[Show abstract][Hide abstract] ABSTRACT: Analysis of radio Doppler data generated by the Deep Space Network with
Mariner 10 during its first and third encounters with Mercury yielded a
quadrupole gravitational field with J2 equal to 6.0 ± 2.0 and C22
equal to 1.0 ± 0.5 in units of 10-5 (Anderson et al.,
Icarus 71, 337, 1987) . However, this underlying global field leaves
systematic Doppler residuals for the third encounter (Esposito et al.,
COSPAR: Space Res. 17, 639, 1978), residuals that are most likely caused
by a large gravity anomaly in the region of closest approach to Mercury
at latitude 67.96° and east longitude 53.09°. We report here a
detailed characterization of the likely sources producing the putative
gravity anomaly. The recovered Doppler residuals and ground track from
Esposito et al. (1978) are fit by a model that includes the spacecraft's
six trajectory initial conditions along with Mercury's mass GM =
22,032.09 ± 0.91 km3 s-2 (Anderson et al.,
1987) and the two quadrupole coefficients. After convergence to the
best-fit trajectory, any remaining residuals represent an unmodeled
signal that is assumed to arise from anomalous mass concentrations on
Mercury plus noise. In order to reduce the noise evident in the Doppler
residuals, we smooth them with a variable-width Gaussian filter (Palguta
et al., Icarus 180, 428, 2006). The filter width in the time domain
increases with the spacecraft altitude, reducing the noise before and
after closest approach. Accelerations along the line of sight (LOS) are
calculated by sampling the differentiated Doppler smoothing curve at a
10-second time interval, the sample interval for the Doppler frequency
data. Multiple spherical-cap disk models are then used to fit the LOS
acceleration data. The spherical-cap disk models not only provide the
locations and magnitudes of anomalous mass concentrations on Mercury,
but also their vertical and horizontal dimensions. We find that a
minimum of four mass anomalies on or near Mercury's surface is required
to explain the LOS acceleration data. Although at least four masses are
required to reproduce all the major acceleration features, we show that
the fit to the data can be improved by including up to seven
spherical-cap disks. The modeled mass anomalies are on the order of
1018 - 1019 kg. Additionally, at least two
negative and two positive mass anomalies are present in each model.
Although there is some variability in the placement of the mass
anomalies among the different models, we find that positive mass
anomalies are typically found near the beginning and end of the Mariner
10 trajectory path (20° N, 25° E) and (55° N, 177° E).
[Show abstract][Hide abstract] ABSTRACT: The physical shape of a giant planet reveals important information about
its rotation and internal structure. We investigate how differential
rotation on cylinders affects Jupiter's shape. We project Jupiter's
measured zonal wind velocities along cylinders to describe its
centrifugal potential which is then used to derive Jupiter's shape using
an equipotential surface theory. The derived shape for different
cylindrical radii is then compared with Jupiter's shape from radio
occultation measurements. It is found that both solid-body rotation
(System III rotation rate) and differential rotation up to a latitude of
~ 20-30 degrees are consistent with Jupiter's measured shape. We next
use a first-order theory that relates the second gravitational
coefficient J_2 to the flattening to calculate the corrections to J_2
for the different rotational configurations. We find that the
contribution of the flattening to J_2 is significant. We therefore
suggest that interior models of the giant planets must account for J_2n
corrections caused by both shape (flattening) and internal dynamics.
[Show abstract][Hide abstract] ABSTRACT: A new approach is presented to evaluate the shape, the gravitational
coefficients J2 and C22 and the normalized axial
moment of inertia C/MR2 for a rotating, axially symmetric
satellite in hydrostatic equilibrium and distorted by synchronous
rotation and tides. C is the axial moment of inertia and M and R are the
mass and radius of the body, respectively. The theory is valid for an
arbitrary interior density profile. Application is made to Rhea and
Titan assuming that these bodies are in hydrostatic equilibrium. A
likely internal structure for Rhea is a uniform mixture of ice and rock,
75% ice and 25% rock, by mass, with an ice I-ice II phase change
occurring within Rhea at a fractional radius of about 0.4 to 0.55.
[Show abstract][Hide abstract] ABSTRACT: Asteroid 21 Lutetia was approached by the Rosetta spacecraft on 10 July 2010. The additional Doppler shift of the spacecraft radio signals imposed by 21 Lutetia's gravitational perturbation on the flyby trajectory were used to determine the mass of the asteroid. Calibrating and correcting for all Doppler contributions not associated with Lutetia, a least-squares fit to the residual frequency observations from 4 hours before to 6 hours after closest approach yields a mass of (1.700 ± 0.017) × 10(18) kilograms. Using the volume model of Lutetia determined by the Rosetta Optical, Spectroscopic, and Infrared Remote Imaging System (OSIRIS) camera, the bulk density, an important parameter for clues to its composition and interior, is (3.4 ± 0.3) × 10(3) kilograms per cubic meter.
[Show abstract][Hide abstract] ABSTRACT: The Rosetta spacecraft encountered its second asteroid target (21)
Lutetia on 10th July 2010. The asteroid perturbed the flyby trajectory
and velocity of the spacecraft (closest approach is at 3168 km). The
mass of the asteroid was determined from the shift of the radio carrier
signal frequency at X-band (8.4 GHz). Altough the flyby geometry as
suboptimal and there is a tracking gap at closest approach, the mass was
determined to 1.7 1018 kg at an uncertainty of 1%. Major driver of the
uncertainty are the frequency noise level, the tropospheric correction
and the uncertainty of the closest approach distance. The Rosetta camera
OSIRIS determined the size and volume of Lutetia. The bulk density is
derived from the determined mass and volume to (3400 +/- 275) kg/m3. The
precision of the bulk density is driven by the precision of the volume
estimate. Knowledge of the mass and bulk density is an important
contributor to understand the asteroid's composition, internal structure
and porosity. Lutetia is considered to be a major perturber of a number
of smaller asteroids and a perturber of the motion of Mars. The derived
mass from the Rosetta flyby will therefore be compared with those mass
values derived from astrometry.
[Show abstract][Hide abstract] ABSTRACT: The moment of inertia of a giant planet reveals important information
about the planet's internal density structure and this information is
not identical to that contained in the gravitational moments. The
forthcoming Juno mission to Jupiter and the Solstice Mission (Cassini
XXM) to Saturn might determine the angular momentum of the planets, and
therefore their moments of inertia NMoI=C/MR2 by measuring the planets'
pole precession, and the Lense-Thirring acceleration of the spacecraft
(C is the axial moment of inertia, and M and R are the planet's mass and
mean radius, respectively). The possible range of NMoI values for
Jupiter and Saturn based on their measured gravitational fields using a
simple core/envelope model are presented. The model suggests that
Jupiter's NMoI lies in the range 0.2629 - 0.2645. Saturn's NMoI is found
to be ∼ 0.218. Constraining Saturn's NMoI value, however, is possible
only if an accurate determination of Saturn's rotation period is
available.
[Show abstract][Hide abstract] ABSTRACT: The present OSS mission continues a long and bright tradition by associating the communities of fundamental physics and planetary sciences in a single mission with ambitious goals in both domains. OSS is an M-class mission to explore the Neptune system almost half a century after flyby of the Voyager 2 spacecraft. Several discoveries were made by Voyager 2, including the Great Dark Spot (which has now disappeared) and Triton's geysers. Voyager 2 revealed the dynamics of Neptune's atmosphere and found four rings and evidence of ring arcs above Neptune. Benefiting from a greatly improved instrumentation, it will result in a striking advance in the study of the farthest planet of the Solar System. Furthermore, OSS will provide a unique opportunity to visit a selected Kuiper Belt object subsequent to the passage of the Neptunian system. It will consolidate the hypothesis of the origin of Triton as a KBO captured by Neptune, and improve our knowledge on the formation of the Solar system. The probe will embark instruments allowing precise tracking of the probe during cruise. It allows to perform the best controlled experiment for testing, in deep space, the General Relativity, on which is based all the models of Solar system formation. OSS is proposed as an international cooperation between ESA and NASA, giving the capability for ESA to launch an M-class mission towards the farthest planet of the Solar system, and to a Kuiper Belt object. The proposed mission profile would allow to deliver a 500 kg class spacecraft. The design of the probe is mainly constrained by the deep space gravity test in order to minimise the perturbation of the accelerometer measurement.
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[Show abstract][Hide abstract] ABSTRACT: The shapes and gravitational fields of rotationally and tidally distorted planets and satellites depend on their interior mass distributions. Measurements of these observable characteristics are therefore used to infer the internal structure of planetary bodies. Interpretations are based on approximate formulae such as the Radau-Darwin relation derivable from the theory of figures or more accurate evaluations of the theory. The exact solution for the shape and gravitational field of the rotationally distorted constant density Maclaurin spheroid has, until now, provided one of the only ways to assess the accuracy and range of validity of approximate theory of figure predictions. We generalize the Maclaurin spheroid solution to a 2-layer core-envelope body, a more realistic model of a real planet or moon. The exact 2-layer Maclaurin spheroid solution, e. g., the shapes of the surface and core-envelope interface, depend on 3 parameters, the core-envelope density ratio, the fractional volume of the core, and Omega2/2piGrho2, where Omega is the rotation rate, G is the gravitational constant, and rho2 is the envelope density. For realistic parameter values, the flattening of the interface is smaller than that of the surface. Results of the exact solution are compared with predictions of the theory of figures up to order 3 in the small rotational parameter of the theory. The exact solution serves as a benchmark for numerical models that attempt to invert gravitational and shape data to infer internal planetary structure.
[Show abstract][Hide abstract] ABSTRACT: The Rosetta spacecraft flew by its second asteroid target (21) Lutetia on 10th July 2010. The flyby and recording of the radio carrier signals went very well. The frequency noise recorded at NASA's DSS-63 70-m antenna near Madrid was lower than expected. The asteroid perturbed the flyby trajectory and velocity of the spacecraft (closest approach was at 3160 km). The mass of the asteroid was determined from the Doppler shift of the radio signal carrier frequencies. A preliminary analysis of the flyby data and the housekeeping data show contributions from the rotation of the spacecraft body during flyby which still need to be corrected (The abstract is written one week after data reception at the institute). Pre-flyby simulations showed that a mass determination of the asteroid to an accuracy of 1% or better shall be possible even tough the flyby geometry was suboptimal and there was a tracking gap at closest approach. The bulk density will be derived from the determined mass and the volume. Volume estimates from the OSIRIS camera and from ground observations will be applied. Knowledge of the mass and bulk density is an important contributor to understand the asteroid's composition, internal structure and porosity and may probably also give clues for the definition of the asteroid's type. Lutetia is considered to be a major perturber of a number of smaller asteroids. The derived mass from the flyby will therefore be compared with those mass values derived from asteroid orbit perturbations.
[Show abstract][Hide abstract] ABSTRACT: The interior structures of Uranus and Neptune are not well constrained. We present 'empirical' models (pressure vs. density) of Uranus and Neptune interiors constrained by the gravitational coefficients J2, J4, and the planetary radii and masses, using Voyager's solid-body rotation periods. The empirical pressure-density profiles are then interpreted in terms of physical equations of state of hydrogen, helium, ice (H2O), and rock (SiO2) to test the physical plausibility of the models. The interiors of Uranus and Neptune are found to be very similar. Our analysis suggests that the concentration of heavier elements inside both Uranus and Neptune interiors could increase gradually towards the planetary centers, without having sharp compositional transitions as typically assumed. Uranus and Neptune (solid-body) rotation periods, 17.24h and 16.11h, respectively, are based on Voyager 2 measurements of variations in the planets' radio signals and on fits to the planets' magnetic fields. The realization that Saturn's radio period does not represent the planet's deep interior rotation and the complexity of the magnetic fields of Uranus and Neptune raise the possibility that the Voyager 2 radio and magnetic periods might not represent the deep interior rotation periods of the planets. We use wind and shape data to investigate the rotation of Uranus and Neptune. Minimization of wind velocities or dynamic heights of the 1 bar isosurfaces, constrained by the single occultation radii and gravitational coefficients of the planets, leads to solid-body rotation periods of 16.58h for Uranus and 17.46h for Neptune. We derive shapes for the planets based on these rotation rates. Wind velocities with respect to these rotation periods are essentially identical on Uranus and Neptune and wind speeds are slower than previously thought. Alternatively, if we interpret wind measurements in terms of differential rotation on cylinders there are essentially no residual atmospheric winds.
[Show abstract][Hide abstract] ABSTRACT: To a first approximation, most planetary objects consist of a dense core region surrounded by a low-density envelope. The flattening of the core-envelope interface and the surface can reveal important information about internal structure and rotation. The flattening of the internal interface is also important for constraining the dynamics of motions in a liquid core. We determine the flattening of the core-envelope boundary and the surface for constant density layers. The flattening is calculated in two ways. One approach makes use of an exact semi-analytical solution. A second approach is based on the theory of figures valid to third order in the standard smallness parameter. We evaluate the accuracy of the approximate theory of figure solutions by comparing these solutions for first, second, and third order in the smallness parameter with the exact solution. We find that the flattening at the core-envelope interface can be significantly different from that at the planetary surface.
[Show abstract][Hide abstract] ABSTRACT: The Rosetta spacecraft will fly by its second target asteroid (21) Lutetia on 10 July 2010. Simulations based on the currently known size of Lutetia and assumptions on the bulk density show that tracking of two radio-carrier frequencies at X-band (8.4 GHz) and S-band (2.3 GHz) during the flyby will determine the mass at less than 1% accuracy. Derivation of the asteroid volume by camera observation will drive the uncertainty in derivation of the bulk density. Mass and bulk density provide valuable clues that might help resolve the difficulties in determining the taxonomic class of the asteroid.
Astronomy and Astrophysics 07/2010; 518. DOI:10.1051/0004-6361/201014325 · 4.38 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: It has been generally believed that a rotation period could be assigned to each of the giant planets. Accepted values of these periods, till now, are 9h 55m 29s, 10h 39m 22s, 17h 14m 24s, and 16h 06m 36s for Jupiter, Saturn, Uranus, and Neptune, respectively. The rotation period of Jupiter is based on the periodic variations in the planet's kilometric radiation and magnetic field, periodicities that have been unchanged since the Voyager flybys. The association of these periodicities with Jupiter's internal rotation period is based on the idea that the radio and magnetic phenomena are tied to the planet's magnetic field lines anchored deep within Jupiter. The periodic variations of the Saturnian Kilometric Radiation (SKR), unlike those of Jupiter, have not been rock solid, however; the periodicity has changed from 10h 39m 22s at the time of Voyager to 10h 45m 45s at the time of Cassini. Clearly, the SKR period does not represent the internal rotation period of Saturn, and it raises the possibility that the rotation periods of the other giant planets are uncertain. In fact, we must seriously reconsider whether the interiors of the giant planets are in solid body rotation with a single period. Even for Jupiter, the 9h 55m 29s rotation period might represent only the rotation of the region in which the magnetic field is generated. The dynamo region could extend from some unknown inner radius out to about 0.9 Jovian radius. The deeper Jovian interior could be rotating with a different period. A recent attempt to model the interior of Jupiter with new equation of state data concluded that the gravitational coefficients of Jupiter could not be fit unless Jupiter's internal rotation rate was constant on cylinders parallel to the rotation axis (Militzer, B., W.B. Hubbard, J. Vorberger, I. Tamblyn, and S.A. Bonev, A massive core in Jupiter predicted from first-principles simulations, 2008, ApJ, 688, L45-L48 [doi: 10.1086/594364]). For Saturn, two studies of the atmospheric motions (Anderson, J.D. and G. Schubert, 2007, Saturn's gravitational field, internal rotation, and interior structure, Science, 317, 1384-1387 [doi: 10.1126/science.1144835]; Read, P.L., T.E. Dowling, and G. Schubert, Saturn's rotation period from its atmospheric planetary-wave configuration, 2009, Nature, 460, 608-610 [doi:10.1038/nature08194 Letter]) have inferred planetary rotation periods significantly shorter than the Voyager period, implying Jovian-like atmospheric winds. The inferred shapes (oblateness) of Uranus and Neptune are inconsistent with heretofore accepted planetary rotation rates. Either the shapes or the rotation periods of the icy giants are not well determined. If the magnetic fields of Uranus and Neptune are generated in relatively thin shells (Stanley, S., and J. Bloxham, Convective-region geometry as the cause of Uranus' and Neptune's unusual magnetic fields, Nature, 428, 151-153 [doi:10.1038/nature02376] Letter), then the periodicities of the fields might not reflect the rotation of the bulk of the planets.
[Show abstract][Hide abstract] ABSTRACT: The constraints on giant planet interior models with density discontinuities, for example, a core-envelope boundary, are more difficult to treat than a continuous density distribution that decreases monotonically and continuously from the center to the surface of the planet. We revise our previous interior calculations (Anderson, J. D., and G. Schubert, Saturn's gravitational field, internal rotation, and interior structure, 2007, Science, 317, 1384-1387, doi: 101126/science.1144835, 2007), which solved a system of integro-differential equations to third order in the smallness parameter omega2a3/GM (omega is the angular velocity of the planet, a is the planet's equatorial radius, G is the gravitational constant, and M is the planet's mass), and introduce Clairaut's differential equation for the flattening f, with appropriate boundary conditions at the planet's surface and at its center. The calculations can be carried to second order in the smallness parameter by solving Darwin's differential equation for k, a parameter that describes a second-order deviation from sphericity. In principle, the calculations can be extended to differential equations of arbitrary order in smallness. As with our earlier method, we apply this revised method to the outer planets with interiors comprising a compressible core, obeying a linear density distribution, and an envelope in which density vs. radius is described by a sixth degree polynomial. This method of gravity sounding, with cores and envelope polynomial density distributions, can yield insights into a class of possible cores that fit the boundary conditions, consisting of the measured even zonal gravitational harmonics, plus the measured size and total mass of the planet. We apply the method to the four outer planets.
[Show abstract][Hide abstract] ABSTRACT: The four giant planets, Jupiter, Saturn, Uranus, and Neptune have strong zonal winds with velocities up to a few hundred meters per second. These zonal wind velocities, however, are based on assumed values of the solid-body rotation periods of the planets. Jupiter's kilometric radiation period of 9h 55m 29s has not changed in many decades, the 10h 39m 22s rotation period of Saturn, however, which derives from the Voyager era periodicity in Saturn's kilometric radiation (SKR), is now known to be variable and Saturn's rotation rate is therefore uncertain. Accordingly, the atmospheric zonal wind velocities with respect to the underlying rotating planet are unknown for Saturn. Uranus and Neptune rotation periods were derived from Voyager 2 radio astronomy observations to be 17h 14m 24s and 16h 06m 36s, respectively. Since we now realize that the kilometric radiation period might not represent the planet's deep interior rotation, we must consider the possibility that the Voyager 2 radio periodicities also do not represent the deep interior rotation of Uranus and Neptune. We use wind and shape data to investigate the rotation periods of Jupiter, Saturn, Uranus and Neptune. We show that Jupiter's and Saturn's shapes are consistent with solid-body rotation periods of 9h 55m and 10h 32m, respectively. We suggest that zonal winds have a minor effect on the planetary shape for both Jupiter and Saturn. Uranus and Neptune shapes (oblateness) are found to be inconsistent with Voyager's solid-body rotation periods. We address the possibility of deep differential rotation on cylinders.
[Show abstract][Hide abstract] ABSTRACT: We summarise the scientific and technological aspects of the SAGAS (Search for Anomalous Gravitation using Atomic Sensors) project, submitted to ESA in June 2007 in response to the Cosmic Vision 2015-2025 call for proposals. The proposed mission aims at flying highly sensitive atomic sensors (optical clock, cold atom accelerometer, optical link) on a Solar System escape trajectory in the 2020 to 2030 time-frame. SAGAS has numerous science objectives in fundamental physics and Solar System science, for example numerous tests of general relativity and the exploration of the Kuiper belt. The combination of highly sensitive atomic sensors and of the laser link well adapted for large distances will allow measurements with unprecedented accuracy and on scales never reached before. We present the proposed mission in some detail, with particular emphasis on the science goals and associated measurements.
[Show abstract][Hide abstract] ABSTRACT: In response to ESA´s Call for proposalsof 5 March 2007 of the COSMIC VISION 2015-2025 plan of the ESA science programme, we propose a M-class satellite mission to test the Equivalence Principle in the quantum domain by investigating the extend free fall of matter waves instead of macroscopic bodies as in the case of GAUGE, MICROSCOPE or STEP. The satellite, called Matter Wave Explorer of Gravity, will carry an experiment to test gravity, namely the measurement of the equal rate of free fall with various isotopes of distinct atomic species with precision cold atom interferometry in the vicinity of the earth. This will allow for a first quantum test of the Equivalence with spin polarized particles and with pure fermionic and bosonic atomic ensembles. Due to teh space conditions, the free fall of Rubidium and Potassium isotopes will be compared with a maximum accelerational sensitivity of 5 x 10^-16 m/s^2 corresponding to an accuracy of the test of the Equivalence Principle of 1 part in 10^16. Besides the primary scientific goal, the quantum test of the Equivalence Principle, the mission can be extended to provide additional information about the gravitational field of the earth or for testing theories of fundamental processes of decoherence which are investigated by various theory groups in the context of quantum gravity phenomenology. In this proposal we present in detail the mission objectives and the technical aspects of the proposed mission.
[Show abstract][Hide abstract] ABSTRACT: The Solar System Odyssey mission uses modern-day high-precision
experimental techniques to test some the laws questions of fundamental physics which determine dynamics in the solar system.
It could lead to a major discoveries by using demonstrated technologies and
could be flown within the Cosmic Vision time frame. The mission proposes to perform a set of precision gravitation experiments from the vicinity
of Earth to the outer. Its scientific objectives can be summarized as follows: (1) test of gravity force law in the Solar System up to and beyond the orbit Saturn; (2) precise investigation of navigation anomalies at the fly-bys, (3) measurement of Eddington´s parameter at occultation, (4) mapping of gravity field in the outer solar system and study of the Kuiper belt. To aim this, the Odyssey mission is built up on a main spacecraft, designed to fly up to 13 AU, with the following components: (a) a high precision accelerometer, with bias-rejection system, measuring the deviation of the trajectory from the geodesics, that is also giving gravitational forces; (b) Ka-band transponders, as for Cassini, for a precise range and Doppler measurement up to 13 AU, with additional VLBI equipment; (c) optional laser equipment, which would allow one to improve the range and Doppler measurement, resulting in particular in an improved measurement (with respect to Cassini) of the Eddington´s parameter. In this baseline concept, the main spacecraft is designed to operate beyond the Saturn
orbit, up to 13 AU. It experiences multiple planetary fly-bys at Earth, Mars, or Venus, and Jupiter. The cruise and fly-by phases allow the mission to schieve its baseline scientific objectives [(1) to (3) in the above list]. In addition to this baseline concept, the Odyssey mission proposes the release of the Enigma radio-beacon at Saturn, allowing one to extend the deep space gravity test up to at least 50 AU, while achieving the scientific objective of a mapping of gravity field in the outer Solar System [(4) in the above list].
[Show abstract][Hide abstract] ABSTRACT: Anderson and Schubert (2007, Science, 317, 1384) proposed that Saturn's rotation period might be ascertained by minimizing the dynamic heights of the 100 mbar isosurface with respect to the geoid; they derived a rotation period of 10h 32m 35s. We investigate the same approach for Jupiter to see if the Jovian rotation period is predicted by minimizing the dynamical heights of its isobaric (1 bar pressure level) surface. The shape of the Jovian pressure isosurface is derived from zonal wind data (García-Melendo and Sanchez-Lavega, 2001, Icarus, 316) and the Jovian geoid. Then, by regarding the Jovian geoid as undetermined in terms of Jupiter's rotation rate, we vary the rotation rate and the geoid and search for the geoid that minimizes the dynamical heights of the pressure isosurface in an rms sense. A rotation period of 9h 54m 29.7s is found to minimize the dynamical heights of the pressure isosurface. This rotation period is only one minute shorter than the measured period of Jupiter. The successful application of the method to Jupiter lends support to its relevance for Saturn. Application of the approach to Neptune and Uranus will also be discussed.
[Show abstract][Hide abstract] ABSTRACT: Interior models of Jupiter and Saturn, with the density profile represented as a 6th degree polynomial, provide a good fit to gravitational and atmospheric data (Anderson & Schubert, Science 317, 1384, 2007; Helled et al., submitted to Icarus, 2008). However, the representation of the density profile by a polynomial function of radius is inadequate to account for a density discontinuity at the surface of a heavy element core. We present interior models of Jupiter and Saturn with density profiles accounting for the existence of a core. The density profile of the planet is represented by a piecewise function, which includes a constant density region (core), and a polynomial for the planetary envelope. The core density and radius, and the polynomial coefficients are a priori unknown and are found by iterating until the gravitational harmonics of the interior models converge to the measured ones. The density profiles, together with an integration of the hydrostatic equation, provide a pressure-density relation, referred to as an empirical equation of state (EOS). The empirical EOS makes no assumption about the planet's composition or how different elements are distributed with depth. It is also independent of any theoretical models of the behavior of hydrogen, helium, and heavier element mixtures at high temperature and pressure. The interior models reveal information on the planets' internal structure and whether interiors with cores are consistent with the gravitational data. The models also improve our understanding of the effect of a central core on the measured gravitational moments.