Launch window study for the highly eccentric orbit satellite HEOS-1

A satellite with a high eccentricitye0.95 is strongly perturbed by the sun and the moon. This fact and mission constraints restrict considerably the possible launch times for such a satellte. The launch window calculations can be performed in two steps in order to save computing time. An approximate analytical solution provides a general survey of the launch opportunities. An accurate numerical approach is then necessary for the exact definition of the launch window. In the case of the orbit of HEOS-1 (satellite 68 10901), moreover the consideration of the injection errors has been of great importance.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969.     

Decay of a highly eccentric satellite

The re-entry phase of a highly eccentric satellite is discussed. Numerical simulations allowing the prediction of the exact date of re-entry of a highly eccentric satellite are exposed.It is shown that under very particular circumstances the life of the satellite can be extended by a few days. The number of final revolutions of the rapidly contracting orbit depends critically on the air density between 70 km and 100 km.Re-entry of the European scientific satellite HEOS-1 predicted for 28 October, 1975 is near such a situation.     

On the perturbation of the anomalistic period of a highly eccentric orbit satellite due to the zonal harmonics

In this note a simple formula is given for the perturbation of the anomalistic period of a highly eccentric orbit due to the zonal harmonics. This perturbation depends essentially only on the semi-major axisa, the eccentricitye (or pericentre radius r =a(1-e)) and the latitude of the pericentre.     

Lunar shape does not record a past eccentric orbit

The Moon has long been known to have an overall shape not consistent with expected past tidal forces. It has recently been suggested (Garrick-Bethell, I., Wisdom, J., Zuber, M.T. [2006]. Science 313, 652-655) that the present lunar moments of inertia indicate a past high-eccentricity orbit and, possibly, a past non-synchronous spin-orbit resonance. Here I show that the match between the lunar shape and the proposed orbital and spin states is much less conclusive than initially proposed. Garrick-Bethell et al. (Garrick-Bethell, I., Wisdom, J., Zuber, M.T. [2006]. Science 313, 652-655) spin and shape evolution scenarios also completely ignore the physics of the capture into such resonances, which require prior permanent deformation, as well as tidal despinning to the relevant resonance. If the early lunar orbit was eccentric, the Moon would have been rotating at an equilibrium non-synchronous rate determined by it eccentricity. This equilibrium supersynchronous rotation would be much too fast to allow a synchronous spin-orbit lock at e = 0.49, while the capture into the 3:2 resonance is possible only for a very constrained lunar eccentricity history and assuming some early permanent lunar tri-axiality. Here I show that large impacts in the early history of the Moon would have frequently disrupted this putative resonant rotation, making the rotation and eccentricity solutions of Garrick-Bethell et al. (Garrick-Bethell, I., Wisdom, J., Zuber, M.T. [2006]. Science 313, 652-655) unstable. I conclude that the present lunar shape cannot be used to support the hypothesis of an early eccentric lunar orbit.     

The formation of an eccentric gap in a gas disc by a planet in an eccentric orbit

We investigate the effect of a planet on an eccentric orbit on a two-dimensional low-mass gaseous disc. At a planet eccentricity above the planet's Hill radius divided by its semimajor axis, we find that the disc morphology differs from that exhibited by a disc containing a planet in a circular orbit. An eccentric gap is created with eccentricity that can exceed the planet's eccentricity and precesses with respect to the planet's orbit. We find that a more massive planet is required to open a gap when the planet is on an eccentric orbit. We attribute this behaviour to spiral density waves excited at corotation resonances by the eccentric planet. These act to increase the disc's eccentricity and exert a torque opposite in sign to that exerted by the Lindblad resonances. The reduced torque makes it more difficult for waves driven by the planet to overcome viscous inflow in the disc.     

Perturbations of the orbit of Explorer 19 due to solar radiation

The orbit of the balloon satellite, Explorer 19, is analysed to determine the effects of direct solar radiation pressure over one revolution of the satellite (111 min) for MJD 42822 and MJD 42966. At the earlier date, the satellite entered the Earth's shadow, presenting an opportunity to examine the effectiveness of two different shadow models. The reflectivity of the surface of the satellite was estimated from analysis of the variations in orbital eccentricity over a period of 236 days.Although many of the parameters associated with the shape and orientation of the satellite are unknown, the theory for a non-spherical satellite is applied using trial and error methods to determine the parameters of best fit. The paper concludes with an examination of the perturbations in orbital eccentricity and inclination due to incident, specularly reflected, and diffusely reflected radiation.     

A note on the tidally induced potential of a satellite in eccentric orbit

A planetary body moving on an eccentric orbit around the primary is subject to a periodic perturbing potential, affecting its internal mass distribution. In a previous paper (Rappaport et al., 1997, Icarus 126, 313), we have calculated the periodic modulation of the gravity coefficients of degree 2, for a body on a synchronous orbit. Here, the previous analysis is extended by considering also non-synchronous orbits, and by properly accounting for the apparent motion of the primary due to the non uniform motion along the elliptical orbit. The cases of Titan and Mercury are briefly discussed.     

Accurate computation of highly eccentric satellite orbits

For computing highly eccentric (e0.9) Earth satellite orbits with special perturbation methods, a comparison is made between different schemes, namely the direct integration of the equations of motion in Cartesian coordinates, changes of the independent variable, use of a time element, stabilization and use of regular elements. A one-step and a multi-step integration are also compared.It is shown that stabilization and regularization procedures are very helpful for non or smoothly perturbed orbits. In practical cases for space research where all perturbations are considered, these procedures are no longer so efficient. The recommended method in these cases is a multi-step integration of the Cartesian coordinates with a change of the independent variable defining an analytical step size regulation. However, the use of a time element and a stabilization procedure for the equations of motion improves the accuracy, except when a small step size is chosen.     

Exospheric zonal winds between 540 and 620 km from the orbit of Explorer 24

Data on the variation of the orbital inclination of the balloon satellite Explorer 24 (1964-76A) from 1964 to 1968 have been used to determine zonal winds between 540 and 620 km. In this height region the effect of zonal winds on the orbital inclination may become very small compared to other perturbations like accelerations due to the geopotential, lunisolar gravitation and the solar radiation pressure. It is demonstrated especially that the solar radiation pressure may become the most significant force changing the orbital inclination. The diurnal mean zonal winds derived from Explorer 24 point to an exospheric rotation rate which is about 6–10% less than the rotation rate of the Earth in the analyzed height region. Since the possible errors of the data analysis are of a similar order of magnitude, it can not be excluded that the exosphere corotates with the Earth. Furthermore, a local time dependence of the zonal winds could be detected. The diurnal varitation of the zonal wind is shown to be in good agreement with the theoretical model of Blum and Harris. Our results are discussed and compared with all previous investigations of orbital inclination changes of satellites above 350 km.     

The eccentric orbit of the short-period eclipsing binary V 1010 Oph

Radial velocity observations of V 1010 Oph obtained at Asiago from 23 spectra combined with those published by Guinan and Koch (1977) reveal a significant eccentricity (e0.20±0.03) with a periastron angle close to 90° at 1 level. This underlines the need for a new analysis of the light curve for photometric elements.On leave from Konkoly Observatory, Budapest, Hungary     

The rectilinear three-body problem as a basis for studying highly eccentric systems

The rectilinear elliptic restricted three-body problem (TBP) is the limiting case of the elliptic restricted TBP when the motion of the primaries is described by a Keplerian ellipse with eccentricity \(e'=1\), but the collision of the primaries is assumed to be a non-singular point. The rectilinear model has been proposed as a starting model for studying the dynamics of motion around highly eccentric binary systems. Broucke (AIAA J 7:1003–1009, 1969) explored the rectilinear problem and obtained isolated periodic orbits for mass parameter \(\mu =0.5\) (equal masses of the primaries). We found that all orbits obtained by Broucke are linearly unstable. We extend Broucke’s computations by using a finer search for symmetric periodic orbits and computing their linear stability. We found a large number of periodic orbits, but only eight of them were found to be linearly stable and are associated with particular mean motion resonances. These stable orbits are used as generating orbits for continuation with respect to \(\mu \) and \(e'<1\). Also, continuation of periodic solutions with respect to the mass of the small body can be applied by using the general TBP. FLI maps of dynamical stability show that stable periodic orbits are surrounded in phase space with regions of regular orbits indicating that systems of very highly eccentric orbits can be found in stable resonant configurations. As an application we present a stability study for the planetary system HD7449.     

Optimal periodic relative orbit and rectilinear relative orbits with eccentric reference orbits

The problem of two-body linearized periodic relative orbits with eccentric reference orbits is studied in this paper. The periodic relative orbit in the target-orbital coordinate system can be used in fly-around and formation-flying orbit design. Based on the closed-form solutions to the Tschauner–Hempel equations, the initial condition for periodic relative orbits is obtained. Then the minimum-fuel periodic-orbit condition with a single impulse is analytically derived for given initial position and velocity vectors. When considering the initial coasting time, the impulse position of the global minimum-fuel periodic orbit is proved to be near to the perigee of the target and can be obtained by numerical optimization algorithms. Moreover, the condition for a special periodic orbit, i.e., the rectilinear relative orbit in the target-orbital frame, is obtained. Numerical simulations are used to demonstrate the efficacy of the method, and show the geometry of the periodic relative orbit and the rectilinear relative orbit.     

Analytical linear perturbation theory for highly eccentric satellite orbits

Kaula's satellite linear perturbation theory has been extended for the case of highly eccentric orbits by using elliptic function expansions.On leave from Institute of Applied Astronomy, St.-Petersburg     

WASP-14b: 7.3 MJ transiting planet in an eccentric orbit

We report the discovery of a 7.3 M _J exoplanet WASP-14b, one of the most massive transiting exoplanets observed to date. The planet orbits the 10th-magnitude F5V star USNO-B1 11118−0262485 with a period of 2.243 752 d and orbital eccentricity   e = 0.09  . A simultaneous fit of the transit light curve and radial velocity measurements yields a planetary mass of 7.3 ± 0.5 M _J and a radius of 1.28 ± 0.08 R _J. This leads to a mean density of about 4.6 g cm⁻³ making it the densest transiting exoplanets yet found at an orbital period less than 3 d. We estimate this system to be at a distance of  160 ± 20  pc. Spectral analysis of the host star reveals a temperature of  6475 ± 100 K, log  g = 4.07  cm s⁻² and   v sin  i = 4.9 ± 1.0  km s⁻¹, and also a high lithium abundance,  log  N (Li) = 2.84 ± 0.05  . The stellar density, effective temperature and rotation rate suggest an age for the system of about 0.5–1.0 Gyr.     

Magnetic instability in a rotating layer at highly eccentric positions of the critical level

This study follows the numerical results presented in Marsenić & Ševčík (2010) that explored the influence of the critical level position on stability of a system. The model was a horizontal fluid layer between z = ±0.5d rotating with an angular velocity Ω = Ω₀ ž about the vertical axis z . The fluid was considered to be inviscid, finitely electrically conducting and incompressible and was permeated by a horizontal magnetic field B ₀ = ℬ︁₀B₀(z) y̌ , where ℬ︁₀ was the magnitude of the field and the function B₀(z) = tanh [γ (z –z₀)]. When γ is large, the field gradient is concentrated near z = z₀, the critical level, the field being almost homogeneous elsewhere. In this way it controls the width of the magnetic shear layer associated with the current sheet. It was found that at conditions when the magnetic field gradient was large enough (γ = O (10)) and the critical level was placed close enough to the (bottom) perfectly conducting boundary (z₀ < –0.388d for γ = 80), magnetically driven convection appeared localized to a close neighbourhood of the critical level – the so called critical layer. Based on the circumstances of its rise and its properties it was identified with the resistive tearing‐mode instability. This paper presents an analytical treatment of the problem assuming γ ≥ 1. The approach consists in separation of the computational domain into an outer region where the diffusionless limit (Elsasser number Λ → ∞) applies and an inner region (the critical layer) of finite conductivity. According to the tearing‐mode theory in classical systems, the solution in the inner region is sought as long‐wavelength with respect to the width of the critical layer. The obtained solution shows features similar to the one obtained numerically and confirmed relevance of the simplifying physical assumptions made in each region. The convection in the critical layer is strictly conditioned by a sharp magnetic shear. If the shear region is removed by further positioning of the critical level towards the perfectly conducting boundary, the localized convection disappears. It is in compliance with the fact that the system is stabilised by a perfectly conducting boundary with respect to the tearing mode. Stability is then checked numerically in the layer bounded by perfectly conducting boundaries where the critical point of the magnetic field lies on one of them. The existence of a magnetically driven instability is confirmed. Depending on the value γ, it may rise as a stationary convection (for γ < 1.5) or as a wave which for γ > 16 exhibits similarity properties (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Length of arc as independent argument for highly eccentric orbits

For analytic step regulation in numerical integration of highly eccentric orbits it is proposed to use the orbital arc length of a moving particle as independent argument.     

Tidal interactions and disruptions of giant planets on highly eccentric orbits

We calculate the evolution of planets undergoing a strong tidal encounter using smoothed particle hydrodynamics (SPH), for a range of periastron separations. We find that outside the Roche limit, the evolution of the planet is well-described by the standard model of linear, non-radial, adiabatic oscillations. If the planet passes within the Roche limit at periastron, however, mass can be stripped from it, but in no case do we find enough energy transferred to the planet to lead to complete disruption. In light of the three new extrasolar planets discovered with periods shorter than two days, we argue that the shortest-period cases observed in the period-mass relation may be explained by a model whereby planets undergo strong tidal encounters with stars, after either being scattered by dynamical interactions into highly eccentric orbits, or tidally captured from nearly parabolic orbits. Although this scenario does provide a natural explanation for the edge found for planets at twice the Roche limit, it does not explain how such planets will survive the inevitable expansion that results from energy injection during tidal circularization.     

The short-period eclipsing binary V1010 Ophiuchi does it have an eccentric orbit?

V1010 Oph is an unusually complicated close binary. Leung and Wilson (1977) found the system had an overcontact configuration. Subsequently, Margoniet al. (1981) claimed that the system had an eccentric orbit withe=0.25 based upon their spectroscopic study. We computed a theoretical radial velocity curve (accounting for tidal, reflection, and eclipse effects) based on the photometric parameters (e.g.,e=0) of Leung and Wilson. The computed curve fits the Asiago and Kitt Peak data very well. It is suggested that the asymmetry in the observed radial velocity arose from tidal, reflection, and eclipse effects rather than from orbital eccentricity. Other arguments against an eccentric orbit for V1010 Oph are also discussed. It is concluded that the eccentricity derived from the spectroscopic study may be spurious.On leave from Beijing University, China.     

Ephemeris of Pluto

The limited period of observational data and the long orbital period have limited the accuracy of the Pluto ephemerides. This condition will continue for some time. Ephemerides of Pluto are compared with the observational data.     

The Parkes Multibeam Pulsar Survey: PSR J1811−1736, a pulsar in a highly eccentric binary system

We are undertaking a high-frequency survey of the Galactic plane for radio pulsars, using the 13-element multibeam receiver on the 64-m Parkes radio telescope. We describe briefly the survey system and some of the initial results. PSR J1811−1736, one of the first pulsars discovered with this system, has a rotation period of 104 ms. Subsequent timing observations using the 76-m radio telescope at Jodrell Bank show that it is in an 18.8-d, highly eccentric binary orbit. We have measured the rate of advance of periastron which indicates a total system mass of 2.6±0.9 M_⊙, and the minimum companion mass is about 0.7 M_⊙. This, the high orbital eccentricity and the recycled nature of the pulsar suggest that this system is composed of two neutron stars, only the fourth or fifth such system known in the disc of the Galaxy.