3 The technique for reduction of photometric observations of mutual phenomena of satellites

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 t₀ the detector receives the light reflected from the passive satellite at time $t_{\rm p}$, and the light reflected from the active satellite at time $t_{\rm a}$, so that $t_{\rm p} < t_{\rm a} < t_0$. During observations of a mutual eclipse of the satellites the observer's detector receives at observing time t₀ the light that travelled in the vicinity of the active satellite at time $t_{\rm a}$ and was reflected from the passive satellite at time $t_{\rm p}$. In this case $t_{\rm a} < t_{\rm p} < t_0$. 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 $t_{\rm p}$ and $t_{\rm a}$, which correspond to observing time t₀, 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 t₀ 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 $t_{\rm a}$.

Let $x_{\rm a}, y_{\rm a}, z\rm _a$ be the planetocentric coordinates of the active satellite at time $t_{\rm a}$, and $x_{\rm p}, y_{\rm p}, z_{\rm p}$, the planetocentric coordinates of the passive satellite at time $t_{\rm p}$. The flux from the passive satellite measured at time t₀ thus depends on $x_{\rm p}, y_{\rm p}, z_{\rm p}$ and $x_{\rm a}, y_{\rm a}, z\rm _a$.

Hereafter we consider the vector $\vec{R}_{\rm ap}$ of the position of the active satellite at time $t_{\rm a}$ relative to the position of the passive satellite at time $t_{\rm p}$. The components of this vector in the coordinate system x, y, z are determined by the following relations:

$$\Delta_x= x_{\rm a} - x_{\rm p}, \;\;\; \Delta_y= y_{\rm a} - y_{\rm p}, \;\;\; \Delta_z= z_{\rm a} - z_{\rm p}.$$ (1)

We now introduce another Cartesian coordinate system X, Y, Z with the origin at the center of the passive satellite. Let us make the Z-axis parallel to the event vector and the X-axis, parallel to the (x, y)-plane. Let the Y-axis always make an acute angle with the zaxis and make up a right-hand side coordinate system together with the X and Z axes. This defines unambiguously the direction of the X-axis. Geometrically it will be possible on condition that the z-axis and Z-axis are not parallel. For the natural satellites of all known planets it is always true.

We now denote the coordinates of the active satellite in the X, Y, Z reference frame at time $t_{\rm a}$ these are the components of the vector $\vec{R}_{\rm ap}$ as $X_{\rm a}, Y_{\rm a}, Z_{\rm a}$. It follows from Emel'yanov's ([2000]) results that the magnitude of the passive satellite, as measured at time t₀, depends on $X_{\rm a}, Y_{\rm a}$, and only slightly on projection $Z_{\rm a}$of the mutual radius-vector of the two satellites on the event vector.

The components of the event vector $\vec{R}_{\rm ap}$ in two reference frames are connected by the following evident relations

$$\begin{array}{c} X_{\rm a}= a_x \Delta_x + a_y \Delta_y + a_... ...\rm a}= b_x \Delta_x + b_y \Delta_y + b_z \Delta_z, \end{array}$$ (2)

where coefficients a_x, a_y, a_z, b_x, b_y, b_z depend only on the direction of the event vector. Note that coefficient a_z is equal to zero.

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 Sproduced 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 Sfrom the occulting and occulted satellites. In both cases flux Soutside eclipse or occultation is equal to unity by definition. During a mutual occultation or eclipse flux S decreases and this decrease depends primarily on $X_{\rm a}$ and $Y_{\rm a}$.

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:

$$E=[ K + Q \; (t - t_{\rm b}) ] \; S(X_{\rm a}, Y_{\rm a}) + P + L \; (t - t_{\rm b}),$$

where t is the time of observation; $t_{\rm b}$ some given instant of time, and K, Q, P, L, the empirical parameters to be inferred from photometric observations. The time $t_{\rm b}$ can be chosen arbitrarily prior to the reduction of the photometric observations. After the reduction both K, P correspond to the same time $t_{\rm b}$. Function $S(X_{\rm a}, Y_{\rm a})$ is determined by the model of mutual phenomena and the theory of satellite motion. The dependence S on $X_{\rm a}, Y_{\rm a}$, has been analyzed by Thuillot & Morando ([1990]), described by Emel'yanov ([1995]), Emel'yanov et al. ([1997]) and Emel'yanov ([1999]).

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 $X_{\rm a}, Y_{\rm a}$ corresponding to a photometric measurement at certain time as $X^{\rm (o)}, Y^{\rm (o)}$. To reduce observations, we use one of the most accurate theories of satellite motion available. We denote theoretically computed values of $X_{\rm a}, Y_{\rm a}$, as $X_{\rm t}, Y_{\rm t}$, respectively. We further assume that the differences

$$D_x = X^{\rm (o)} - X_{\rm t} , \;\;\; D_y = Y^{\rm (o)} - Y_{\rm t} ,$$

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

$$\begin{array}{c} \vspace{1em} E_i=[K + Q \; (t_i - t_{\rm b}... ..._{\rm t} + D_y) + \\ + P + L \; (t_i - t_{\rm b}) \end{array}$$

for corrections D_x, D_y, where $X_{\rm t}, Y_{\rm t}$ are computed theoretically for the time of observation, t_i. The set of conditional equations composed for all photometric measurements of the same event can then be solved using the least squares method. The solution yields parameters D_x, D_y, K, Q, L, and P providing the best fit to the observations made.

After doing this and to present the results of reduction, we choose arbitrarily one of the observing times, t₀, and compute $X^{\rm (o)}, Y^{\rm (o)}$ for this time using formulas

$$X^{(o)} = X_{\rm t} + D_x , \;\;\; Y^{\rm (o)} = Y_{\rm t} + D_y .$$ (3)

These values are the main result of reduction. $X^{\rm (o)}$ and $Y^{\rm (o)}$ can be used only if some other quantities considered here are known. In particular, we must know the time instants $t_{\rm a}$ and $t_{\rm p}$, corresponding to the chosen time t₀. With the reduction we save the values of $t_{\rm a}, t_{\rm p}, a_x, a_y, a_z, b_x, b_y, b_z$ computed for the chosen observing time, t₀.