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# 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.  $$\text{FIGURE XIII.3}$$