D.G. King-Hele

The effect of the earth’s oblateness on the orbit of a near satellite

The equations of motion of a satellite in an orbit over an oblate earth in vacuo are solved analytically, by a perturbation method. The solution applies primarily to orbits of eccentricity 0⋅05 or less. The accuracy of the solution for radial distance should then be about 0⋅001%, and the error in angular travel about 0⋅001% per revolution. The earth’s oblateness has four main effects on the motion: (1) The orbital plane, instead of remaining fixed, rotates about the earth’s axis in the opposite direction to the satellite, at a rate of 10⋅00(R/r̄)3.5 cos α deg./day, where α is the inclination of the orbital plane to the equator, R the earth’s equatorial radius and r̄ the satellite’s mean distance from the earth’s centre. (2) The period of revolution of the satellite, from one northward crossing of the equator to the next, is 14⋅5 √(R/r̄) sin2 α sec greater for an inclined orbit than for an equatorial orbit. (3) The radial distance r from the earth’s centre changes. For a given angular momentum the mean r is 14⋅5 R/r̄ nautical miles greater for a polar orbit than an equatorial one. Also, during each revolution r oscillates twice, the amplitude of the oscillation being 0⋅94 (R/r̄) sin2 αn. miles. (4) The major axis of the orbit rotates in the orbital plane at a rate of 5⋅00(R/r̄)3.5 (5 cos2 α —1) deg./day. Thus it rotates in the same direction as the satellite if α < 63⋅4°, or in the opposite direction if α > 63⋅4°. A brief comparison is made between theory and observation for Sputniks 1 and 2.

Upper-Atmosphere Zonal Winds from Satellite Orbit Analysis.

Abstract In this paper we review and interpret the values of upper-atmosphere rotation rate (zonal winds) obtained by analysing satellite orbits determined from observations. The history of the method is briefly reviewed, the basic principles are explained, objections to the method are answered, and three examples are given. Existing analyses of the atmospheric rotation rate A are critically reviewed, and, after rejecting some and revising others, we are left with 85 values. These are divided according to local time and season, to give the variation of A with height in nine situations—namely morning, evening and average local time, for summer, winter and average season. These observational results indicate that the value of Λ (in rev/day), averaged over both local time and season, increases from 1.0 at 125 km to 1.22 at 325 km and then decreases to 1.0 at 430 km and 0.82 at 600 km. The value of Λ is higher in the evening (18–24 h), with a maximum value (near 1.4) corresponding to a West-to-East wind of 150 m s −1 at heights near 300 km. The value of Λ is lower in the morning (06–12 h), with East-to-West winds of order 50 m s −1 at heights of 200–400 km. There is also a consistent seasonal variation, the values of Λ being on average 0.15 higher in winter and 0.1 lower in summer than the average seasonal value. No significant variation with solar activity is found, but there is a slight tendency for a greater rotation rate at lower latitudes for heights above 300 km. Unexpectedly, the values for the 1960s are found to be significantly higher than those for the 1970s. Finally, these observational values are compared with the theoretical global model of Fuller-Rowell and Rees: there is complete agreement on the trends, though there are some differences in the mean values.

A revaluation of the rotational speed of the upper atmosphere

Abstract In this paper the rotational speed of the upper atmosphere, mainly at heights of 200–300 km, is evaluated from the changes in the orbital inclinations of thirteen satellites. The values obtained represent the mean rotational speed over the latitudes covered by the satellites, at dates between late 1962 and early 1966, i.e. when solar activity was low. If the angular velocity of the atmosphere is taken as Λ times that of the Earth, the values of Λ found are mostly between 1.0 and 1.6 with estimated S.D. between 0.1 and 0.25. If we exclude two values at heights above 300 km and one anomalous value, the mean of the remaining ten values of Λ obtained is 1.27, with r.m.s. scatter 0.18: this would correspond to an average west-to-east wind of about 100 m/sec in mid-latitudes.

Evaluation of odd zonal harmonics in the geopotential, of degree less than 33, from the analysis of 22 satellite orbits☆

Coefficients of the odd zonal harmonics in the Earths gravitational potential—J3, J5, J7, etc.—are evaluated by analysing the variations in orbital eccentricity of 22 satellites, chosen to give the widest and most uniform possible distribution in semi major axis and inclination. These satellites provide 22 simultaneous equations for the coefficients J3, J5, etc., and the equations are solved by the least-squares method for sets of coefficients of successively higher order. The solutions show that J9 may be taken as zero, and that, for 9 < n < 33, the odd Jn do not differ significantly from zero unless n is a multiple of 3. Consequently J11, J13, J17, J19, J23, J25, J29 and J31 can be taken as zero, and it is feasible to carry the solutions to harmonics of much higher degree than was previously possible. The best representation of the odd zonal harmonics in the geopotential is provided by the following set of values: 106J3 = −2·54 ± 0·01 106J5 = −0·21 ± 0·01 106J7 = −0·40 ± 0·01 106J15 = −0·20 ± 0·03 106J21 = 0·26 ± 0·05 106J27 = −0·15 ± 0·10 J9 = J11 = J13 = J17 = J19 = J23 = J25 = J29 = J31 = 0.

The gravity field of the Earth

In recent years the Earth’s gravitational field has been determined with continually improving accuracy, by using hundreds of thousands of observations of Earth satellites, chiefly optical, laser and Doppler, together with surface gravimetry and, most recently, altimeter measurements from the Geos 3 satellite. The geopotential is usually expressed as a double series of tesseral harmonics, and several hundred of the harmonic coefficients are evaluated. Progress in this work during the 1970s is briefly outlined, and some attempt is made to assess the accuracy of current geoid maps and sets of harmonic coefficients, as exemplified in the latest models derived at the Goddard Space Flight Center. The harmonic coefficients of order 14, 15 and 30 in the Goddard Earth Model 10B are compared with values obtained independently by analysis of resonant orbits: the results suggest that the values in GEM 10B are realistic for these orders, and presumably others. It appears that the accuracy of the geoid maps is now approaching 1 m.

Air density at heights near 190 km in 1966–67, from the orbit of Secor 6

Abstract The air density at a height of 191 km between 14 June 1966 and 5 July 1967 has been evaluated by analysing the orbit of the satellite Secor 6 (1966-51B). The 154 values of density obtained show that nearly all geomagnetic disturbances are accompanied by increases in density, of larger magnitude than expected. Surprisingly, however, the strong 27-day recurrence tendencies in solar activity, as represented by the 10.7-cm radiation energy, are quite absent from the density variations. When the effect of the day-to-night variation is removed, the most important variation in air density at 191 km during 1966–7 is found to be a semi-annual oscillation, with the maximum density, in October 1966 and April 1967, exceeding the minimum, in July 1966 and January 1967, by a factor of 1.45. This semi-annual variation, which is very similar in form to that found at a height of 1130 km from Echo 2, has during 1966–67 displaced solar activity as the main source of atmospheric variation at heights near 200 km.

The effect of atmospheric rotation on a satellite orbit, when scale height varies with height

Abstract The upper atmosphere at heights of 200–300 km rotates at least as fast as the Earth, and a satellite in this region therefore suffers a lateral aerodynamic force, which slightly reduces the inclination of the orbit to the equator. The theoretical expressions previously obtained for the changes in inclination and in longitude of the node apply to an exponential atmosphere, in which the density scale height H is constant. The present paper develops the theory for orbits of eccentricity less than 0.2 with a more realistic atmospheric model in which H varies linearly with height y: it is assumed that μ = dH/dy is less than 0.2. The results show that the values given by the constant-H theory can be altered by up to 9 per cent when μ is included, if perigee is near the equator. But the effect of μ can be nearly eliminated if H is evaluated at a height 0.75H above perigee. Graphs are given to show the effective average height at which the inclination-changing forces act, i.e. the effective height at which the atmospheric rotational speed is being sampled. This height depends on the eccentricity e and the argument of perigee: for e > 0.01 the height is between 0.5H and 1.2H above perigee.

The Contraction of Satellite Orbits under the Influence of Air Drag V. with Day-To-Night Variation in Air Density

The effect of air drag on satellite orbits of small eccentricity (< 0-2) was studied in part I on the assumption that the atmosphere was spherically symmetrical. In reality the density of the upper atmosphere depends on the elevation of the Sun above the horizon and has a maximum when the Sun is almost overhead. In the present paper the theory is extended to an atmosphere in which the air density at a given height varies sinusoidally with the geocentric angular distance from the maximum-density direction. Equations are derived which show how perigee distance and orbital period vary with eccentricity throughout the satellite’s life, and how eccentricity varies with time. Expressions are also obtained for lifetime and air density at perigee in terms of the rate of change of orbital period. The main geometrical parameter determining the long-term effect of this day-to-night variation is the angular distance p of perigee from the maximum-density direction. Results are obtained for <})pconstant and

Upper-atmosphere zonal winds: Variation with height and local time

Abstract The average rotation rate of the upper atmosphere can be found by analysis of the changes in the orbital inclinations of satellites, and results previously obtained have indicated that the atmospheric rotation rate appreciably exceeds the Earths rotation rate at heights between 200 and 400 km. We have examined all such results previously published in the light of current standards of accuracy: some are accepted, some revised, and some rejected as inadequate in accuracy. We also analyse a number of fresh orbits and, adding these to the accepted and revised previous results, we derive the variation of zonal wind speed with height and local time. The rotation rate (rev/day) averaged over all local times increases from near 1.0 at 150 km height to 1.3 near 350 km (corresponding to an average west-to-east wind of 120 m/s), and then decreases to 1.0 at 400 km and probably to about 0.8 at greater heights. The maximum west-to-east winds occur in the evening hours, 18–24 h local time: these evening winds increase to a maximum of about 150 m/s at heights near 350 km and decline to near zero around 600 km. In the morning, 4–12 h local time, the winds are east to west, with speeds of 50–100 m/s above 200 km. We also tentatively conclude that, at heights above 350 km, the average rotation rate is greater in equatorial latitudes (0–25°) than at higher latitudes.

Air density at heights near 150 km in 1970, from the orbit of Cosmos 316 (1969-108A)

Abstract Cosmos 316 (1969-108A) was launched on 23 December 1969 into an orbit with an initial perigee height of 154 km at an inclination of 49.5° to the equator. Being very massive, Cosmos 316 had a longer lifetime than any previous satellite with such a low initial perigee: it remained in orbit until 28 August 1970. Because of its interest for upper-atmosphere research, the satellite was intensively observed, and accurate orbits are being determined at RAE from all available observations. Using perigee heights from the RAE orbits so far computed, and decay rates from Spacetrack bulletins, 102 values of air density have been obtained, giving a detailed picture of the variations in density at heights near 150 km between 24 December 1969 and 28 August 1970. The three strongest geomagnetic storms, on 8 March, 21 April and 17 August 1970, are marked by sudden increases in density of at least 23, 15 and 24 per cent respectively. With values of density extending over eight months, it is possible for the first time to examine a complete cycle of the semi-annual variation at a height near 150 km: the values of density, when corrected to a fixed height, exhibit minima in mid January and early August; at the intervening maximum, in April, the density is 30 per cent higher than at the minima.