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.     

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.     

Analysis of the orbit of 1970-65D,cosmos 359 rocket

Cosmos 359 rocket 1970-65D, was launched on 22 August 1970 into an orbit inclined at 51·2° to the Equator, with an initial perigee height of 209 km: it decayed on 6 October 1971 after a lifetime of 410 days. The orbit has been determined at 42 epochs during the lifetime, using the RAE orbit refinement program, PROP, with over 2600 observations. Observations from the Hewitt cameras at Malvern and Edinburgh were available for 10 of the 42 orbits.Ten values of density scale height, at heights between 185 and 261 km, have been determined from analysis of the variations in perigee height.Upper-atmosphere zonal winds and 15th-order harmonics in the geopotential have been evaluated from the changes in orbital inclination. The average atmospheric rotation rate, for heights near 220 km, is found to be 1·04 rev/day; but there are striking departures from the average, with well-established values of 1·30, 0·75, 1·35 and 0·95 over four successive 75-day intervals. The changes in inclination at the 15th-order resonance in November 1970 give values of lumped 15th-order harmonics, which will provide equations for evaluating coefficients of order 15 and even degree (16,18,…) and also show that useful results on the geopotential can be obtained from satellites with perigee as low as 200 km.     

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.     

Geopotential harmonics of order 15 and 30, from analysis of the orbit of satellite 1971-10B

The satellite 1971-10B passed through exact 15th-order resonance on 30 March 1981 and orbital parameters have been determined at 52 epochs from some 3500 observations using the RAE orbit refinement program, PROP, between September 1980 and October 1981. The variations in inclination and eccentricity during this time have been analysed, and six lumped 15th-order harmonic coefficients and two 30th-order coefficients have been evaluated. The 15th-order coefficients are the best yet obtained for an orbital inclination near 65°; and previously there were no 30th-order coefficients available at this inclination. The lumped coefficients have been used to test the Goddard Earth Model GEM 10B: there is good agreement for seven of the eight coefficients.     

Orbit determination and analysis for Cosmos 58 at 15th-order resonance, to evaluate geopotential harmonics of order 15 and 30

The orbital parameters of Cosmos 58 have been determined at 65 epochs from some 4500 observations, between March 1982 and September 1983, using the RAE orbit refinement program, PROP. During this time, the satellite passed slowly through 15th-order resonance, and the orbital inclination and eccentricity have been analysed. Six lumped 15th-order geopotential harmonic coefficients have been evaluated, with an accuracy equivalent to between 0.8 and 2.0cm in geoid height. Six 30th-order coefficients have also been determined, with accuracies between 2 and 7 cm in geoid height. The coefficients have been compared with those from the GEM 10B and 10C models. There is good agreement for nine of the twelve coefficients.     

29Th-order harmonics in the geopotential from the orbit of ariel 1 at 29: 2 resonance

In analysing the orbit of Ariel 1 to determine upper-atmosphere winds, it was observed that the orbital inclination underwent a noticeable perturbation in November 1969 at the 29:2 resonance with the Earth's gravitational field, when the satellite track over the Earth repeats every 2 days after 29 revolutions. The variations in the inclination and eccentricity of the orbit between July 1969 and February 1970 have now been analysed, using 35 US Navy orbits, and fitted with theoretical curves to obtain lumped values of 29th-order harmonic coefficients in the geopotential.     

Analysis of the orbital inclination of 1963-27A

The Agena B upper-stage rocket 1963-27A was launched into near-circular orbit, inclined at 82.3° to the Equator, on 29 June 1963. Its orbital elements were available at 52 epochs over the 16 month interval prior to its decay on 26 October 1969. During this period the satellite passed through 31:2 resonance and the variation in orbital inclination near this event was analysed to obtain two lumped geopotential harmonics of order 31. Since the resonance was a weak feature in the data, the resulting values are poorly defined.Either side of the resonance period, the inclination was used to estimate the mean atmospheric rotation rate Λ rev day^?1. The values obtained were Λ = 0.85 ± 0.18 at a height of 440 km for the period June 1968 to February 1969 and Λ = 1.13 ± 0.10 at 338 km for the period June to October 1969.     

Individual geopotential coefficients of order 15 and 30, from resonant satellite orbits

The analysis of variations in satellite orbits when they pass through 15th-order resonance (15 revolutions per day) yields values of lumped geopotential harmonics of order 15, and sometimes of order 30. The 15th-order lumped harmonics obtained from 24 such analyses over a wide range of orbital inclinations are used here to determine individual harmonic coefficients of order 15 and degree 15,16,…35; and the 30th-order lumped harmonics (from eight of the analyses) are used to evaluate individual coefficients of order 30 and degree 30,32,…40. The new values should be more accurate than any previously obtained. The accuracy of the 15th-order coefficients of degree 15, 16,…23 is equivalent to 1 cm in geoid height, while the 30th-order coefficients of degree 30, 32 and 34 are determined with an accuracy which is equivalent to better than 2 cm in geoid height. The results are used to assess the accuracy of the Goddard Earth Model 10B.     

Analysis of the orbit of 1971-30B at 15th-order resonance

The orbit of the satellite 1971-30B (Tournesol rocket) has been determined from more than 2000 observations at 34 epochs spaced at 8-day intervals between March and November 1978 when the orbit was experiencing 15th-order resonance. The variations in the orbital inclination, which was near 46.4°, and in the eccentricity, which was near 0.01, have been analysed to determine values of six lumped harmonics of order 15. In view of the fact that the orbit passed through resonance quite rapidly, the results are very satisfactory: the standard deviations of the lumped harmonics correspond to accuracies between 1 and 3 cm in geoid height.     

Analysis of 208 U.S. Navy orbits for the satellite 1970-47B at 14th-order resonance

The orbit of 1970-47B passed very slowly through 14th-order resonance, and the changes in orbital inclination and eccentricity have been analysed over a 4-year period, from January 1977 to January 1981, using 208 U.S. Navy orbits. The analysis has yielded values for three pairs of lumped harmonic coefficients of 14th order, which have accuracies equivalent to 0.4, 1.5 and 2.0 cm in geoid height. Three pairs of values of 28th-order lumped harmonic coefficients were also obtained, and the best of these has a standard deviation (S.D.) corresponding to an accuracy of 0.7 cm in geoid height. The lumped harmonic coefficients have been compared with the corresponding values from the latest geopotential models, and agreement is satisfactory.     

Lumped geopotential harmonics of order 29, from analysis of the orbit of Cosmos 837 rocket

The orbit of Cosmos 837 rocket (1976-62E) has been determined at 36 epochs between January and September 1978, using the RAE orbit refinement program PROP 6 with about 3000 observations. The inclination was 62.7° and the eccentricity 0.039. The orbital accuracy achieved was between 30m and 150m, both radial and crosstrack. The orbit was near 29:2 resonance in 1978 (exact resonance occurred on 14 May) and the values of orbital inclination obtained have been analysed to derive lumped 29th-order geopotential harmonic coefficients, namely:

$\text{10}{}^9\text{C}{}^{\mathrm{0,2}}{}_{29}\text{= ? 10 ± 15}$

and

$\text{10}{}^9\text{S}{}^{\mathrm{0,2}}{}_{29}\text{= ?76 ± 12}$

. These will be used in future, when enough results at different inclinations have accumulated, to determine individual coefficients of order 29. The values of lumped harmonics obtained from analysis of the values of eccentricity were not well defined, because of the high correlations between them and the errors in removing the very large perturbation (31 km) due to odd zonal harmonics.     

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.     

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.     

The orbit of proton 4 redetermined,with geophysical implications

The orbit of Proton 4, 1968-103A, has been redetermined, in greater detail and with better accuracy, in order to clarify previously puzzling features in the variation of orbital inclination. Orbital parameters have been determined at 25 epochs between December 1968 and July 1969, using about 1600 optical and radar observations with the RAE orbit refinement program PROP 6.During January 1969 the orbit passed through 31:2 resonance—when the ground track over the Earth repeats every two days after 31 revolutions of the satellite. A simultaneous least-squares fitting of theoretical curves to the values of inclination and eccentricity between 14 December 1968 and 6 March 1969 has yielded values for two pairs of lumped 31st-order geopotential coefficients, appropriate to an inclination of 51.5°. This is the first specific evaluation of 31st-order coefficients.The 15 values of inclination after the resonance, from March to near decay in July 1969, have been used to determine mean, morning and afternoon-evening values for the rotation rate of the atmosphere at a height near 260 km; the values of rotation rate, namely 1.1, 0.9 and 1.3 rev/day respectively, confirm the trends already established from analysis of other satellite orbits.     

Upper-atmosphere features revealed by the orbit of 1980-43A

The satellite NOAA-B (1980-43A) was launched in May 1980 into an orbit with perigee height near 260 km and apogee height 1440 km, at an inclination of 92.2°.The lifetime was 11 months. The orbit has been determined at 40 epochs between October 1980 and May 1981 from about 3000 radar and optical observations. The average orbital accuracy, radial and cross-track, was about 100 m, with rather better accuracy in the final 14 days.The variation of orbital inclination has been analysed to determine two good values of atmospheric rotation rate, namely 1.10 ± 0.10 rev day^?1 at 300 km (average local time) and 1.15 ± 0.06 rev day^?1 at 225 km (evening).The effect of atmospheric rotation on the precession of the orbital plane of an actual satellite has never previously been detected; it is clearly apparent for 1980-43A in its last days and conforms to the expected theoretical change.The variation of perigee height has been analysed to determine ten values of atmospheric density scale height, for heights of 280–370 km. These values, accurate to about 3%, exceed by 15% the values indicated by the COSPAR International Reference Atmosphere. Solar activity was higher in the years 1980–1981 than at any time since early 1958 and it appears that the CIRA model underestimates the density and density scale height at high levels of solar activity.     

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.     

Analysis of the orbit of 1966-51C

The satellite 1966-51C was launched in June 1966 into a polar orbit with perigee height 180 km, apogee height 3600 km, and orbital period 125 min. The orbit contracted rapidly under the influence of air drag, and the satellite decayed in March 1967. The only available observations are from the NASA Minitrack system, and 825 of these have been used with PROP6 orbit refinement program, to determine orbital parameters at 20 epochs. For most of these orbits the standard deviations in inclination and right ascension of the node are less than 0.002° (corresponding to about 200 m cross-track) and the standard deviations in eccentricity are less than 0.00002 (150 m in height).The variation in inclination is analysed to determine upper-atmosphere zonal wind speeds, with excellent resolution in local time. The results, for heights near 210 km and a representative latitude near 30°, indicate west-to-east winds of 100 ± 40 m/s for local time 18–21 h, and east-to-west winds of 80 ± 25 m/s for 02–04 h and 09–12 h local time. The values of the right ascension of the node are also analysed, and provide independent confirmation of the wind speeds obtained from the inclination. Analysis of the decrease in perigree distance indicates values of density scale height within 5% of those predicted by the COSPAR International Reference Atmosphere 1972, for the conditions experienced by 1966-51C.     

Analysis of the orbit of 1970-114F in its last 20 days

The satellite 1970-114F, the final-stage rocket of the Molniya 1S communications satellite, decayed in the atmosphere on 3 March 1973. During the last 20 days of its life the orbit suffered exceptionally rapid decay, with the apogee height decreasing from 7000 to 1000 km while the perigee height remained near 110 km. About 650 observations, made by visual observers in Britain and by U.S. Navy sensors, have been used with the PROP6 orbit refinement program to determine orbits at 14 epochs. Although the decay rate was more than ten times greater than in any previous orbit determination with PROP, good orbits were obtained, the standard deviation in inclination being less than 0.002° on eight orbits.The combination of high drag and good accuracy allows three techniques in orbital analysis to be successfully applied for the first time. Since zonal winds have little effect on the orbit, the changes in inclination are analysed to determine meridional winds near perigee, at heights of 110–120 km, latitudes of 63–65°S, and 6–12 hr LT. The changes in right ascension of the node are also successfully analysed for the same purpose. The two methods agree in indicating a south-to-north wind of 40 ± 30 m/sec from 11 to 21 February, a geomagnetically quiet period, and a south-to-north wind averaging 150 ± 30 m/sec from 22 February to 3 March, a geomagnetically disturbed period. Thirdly, the changes in the argument of perigee are analysed to determine atmospheric oblateness, which is found to be equal to the Earth's oblateness, to within ±20%. Lastly, the drag coefficient in transition flow is evaluated and found to be 0.85 ± 0.20.     

Geopotential harmonics of order 29, 30 and 31 from analysis of resonant orbits

The Earth's gravitational potential is usually expressed as an infinite series of tesseral harmonics, and it is possible to evaluate “lumped harmonics” of a particular order m by analyses of resonant satellite orbits—orbits with tracks over the Earth that repeat after m revolutions. In this paper we review results on 30th-order harmonics from analyses of 15th-order resonance, and results on 29th- and 31st-order harmonics from 29:2 and 31:2 resonance.The values available for 30th-order lumped harmonics of even degree are numerous enough to allow a solution for individual coefficients of degree up to 40. The best-determined coefficients are those of degree 30, namely

$\text{10}{}^9\text{C}{}_{\mathrm{30,30}}\text{= ?1.2±1.1 10}{}^9\text{S}{}_{\mathrm{30,30}}\text{= 9.6±1.3}$

The standard deviations here are equivalent to 1 cm in geoid height.For the 29th- and 31st-order harmonics, and for the 30th-order harmonics of odd degree, there are not enough values to determine individual coefficients, but the lumped values from particular satellites can be used for “resonance testing” of gravity field models, particularly the Goddard Earth Model 10B (up to degree 36) and 10C (for degree greater than 36). The results of applying these tests are mixed. GEM 10B/C emerges well for order 30, with s.d. about 3×10^?9; for order 31, the GEM 10B values are probably good but the GEM 10C values are probably not; for order 29, the test is indecisive.