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.     

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 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.     

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.     

Efficient Determination of Global Gravity Field from Satellite-to-satellite Tracking Mission

The gravity field dedicated satellite missions like CHAMP, GRACE, and GOCE are supposed to map the Earth's global gravity field with unprecedented accuracy and resolution. New models of the Earth's static and time-variable gravity fields will be available every month as one of the science products from GRACE. A method for the efficient gravity field recovery is presented using in situ satellite-to-satellite observations at altitude and results on static as well as temporal gravity field recovery are shown. Considering the energy relationship between the kinetic energy of the satellite and the gravitational potential, the disturbing potential observations can be computed from the orbital state vector, using high-low GPS tracking data, low–low satellite-to-satellite GRACE measurements, and data from 3-axis accelerometers. The solution method is based on the conjugate gradient iterative approach to efficiently recover the gravity field coefficients and approximate error covariance up to degree and order 120 every month. Based on the monthly GRACE noise-only simulation, the geoid was obtained with an accuracy of a few cm and with a resolution (half wavelength) of 160 km. However, the geoid accuracy can become worse by a factor of 6–7 because of spatial aliasing. The approximate error covariance was found to be a very good accuracy measure of the estimated coefficients, geoid, and gravity anomaly. The temporal gravity field, representing the monthly mean continental water mass redistribution, was recovered in the presence of measurement noise and high frequency temporal variation. The resulting recovered temporal gravity fields have about 0.3 mm errors in terms of geoid height with a resolution of 670 km.     

The accuracy of geopotential models

Extensive tests of two recent geopotential models (GEM 7 and 8) have been made with observations not used in the solutions. Several other recent models are also evaluated. These tests show the accuracy of the satellite derived model (GEM 7, with 400 coefficients) to be about 4.3 m (r.m.s.) with respect to the global geoid surface. The corresponding accuracy of the combined satellite and surface gravimetry model (GEM 8, with 706 coefficients) is found to be 3.9m (r.m.s.). These results include a calibration for the commission errors of the coefficients in the models and an estimate of the errors from omitted coefficients. For GEM 7, the formal precision (commission errors) of the solution gives 0.7 m for the geoid error which after calibration increases to 2.4 m.

Independent observations used in this assessment include: 159 lumped coefficients from 35 resonant orbits of 1 and 9 through 15 revolutions per day, two sets of (8, 8) fields derived from optical-only and laser-only data, sets of zonal and resonant coefficients derived from largely independent sources and geoid undulations measured by satellite altimetry. In addition, the accuracy of GEM 7 has been judged by the gravimetry in GEM 8. The ratio of estimated commission to formal error in GEM 7 and 8 ranges from 2 to 5 in these tests.     

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 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 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 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.     

Future Benefits of Time-Varying Gravity Missions to Ocean Circulation Studies

A summary is offered of the potential benefits of future measurements of temporal variations in gravity for the understanding of ocean dynamics. Two types of process, and corresponding amplitudes are discussed: ocean basin scale pressure changes, with a corresponding amplitude of order 1 cm of water, or 1 mm of geoid height, and changes in along-slope pressure gradient, at cross-slope length scales corresponding to topographic slopes, with a corresponding amplitude of order 1 mm of water, or a maximum of about 0.01 mm of geoid. The former is feasible with current technology and would provide unprecedented information about abyssal ocean dynamics associated with heat transport and climate. The latter would be a considerable challenge to any foreseeable technology, but would provide an exceptionally clear, quantitative window on the dynamics of abyssal ocean currents, and strong constraints on ocean models. Both options would be limited by the aliassing effect of rapid mass movements in the earth system, and it is recommended that any future mission take this error source explicitly into account at the design stage. For basin-scale oceanography this might involve a higher orbit than GRACE or GOCE, and the advantages of exact-repeat orbits and multiple missions should be considered.     

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.     

Topography of polygonal structures at the Phoenix landing site on mars through the relief retrieval from the HiRISE images with the improved photoclinometry method

The relief of polygonal structures at the Phoenix landing site on Mars has been determined with the improved photoclinometry method from the images acquired with the HiRISE camera on board the Mars Reconnaissance Orbiter. The investigations showed that, within 1 km from the landing site, the topography amplitude of the relief on the surface scales of 5.5–65 m varies within the range of ～40 to 70 cm. The polygonal structures of 2–6 m across correspond to the small-scale relief with the topography amplitude ranging from 20 to 30 cm and the standard deviation of about 3 cm. Within 1 km from the landing site, the variations of these characteristics are small. For the small polygons that are less than 5.5 m in size, the typical height is 10–15 cm. The polygons of 18–22 m in size are up to 28 cm in height, while the polygons of 60–90 m in size reach about 44 cm in height. The error in determining the relief heights was ±5.5%. The investigations showed that the improved photoclinometry method is promising for the study of small-scale features of the Martian surface.     

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.     

Ice Mass Balance and Ice Dynamics from Satellite Gravity Missions

An overview of advances in ice research which can be expected from future satellite gravity missions is given. We compare present and expected future accuracies of the ice mass balance of Antarctica which might be constrained to 0.1–0.3 mm/year of sea level equivalent by satellite gravity data. A key issue for the understanding of ice mass balance is the separation of secular and interannual variations. For this aim, one would strongly benefit from longer uninterrupted time series of gravity field variations (10 years or more). An accuracy of 0.01 mm/year for geoid time variability with a spatial resolution of 100 km would improve the separability of ice mass balance from mass change due to glacial isostatic adjustment and enable the determination of regional variations in ice mass balance within the ice sheets. Thereby the determination of ice compaction is critical for the exploitation of such high accuracy data. A further benefit of improved gravity field models from future satellite missions would be the improvement of the height reference in the polar areas, which is important for the study of coastal ice processes. Sea ice thickness determination and modelling of ice bottom topography could be improved as well.     

Intercomparisons of earth gravity models by means of lumped coefficients: Terms for eccentricity and longitude

The comparisons of the Earth gravity field models by the order of their harmonic coefficients, performed with the basic lumped coefficients (Planet. Space Sci.29, 653, 1981, Paper I) are here extended to cover all harmonic coefficients (both odd and even degree). The lumped coefficients (the “e-terms” and “longitudinal” terms), corresponding to 18 Earth models, are compared mutually (Figs. 2–15). The large differences, observed for various models and orders, are of particular interest: they are gathered into Table 1. The result of Paper I was a little pessimistic. The same is true here: various inhomogeneities, sometimes very large, in the accuracy of the harmonic coefficients must exist—even for low orders. Most of our comments and objections, however, relate to the older Earth models, which have only a historical value now. Our comparisons are only relative ones; an actual test of the accuracy of the models (their calibration) is possible via data with independent status (Kloko?ník, 1982, 1983).     

Polar wander on Mars: Evidence in the geoid

The geoid of Mars is dominated by its equilibrium figure and by the effect of the Tharsis rise. To investigate the rotational stability of Mars prior to the rise of Tharsis, we produced a residual non-hydrostatic geoid without Tharsis. First the hydrostatic component of the present-day flattening was removed. This procedure was performed using a 6% non-hydrostatic component of flattening, a value set by the spin axis precession rate of Mars. Then zonal spherical harmonics up to degree 6 centered on Tharsis were removed. Finally, the resultant residual geoid was evaluated for rotational stability by comparing polar and equatorial moments at 4050 trial pole positions. If the spin axis of ancient Mars was secularly stable, our analysis indicates that substantial polar wander has occurred with the rise of Tharsis. Stable spin axis positions on the non-hydrostatic residual figure of Mars are 15° to 90° from the present-day poles. This result is consistent with previously proposed paleopoles based on magnetic anomalies, geomorphology, and grazing impacts.     

一种计算垂线偏差的抗差最小二乘严密解法

针对当前利用大地水准面模型求解垂线偏差精度不高、稳健性差的问题,设计了一种严密的垂线偏差抗差最小二乘解法.首先,基于大地水准面与垂线偏差的关系,采用EGM2008 (Earth Gravity Model 2008)重力场模型计算参数初始解;然后,引入中位数抗差法,并选用Huber权函数计算等价权,迭代计算出稳健的垂线偏差最小二乘解;最后,结合两个实测算例对设计方法进行验证.试验结果表明,该方法计算的垂线偏差分量与约定真值最大偏差在0.5′′左右,相较于对比方法精度更高;同时,该方法能有效抵抗粗差值的影响,具有较强的稳健性.     

An Analytical Solution of the Lagrange Equations Valid also for Very Low Eccentricities: Influence of a Central Potential

This paper presents an analytic solution of the equations of motion of an artificial satellite, obtained using non singular elements for eccentricity. The satellite is under the influence of the gravity field of a central body, expanded in spherical harmonics up to an arbitrary degree and order. We discuss in details the solution we give for the components of the eccentricity vector. For each element, we have divided the Lagrange equations into two parts: the first part is integrated exactly, and the second part is integrated with a perturbation method. The complete solution is the sum of the so-called “main” solution and of the so-called “complementary” solution. To test the accuracy of our method, we compare it to numerical integration and to the method developed in Kaula (Theory of Satellite Geodesy, Blaisdell publ. Co., New York. 1966), expressed in classical orbital elements. For eccentricities which are not very small, the two analytical methods are almost equivalent. For low eccentricities, our method is much more accurate.