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\vfill\eject
\centerline{\bf New Constants for Sampson-Lieske Theory of the}
\centerline{\bf Galilean Satellites from mutual occultations data}
\vskip.5cm
\centerline{R. Vasundhara$^2$, J.E.Arlot$^1$, P.Descamps$^1$}
\vskip.2cm
\centerline{$^1$ Bureau des Longitudes, URA 0707 CNRS,}
\centerline{77 avenue Denfert-Rochereau, F. 75014 Paris}
\vskip.2cm
\centerline{$^2$ Indian Institute of Astrophysics}
\centerline{Bangalore 560034, India}
\vskip.5cm
\centerline{\bf Abstract}
\vskip.2cm
New constants for Sampson-Lieske theory of the Galilean
Satellites have been derived using 6360 individual photographic
positions (1891-1990) and 438 pseudo astrometric positions from
mutual occultations during 1973, 1979, 1985 and 1991.  Using
these new sets of constants, although no significant improvement
is noticed in the O-C values of the sky plane co-ordinates
compared to G5 and E 2 $\times$ 3, residuals in longitude for Io
and Euyropa are found to improve.   Problems concerning the
inclusion of mutual event data in attempting evaluation of
secular variations of the mean motions of the satellites are
discussed. 
\vskip.5cm
\noindent Keywords : Galilean Satellites - Mutual occultations -
ephemerides - Jupiter
\vfill\eject

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\noindent {\bf 1. Introduction}
\vskip.2cm
The constants of motion of the Galilean Satellites are now 
efficiently updated using Lieske's (1974, 1977) technique by computing
the corrections ($\varepsilon, \beta$) to the basic sets of constants.
Lieske progressively updated these constants from E1 (Lieske,
1978) using the eclipse data of  1878-1903 to E2 (Lieske, 1980)
by supplementing visual eclipses between 1903 and 1972, photographic
data between 1967 and 1978, and mutual events of 1973.  The E2x3
(Lieske, 1987) ephemeris was derived by further additions of more
mutual event pairs from 1973 and 1979, 183 pairs of data from
Voyager optical navigation images and 15711 classical eclipses from
1652-1983.  Arlot (1982, hereafter paper I) used 8856 individual
photographic observations between 1891 to 1978 to derive the G5
ephemeris.  The present investigations relate to determinations
of a new sets of constants using photographic data between 1891
to 1990 and astrometric positions derived from mutual
occultations during 1973, 1979, 1985 and 1991.
\vskip.4cm
\noindent {\bf 2. The observational material}
\vskip.2cm
\noindent {\bf 2.1 The photographic data}
\vskip.2cm
Most of the photographic data that were used in paper I. were
utilized in this study.  Additional excellent unpublished photographic
material covering the period between 1986 and 1990 (Pascu, 1992)
have been included.  Table 1 gives the details of the photographic
observations.
\vskip.4cm
\noindent {\bf 2.2 The mutual occultation data}
\vskip.2cm
Two sets of mutual occultation data were used.  The data set (1)
contains the astrometric positions derived by Descamps and
Thuillot 
(1993), and Descamps (1992, 1994) from the mutual occultation light
curves of 1985 and 1991 assuming Hapke's light scattering law
for a macroscopically rought surface (Hapke, 1984).  The data
set (2) consists of astrometric positions derived from the mutual
occultation light curves of 1985 (Arlot et al. 1992) and 1991
(Arlot et al, 1991) 
using the model by Vasundhara (1993).  While Lommel-Seeliger's
law was considered adequate for the light curves of Io during
1991, the Hapke's parameters derived by Descamps and Thuillot were (1992) 
used for rest of the light curves.  The published astrometric positions
of 1985 (GSO and Franklin, 1991) which were derived using
Lambert's law were reconstructed to adopt Hapke's law and included
in Set (2).  Both sets (1) and (2) contain reconstructed
values of the published positions of 1973 (Aksnes and
Franklin 1976) and 1979 (Aksnes et al. 1984), to account for 
the phase corrections (Aksnes, Franklin and Magnusson 1986).  
Tables 2(a \& b) give the statistics of the events for these 
two data sets.
\vskip.4cm
\noindent {\bf 3. Corrections to the constants}
\vskip.2cm
The method developed in paper I has been used in the present
study.  Since the mutual occultations provide relative position
of the two satellites, positions of the satellites in the
photographic data set were also considered with respect of their
center of mass.  If Ci (i = 1, 29 for $\varepsilon$, and i = 1 to 22 for $\beta$)
are the constants of motion, Q$_o$ and Q$_c$ the observed
and calculated sky plane positions X or Y, and $ \frac {\partial
Q} {\partial C_i}$ the partial derivatives of these observables
with respect to C$_i$ we have

$$Q_o - Q_c = \Sigma_i  \frac {\partial Q} {\partial
C_i} \triangle C_i \eqno$$

The inherent accuracies of the photographic observations with
long focal length instruments were included as in Paper I in an 
implicit manner as by considering the angular distances in microns
instead of arcsecs.   The weights of observations of
Helsingfors (1891-1897) and Pulkovo (1895-1898, 1904-1910) were
increased twofolds since each of their observations
corresponds to several exposures on a plate.  The mutual
occultations have residuals of about 0.02 arcsec and therefore were 
assigned weights of 50. The constants that were updated were the
mean motions ($\varepsilon_6 - \varepsilon_8$),  primary eccentricities
($\varepsilon_{16} - \varepsilon_{19})$, primary sine inclinations 
($\varepsilon_{21} - \varepsilon_{24})$, mean longitudes ($\beta_1, \beta_2, \beta_4$), 
proper perijoves ($\beta_6 -\beta_9)$, proper nodes ($\beta_{11} - \beta_{14}$),
libration amplitude ($\epsilon_9$) and the libration
phase angle ($\beta_5$).  The starting ephemeris for the initial run was E1,
three more iterations were carried out by progressively adjusting the
($\epsilon, \beta)$ values.  The resulting set of ($\epsilon, \beta$)
values labled I-32 and another set of constants I-33 derived 
by assigning a weight of 10 to the mutual occultation data sets are given in
table 3 along with G5 for comparison.

The (O-C) in the sky plane co-ordinates of the four satellites
from their centre of masss are shown in Fig.1 (a-b) using I32, 
Fig.2 (a-b) using I33, Fig.3 (a-b) using G5 and Fig.4 (a-b) using 
E2$\times$3.  The r.m.s. of the residuals in the sky plane co-ordinates
calculated using the expression.
\vskip.4cm

$$\sigma = \sqrt {\frac {\sum^N_{i = 1} \sum^4_{j = i} [( \times^j_i)^2 
+ (Y_i^j)^2} {2 \times 4 \times N}}\eqno$$


are given in Tables [1,2 (a \& b)] for the four ephemerides, 
where j refers to the satellites and i the running index of the data point.
\vskip.4cm
\noindent {\bf 4. (O - C) in longitude}
\vskip.2cm
The error in the longitude of the satellite using G5, I32 and E2$\times$2
ephemerides are given table 4.  The residuals in longitudes were
computed by considering (i) only the photographic (Ph) data (ii)
Photographic and mutual event data (Ph+ME) weighted as 50.   The errors in 
for Callisto have increased using I32 because of limited  number of mutual
event observations for this satellites.
 
 The (O-C) of the individual data point from 1891 through 1991 were 
 analysed to look for co-efficient of the T$^2$ terms.  The derived
 values of $ \dot n /n $ using I32, I33, G5 and E2$\times$3 are given in
 Table 5.  Since the weight for the ME data set for determination
 of I33 was set to 10, only the same has been used for determining
 the $ \dot n /n $ values.
 \vskip.4cm
 \noindent {\bf 5. Results and discussions}
 \vskip.2cm
 The (O-C) in the sky plane, co-ordinates of the  satellites in 
 Figs (1-4) and Tables (2a \& b) clearly justify higher weights
 of 50 for the ME data set.  Effects of inclusion of the ME data set lowers
 the longitude residual compared to G5 and E2$\times$3 for J1 and J3 (Table 4). 
 The new sets of constants in I32 produce significantly lower residual for J1 
 but not for the other satellites.   A curious trend for J2 is
 noticed in the estimates of $ \dot n_2 /n_2$. G5, derived from  most of the photographic data
 of the present analysis sans the ME data sets leads to a positive
 value for $ \dot n_2 /n_2$ whereas I32 and I33 derived after including
the  ME data indicate 
 decelaration of Europa (Table 5).  Again, both G5 and E2$\times$3 ephermerides yield
 progressively negative values for $ \dot n_2 /n_2$ when the weight of ME data set is 
 increased. Excellent photographic data by Pascu during 1974-1990 with residuals
 only 3-5 times that of the ME but outnumbering in number and
spanning almost the same period yield a positive $ \dot n_2 /n_2 $. This apparent 
 deceleration of J2 when ME data set is used may therefore be
due to of the
 unexplained progressively increasing longitude residuals of -12 km, -95 km,
 - 211 km and -308 km during 1973, 1979, 1985 and 1991 respectively (Mallama,
 1992). Predominance of the J1-J2 events and limited longitude coverage
 during a mutual event season preclude reliable estimation of the secular
 changes in the mean motion of the Galilean Satellites.  A larger data set 
 consisting of the mutual eclipses and the data from the forthcoming
 mutual event season in 1996-1997 could help in resolving this apparent 
 deceleration of Europe in the ME data set.
\vskip.4cm
\noindent {\bf Acknowledgements. }
\vskip.2cm
We would like to thank Dr.J.H.Lieske from JPL for the computer codes for
calculating the revised Sampson theory.  We are indebted to Dr.D.
Pascu from USNO for allowing us to use his yet unpublished excellent data
of 1986-1990.

\vskip 1.5cm
\noindent {\bf Fig. Captions:  } ( O -- C ) in 
$ \Delta \alpha \cos \delta (a) $,  $\Delta \delta (b) $ relative to the
center of mass of the observed satellites. The filled triangles and crosses
correspond to photographic and mutual event data sets respectively.

\vfill
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\centerline{\bf References}
\item{} Aksnes, K., Franklin, F., Millis, R., Birch, P., Blanco, C.,
Catalano, S., Piironen, J. 1984, {\it Astron. J.} {\bf 89}, 280. 
\item{} Aksnes, K., Franklin, F., Magnusson, P. 1986, {\it Astron. J.} {\bf
92}, 1436. 
\item{} Arlot, J.E., 1982, {\it Astron. Astrophys.} {\bf 107}, 305.
\item{} Arlot et al. 1991, {\it Data bank of collections of mutual 
events light curves of 1990 -- 1991 through PHEMU 91 }.
\item{} Arlot, J.E., Thuillot, W., Barroso J. Jr, Bagenal, L., Blanco, C. et al
 1992, {\it Astron. Astrophys. Suppl. Ser.} {\bf 92}, 151.
\item{} Decamps, P. 1992a, Ph.D. Thesis.
\item{} Decamps, P. 1994, ICARUS.
\item{} Descamps, P., and Thuillot, W., 1993, {\it Icarus}, in press.
\item{} GSO \& Franklin, F., 1991, {\it Astron. J.} {\bf 102}, 806.
\item{} Hapke, B.W. 1984, {\it Icarus} {\bf 59}. 41.
\item{} Lieske, J.H., 1974, {\it Astron. Astrophys.} {\bf 31}, 137.
\item{} Lieske, J.H. 1977, {\it Astron. Astrophys.} {\bf 56}, 333.
\item{} Lieske, J.H. 1978, {\it Astron. Astrophys.} {\bf 65}, 83.
\item{} Lieske, J.H. 1980, {\it Astron. Astrophys.} {\bf 82}, 340.
\item{} Mallama, A., 1992, {\it Icarus} {\bf 95}, 309.
\item{} Pascu, 1992, Private Communication.
\item{} Vasundhara, R. 1993  {\it Ph.D Thesis}
\item{} Vasundhara, R. 1994 {\it Astron. Astrphys.}{\bf 281}, 565.
\vfill\eject

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\nopagenumbers
\centerline{\bf Table 3(a) $\varepsilon$ values for I32 and I33 
and G5 ephemerides$^1$}
\vskip.5cm
\settabs\+{$\varepsilon$0000} &{0.1944073 $\pm$ 0.070235.10$^{-8}$00000}
&{0.2113185 $\pm$.10$^{-8}$00000} &{-0.1835945 $\pm$ 
0.08513.10$^{-8}$00} \cr
\vskip.2cm\hrule\vskip.2cm
\+$\epsilon$  & \ \ \ \quad I 32  & \ \ \quad I 33  &\ \ \ \quad G 5  \cr
\vskip.2cm\hrule\vskip.2cm
\+1 &(0.041590 $\pm$ 0.0098) & & \cr
\+2 &(-0.002461 $\pm$ 0.0200) & & \cr
\+3 &(-0.231890 $\pm$ 0.0076) & & \cr
\+4 &(0.256938 $\pm$ 0.0084) & & \cr
\+5 &($\{$2.067082138 $\pm$ 0.095$\}$.10$^{-4}$) & & \cr
\+6 &$\{$0.1944073 $\pm$ 0.070235$\}$.10$^{-8}$ &$\{$0.2113185 $\pm$ $\}$.10$^{-8}$ 
&$\{$-0.1835945 $\pm$ 0.08513$\}$.10$^{-8}$ \cr
\+7 &$\{$1.0642537 $\pm$ 0.044510$\}$.10$^{-8}$ &$\{$0.11012828 $\pm$$\}$.10$^{-8}$ 
&$\{$-0.1054947 $\pm$ 0.05523$\}$.10$^{-8}$ \cr
\+8 &$\{$-4.3848824 $\pm$ 0.122708$\}$.10$^{-8}$ &$\{$-4.4122061 $\pm$$\}$.10$^{-8}$ 
&$\{$-5.5560148 $\pm$ 0.15354$\}$.10$^{-8}$ \cr
\+9 &$\{$14.0045547 $\pm$ 0.644048$\}$.10$^{-4}$ &$\{$14.1673386 $\pm$$\}$.10$^{-4}$ 
&$\{$14.9959 $\pm$ 0.086918$\}$.10$^{-4}$ \cr
\+10 &($\{$0.1620 $\pm$ 0.0633.10$\}$$^{-4}$) & & \cr
\+11 &(-0.007778 $\pm$ 0.00054) & & \cr
\+12 &(-0.275934 $\pm$ 0.024) & & \cr
\+13 &(-0.000308 $\pm$ 0.0006) & & \cr
\+14 &($\{$0.0950 $\pm$ 2.80$\}$.10 $^{-4}$) & & \cr
\+15 &(0.00 $\pm$ 0.15) & & \cr
\+16 &-0.7697975 $\pm$ 0.070239 &-0.8137604 $\pm$ &-0.473172 $\pm$ 0.098801 \cr
\+17 &-1.0132804 $\pm$ 0.060432 &+0.8728083 $\pm$ &+0.463707 $\pm$ 0.108628 \cr
\+18 &-0.0547743 $\pm$ 0.0033858 &-0.0545148 $\pm$ &-0.043680 $\pm$ 0.004671 \cr
\+19 &-0.0019453 $\pm$ 0.000415 &-0.0017829 $\pm$ &-0.003586 $\pm$ 0.000542 \cr
\+20 &(0.002781 $\pm$ 0.0005) & & \cr
\+21 &+0.4601799 $\pm$ 0.040741 &+0.4752529 $\pm$ &+0.492066 $\pm$ 0.060640 \cr
\+22 &-0.0093312 $\pm$ 0.001389 &-0.0092887 $\pm$ &-0.006578 $\pm$ 0.002110 \cr
\+23 &+0.0484492 $\pm$ 0.002269 &-0.0485697 $\pm$ &+0.050194 $\pm$ 0.003248 \cr
\+24 &-0.0784726 $\pm$ 0.000910 &-0.0784863 $\pm$ &-0.070668 $\pm$ 0.001315 \cr
\+25 &(0.005384 $\pm$ 0.011) & & \cr
\+26 &(-0.000133 $\pm$ 0.0011) & & \cr
\+27 &($\{$0.0 $\pm$ 0.0047$\}$.10$^{-4}$) & & \cr
\+28 &($\{$0.0 $\pm$ 0.0010$\}$.10$^{-4}$) & & \cr
\vskip1cm
$^1$ Parameters within the paranthesis were not determined.
\vfill\eject
\centerline{\bf Table 3(b) $\beta$ values for I 32, I 33 and G 5 
ephemerids$^1$}
\vskip.5cm
\settabs\+{$\beta$0000} &{0.1944073 $\pm$ 0.070235.10$^{-8}$00000}
&{0.2113185 $\pm$.10$^{-8}$00000} &{-0.1835945 $\pm$ 
0.08513.10$^{-8}$00} \cr
\vskip.2cm\hrule\vskip.2cm
\+$\beta$  &\ \ \ \quad I 32  &\ \ \quad I 33  &\ \ \ \quad G 5 \cr
\vskip.2cm\hrule\vskip.2cm
\+1 &(0.042583 $\pm$ 0.004717 &0.046863 &0.046245 $\pm$ 0.04819 \cr
\+2 &-0.014971 $\pm$ 0.000945 &-0.013741 &-0.014445 $\pm$ 0.000877 \cr
\+3 &\quad $-$ &\quad $-$ &\quad $-$ \cr
\+4 &-0.068665 $\pm$ 0.000273 &-0.068782 &-0.066536 $\pm$ 0.000443 \cr
\+5 &182.841072 $\pm$ 0.003566 &182.824371 &182.797225 $\pm$ 0.005084 \cr
\+6 &81.1687813 $\pm$ 21.2911 &58.811159 &62.231560 $\pm$ 7.608520 \cr
\+7 &86.778098 $\pm$ 2.060911&87.679983  
&86.403965 $\pm$ 4.320290 \cr
\+8 &14.487234 $\pm$ 0.185102 &14.621422  &14.181513 $\pm$ 0.236000 \cr
\+9 &-0.689518 $\pm$ 0.023084 &-0.698253 &-0.747260 $\pm$ 0.028121 \cr
\+10 &(0.166755 $\pm$ 0.05) & & \cr
\+11 &62.922453 $\pm$ 1.529083  &63.057544 &60.834877 $\pm$ 3.490252 \cr
\+12 &5.785080 $\pm$ 0.084104 &5.827361  &6.075500 $\pm$ 0.125467 \cr
\+13 &-5.423444 $\pm$ 0.128188 &-5.305660 &-5.450563 $\pm$ 0.210661 \cr
\+14 &4.468647 $\pm$ 0.055347 &4.447649 &+4.903778 $\pm$ 0.077832 \cr
\+15 &(-0.233589 $\pm$ 0.19 & & \cr
\+16 &(0.0 $\pm$ 0.4   & & \cr
\+17 &(-0.140391 $\pm$ 0.03) & & \cr
\+18 &(15.610 $\pm$ 3.1 & & \cr
\+19 &(5.135 $\pm$ 4.2) & & \cr
\+20 &(-1.719 $\pm$ 2.1) & & \cr
\+21 &-7.784 $\pm$ 1.8) & & \cr
\+22 &(0.4428 $\pm$ 0.007 & & \cr
\vskip1cm
$^1$ Parameters within the paranthesis were not determined.
\vfill\eject
\centerline{\bf Table 1 : Details of photographic data}
\vskip.2cm\hrule\vskip.2cm
\settabs\+{Biesbrock00} &{Observatory} &{1977/19780}
&{2840} &{of00} &{posit0} &{3070} &{Residual} &{arcsec0}
&{0.11100} &{0.391} \cr
\+Observer &Observatory &\ \ Year &No. &of &positions & &Residual &(arcsec)  
& & \cr
\+ & & &\quad J1 &\quad J2 &\quad J3 &\quad J4 &\quad G-5 &I-32 &I-33 &E2$\times$3 \cr
\vskip.2cm\hrule\vskip.2cm
\+Renz &Helsingfors &1891 & & & & &0.117 & & & \cr
\+Renz &Helsingfors &1892-1897 &171 &175 &184 &174 & &0.110
&0.111 &0.114 \cr 
\+Renz &Pulkovo &1895-1898 & & & & & & & & \cr
\vskip.1cm
\+Balanovsky &Pulkovo &1904-1910 & & & & & & & & \cr
\+Chevalier &Zo-Se &1917/1918 &132 &132 &133 &121 &0.148 &0.151
&0.151 &0.158 \cr
\+De Sitter &Greenwich &1918/1919 & & & & & & & & \cr
\vskip.1cm
\+De Sitter &The Cape &1924 & & & & & & & & \cr
\+Petrescu &Bucarest  &1934 &85 &104 &106 &118 &0.217 &0.223
&0.222 &0.226 \cr
\+Petrescu &Paris &1936 & & & & & & & & \cr
\vskip.1cm
\+Van  &Yerkes &1961-1963 & & & & & & & & \cr
\+Biesbrock & & & & & & & & & & \cr
\+Gorel &Nicolaiev &1962-1966 & & & & & & & & \cr
\+Soelie &Bordeaux &1966-1974 & & & & & & & & \cr
\+Gorel &Nicolaiev &1973-1974 &284 &282 &318 &307 &0.390 &0.390
&0.390 &0.391 \cr
\+Debehogne &Rio de  &1977/1978 & & & & & & & & \cr
\+ &Janeiro & & & & & & & & & \cr
\+\quad ''   &Uccle &1977/1978 & & & & & & & & \cr
\+\quad ''   &La Silla &1977/1978 & & & & & & & & \cr
\vskip.1cm
\+Ianna  &Mc Cormick &1977/1978 &70 &100 &95 &108 &0.104 &0.104
&0.104 &0.107 \cr
\+et al & & & & & & & & & & \cr
\+Pascu &Mc Cormick &1967/1968 &88 &87 &89 &95 &0.095 &0.095
&0.095 &0.096 \cr
\+ \quad '' &USNO  &1973 &72 &65 &62 &67 &0.110 &0.111
&0.111  &0.112 \cr
\+ &(Wash.) & & & & & & & & & \cr
\+ \quad '' &\quad '' &1974 &107 &115 &120 &123 &0.082 &0.085
&0.085 &0.084 \cr
\+ \quad '' &\quad '' &1975-1977 &107 &116 &109 &119 &0.085 &0.086
&0.086 &0.086 \cr
\+\quad '' &\quad '' &1977/1978 &59 &59 &53 &59 &0.102 &0.103
&0.103  &0.104 \cr
\+Pascu$^1$ &\quad '' &1986 &84 &74 &82 &80 &0.077 &0.074 &0.074
&0.077 \cr
\+\quad '' &\quad '' &1987 &116 &133 &129 &133 &0.074 &0.071
&0.071 &0.074 \cr
\+\quad '' &\quad '' &1988/1989 &52 &54 &60 &60 &0.059 &0.056
&0.056 &0.061 \cr
\+\quad '' &\quad '' &1990 &79 &72 &84 &96 &0.068 &0.067 &0.067 
&0.069 \cr
\vskip.2cm\hrule\vskip.2cm
\noindent $^1$ Private communications
\vfill\eject
\centerline{\bf Table 2(a) Details of mutual occultation data set (1)}
\vskip.5cm
\settabs\+{1990-9100} &{Total00} &{J1000} &{No. of} &{posi00} 
&{J4000} &{Residual} &{(arcsec)0000} &{I330000} &{E2 $\times$ 3} \cr
\hrule\vskip.2cm
\+Year &Total &  &No. of  &posi- &  & &Residual &(arcsec) & \cr
\+ & & & &tions & & & & & \cr
\+ &events &\quad J1 &\quad J2 &\quad J3 &\quad J4 &\quad G5 &I32 &I33 &E2$\times$3 \cr
\vskip.2cm\hrule\vskip.2cm
\+1973 &21 &20 &21 &1 &0 &0.022 &0.019 &0.019 &0.018 \cr 
\vskip.2cm
\+1979 &3 &1 &3 &2 &0 &0.006 &0.008 &0.008  &0.010 \cr
\vskip.2cm
\+1985 &23 &11 &14 &15 &6 &0.019 &0.020 &0.020 &0.022 \cr
\vskip.2cm
\+1990-91 &34 &29 &34 &5 &0 &0.036 &0.029 &0.030 &0.032 \cr
\vskip.2cm\hrule\vskip.5cm
1. Source : Decamps (1992a)
\vskip2cm
\centerline{\bf Table 2(b) Details of mutual occultation data set(2)}
\vskip.5cm
\hrule\vskip.2cm
\+Year &Total &No. &of &posi- & & &Residuals &(arcsec) & & \cr
\+ & & & &tions & & & & & & \cr
\+ &events &J1 &J2 &J3 &J4 &G5 &I32 &I33 &E2$\times$3 \cr
\vskip.2cm\hrule\vskip.2cm
\+1973$^1$ &27 &25 &27 &2 &0 &0.011 &0.009 &0.009 &0.011 \cr 
\vskip.2cm
\+1979$^2$ &4 &3 &4 &1 &0 &0.012 &0.013 &0.013  &0.013 \cr
\vskip.2cm
\+1985$^3$ &16 &7 &9 &10 &6 &0.022 &0.022 &0.023 &0.021 \cr
\vskip.2cm
\+1985$^4$ &71 &52 &41 &56 &21 &0.025 &0.025 &0.026 &0.023 \cr
\vskip.2cm
\+1990-91$^{5,6}$ &20 &16 &19 &4 &1 &0.034 &0.027 &0.028 &0.032 \cr
\vskip.2cm\hrule\vskip.5cm
\noindent Sources :
\vskip.2cm
\item{1.} Aksnes and Franklin, 1976
\item{2.} Aksnes et al., 1984
\item{3.} Arlot et al., 1992
\item{4.} Franklin and GSO, 1991
\item{5.} Vasundhara, 1994
\item{6.} Arlot et al., 1995
\vfill\eject
\centerline{\bf Table 4 : Error in Longitude (min of time)}
\vskip.4cm
\hrule\vskip.2cm
\settabs\+{Data set00} &{0.060 $\pm$ 0.011000} &{0.009 $\pm$ 0.012000}
&{0.078 $\pm$ 0.015000} &{0.201 $\pm$ 0.02000} &{Ephem.00} \cr
\+Data set &\ \quad J1 &\ \quad J2 &\ \quad J3 &\ \quad J4
&Ephem. \cr
\vskip.2cm\hrule\vskip.2cm

\+ Ph &0.060 $\pm$ 0.011 &0.009 $\pm$ 0.012 &0.053 $\pm$ 0.015
&0.073 $\pm$ 0.020 &G5 \cr
\+Ph+ME &0.058 $\pm$ 0.010 &-0.021 $\pm$ 0.011 &0.041 $\pm$
0.014 &0.076 $\pm$ 0.019 &G5 \cr
\vskip.4cm
\+ Ph &0.019 $\pm$ 0.011 &0.039 $\pm$ 0.012 &0.078 $\pm$ 0.015
&0.201 $\pm$ 0.020 &I32 \cr
\+Ph+ME &0.018 $\pm$ 0.010 &0.017 $\pm$ 0.011 &0.067 $\pm$
0.014 &0.203 $\pm$ 0.019 &I32 \cr
\vskip.4cm
\+ Ph &0.041 $\pm$ 0.011 &0.057 $\pm$ 0.013 &0.165 $\pm$ 0.016
&0.042 $\pm$ 0.020 &E2$\times$3 \cr
\+Ph+ME &0.038 $\pm$ 0.010 &0.033 $\pm$ 0.011 &0.157 $\pm$
0.014 &0.035 $\pm$ 0.019 &E2$\times$3 \cr
\vskip.4cm
\+ Ph &\quad 1508 &\quad 1568 &\quad 1624
&\quad 1660  &Data points  \cr
\+Ph+ME &1508 + 153 &1568 + 170  &1624 + 88
&1660 + 27 &Data points \cr
\vskip.2cm\hrule
\vskip1cm
\centerline{\bf Table 5 : Secular accelerations of the Satellite Io-Ganymede}
\vskip.5cm
\hrule\vskip.2cm
\settabs\+{Data set0000000} &{0.060 $\pm$ 0.01100000} &{0.009 $\pm$ 0.01200000}
&{0.078 $\pm$ 0.01500000}  &{Ephemeris00} \cr
\+\ Data &\ \quad J1 &\ \quad J2 &\ \quad J3 
&Ephemeris \cr
\+ &$ \dot n_1/n_1 \times 10^{11}$ &$ \dot n_2/n_2 \times 10^{11}$
&$ \dot n_3/n_3 \times 10^{11}$ & \cr
\vskip.2cm\hrule\vskip.2cm

\+ Ph &24.69 $\pm$ 7.89 &-6.08 $\pm$ 9.29 &16.02 $\pm$ 11.15
&I32 \cr
\+Ph+ME (50)$^1$ &22.02 $\pm$ 7.48 &-25.26 $\pm$ 8.42 &10.56 $\pm$
10.64  &I32 \cr
\+Ph+ME(10)$^2$ &24.56 $\pm$ 7.62 &-7.04 $\pm$ 8.96 &15.77
$\pm$ 10.78 &I32 \cr
\vskip.4cm
\+Ph &24.28 $\pm$ 7.89 &-4.49 $\pm$ 9.29 &14.52 $\pm$ 11.15
&I33 \cr
\+Ph+ME(10) &24.13 $\pm$ 7.62 &-5.15 $\pm$ 8.96 &14.26 $\pm$
10.78  &I33 \cr
\vskip.4cm
\+Ph &27.34 $\pm$ 7.91 &11.11 $\pm$ 9.31 &3.89 $\pm$ 11.18
&G5 \cr
\+Ph+ME(50) &24.57 $\pm$ 7.29 &-12.69 $\pm$ 8.44 &-0.22 $\pm$
10.67  &G5\cr
\+Ph+ME(10) &27.19 $\pm$ 7.64 &9.91 $\pm$ 8.98 &3.69 $\pm$
10.08 &G5 \cr
\vskip.4cm
\+Ph &32.92 $\pm$ 7.94 &26.14 $\pm$ 9.35 &-28.86 $\pm$ 11.23
&E2$\times$3 \cr
\+Ph+ME(50) &29.69 $\pm$ 7.53 &-2.71 $\pm$ 8.48 &-30.66
$\pm$ 10.71 &E2$\times$3 \cr
\+Ph +ME(10) &32.75 $\pm$ 7.67 &24.72 $\pm$ 9.02 &-28.96
$\pm$ 10.85 &E2$\times$3 \cr
\vskip.2cm\hrule\vskip.2cm
\item{1.} Weight assigned for the mutual event set = 50
\item{2.} Weight assigned for the mutual event set = 10
\vfill
\eject

\end
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