15th-order Resonance Research Articles

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 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 ( Ω and ω ) has also been analysed with partial success.

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.

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 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 Cosmos 316 (1969—108 A)

Cosmos 316 (1969-108A) was a most unusual satellite. It was launched on 23 December 1969 into an orbit having a very low perigee (height 154 km ) and inclined at 49.5° to the equator. But, because of its high mass/area ratio, Cosmos 316 remained in orbit for 8 months, thus offering a unique opportunity to study the atmosphere at heights near 150 km by analysis of orbital changes. Orbits have been determined at 36 epochs during the satellite’s life with the aid of the R .A .E . orbit determination program Prop ; about 1600 observations were used, mainly radar and visual, but also some accurate photographic observations. The orbits obtained, which utilize the full accuracy of the observations, are the most accurate and consistent yet pub­lished for a satellite with such a low perigee. The perigee and apogee distances are accurate to about 150 m , and the s.d. in inclination is generally between 0.0005° and 0.0025°. The final orbit, at an epoch 20 h before re-entry and with a perigee height of 118 km, is among the most accurate, and indicates how the drag coefficient decreases as the atmospheric mean free path decreases to less than the satellite’s dimensions. The changes in inclination are analysed to determine upper-atmosphere winds, which prove to be very variable: the most extreme is a west-to-east wind of order 300 m /s in the afternoon in near-equatorial regions of the southern hemisphere between 24 July and 7 August 1970. The perigee distances have been used previously in determining variations of upper-atmosphere density, and are now also used to evaluate scale height. It proved possible to detect the change in inclination at the time of 15th-order resonance with the Earth’s gravitational field, when successive ground track s are 24° apart, and an equation for the 15th-order coefficients in the geopotential was obtained which confirms recent results from Transit 1B .

Equations for 15th-order geopotential coefficients from the orbit of transit 1B

The RAE computer program PROP has been used to re-determine in greater detail the orbit of satellite Transit 1B, 1960 γ 2, using a modified method on occasions when the number of observations was previously thought to be too few to determine the orbit. The satellite had an oibital inclination of about 51°, and it was found possible to determine inclination accurately using only a small number of visual observations from latitudes near 51°N. The main object of the study is to investigate the region of 15th-order resonance revealed by a steep decrease in inclination of about 0.04° between March and November 1962. The analysis leads to the following equations for the 15th-order geopotential coefficients: 0.56 C ̄ 17,15 − C ̄ 19,15 + 0.74 C ̄ 21,15 = (36 ± 4) × 10 −9 and 0.56 S ̄ 17,15 − S ̄ 19,15 + 0.74 S ̄ 21,15 = (35 ± 6) × 10 −9 In addition, an improved value has been obtained for the average rotation rate Λ of the upper atmosphere at a mean height of 380 km between 1963 and 1967, namely Λ = 1.35 ± 0.10 rev/ day

Evaluation of 15th-order Harmonics in the Geopotential

AN Earth satellite with initial orbital period greater than about 96 minutes will, as its orbit decays under the influence of air drag, pass through the condition of 15th-order resonance, when successive equatorial crossings are exactly 24° apart, so that the satellite follows a fixed set of 15 ground tracks each day. If the decay is slow enough, the inclination of the orbit to the equator changes appreciably1,2 as the satellite passes through resonance, because of the build-up of perturbations caused by tesseral harmonics in the geopotential of order 15 and degree 15,17,19…. By observing a particular satellite, determining its orbit and analysing the change in inclination, it is possible to derive linear equations between the coefficients of these 15th-order harmonics; and the equations can be solved for individual coefficients when results have been obtained for a number of satellites at different orbital inclinations.

Equations for 15th-Order Harmonics in the Geopotential

BY analysing the orbit of Ariel 3, Gooding1 obtained two equations between 15th-order coefficients in the Earth's gravitational potential from the changes in orbital inclination as the satellite passed through 15th-order resonance (when the successive tracks over the Earth are exactly 24° apart). Gooding's equations, simplified by omitting terms which should be small, are: illustration where the quantities C and S are normalized coefficients in the infinite series expansion of the geopotential, as defined, for example, by Gaposchkin and Lambeck2. Open image in new window