Measurement and implications of
Saturn's gravity field and ring mass
Item Type Article
Authors Iess, L; Militzer, B; Kaspi, Y; Nicholson, P; Durante, D; Racioppa,
P; Anabtawi, A; Galanti, E; Hubbard, W; Mariani, M J; Tortora, P;
Wahl, S; Zannoni, M
Citation Iess, L., Militzer, B., Kaspi, Y., Nicholson, P., Durante, D.,
Racioppa, P., ... & Tortora, P. (2019). Measurement and
implications of Saturn’s gravity field and ring mass. Science,
364(6445), eaat2965.
DOI 10.1126/science.aat2965
Publisher AMER ASSOC ADVANCEMENT SCIENCE
Journal SCIENCE
Rights Copyright © 2019 The Authors, some rights reserved; exclusive
licensee American Association for the Advancement of Science.
No claim to original U.S. Government Works.
Download date 09/08/2022 22:41:00
Item License http://rightsstatements.org/vocab/InC/1.0/
Version Final accepted manuscript
Link to Item http://hdl.handle.net/10150/633328
Submitted Manuscript
Measurement and implications of Saturn’s gravity field and ring mass
Authors: L. Iess
1*
, B. Militzer
2
, Y. Kaspi
3
, P. Nicholson
4
, D. Durante
1
, P. Racioppa
1
, A.
Anabtawi
5
, E. Galanti
3
, W. Hubbard
6
, M. J. Mariani
1
, P. Tortora
7
, S. Wahl
2
, M. Zannoni
7
Affiliations:
1
Sapienza Università di Roma, Rome 00184, Italy
2
University of California, Berkeley, CA 94720, USA
3
Weizmann Institute of Science, Rehovot 76100, Israel
4
Cornell University, Ithaca, NY 14853, United States
5
Jet Propulsion Laboratory/Caltech, Pasadena, CA 91109, USA
6
Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, USA
7
Università di Bologna, Forlì 47100, Italy
*Correspondence to: luciano.iess@uniroma1.it
Abstract:
The interior structure of Saturn, the depth of its winds and the mass and age of its rings constrain
its formation and evolution. In the final phase of the Cassini mission, the spacecraft dived
between the planet and the innermost ring, at altitudes 2600-3900 km above the cloud tops.
During six of these crossings, a radio link with Earth was monitored to determine the
gravitational field of the planet and the mass of its rings. We find that Saturn’s gravity deviates
from theoretical expectations and requires differential rotation of the atmosphere extending to a
depth of at least 9000 km. The total mass of the rings is 0.41 ± 0.13 of the Saturnian moon
Mimas, indicating that they may have formed only 10
7
-10
8
years ago.
One Sentence Summary:
Measurements of the radial velocity of the Cassini spacecraft indicate a strong differential
rotation inside the planet, a substantial core and a low mass – and thus a young age – for its
rings.
Main text:
The mass distribution inside a fluid and rapidly rotating planet, such as Saturn, is largely driven
by the ratio between centrifugal and gravity forces. In the absence of internal dynamics, axial
and hemispherical symmetry is expected, implying that in the decomposition of the gravitational
potential into spherical harmonics (an orthonormal basis for functions defined over the sphere)
only even zonal harmonics appear (zonal harmonics are longitude-independent). Assuming
hydrostatic equilibrium, interior models of gas giant planets indicate that the zonal coefficients
J
2k
can be approximated by !
"#
$ %
#
&
#
, where q is the ratio of the centrifugal and gravity
acceleration at the equator (about 0.16 for Saturn),k is an integer positive number, and the
coefficients a
k
depend on the density profile inside the planet (1).
Optical tracking of clouds indicates that dynamical phenomena operate on Saturn and Jupiter.
The measured zonal (west-east) wind velocity field suggests a state of differential rotation,
whereby the angular velocity at any location depends on its distance from the axis of rotation and
the depth along this axis (2, 3). If the velocity field seen at the cloud top level (conventionally
defined as the 1 bar level) continues into the interior, then internal dynamics are expected to
affect the gravitational field in two ways. Firstly, the equipotential surfaces are perturbed
symmetrically, redistributing mass in such a way that the even zonal coefficients deviate from
the relation !
"#
'&
#
(2). Secondly, any north-south asymmetry in the velocity field leads to
nonzero values of the odd zonal harmonics (4). These theoretical expectations have been
confirmed by the Juno mission at Jupiter (5-7), where gravity measurements showed that zonal
winds are 2000-3000 km deep and suggest that the heavy element core is diffuse (8).
Gravity measurements at Saturn can be used to determine the mass of the rings, which dynamical
and compositional dating methods show is related to the rings’ age (9-11). Prior to the Grand
Finale phase of the mission, the pericenter of Cassini’s orbit was always outside Saturn’s A ring,
so that the gravitational effects of the rings could not be separated from those of the oblateness of
the planet. During the Grand Finale, Cassini flew between the planet and the rings. This
geometry effectively breaks the degeneracy between the even zonal field and the mass of the
rings, providing a direct, dynamical estimate of the ring mass.
Cassini gravity measurements
We determined Saturn’s gravitational field by reconstruction of Cassini’s trajectory during the
Grand Finale, using a coherent microwave link between Earth tracking stations and the
spacecraft. Range-rate measurements were obtained from the Doppler shift of a carrier signal
sent from the ground at 7.2 GHz (X-band) and retransmitted back to Earth by Cassini’s onboard
transponder at 8.4 GHz. An auxiliary downlink at Ka-band (32.5 GHz) was also recorded.
In April 2017, Cassini was inserted into a series of inclined, highly eccentric orbits, grazing
Saturn’s cloud tops at each pericenter (Table S1). The orbit nodes were chosen such that the
angle between the orbit normal and the direction to Earth is close to 90°. This edge-on condition
provides the maximum projection of the spacecraft velocity along the line of sight, so is optimal
for range-rate measurements and gravity estimations.
Of the 22 Grand Finale orbits (labelled Rev 271 through Rev 293), six were selected for gravity
measurements. Five orbits (Revs 273, 274, 278, 280 and 284) provided useful data (data from
Rev 275 were lost due to a station configuration error). These orbits were selected to minimize
neutral particle drag and maximize spacecraft view period around closest approach (C/A).
We produced an orbital solution based on 5 data arcs, using Doppler observations with count
times of 30 s, spanning a period of 24-36 hours about each closest approach. Tracking data were
acquired by the antennas of the three complexes (Goldstone, USA; Madrid, Spain; and Canberra,
Australia) of NASA’s Deep Space Network (DSN), and two deep space antennas from ESA’s
ESTRACK network, located in the southern hemisphere (Malargüe, Argentina, and New Norcia,
Australia).
Two-way Doppler measurements at X-band make up 93% of the data set (12), with the addition
of a few three-way passes to fill gaps during ground station handovers. Available X-band range
observables are also included. All data with either uplink or downlink elevation below 15° were
discarded to avoid systematic measurement errors due to imperfect calibration of tropospheric
path delays. Calibrations of Earth’s tropospheric and ionospheric path delays were provided by
the DSN based on pressure, humidity and Global Positioning System data. Noise from solar
plasma, which can strongly affect X-band radio links, was low due to the large solar elongation
angle (>142° on all 6 gravity orbits) (13). The data quality was statistically equivalent in X- and
Ka-band data, with a root-mean-square (RMS) Doppler noise between 0.020 and 0.088 mm·s
-1
at
30 s integration time (Fig. S1 and Table S1). For comparison, the Doppler signals due to the
weakest measurable harmonics (J
3
, J
10
) and Saturn’s ring are 40-200 times larger than the
average Doppler noise (Fig. S2).
The X-band Doppler data were favoured in our analysis because of the higher signal-to-noise
ratio during the unavoidable ring occultation periods. The radio link was maintained during the
occultations, except for a blockage of about 10 minutes on each orbit when the signal crossed the
opaque core of the optically thick B ring (region B3). A second, longer blockage period due to
ring occultations occurred on the outbound leg of the gravity orbits. Diffraction and near-forward
scattering of the radio signal by ring particles caused an increase in the Doppler noise at X-band
by a factor of 2-3 during the occultation periods. Ka-band is more sensitive to this effect and
suffers repeated signal losses.
Dynamical model
Our orbital fitting is based on the dynamical model previously adopted and tested in the
determination of the gravity fields of Titan (14) and Enceladus (15), implemented in the Jet
Propulsion Laboratory (JPL) navigation code MONTE (16). The model was extended to include
Saturn’s gravitational parameter GM, the zonal harmonic coefficients J
2
-J
20
, the tesseral
(longitude-dependent) field of degree 2 (to account for possible non-principal axis rotation), and
the mass of the rings. The truncation of the zonal field was set to twice the degree of the highest
gravity harmonic whose central value is above the uncertainty (degree 10). The estimate of the
even zonal harmonics used in our interpretation (degree ≤10) is insensitive to changes in the
truncation. Although this model was adequate for fitting Juno gravity data at Jupiter (5), obtained
in an orbital configuration similar to Cassini’s Grand Finale, it could not reduce Cassini’s
Doppler residuals to the noise level. Instead, small stochastic accelerations were added to the
model (see below) to produce signature-free residuals (Fig. S1).
Although all the rings are included in the dynamical model, only rings A, B and C can produce
an acceleration potentially detectable by Cassini. The rings are assumed to be coplanar with
Saturn’s equator and each is assumed to have a constant surface mass density (the data are
insensitive to radial variations of the density). Space- and ground-based measurements of ring
occultations provide the determination of Saturn’s spin axis position and precession rate (17).
Although Doppler data are sensitive to the orientation of Saturn’s equatorial plane, we adopt the
prior determination because it is about ten times more accurate than that obtained from our
orbital fitting.
Accelerations that are known to be large enough to produce noticeable signatures on range-rate
data were accounted for. These include the point-mass gravitational accelerations from Saturn
and its satellites (including the ring moons), computed from the JPL planetary and satellite
ephemerides DE430, SAT389 and SAT393 (18), the acceleration from the Sun, the planets and
satellites of the Solar System, and Saturn’s tidal response to its satellites (12).
The acquired Doppler data were combined in a multi-arc, weighted least-squares estimation
filter. In the multi-arc approach, the entire time span of the observations is decomposed into
shorter intervals and a distinction is made between global and local estimated parameters. Global
parameters are common to all arcs and estimated using all available observables. These include
Saturn’s GM, J
2
through J
20
, C
21
, C
22
, S
21
, S
22
, and the masses of the A, B and C rings. Although
the estimates of the masses of the three rings are highly correlated, their sum is well determined.
The ring masses were initially set to the current best estimates of their values from ring
occultation data (19-21), with an a priori uncertainty of 100% (see Table S2). The mass of the B
ring had a large uncertainty of 10 Mimas masses (Saturn’s moon Mimas GM is 2.5026 km
3
·s
-2
).
The sum of the masses of Saturn and its rings was constrained to be equal to the value (and
uncertainty) estimated from satellite ephemerides (18).
Local parameters are those that belong only to a single arc and a different value was estimated
for each arc. These included the Cassini position and velocity when the gravity observations
began in each of the five orbits, at least 12 hours before transit at pericenter. The a priori
uncertainties for position and velocity were set at 100 km and 1 m·s
-1
, respectively.
In the estimation filter, parameters whose uncertainties are large enough to contribute to the final
covariance matrix have been included as consider parameters. These include the Love number
k
22
(determining Saturn’s tidal response), Saturn’s pole direction, accelerations due to thermal
emission from Cassini’s radioisotope thermoelectric generators (RTG), and solar radiation
pressure. The nominal value and a priori uncertainty of the Love number are taken from previous
Cassini constraints (22), as were the pole position and precession rate (17). The anisotropic
thermal acceleration from the RTGs was determined to 5% or better during the Cassini mission
by the Navigation Team. The relative uncertainty associated with solar radiation pressure
acceleration was set to 20%.
Gravity determination
Our deterministic model, based on the geophysical expectations for the gravity field of a gas
giant like Saturn, can adequately fit the Doppler data if each arc is analysed separately. However,
the same model cannot jointly fit all passes in a combined, multi-arc, gravity and orbital solution.
The signatures in range-rate residuals are as large as 0.2 mm·s
-1
over time scales of 20-60
minutes, corresponding to radial accelerations of the order of 10
-7
m·s
-2
. This value is an
underestimate of the real unmodeled accelerations, as a large fraction of them are aliased in those
associated with estimated parameters. The unmodeled accelerations acting on Cassini must be
compensated for, to avoid biases in the estimates of the gravity harmonics and ring mass.
The missing accelerations could be due to longitudinally-varying density anomalies resulting
from wind dynamics or convection in Saturn’s deep interior (23). For a rocky planet, the
corresponding gravitational field would be static and described by tesseral harmonics. However,
this approach is not immediately applicable to a fluid planet like Saturn, because vortices move
longitudinally with different speeds depending on latitude. The resulting gravity disturbances
(caused by the non-zonal wind dynamics) would not be static in any reference frame over the 60-
