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 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 ariel 1, 1962-15A,near 15th-order resonance

Ariel 1, the first international satellite, was launched on 26 April 1962, into an orbit inclined at 53.85° to the equator, with an initial perigee height near 390 km. On 8 May 1973 the orbit passed through 15th-order resonance and has been determined, with the RAE orbit refinement program PROP, at eight epochs between February and August 1973 using 500 observations.The orbital inclinations during the time of 15th-order resonance, as given by these eight orbits and 31 U.S. Navy orbits, were fitted with a theoretical curve using the THROE computer program, the best fit giving $\text{10}{}^9\text{C}\text{?}{}_{15}\text{= ?370 ± 14}$ and 10⁹S₁₅ = ?114 ± 31.The values of eccentricity were also successfully fitted using THROE, and the results are discussed.     

Analysis of the orbit of 1970-97b (cosmos 378 rocket)

Cosmos 378 rocket, 1970-97B, entered orbit on 17 November 1970, with orbital inclination 74.0°, period 105 min and perigee height 230 km, and decayed on 30 September 1972 after 683 days in orbit. The RAE computer program PROP was used, with more than 1900 observations from 64 stations, to determine the orbit at 39 epochs between February 1971 and September 1972.The main aim of the analysis was to determine the atmospheric rotation rate from the decrease in orbital inclination, which was determined with a mean standard deviation of 0.0010° and a best standard deviation of 0.0003°. After removal of relevant perturbations, analysis of the variation in inclination between July 1971 and April 1972 yields the surprisingly low average atmospheric rotation rate of 0.75 ± 0.05 rev/day, at a mean height of 250 km. The local time at perigee is however strongly biassed towards daytime values (07–16 hr), so the results lend support to the picture of east-to-west winds by day and west-to-east winds by night.Values of scale height are obtained by analysis of the change in perigee height.     

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 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 268 rocket (1969-20B)

The orbit of Cosmos 268 rocket (1969-20B) has been determined at 28 epochs during its 342-day life, with the aid of the PROP5 orbit refinement program. All available observations were used, including 16 from the Hewitt camera at Malvern, 28 from the 200-mm camera at Meudon, 56 from the kinetheodolite at the Cape Observatory, 700 visual observations from volunteer observers, 500 US Navy observations and 200 British radar observations. The orbits are of very good accuracy for such a high-drag satellite, most of the values of inclination having standard deviations less than 0.002°. The most accurate orbits are those utilizing photographic observations, and the best of these has standard deviations of 0.00001 in eccentricity and 0.0001° in inclination.

The values of inclination obtained, after correction to allow for the effects of other perturbing forces, have been analysed to determine zonal wind speeds in the upper atmosphere at heights a little above perigee (230–250 km) averaged over latitudes up to about 25°. The results show a clear distinction between the wind at night (21 to 03 hr local time), which is west-to-east with an average speed of 140 ± 50 m/sec, and the wind by day (08 to 17 hr), which is east-to-west with an average speed of 110 ± 50 m/sec.     

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.     

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

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.     

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.     

Air density at heights near 200km from the orbit of 1970-65D

Cosmos 359 rocket, 1970-65D, entered orbit on 22 August 1970, with an initial perigee height of 209 km and inclination 51·2°, and decayed on 6 October 1971. Using the values of perigee height from RAE orbits and decay rates from USAF Spacetrack bulletins, 146 values of air density have been calculated between August 1970 and September 1971, mainly at heights between 180 and 230 km.On ten occasions in 1971 when there were substantial geomagnetic disturbances there were sudden increases in density, the largest being about 32 per cent.When the density was corrected to a fixed height and allowance was made for the day-tonight variation and the effects of solar activity, the dominant feature was a semi-annual variation, with maxima in density centred at 6 November 1970 and 7 April 1971, and minima centred at 5 January and 28 July 1971. The maxima in density are nearly equal and exceed the minima by about 50 per cent.     

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 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.     

Periodic orbit families near the 4:1 jovian resonance

An enlarged averaged Hamiltonian is introduced to compute several families of periodic orbits of the planar elliptic 3-body problem, in the Sun–Jupiter–Asteroid system, near the 4:1 resonance. Four resonant critical point families are found and their stability is studied. The families of symmetric periodic orbits of the elliptic problem appear near the corresponding fixed points computed in this model. There is a good agreement for moderate eccentricity of the asteroid for three of these families, whereas the remaining family cannot be considered as a family of periodic orbits of the real model. This revised version was published online in July 2006 with corrections to the Cover Date.     

Air density at heights near 435 km from the orbit of skylab 1 (1973-27A)

On February 8, 1974, Skylab 1 was manoeuvred into a near circular orbit of inclination 50.04° and perigee near 420km. Orbital parameters have been computed at forty-six epochs thereafter using all available observations. Using these orbital elements, supplemented by orbital decay rates derived from NORAD bulletins, 193 values of air density were determined between 23 February, 1974 and 11, August, 1976. Corrected to a fixed height and normalised with respect to exospheric temperature these values reveal the semi-annual variation, exhibiting maxima in March–April and October–November, and minima in January–February and July–August. For 1974–1976 the July minima are more pronounced than the January minima whilst the April and October maxima appear equal. Overall the variation is greater than that indicated by CIRA 1972.     

Determination of zonal wind speeds at heights near 530 km from the orbit of Explorer 24 (1964-76A)

The orbit of Explorer 24 (1964–1976A) has been determined at 18 epochs during the five month period prior to its decay in October 1968, using the RAE orbit refinement computer program PROP6. As a balloon, the satellite was strongly influenced by atmospheric perturbations, despite its high perigee altitude near 490 km. It therefore provided an opportunity of determining atmospheric rotation rates at high altitude. The rotation rate, Λ rev day^?1, was estimated from the observed variation in orbital inclination, after the removal of perturbations including those due to solar radiation pressure.The mean rotation rates, averaged over local time, are Λ = 0.98 for 18 May to 18 August 1968 at 542 km; Λ = 1.06 for 18 May to 13 October 1968 at 533 km.For morning conditions, Λ = 0.9 for 22 June to 20 July 1968 at 540 km; Λ = 0.8 during September 1968 at 513 km.For evening conditions, Λ = 1.1 for 18 May to 15 June 1968, and for 26 July to 7 September 1968, at 540 km and 536 km respectively; Λ = 1.3 for 28 September to 13 October 1968 at 484 km.Further, the maximum W to E zonal wind has been estimated to occur at 20.5 h local time, during the period of the analysis.     

15th order resonance terms using the decaying orbit of TETR-3

The orbit of TETR-3 (1971-83B), inclination: 33°, passed through resonance with 15th order geopotential terms in February 1972. The resonance caused the orbit inclination to increase by 0.015°. Analysis of 48 sets of mean Kepler elements for this satellite in 1971–1972 (across the resonance) has established the following strong constraint for high degree, 15th order gravitational terms (normalized):

$\text{10}{}^9\text{(}\text{C, S}\text{)}{}_{15}\text{= (28.3 ± 3.0, 7.4 ± 3.0) =}\text{0.001(}\text{C, S}\text{)}{}_{\mathrm{15,15}}\text{?0.015(}\text{C, S}\text{)}{}_{\mathrm{17,15}}\text{+0.073(}\text{C, S}\text{)}{}_{\mathrm{19,15}}\text{?0.219(}\text{C, S}\text{)}{}_{\mathrm{21,15}}\text{+0.477(}\text{C, S}\text{)}{}_{\mathrm{23,15}}\text{?0.781(}\text{C, S}\text{)}{}_{\mathrm{25,15}}\text{+1.000(}\text{C, S}\text{)}{}_{\mathrm{27,15}}\text{?0.0963(}\text{C, S}\text{)}{}_{\mathrm{29,15}}\text{+0.622(}\text{C, S}\text{)}{}_{\mathrm{31,15}}\text{?0.119(}\text{C, S}\text{)}{}_{\mathrm{33,15}}\text{?0.290(}\text{C, S}\text{)}{}_{\mathrm{35,15}}\text{+0.403(}\text{C, S}\text{)}{}_{\mathrm{37,15}}\text{?0.223(}\text{C, S}\text{)}{}_{\mathrm{39,15}}\text{?0.058(}\text{C, S}\text{)}{}_{\mathrm{41,15}}\text{+…}$

This result combined with previous results on high inclination 15th order and other resonant orbits suggests that the coefficients of the gravity field beyond the 15th degree are smaller than Kaula's rule ($\text{10}{}^{\mathrm{?5}}\text{l}{}^2$).     

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.     

Air density at heights near 400km from the orbit of 1963-27A

The Agena B upper-stage rocket 1963-27A was launched into a near-circular orbit, inclined at 82.3° to the Equator, on 29 June 1963. Its orbit is determined at 52 epochs over the 16 month interval prior to its decay on 26 October 1969. The resulting orbital elements are used to obtain 95 atmospheric density values, at heights near 400km. Corrected to fixed heights, and normalised to a common exospheric temperature, these values reveal the semi-annual variation in density. A comparison between the observed variation and that of a recent model atmosphere is made. Although agreement between the two is generally good, their principal differences are discussed.     

Features of the upper atmosphere revealed by analysis of the orbit of cosmos 1009 rocket, 1978-50B

COSMOS 1009 rocket was launched on 19 May 1978 into an orbit with initial perigee height 150 km and apogee 1100 km: its lifetime was only 17 days. The orbit has been determined daily during the final 14 days of its life, using the RAE orbit refinement program PROP6,with about 1100 observations supplied by NORAD. An average accuracy of about 60 m, radial and cross-track, was achieved.The orbits were analysed to reveal three features of the upper atmosphere at heights between 125 and 175 km. From the decrease in perigee height, five values of density scale height, accurate to ±4%, were obtained. The first three were within 10% of those from CIRA 1972; the fourth, after a magnetic storm, was higher than expected; the fifth gave evidence of the decrease in drag coefficient at heights below 130 km.Atmospheric oblateness produced a change of 4° in perigee position during the last four days of the life. Analysis showed that the ellipticity of the upper atmosphere was approximately equal to that of the Earth, f, for the first two of the four days, and about $\text{1}\text{2}\text{f}$ in the last two.The orbital inclination decreased during the 14 days by about 50 times its standard deviation, and the observed variation was analysed to determine zonal winds at heights of 150–160 km at latitudes near 47° north. The zonal wind was very weak (0±30 m/s) for 23–28 May at local times near 03h; and 90±30 m/s east-to-west for 29 May to 4 June at local times near 01 h.