The three inner Galilean satellites of Jupiter—Io, Europa, and Ganymede—are observed to move in a particular dynamical configuration, which is commonly known as the Laplace resonance. These satellites are characterized by a 2:1 ratio between the mean longitudes of Io-Europa and Europa-Ganymede. Another dynamical configuration, known as the de Sitter resonance, occurs when the longitude of Ganymede is fixed, instead of rotating like in the Laplace resonance. Besides studying the Laplace and de Sitter resonances, we also consider their generalizations to the case in which the mean longitudes of the first two satellites are in a ratio k:j, while those of the second and third satellites are in a ratio m:n with k,j,m,nZ+ and |jk|, |nm|2. We derive a model apt to describe such resonant configurations. We make an extensive study of the structural stability of the resonances; we show that the libration of the Laplace resonant angle is deeply affected by small variations of some quantities, most notably the semimajor axes and the oblateness. A remarkable result is that in several cases, the standard Laplace resonance of the Galilean satellites displays a regular behavior in comparison to other resonances characterized by different mean longitude ratios, which instead show a rather chaotic behavior even on short time scales. This result provides a motivation to support why the Galilean satellites are found in the actual Laplace resonance.

1.
J.
Lieske
, “
Galilean satellites ephemerides E5
,”
Astron. Astrophys. Suppl. Ser.
129
,
205
217
(
1997
).
2.
F.
Paita
,
A.
Celletti
, and
G.
Pucacco
, “
Element history of the Laplace resonance: A dynamical approach
,”
Astron. Astrophys.
617
,
A35
(
2018
).
3.
P.-S.
Laplace
, “Traité de mecanique celeste, 4,” in Paris: Crapelet; Courcier; Bachelier, 1805; 1 v. ; in 4.; DCC.f.80 (Crapelet; Courcier; Bachelier, Paris, 1805).
4.
R.
Sampson
, “
Theory of the four great satellites of Jupiter
,”
Memoirs R. Astron. Soc.
63
,
1
270
(
1921
).
5.
B.
Marsden
, “The motions of the Galilean satellites of Jupiter,” Ph.D. thesis (Yale University, University Microfilms, Inc., Ann Arbor, MI, 1966), Chap. X, pp. 65–92.
6.
S.
Ferraz-Mello
, “
Problems on the Galilean satellites of Jupiter
,”
Celestial Mech.
12
,
27
37
(
1975
).
7.
J.
Henrard
, “
Libration of Laplace’s argument in the Galilean satellites theory
,”
Celestial Mech.
34
,
255
262
(
1984
).
8.
R.
Malhotra
, “
Tidal origin of the Laplace resonance and the resurfacing of Ganymede
,”
Icarus
94
,
399
412
(
1991
).
9.
A.
Showman
and
R.
Malhotra
, “
Tidal evolution into the Laplace resonance and the resurfacing of Ganymede
,”
Icarus
127
,
93
111
(
1997
).
10.
J.
Lieske
, “
Theory of motion of Jupiter’s Galilean satellites
,”
Astron. Astrophys.
56
,
333
352
(
1977
).
11.
S.
Musotto
,
F.
Varadi
,
W.
Moore
, and
G.
Schubert
, “
Numerical simulations of the orbits of the Galilean satellites
,”
Icarus
159
,
500
504
(
2002
).
12.
V.
Lainey
,
L.
Duriez
, and
A.
Vienne
, “
New accurate ephemerides for the Galilean satellites of Jupiter (I)
,”
Astron. Astrophys.
420
,
1171
1183
(
2004
).
13.
G.
Kosmodamianskii
, “
Numerical theory of the motion of Jupiter’s Galilean satellites
,”
Solar Syst. Res.
43
,
465
474
(
2009
).
14.
W.
de Sitter
, “
Jupiter’s Galilean satellites
,”
Mon. Not. R. Astron. Soc.
91
,
706
738
(
1931
).
15.
F.
Tisserand
, Traité de mécanique céleste (Gauthier Ed., 1896), Chap. X.
16.
C.
Yoder
, “
How tidal heating in Io drives the Galilean orbital resonance locks
,”
Nature
279
,
767
770
(
1979
).
17.
R.
Greenberg
, “
Tidal evolution of the Galilean satellites: A linearized theory
,”
Icarus
46
,
415
423
(
1981
).
18.
R.
Greenberg
, “Orbital evolution of the Galilean satellites,” in Satellites of Jupiter (University of Arizona Press), pp. 65–92.
19.
J.
Henrard
, “
Orbital evolution of the Galilean satellites: Capture into resonance
,”
Icarus
53
,
55
67
(
1983
).
20.
C.
Yoder
and
S.
Peale
, “
The tides of Io
,”
Icarus
47
,
1
35
(
1981
).
21.
G.
Lari
, “
A semi-analytical model of the Galilean satellites’ dynamics
,”
Celest. Mech. Dyn. Astron.
130
,
50
(
2018
).
22.
H. W.
Broer
and
L.
Zhao
, “
De Sitter’s theory of Galilean satellites
,”
Celest. Mech. Dyn. Astron.
127
,
95
119
(
2017
).
23.
H. W.
Broer
and
H.
Hanßmann
, “
On Jupiter and his Galilean satellites: Librations of de Sitter’s periodic motions
,”
Indagat. Math.
27
,
1305
1336
(
2016
).
24.
A.
Celletti
,
F.
Paita
, and
G.
Pucacco
, “
The dynamics of the de Sitter resonance
,”
Celest. Mech. Dyn. Astron.
130
,
15
(
2018
).
25.
C. D.
Murray
and
S. F.
Dermott
, in Solar System Dynamics (Cambridge University Press, Cambridge, UK, 1999), ISBN: 0-521-57295-9 (hc.), ISBN: 0-521-57297-4 (pbk.).
26.
See https://naif.jpl.nasa.gov/naif/toolkit.html for information about the NASA SPICE toolkit.
27.
C.
Froeschlé
,
E.
Lega
, and
R.
Gonczi
, “
Fast Lyapunov indicators. Application to asteroidal motion
,”
Celest. Mech. Dyn. Astron.
67
,
41
62
(
1997
).
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