Diana W. Scott

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 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.

Upper-atmosphere rotational speed and its variation with height

Abstract The average rotational speed of the upper atmosphere can be determined by analysing the changes in the orbital inclinations of satellites. This procedure is applied to eleven new orbits to give values of Λ, the ratio of atmospheric angular velocity to the Earths angular velocity, at heights between 250 and 380 km. When these new results are combined with 21 previous values the variation of Λ with height can be determined over a much wider height range than was previously possible, namely from 200 to 400 km. It is found that Λ increases from about 1.1 at 200 km height to about 1.35 at 300 km, and about 1.45 at 400 km. This implies that the wind is, on average, from west-to-east, with its mean speed increasing from 40 m/sec at 200 km height to 180 m/sec at 400 km (for latitudes near 30°). There is no evidence that Λ varies from year to year or with solar activity.

EXOSPHERIC DENSITIES NEAR SOLAR MINIMUM DERIVED FROM THE ORBIT OF ECHO 2.

Abstract The density of the upper atmosphere at heights between 1080 and 1170 km is evaluated from the change in the orbital period of Echo 2 for dates between February 1964 and December 1965. The air density shows a pronounced semi-annual variation, while the variation between day and night is unlikely to exceed a factor of 2. A shortened version of this paper was presented at the COSPAR Seventh International Space Science Symposium, Vienna, May 1966.

The odd zonal harmonics in the Earth's gravitational potential

Abstract The odd zonal harmonics in the Earths gravitational potential are determined by analysing the changes in the eccentricities of six satellites having orbital inclinations spaced as uniformly as possible between 28° and 96°. The most satisfactory representation of the potential is found to be in terms of four coefficients, and their values are, in the usual notation: 10 6 J 3 = −2.56, 10 6 J 5 = −0.15, 10 6 J 7 = −0.44, 10 6 J 9 = 0.12. The resulting potential is compared with that obtained by other authors. Three and five-coefficient solutions are also presented.

Further determinations of upper-atmosphere rotational speed from analysis of satellite orbits

Abstract The average angular velocity of the upper atmosphere, which we take as Λ times the Earths angular velocity, can be evaluated by analysing the changes in the orbital inclinations of satellites. In this paper the nine most suitable orbits now available are analysed and values of Λ are found for heights between 200 and 260 km. The results, which are more accurate than in our previous studies, confirm that Λ ⪢ 1, i.e. that the atmosphere rotates faster than the Earth at these heights, and show that Λ increases with height, from 1.1 at 210 km to 1.4 at 260 km. This corresponds to mean west-to-east winds of 30 m/s at 210 km, increasing to 130 m/s at 260 km height. Results from one satellite indicate that the wind is probably strongest at times near sunset, with Λ = 1.5 ± 0.1 at 200 km height in August 1966. Comparisons are made with previous observational results and some of the suggested theoretical explanations are outlined.

Odd zonal harmonics in the geopotential, determined from fourteen well-distributed satellite orbits

Abstract Coefficients of the odd zonal harmonics in the Earths gravitational potential are evaluated by analysing the oscillations in orbital eccentricity of seventeen satellites, chosen to give the widest and most uniform possible distribution in inclination and semi major axis. Three of the satellites are excluded for various reasons; the other fourteen yield various sets of values for the odd zonal harmonic coefficients J3, J5, J7… The best representations of the odd harmonics in the potential appear to be in terms of either seven or ten coefficients, assuming that higher-degree coefficients are zero. The two sets of values are: 7-Coefficient 10-Coefficient 106J3 −2.53 ± 0.02 −2.50 ± 0.01 106 J 5 −2.22 ± 0.04 −0.26 ± 0.01 106 J 7 −0.41 ± 0.06 −0.40 ± 0.02 106 J 9 +0.09 ± 0.06 0 ± 0.06 106 J 11 −0.14 ± 0.05 −0.27 ± 0.06 106 J 13 +0.29 ± 0.06 +0.36 ± 0.08 106 J 15 −0.40 ± 0.06 −0.65 ± 0.10 106 J 17 +0.30 ± 0.08 106 J 19 0 ± 0.11 106 J 21 +0.58 ± 0.11 The errors in the last few coefficients of each set may be slightly greater than the standard deviations suggest, because no allowance is made for the neglected higher harmonics. The large magnitudes of J15 and J21 are noteworthy: neither of these coefficients has been determined before. Previously accepted values of J3 to J9, are in agreement with those in the more complete sets of coefficients above.

Variations in exospheric density at heights near 1100km, derived from satellite orbits

Abstract In an earlier paper, values of exospheric density were obtained from the orbit of Echo 2 for the years 1964–1965. The results indicated a semi-annual variation in density by a factor of between 2 and 3, considerably larger than predicted by existing atmospheric models. These studies have now been extended to the beginning of 1967, using both Echo 2 and Calsphere 1, to show how the density is responding to increasing solar activity. Variations in density during 1964 have been analysed in more detail. The long-term variation associated with the solar cycle and the short-term variations associated with magnetic and solar disturbances agree with the variations expected on the basis of current models. The semi-annual variation is persisting to higher levels of solar activity, and although its amplitude is diminishing the factor of variation was still 1.6 in 1966.

Lifetimes of satellites in large-eccentricity orbits

Abstract A method has been developed for predicting the long-term development of orbits having moderate to large eccentricities (0.33