R₀, and $\gamma_0$ is the origin of right ascensions, while $\gamma$ is replaced by the CEO $\sigma$. Thus, the first two rotations $-\zeta_{\rm A}$ and $+\theta _{\rm A}$ are similar. Only the third, $-z_{\rm A}$, is changed, and is now equal to the angle $[\sigma ,H]$.

As for $z_{\rm A}$, $[\sigma ,H]$ will be provided by the precession-nutation expressions (which include the geodesic precession-nutation). Actually, the transformation can be made in a much simpler manner. Now, continuous VLBI observations systematically provide the position of P in the ICRF, just as is the case for many years of the polar motion. So, just as one used to correct for polar motion using the positions of the terrestrial pole, similarly, one should compute the coordinates in the geocentric ICRF using the observed position of P ( $-\zeta_{\rm A}$ and $+\theta _{\rm A}$) and of the CEO ($\sigma$). It is necessary that the method and values for the reduction of observations be documented and referenced.