G.E. Cook

SATELLITE DRAG COEFFICIENTS

Abstract The drag of artificial earth satellites is reconsidered in the light of recent studies of gas-surface interactions and atmospheric composition. Between heights of 140 and 400 km at times of low solar activity, or heights of 140 and 600 km at times of high solar activity, the drag coefficient is almost independent of height, and at present, because of the lack of decisive evidence, there is perhaps not sufficient reason to abandon the value of 2.2 which has been widely used in recent years for the drag coefficient. It must be recognized, however, that this value is subject to some uncertainty and may be too low, perhaps by as much as 10–15percent. At heights above 400 km (low solar activity) or 600 km (high solar activity) the drag coefficient increases with height, since both the degree of energy transfer and the molecular speed ratio decrease as the molecular weight of the atmosphere decreases.

Perturbations of near-circular orbits by the earth's gravitational potential

Abstract The behaviour of near-circular orbits in the Earths gravitational potential is discussed in detail. The major axis rotates at a non-uniform rate or oscillates about a position of equilibrium, depending on the initial conditions.

THE LARGE SEMI-ANNUAL VARIATION IN EXOSPHERIC DENSITY: A POSSIBLE EXPLANATION

Abstract An analysis of the orbit of the satellite Calsphere 1 confirms the large semi-annual variation in density at heights near 1100 km previously found from the orbit of Echo 2, and not predicted by present atmospheric models. This variation is probably due to relatively smaller variations at much lower altitudes, in particular variations at heights near 120 km, which is taken as the lower boundary for the construction of upper-atmosphere models.

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.

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.

Analysis of 27 Satellite Orbits to Determine Odd Zonal Harmonics in the Geopotential.

Abstract The odd zonal harmonics in the geopotential are the terms independent of longitude and antisymmetric about the Equator: they define the ‘pear-shape’ effect. The coeffecients J 3 , J 5 , J 7 ,…of these harmonics have been evaluated by analysing the variations in eccentricity of 27 orbits covering wide range of inclinations. We use again most of the orbits from our previous (1969) evaluations, but we now have the advantage of 3 accurate orbits at inclinations between 60° and 66°, where the variations in eccentricity become very large, and 3 near-equatorial orbits, at inclinations between 3° and 15°, whereas previously there were none at inclinations lower than 28°. The new data lead to much more accurate and reliable values for the coeffecients. Our recommended set, which terminates at J 17 , is 10 9 J 3 = −2531 ± 7 10 9 J 11 = 159 ± 16 J 5 = −246 ± 9 J 13 = −131 ± 22 J 7 = −326 ± 11 J 15 = −26 ±24 J 9 = −94 ± 12 J 17 = −258 ± 19 . With this new set of values the pear-shape tendency of the Earth amounts to 44.7 m at the poles, instead of the previous 40 m, though the new geoid is within 1 m of the old at latitudes away from the poles.

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.

The semi-annual variation in the upper atmosphere during 1967 and 1968

Abstract The orbits of five satellites have been analysed to reveal the semi-annual variation in air density at heights between 300 and 505 km during the period of high solar activity in 1967 and 1968, and the results combined with data from other satellites. The semi-annual effect shows considerable variation from cycle to cycle, both in magnitude and in the form of the height dependence. The dates of the maxima and minima given by the individual satellites each year are in very good agreement with each other and are close o the mean values obtained by Jacchia, Slowey and Campbell for the years 1958–1966; this suggests the effect is more regular in its timing than has previously been assumed. The main difficulty in interpreting the results is the lack of data at heights below 250 km. At times the observed magnitudes are compatible with a variation solely in exospheric temperature due to temperature changes in the thermosphere, whereas at other times a variation in conditions at the lower boundary near the turbopause (about 110 km) is required.

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.