Spherical harmonic representation of the gravity field from dynamic satellite data

Spherical harmonics are the natural parameters for the Earth's gravity field as sensed by orbiting satellites, but problems of resolution arise because the spectrum of effects is narrow and unique to each orbit. Comprehensive gravity models now contain many hundreds of thousands of observations from more than thirty different near-Earth artificial satellites. With refinements in tracking systems, newer data is capable of sensing the spherical harmonics of the field experienced by these satellites to very high degree and order. For example, altimeter, laser and satellite-tracking-satellite systems contain gravitational information well above present levels of satellite gravity field recovery (l = 20), but significant aliasing results because the orbital parameters are too restricted compared to the large number of spherical harmonics.It is shown however that the unique spectrum of information for each satellite contained within a comprehensive spherical harmonic model can be represented by simple gravitational constraint equations (lumped harmonics). All such constraints are harmonic in the argument of perigee (ω) with constants determinable directly from tracking data or reconstituted from the comprehensive solution:

$\text{(C}{}^1\text{, S}{}^1\text{) = (C}{}_{\mathrm{o}}\text{, S}{}_{\mathrm{o}}\text{) + Σ}{}_{\mathrm{i}}\text{= 1 (C}{}_{\mathrm{Ci}}\text{, S}{}_{\mathrm{Ci}}\text{) cos i ω + (C}{}_{\mathrm{Si}}\text{, S}{}_{\mathrm{Si}}\text{) sin i ω}$

. The constants are simple linear combinations of the geopotential harmonics. Through these lumped harmonics any satellite gravity field can be decomposed and then uniformly extended to any degree or tailored to a given orbit without reintegration of the trajectory and variational equations. They also make possible the inclusion of information into the field from special deep resonance passages, long arc zonal analyses, and satellites unique to other models. Numerous examples of the derivation, combination, extension and tailoring of the harmonics are presented. The importance of using data spanning an apsidal period is emphasized.     

Doppler searches for planetary systems

Despite extensive Doppler searches, no companions of Jupiter, or Brown Dwarf, mass have been found within a few AU of more than 100 late-type stars and, for many, astrometry suggests that there are none in larger orbits either. Terrestrial-mass companions have been found for the millisecond pulsar PSR1257+12. I also consider what improvements in optical spectroscopic techniques would allow us to detect stellar companions of Uranus-mass.     

Lunar crustal density profile from an analysis of doppler gravity data

Low altitude line-of-sight gravity data obtained by CSM and LM radio tracking during several Apollo missions are used to construct an equispaced normalized vertical gravity net 30 km above selected lunar highland regions. Correlation of local vertical gravity anomalies with craters of different depth reveals a density increase with depth in the upper lunar highland crust. Crustal densities determined in this fashion are in good agreement with other, previously published crustal density values. The nature of the density increase implies a lunar crust consisting of fractured rather than competent rock.     

A one-year seasonal analysis of martian gravity waves using MCS data

Temperature measurements from the Mars Climate Sounder on NASA’s Mars Reconnaissance Orbiter are examined for gravity wave signals using the Stockwell transform, a technique previously applied to terrestrial temperature profiles. An analysis is presented for internal-gravity waves throughout the martian atmosphere for the period July 2007–May 2009, representing a full martian year of data, divided by season. Momentum fluxes observed in the altitude range 200–20 Pa are measured as ～10^?5–10^?3 Pa, significantly higher than at equivalent altitudes on Earth, and are primarily peaked around the tropics. Observed orographic effects are minimal; the primary observed effects are seasonal. Waves are shown to vary significantly in wavelength with latitude, but are generally approximately zonally symmetric. Horizontal wavelengths are consistently much shorter in the northern hemisphere than the southern.     

Small body surface gravity fields via spherical harmonic expansions

Conventional gravity field expressions are derived from Laplace’s equation, the result being the spherical harmonic gravity field. This gravity field is said to be the exterior spherical harmonic gravity field, as its convergence region is outside the Brillouin (i.e., circumscribing) sphere of the body. In contrast, there exists its counterpart called the interior spherical harmonic gravity field for which the convergence region lies within the interior Brillouin sphere that is not the same as the exterior Brillouin sphere. Thus, the exterior spherical harmonic gravity field cannot model the gravitation within the exterior Brillouin sphere except in some special cases, and the interior spherical harmonic gravity field cannot model the gravitation outside the interior Brillouin sphere. In this paper, we will discuss two types of other spherical harmonic gravity fields that bridge the null space of the exterior/interior gravity field expressions by solving Poisson’s equation. These two gravity fields are obtained by assuming the form of Helmholtz’s equation to Poisson’s equation. This method renders the gravitational potentials as functions of spherical Bessel functions and spherical harmonic coefficients. We refer to these gravity fields as the interior/exterior spherical Bessel gravity fields and study their characteristics. The interior spherical Bessel gravity field is investigated in detail for proximity operation purposes around small primitive bodies. Particularly, we apply the theory to asteroids Bennu (formerly 1999 RQ36) and Castalia to quantify its performance around both nearly spheroidal and contact-binary asteroids, respectively. Furthermore, comparisons between the exterior gravity field, interior gravity field, interior spherical Bessel gravity field, and polyhedral gravity field are made and recommendations are given in order to aid planning of proximity operations for future small body missions.     

Inversion of gravity data for giant planets

Arguments based on changes in viscosity of the lunar outer layers, changes in crater shapes due to a cold flow process and changes in the flux of crater forming planetesimals indicate that the period of premare time during which the vast majority of craters, large and small, were formed was several hundred million years long. The craters of all sizes were formed in exponentially declining numbers during this period with a half life of about 88×10⁶ yr.Recent work done by numerous students of RbSr, UPb and KAr ages of lunar rocks are beginning to recognize numerous events as having occured prior to the formation of Mare Imbrium and after the crust of the moon solidified.There was no major series of events which produced the “terminal lunar cataclysm” approximately 3.95×10⁹ yr ago. The magnitude and timing of the Imbrium collision was the single overwhelming event at that time.     

Lunar orbiter gravity analysis

This paper presents the results to date of the Jet Propulsion Laboratory (JPL) effort at analyzing the tracking data from the five Lunar Orbiter spacecraft. Emphasis is placed on the long-arc evaluation, to which most of the work was directed, rather than on the mascon analysis, which will be reported separately.The model of the Moon's gravity field that has been derived from the long-arc analysis is interpreted in terms of the Moon's mass distribution. The results are evaluated from the point of view of consistency and of statistical significance. In particular, the various factors contributing to the uncertainty in the estimation process are discussed.     

Lunar gravity parameters from line-of-sight acceleration data

A new approach is discussed for determination of lunar gravity parameters. Estimations were made with the line-of-sight (LOS) accelerations of artificial satellites. Tests indicate the advantages of such an approach over the classical one.     

Local Group velocity versus gravity: the coherence function

In maximum-likelihood analyses of the Local Group (LG) acceleration, the object describing non-linear effects is the coherence function (CF), i.e. the cross-correlation coefficient of the Fourier modes of the velocity and gravity fields. We study the CF both analytically, using perturbation theory, and numerically, using a hydrodynamic code. The dependence of the function on Ω_m and the shape of the power spectrum is very weak. The only cosmological parameter that the CF is strongly sensitive to is the normalization σ ₈ of the underlying density field. A perturbative approximation for the function turns out to be accurate as long as σ ₈ is smaller than about 0.3. For higher normalizations we provide an analytical fit for the CF as a function of σ ₈ and the wavevector. The characteristic decoherence scale which our formula predicts is an order of magnitude smaller than that determined by Strauss et al. This implies that present likelihood constraints on cosmological parameters from analyses of the LG acceleration are significantly tighter than hitherto reported.     

A far-ultraviolet photometer for planetary surface analysis

The measurement of local variations in the far-ultraviolet albedo is explored as a means of detecting changes in the refractive index of rocks and dust on the surface of atmosphereless planets and satellites. Far-ultraviolet spectrophotometric measurements of the lunar surface which were obtained on the Apollo 17 orbital mission are presented to demonstrate that significant albedo variations occur in the spectral range 120 to 170 nm. These data also confirm the hypothesis that the albedo variations represent refractive index differences in the surface materials. A three-band photometer is described which, when put in orbit around a solar system object, is capable of providing refractive index maps with a sensitivity of 1 part in the second decimal place and with kilometer resolution. Comparative surface composition and surface history analyses based on such maps are discussed.Alfred P. Sloan Research Fellow.     

A well-behaved f(R) gravity model in planetary motions

In this paper we consider asymptotic behavior of a hybrid action of f(R) gravity model which proposed by Saffari and Rahvar (2008), in the Solar system scale, which can explain Pioneer anomalous acceleration. We use the resultant weak field gravitational potential which comes from the hybrid action to test its impacts on the Solar system dynamics, by comparing theoretical precession of perihelion of a test particle, , with corrections to the standard Newtonian-Einstenian precessions of perihelia of some planets, which recently estimated by [Pitjeva, 2005a], [Pitjeva, 2005b], [22] and [23]. Here we show that the asymptotic behavior of hybrid action is in more accordance with observation relative to the other modifications such as power law and logarithmic corrections (Iorio, 2008). We also show that an extra additional lensing of the prediction of General Relativity is reproduced. Finally we consider the stability condition of planetary orbits in the presence of the hybrid action.     

Spherical harmonic analysis on hemisphere

The spherical harmonic analysis on hemisphere is reexamined to apply in helioseismology. New methods of mode identification are proposed to analyze the line-of-sight component which is expressed in terms of six spherical harmonics.     

A harmonic analysis of lunar topography

A global lunar topographic map has been derived from existing Earth-based and orbital observations supplemented in areas without data by a linear autocovariance predictor. Of 2592 bins, each 5° square, 1380 (64.7% by area) contain at least one measurement. A spherical harmonic analysis to degree 12 yields a mean radius of 1737.53 ± 0.03 km (formal standard error) and an offset of the center of figure of 1.98±0.06 km toward (19±2)°S, (194±1)°E. A Bouguer gravity map, derived from a 12-degree free-air gravity model and the present topography data, is presented for an elevation of 100 km above the mean surface. It is confirmed that the low-degree gravity harmonics are determined primarily by surface height variations and only secondarily by lateral density variations.     

Impact-generated dust clouds around planetary satellites: model versus Galileo data

This paper focuses on tenuous dust clouds of Jupiter's Galilean moons Europa, Ganymede and Callisto. In a companion paper (Srem?evi? et al., Planet. Space Sci. 51 (2003) 455-471) an analytical model of impact-generated ejecta dust clouds surrounding planetary satellites has been developed. The main aim of the model is to predict the asymmetries in the dust clouds which may arise from the orbital motion of the parent body through a field of impactors. The Galileo dust detector data from flybys at Europa, Ganymede and Callisto are compatible with the model, assuming projectiles to be interplanetary micrometeoroids. The analysis of the data suggests that two interplanetary impactor populations are most likely the source of the measured dust clouds: impactors with isotropically distributed velocities and micrometeoroids in retrograde orbits. Other impactor populations, namely those originating in the Jovian system, or interplanetary projectiles with low orbital eccentricities and inclinations, or interstellar stream particles, can be ruled out by the statistical analysis of the data. The data analysis also suggests that the mean ejecta velocity angle to the normal at the satellite surface is around 30°, which is in agreement with laboratory studies of the hypervelocity impacts.     

A dynamical analysis of the 14 Herculis planetary system

Precision radial velocity measurements of the Sun-like dwarf 14 Herculis published by Naef et al., Butler et al. and Wittenmyer, Endl & Cochran reveal a Jovian planet in a 1760-d orbit and a trend indicating the second distant object. On the grounds of dynamical considerations, we test a hypothesis that the trend can be explained by the presence of an additional giant planet. We derive dynamical limits to the orbital parameters of the putative outer Jovian companion in an orbit within ∼13 au. In this case, the mutual interactions between the Jovian planets are important for the long-term stability of the system. The best self-consistent and stable Newtonian fit to an edge-on configuration of Jovian planets has the outer planet in 9-au orbit with a moderate eccentricity of ∼0.2 and confined to a zone spanned by the low-order mean motion resonances 5:1 and 6:1. This solution lies in a shallow minimum of (χ²_ν)^1/2 and persists over a wide range of the system inclination. Other stable configurations within 1σ confidence interval of the best fit are possible for the semimajor axis of the outer planet in the range of (6,13) au and the eccentricity in the range of (0, 0.3). The orbital inclination cannot yet be determined but when it decreases, both planetary masses approach ∼10 m _J and for i ∼ 30° the hierarchy of the masses is reversed.     

Spherical harmonic analysis of steady photospheric flows

Steady photospheric flows can be represented by a spectrum of spherical harmonic modes. A technique is described in which full disc doppler velocity measurements are analysed using the spherical harmonic functions to determine the characteristics of this spectrum and the nature of these flows. Synthetic data is constructed for testing this technique. This data contains limb shift, rotation, differential rotation, meridional circulation, supergranules, giant cells and various levels of noise.The data is analysed in several steps. First, the limb shift is calculated by finding the average velocity in concentric rings about disc center. A polynomial representation of the limb shift is then removed from the data. Secondly, the rotation profile is calculated by finding an average slope in the velocity across the disc at each latitude position. This rotation profile is fit with Legendre polynomials and removed from the data. The third step is to find the meridional circulation by calculating the spherical harmonic transform for the axisymmetric poloidal modes and correcting for the effects of the limb shift analysis. The final step is to calculate the full spectrum of spherical harmonic components for the convective flows. Supergranules are separated from giant cells by spectral filtering for high (l >32) and low (l <32) wavenumbers, respectively.Some information about the spectrum is lost because only one hemisphere is seen, only the line-of-sight velocity is measured and the measurements contain noise. The lack of information about the motions on the backside of the Sun produces a broad smearing of the spectrum into nearby modes. The lack of information about the transverse velocity component produces a mixing between modes whose longitudinal wavenumbers differ by two and between the poloidal and toroidal components with the same wavenumber. In spite of this mode mixing much can be learned from this analysis. Solar rotation and differential rotation can be accurately measured and monitored for secular changes. Meridional circulations with small amplitudes can be measured and monitored and giant cells can be separated from supergranules.     

Turbulence in planetary occultations: II. Effects on atmospheric profiles derived from Doppler measurements

Turbulence in planetary atmospheres leads to both fluctuating and systematic errors in atmospheric profiles derived from Doppler measurements during radio occultation. If the upper atmospheres of Venus and Jupiter are about as turbulent as the earth's troposphere, we deduce rms fractional errors in temperature and pressure of less than ～ 10^?2 for the Mariner 10 and Pioneer 10/11 occultation experiments. Fractional systematic errors are typically of the order of 10^?6. These estimates depende rather weakly on quantities characterizing the atmosphere and the occultation, and it is conjectured that turbulence-induced errors in atmospheric profiles derived from Doppler measurements are always very small in the weak scattering limit     

Mars: Hellas Planitia gravity analysis

Doppler radio tracking data from Viking Orbiter 1 has provided new detailed observations of gravity variations over Hellas Planitia. Line-of-sight Bouguer gravity definitely indicates that isostatic adjustment has occurred. Two theoretical models were tested to obtain fits to the gravity data. Results for a surface deficit model, and a model with a surface deficit and a mass excess at depth are displayed. The mass-at-depth model produced very marked improvement in the data fit as compared to the surface deficit model. The optimum depth for the mass excess is 130 km.     

A new analysis of the GJ581 extrasolar planetary system

We have done a new analysis of the available observations of the GJ581 exoplanetary system. Today this system is controversial due to choices that can be done in the orbital determination. The main ones are the occurrence of aliases and the additional bodies??the planets f and g??announced in Vogt et?al. (Astrophys J 723:954?C965, 2010). Any dynamical study of exoplanets requires the good knowledge of the orbital elements and the investigations involving the planet g are particularly interesting, since this body would lie in the habitable zone (HZ) of the star GJ581. This region, for this system, is very attractive of the dynamical point of view due to several resonances of two and three bodies present there. In this work, we investigate the conditions under which the planet g may exist. We stress the fact that the planet g is intimately related with the orbital elements of the planet d; more precisely, we conclude that it is not possible to disconnect its existence from the determination of the eccentricity of the planet d. Concerning the planet f, we have found one solution with period ??450?days, but we are judicious about any affirmation concerning this body because its signal is in the threshold of detection and the high period is in a spectral region where the occurrence of aliases is very common. Besides, we outline some dynamical features of the HZ with the dynamical map and point out the role played by some resonances laying there.