13.16: Topocentric-Geocentric Correction

In section 13.1 I indicated two small (but not negligible) corrections that needed to be made, namely the $∆T$ correction (which can be made at the very start of the calculation) and the light-time correction, which can be made as soon as the geocentric distances have been determined – after which it is necessary to recalculate the geocentric distances from the beginning! I did not actually make these corrections in our numerical example, but I indicated how to do them.

There is another small correction that needs to be made. The diameter of Earth subtends an angle of $17".6$ at $1 \ \text{au}$, so the observed position of an asteroid depends appreciably on where it is observed from on Earth’s surface. Observations are, of course, reported as topocentric – i.e. from the place ($τοπος$) where the observer was situated. They must be corrected by the computer to geocentric positions – but of course that can’t be done until the distances are known. As soon as the distances are known, the light-time and the topocentric-geocentric corrections can be made. Then, of course, one has to return to the beginning and recompute the distances – possibly more than once until convergence is reached. This section shows how to make the topocentric-geocentric correction.

We have used the notation $\mathfrak{x, y , z}$ for geocentric coordinates, and I shall use $\mathfrak{x^\prime , \ y^\prime , \ z^\prime}$ for topocentric coordinates. In figure $\text{XIII.3}$ I show Earth from a point in the equatorial plane, and from above the north pole. The radius of Earth is $R$, and the radius of a small circle of latitude $\phi$ (where the observer is situated) is $R \cos \phi$. The $\mathfrak{x}$- and $\mathfrak{x}^\prime -$ axes are directed towards the first point of Aries, $\Upsilon$.

It should be evident from the figure that the corrections are given by

$$\mathfrak{x}^\prime = \mathfrak{x} - R \cos \phi \cos \text{LST}, \label{13.16.1} \tag{13.16.1}$$

$$\mathfrak{y}^\prime = \mathfrak{y} - R \cos \phi \sin \text{LST} \label{13.16.2} \tag{13.16.2}$$

and $$\mathfrak{z}^\prime = \mathfrak{z} - R \sin \phi . \label{13.16.3} \tag{13.16.3}$$

Any observer who submits observations to the Minor Planet Center is assigned an Observatory Code, a three-digit number. This code not only identifies the observer, but, associated with the Observatory Code, the Minor Planet Center keeps a record of the quantities $R \cos \phi$ and $R \sin \phi$ in $\text{AU}$. These quantities, in the notation employed by the $\text{MPC}$, are referred to as $−∆_{xy}$ and $−∆_{z}$ respectively. They are unique to each observing site.

$\text{FIGURE XIII.3}$