New tidal paradigm in giant planets supported by rapid
orbital expansion of Titan
Val
́
ery Lainey
1
,
2
⇤
, Luis Gomez Casajus
3
, Jim Fuller
4
, Marco Zannoni
3
, Paolo Tortora
3
, Nicholas
Cooper
5
, Carl Murray
5
, Dario Modenini
3
, Ryan Park
1
, Vincent Robert
2
,
6
, Qingfeng Zhang
7
1
Jet Propulsion Laboratory, California Institute of Technology,
4800 Oak Grove Drive, Pasadena, CA 91109-8099, United States
2
IMCCE, Observatoire de Paris, PSL Research University, CNRS,
Sorbonne Universits, UPMC Univ. Paris 06, Univ. Lille, 77
3
Dipartimento di Ingegneria Industriale, Universit
`
a di Bologna, 47121 Forl
`
ı, Italy
4
TAPIR, Walter Burke Institute for Theoretical Physics
Mailcode 350-17, Caltech, Pasadena, CA 91125, USA
5
Queen Mary University of London, Mile End Rd, London E1 4NS, United Kingdom
6
IPSA, 63 bis boulevard de Brandebourg, 94200 Ivry-sur-Seine, France
7
Department of Computer Science, Jinan University, Guangzhou 510632, P. R. China
⇤
To whom correspondence should be addressed; E-mail: lainey@imcce.fr.
Copyright 2019. All rights reserved.
1
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Tidal effects in planetary systems are the main driver in the orbital migration of natural
satellites. They result from physical processes occurring deep inside celestial bodies, whose
effects are rarely observable from surface imaging. For giant planet systems, the tidal mi-
gration rate is determined by poorly understood dissipative processes in the planet, and
standard theories suggest an orbital expansion rate inversely proportional to the power
11
/
2
in distance
1
, implying little migration for outer moons such as Saturn’s largest moon,
Titan. Here, we use two independent measurements obtained with the
Cassini
spacecraft
to measure Titans orbital expansion rate. We find Titan migrates away from Saturn at
11
.
3
±
2
.
0
cm/year, corresponding to a tidal quality factor of Saturn of
Q
'
100
, and a mi-
gration timescale of roughly 10 Gyr. This rapid orbital expansion suggests Titan formed
significantly closer to Saturn and has migrated outward to its current position. Our re-
sults for Titan and five other moons agree with the predictions of a resonance locking tidal
theory
2
, sustained by excitation of inertial waves inside the planet. The associated tidal
expansion is only weakly sensitive to orbital distance, motivating a revision of the evolu-
tionary history of Saturns moon system. The resonance locking mechanism could operate
in other systems such as stellar binaries and exoplanet systems, and it may allow for tidal
dissipation to occur at larger orbital separations than previously believed.
Saturn is orbited by 62 moons, and the intricate dynamics of this complex system provide
clues about its formation and evolution. Of crucial importance are tidal interactions between the
moons and the planet. Each moon raises a tidal bulge in the planet, and because Saturn rotates
faster than the moons orbit, frictional processes within the planet cause the tidal bulge to lead in
front of each moon. Each moon’s tidal bulge pulls the moon forward such that it gains angular
momentum and migrates outward, similar to the tidal evolution of the Earth-Moon system.
However, in giant planets such as Saturn, the dissipative processes that determine the bulge lag
2
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angle and corresponding tidal migration timescale remain poorly understood.
Prior monitoring of the mid-sized inner moons’ orbital locations suggests that they are
migrating outward faster than allowed if they formed at the same time as Saturn
3,4
. These
observations motivated two new formation scenarios. One possibility is that Saturn’s rings have
viscously spread over time, steadily forming mid-sized moons at their outer edge defined by the
Roche limit
5
. Another possibility is that resonances between the orbits of Saturn’s mid-sized
moons and the pull of the Sun can lead to collisions within the satellite system, after which new
moons conglomerate from the debris disk
6
. While each scenario predicts different ages for the
satellites, these prior studies have assumed a constant tidal lag angle
✓
for each moon’s tidal
bulge, parameterized by a tidal quality factor
Q
'
1
/
✓
. The
Q
governing the tidal interaction
with each moon is inversely proportional to the tidal energy dissipation rate within Saturn
7
.
Denoting the semi-major axis
a
, the orbital expansion rate
t
1
tide
of each moon is
t
1
tide
=
̇
a
a
=
3
k
2
Q
M
moon
M
✓
R
a
◆
5
n,
(1)
where
M
moon
is the mass of the moon,
R
and
M
are the radius and mass of Saturn,
k
2
is the
Love number of degree two, and
n
=2
⇡
/P
orb
is the moon’s mean motion. Because of the
strong dependence on
a
, most tidal theories predict slower migration for outer moons such as
Titan.
To help explain the rapid migration of the mid-sized moons measured by
3,4
, a new
paradigm for the tidal evolution of moons, known as resonance locking, was proposed by
2
.
Tidal dissipation due to inertial waves in Saturn’s convective envelope
8
or gravity modes in
Saturn’s deep interior
9
is enhanced at discrete resonances with planetary oscillations. The res-
onant frequencies are determined by Saturn’s internal structure, which is slowly evolving due
to processes such as gravitational contraction, helium rain out
10
, and core erosion
11
. Moons
3
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can get caught in these resonances as Saturn’s structure evolves, causing the moons to migrate
outward on a timescale determined by Saturn’s internal evolution. While explaining the fast
orbital expansion of Rhea,
2
predicted a similar expansion rate (and smaller tidal Q) for Titan,
making the monitoring of Titan’s orbit a strong case for testing their model. Contrary to most
tidal theories where the tidal
Q
is constant for all moons, resonance locking predicts the tidal
Q
for outer moons is much smaller.
To measure the migration rates of Saturn’s moon, we use two independent methods. In
the first approach, a coherent orbit of Titan was determined by reconstructing the trajectory
of the
Cassini
spacecraft during 10 close encounters of the moon between February 2006 and
August 2016. During each Titan encounter, we are sensitive to the relative position of the
Cassini
spacecraft with respect to both the moon and Saturn, providing indirect information
on the orbit of Titan during the timespan of the
Cassini
mission. Our data sets encompass
only radio tracking data acquired by the ground antennas of the Deep Space Network, namely
Doppler observables at X- and Ka-band (8.4 GHz and 32.5 GHz, respectively), and range data
at X-band. Due to limited temporal coverage of radiometric data in the vicinity of the other
moons, it was not possible to obtain a reliable estimation of their orbits, which were instead
retrieved from the latest satellite ephemerides released by JPL (see Methods).
The radiometric data analysis strategy was based on the classical approach used by the
Cassini
Radio Science Team in the past for gravity science experiments
12–17
. Our solution was
obtained using JPL’s orbit determination program MONTE
18
, using a linearized weighted least
squares filter that allowed us to determine corrections to an a-priori dynamical model taking
into account all the relevant accelerations that affected the orbit of Titan and the trajectory
of the
Cassini
spacecraft. The least squares information filter used a multi-arc approach, in
4
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which radiometric data obtained during non-contiguous orbital segments, called arcs, are jointly
analyzed to produce a single solution of a set of global parameters, which affect all the arcs.
Our global parameters include the initial state vector of Titan, its gravity field up to the 5th
degree and order, Saturn’s gravity field up to
J
6
, Saturn’s tidal parameters
Re
(
k
2
)
and
Im
(
k
2
)
at Titan’s frequency, and the
Cassini
’s thermal recoil acceleration.
Our second method is based solely on classical astrometry data. Similar to
4
, we used
more than a century of observations, starting in 1886 through the whole
Cassini
mission. New
observations of the main moons from
19,20
were added to supplement those from
4
. Our model
solved the equations of motion of the eight main moons of Saturn, with the addition of the
four Lagrangian moons of Dione and Tethys, as well as Methone and Pallene. The Lagrangian
moons are useful to obtain Saturn’s Love number
k
2
, while Methone and Pallene are very sensi-
tive to Mimas’ mass and Saturn’s gravity field. The perturbation of the four innermost moons of
Saturn, Prometheus, Pandora, Janus and Epimetheus is introduced by ephemerides. We checked
that the chaos affecting the orbits of these moons, as well as a possible secular variation of Sat-
urn’s
J
2
, did not affect our results (Methods).
In addition to the initial state vectors of the moons, we fitted the masses of the moons and
their primary, the
J
2
,
J
4
and
J
6
of Saturn’s gravity field, the orientation and precession of the
Saturn’s pole, Saturn’s
k
2
(that assumes
k
20
=
k
21
=
k
22
), and the tidal ratio
k
2
/Q
at the tidal
frequencies of Mimas, Enceladus, Tethys, Dione, Rhea and Titan. Due to the large uncertainty
in Enceladus’ current tidal dissipation rate
21,22
, we performed three independent fits, assuming
a broad range of values of 10, 33 and 55 GW.
Our results are shown in Figure 1 and Table 1. From the radio tracking data, we measure
the tidal quality factor of Titan’s tidal bulge to be
Q
=124
±
22
(
3
uncertainties), assuming
5
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a fixed
Re
(
k
2
)
equal to 0.382. From astrometry, we find a slightly smaller value
Q
=61
+240
31
,
but the two are consistent within
2
uncertainties. We tested the reliability of these results by
performing many trials with different parameters (Methods), finding no substantial variation
in our result. This unexpectedly small value of
Q
for Saturn’s tidal bulge raised by Titan is
much smaller than our astrometric measurements of the
Q
s for the tidal bulges of other moons,
which range from
Q
⇠
300
for Rhea to
Q
&
3000
for Tethys. Each tidal bulge clearly has a
different value of
Q
, and all well-constrained values lie below the minimum value
Q
=1
.
8
⇥
10
4
predicted if the moons formed at the same time as Saturn and
Q
is constant
23
. Hence, our
results show that most of Saturn’s moons, including Titan, are migrating outward more rapidly
than expected.
The rapid migration of Titan is unexpected for all tidal dissipation mechanisms, except for
resonance locking, which predicted the observed migration. Figure 1 shows the predicted tidal
Q
for a resonance locking model with inertial waves with planetary spin evolution timescale
t
p
=6
Gyr (supplementary information), where the timescale
t
p
is a parameter of the model
that is expected to be comparable to the age of the solar system. In this model, the migration
timescale
t
tide
=
a/
̇
a
⇡
3
t
p
/
2
of each moon is roughly the same and is driven by the rate at
which inertial wave “resonant” frequencies evolve along with Saturn’s spin and structure. How-
ever, the predicted
Q
is different for each moon, with smaller values for outer moons, similar to
the measurements. Despite the different values of
Q
, Figure 2 shows that the observed migra-
tion timescales for each of Saturn’s moons are indeed very similar, with
t
tide
⇡
10 Gyr
. Hence,
we interpret our observations as strong evidence that a resonance locking process is driving
the migration of many of Saturn’s moons. The nearly constant migration time
t
tide
for sev-
eral of Saturn’s moons suggests that resonance locking with inertial waves, rather than gravity
modes (which predicts smaller
t
tide
for outer moons), is the most probable explanation for the
6
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Table 1:
Top:
Retrieved tidal parameters of Titan’s tidal bulge, and their associated
3
uncertainties, using
Cassini
radio tracking data.
Bottom:
Saturn’s Love number
k
2
and inverse quality factor
1
/Q
governing the migration of each moon, based on
astrometric data. To account for tidal dissipation inside Enceladus, we performed four
independent fits, assuming a heating rate of 3, 10, 33 and 55 GW. Error bars are 3
formal uncertainties.
Parameter
Value
Uncertainty
Re
(
k
2
)
0
.
33
0
.
20
Im
(
k
2
)
3
.
08
⇥
10
3
0
.
55
⇥
10
3
Enceladus heating rate
3 GW
10 GW
33 GW
55 GW
Saturn’s
k
2
0
.
382
±
0
.
017
0
.
382
±
0
.
017
0
.
382
±
0
.
017
0
.
382
±
0
.
017
1
/Q
⇥
10
4
(Mimas)
1
.
4
±
2
.
6
1
.
4
±
2
.
6
1
.
4
±
2
.
6
1
.
4
±
2
.
6
1
/Q
⇥
10
4
(Enceladus)
3
.
1
±
1
.
2
4
.
7
±
1
.
2
8
.
7
±
1
.
3
13
.
0
±
1
.
3
1
/Q
⇥
10
4
(Tethys)
1
.
45
±
0
.
61
1
.
45
±
0
.
60
1
.
46
±
0
.
63
1
.
45
±
0
.
60
1
/Q
⇥
10
4
(Dione)
3
.
9
±
2
.
2
3
.
6
±
2
.
2
3
.
0
±
2
.
2
2
.
3
±
2
.
2
1
/Q
⇥
10
3
(Rhea)
3
.
67
±
0
.
88
3
.
67
±
0
.
87
3
.
68
±
0
.
90
3
.
68
±
0
.
87
1
/Q
⇥
10
2
(Titan)
1
.
8
±
1
.
5
1
.
8
±
1
.
5
1
.
8
±
1
.
5
1
.
8
±
1
.
5
7
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moons’ migration (supplementary information). This may also help explain why mean motion
resonances between moons have survived, as resonance locking with gravity modes typically
results in divergent migration that can disrupt mean motion resonances between moons.
Figure 3 shows a possible orbital evolutionary history for Saturn’s moons in the resonance
locking framework, using a migration timescale for all moons of
t
tide
=3
t
Sa
, where
t
Sa
is the
changing age of Saturn (see supplementary material for more details). This simple model is
roughly consistent with our data, and it incorporates the fact that we expect shorter migration
time scales in the past when Saturn was evolving more quickly. Our model implies that Titan’s
semi-major axis has likely increased by a factor of a few over the age of the solar system, much
farther than prior expectations. The substantial migration of Titan may explain how it was able
to capture Hyperion into mean motion resonance
24
, and a previous resonance crossing with
Iapetus may explain the latter’s eccentricity and inclination
25
. Due to the changing values of
Q
associated with resonance locking, Saturn likely had larger
Q
values in the past, such that
moons migrated more slowly than they would by assuming a constant
Q
. Hence, our results
reconcile the rapid migration of the inner moons (and large tidal heating rate of Enceladus) with
an age of at least a few Gyr, though we still tentatively conclude the inner moons could have
formed well after Saturn (supplementary information).
While dissipation of the dynamical tide within Saturn’s convective envelope or stably
stratified core
26
seems to be the most significant mechanism of tidal friction, dissipation of the
equilibrium tide must also contribute. Assessing tidal dissipation in a solid core using a two-
layer model,
27
showed that a large range of effective equilibrium tidal dissipation (denoted
by
Q
e
) is possible depending on geophysical parameters. Looking at Figure 1 and Table 1,
we constrain
Q
e
&
5000
, but it remains possible that the migration of Mimas, Tethys, and
8
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Dione are driven by equilibrium tidal dissipation within Saturn’s core or envelope. While our
results indicate a small
Q
for Enceladus that favors an active resonance lock, it is possible
that its migration is due to equlibrium tidal dissipation, though this requires an extremely low
current dissipation rate (a few GW) within Enceladus. Future astrometric measurements will
help reduce the error bar on
Q
e
, clarifying the different energy dissipation mechanisms at work,
and allowing for a more accurate picture of the dynamical evolution of Saturn’s moons.
9
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Figure 1: -Saturnian tidal quality factor-
Effective tidal quality factor
Q
of Saturn for the tidal bulge raised by each of its moons measured
in this work. Purple points are measurements from astrometric solutions, with 3
error bars,
which extend to encompass results from each of the considered energy dissipation rates in
Enceladus. Titan’s red point is measured using
Cassini
radio tracking data. Blue shaded regions
are the predicted tidal quality factors from a resonance locking model with a Saturn evolution
time of
t
p
=6
Gyr. The vertical extent of the blue bars of Mimas, Enceladus, Tethys, and Dione
is due to their mean-motion resonances and is determined by their relative migration rates
2
. The
horizontal dashed line is the minimum value of
Q
that allows for coeval formation of Mimas and
Saturn, assuming
Q
is constant
23
. The background is shaded by relative migration timescale,
with faster migration in the lower left and slower migration in the upper right.
10
Figure 2: - Tidal migration timescale -
Tidal migration timescale for each of Saturn’s moons based on the measurements from Figure
1, in the absence of mean-motion resonances. Blue points show the same resonance locking
model as Figure 1, and the background is again shaded by relative migration timescale. The
actual migration timescales of Mimas and Enceladus may be longer than the measured values
because of mean-motion resonances with Tethys and Dione, whose actual migration timescales
may be shorter because they are pushed out by Mimas and Enceladus. The tidal migration
timescale with a constant
Q
(blue dashed line) corresponds to
Q
=1
.
8
⇥
10
4
as in Figure 1.
The migration timescale of each moon is within a factor of
⇡
2
of 10 Gyr.
11
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0
1
2
3
4
5
Time (Gyr)
0
5
10
15
20
25
Semi-major axis (
R
S
)
Mimas
Enceladus
Tethys
Dione
Rhea
Titan
F Ring
Saturn
Resonance Locking
Consant Q
2
3
4
5
6
log
(
Q
ef
)
Figure 3: - Moon orbital evolution -
A possible evolutionary history of the orbital distance of Saturn’s moons as a function of time,
for both a resonance locking model with inertial waves (solid colored lines) and a constant
Q
=5000
model (black dashed lines). While mean-motion resonances (not accounted for
here) can alter these histories, these models illustrate the qualitatively different behaviors of
resonance locking and constant
Q
models. The resonance locking models are shaded by the
effective tidal quality factor,
Q
ef
, at a given moment in time. Our results indicate that Titan
and Rhea have migrated farther than previously expected, while the inner moons may have a
markedly different history than they would in constant
Q
models.
12
Methods
*
Radiometric data selection and calibration
To measure a precise orbit of Titan, we analyzed the
Cassini
radiometric data acquired
during 10 close encounters with Titan (T11, T22, T33, T45, T68, T74, T89, T99, T110, and
T122) throughout the
Cassini
mission. To increase the sensitivity we selected only the encoun-
ters with data coverage around the closest approach.
The main observable used in the reconstruction of
Cassini
’s trajectory is the spacecraft
range rate, obtained from the Doppler shift of a microwave carrier transmitted from ground at
X-band (7.2 GHz) and sent back coherently at both X- (8.4 GHz) and Ka-band (32.5 GHz).
Doppler observables were integrated over a count time of 60 s. In addition, in order to study the
orbital evolution of Titan, we used range data at X-band.
We preferred two-way Doppler data over three-way, because of the intrinsic higher sta-
bility. When two-way observables were unavailable, we used three-way, adding in our filter
the necessary bias to correct for possible DSN inter-station clock offset. When available, X/Ka
measurements were preferred over X/X, as they are less sensitive to the dispersive effects, like
Earth’s ionosphere and solar and interplanetary plasma. We corrected the tracking data for the
effects caused by the Earth’s troposphere and ionosphere, using Global Positioning System data
and microwave radiometer data, when available. Tracking data acquired at ground station eleva-
tions lower than 15 degrees were discarded in order to avoid errors due to inaccurate calibration
of the tropospheric and ionospheric induced delays. Furthermore, we generated corrections to
take into account the additional Doppler shift induced by the spin of the
Cassini
spacecraft.
13
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Dynamical model
The dynamical model included the relativistic point-mass gravitational acceleration from the
Sun, the planets, the Moon, Pluto, and the main Saturn satellites. In addition, the setup included
the gravity field of Saturn and its planetary rings resulting from the analysis of data from the
Grand Finale orbits
16
. Saturn’s response to the tides raised by Titan was modelled using a
complex Love number
k
2
. Furthermore, the model included the following non-gravitational
accelerations for
Cassini
: the solar radiation pressure, the drag induced by the upper-layer of
Titan’s atmosphere, and the acceleration due to the non-isotropic thermal emission, mainly
generated by the three onboard Radioisotope Thermoelectric Generators.
No constraints were applied to the estimation of the global parameters, because the a priori
uncertainties were chosen to be at least one order of magnitude larger than the obtained formal
uncertainties or the formal uncertainty currently available from the literature
4,16
. Besides global
parameters, we estimated also local parameters, which means that they affect only a single arc.
For each encounter, they include the initial state of
Cassini
, the drag perturbation during the
low-altitude flybys, the low gain antenna phase-centre position during T110, constant Doppler
bias for the three-way passes, and constant range biases per station and pass. The a priori
uncertainties for
Cassini’s
position and velocity were 20 km and 0.2 m/s, respectively.
*
Measurement of Titan’s orbit
The inclusion of the Saturn’s tidal dissipation at Titan’s frequency to our dynamical model
allowed for a fit to the noise level, as shown by the range rate residuals (Figures 4 and 5). The es-
timated gravity field of Titan and Saturn are both fully compatible with the latest measurements
published by the
Cassini
RS team
16,17
.
14
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