5 Solar system objects

 

 

Figure 3:  Definition of the abscissa offset ${\vec{v}}$ for a predicted position ($\alpha_m$, $\delta_m$), with respect to the catalogue position ($\alpha_0$, $\delta_0$) and RGC orientation $\theta$

The solar system objects observed by Hipparcos (48 minor planets, two of Jupiter's moons, Callisto and Europa, and one of Saturn's moons, Titan), do not lend themselves for a simple parametrized representation. They could only be presented in the form of the abscissa results, with one observation per field of view crossing. Although an actual position is given, the only value of this position is to serve as a reference point for the abscissa measurement. The actual data-point could be found anywhere close to this line. Figure 3 shows how from the published data (described in Vol. 1, Sect. 2.7) and a predicted position ($\alpha_m$, $\delta_m$) the abscissa residual v can be derived. Assuming that the distances between the published and the predicted positions are small (generally less than 1 arcsec), the Euclidian approximations can be made:  

$$v = (\alpha_m-\alpha_0)\cos\delta\sin\theta + (\delta_m-\delta_0)\cos\theta .$$ (13)

The handling of these measurements is very similar to the handling of the star abscissae: a model predicts abscissa positions and their dependence on the model parameters. We take e.g. the best available ephemerides of an object to calculate predicted positions ($\alpha_{m\vert i}$, $\delta_{m\vert i}$). The abscissa residuals v_i for these predicted positions (Eq. 13) can be represented in various ways: either as a function of some critical parameters in the model, or simply as observed offsets in coordinates as a function of time:  

$${\rm d} v = {\rm d}\alpha\cos\delta\sin\theta + {\rm d}\delta\cos\theta + \epsilon,$$ (14)

with $<\epsilon^2\gt^{1\over 2} = \sigma_v$.By representing ${\rm d}\alpha$ and ${\rm d}\delta$ as functions of parameters a_i, such as time or orbital elements, Eqs. (14) are solved through minimizing $\sum_i\left({v_i - {\rm d}v_i\over\sigma_{v_i}}\right)^2$.Here, as in all other similar solutions, no correlations are assumed to exist between data on different reference great circles.

It should be realized, however, that for the determination of both ${\rm d}\alpha$ and ${\rm d}\delta$, measurements are required from different reference great circles, providing the different angles $\theta$ needed to remove their correlation.