The critical inclination in artificial satellite theory

Certain it is that the critical inclination in the main problem of artificial satellite theory is an intrinsic singularity. Its significance stems from two geometric events in the reduced phase space on the manifolds of constant polar angular momentum and constant Delaunay action. In the neighborhood of the critical inclination, along the family of circular orbits, there appear two Hopf bifurcations, to each of which there converge two families of orbits with stationary perigees. On the stretch between the bifurcations, the circular orbits in the planes at critical inclinmation are unstable. A global analysis of the double forking is made possible by the realization that the reduced phase space consists of bundles of two-dimensional spheres. Extensive numerical integrations illustrate the transitions in the phase flow on the spheres as the system passes through the bifurcations.A delicacy so very susceptible of offence...—Hester Lynch PIOZZI,Observations and Reflections made in the Course of a Journey through France, Italy and Germany (1789)NAS/NRC Postgraduate Research Associate in 1984–1985.     

Magnetic field inclination angle at geosynchronous orbit

Simultaneous magnetic field data from the geostationary satellites GOES 2 and GOES 3 have shown significant differences between the inclination angles measured at these two satellites. The relationship between the two inclination angles is examined for three magnetic activity groups and eight local time sectors by using data from GOES 2 and GOES 3 for the 12-month period from March 1979 to February 1980. During this time GOES 2 was 1.09 R_E off the geomagnetic equator in the northern geomagnetic hemisphere, while GOES 3 was separated from GOES 2 by 2 h L.T. and 0.47 R_E off the geomagnetic equator, also in the northern geomagnetic hemisphere. The average inclination angle is found to differ substantially from the predictions based on the previously published magnetic field models. The relationship between the two inclination angles is fitted by a simple model of magnetic field perturbation that varies as a power of distance from the Equator. During magnetic disturbances, the spatial variation of magnetic field perturbation on the nightside suggests that the ring current density increases with distance from the geomagnetic equator.     

Relativistic effects in the critical inclination problem in artificial satellite theory

It is well known that in artifical satellite theory special techniques must be employed to construct a formal solution whenever the orbital inclination is sufficiently close to the critical value cos^–1 (1/5). In this article the authors investigate the consequences of introducing certain relativistic effects into the motion of a satellite about an oblate primary. Particular attention is paid to the critical inclination(s), and for such critical motions an appropriate method of solution is formulated.     

Luni-solar effects of geosynchronous orbits at the critical inclination

The luni-solar effects of a geosynchronous artificial satellite orbiting near the critical inclination is investigated. To tackle this four-degrees-of-freedom problem, a preliminary exploration separately analyzing each harmonic formed by a combination of the satellite longitude of the node and the Moon longitude of the node is opportune. This study demonstrates that the dynamics induced by these harmonics does not show resonance phenomena. In a second approach, the number of degrees of freedom is halved by averaging the total Hamiltonian over the two non-resonant angular variables. A semi-numerical method can now be applied as was done when considering solely the inhomogeneity of the geopotential (see Delhaise et Henrard, 1992). Approximate surfaces of section are constructed in the plane of the inclination and argument of perigee. The main effects of the Sun and Moon attractions compared to the terrestrial attraction alone are a strong increase in the amplitude of libration in inclination (from 0.6° to 3.2°) and a decrease of the corresponding libration period (from the order of 200 years to the order of 20 years).Research Assistant for the Belgian National Fund for Scientific Research     

The problem of the critical inclination revisited

The behaviour of the argument of the pericentre is investigated for the orbit of an artificial satellite which is moving under the potential when the inclination of the orbit is close to thecritical value tan^?1 2. The theory is developed to first order and it is applicable to all values of the eccentricity, with the exception of those in the neighbourhood of zero and unity. Four principal types of behaviour are noted and these are illustrated in appropriate phase-plane diagrams. It is shown that the two types which exhibit double libration in the argument of the pericentre are restricted to a relatively small domain in the (a, e)-plane of possible motions. Moreover, it is demonstrated that for double libration to occur it is necessary, but not sufficient, that \(e > \sqrt 6/13\) . The ranges of values of the inclination for which libration of the pericentre is a possibility are given for the more important cases. The general results are applied to the specific case of artificial Earth satellites whose orbits are inclined to the equator at angles close to the value of the critical inclination.     

How critical is the critical inclination?

It has been claimed that the representation of satellite motion in the vicinity of the critical inclination is a matter of practical, as well as theoretical interest, since the perturbations in the coordinates are of the order of 25 times greater near the critical inclination than away from it (Messageet al., 1962). In this paper we show, using Encke's method of numerical integration for satellites which are at, near, and away from the critical inclination, that there are no discernible features in the coordinate perturbations which distinguish the critical inclination from any other.     

Extension of the critical inclination

The critical inclination is of special interest in artificial satellite theory. The critical inclination can maintain minimal deviations of eccentricity and argument of pericentre from the initial values, and orbits at this inclination have been applied to some space missions. Most previous researches about the critical inclination were made under the assumption that the oblateness term J ₂ is dominant among the harmonic coefficients. This paper investigates the extension of the critical inclination where the concept of the critical inclination is different from that of the traditional sense. First, the study takes the case of Venus for instance, and provides some preliminary results. Then for general cases, given the values of argument of pericentre and eccentricity, the relationship between the multiplicity of the solutions for the critical inclination and the values of J ₂ and J ₄ is analyzed. Besides, when given certain values of J ₂ and J ₄, the relationship between the multiplicity of the solutions for the critical inclination and the values of semimajor axis and eccentricity is studied. The results show that for some cases, the value of the critical inclination is far away from that of the traditional sense or even has multiple solutions. The analysis in this paper could be used as starters of correction methods in the full gravity field of celestial bodies.     

The problem of critical inclination combined with a resonance in mean motion in artificial satellite theory

In this paper the two-degree of freedom problem of a geosynchronous artificial satellite orbiting near the critical inclination is studied. First a local approach of this problem is considered. A semi-numerical method, well suited to describe the perturbations of a non-trivial separable system, is then applied such that surfaces of section illustrating the global secular dynamics are obtained. The results are confirmed by numerical integrations of the full Hamiltonian.Research Assistant for the Belgian National Fund for Scientific Research     

The critical inclination problem with small eccentricity

The general theory described in an earlier paper is used to investigate the long-term behavior of the eccentricity and argument of the perigee for Earth satellites which have nearly circular orbits and orbital inclinations in the neighbourhood of the critical value 63°.4. Taking into account the effect of all the zonal harmonics, it is shown that there are just twoTypes of behaviour possible, compared with the six described in the earlier general theory and the four found by Aoki (1963).     

The critical inclination problem with small eccentricity

A qualitative solution is presented of the critical inclination problem in artificial satellite theory for motions in which the orbits are nearly circular. The effects of all the zonal harmonics are taken into account, and bothshallow anddeep resonance regimes are considered. An investigation of the (e sing,e cosg)-plane reveals that six fundamentally different types of phase-plane portraits exist. These portraits illustrate the long-term behaviour of the eccentricity and line of apsides.     

The determination of the satellite orbit of Mariner 9

This paper presents a comprehensive analysis of the Mars orbital phase of the Mariner 9 trajectory as determined from Earth based radio data. Both the method and accuracy of the orbit determination process are reviewed. Analysis is presented to show the effects of Mars gravity model and node in the plane of the sky errors on the accuracy of orbit determination. In addition the long term evolution of the orbit from insertion through the first 500 revolutions is presented, and decomposed into effects from the Mars garvity field,n-body perturbations, and solar radiation pressure. Since the orbit period is nearly commensurable with the Mars rotational period, the orbit experiences significant resonance perturbations. The primary perturbation is in-track with a maximum amplitude of 1000 km and a wavelength of 39 spacecraft revolutions.This paper was presented at the AIAA/AAS Astrodynamics Conference, Palo Alto, California, September 11 and 12, 1972. At this time Mariner 9 operations were still underway. The operational life of Mariner 9 ended October 27, 1972, when the supply of nitrogen gas, used for attitude stabilization, was depleted. This paper represents one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, under NASA Contract No. NAS 7-100.     

The global solution of the problem of the critical inclination

The publication of the solution of the Ideal Resonance Problem (Garfinkelet al., 1971) has opened the way for a complete first-orderglobal theory of the motion of an artificial satellite, valid for all inclinations. Previous attempts at such a theory have been only partially successful. With the potential function restricted to $$V = - 1/r + J_2 P_2 (\sin \theta )/r^3 + J_4 P_4 (\sin \theta )/r^5 ,$$ the paper constructs aglobal solution of the first order in √J ₂ for the Delaunay variablesG, g, h, l and for the coordinatesr, θ, and ?. As a check, it is shown that this solution includes asymptotically theclassical limit with the critical divisor 5 cos² i?1. The solution is subject to thenormality condition $$eG^2 /(1 + \frac{{45}}{4}e^2 ) \geqslant O\left[ {\left| {\frac{1}{5}(J_2 + J_4 /J_2 )} \right|^{1/4} } \right],$$ which bounds the eccentricitye away from zero in deep resonance. A historical section orients this work with respect to the contributions of Hori (1960), Izsak (1962), and Jupp (1968).     

The critical inclination problem: A simple treatment

A short analysis is presented in the hope of clarifying the situation.     

Sun-synchronous orbits near critical inclination

The long period dynamics of Sun-synchronous orbits near the critical inclination 116.6° are investigated. It is known that, at the critical inclination, the average perigee location is unchanged by Earth oblateness. For certain values of semimajor axis and eccentricity, orbit plane precession caused by Earth oblateness is synchronous with the mean orbital motion of the apparent Sun (a Sun-synchronism). Sun-synchronous orbits have been used extensively in meteorological and remote sensing satellite missions. Gravitational perturbations arising from an aspherical Earth, the Moon, and the Sun cause long period fluctuations in the mean argument of perigee, eccentricity, inclination, and ascending node. Double resonance occurs because slow oscillations in the perigee and Sun-referenced ascending node are coupled through the solar gravity gradient. It is shown that the total number and infinitesimal stability of equilibrium solutions can change abruptly over the Sun-synchronous range of semimajor axis values (1.54 to 1.70 Earth radii). The effect of direct solar radiation pressure upon certain stable equilibria is investigated.     

The optimization of satellite orbit for Space-VLBI observation

By sending one or more telescopes into space,Space-VLBI(SVLBI)is able to achieve even higher angular resolution and is therefore the trend of the VLBI technique.For the SVLBI program,the design of satellite orbits plays an important role for the success of planned observation.In this paper,we present our orbit optimization scheme,so as to facilitate the design of satellite orbits for SVLBI observation.To achieve that,we characterize the uv coverage with a measure index and minimize it by finding out the corresponding orbit configuration.In this way,the design of satellite orbit is converted to an optimization problem.We can prove that,with an appropriate global minimization method,the best orbit configuration can be found within the reasonable time.Besides that,we demonstrate that this scheme can be used for the scheduling of SVLBI observations.     

Balanced earth satellite orbits

An analytic model for third-body perturbations and for the second zonal harmonic of the central body's gravitational field is presented. A simplified version of this model applied to the Earth-Moon-Sun system indicates the existence of high-altitude and highly-inclined orbits with their apsides in the equator plane, for which the apsidal as well as the nodal motion ceases. For special positions of the node, secular changes of eccentricity and inclination disappear too (balanced orbits). For an ascending node at vernal equinox, the inclination of balanced orbits is 94.56°, for a node at autumnal equinox 85.44°, independent of the eccentricity of the orbit. For a node perpendicular to the equinox, there exist circular balanced orbits at 90° inclination. By slightly adjusting the initial inclination as suggested by the simplified model, orbits can be found — calculated by the full model or by different methods — that show only minor variations in eccentricity, inclination, argument of perigee, and longitude of the ascending node for 10⁵ revolutions and more. Orbits near the unstable equilibria at 94.56° and 85.44° inclination show very long periodic librations and oscillations between retrogade and prograde motion.Retired from IBM Vienna Software Development Laboratory.     

On the true circular orbit of a satellite

The osculating orbit of a planetary satellite moving in the equatorial plane of the central body under the influence of a rotational symmetric perturbation force is elliptical in first order approximation even if the true orbit is always circular. The satellite motion is influenced by a resonance effect due to this perturbing force. An inclined true satellite orbit cannot be circular.     

Critical inclination in the main problem of a massive satellite

The classical problem of the critical inclination in artificial satellite theory has been extended to the case when a satellite may have an arbitrary, significant mass and the rotation momentum vector is tilted with respect to the symmetry axis of the planet. If the planet’s potential is restricted to the second zonal harmonic, according to the assumptions of the main problem of the satellite theory, two various phenomena can be observed: a critical inclination that asymptotically tends to the well known negligible mass limit, and a critical tilt that can be attributed to the effect of transforming the gravity field harmonics to a different reference frame. Stability of this particular solution of the two rigid bodies problem is studied analytically using a simple pendulum approximation.