# Precision of natural satellite ephemerides from observations of different types

N. Emelyanov

► To cite this version:

N. Emelyanov. Precision of natural satellite ephemerides from observations of different types. Monthly Notices of the Royal Astronomical Society, 2017, 469 (4), pp.4889-4898. 10.1093/mnras/stx1013. obspm-02269615

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https://hal-obspm.ccsd.cnrs.fr/obspm-02269615v1

Submitted on 20 Aug 2025

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# Precision of natural satellite ephemerides from observations of different types

N. V. Emelyanov<fnref n="1" />,<fnref n="2" />*

Accepted 2017 April 25. Received 2017 April 18; in original form 2017 January 11

## ABSTRACT

Currently, various types of observations of natural planetary satellites are used to refine their ephemerides. A new type of measurement – determining the instants of apparent satellite encounters – has recently been proposed by Morgado and co-workers. The problem that arises is which type of measurement to choose in order to obtain an ephemeris precision that is as high as possible. The answer can be obtained only by modelling the entire process: observations, obtaining the measured values, refining the satellite motion parameters, and generating the ephemeris. The explicit dependence of the ephemeris precision on observational accuracy as well as on the type of observations is unknown. In this paper, such a dependence is investigated using the Monte Carlo statistical method. The relationship between the ephemeris precision for different types of observations is then assessed. The possibility of using the instants of apparent satellite encounters to obtain an ephemeris is investigated. A method is proposed that can be used to fit the satellite orbital parameters to this type of measurement. It is shown that, in the absence of systematic scale errors in the CCD frame, the use of the instants of apparent encounters leads to less precise ephemerides. However, in the presence of significant scale errors, which is often the case, this type of measurement becomes effective because the instants of apparent satellite encounters do not depend on scale errors.

**Key words:** astrometry – ephemerides – planets and satellites: individual: Jupiter satellites.

## 1 INTRODUCTION

The production of precise ephemerides for natural planetary satellites is important for a number of reasons. First and foremost, they are used when preparing space missions. Furthermore, disagreement between observations and ephemerides can provide new information on the factors affecting satellite motion. Ephemerides can also be used to investigate problems regarding the orbital evolution and origin of planets and satellites.

Satellite ephemerides are generated on the basis of observations, and thus the ephemeris precision is determined by observational accuracy. The precision also depends heavily on the time interval of observations and on the distribution of observed satellite positions along the orbit.

In this paper, the problem of choosing the optimal method of measuring the satellite coordinates in order to achieve the most precise ephemeris is systematically investigated. To do this, we first consider what types of measurements there are and how satellite ephemerides are generated from observations.

Differential equations of motion of planetary satellites are composed for their planetocentric rectangular coordinates x, y, z. The

parameters to be determined are either the initial conditions or the orbital elements. We designate the latter as $p_1, p_2, ..., p_n$. Having solved the equations, we obtain the rectangular coordinates as functions of time t and the parameters of satellite motion. Thus we have

$$ 
\begin{align} 
x &= x(t, p_1, p_2, ..., p_n), \nonumber \\ 
y &= y(t, p_1, p_2, ..., p_n), \nonumber \\ 
z &= z(t, p_1, p_2, ..., p_n). \tag{1} 
\end{align} 
$$

These functions describe the model of motion of the satellites.

The applied dynamic model is described in the next section, where the specific meaning of the parameters is given. We follow the usual method of adjusting parameters from observations. In order to show what kind of problems are encountered, it is necessary to give a general description of the method. So here we give general designations for the motion parameters $p_1, p_2, ..., p_n$.

When a satellite is observed, various values are obtained as the measured ones. Any measured value $\xi$ depends on time t and on the satellite coordinates. That is, we have

$$ \xi = \xi(t, x, y, z). \tag{2} $$

<footnote n="1">¹M. V. Lomonosov Moscow State University – Sternberg Astronomical Institute, 13 Universitetskij Prospect, 119992 Moscow, Russia</footnote>

<footnote n="2">²Institut de mécanique céleste et de calcul des éphémérides – Observatoire de Paris, UMR 8028 du CNRS, 77 avenue Denfert-Rochereau, 75014 Paris, France</footnote>

<footnote>* E-mail: emelia@sai.msu.ru</footnote>
This function is determined by the type of observation and the type of measured value. Substituting the formulae (1), namely the equations of motion, we have the relationship

$$ \xi = \xi(t, p_1, p_2, \dots, p_n). \qquad (3) $$

The measured values are given for the dates of observations $t_i$ with the index $i = 1, 2, \dots, m$, where $m$ is the total number of observations. The values of $\xi$ obtained from observations are denoted as $\xi_1^{(0)}, \xi_2^{(0)}, \dots, \xi_m^{(0)}$. This allows us to compose a system of equations relative to the parameters in question:

$$ \xi_i^{(0)} = \xi(t_i, p_1, p_2, \dots, p_n) \quad (i = 1, 2, \dots, m). \qquad (4) $$

These equations are incompatible, as the left-hand sides include observational errors while the right-hand sides include the errors of the motion model. In addition, the errors of different observations are generally independent. Thus, the exact solution of equations (4) cannot be obtained. However, the unknown parameters can be evaluated from the equations by using the least-squares method (LSM).

After the parameters are found, the satellite ephemeris can be calculated for any date by the formulae (1) or (3), depending on whether we need the coordinates or the measured values.

The subject of discussion in this paper is the function (2), which is determined by the type of observation.

We assume that ground-based observations give us satellite celestial angular coordinates as the measured values. It is each of these coordinates that is the function $\xi$. In practice, however, the measured values are the linear coordinates of celestial bodies in the instrumental frame of the photographic plate or CCD placed at the focus of the telescope (the coordinates being expressed in millimetres or pixels). Transforming the instrumental frame coordinates of the satellite into celestial angular coordinates is a complex problem in astrometry. It is necessary to take into consideration the imaging distortion errors caused by the optics of the telescope and the refraction of light in the Earth's atmosphere. A particular problem is to define the scale of the image and the orientation of the coordinate system of the frame with respect to the standard celestial coordinate system, the ICRS. It is this astrometric problem that we are dealing with in this paper.

The relationship between the measured coordinates of a celestial body in the instrumental frame and its celestial coordinates is determined by the type of measurement. If there are stars in the frame, the satellite's celestial coordinates can be obtained by measuring its frame position relative to the stars, whose coordinates are taken from catalogues. These are referred to here as absolute observations. Obviously, the satellite's celestial coordinates obtained in this way contain errors from the catalogue's star coordinates. Moreover, such measurements can only be effective when the brightness of stars in the frame differs only slightly from that of the satellite; otherwise, the stars appear very underexposed or even are undetected (short exposures), or the satellite images become saturated (through long exposures). For the particular case of the Galilean moons of Jupiter, this happens very rarely. For this reason, we do not consider the absolute observations further in this paper.

If two or more satellites appear in the field of the frame, the measured values can be the differences between the satellite coordinates in the frame. In order to obtain the differences between the celestial coordinates of two satellites, we need to know two parameters: the image scale and the orientation of the frame coordinate system relative to the direction to the celestial pole. There are various ways to define these parameters. In any case, however, orientation errors and scale errors remain.

The measured value can simply be the distance between two satellites. To determine the angular distance, it is necessary to know only the scale of the image in the frame, expressed, for example, in arcseconds per pixel. The orientation error does not affect the results of these kinds of measurements. The scale depends on the focal length of the telescope, which can vary from night to night. The scale errors, obviously, have an effect on the accuracy of the measurements.

In order to exclude scale and orientation errors, Morgado et al. (2016) proposed an original method of observing the major planetary satellites. The idea is that, when observing the relative motion of two satellites, it often happens that at a certain small time interval the apparent trajectory of one satellite relative to another turns out to be almost rectilinear. At some instant, the distance between the two satellites reaches its minimum. At this time interval, which is called the encounter time interval, the velocity along the relative trajectory does not change significantly. Frequent measurements of the apparent distance between two satellites allows us to determine the instant $t_0$ of the closest approach, termed the encounter instant. Because the encounter instant depends on the parameters of satellite motion, it can be used as a measured value $\xi$ in equation (3), where $p_1, p_2, \dots, p_n$ are the parameters of motion of both satellites. The expected advantages of this method of observation are that the encounter instant $t_0$ does not depend on the frame orientation, or even on the pixel scale, and thus it does not depend on their systematic errors. In particular, if we add a certain value to all measured mutual distances at the encounter time interval, the value of $t_0$ does not change. Morgado et al. (2016) give examples of observations of apparent encounters of the Galilean satellites of Jupiter. However, the authors did not explain how these observations can be used to determine the parameters of motion of the satellites. A method to evaluate the ephemeris precision obtained using the new method was not proposed either.

As noted above, in this paper we try to evaluate the precision of planetary satellite ephemerides obtained from observations of different types. The observations obtained when the instants of apparent encounters are measured are of particular interest.

We dealt with the problem of evaluating the ephemeris error in the following manner. We do not have many measurements of the apparent encounters of satellites – only a few examples given by Morgado et al. (2016). It turned out that determining the satellite motion parameters on the basis of such observations is very time-consuming. The aim of this research was to evaluate the precision of the ephemeris obtained from models of motion based on observations of the major planetary satellites. However, the observations we have included a fixed set of unknown errors for each observation. Therefore, for a more general and systematic investigation, we evaluated the ephemeris precision by simulating observational errors using the Monte Carlo method. Moreover, in the case of the major satellites of Jupiter, Saturn, Uranus and Neptune, the number of revolutions made during the time intervals of their actual observations is large. This means that the numerical integration of the differential equations of motion of these satellites would be very time-consuming.

Under these circumstances, the investigation was conducted using artificial sets of relative observations and the Keplerian model of the motion of satellites, which is as close as possible to their actual orbits. The orbital motions of the Earth and Jupiter were modelled by the Keplerian orbits with the mean elements taken from Simon et al. (1994). The configurations of the satellites' apparent motions and encounters were close to those that actually occur. This allowed us to use the Monte Carlo method to evaluate the precision of the
resulting satellite ephemerides. The objects of the study were the four Galilean moons of Jupiter, namely J1 (Io), J2 (Europa), J3 (Ganymede) and J4 (Callisto).

The simulated observations of the satellites and subsequent ephemerides were generated using this dynamic model. The geometric configuration of mutual encounters between satellites is quite similar to that of mutual occultations between the Galilean moons of Jupiter, for which a model of motion for the satellites was analysed by Emelyanov (2009). This model is used both to generate pseudo-observations and to perform fits on them.

An evaluation of the precision of the ephemerides was made using the method described in detail by Emelyanov (2010).

## 2 ARTIFICIAL SETS OF OBSERVATIONS

So, in our research, the ground-based observations of satellites during their apparent encounters are of major importance. For different pairs of satellites, we modelled the mutual apparent geocentric distances *d* at the time intervals of apparent encounters. These distances were used as the results of observations of one of three types. Then they were used to define the instants *t*₀ of the satellites' apparent encounters. A series of encounter instants was the result of observations of another type. For the dates for which mutual distances were modelled, the differences between the apparent geocentric equatorial coordinates of the satellites were also calculated. In other words, a series of measured values *X* = Δα cos δ and *Y* = Δδ was composed, where Δα and Δδ are differences in right ascension and declination of both satellites and δ is the declination of one of the satellites. The differences of the apparent geocentric equatorial coordinates were modelled at the time intervals of the mutual encounters of the satellites. This corresponds to the optimal choice of the instants of such observations, because, when the mutual apparent distances are at a minimum, the contribution of the frame orientation error is also minimal.

From here on, we adopt the following numbering of the types of observations: the measurements of *X*, *Y* are observations of the first type; the measurements of the apparent separation *d* hereinafter referred to the mutual distance are those of the second type; and the instants of apparent satellite encounters *t*₀ are those of the third type.

The composition of the series of observations was made in the following way. First, a complete interval (*t*<sub>b</sub>, *t*<sub>e</sub>) and a step *Δt* were defined. Then, for successive instants of time with the step *Δt*, the six mutual distances (that is, for the pairs J1-J2, J1-J3, J1-J4, J2-J3, J2-J4, J3-J4) were calculated. After that, for each pair of satellites and for the given dates, the mutual distances were scanned, which is necessary to find time intervals at which encounters may take place. The selection of observations was made under the condition that the mutual distance *d* should not exceed the value of *d*<sub>lim</sub>; that is,

$$d < d_{\text{lim}}.$$

As results, we obtained a series of time instants and mutual distances, which we denote as $t_i^{(k)}$, $d_i^{(k)}$ ($i = 1, 2, \dots, N^{(k)}$), where *k* is the number of encounter time interval, *i* is the number of measurements at this interval, and $N^{(k)}$ is the number of measurements at the interval with the index *k*. Selecting the observations, we also set the conditions

$$d_2^{(k)} < d_1^{(k)}, \quad d_{N^{(k)}-1}^{(k)} < d_{N^{(k)}}^{(k)} \quad d_{i-1}^{(k)} - 2d_i^{(k)} + d_{i+1}^{(k)} < 0.$$

These conditions correspond to a time interval (*t*₁, *t*₂), where the mutual distance *d*(*t*), being a function of time, satisfies the following inequalities:

$$\frac{d(d)}{dt}|_{t=t_1} < 0, \quad \frac{d(d)}{dt}|_{t=t_2} > 0, \quad \frac{d^2(d)}{dt^2} > 0.$$

Given this, the function *d*(*t*) has a unique minimum at the interval (*t*₁, *t*₂).

When detecting the encounters in this way, it sometimes happens that at the beginning of the event the velocity of one satellite relative to the other is directed towards the target satellite, while at the end of the encounter the velocity reverses its direction. Such encounters were omitted from consideration. In addition, we rejected cases for which the number of observations $N^{(k)}$ was lower than a given number $N_{\text{min}}$; that is, the following condition was imposed:

$$N^{(k)} > N_{\text{min}}.$$

The instants $t_i^{(k)}$ selected in this way were taken to compose the series of observations. For these instants, the relative coordinates *X*, *Y* were calculated as the results of observations of the first type, and mutual distances *d* as those of the second type. The calculation of the instants of apparent encounters *t*₀ as the results of observations of the third type is discussed in the next section.

Furthermore, in order to realize the Monte Carlo method, the so-called observation samples were generated for the same instants of time. This was achieved by adding some artificial errors (obtained by random number generation) to the results of observations. In each sample, a particular set of random errors was added to the measurements. The random errors were generated for a given rms value $\sigma_{\text{obs}}$ of observational errors.

Evaluation of the ephemeris precision was then made using the method described by Emelyanov (2010). First, the values of some of the parameters of the motion model of the satellites are fixed. Using these parameters, the reference ephemeris is calculated, namely the apparent geocentric satellite coordinates relative to the planet obtained for a series of dates that follows the interval of observations (*t*<sub>b</sub>, *t*<sub>e</sub>), that is, after *t*<sub>e</sub>. Then, observation samples are composed in the way described above. For each sample, the parameters of satellite motions are refined. The refined parameters are then used to calculate the sample ephemeris, namely the apparent geocentric satellite coordinates relative to the planet at the dates for which the reference ephemeris was calculated earlier. Then, for each date, the differences between the coordinates of the refined ephemeris and the reference ephemeris (without adding the errors) are calculated. These differences are the errors of the ephemeris generated on the basis of the given observation sample.

The same procedure is then repeated for a large set of observation samples. Finally, for all observation samples, we define the rms values of the ephemeris errors for each date. This value, which we denote as $\sigma_{\text{eph}}$, is adopted as the ephemeris precision.

Note that the observations that were actually made represent only one observation sample. Other real samples do not exist: that is why they had to be created artificially.

When modelling the observational errors, the following three sources of errors should be considered. The first one is the error in the satellite's position in the frame, which results either in errors of the differences *X*, *Y* in the apparent angular coordinates or in the error of the mutual angular distances *d*. This is the error in measuring the photocentres of the satellites in the frame. Each measurement obviously has its own error. The predetermined value of the rms of these errors is denoted as $\sigma_{\text{meg}}$ (measured in arcseconds).
<figure><img src="image_2.png" /><figcaption>Figure 1. Trajectory of the relative apparent motion of the satellites.</figcaption></figure>

The second source of error comes from the frame orientation. However, it only affects the measured values of X, Y. We assume that the orientation error is constant for all measurements made during one encounter of satellites, but for other encounters the errors are different. The orientation error reveals itself in the parasitic turn of the frame around the optical axis of the telescope at a certain angle. The given rms of these angles is denoted as $\sigma_{or}$ (measured in degrees).

The third source of error comes from the frame scale. Its presence causes the measured mutual distances to be multiplied by a factor of $(1 + \varepsilon_{sc})$, where $\varepsilon_{sc}$ is some very small dimensionless quantity. We suppose that the scale error is constant for all measurements within one encounter but that for other encounters the errors differ. The given rms value of the error $\varepsilon_{sc}$ is denoted as $\sigma_{sc}$.

### 3 DETERMINING THE INSTANTS OF APPARENT SATELLITE ENCOUNTERS

Let us consider the motion of a satellite B relative to a satellite A during their apparent encounter. Fig. 1 shows a possible relative trajectory and the position of satellite A at an instant $t$.

Accepting that the velocity $V$ of the satellite along the relative trajectory can change, we assume that on the given section of the trajectory the velocity does not differ much from its value $V_0$ at point C.

In view of the limitations imposed in the previous section, the apparent encounters under consideration should look like exactly as in Fig. 1.

Let $d$ be the mutual distance between the satellites at an arbitrary time instant $t$. At some point, at the instant $t = t_0$, the satellites reach the minimum apparent mutual distance $d_0$ (in Fig. 1 this is the point C).

Let $f(t)$ be the function representing the square of the mutual distance, namely $f(t) = d^2$. With the limitations given above, at the encounter time interval, the function $f(t)$ has a unique minimum achieved at the instant $t = t_0$.

We introduce the following notation:

$$x = t - t_1, \quad x_0 = t_0 - t_1, \quad F(x) = f(t_1 + x).$$

Suppose that the observations made at the instants $t_i$ ($i = 1, \dots, N$) belonging to the range ($t_1, t_N$) provide us with the values of the mutual apparent distances $d_i$. (Here, for ease of description, we omit the index $k$, which indicates the number of the encounter.)

We approximate the function $F(x)$ by the polynomial

$$P_n(x) = \sum_{i=0}^{n} a_i x^i. \qquad (5)$$

In order to represent the function accurately, it is necessary that the number $N$ of the measured distances be as high as possible. The coefficients of the polynomial can then be determined by the LSM.

The instant $t_0$ at which the minimum mutual distance is reached can be found by solving the equation

$$\frac{dP_n(x)}{dx} = 0. \qquad (6)$$

The root $x_0$ gives us the instant we are looking for: $t_0 = t_1 + x_0$.

The roots are found using the bisection method. In order to know the root with precision $\varepsilon$, it is necessary to calculate the function $dP_n(x)/dx$ about $M$ times, where

$$M = \log_2 \frac{t_2 - t_1}{\varepsilon}.$$

The minimum distance is then determined from the relationship

$$d_0 = \sqrt{P_n(x_0)}.$$

In subsequent calculations, we need to know the relative apparent velocity $V_0$ of the satellite along the trajectory at the instant $t_0$. Using the representation of the mutual distance by the polynomial $P_n(x)$, the required velocity can be calculated by the formula

$$V_0 = \sqrt{\sum_{n=2}^{N} a_n \frac{n(n-1)}{2} x_0^{n-2}}.$$

If the measurements of the apparent mutual distances of satellites had been performed without errors, then, having chosen a sufficiently large degree of the polynomial $P_n(x)$, it would have been possible to find $t_0$ with any required accuracy. However, the measured distances have errors, which results in errors in the value of $t_0$. In making our model calculations, we assume that the only sources of errors in $t_0$ are errors in the mutual distances $d$. Hence, when composing the artificial samples of observations, no additional errors in the values of $t_0$ are added.

When comparing the ephemeris precision derived from observations of different types, it is important that the series of observations of the second ($d$) and third ($t_0$) types are simulated with errors from the same sources, namely the random errors of measured distances and scale errors.

### 4 FITTING THE SATELLITE MOTION PARAMETERS TO THE INSTANTS OF APPARENT ENCOUNTERS

The methods for determining satellite motion parameters from positional observations are well known and widely used. Examples of their use can be seen in, for instance, Emelyanov & Nikonchuk (2013) and Emelyanov & Samorodov (2015).

These methods can be applied to any dynamical model of any natural satellite. Here they are applied to the dynamical model of the Galilean moons of Jupiter described above. For the sake of brevity, we use the general notation for the parameters of satellite motion. Note that we make the fit of the parameters to the observations for all four Galilean moons together. Thus the number of parameters is $n = 24$.

According to these methods, the original non-linear equations with respect to unknown parameters (4) are replaced by the linear equations

$$\Delta\xi_i = \sum_{j=1}^{n} \left( \frac{\partial\xi_j}{\partial p_j} \right)_i \Delta p_j \quad (i = 1, 2, \dots, m), \qquad (7)$$

where

$$\Delta\xi_i = \xi_i^{(0)} - \xi(t_i, p_1^{(0)}, p_2^{(0)}, \dots, p_n^{(0)}), \quad \Delta p_j = p_j - p_j^{(0)},$$
and $p_1^{(0)}, p_2^{(0)}, \dots, p_n^{(0)}$ are some preliminary values of the parameters determined in earlier research. The partial derivatives in (7) are calculated for the dates of observations $t_i$ and with the parameters equal to their preliminary values $p_1^{(0)}, p_2^{(0)}, \dots, p_n^{(0)}$.

Note that the dissipation parameters found from observations in Emelyanov & Nikonchuk (2013) were then recalculated, and the corresponding Erratum communication is published in Emelyanov & Nikonchuk (2016).

In order to find the partial derivatives, the following expressions are used:

$$ \frac{\partial \xi}{\partial p_j} = \frac{\partial \xi}{\partial x} \frac{\partial x}{\partial p_j} + \frac{\partial \xi}{\partial y} \frac{\partial y}{\partial p_j} + \frac{\partial \xi}{\partial z} \frac{\partial z}{\partial p_j}. $$

The factors $\partial\xi/\partial x$, $\partial\xi/\partial y$, $\partial\xi/\partial z$ are obtained by differentiating the relationships (2). The partial derivatives of the coordinates by the parameters,

$$ \frac{\partial x}{\partial p_j}, \quad \frac{\partial y}{\partial p_j}, \quad \frac{\partial z}{\partial p_j}, \qquad (8) $$

are calculated either using the known formulae of the analytic theory of satellite motion or by numerical integration of differential equations for these derivatives together with the equations of motion.

The equations (7) are not compatible because of the presence of observational errors and the errors of the motion model. Estimation of the corrections $\Delta p_1, \Delta p_2, \dots, \Delta p_n$ can be made by the LSM. The refined parameters are then obtained from the relationships

$$ p_j = p_j^{(0)} + \Delta p_j \quad (j = 1, 2, \dots, n). $$

Because the equations (7) are approximate, the process of fitting the parameters has to be repeated several times until the corrections $\Delta p_j$ become small enough.

Morgado et al. (2016) proposed using the instants of apparent satellite encounters as observational data. However, they do not indicate how these observations can be used to determine the parameters of satellite orbits.

In this case, equations (7) have the form

$$ \Delta t_0^{(k)} = \sum_{j=1}^{n} \left( \frac{\partial t_0}{\partial p_j} \right)_k \Delta p_j \quad (k = 1, 2, \dots, m), \qquad (9) $$

where

$$ \Delta t_0^{(k)} = t_0^{(k)} - \bar{t}_0^{(k)}, $$

$t_0^{(k)}$ are the instants of apparent encounters found from observations, $\bar{t}_0^{(k)}$ are the instants of apparent encounters corresponding to the motion model of the satellites the parameters $p_1^{(0)}, p_2^{(0)}, \dots, p_n^{(0)}$, and the index $k$ is the number of apparent encounter. Equations (9) give the partial derivatives of the instants of apparent encounter with respect to the parameters of satellite motion.

The instant of apparent encounter $\bar{t}_0^{(k)}$ is found from the measured mutual distances $d_i^{(k)}$ as described in the previous section. Thus, two problems have to be solved: finding $\bar{t}_0^{(k)}$ and calculating the partial derivatives $(\partial t_0 / \partial p_j)_k$.

Obviously, in order to find $\bar{t}_0^{(k)}$, it is necessary to calculate the mutual distances $\bar{d}_i^{(k)}$ obtained from the model of motion at the dates of observations. Then, the instants $\bar{t}_0^{(k)}$ are defined from $\bar{d}_i^{(k)}$ in the same way as $t_0^{(k)}$ were calculated from observations.

The computational process for the derivatives $(\partial t_0 / \partial p_j)_k$ turns out to be more complicated. There are no analytic formulae for the calculation of these derivatives, and no differential equations with

respect to these derivatives analogous to those for the derivatives (8). Therefore, we have to find $(\partial t_0 / \partial p_j)_k$ approximately and numerically as the ratio of a finite increment of the function to that of the argument.

Note that, when using the instants of apparent encounters of two satellites as the measured values, the parameter list $p_1, p_2, \dots, p_n$ contains six parameters of motion of the first satellite and six parameters of the second one. Hence, in order to compute the partial derivatives $(\partial t_0 / \partial p_j)_k$, we need to give an increment to each of these 12 parameters in turn. For each of these 12 variants, the mutual distances should be calculated for the dates of observations on the basis of the input parameters $p_1^{(0)}, p_2^{(0)}, \dots, p_n^{(0)}$, among which, one, with the index $j'$, has a finite increment $\Delta' p_{j'}$. Having obtained these mutual distances, we find $t_0^{(k,j')}$ in the way described in the previous section. Then the derivative under consideration $(\partial t_0 / \partial p_{j'})_k$ can be calculated approximately as the ratio

$$ \frac{t_0^{(k,j')} - \bar{t}_0^{(k)}}{\Delta' p_{j'}}. $$

This operation is to be carried out 12 times, once for each of the 12 parameters.

This research presents the realization of the procedure described above. Note that it can be used with any accurate model of the motion of satellites. For instance, for the Galilean satellites of Jupiter, the numerical model by Lainey et al. (2009) can be taken. We have applied this procedure of evaluating the precision of the ephemeris generated on the basis of artificially composed observations of the Galilean satellites and the Keplerian model of their motion.

## 5 ALTERNATIVE METHODS OF COMPOSING THE CONDITIONAL EQUATIONS WHEN THE INSTANTS OF MUTUAL ENCOUNTERS ARE USED

The method of composing the conditional equations for the case in which the instants of mutual encounters are measured, considered above, results in very time-consuming calculations. It is desirable to find a method that is more effective. Let us consider two possible alternatives.

The first one is our original method based on the following considerations. Suppose that we have observations made at the dates $t_0^{(i)}$ ($i = 1, 2, \dots, m$) when the apparent distance between the satellites reached its minimum. For the sake of brevity, hereafter we omit the subscript $i$.

Let $X_1, Y_1$ be the topocentric apparent coordinates of satellite 1 relative to the planet, and $X_2, Y_2$ be the equivalent coordinates of satellite 2. These coordinates are measured in the tangent plane as observed from the Earth. The X-axis is directed to the east, and the Y-axis to the north. We use the notation

$$ X_{12} = X_1 - X_2, \quad Y_{12} = Y_1 - Y_2. $$

The rates of change in time are $\dot{X}_{12}, \dot{Y}_{12}$.

The values $X_{12}, Y_{12}$ are functions of $t$ and the parameters of motion of both satellites $p_1, p_2, \dots, p_n$. Suppose that the instant of the minimum mutual distance between the satellites $t_0$ deduced from observations and the true values of parameters $p_1, p_2, \dots, p_n$ give us true values $X_{12} = X_{12}^{(\text{true})}$, $Y_{12} = Y_{12}^{(\text{true})}$. That is,

$$ X_{12}(t_0, p_1, p_2, \dots, p_n) = X_{12}^{(\text{true})}, \\ Y_{12}(t_0, p_1, p_2, \dots, p_n) = Y_{12}^{(\text{true})}. \qquad (10) $$
We suppose also that the same true values can be obtained from the theory of satellite motions with some given preliminary values of the parameters $p_1^{(0)}$, $p_2^{(0)}$, ..., $p_n^{(0)}$ but at another time instant $\bar{t}_0$ of the minimum mutual distance between the satellites. Thus we have

$$ 
\begin{aligned}
X_{12}(\bar{t}_0, p_1^{(0)}, p_2^{(0)}, \dots, p_n^{(0)}) &= X_{12}^{(\text{true})}, \\
Y_{12}(\bar{t}_0, p_1^{(0)}, p_2^{(0)}, \dots, p_n^{(0)}) &= Y_{12}^{(\text{true})}.
\end{aligned} 
\quad (11) $$

The equations (10) and (11) give us

$$ X_{12}(\bar{t}_0, p_1^{(0)}, p_2^{(0)}, \dots, p_n^{(0)}) = X_{12}(t_0, p_1, p_2, \dots, p_n), \quad (12) $$

$$ Y_{12}(\bar{t}_0, p_1^{(0)}, p_2^{(0)}, \dots, p_n^{(0)}) = Y_{12}(t_0, p_1, p_2, \dots, p_n). \quad (13) $$

Assuming that the differences

$$ \Delta t = t_0 - \bar{t}_0, \quad \Delta p_j = p_j - p_j^{(0)} \quad (j = 1, 2, \dots, n) $$

are small enough, we obtain the following approximate conditional equations:

$$ -\dot{X}_{12}\Delta t = \sum_{j=0}^{n} \left( \frac{\partial X_{12}}{\partial p_j} \right)_0 \Delta p_j, $$

$$ -\dot{Y}_{12}\Delta t = \sum_{j=0}^{n} \left( \frac{\partial Y_{12}}{\partial p_j} \right)_0 \Delta p_j, $$

where the partial derivatives are calculated with the preliminary values of the parameters ($p_1^{(0)}, p_2^{(0)}, \dots, p_n^{(0)}$) and for $t = \bar{t}_0$. The instant of mutual encounter of the satellites, $\bar{t}_0$, can be calculated from the model of satellite motions with the given values of the parameters ($p_1^{(0)}, p_2^{(0)}, \dots, p_n^{(0)}$) by the bisection method. We realized and tested this method of composing the conditional equations. It proved to be less time-consuming than the method described above. The entire analysis was repeated using this second method of calculating the partial derivatives. All the results were the same.

Another method of composing the conditional equations for the case under investigation was proposed by Vieira-Martins (private communication).

It was proposed to use as an observed value $\xi$ the function

$$ \xi = S(t, p_1, p_2, \dots, p_{12}) = X_{12} \dot{X}_{12} + Y_{12} \dot{Y}_{12}, $$

where $p_1, p_2, \dots, p_6$ are the parameters of motion of the first satellite, and $p_7, p_8, \dots, p_{12}$ are those of the second one.

The function $S$ is the scalar product of the position vector {$X_{12}, Y_{12}$} and the velocity vector {$\dot{X}_{12}, \dot{Y}_{12}$} of satellite 1 relative to satellite 2 in the tangent plane. In this case, the observed values $\xi^{(0)} = S^{(0)}$ are equal to zero at the instant of the minimum mutual distance between the satellites; that is,

$$ \xi^{(0)} = S^{(0)} = 0. $$

The equations (7) can then be written as

$$ -S(t_0, p_1^{(0)}, p_2^{(0)}, \dots, p_{12}^{(0)}) = \sum_{j=1}^{n} \left( \frac{\partial S}{\partial p_j} \right)_0 \Delta p_j, \quad (14) $$

where the derivatives can be expressed by the following relationships:

$$ \frac{\partial S}{\partial p_j} = X_{12} \frac{\partial \dot{X}_{12}}{\partial p_j} + \frac{\partial X_{12}}{\partial p_j} \dot{X}_{12} + Y_{12} \frac{\partial \dot{Y}_{12}}{\partial p_j} + \frac{\partial Y_{12}}{\partial p_j} \dot{Y}_{12}. $$

These derivatives and the values $S(t_0, p_1^{(0)}, p_2^{(0)}, \dots, p_{12}^{(0)})$ can be calculated using the motion model of the satellites.

We realized and tested this method of composing the conditional equations. In order to carry out the test, a series of simulated observations giving the instants of mutual encounters of satellites, $t_0$, was produced in the way described in the previous sections. However, we could not achieve convergence of the solution with this method if we used only the instants of mutual encounters of satellites as observations. Nevertheless, if we fix the inclinations and longitudes of the ascending nodes of both satellites, fitting only four parameters for each of them, the process of the parameter refinement ends very quickly. This proves correctness of the way in which we realized the method. It is evident that the instants of mutual encounters of satellites depend very weakly on small variations in orbital inclinations.

It should be noted that this fact does not hamper the practical implementation of the method proposed by Vieira-Martins (private communication) of composing the conditional equations. The measurements of the instants of mutual encounters of satellites are always used in combination with other types of observations. In such cases, the proposed method could be very efficient.

## 6 EVALUATION OF THE PRECISION OF THE SATELLITE EPHEMERIS OBTAINED FROM OBSERVATIONS OF VARIOUS TYPES

As noted earlier, the aim of our study was to evaluate the precision of the ephemeris based on observations. The measured values can belong to one of three types. We suppose that we have a series of time instants when observations were made; that is, we have satellite images in which the relative coordinates and distances between the satellites can be measured. After that, we have a choice of what type of measurement to use. Obviously, for the ephemeris precision to depend only on the type of measurement, the different types of measurement are to be made for the same instants of time. Thus we can draw conclusions and make recommendations concerning what type of measurement is the most appropriate for obtaining the best ephemeris precision.

Using the procedures described above, we made calculations with the following initial data. We considered the motions of the four Galilean moons of Jupiter. Table 1 gives the adopted values of the model parameters of the satellite orbits. Orbital inclinations and longitudes of ascending nodes are referred to the equator of Jupiter. The coordinates of the pole of Jupiter referred to the Earth's equator are taken as follows: right ascension $\alpha_0 = 268:056595$, declination $\delta_0 = 64:495303$. The mean anomaly epoch is $t_M = 57 023.0$ MJD (Modified Julian Days: MJD=JD-2 400 000.5). The gravitational parameter of the planet was determined separately for each satellite from mean motions and semi-major axes.

The complete time interval of observations expressed in MJD spans from $t_b = 57 324.0$ to $t_e = 57 924.0$; that is, 600 d. Measurements were made with a step of 0.005 d.

When choosing the intervals when encounters occur, the following restrictions were adopted: the maximum mutual distance is $d_{\text{lim}} = 60.0$ arcsec, and the minimum number of instants in the encounter time interval is $N_{\text{min}} = 30$.

The time interval for which the ephemeris precision was estimated followed directly after the interval of observations and ended at the date $t_l = 58 924.0$ (MJD). Thus, the observations cover an interval of 600 d, while the ephemeris precision was estimated at the interval of 1000 d after the date of the last observation. The time step for the ephemeris calculations was taken to be 1 d.

The rms value of the measurement error was taken to be $\sigma_{\text{meg}} = 0.05$ arcsec. The rms values of the orientation errors $\sigma_{\text{or}}$
<table><caption>**Table 1.** Orbital parameters of the satellites J1 (Io), J2 (Europa), J3 (Ganymede), and J4 (Callisto).</caption><thead><tr><th rowspan="2">Parameter</th><th colspan="4">Satellite</th></tr><tr><th>J1</th><th>J2</th><th>J3</th><th>J4</th></tr></thead><tbody><tr><td>Mean motion [deg d<sup>-1</sup>]</td><td>203.318566151</td><td>101.356097472</td><td>50.320500158</td><td>21.553257856</td></tr><tr><td>Semimajor axis [km]</td><td>421 942.391</td><td>671 121.962</td><td>1 070 399.063</td><td>1 881 648.669</td></tr><tr><td>Eccentricity</td><td>0.00415385</td><td>0.00946777</td><td>0.00178730</td><td>0.00685849</td></tr><tr><td>Inclination [deg]</td><td>0.029205</td><td>0.479934</td><td>0.139372</td><td>0.236175</td></tr><tr><td>Mean anomaly [deg]</td><td>20.205681</td><td>338.579279</td><td>174.070216</td><td>172.702644</td></tr><tr><td>Argument of pericentre [deg]</td><td>41.714553</td><td>129.774956</td><td>271.132265</td><td>199.402257</td></tr><tr><td>Longitude of asc. node [deg]</td><td>272.279196</td><td>7.564293</td><td>41.599430</td><td>165.952127</td></tr></tbody></table>

and of the scale error $\sigma_{sc}$ were incremented, starting from zero (see their values in the tables below). Regarding the calculation of $\sigma_{sc}$, see the end of the Section 2.

With the parameters and conditions given above, the time intervals of mutual encounters were obtained for the six pairs of satellites. For the time instants within these intervals, the differences in the satellites' coordinates X, Y and the mutual distances d were calculated. Then, with predetermined rms values of the observational errors, we computed the random errors with the Gaussian distribution of the probability density. The errors were added to the values of X, Y, d as described above. As is the usual practice, we discarded a random error if its modulus exceeded $3\sigma$.

Based on the obtained values of mutual distances to which errors were added, the instants of apparent encounters $t_0$ were computed. The degree $n$ of the polynomial (5) was taken to be 6.

As a result of calculations, for the predetermined time interval of 600 d, a total of 889 intervals and instants of apparent encounters were found for all six pairs of satellites. At these intervals, the differences in the coordinates X, Y and mutual distances d for 41 524 time instants were calculated. The number of instants when the distances were measured to determine the instants of apparent encounters turned out to be from 30 to 142 for different intervals and for different pairs of satellites. The minimum distances themselves varied in the range from 1 to 45 arcsec.

We also estimated the accuracy with which the polynomials of the 6th degree represent the model values of the measured mutual distances before random errors were added. In order to do this, the rms value of the deviations of the distances from their polynomial approximation was calculated for each interval. These estimations turned out to vary greatly for different intervals. In most cases, they did not exceed 0.00001 arcsec. In about 20 cases they were greater than 0.0001 arcsec. The maximum value detected among all intervals and all pairs of satellites was 0.001 arcsec. These evaluations reveal that the errors of the polynomial representation of mutual distances are significantly lower than the modelled errors of measurements of the distances.

The determination of the instants of apparent satellite encounters based on the polynomial representation was made, as noted above, by the bisection method, with an accuracy of $10^{-9}$ of days.

At the instant of encounter, the velocities with which one satellite moved relative to another were 200–700 arcsec $d^{-1}$. Sometimes, the relative velocity was only about 20 arcsec $d^{-1}$; that is, the encounter was very slow, although in fact the satellites were moving in the sky plane in approximately the same direction with close velocities.

All calculations were made using a computer program written in the C programming language.

In order to estimate the ephemeris precision with the Monte Carlo method, calculations were carried out for 100 samples of

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Ephemeris precision $\sigma_{eph}$ as a function of time for observations of the first type (the measurements of the differences X, Y). Points show the precision along the trajectory, and circles that across the trajectory.</figcaption></figure>

model observations. The deviations of satellite positions computed by the ephemeris were calculated both across and along the apparent trajectories.

In the first variant, the systematic errors of orientation and scale were not introduced. The results of the ephemeris precision evaluations are shown in Figs 2, 3 and 4 for all three types of observations. The figures show that the best ephemeris precision is obtained when the measurements of differences in the satellites' coordinates X, Y are used. A slightly worse ephemeris is obtained with the measurements of the relative distances d. In both cases, the errors along the apparent trajectory are several times greater than those across the path.

When using the instants of apparent encounters of satellites, the resulting ephemerides are the least precise. Moreover, unlike the measurements of the first and second types, the errors across the trajectory are several times greater than those along it. This is easily explained by the fact that the instants of apparent encounters fix the relative position of satellites along the trajectories, and weakly depend on the displacements across them.

In order to simplify the further comparison of the ephemeris precision, we calculated the averaged rms values of deviations from the reference ephemeris at an interval of 100 d located just before the end of the ephemeris time interval.
<figure><img src="image_4.png" /><figcaption>**Figure 3.** Ephemeris precision $\sigma_{eph}$ as a function of time for observations of the second type (the measurements of mutual distances $d$). Points show the precision along the trajectory, and circles that across the trajectory.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 4.** Ephemeris precision $\sigma_{eph}$ as a function of time for observations of the third type (the measurements of the instants $t_0$ of apparent encounters of satellites). Points show the precision along the trajectory, and circles that across the trajectory.</figcaption></figure>

In the following considerations we took into consideration the systematic errors of orientation $\sigma_{or}$ and scale $\sigma_{sc}$ in the way described above. The results of the comparison of ephemeris precision for various values of these errors are given in the tables. Table 2 gives averaged rms values when the measurements of differences X, Y in satellite coordinates are used. Table 3 contains averaged rms values when the distances between satellites, $d$, and the instants of apparent encounters, $t_0$, are measured.

As a result of evaluations of the precision of satellite ephemerides generated on the basis of all three types of observations, we come to the following conclusions. In the absence of systematic errors in orientation and scale, the best ephemeris precision is achieved by using measurements of differences in satellite coordinates. The use

<table><caption>**Table 2.** Ephemeris errors for various values of orientation errors and scale errors when the measurements of the differences X, Y in satellite coordinates are used. Mean values on the 100-d time interval of the rms of the satellite position deviations along and across the apparent relative satellite path are given. The rms values are calculated through samples of the Monte Carlo method.</caption><thead><tr><th colspan="2">Systematic errors set</th><th colspan="2">Mean errors of ephemeris [arcsec]</th></tr><tr><th>σ<sub>or</sub> [deg]</th><th>σ<sub>sc</sub></th><th>Along</th><th>Across</th></tr></thead><tbody><tr><td>0.0</td><td>0.0</td><td>0.0026</td><td>0.0007</td></tr><tr><td>0.0625</td><td>0.0</td><td>0.0068</td><td>0.0016</td></tr><tr><td>0.125</td><td>0.0</td><td>0.0129</td><td>0.0029</td></tr><tr><td>0.25</td><td>0.0</td><td>0.0255</td><td>0.0057</td></tr><tr><td>0.5</td><td>0.0</td><td>0.0508</td><td>0.0114</td></tr><tr><td>1.0</td><td>0.0</td><td>0.1016</td><td>0.0228</td></tr><tr><td>0.0</td><td>0.0005</td><td>0.0033</td><td>0.0011</td></tr><tr><td>0.0</td><td>0.001</td><td>0.0048</td><td>0.0019</td></tr><tr><td>0.0</td><td>0.002</td><td>0.0086</td><td>0.0036</td></tr><tr><td>0.0</td><td>0.004</td><td>0.0166</td><td>0.0070</td></tr><tr><td>0.0</td><td>0.008</td><td>0.0331</td><td>0.0140</td></tr><tr><td>0.0</td><td>0.016</td><td>0.0661</td><td>0.0279</td></tr><tr><td>0.0625</td><td>0.002</td><td>0.0109</td><td>0.0040</td></tr><tr><td>0.0625</td><td>0.008</td><td>0.0345</td><td>0.0146</td></tr><tr><td>0.0625</td><td>0.016</td><td>0.0680</td><td>0.0291</td></tr><tr><td>0.25</td><td>0.002</td><td>0.0275</td><td>0.0070</td></tr><tr><td>0.25</td><td>0.008</td><td>0.0425</td><td>0.0158</td></tr><tr><td>0.25</td><td>0.016</td><td>0.0722</td><td>0.0297</td></tr><tr><td>1.00</td><td>0.002</td><td>0.1051</td><td>0.0234</td></tr><tr><td>1.00</td><td>0.008</td><td>0.1097</td><td>0.0276</td></tr><tr><td>1.00</td><td>0.016</td><td>0.1265</td><td>0.0352</td></tr></tbody></table>

<table><caption>**Table 3.** Ephemeris errors for various values of the scale error. The columns $d$ and $t_0$ correspond to the second (distances $d$) and third (the instants of apparent encounters $t_0$) types of observations, respectively. See also the explanations for Table 2.</caption><thead><tr><th rowspan="2">Error of<br>scale<br>σ<sub>sc</sub></th><th colspan="4">Mean rms values [arcsec]</th></tr><tr><th>Along</th><th>d</th><th>Across</th><th>Along</th><th>t<sub>0</sub></th><th>Across</th></tr></thead><tbody><tr><td>0</td><td>0.0035</td><td></td><td>0.0011</td><td>0.0284</td><td></td><td>0.1463</td></tr><tr><td>0.0005</td><td>0.0049</td><td></td><td>0.0022</td><td>0.0282</td><td></td><td>0.1472</td></tr><tr><td>0.0010</td><td>0.0076</td><td></td><td>0.0041</td><td>0.0283</td><td></td><td>0.1472</td></tr><tr><td>0.0020</td><td>0.0139</td><td></td><td>0.0080</td><td>0.0283</td><td></td><td>0.1472</td></tr><tr><td>0.0040</td><td>0.0272</td><td></td><td>0.0158</td><td>0.0285</td><td></td><td>0.1470</td></tr><tr><td>0.0080</td><td>0.0540</td><td></td><td>0.0316</td><td>0.0286</td><td></td><td>0.1465</td></tr></tbody></table>

of measurements of mutual distances slightly increases the errors. If the instants of apparent encounters are measured, the ephemeris turn out to be the least precise.

In the absence of scale errors, the use of the instants of mutual encounters gives less precise ephemerides than in the case when mutual distances are measured. The introduction of scale errors increases the ephemeris errors if the measurements of mutual distances are used. A remarkable effect is achieved when the instants of mutual encounters are taken as measured values: the ephemeris precision does not depend on the scale errors at all. If the relative scale error exceeds 0.004, the ephemeris error obtained using the measurements of mutual distances exceeds that of the ephemeris generated on the basis of measurements of the instants of mutual encounters, so that the use of this type of observation proves to be effective.

When systematic errors of frame orientation are present, the ephemerides generated on the basis of the differences in satellites' coordinates become less precise.
In the case of the measurement of differences in satellite coordinates we have not performed an analysis of the dependence of ephemeris precision on the scale errors. This relationship is similar to the case of mutual distance measurement.

## 7 THE REALISTIC ORDER OF FITTING THE SATELLITE MOTION PARAMETERS

Since the discovery of the Galilean satellites of Jupiter in 1610, many observations have been accumulated. New observations are carried out regularly, replenishing the data already available. This allows the parameters of motion to be refined and new, more precise ephemerides of these satellites to be generated. It would be interesting to ascertain the benefits that the measurements of mutual encounter instants could provide in this situation.

In order to imitate the realistic procedure of fitting the orbital parameters of Jupiter's Galilean satellites, we artificially composed series of observations. Suppose that we have new observations that we can add to those already available with the aim of refining the model of motion and satellite ephemeris on the basis of upgraded observational data. Let us take several adjacent time intervals. At the first one, we have earlier observations made by many observers. In our model, we use an interval of 57 300 d, which is about 157 yr. For this interval, we will simulate observations of the Galilean satellites, providing measurements of the apparent coordinate differences X, Y for satellite pairs. We consider two versions of such data. In the first one, the observations are simulated with a time-step of 100 d, so that we have 573 observations for each pair. In the second one, the observations have a time-step of 50 d, so that there are 1146 in total. In both cases, the random errors with the probability density distributed according to the normal (Gaussian) law and given rms of 0.05 arcsec were added to the exact model values of the measured coordinates.

Then follows the time interval containing new observations that we want to add to the existing data base to update the ephemeris. This interval, as in the previous section, has a duration of 600 d. Directly after it follows a 1000-d interval at which the new ephemerides are computed. Suppose that the new observations give us mutual distances for all six pairs of satellites. From these measurements we choose groups that have apparent encounters of satellites. The measurements of mutual distances in any case are separated by 0.005-d intervals. To these measurements we also add random errors with rms values of 0.05 arcsec. All new observations have the same characteristics as described in the previous section.

Now we have a choice: either to use directly the series of mutual distances *d* to update the parameters of satellite motions or to determine for each interval the instant *t*₀ when the distance is minimal and to use these data instead of the mutual distances *d*. The choice is to be made in the situation when the measured distances have a scale error so that the real scale differs from its expected value by a factor of (1 + ε<sub>sc</sub>), which is assumed to be constant within the time interval of one mutual encounter but is different for other intervals. We also model the error ε<sub>sc</sub> as a random variable with a given rms value of σ<sub>sc</sub>. Let us carry out calculations with different values of σ<sub>sc</sub>, computing each time the ephemeris precision both across and along the apparent trajectory of the satellite relative to the planet. For each instant of time and for each sample of the Monte Carlo method, we generate the ephemeris and then calculate the rms value of deviations from the reference ephemeris using all samples for all four satellites. As an indicator of precision, we take the average of these estimations over all instants of the last 100 d of the interval following that of the new observations, that is, from the 900th to the

<table><caption>**Table 4.** Ephemeris errors for various values of scale errors in the case for which the preceding observations follow with a step of 100 d 100 d (573 observations). The columns *d* and *t*₀ correspond to the second (distances *d*) and the third (the instants of apparent encounters *t*₀) types of observations, respectively. See also the explanations in the legend of Table 2.</caption><thead><tr><th rowspan="2">Scale error<br>σ<sub>sc</sub></th><th colspan="4">Mean errors of ephemeris [arcsec]</th></tr><tr><th>Along</th><th>d</th><th>Across</th><th>t<sub>0</sub></th></tr></thead><tbody><tr><td>0</td><td>0.0013</td><td>0.0008</td><td>0.0032</td><td>0.0017</td></tr><tr><td>0.0002</td><td>0.0014</td><td>0.0010</td><td>0.0032</td><td>0.0017</td></tr><tr><td>0.0004</td><td>0.0018</td><td>0.0013</td><td>0.0032</td><td>0.0017</td></tr><tr><td>0.0008</td><td>0.0027</td><td>0.0021</td><td>0.0032</td><td>0.0017</td></tr><tr><td>0.0016</td><td>0.0049</td><td>0.0040</td><td>0.0032</td><td>0.0017</td></tr><tr><td>0.0032</td><td>0.0096</td><td>0.0079</td><td>0.0032</td><td>0.0017</td></tr></tbody></table>

<table><caption>**Table 5.** Ephemeris errors for various values of scale errors in the case for which the preceding observations follow with a step of 50 d (1146 observations). The columns *d* and *t*₀ correspond to the second (distances *d*) and the third (the instants of apparent encounters *t*₀) types of observations, respectively. See also the explanations in the legend of Table 2.</caption><thead><tr><th rowspan="2">Scale error<br>σ<sub>sc</sub></th><th colspan="4">Mean errors of ephemeris [arcsec]</th></tr><tr><th>Along</th><th>d</th><th>Across</th><th>t<sub>0</sub></th></tr></thead><tbody><tr><td>0</td><td>0.0012</td><td>0.0007</td><td>0.0023</td><td>0.0012</td></tr><tr><td>0.0002</td><td>0.0013</td><td>0.0008</td><td>0.0023</td><td>0.0012</td></tr><tr><td>0.0004</td><td>0.0015</td><td>0.0011</td><td>0.0023</td><td>0.0012</td></tr><tr><td>0.0008</td><td>0.0023</td><td>0.0017</td><td>0.0023</td><td>0.0012</td></tr><tr><td>0.0016</td><td>0.0040</td><td>0.0032</td><td>0.0023</td><td>0.0012</td></tr><tr><td>0.0032</td><td>0.0070</td><td>0.0063</td><td>0.0023</td><td>0.0012</td></tr></tbody></table>

1000th day. Like all evaluations of the ephemeris precision, these estimations will characterize the ephemeris precision both along and across the satellites' apparent trajectories relative to the planet.

The results of these calculations for the variants in which the preceding 573 observations are simulated with a step of 100 d are given in Table 4. Table 5 gives the results for the variants in which the preceding 1146 observations are simulated with a step of 50 d.

Comparison of the results presented in the two tables reveals that the ephemeris errors depend on the ratio of the number of new observations to that of the observations made earlier. The greater the number of earlier observations, the greater the weight they have in the data set as a whole, reducing in this way the influence of newer observations.

It should be noted that the above-described process of evaluating the ephemeris precision is based on the type of observation. Hence, in fitting the satellites' orbital parameters to real observations, the evaluations of the precision will be different. The conclusions based on the results presented in this paper concern only the evaluations of the precision of ephemerides generated from observations of the three types considered here.

## 8 CONCLUSIONS

We simulated observations and refined ephemerides by modelling the entire process of ephemeris production using the Monte Carlo statistical method. We did this in order to evaluate the dependence of ephemeris precision on three types of ground-based observations.

We found that, when simulating additional frame scale errors, the increase of these errors from zero to a certain sufficiently large limit
does not affect the ephemeris precision if the instants of satellite
apparent encounters are used as new observations. For sufficiently
small scale errors, however, it is better to fit the satellite motion
parameters on the basis of the measurements of mutual distances,
because in this case more precise ephemerides are obtained.

The proposal by Morgado et al. (2016) to use the instants of
apparent encounters of satellites instead of measuring the mutual
distances requires critical evaluation. If we are sure that, in addition
to the errors of measuring the distances themselves, there are no
additional scale errors in the frame, the direct measurements of the
distances are to be used to obtain the best ephemeris precision.
However, if there are scale errors, the use of the instants of mutual
encounters can result either in an improvement of the precision or in
its deterioration. Deciding which method to use in individual cases
requires knowledge of scale errors and additional estimations of the
ephemeris precision similar to those presented in this paper.

ACKNOWLEDGEMENTS

This work was supported by the Russian Foundation for Basic
Research, project no. 16-52-150005-CNRS-a. The author thanks

the anonymous reviewer, whose comments helped to improve this
paper.

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