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.     

Evaluation of harmonics in the geopotential of order 15 and odd degree

When a satellite orbit decaying slowly under the action of air drag experiences 15th-order resonance with the Earth's gravitational field, so that the ground track repeats after 15 rev, the orbital inclination suffers appreciable changes due to the perturbations from the harmonics in the geopotential of order 15 and odd degree (15,17,19 …). In this paper the changes in inclination at resonance of 11 satellites at inclinations between 30° and 90° have been analysed to determine values of the geopotential coefficients of order 15 and degree $\text{l,}\text{C}\text{?}{}_{\mathrm{l,15}}$ and $\text{S}\text{?}{}_{\mathrm{l,15}}$ in the usual notation. The recommended solution, going up to l = 31, is:

     

3.

Orbit determination and analysis for Cosmos 482 from 1978 to 1981     

D.G. King-Hele  A.N. Winterbottom 《Planetary and Space Science》1985,33(10):1125-1143

The orbit of the satellite Cosmos 482 (1972-23A) has been determined at 77 epochs between 8 November 1977 and 18 April 1981 from 5650 optical and radar observations. The computations were made with the RAE orbit determination program PROP 6, and an average accuracy of 150 m radial and cross-track was achieved.Cosmos 482 was a high-drag satellite in an eccentric orbit and, between the first epoch and the last, the orbital period decreased from 157 to 94 min, the eccentricity decreased from 0.32 to 0.04, and the orbital inclination decreased from 52.14° to 51.95° due to the transverse forces caused by atmospheric rotation. The orbit was therefore ideal for determining the atmospheric rotation rate from the decrease in inclination, and seven accurate values of rotation rate have been obtained. The new values strengthen the existing overall picture of upper-atmosphere winds, and are generally in good accord with the previous results.An improved equation has been derived for calculating density scale height H from the decrease in perigee distance, and has been applied to determine seven values of H. The corresponding values of H from the COSPAR International Reference Atmosphere are on average 5% lower than the observational values, for 1980–1981.     

4.

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

D. G. King-Hele 《Planetary and Space Science》1986,34(12):1319-1328

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.     

5.

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

D.G. King-Hele  Doreen M.C. Walker 《Planetary and Space Science》1985,33(2):223-238

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.     

6.

Geopotential harmonics of order 15 and even degree,from changes in orbital eccentricity at resonance     

D.G. King-Hele  Doreen M.C. Walker  R.H. Gooding 《Planetary and Space Science》1975,23(2):229-246

When a satellite orbit decaying slowly under the action of air drag experiences 15th-order resonance with the Earth's gravitational field, so that the ground track repeats after 15 rev, the orbital eccentricity may suffer appreciable changes due to perturbations from the gravitational harmonics of order 15 and even degree (16, 18, 20…). In this paper the changes in eccentricity at resonance for six satellites in near-circular orbits at inclinations between 56 and 90° have been analysed to derive 11 pairs of equations linking the harmonic coefficients of order 15 and (even) degree l, $\text{C}{}_{\mathrm{l,15}}\text{and}\text{S}{}_{\mathrm{l,15}}$ in the usual notation. These equations (together with eight constraint equations) are solved to give:

l	109C?l,15	109S?l,15
15	?21.5 ± 0.9	?8.4 ± 0.9
17	4.4 ± 1.6	9.0 ± 1.5
19	?15.6 ± 2.6	?14.1 ± 2.7
21	10.4 ± 3.0	7.3 ± 3.5
23	22.5 ± 2.8	1.2 ± 4.4
25	?0.9 ± 4.7	?3.8 ± 5.3
27	?11.2 ±3.3	9.1 ± 3.2
29	?20.5 ± 5.4	?1.2 ± 6.1
31	17.7 ± 6.6	?1.0 ± 7.1

     

7.

Analysis of the orbit of Cosmos 72 (1965-53B) near 15th-order resonance     

D.G. Keng-Hele 《Planetary and Space Science》1974,22(9):1269-1277

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.     

8.

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

Doreen M.C. Walker 《Planetary and Space Science》1985,33(1):97-107

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.     

9.

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

《Planetary and Space Science》1987,35(1):79-90

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.     

10.

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

D.G. King-Hele  Doreen M.C. Walker 《Planetary and Space Science》1982,30(4):411-425

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.     

11.

Lumped geopotential harmonics of order 14, from the orbit of 1967-11G     

D.G. King-Hele 《Planetary and Space Science》1985,33(10):1145-1153

The satellite 1967-11G, which had an orbital inclination of 40°, passed through the 14th-order resonance with the Earth's gravitational field in 1974. The changes in its orbital inclination at resonance have been analysed to obtain values for four lumped 14th-order harmonics in the geopotential, with accuracies equivalent to about 5 cm in geoid height. Analysis of the eccentricity was also attempted, but did not yield useful results.As no previous satellite analysed at 14th-order resonance has had an inclination near 40°, the results have proved to be valuable in determining individual 14th-order harmonics in the geopotential.     

12.

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

Doreen M.C. Walker 《Planetary and Space Science》1977,25(4):337-342

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.     

13.

Geopotential harmonics of order 15 and odd degree from analysis of resonant orbits     

D.G. King-Hele  Doreen M.C. Walker  R.H. Gooding 《Planetary and Space Science》1975,23(9):1239-1256

We have analysed the variations of inclination in 13 satellite orbits as they pass slowly, under the action of air drag, through 15th-order resonance with the geopotential, when successive equatorial crossings are 24° apart and the ground track repeats after 15 rev. The size and form of the change in inclination are determined mainly by the values of the geopotential harmonics of 15th order and odd degree, $\text{C}\text{?}{}_{\mathrm{l,15}}$ and $\text{S}\text{?}{}_{\mathrm{l,15}}$ (with l = 15, 17, 19, …) in the usual notation. Our analysis gives values of these coefficients up to l = 33 as follows:

$\text{l}$	109$\text{C}{}_{\mathrm{l,15}}$	109$\text{S}{}_{\mathrm{l,15}}$
16	?13.7 ± 1.3	?18.5 ± 2.7
18	?42.3 ± 1.8	?34.7 ± 3.4
20	10.5 ± 3.1	29.8 ± 5.2
22	?8.6 ± 3.8	?20.2 ± 7.4

     

14.

Analysis of the orbit of ariel 1, 1962-15A,near 15th-order resonance     

Doreen M.C. Walker 《Planetary and Space Science》1975,23(4):565-574

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.     

15.

Samos 2 (1961 α 1): Orbit determination and analysis at 31 : 2 resonance     

Doreen M.C. Walker 《Planetary and Space Science》1980,28(11):1059-1072

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.     

16.

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

Doreen M.C. Walker 《Planetary and Space Science》1985,33(12):1439-1449

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.     

17.

Position and velocity perturbations for the determination of geopotential from space geodetic measurements   总被引：10，自引：0，他引：10  

Peiliang Xu 《Celestial Mechanics and Dynamical Astronomy》2008,100(3):231-249

Although space geodetic observing systems have been advanced recently to such a revolutionary level that low Earth Orbiting (LEO) satellites can now be tracked almost continuously and at the unprecedented high accuracy, none of the three basic methods for mapping the Earth’s gravity field, namely, Kaula linear perturbation, the numerical integration method and the orbit energy-based method, could meet the demand of these challenging data. Some theoretical effort has been made in order to establish comparable mathematical modellings for these measurements, notably by Mayer-Gürr et al. (J Geod 78:462–480, 2005). Although the numerical integration method has been routinely used to produce models of the Earth’s gravity field, for example, from recent satellite gravity missions CHAMP and GRACE, the modelling error of the method increases with the increase of the length of an arc. In order to best exploit the almost continuity and unprecedented high accuracy provided by modern space observing technology for the determination of the Earth’s gravity field, we propose using measured orbits as approximate values and derive the corresponding coordinate and velocity perturbations. The perturbations derived are quasi-linear, linear and of second-order approximation. Unlike conventional perturbation techniques which are only valid in the vicinity of reference mean values, our coordinate and velocity perturbations are mathematically valid uniformly through a whole orbital arc of any length. In particular, the derived coordinate and velocity perturbations are free of singularity due to the critical inclination and resonance inherent in the solution of artificial satellite motion by using various types of orbital elements. We then transform the coordinate and velocity perturbations into those of the six Keplerian orbital elements. For completeness, we also briefly outline how to use the derived coordinate and velocity perturbations to establish observation equations of space geodetic measurements for the determination of geopotential.     

18.

A long hard look at MCG–6-30-15 with XMM–Newton– II. Detailed EPIC analysis and modelling     

S. Vaughan  A. C. Fabian 《Monthly notices of the Royal Astronomical Society》2004,348(4):1415-1438

     

19.

Observations of solar filaments at 8, 15, 22 and 43 GHz     

E. J. Schmahl  M. Bobrowsky  M. R. Kundu 《Solar physics》1981,71(2):311-328

On April 3, 4, 6, and 8, 1978, solar observations were made using the Haystack 120 ft telescope at 8, 15, 22, and 43 GHz. H filtergrams obtained at the Sacramento Peak Observatory on the same days showed an average of more than 30 filaments or filament fragments (per day) on the disk. Most of these appeared as depressions in brightness temperature at 15 and 22 GHz. Because of the relatively low spatial resolution at 8 GHz, only a few appeared at that frequency, and presumably because of lower opacity in filaments at higher frequencies, few depressions were visible at 43 GHz. At 15 and 22 GHz, more depressions appeared than H filaments, but virtually all the radio depressions overlay magnetic neutral lines. Taking the data sets for each day as independent samples, we found that at 22 GHz, 46 of the 77 radio depressions were associated with H filaments; at 15 GHz the correlation was smaller; only 27 out of 48 being associated with the H filaments. The data imply that the microwave depression features are the result of absorption by filaments and perhaps also the result of other effects of the associated filament channel, but not necessarily coronal depletion. The effects of filament absorption are, statistically, about twice as effective as other phenomena (such as absorption by material invisible in H, for example) in creating the radio depression. A center-to-limb study of a single large filament clearly showed that at 15 and 22 GHz the absorption by cool hydrogen supported above the neutral line was the predominant factor in producing the observed depression at radio frequencies.     

20.

Reduction in the nuclear quadrupole resonance signal level at resonsance frequency of 30 MHz, concomitant with impact of comet fragment with Jupiter     

E. P. A. Sullivan 《Earth, Moon, and Planets》1996,74(2):175-182

     

l	109C?l,15	109S?l,15
15	?23.5 ± 0.8	?7.7 ± 0.8
17	6.3 ± 1.5	5.6 ± 1.5
19	?25.1 ± 2.5	?7.3 ± 2.3
21	27.8 ± 3.6	?0.7 ± 3.4
23	17.1 ± 4.1	13.9 ± 4.8
25	?1.1 ± 3.0	8.5 ± 4.2
27	10.0 ± 3.3	6.7 ± 2.7
29	?9.4 ± 3.5	0.1 ± 4.7
31	10.1 ± 5.4	3.8 ± 5.6
33	1.1 ± 5.7	3.1 ± 5.8

Orbit determination and analysis for Cosmos 482 from 1978 to 1981

The orbit of the satellite Cosmos 482 (1972-23A) has been determined at 77 epochs between 8 November 1977 and 18 April 1981 from 5650 optical and radar observations. The computations were made with the RAE orbit determination program PROP 6, and an average accuracy of 150 m radial and cross-track was achieved.Cosmos 482 was a high-drag satellite in an eccentric orbit and, between the first epoch and the last, the orbital period decreased from 157 to 94 min, the eccentricity decreased from 0.32 to 0.04, and the orbital inclination decreased from 52.14° to 51.95° due to the transverse forces caused by atmospheric rotation. The orbit was therefore ideal for determining the atmospheric rotation rate from the decrease in inclination, and seven accurate values of rotation rate have been obtained. The new values strengthen the existing overall picture of upper-atmosphere winds, and are generally in good accord with the previous results.An improved equation has been derived for calculating density scale height H from the decrease in perigee distance, and has been applied to determine seven values of H. The corresponding values of H from the COSPAR International Reference Atmosphere are on average 5% lower than the observational values, for 1980–1981.     

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.     

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.     

Geopotential harmonics of order 15 and even degree,from changes in orbital eccentricity at resonance

When a satellite orbit decaying slowly under the action of air drag experiences 15th-order resonance with the Earth's gravitational field, so that the ground track repeats after 15 rev, the orbital eccentricity may suffer appreciable changes due to perturbations from the gravitational harmonics of order 15 and even degree (16, 18, 20…). In this paper the changes in eccentricity at resonance for six satellites in near-circular orbits at inclinations between 56 and 90° have been analysed to derive 11 pairs of equations linking the harmonic coefficients of order 15 and (even) degree l, $\text{C}{}_{\mathrm{l,15}}\text{and}\text{S}{}_{\mathrm{l,15}}$ in the usual notation. These equations (together with eight constraint equations) are solved to give:

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.     

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.     

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.     

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.     

Lumped geopotential harmonics of order 14, from the orbit of 1967-11G

The satellite 1967-11G, which had an orbital inclination of 40°, passed through the 14th-order resonance with the Earth's gravitational field in 1974. The changes in its orbital inclination at resonance have been analysed to obtain values for four lumped 14th-order harmonics in the geopotential, with accuracies equivalent to about 5 cm in geoid height. Analysis of the eccentricity was also attempted, but did not yield useful results.As no previous satellite analysed at 14th-order resonance has had an inclination near 40°, the results have proved to be valuable in determining individual 14th-order harmonics in the geopotential.     

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.     

Geopotential harmonics of order 15 and odd degree from analysis of resonant orbits

We have analysed the variations of inclination in 13 satellite orbits as they pass slowly, under the action of air drag, through 15th-order resonance with the geopotential, when successive equatorial crossings are 24° apart and the ground track repeats after 15 rev. The size and form of the change in inclination are determined mainly by the values of the geopotential harmonics of 15th order and odd degree, $\text{C}\text{?}{}_{\mathrm{l,15}}$ and $\text{S}\text{?}{}_{\mathrm{l,15}}$ (with l = 15, 17, 19, …) in the usual notation. Our analysis gives values of these coefficients up to l = 33 as follows:

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.     

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 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.     

Position and velocity perturbations for the determination of geopotential from space geodetic measurements

Although space geodetic observing systems have been advanced recently to such a revolutionary level that low Earth Orbiting (LEO) satellites can now be tracked almost continuously and at the unprecedented high accuracy, none of the three basic methods for mapping the Earth’s gravity field, namely, Kaula linear perturbation, the numerical integration method and the orbit energy-based method, could meet the demand of these challenging data. Some theoretical effort has been made in order to establish comparable mathematical modellings for these measurements, notably by Mayer-Gürr et al. (J Geod 78:462–480, 2005). Although the numerical integration method has been routinely used to produce models of the Earth’s gravity field, for example, from recent satellite gravity missions CHAMP and GRACE, the modelling error of the method increases with the increase of the length of an arc. In order to best exploit the almost continuity and unprecedented high accuracy provided by modern space observing technology for the determination of the Earth’s gravity field, we propose using measured orbits as approximate values and derive the corresponding coordinate and velocity perturbations. The perturbations derived are quasi-linear, linear and of second-order approximation. Unlike conventional perturbation techniques which are only valid in the vicinity of reference mean values, our coordinate and velocity perturbations are mathematically valid uniformly through a whole orbital arc of any length. In particular, the derived coordinate and velocity perturbations are free of singularity due to the critical inclination and resonance inherent in the solution of artificial satellite motion by using various types of orbital elements. We then transform the coordinate and velocity perturbations into those of the six Keplerian orbital elements. For completeness, we also briefly outline how to use the derived coordinate and velocity perturbations to establish observation equations of space geodetic measurements for the determination of geopotential.     

Observations of solar filaments at 8, 15, 22 and 43 GHz

On April 3, 4, 6, and 8, 1978, solar observations were made using the Haystack 120 ft telescope at 8, 15, 22, and 43 GHz. H filtergrams obtained at the Sacramento Peak Observatory on the same days showed an average of more than 30 filaments or filament fragments (per day) on the disk. Most of these appeared as depressions in brightness temperature at 15 and 22 GHz. Because of the relatively low spatial resolution at 8 GHz, only a few appeared at that frequency, and presumably because of lower opacity in filaments at higher frequencies, few depressions were visible at 43 GHz. At 15 and 22 GHz, more depressions appeared than H filaments, but virtually all the radio depressions overlay magnetic neutral lines. Taking the data sets for each day as independent samples, we found that at 22 GHz, 46 of the 77 radio depressions were associated with H filaments; at 15 GHz the correlation was smaller; only 27 out of 48 being associated with the H filaments. The data imply that the microwave depression features are the result of absorption by filaments and perhaps also the result of other effects of the associated filament channel, but not necessarily coronal depletion. The effects of filament absorption are, statistically, about twice as effective as other phenomena (such as absorption by material invisible in H, for example) in creating the radio depression. A center-to-limb study of a single large filament clearly showed that at 15 and 22 GHz the absorption by cool hydrogen supported above the neutral line was the predominant factor in producing the observed depression at radio frequencies.