15th-order Resonance Research Articles

Taking advantage of inclination variation in resonant remote-sensing satellite orbits

The inclination of remote-sensing satellites, which are generally placed in sun-synchronous orbits, varies as a function of the nominal equatorial crossing local mean solar time selected for a given mission. The Indian Remote-Sensing satellites will have an inclination reduction of about 0.034° per year and for most of the satellites, the local time chosen was around 10:30 hours at descending node. In practice, the initial inclination is biased appropriately so that the expensive out-of-plane maneuvers could be taken up after few years of mission operations, depending on the deviations permitted in the local time for a given mission. However, the scenario differs when the mission objectives require an almost exact repeat orbit of 14 or 15 per day. In such a situation, the satellite orbit, which passes through a 14th or 15th order resonance, undergoes a nearly secular increase in orbit inclination. This paper presents a detailed analysis carried out for such an orbit, based on Cowell's approach. Long-term predictions have been carried out by considering all major forces that perturbs the satellite orbit. Observed behavior of orbit, based on the daily definitive orbit determination is also presented. The variation in inclination and the cause is clearly brought out. Further, it is demonstrated that the selection of longitude for nominal ground track pattern has an impact on the inclination variation. A proposal is made to take advantage of such expected inclination variation so that initial inclination bias can be chosen appropriately. Ground track longitude can be chosen to take advantage, subject to the mission coverage requirements. The paper contains the results of an exhaustive analysis of the actually observed orbit resonance. It is felt that the work has both theoretical and operational importance for remote-sensing missions.

Evaluation of 14th-order harmonics in the geopotential from 18 resonant satellite orbits

The values of lumped harmonics from the analyses of 18 satellite orbits at 14th-order resonance are used to determine individual tesseral harmonics of order 14 in the Earth’s gravity field. Eight C and five S coefficients are determined for odd degree and six C , S pairs for even degree. The results are most useful in providing the only independent test of the values obtained in recent comprehensive geopotential models. The models of the 1980s are generally consistent with the resonance values, which have lower or comparable standard deviations. The new Goddard Earth Model, GEM-T3, and European model, GRIM4-C1 have much lower standard deviations and show significant differences from our values.

Determination and analysis of the orbit of Meteor 28 (1977-57A) at 154 epochs during 15th-order resonance

The orbit of Meteor 28 (1977-57A) has been determined at 154 epochs between 8 January 1984 and 18 July 1989 during which the orbit was perturbed by 15th-order resonance with the geopotential. In total 9521 observations were used in the orbit determination process, the majority of which were supplied by the U.S. Navy. The remainder were British radar, visual and Hewitt camera observations. The variations in the inclination and eccentricity have been analysed to determine six lumped harmonics of order 15 and four of order 30 with accuracies of between 0.4 and 4.4 cm in geoid height. Since the orbit of Meteor 28 was Sun-synchronous, many sources of perturbation of the orbit (solar gravity, tidal effects, solar radiation pressure and drag) were enhanced by solar resonance. The removal of perturbations from the inclination and eccentricity to isolate the geopotential resonance effects posed a significant challenge in the analysis.

Analysis of the orbit of Transit 9 (1964-26 A) at 14th-order resonance

The orbit of Transit 9 (1964-26 A) passed through exact 14th-order geopotential resonance during January 1989. The orbit was determined at 52 epochs during the period August 1988–May 1990 when the perturbing effects of the resonance were significant. The resulting variations in inclination and eccentricity were analysed to give the values of six lumped harmonic coefficients corresponding to a mean orbital inclination of 90.468°. A detailed perturbation analysis was required in order to isolate the resonance perturbation in the orbit elements.

Analysis of the resonance angle of Cosmos 1603 (1984-106 A) near 14:1 resonance

During 1987 the near-circular orbit of the U.S.S.R. satellite Cosmos 1603 (1984-106 A) was strongly perturbed by 14th-order resonance terms in the Earth's gravity field. Due to near minimum solar activity, aerodynamic effects were subsidiary to resonance to the extent that, in a previous paper, lumped harmonics were recovered from the mean motion. The study is now taken a step further with lumped harmonics sought from the variation of the resonance parameter, Φ 1,14. An analytical expression is developed for the variation of Φ 1,14 with time due to the principal resonance term and then extended to include the second-order resonance effect. Air-drag is modelled by a small correction term. Lumped harmonics of order 14 and 28 are extracted from the resonance variable and compared against previous values from Cosmos 1603 and those from global gravity fields.

Orbit determination and analysis of Meteor 3 (1970−19A) at 15th-order resonance

The orbit of Meteor 3 (1970−19 A) has been determined at 52 epochs between August 1978 and January 1980 during which the orbit was perturbed by 15th-order resonance with the geopotential. U.S. Navy, together with Hewitt camera, theodolite, kinetheodolite and visual observations were used in the orbit determination process. The variations in the inclination and eccentricity have been analysed to determine six lumped harmonics of order 15 and two of order 30. In the analysis of the eccentricity an air drag model incorporating an oblate atmosphere with a diurnal density variation was used. The variations in the inclination and eccentricity have previously been analysed by King-Hele and Walker (1982, Proc. R. Soc. A379, 247) using orbits determined by the U.S. Naval Research Laboratory. The aim of this work is to find improved values of the lumped harmonics by using more accurately determined orbits.

Geopotential harmonics of order 15 and 30 from analysis of 50 orbit determinations of 1967-102 B

The orbital parameters of the satellite 1967-102 B, Cosmos 184 rocket, have been determined at 50 epochs from some 3900 observations. For 21 of these determinations, Hewitt camera observations were available. During the time of the orbit determinations the satellite passed through 15th-order resonance. The variations in inclination and eccentricity have been analysed to obtain six lumped 15th-order geopotential harmonics, with accuracies equivalent to between 0.2 and 3.3 cm in geoid height and four lumped 30th-order geopotential harmonics, with accuracies better than 1 cm in geoid height. The lumped harmonics obtained in this paper have been compared with those from the GEM 10B and 10C models and with those from GRIM3-L1 and the RAPP 1981 models. A comparison has also been made with the lumped harmonics obtained from a new PGS-3337 model published by NASA, and the agreement with this model is very good.

Analysis of the 15th-order resonance perturbation in the orbit of Tansei 4 rocket (1980-15 B)

The orbit of Tansei 4 rocket passed through exact 15th-order resonance during February 1982. The orbit was determined at 21 epochs over a 6-month period during which the main effects of the resonant perturbation were manifested. The determined values of inclination have been processed to reveal the change due to resonance, and this has been used to estimate the values of two lumped geopotential harmonic coefficients corresponding to an inclination of 38.67°. The usefulness of the results with respect to the future evaluation of tesseral harmonic coefficients is discussed, and the values are compared with those of standard Earth gravity models.

Analysis of the orbital elements of the satellite COSMOS 1603 (1984-106 A) at 14th-order resonance

The orbital parameters of Cosmos 1603 (1984-106 A) have been determined at 43 epochs from over 2900 observations, of which over 80% were supplied by the U.S. Naval Research Laboratory. Hewitt camera observations were available for 25 of the determinations. Orbital elements were determined between January and December 1987, during which time the satellite was close to 14:1 resonance. The satellite experienced the interesting property of being temporarily trapped with respect to a secondary resonance parameter due to the low air-drag in 1987. This effect gave rise to a quasi-secular increase in the eccentricity and libration of the secondary resonance variable. Analysis of the inclination and eccentricity yielded six lumped harmonic coefficients of order 14. Analysis of the mean motion yielded additional pairs of lumped harmonics of orders 14, 28 and 42, the 14th-order harmonics superseding those obtained from analysis of the inclination. The derived values were used to test the Goddard Earth Models, GEM-T1 and GEM-T2, at high order.

New values for geopotential harmonic coefficients of order 30 and even degree

A recent analysis of the orbit of the satellite 1967-102B as it passed through 15th-order resonance has yielded new and more accurate values of lumped 30th-order harmonics in the geopotential. These values have been used, together with eleven sets of values previously available, to derive improved values for geopotential harmonics of order 30 and degree 30, 32, 34,…, 40 : their average standard deviation is equivalent to an error in geoid height of 1.2 cm.

Orbit determination and analysis for cosmos 236 at 15th-order resonance

Cosmos 236 (1968-70A) was launched on 27 August 1968 into a near-circular orbit of inclination 56° and is predicted to decay towards the end of 1989. The orbit has been determined from observations for 77 epochs between July 1983 and October 1984 over the time interval when the orbit was expected to be significantly influenced by the effects of 15th-order resonance with the Earth's gravitational field: exact resonance occurred on 13 March 1984. The observations numbered over 4700, including 284 from the Hewitt cameras of the University of Aston sited at Herstmonceux in England and at Siding Spring in Australia. The orbital accuracy achieved was fairly consistent throughout, with the standard deviation in orbital inclination and eccentricity corresponding on average to positional accuracies of 85 and 65 m, respectively. Analysis of the changes in inclination and in eccentricity at resonance has given values of three pairs of lumped harmonics of order 15 and three pairs of order 30, one pair of each from inclination and two from eccentricity. The values from inclination had standard deviations equivalent to accuracies in geoid height of 0.6 and 2.0 cm for orders 15 and 30, respectively while the equivalent accuracies for the values from eccentricity were 1.6 and 6.0 cm.

Evaluation of 15th- and 30th-order geopotential harmonic coefficients from 26 resonant satellite orbits

The Earth's gravitational potential is usually expressed as an infinite series of harmonics, and the values of harmonic coefficients of order 15 and 30 can be determined most accurately by analysis of satellite orbits which experience 15th-order resonance. The results from two recent resonance analyses, for 1965-09A and 1968-70A, have here been added to those previously available, to produce an improved evaluation of 44 coefficients of order 15 and degree 15–36, and 12 coefficients of order 30 and even-degree 30, 32, …, 40. Compared with previous results, the new evaluation shows a great improvement in the standard deviations of many of the 15th-order coefficients of even-degree, thanks largely to the contribution of 1965-09A at inclination 31.8°: for the coefficients of degree 24, 26, 28 and 30, the standard deviation has been reduced by a factor of 3.1 on average; and for degree 32, 34 and 36 by a factor of 1.4 on average. For the other coefficients—those of 30th-order, and odd-degree 15th-order—the changes are relatively small. In the new 15th-order solution, all the 30 coefficients of degree 15–29 have standard deviations ⩽ 2.0 × 10 −9, and the average standard deviation of these 30 values is equivalent to an error in geoid height of 0.7 cm. Comparison of our values with those in comprehensive geoid models, which usually have larger standard deviation, lead us to conclude that, for order 15 and 30, the nominal standard deviations of the comprehensive models are quite realistic.

The analysis of the orbit of 1975-22B at 14th-order resonance

The orbit of Intercosmos 13 rocket (1975-22B) passed through exact 14th-order resonance during December 1977. The variations in orbital eccentricity and inclination are analyzed to evaluate four lumped geopotential harmonics, using 21 orbits determined by Brookes and Swinerd (1986, Planet. Space Sci. 34, 1263). In the analysis of eccentricity an acceptable result was recoverable only by the careful removal of the drag perturbation, due to an increase in solar activity during the latter half of the period considered. The inclination variation had been analyzed previously by Brookes and Swinerd, but the discovery of an inaccuracy in their results led to a re-evaluation of the lumped harmonic values in this paper.

Orbit determination of 1965-09A for analysis of the eccentricity at 15th-order resonance

A satellite experiences 15th-order resonance with the Earth's gravitational field when its track over the Earth repeats each day after 15 revolutions : the orbit is then very sensitive to the effects of 15th-order harmonics in the geopotential. By analysis of many such orbits, 15th-order harmonics of degree 15–36 have been evaluated with an accuracy equivalent to 0.6 cm in geoid height for degree 15–23, but with much poorer accuracy for the even degrees 24, 26, … 36, because of the absence of results for even degree from satellites with orbital inclination less than 43 . The satellite Pegasus 1, 1965-09A, at inclination 31.8° offers the chance to fill the gap: it passed slowly through the 15th-order resonance between November 1973 and January 1976. Orbits have been determined at 73 epochs from 4057 observations, chiefly U.S. Navy, and the variations in orbital eccentricity have been analysed to determine values of four lumped 15th-order harmonics with an accuracy equivalent to between 2 and 6 cm in geoid height. By adding these new results, the solution for the individual harmonics is greatly improved in accuracy, by a factor of 3.0 on average for degree 24, 26, 28 and 30, and a factor of 1.4 on average for degree 32, 34 and 36.

Geopotential harmonics of order 15 and 30, derived from the orbital resonances of 25 satellites

The recent accurate analysis of the satellite 1965-14A at 15th-order resonance has allowed significantly improved solutions to be derived for the individual harmonic coefficients in the geopotential of order 15 and 30. For order 15, coefficients of degree 15–36 have been evaluated (Tables 3 and 5); for degree 15–23, the mean accuracy is equivalent to 0.6 cm in geoid height; but the accuracy is poorer for degree 24–36, averaging 2.4 cm. For order 30, only the coefficients of even degree, from 30 to 40, have been evaluated (Table 8): for degree 30 and 32 the accuracy is equivalent to 1 cm in geoid height, but deteriorates to 2 cm for higher degree. The accuracies for 15th order, though in need of improvement for high degree, are better than tl ose available for any other order, and are already of the standard required for achieving in the 1990s the very difficult goal of a comprehensive geoid accurate to 10cm.

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.

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 variation in inclination of Intercosmos 13 Rocket, 1975-22B

The orbit of Intercosmos 13 rocket (1975-22B) has been determined at 103 epochs between 30 April 1975 and 10 April 1980 from almost 7000 observations. One hundred and three values of inclination have been determined and corrections incoporated for the effects due to zonal harmonic, lunisolar and tesseral harmonic perturbations, precession, and solid Earth tides. The modified data have been analysed to yield values of the atmospheric rotation rate, Λ rev day −1, viz. Λ = 0.94 ± 0.10 at an average height of 322 ± 6 km and Λ = 1.27 ± 0.02 at 288 km. Analysis of the inclination near 14th-order resonance has indicated lumped harmonic values 10 9 C 1.0 1.4 = − 76.13 ± 12.47, 10 9 S 1,0 14 = − 29.89 ± 32.64, 10 9 C −1.2 14 = − 63.11 ± 15.44 10 9 S −1.2 14 = − 32.52 ± 26.96, for inclination 82.952°.

14th-order harmonics in the geopotential from analysis of satellite orbits at resonance

The Earth's gravitational potential is usually expressed as a double series of tesseral harmonics of degree l and order m, and values of coefficients for l and m up to 36 or more are determined in recent models. The individual harmonic coefficients of a particular order m can most accurately be evaluated by analysing the perturbations on satellites which experience mth-order resonance (i.e. repeat their ground track after m revolutions). Here we use results from analyses of 15 satellite orbits at 14th-order resonance to evaluate 7 pairs of individual coefficients of 14th order and odd degree ( l = 15, 17,… 27), and 6 pairs of coefficients of even degree ( l = 14, 16,… 24). The five most accurate pairs of values, for l = 14, 15, 16, 17 and 19, have a mean standard deviation corresponding to an accuracy of 0.9 cm in geoid height. These values are used as a test of the accuracy of three recent comprehensive models of the geopotential : the agreement is generally satisfactory, the differences being consistent with the expected errors (about 3 × 10 −9) in the models. We also obtain values for the pair of coefficients of order 28 and degree 28, again with standard deviation equivalent to 0.9 cm in geoid height.

Orbit determination and analysis for 1970-97B at 14th-order resonance

The orbit of the satellite 1970-97B, the rocket of Cosmos 378, at inclination 74°, has been determined at 18 epochs between April and August 1971, when the effects of 14th-order resonance with the Earth's gravitational field were appreciable. The orbits were determined with the PROP 6 program from Hewitt camera, kinetheodolite, U.S. Navy and visual observations, and an average accuracy of 90 m in perigee distance was achieved, despite the low perigee height (near 230 km) and the consequent high drag. The orbits, together with 13 previously evaluated, have been analysed to reveal the effects of the 14th-order resonance and to evaluate six lumped geopotential harmonics of order 14. Because the orbit passed through resonance rapidly, the values are not as accurate as those from slow resonances; but they are more accurate than any others available for an inclination near 74°, and have proved their worth in a recent determination of individual 14th-order coefficients.