Frozen orbits for satellites close to an Earth-like planet

We say that a planet is Earth-like if the coefficient of the second order zonal harmonic dominates all other coefficients in the gravity field. This paper concerns the zonal problem for satellites around an Earth-like planet, all other perturbations excluded. The potential contains all zonal coefficientsJ ₂ throughJ ₉. The model problem is averaged over the mean anomaly by a Lie transformation to the second order; we produce the resulting Hamiltonian as a Fourier series in the argument of perigee whose coefficients are algebraic functions of the eccentricity — not truncated power series. We then proceed to a global exploration of the equilibria in the averaged problem. These singularities which aerospace engineers know by the name of frozen orbits are located by solving the equilibria equations in two ways, (1) analytically in the neighborhood of either the zero eccentricity or the critical inclination, and (2) numerically by a Newton-Raphson iteration applied to an approximate position read from the color map of the phase flow. The analytical solutions we supply in full to assist space engineers in designing survey missions. We pay special attention to the manner in which additional zonal coefficients affect the evolution of bifurcations we had traced earlier in the main problem (J ₂ only). In particular, we examine the manner in which the odd zonalJ ₃ breaks the discrete symmetry inherent to the even zonal problem. In the even case, we find that Vinti's problem (J ₄+J ₂ ² =0) presents a degeneracy in the form of non-isolated equilibria; we surmise that the degeneracy is a reflection of the fact that Vinti's problem is separable. By numerical continuation we have discovered three families of frozen orbits in the full zonal problem under consideration; (1) a family of stable equilibria starting from the equatorial plane and tending to the critical inclination; (2) an unstable family arising from the bifurcation at the critical inclination; (3) a stable family also arising from that bifurcation and terminating with a polar orbit. Except in the neighborhood of the critical inclination, orbits in the stable families have very small eccentricities, and are thus well suited for survey missions.     

Secular resonances between bodies on close orbits II: prograde and retrograde orbits for irregular satellites

In extending the analysis of the four secular resonances between close orbits in Li and Christou (Celest Mech Dyn Astron 125:133–160, 2016) (Paper I), we generalise the semianalytical model so that it applies to both prograde and retrograde orbits with a one-to-one map between the resonances in the two regimes. We propose the general form of the critical angle to be a linear combination of apsidal and nodal differences between the two orbits \( b_1 \Delta \varpi + b_2 \Delta \varOmega \), forming a collection of secular resonances in which the ones studied in Paper I are among the strongest. Test of the model in the orbital vicinity of massive satellites with physical and orbital parameters similar to those of the irregular satellites Himalia at Jupiter and Phoebe at Saturn shows that \({>}20\) and \({>}40\%\) of phase space is affected by these resonances, respectively. The survivability of the resonances is confirmed using numerical integration of the full Newtonian equations of motion. We observe that the lowest order resonances with \(b_1+|b_2|\le 3\) persist, while even higher-order resonances, up to \(b_1+|b_2|\ge 7\), survive. Depending on the mass, between 10 and 60% of the integrated test particles are captured in these secular resonances, in agreement with the phase space analysis in the semianalytical model.     

An analytic method to account for drag in the Vinti satellite theory

In order to retain separability in the Vinti theory of Earth satellite motion when a nonconservative force such as air drag is considered, a set of variational equations for the orbital elements are introduced, and expressed as functions of the transverse, radial, and normal components of the nonconservative forces acting on the system. In this approach, the Hamiltonian is preserved in form, and remains the total energy, but the initial or boundary conditions and hence the Jacobi constants of the motion advance with time through the variational equations. In particular, the atmospheric density profile is written as a fitted exponential function of the eccentric anomaly, which adheres to tabular data at all, altitudes and simultaneously reduces the variational equations to definite integrals with closed form evaluations whose limits are in terms of the eccentric anomaly. The values of the limits for any arbitrary time interval are obtained from the Vinti program.Results of this technique for the case of the intense air drag satellites San Marco-2 and Air Force Cannonball are given. These results indicate that the satellite ephemerides produced by this theory in conjunction with the Vinti program are of very high accuracy. In addition, since the program is entirely analytic, several months of ephemerides can be obtained within a few seconds of computer time.     

Life near the Roche limit: Behavior of ejecta from satellites close to planets

The distribution of ejecta from impact craters significantly affects the surface characters of satellites and asteroids. In order to understand better the distinctive features seen on Phobos, Deimos, and Amalthea, we study the dynamics of nearby debris but include several factors — planetary tides plus satellite rotation and nonspherical shape-that complicate the problem. We have taken several different approaches to investigate the behavior of ejecta from satellites near planets. For example, we have calculated numerically the usual pseudoenergy (Jacobi) integral. This is done in the framework of a restricted three-body problem where we model the satellites as triaxial ellipsoids rather than point masses as in past work. Iso-contours of this integral show that Deimos and Amalthea are entirely enclosed by their Roche lobes, and the surfaces of their model ellipsoids lie nearly along equipotentials. Presumably this was once also the case for Phobos, before tidal evolution brought it so close to Mars. Presently the surface of Phobos overflows its Roche lobe, except for the regions within a few kilometers of the sub- and anti-Mars points. Thus most surface material on Phobos is not energetically bound: nevertheless it is retained by the satellite because local gravity has an inward component everywhere. Similar situations probably prevail for the newly discovered satellite of Jupiter (J14) and for the several objects found just outside Saturn's rings. We have also examined the fate of crater ejecta from the satellites of Mars by numerical integration of trajectories for particles leaving their surfaces in the equatorial plane. The ejecta behavior depends dramatically on the longitude of the primary impact, as well as on the speed and direction of ejection. Material thrown farther than a few degrees of longitude remains in flight for an appreciable time. Over intervals of an hour or more, the satellites travel through substantial arcs of their orbits, so that the Coriolis effect then becomes important. For this reason the limit of debris deposition is elongated toward the west while debris thrown to the east escapes at lower ejection velocities. We display some typical trajectories, which include many interesting special effects, such as loops, cusps, “folded” ejecta blankets, and even a temporary satellite of Deimos. Besides being important for understanding the formation of surface features on satellites, our work is perhaps pertinent to regolith development on small satellites and asteroids, and also to the budgets of dust belts around planets.     

An averaging method to study the motion of lunar artificial satellites I: Disturbing Function

We describe a semi-analytical averaging method aimed at the computation of the motion of an artificial satellite of the Moon. In this paper, the first of the two part study, we expand the disturbing function with respect to the small parameters. In particular, a semi-analytic theory of the motion of the Moon around the Earth and the libration of the lunar equatorial plane using different reference frames are introduced. The second part of this article shows that the choice of the canonical Poincaré variables lead to equations in closed form without singularities in e = 0 or I = 0. We introduce new expressions that are sufficiently compact to be used for the study of any artificial satellite. This revised version was published online in July 2006 with corrections to the Cover Date.     

An averaging method to study the motion of lunar artificial satellites II: Averaging and Applications

On this, the second part of a two part study (Steichen, 1998) we further develop a semi-analytical theory for a lunar artificial satellite. This theory is obtained by averaging analytically the Hamiltonian function over period up to a month. The averaged equations are then numerically integrated. The solution is free from singularities at e = 0 and I = 0 and is not expanded in powers of these variables. In the last section, the analytic work is applied to characteristic examples to validate the method used. This revised version was published online in July 2006 with corrections to the Cover Date.     

Secular resonances between bodies on close orbits: a case study of the Himalia prograde group of jovian irregular satellites

The gravitational interaction between two objects on similar orbits can effect noticeable changes in the orbital evolution even if the ratio of their masses to that of the central body is vanishingly small. Christou (Icarus 174:215–229, 2005) observed an occasional resonant lock in the differential node \(\varDelta \varOmega \) between two members in the Himalia irregular satellite group of Jupiter in the N-body simulations (corresponding mass ratio \(\sim 10^{-9}\)). Using a semianalytical approach, we have reproduced this phenomenon. We also demonstrate the existence of two additional types of resonance, involving angle differences \(\varDelta \omega \) and \(\varDelta (\varOmega +\varpi )\) between two group members. These resonances cause secular oscillations in eccentricity and/or inclination on timescales \(\sim \)1 Myr. We locate these resonances in (a, e, i) space and analyse their topological structure. In subsequent N-body simulations, we confirm these three resonances and find a fourth one involving \(\varDelta \varpi \). In addition, we study the occurrence rates and the stability of the four resonances from a statistical perspective by integrating 1000 test particles for 100 Myr. We find \(\sim \)10 to 30 librators for each of the resonances. Particularly, the nodal resonance found by Christou is the most stable: 2 particles are observed to stay in libration for the entire integration.     

The astrometric observations of planetary satellites, close approaches and occultations of stars by asteroids and mutual events in the systems of planetary satellites with the 26-in. refractor of Pulkovo observatory in 1995-2006

Astrometric and photometric observations of major planets, their satellites and asteroids have been made with the 26-in. refractor of the Pulkovo observatory during the period from 1995 to 2006. The CCD (ST6) and photographic observations were carried out. Accurate relative position of satellites of Jupiter and Saturn have been derived. The positions of Saturn have been calculated using the theoretically predicted coordinates of satellites relative to the planet without measurements of the photographic images of the planet. Also the observations of Hale-Bopp comet and Mercury transit have been made. The 26-in. refractor has been included into the international campaign PHEMU-2003: photometric CCD observations of mutual occultations and eclipses of Galilean satellites. The light curves of the events have been obtained and parameters of the events have been determined.     

An analytic column density profile to fit prestellar cores

We present a new analytical three-parameter formula to fit observed column density profiles of prestellar cores. It represents a line-of-sight integral through a spherically symmetric or disc-like isothermal cloud. The underlying model resembles the Bonnor–Ebert model in that it features a flat central region leading into a power-law decline  ∝ r ⁻²  in density, and a well-defined outer radius. However, we do not assume that the cloud is in equilibrium, and can instead make qualitative statements about its dynamical state (expansion, equilibrium, collapse) using the size of the flat region as a proxy. Instead of having temperature as a fitting parameter, our model includes it as input, and thus avoids possible inconsistencies. It is significantly easier to fit to observational data than the Bonnor–Ebert sphere. We apply this model to L1689B and B68. We show that L1689B cannot be in equilibrium but instead appears to be collapsing, while our model verifies that B68 is not far from being a hydrostatic object.     

A geometric method to determine the distances of galaxies

Es wird die Möglichkeit diskutiert, die Größenverteilungsfunktion von Kugelsternhaufen für die Entfernungsbestimmung von Galaxien zu nutzen.     

Forthcoming close angular approaches of planets to radio sources and possibilities to use them as GR tests

During close angular approaches of solar system planets to astrometric radio sources, the apparent positions of these sources shift due to relativistic effects and, thus, these events may be used for testing the theory of general relativity; this fact was successfully demonstrated in the experiments on the measurements of radio source position shifts during the approaches of Jupiter carried out in 1988 and 2002. An analysis, performed within the frames of the present work, showed that when a source is observed near a planet’s disk edge, i.e., practically in the case of occultation, the current experimental accuracy makes it possible to measure the relativistic effects for all planets. However, radio occultations are fairly rare events. At the same time, only Jupiter and Saturn provide noticeable relativistic effects approaching the radio sources at angular distances of about a few planet radii. Our analysis resulted in the creation of a catalog of forthcoming occultations and approaches of planets to astrometric radio sources for the time period of 2008–2050, which can be used for planning experiments on testing gravity theories and other purposes. For all events included in the catalog, the main relativistic effects are calculated both for ground-based and space (Earth-Moon) interferometer baselines.     

An analytic solution to the classical two-body problem with drag

An analytic solution to the two-body problem with a specific drag model is obtained. The model treats drag as a force proportional to the vector velocity and inversely proportional to the square of the distance to the center of attraction. The solution is expressed in terms of known functions and is of a simple and compact form. The time-of-flight is expressed as a quadrature in the ‘true anomaly’.     

An orbiting clock experiment to determine the gravitational red shift

Plans are presented for an experiment to measure the gravitational red shift of the Earth by comparing a ground-based and satellite-borne hydrogen maser clock. The limiting accuracy is estimated to yield a determination of the red shift to 1 part in 10⁵, corresponding to a clock stability of 3 parts in 10¹⁵.This work was supported by the National Aeronautics and Space Administration.     

An analytic solution to the hypersonic, radiative blunt body problem

We present an analytic model for the thin-shell, radiative interaction between a hypersonic, plane-parallel wind and a rigid, spherical obstacle. This problem has clear applications, e.g., to the interaction of winds from young stars and dense cloudlets, and to the interaction of the wind from a binary partner with the photosphere of the second star. We also present a comparison of the analytic model with a full, axisymmetric numerical simulation. We find only a partial agreement between the numerical simulation and the analytic model, apparently as a result of the very strong 'thin-shell' instabilities of the post-bow shock flow. Our analytic model predicts the surface density, flow velocity and the energy radiated per unit area, as well as the total luminosity of the bow shock. The model can therefore be used directly for carrying out comparisons with observations of different astrophysical objects.     

An analytic approach to the connection between stellar structure parameters and the neutrino emission rate in a simple stellar model

In order to obtain the internal structure of a main-sequence star such as the Sun usually one has to solve the detailed structure equations numerically. This paper is an attempt to construct analytic models for the stellar nuclear energy generation. We give closed-form analytic results for the stellar luminosity and stellar neutrino emission rate when the radial matter density of the spherical star under consideration is linear. For the numerical estimation of the neutrino flux of a specified stellar nuclear reaction we take into account parameters of the standard solar model. The present paper gives for the first time the connection between stellar structure parameters and neutrino fluxes in an analytic stellar model.     

A method to determine regional lunar gravity fields from earth-based satellite tracking data

A method to determine regional gravity fields of the Moon from Earth-based Doppler and range satellite tracking data residuals of a low Moon-orbiting satellite has been developed and thoroughly tested in a controlled simulation environment. A short-arc approach, where one arc consists of the time it takes the satellite to cross the grid of interest on the lunar surface, is used in order to filter out most long-wavelength signal that can still be present in the residuals. Simulation results where the data are contaminated with either typical systematic or stochastic noise show that recovery of the local gravity field down to the level of several mGal is possible. The inclusion of extremely low-altitude data also means that regularisation in the sense of including a priori information in the form of a regularisation matrix is not necessary in order to obtain a good solution at high resolution.     

The angular width of the coherent back-scatter opposition effect: An application to icy outer planet satellites

It has been suggested recently that coherent back-scattering of light from powder-like regolithic surfaces can explain remarkable opposition brightening of some atmosphereless solar system bodies. In this paper, a dense-medium light-scattering theory is used to calculate the half-width at half-maximum (HWHM) of the coherent back-scattering peak for a number of scattering models. We demonstrate that HWHM strongly depends on the optical properties of the scattering medium and can serve as a critical test in comparing alternative models. It is shown that coherent back-scaterring may be a likely explanation of the opposition effect exhibited by icy outer planet satellites.     

A method to determine the atmospheric temperature profile from in situ pressure data: Application to Titan

We have developed a new method to analyse in situ observations of atmospheric variables of state: the reconstruction of the vertical temperature profile from pressure measurements accompanied by rough knowledge of the atmospheric composition and the aerodynamical response properties of the descent vehicle. We can use the method to construct the temperature profile when no direct measurements are available, as well as to analyse the consistency between data from different instruments. We applied the method to the Huygens measurements of Titan's atmosphere, determining the aerodynamical drag properties from radar altimeter data. We discovered that the temperature profile computed in this manner differs from the profile from the temperature sensor (TEM) of the probe by up to 5% in the altitude range of 0–60 km, and up to 10% at higher altitudes due to increased noise. The method gives a tropopause altitude of about 50 km and a surface temperature of about 98 K, in contrast to the TEM temperature measurements. Our error analysis shows that these differences are caused by the known discrepancy in the Huygens altimeter data, with the estimates made by the reconstruction algorithm contributing only 1–2% of error.