We first consider the adopted model of mutual occultations and eclipses of natural planetary satellites as observed from the Earth. A detailed description of this model can be found in Emel'yanov ([1999]). Concerning the satellites they were considered as spheres with uniform surfaces.

We refer to the occulting or eclipsing satellite as the active satellite using subscript a with all quantities that refer to this satellite. We further refer to the occulted or eclipsed satellite as the passive satellite and associate subscript p with it.

In the case of mutual occultation at observing time *t*_{0} the
detector receives the light reflected from the passive satellite
at time ,
and the light reflected from the active satellite
at time ,
so that
.
During observations of a
mutual eclipse of the satellites the observer's detector receives
at observing time *t*_{0} the light that travelled in the vicinity of
the active satellite at time
and was reflected from the
passive satellite at time .
In this case
.
The differences between these three instants of time are
determined by the time of light propagation between the satellites
and the observer.

Hereafter we always refer to instants
of time
and ,
which correspond to observing time *t*_{0},
because these three instants of time are connected by the
propagation of the same photon between the satellites and the
observer.

The theoretical position of the satellite is given by
planetocentric coordinates *x*, *y*, *z*. The (*x*,*y*)-plane is chosen to
be parallel to the Earth equator for the epoch of J2000.
The *x*-axis points to the vernal equinox of the same epoch.

We now consider a vector in space, which we refer to as
*the event vector*
for the sake of convenience. This vector starts at the
observing point at time *t*_{0} in the case of mutual occultation or
at the solar center in the case of mutual eclipse, and ends at the
center of the passive satellite at time .

Let
be the planetocentric coordinates of the active
satellite at time ,
and
,
the planetocentric
coordinates of the passive satellite at time .
The flux from
the passive satellite measured at time *t*_{0} thus depends on
and
.

Hereafter we consider the vector
of the position of the
active satellite at time
relative to the position of the
passive satellite at time .
The components of this vector in
the coordinate system *x*, *y*, *z* are determined by the following
relations:

(1) |

We now introduce another Cartesian coordinate system

We now denote the coordinates of the active satellite in the *X*, *Y*,
*Z* reference frame at time
these are the components of the vector
as
.
It follows from Emel'yanov's ([2000]) results that the magnitude
of the passive satellite, as measured at time *t*_{0},
depends on
,
and only slightly on projection of the mutual radius-vector of the two satellites on the event vector.

The components of the event vector
in two
reference frames are connected by the following evident relations

(2) |

where coefficients

While observing a mutual phenomenon of two satellites, the
measuring system records some quantity *E*. It can be measured in
arbitrary units inherent to the particular detector. In addition
to the directly measured quantity *E* we also introduce flux *S*produced in the space by the eclipsed or occulting and occulted
satellites. This flux passes then through the atmosphere and
the telescope aperture up to the detector.
The flux *S* is measured in the units of the satellite flux when
it is close to the eclipse and still lies outside the shadow
of the occulting satellite in the case of mutual eclipse.
In the case of mutual
occultation the flux is measured in the units of the total flux *S*from the occulting and occulted satellites. In both cases flux *S*outside eclipse or occultation is equal to unity by definition.
During a mutual occultation or eclipse flux *S* decreases
and this decrease depends primarily on
and .

Dependence of *E* on *S* is usually unknown. First, we do not know
beforehand what will be the reaction of the detector output on the
satellite flux. Second, besides the satellite light, the detector
is illuminated also by outside light (sky background, the light of
the planet scatterred in the telescope). Finally it is difficult to
do a forecast about the atmosphere transparency.
The sometimes adopted photometer calibration using known sky light
sources is often unreliable.
We were led to suggest and adopt the following dependence of *E* on
*S*:

where

In general the parameters
*K*, *Q*, *P*, *L* may take into account
the above mentioned effects. For example the parameter
*Q* corresponds to a linear variation of the atmosphere transparency
and the parameters *P*, *L* describe the light of
the planet scatterred in the telescope.
In practice although it is not a good idea to determine
the parameters *Q*, *P*, *L* from observations because of
a great correlation between the errors.
We do it in forced cases only when exclusion of the relevant
effects by the method of observation is not possible.

The basic idea of the reduction of photometric measurements
considered is as follows (Emel'yanov [2000]).
Denote the true values of
corresponding to a photometric
measurement at certain time as
.
To reduce observations, we use one of the
most accurate theories of satellite motion available. We denote
theoretically computed values of
,
as
,
respectively. We further assume that the differences

vary negligibly within the same mutual occultation or eclipse. This allows us to compose for any photometric measurement made at time

for corrections

After doing this and to present the results of reduction,
we choose arbitrarily one of the
observing times, *t*_{0}, and compute
for this time
using formulas

(3) |

These values are the main result of reduction. and can be used only if some other quantities considered here are known. In particular, we must know the time instants and , corresponding to the chosen time

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