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 *N*mutual 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
,
determined by relations (3).

These parameters play the part of measured values of .

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 , which correspond to observing times . We denote this series of quantities as . These quantities play the part of computed theoretical values of , .

We now denote the orbital elements of the passive satellite as
*p*_{1}, *p*_{2}, ... , *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
*q*_{1},
*q*_{2}, ... , *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:

where are the corrections to be found for the preliminary values of the elements. Here partial derivatives of are computed at time , and partial derivatives of , at time corresponding to time

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.

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