4 The use of the obtained data

We now describe how to use the data obtained by reducing photometric observations of mutual phenomena of satellites in order to refine the theory of satellite motion.

Let us assume that we have reduced photometric observations of Nmutual events. For each event we then choose an instant of time as is described above and thereby obtain a series of such instants, t⁽¹⁾, t⁽²⁾, ..., t^(N)and the corresponding series of parameters $\;X^{\rm (o)}_k, Y^{\rm (o)}_k\;\; (k=1, 2, ...,N)$, determined by relations (3).

These parameters play the part of measured values of $X_{\rm a}, Y_{\rm a}$.

We further assume that before carrying out the second stage we have at our disposal an approximate theory of satellite motion and approximate elements of satellite orbits inferred from earlier observations. It may be another theory, which differs from that used to reduce photometric observations. The formulas of this theory and relations (1), (2) can be used to compute $X_{\rm a}, Y_{\rm a}$, which correspond to observing times $\; t^{(1)}, t^{(2)}, ..., t^{(N)}$. We denote this series of quantities as $\; X^{\rm (c)}_k, Y^{\rm (c)}_k \;\; (k=1, 2, ...,N)$. These quantities play the part of computed theoretical values of $X_{\rm a}$, $Y_{\rm a}$.

We now denote the orbital elements of the passive satellite as p₁, p₂, ... , p_n. Depending on the theory used, they can be different from the elements of the Keplerian orbit and the number n of these parameters can exceed six. We denote the corresponding orbital elements of the active satellite as q₁, q₂, ... , q_n.

We can now write the conditional equations which are used to refine the elements of satellite orbits. According to the general formulation of the method of differential adjusting and in view of relations (1), (2) the conditional equations can be written in the following form:

$$X^{\rm (o)}_k - X^{\rm (c)}_k$$

$$= \sum_{j=1}^n \left[ a_{x} \left(\frac{\partial x_{\rm a}}... ...\partial z_{\rm a}}{\partial q_j}\right)_0 \right] \Delta q_j$$

$$- \sum_{j=1}^n \left[ a_x \left(\frac{\partial x_{\rm p}}{\... ...artial z_{\rm p}}{\partial p_j}\right)_0 \right] \Delta p_j ,$$

$$Y^{\rm (o)}_k - Y^{\rm (c)}_k$$

$$= \sum_{j=1}^n \left[ b_x \left(\frac{\partial x_{\rm a}}{\... ...\partial z_{\rm a}}{\partial q_j}\right)_0 \right] \Delta q_j$$

$$- \sum_{j=1}^n \left[ b_x \left(\frac{\partial x_{\rm p}}{\... ...artial z_{\rm p}}{\partial p_j}\right)_0 \right] \Delta p_j ,$$

$$(k=1,2,\;...\;,\;N).$$

where $\Delta p_j, \Delta q_j$ are the corrections to be found for the preliminary values of the elements. Here partial derivatives of $x_{\rm a}, y_{\rm a}, z_{\rm a}$ are computed at time $t_{\rm a}$, and partial derivatives of $x_{\rm p}, y_{\rm p}, z_{\rm p}$, at time $t_{\rm p}$corresponding to time t^(k). To compute these derivatives, we use the formulae of the theory of motion of satellites and preliminary orbital elements. The necessary values of $t_{\rm a}, t_{\rm p}, a_x, a_y, a_z, b_x, b_y, b_z$ are known and saved from the observation reduction.

The conditional equations thus constructed can be used, if combined with other conditional equations inferred from other types of observations with convenient weights, to refine the elements of satellite orbits.