Analysis of the orbit of 1968-90A (Cosmos 248)

The satellite 1968-90A (Cosmos 248), was launched in October 1968 into an orbit inclined at 62.25° to the equator, with an initial perigee height of 475 km, apogee height 543 km, and orbital period 94.8 min. The orbit has been determined at 57 epochs over nearly one and a quarter cycles of the argument of perigee from January 1972 until December 1975 with the aid of the RAE orbit refinement program PROP, using nearly 3000 observations. For most of these orbits the standard deviations in inclination are less than 0.0009° (corresponding to about 100m in cross-track distance). The values of eccentricity give perigee heights accurate to between 30 and 120m.The main purpose of the orbit determination was to provide accurate values of the eccentricity for use in determining the odd zonal harmonics in the Earth's gravitational potential. These values have been analysed to determine the amplitude of the oscillation in eccentricity, which is found to be 0.00433 ± 0.00001.     

Lunar perturbations of artificial satellites of the earth

Lunisolar perturbations of an artificial satellite for general terms of the disturbing function were derived by Kaula (1962). However, his formulas use equatorial elements for the Moon and do not give a definite algorithm for computational procedures. As Kozai (1966, 1973) noted, both inclination and node of the Moon's orbit with respect to the equator of the Earth are not simple functions of time, while the same elements with respect to the ecliptic are well approximated by a constant and a linear function of time, respectively. In the present work, we obtain the disturbing function for the Lunar perturbations using ecliptic elements for the Moon and equatorial elements for the satellite. Secular, long-period, and short-period perturbations are then computed, with the expressions kept in closed form in both inclination and eccentricity of the satellite. Alternative expressions for short-period perturbations of high satellites are also given, assuming small values of the eccentricity. The Moon's position is specified by the inclination, node, argument of perigee, true (or mean) longitude, and its radius vector from the center of the Earth. We can then apply the results to numerical integration by using coordinates of the Moon from ephemeris tapes or to analytical representation by using results from lunar theory, with the Moon's motion represented by a precessing and rotating elliptical orbit.     

J2 Invariant Relative Orbits for Spacecraft Formations

An analytic method is presented to establish J ₂ invariant relative orbits. Working with mean orbit elements, the secular drift of the longitude of the ascending node and the sum of the argument of perigee and mean anomaly are set equal between two neighboring orbits. By having both orbits drift at equal angular rates on the average, they will not separate over time due to the J₂ influence. Two first order conditions are established between the differences in momenta elements (semi-major axis, eccentricity and inclination angle) that guarantee that the drift rates of two neighboring orbits are equal on the average. Differences in the longitude of the ascending node, argument of perigee and initial mean anomaly can be set at will, as long as they are setup in mean element space. For near polar orbits, enforcing both momenta element constraints may result in impractically large relative orbits. It this case it is shown that dropping the equal ascending node rate requirement still avoids considerable relative orbit drift and provides substantial fuel savings.This revised version was published online in October 2005 with corrections to the Cover Date.     

Analytical direct planetary perturbations on the orbital motion of satellites

An analytical expansion of the disturbing function arising from direct planetary perturbations on the motion of satellites is derived. As a Fourier series, it allows the investigation of the secular effects of these direct perturbations, as well as of every argument present in the perturbation. In particular, we construct an analytical model describing the evection resonance between the longitude of pericenter of the satellite orbit and the longitude of a planet, and study briefly its dynamic. The expansion developed in this paper is valid in the case of planar and circular planetary orbits, but not limited in eccentricity or inclination of the satellite orbit.     

Analysis of the orbit of cosmos 387 (1970-111A) near 15th-order resonance

Cosmos 387 (1970-111A) was launched on 16 December 1970 into a near-circular orbit with an average height of 540 km and an inclination of 74.0°. On 5 November 1971 the orbit, in its slow contraction under the influence of air drag, passed through 15th-order resonance, when the ground track repeats after 15 revolutions. The orbit has been determined with the aid of the RAE orbit refinement program PROP at 19 epochs between May 1971 and June 1972, using 1500 optical and radar observations. The average accuracy is about 70 m in perigee height and 0.001° in inclination.The variation of orbital inclination while the satellite was experiencing 15th-order resonance, as given by these 19 orbits and 55 U.S. Navy orbits, has been analysed to obtain equations accurate to 4 per cent for the geopotential coefficients of order 15 and odd degree (15, 17, 19 …). These equations have subsequently been used (with others) in determining individual coefficients of order 15 and odd degree.The variation of eccentricity with argument of perigee showed unexpected complexity, including a tight loop near resonance (Fig. 4). Analysis of the variation in eccentricity has yielded, for the first time, accurate equations for the geopotential coefficients of order 15 and even degree (16, 18 …), thus opening the way to the evaluation of individual coefficients of this type. The variations in the argument of perigee and right ascension of the node have also been analysed.     

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.     

Analytical study of the Earth's shadowing effects on satellite orbits

In this paper a new mathematical model is proposed for the study of the effects of the direct solar radiation pressure on the orbit of an artificial Earth satellite. The equations for the first order effects become canonical when a different definition for the orders of magnitude is adopted. This enables us the utilization of the method of Von Zeipel to eliminate all periodic terms. The model leads to the non-existence of pure secular perturbations owing to the direct solar radiation pressure on the metric elements: semi-major axis, eccentricity and inclination. Numerical examples built with an approximation for the shadow function show that the secular inequalities on the angle variables—longitude, perigee and node—are very small.     

Analysis of the orbit of Cosmos 72 (1965-53B) near 15th-order resonance

Cosmos 72 (1965-53B) was launched on 16 April 1965 into a near-circular orbit with an average height of 570 km and inclination 56°. Over the years, the orbit has contracted slowly under the influence of air drag, and On 27 June 1972 passed through exact 15th-order resonance, when successive equator crossings are 24° apart in longitude and the ground track repeats after 15 rev. The orbit has been determined at seven epochs between April 1972 and February 1973, using the RAE orbit refinement program PROP, with 544 optical and radar observations: the average orbital accuracy is about 50 m in height and 0.0008° in inclination.For Cosmos 72 the change in inclination at 15th-order resonance, due to perturbations by 15th-order harmonics in the geopotential, is greater than for any satellite previously analysed— nearly 0.07°—and analysis of the change, using the seven PROP orbits and 45 U.S. Navy orbits, yields equations accurate to 4 per cent for the geopotential coefficients of order 15 and odd degree (15, 17, 19 …). A similar analysis of the variation in eccentricity gives less accurate equations for coefficients of order 15 and even degree (16, 18 …). The variations in right ascension of the node and argument of perigee have also been analysed.     

在电离中电感应阻力对带电卫星轨道根数的摄动影响

研究了在高空电离层中运动的带电荷的卫星受电感应阻力后对轨道根数产生的摄动影响。研究结果表明 ,电感应阻力对带电卫星的轨道半长轴、轨道偏心率、近地点赤经、历元平赤经均有周期摄动影响 ,但除对半长轴有长期摄动效应外对其它轨道根数均无长期摄动。轨道倾角和升交点赤经不受摄动影响。文中以飞行在高度 1 50 0km的电离层中的导体卫星作为算例。计算结果显示 :带电导体卫星在高空电离层中带有一定电量时电感应阻力对轨道半长轴的缩短产生显著效应     

Analysis of the orbit of cosmos 395 rocket (1971-13B) near 15th-order resonance

Cosmos 395 rocket (1971-13B) is moving in a near-circular orbit inclined at 74° to the equator. Its average height, near 540 km after launch in February 1971, slowly decreased under the action of air drag and on 24 March 1972 it experienced exact 15th-order resonance, with the successive equator crossings 24° apart in longitude. Its orbit has been determined at 21 epochs between September 1971 and September 1972 using 1100 observations, including 55 from the Malvern Hewitt camera: the mean S.D. in inclination is 0.001° and in eccentricity 0.00001.The variations in inclination i, eccentricity e, right ascension of the node $\text{Ω}$, and argument of perigee ω, near 15th-order resonance are analysed to determine values of lumped 15th-order harmonic coefficients in the geopotential. The inclination yields equations accurate to 4 per cent for coefficients of order 15 and degree 15,17,19..., which are in excellent agreement with those from Cosmos 387 (1970-111A) in an orbit of similar inclination but different resonant longitude. Analysis of the variations in e gives two pairs of equations for the coefficients of order 15 and degree 16, 18..., which are used to obtain tentative values of the (16,15) coefficients. For the first time the resonant variation of other elements ($\text{Ω}\text{and}\text{ω}$) has also been analysed with partial success.     

Analysis of the orbit of 1970-87a (Cosmos 373)

Cosmos 373, 1970-87A, was launched on 20 October 1970 into an orbit inclined at 62.9° to the Equator, with an initial perigee height of 472 km. The orbit has been determined at 25 epochs covering a period of just over 4 yr using the RAE orbit refinement program PROP, with over 1500 observations. Observations from the Hewitt camera at Malvern were available for all 25 orbits.The main purpose of the orbit determination was to provide accurate values of the eccentricity for use in determining the odd zonal harmonics in the Earth's gravitational potential. The analysis has resulted in extremely accurate values of e with S.D.'s down to 0.000005 and has indicated an amplitude of the oscillation in eccentricity of 0.0085, equivalent to almost 60 km in perigee height—the largest yet recorded for any near-Earth orbit of high accuracy.     

A numerical investigation on the eccentricity growth of GNSS disposal orbits

We present the results of an extensive numerical exploration performed on the eccentricity growth in MEO associated with two possible end-of-life disposal strategies for GNSS satellites. The study calls attention to the existence of values of initial inclination, longitude of ascending node, and argument of perigee that are more advantageous in terms of long-term stability of the orbit. The important role of the initial epoch and a corresponding periodicity are also shown. The present investigation is influential in view of recent analytical and numerical developments on the chaotic nature of the region due to lunisolar perturbations, but also for the upcoming Galileo and BeiDou constellations.     

LAGEOS II perigee rate and eccentricity vector excitations residuals and the Yarkovsky-Schach effect

We have analysed LAGEOS II perigee rate and eccentricity vector excitation residuals over a period of about 7.8 years, adjusting and computing the satellite orbit with the full set of dynamical models included in the GEODYN II software code. The long-term behaviour of these orbital residuals appears to be characterised by several distinct frequencies which are a clear signature of the Yarkovsky-Schach perturbing effect. This non-gravitational perturbation is not included in the GEODYN II models for the orbit determination and analysis. Through an independent numerical analysis, and using the new LOSSAM model to represent the spin-axis behaviour of the satellite, we propagated the Yarkovsky-Schach effect on LAGEOS II perigee rate and compared the results obtained with the orbital residuals. We have thus been able to satisfactorily fit the amplitude of the Yarkovsky-Schach effect to the observed residuals. Our approach here has proven very successful with very positive results. We have been able to obtain a fractional reduction of about 40% of the post-fit rms with respect to the pre-fit value. When analysing the eccentricity vector residuals, we have been able to obtain a better result in the case of the real component, with a fractional reduction of the post-fit rms of about 49% of the initial value. The analysis of the effect's imaginary component in the eccentricity vector rate is more complicated and deserves additional scrutiny. In this case we need a deeper study which includes the analysis of other unmodelled and mismodelled effects acting on the imaginary component. The study performed in this paper will be of significant relevance not only for the geophysical applications involving LAGEOS II orbit analysis, but also for a refined re-analysis of the general relativistic precession produced by the Earth angular momentum, i.e., the Lense-Thirring effect.     

Samos 2 (1961 α 1): Orbit determination and analysis at 31 : 2 resonance

Samos 2, 1961 α 1, launched on 31 January 1961, was the first satellite to enter a sun-synchronous orbit at an inclination of 97.4°. The initial perigee and apogee heights were 474 km and 557 km respectively, the initial period was 94.97 min and the satellite decayed on 21 October 1973 after more than 12 years in orbit.Samos 2 passed through the condition of 31 : 2 resonance in June 1971 and orbital parameters have been determined at 22 epochs from 1674 observations using the RAE orbit refinement program, PROP, between mid-April and Mid-September 1971. The variations of inclination and eccentricity during this time have been analysed and values for six lumped 31st-order harmonic coefficients in the geopotential have been obtained. These have been compared with those derived from the individual coefficients, of order 31 and appropriate degrees, from the most recent Goddard Earth Model, GEM 10C.The decrease in inclination between launch and 1971 has been examined: it is found to be caused mainly by a near-resonant solar gravitational perturbation.     

Long-Period Variations of the Eccentricity Vector Valid also for Near Circular Orbits around a Non-Spherical Body

This paper studies the long period variations of the eccentricity vector of the orbit of an artificial satellite, under the influence of the gravity field of a central body. We use modified orbital elements which are non-singular at zero eccentricity. We expand the long periodic part of the corresponding Lagrange equations as power series of the eccentricity. The coefficients characterizing the differential system depend on the zonal coefficients of the geopotential, and on initial semi-major axis, inclination, and eccentricity. The differential equations for the components of the eccentricity vector are then integrated analytically, with a definition of the period of the perigee based on the notion of “free eccentricity”, and which is also valid for circular orbits. The analytical solution is compared to a numerical integration. This study is a generalization of (Cook, Planet. Space Sci., 14, 1966): first, the coefficients involved in the differential equations depend on all zonal coefficients (and not only on the very first ones); second, our method applies to nearly circular orbits as well as to not too eccentric orbits. Except for the critical inclination, our solution is valid for all kinds of long period motions of the perigee, i.e., circulations or librations around an equilibrium point.     

Multiple resonance in one problem of the stability of the motion of a satellite relative to the center of mass

We investigate the stability of the periodic motion of a satellite, a rigid body, relative to the center of mass in a central Newtonian gravitational field in an elliptical orbit. The orbital eccentricity is assumed to be low. In a circular orbit, this periodic motion transforms into the well-known motion called hyperboloidal precession (the symmetry axis of the satellite occupies a fixed position in the plane perpendicular to the radius vector of the center of mass relative to the attractive center and describes a hyperboloidal surface in absolute space, with the satellite rotating around the symmetry axis at a constant angular velocity). We consider the case where the parameters of the problem are close to their values at which a multiple parametric resonance takes place (the frequencies of the small oscillations of the satellite’s symmetry axis are related by several second-order resonance relations). We have found the instability and stability regions in the first (linear) approximation at low eccentricities.     

Resonant and non-resonant gravity-gradient perturbations of a tumbling tri-axial satellite

Gravity-gradient perturbations of the attitude motion of a tumbling tri-axial satellite are investigated. The satellite center of mass is considered to be in an elliptical orbit about a spherical planet and to be tumbling at a frequency much greater than orbital rate. In determining the unperturbed (free) motion of the satellite, a canonical form for the solution of the torque-free motion of a rigid body is obtained. By casting the gravity-gradient perturbing torque in terms of a perturbing Hamiltonian, the long-term changes in the rotational motion are derived. In particular, far from resonance, there are no long-period changes in the magnitude of the rotational angular momentum and rotational energy, and the rotational angular momentum vector precesses abound the orbital angular momentum vector.At resonance, a low-order commensurability exists between the polhode frequency and tumbling frequency. Near resonance, there may be small long-period fluctuations in the rotational energy and angular momentum magnitude. Moreover, the precession of the rotational angular momentum vector about the orbital angular momentum vector now contains substantial long-period contributions superimposed on the non-resonant precession rate. By averaging certain long-period elliptic functions, the mean value near resonance for the precession of the rotational angular momentum vector is obtained in terms of initial conditions.     

Artificial frozen orbits around Mercury

Orbits around Mercury are influenced by the strong elliptic third-body perturbation, especially for high eccentricity orbits, the periapsis altitude changes dramatically. Frozen orbits whose mean eccentricity and argument of perigee remain constants are obviously a good choice for space missions, but the forming conditions are too harsh to meet practical needs. To deal with this problem, a continuous control method that combines analytical theory and parameter optimization is proposed to build an artificial frozen orbit. The artificial frozen orbits are investigated on the basis of double averaged Hamiltonian, of which the second and third zonal harmonics and the perturbation of elliptic third-body gravity are considered. In this paper, coefficients of perturbations which satisfy the conditions of frozen orbits are involved as control parameters, and the relevant artificial perturbations are compensated by the control strategy. So probes around Mercury can be kept on frozen orbit under the influence of continuous control force. Then complex method of optimization is used to search for the energy optimized artificial frozen orbits. The choosing of optimal parameters, the objective function setting and other issues are also discussed in the study. Evolution of optimal control parameters are given in large ranges of semi-major axis and eccentricity, through the variation of these curves, the fuel efficiency is discussed. The result shows that the control method proposed in this paper can effectively maintain the eccentricity and argument of perigee frozen.     

The Use of Lie Series in the Construction of a Perturbation Theory and Some Recent Results in the Theory of the Motion of Hyperion

This paper begins with a brief review of a form of the Lie series transformation, and then reports some new results in the study, using Lie series methods, of the orbit of Saturn's satellite Hyperion. In particular, improved expressions are given for the long-period perturbations of the orbital elements which describe the motion in the orbit plane, and also first results for expressions for the short-period perturbations in the apse longitude, derived from the Lie series generating function. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analysis of the orbit of 1965-11D (COSMOS 54 rocket)

The satellite 1965-11D was the final-stage rocket used to launch Cosmos 54, 55 and 56 into orbit on 21 February 1965. The orbit of 1965-11D was inclined at 56° to the Equator, with an initial perigee height of 280 km; the lifetime was nearly 5 yr, with decay on 23 December 1969. The orbit has been determined at 75 epochs during the life, using the RAE orbit determination program PROP with over 4000 observations, photographic, visual and radar. Observations from the Hewitt camera at Malvern were available for 34 of the 75 orbits and typical accuracies for these orbits are 0.0005° in inclination and 100 m in perigee height.The variations in perigee height have been analyzed to determine reliable values of density scale height, at heights between 240 and 360 km. The analysis also revealed a rapid decrease of 5 km in perigee distance early in 1966, attributed to the escape of residual propellants.The variations in orbital inclination have been analyzed to determine upper-atmosphere zonal winds and 15th-order harmonics in the geopotential. The region of the upper atmosphere traversed by 1965-11D near its perigee is found to have had an average rotation rate of 1.10 ± 0.05 rev/day in 1966–1967, and 1.00 ± 0.03 rev/day between March 1968 and May 1969. In late 1969 there were probably wide variations in zonal winds, with east-to-west winds of order 100 m/s followed by west-to-east winds of order 200 m/s. The changes in inclination at the 15th-order resonance in July 1969 have been analyzed to give the first accurate values of lumped 15th-order harmonics obtained from a high-drag satellite. This success points the way towards similar analyses of the many other high-drag satellites that pass through 15th-order resonance, to evaluate individual geopotential coefficients of order 15 and even degree.