Determination of magnetic field direction in a comet's head

The Stokes parameters of resonance radiation scattered by a Na atom with the angular momentum F aligned by directed unpolarized radiation in a magnetic field H ～ 10^?5?10^?1 Oe are presented. An influence of the orientation of the magnetic field on these parameters are studied; the intensity ratio $\text{I(D}{}_2\text{)}\text{I(D}{}_1\text{)}$ changes within ±5%, and the polarization degree P(D₂) within ±25%. Measurements of $\text{I(D}{}_2\text{)}\text{I(D}{}_1\text{)}$ and P(D₂), if the geometry of scattering is known, may give information on the direction of the magnetic field in the sodium atmospheres of comets, as well as Io's sodium cloud or man-made cosmic clouds.     

Analysis of 27 satellite orbits to determine odd zonal harmonics in the geopotential

The odd zonal harmonics in the geopotential are the terms independent of longitude and antisymmetric about the Equator: they define the ‘pear-shape’ effect. The coeffecients J₃, J₅, J₇,…of these harmonics have been evaluated by analysing the variations in eccentricity of 27 orbits covering wide range of inclinations. We use again most of the orbits from our previous (1969) evaluations, but we now have the advantage of 3 accurate orbits at inclinations between 60° and 66°, where the variations in eccentricity become very large, and 3 near-equatorial orbits, at inclinations between 3° and 15°, whereas previously there were none at inclinations lower than 28°. The new data lead to much more accurate and reliable values for the coeffecients. Our recommended set, which terminates at J₁₇, is

$\text{10}{}^9\text{J}{}_3\text{= ?2531 ± 7}\text{10}{}^9\text{J}{}_{11}\text{= 159 ± 16}\text{J}{}_5\text{= ?246 ± 9}\text{J}{}_{13}\text{= ?131 ± 22}\text{J}{}_7\text{= ?326 ± 11}\text{J}{}_{15}\text{= ?26 ±24}\text{J}{}_9\text{= ?94 ± 12}\text{J}{}_{17}\text{= ?258 ± 19}$

. With this new set of values the pear-shape tendency of the Earth amounts to 44.7 m at the poles, instead of the previous 40 m, though the new geoid is within 1 m of the old at latitudes away from the poles.     

Orientation and shape of the Earth's bow shock in three dimensions

Nearly 2500 shock crossings from HEOS-1, HEOS-2 and 5 IMP spacecraft, covering most of the northern and part of the southern bow shock surface for X values X > ? 20 R_E, have been used to carry out a detailed study of the three-dimensional shape and location of the bow shock. The influence of the different solar wind conditions has been reduced by normalising the observed crossings to an average solar wind dynamical pressure (N₀ = 9.4 cm^?3, V₀ = 450 kms^?1). It has been shown that the shock surface is symmetric with respect to the ecliptic plane and intersects the coordinate axes at 11.9 R_E (X), + 27.0 and ? 22.9 R_E (Y), + 23.9 and ? 24.5 R_E (Z) for the average dynamical pressure (N₀ = 9.4 cm^?3, V₀ = 450kms^?1, with $\text{M}{}_{\mathrm{A}}\text{= 9.3}$, $\text{M}{}_{\mathrm{MS}}\text{= 6.1)}$. The observed aberration of the shock surface is 8.9° ± 1°, i.e. 5.1° larger than the aberration predicted from the Earth's motion. This asymmetry around the solar wind apparent direction is described by equation (6) for different Mach numbers M_A and confirms the predictions of Walters [J. geophys. Res. 71, 1319 (1964)] and Michel [J. geophys. Res. 70, 1 (1965)].The magnetosheath thickness is 3.3 R_E along the X-axis, 11.4 R_E (+ Y), 8.7 R_E (? Y), 9.9 R_E (+Z) and 10.9 R_E along the negative Z axis.     

New high-speed photometry of EH Lib and a binary interpretation of its period change

Six times of maxima of the ultrashort-period cepheid variable EH Librae were measured in 1980 May to June and in 1981 January, with a three-channel photocounting high-speed photoelectric photometer. These, together with all the photoelectric times of maxima over the past 30 years, are used to re-examine the nature of the change of the period. We found that we can fix the times of maxima by the following formula

$\text{T}{}_{\max }\text{= T}{}_0\text{+P}{}_0\text{E+}\text{1}\text{2}\text{βE}{}^2\text{+A}\text{sin}\text{2π}\text{EP}{}_0\text{E}{}_0$

where T₀ = HJD 2433438.6088 and P₀ = 0.0884132445 d represent the initial maximum epoch and the pulsation period, β = ?2.8 × 10^?8/yr; A = 0.0015 d, P₀ = 6251 d = 17.1 yr are the semi-amplitude and the period of the sine curve, and E is the number of periods elapsed since T₀, and (E₀ = 70700).If we interpret this 17.1 year periodicity as a modulation of the phase of maximum by binary motion, then the semi-amplitude of the orbital radial velocity variation is K = 2πasini/E₀ = 0.45 km/s and the mass function is

$\text{f(m)=}\text{m}{}^3{}_2\text{sin}{}^3\text{i}\text{(m}{}_1\text{m}{}_2\text{)}{}^2\text{=}\text{(a}\text{sin}\text{i)}{}^3\text{E}{}^2{}_0\text{=6 x 10}{}^{\mathrm{?5}}\text{M}{}_{\mathrm{\odot }}$

     

On the molecular scattering in the terrestrial atmosphere : An empirical formula for its calculation in the homosphere

A recent determination by D. R. Bates of the Rayleigh scattering cross section (σ_RS) for air from 0.2 to 1 μm leads to a simple empirical formula (λ in μm) $\text{σ}{}_{\mathrm{RS}}\text{=}\text{4.02 × 10}{}^{\mathrm{?28}}\text{λ}{}^{\mathrm{4+x}}\text{cm}{}^2$ where $\text{x =}\text{0.389λ + 0.09426}\text{λ ? 0.3228}$ for the spectral region 0.2 μm < λ < 0.55 μm ; the accuracy is within ±0.5%. From the visible at 0.55 μm to the infrared (i.r.) at 1 μm, the same accuracy can be obtained using a constant value, x = 0.04. The formula accounts for the degree of depolarization which varies with the wavelength according to the latest determination by Bates.     

Impact fragmentation experiments of basalts and pyrophyllites

Results of impact fragmentation experiments for basalts and pyrophyllites are reported. Aluminum cylindrical projectiles were impacted on cubic basalt and pyrophyllite targets at velocities of 70 to 990 m/sec. The targets and projectiles were 20 g to 3.3 kg and 2 to 20 g in weight respectively. Weights of the fragments produced by impacts were measured and the size distributions of fragments were examined. Data of the largest fragment mass (m_L) normalized to the original target mass (M_t), m_L/M_t, correlate better with the nondimensional impact stress, P_I, a new scaling parameter introduced by H. Mizutani, Y. Takagi, and S. Kawakami (1984, in preparation) than the conventional projectile's kinetic energy per unit target mass, E/M_t, used in the previous studies. All the m_L/M_t data for basalts obtained in the present study are summarized by m_L/M_t = 2.95 × 10^?2P_I^?1 where P_I = P₀L³/YR³, P₀ = peak shock pressure, L = projectile size, R = target size and Y = material strength of target. For aluminum targets, however, the m_L/M_t is 2.5 orders of magnitude larger than that for brittle targets at impacts with the same P_I. Size distributions of fragments expressed in a log N - log (m/M_t) diagram divided into three regimes bounded by two inflection points. In each regime the curve is expressed by $\text{N (>}\text{m}\text{M}{}_{\mathrm{t}}\text{) = A (}\text{m}\text{M}{}_{\mathrm{t}}\text{)}{}^{\mathrm{?a}}$. The slopes, a, of the log $\text{N -}\text{log}\text{(}\text{m}\text{M}{}_{\mathrm{t}}\text{)}$ curves in the regimes of a large and a medium size range are positively correlated with the nondimensional impact stress, P_I, and expressed as a = C₃ + a₃log P_I. The slopes, a, in the smallest size range are, on the other hand, nearly constant and have values of $\text{0.5}\text{to}\text{0.7 (}\text{1}\text{2}\text{?}\text{2}\text{3}\text{)}$. Present results indicate that the impact fragmentation is scaled well by the new scaling parameter, P_I, of Mizutani, Takagi, and Kawakami and that the present experimental data may shed new light on planetary impact processes.     

On fourier-analysis of geomagnetic pulsations pc-1, “two-fold” and “three-fold” pulsations pc-1 and fundamental frequencies of the magnetospheric cavity—I

The paper gives the results of detailed studies of the frequency spectra $\text{S}{}_{\mathrm{s}}\text{(?)}$ of the chain of the wave packets F_s(t) of geomagnetic pulsations PC-1 recorded at the Novolazarevskaya station. The bulk of the energy of F_s(t) is concentrated in the vicinity of the central frequencies $\text{?}{}_{\mathrm{s0}}$ of spectra—the carrier frequencies of the signals. The velocity $\text{V}{}_0\text{≌ 6.10}{}^3\text{km s}{}^{\mathrm{?1}}$ of the flux of protons generating these signals correspond to them. The spectra of the signals have oscillations—“satellites” irregularly distributed in frequency. These satellites, as the authors believe, testify to the presence of the individual groups of protons of low concentration whose velocities vary within 10³–10⁴ km s^?1.Their energy is only of the order of 10^?2–10^?3 of the energy of the main proton flux. Clearly pronounced maxima on double and triple frequencies $\text{? = 2?}{}_{\mathrm{s0}}\text{and}\text{3?}{}_{\mathrm{s0}}$ are detected. They show that the generation of pulsations PC-1 is accompanied by the generation on the overtones of wave packets called in this paper “two-fold” and “three-fold” pulsations PC-1. Intensive symmetrical satellites of a modulation character have been discovered on frequencies $\text{?}{}^{\mathrm{\pm }}{}_{\mathrm{sK}}$. Frequency differences $\text{Δ?}{}_{\mathrm{sK}}{}^{\mathrm{\pm }}\text{=}\text{¦?}{}_{\mathrm{s0}}\text{? ?}{}_{\mathrm{sK}}{}^{\mathrm{\pm }}\text{¦}\text{= (0.011,0.022}\text{and}\text{0.035)}\text{Hz}$ correspond to them. The authors believe that the values of Δ?^±_sK are resonance frequencies of the magnetospheric cavity in which geomagnetic pulsations PC-1 are generated. It is established that the values of Δ?^±_sK coincide closely with the carrier frequencies of geomagnetic pulsations PC-3 and PC-4 generated in the magnetosphere. This leads to the conclusion that the resonance oscillations of the magnetospheric cavity are their source. Thus, the generation of geomagnetic pulsations of different types and resonance oscillations in the magnetosphere are integrated into a unified process. The importance of the results obtained and the necessity to check further their trustworthiness and universality, using experimental data gathered in different conditions, is stressed.     

Small-scale turbulence in the atmosphere of Venus

Previous studies based on radio scintillation measurements of the atmosphere of Venus have identified two regions of small-scale temperature fluctuations located in the vicinity of 45 and 60 km. A global study of the fluctuations near 60 km, which are consistent with wind-shear-generated turbulence, was conducted using the Pioneer Venus measurements. The structure constants of refractive index fluctuations c_n² and temperature fluctuations c_T² increase poleward, peak near 70° latitude, and decrease over the pole; c_n² varies from 2 × 10^?15 to 1.5 × 10^?14m^{$\text{2}\text{3}$} and c_T² from 4 × 10^?3 to 7 × 10^?2°K²m^{$\text{?2}\text{3}$}. These results indicate greater turbulent activity at the higher latitudes. In the region near 45 km the refractive index fluctuations and the corresponding temperature fluctuations are substantially lower. Based on the analysis of one representative occultation measurement, $\text{c}{}_{\mathrm{<mtext>n</mtext>}}{}^2\text{= 2 × 10}{}^{\mathrm{?16}}\text{m}{}^{\mathrm{<mtext>?2</mtext><mtext>3</mtext>}}\text{and}\text{c}{}_{\mathrm{<mtext>T</mtext>}}{}^2\text{= 7.3 × 10}{}^{\mathrm{?4}}\text{°}\text{K}{}^2\text{m}{}^{\mathrm{<mtext>?2</mtext><mtext>3</mtext>}}$ in the 45-km region. The fluctuations in this region also appear to be consistent with wind-shear-generated turbulence. The turbulence level is considerably weaker than that at 60 km; the energy dissipation rate ε is 4.9 × 10^?5m²sec^?3 and the small-scale eddy diffusion coefficient K is 2 × 10³ cm² sec^?1.     

Possibility of the investigation of interplanetary shock interaction with the earth's bow shock,magnetopause and plasmapause by means of a non-coherent response method

The problem of interaction between the interplanetary shock of 8 March, 1970 and the Earth's bow shock, magnetopause and plasmapause is considered. Estimates are made using existing models of the moments of initial impulsive interaction of interplanetary shocks with the bow shock and of the secondary interaction of the resulting split discontinuities with the magnetopause, plasmapause and a modified bow shock. Using computed data on the plasma's concentration jumps at discontinuities and on the latters' velocities, estimates have been carried out of remote sounding and the response signals' phase difference change rates Δf (which were found to be of the order of ～ 10^?3?10^?2Hz) appearing on the radio path with a non-coherent response near the subsolar region. It has been ascertained that the non-coherent response method permits, by using generators with a stability of $\text{ε =}\text{δ}{}_{\mathrm{<mtext>r</mtext>}}\text{f}\text{f}{}_{\mathrm{<mtext>O</mtext>}}\text{= 10}{}^{\mathrm{?11}}\text{?10}{}^{\mathrm{?10}}$, effective investigation (with a good time resolution) of the impulsive interaction of interplanetary shocks with the plasma discontinuities of the bow shockmagnetopause-plasmapause system.     

Pitch angle dependence of the charge-exchange lifetime of ring current ions

Previous work has parameterized the pitch angle dependence of the charge-exchange lifetime τ of ring current ions in terms of γ, the power of the cosine of the mirror latitude λ_m of the particles, such that $\text{τ(λ}{}_{\mathrm{m}}\text{)}\text{τ(0) ≌}\text{cos}{}^{\mathrm{\gamma }}\text{λ}{}_{\mathrm{m}}$ at given L. Using the atomic hydrogen density model of Johnson and Fish, previous authors have suggested values of γ = 5 or 6. We here evaluate γ as a function of λ_m and L using the more recent Chamberlain density models, and show that γ = 3?4 is more appropriate over most of the pitch angle and L range. Consequently, ion distributions in the ring current decay phase are expected to become rather less anisotropic in pitch angle due to chargeexchange than previously believed. We have also investigated the use of several other simple approximate analytic forms for $\text{τ(λ}{}_{\mathrm{m}}\text{)}\text{τ(0)}$, one of which gives far better agreement with the numerical results than the cos^γ λ_m, variation, and should hence be used in future studies.     

VLF-emissions associated with ring current electrons: Off-equatorial observations

The combination of a small inclination of the orbit (～4°) with the tilt angle (～11°) of the Earth's magnetic dipole axis enabled the S³-A satellite (Explorer 45) to make simultaneous observations of magnetospheric VLF-emissions and the associated enhancement of ring current electrons not only at the magnetic equator but also up to 15° geomagnetic latitudes. Microdensitometer scanning of the wideband data of these emissions reveals that the band of missing emission in the off-equatorial whistler mode emissions (chorus) appears at $\text{f}{}_{\mathrm{<mtext>Ho</mtext>}}\text{2}$ and that the intensities of the off-equatorial emission above $\text{f}{}_{\mathrm{<mtext>Ho</mtext>}}\text{2}$ are very weak in contrast to those of the near equatorial emissions, where $\text{f}{}_{\mathrm{<mtext>Ho</mtext>}}\text{2}$ is the equatorial electron gyrofrequency corresponding to the local gyrofrequency f_H at the satellite. Ray-tracing of whistler mode waves produced by the enhanced ring-current electrons at the geomagnetic equator just outside of the plasmapause has shown that some of these waves are reflected from high latitudes back to the Equator inside the source region. This process had been previously speculated to explain the formation of the bimodal intensity distribution with a gap at half the gyrofrequency (the two-band chorus) in the equatorial emission data. The intensities of those reflected waves, however, are shown to be insufficient to explain the observed emissions below $\text{f}{}_{\mathrm{<mtext>Ho</mtext>}}\text{2}$ at the Equator. These results indicate that the superposition of two types of emissions produced by the same processes but from different locations is not the main mechanism for the formation of the two-band chorus and that the dominant sources of these choruses are located around ± 5° geomagnetic latitude.     

A third-order solution of Vinti's problem with explicit expressions for the poisson brackets

We have derived an explicit third-order solution of Vinti's problem including J₃. Poisson brackets for the elements a, e, s, M_s, ψ_s and $\text{Ω}{}_{\mathrm{s}}$ are given. These may be used in the construction of a third-order theory of artificial satellites.     

Theoretical interpretation of the ground-based photometry of Saturn's B ring

New photographic photometry at small tilt angles during the 1979 and 1981 apparitions is combined with earlier data to yield several physical parameters for Saturn's B ring in red and blue colors. Phase curves are obtained for a mean tilt angle B ? 6°. The value of the volume density D is 0.020±0.004 with no indication of dependence on either the color or the tilt angle for 6°<B<26°. This conclusion is not altered significantly if the individual ring particles have a phase function similar to the phase curves of bright solar system objects. For the geometric albedo of a single particle we derive 0.61±0.04 (red) and 0.41±0.03 (blue), which are superior to earlier estimates because of the additional data now available. These values and the derived amount of multiple scattering as a function of tilt angle constrain the particle phase function in the red to be moderately backscattering. Inferred values of the particle single-scattering albedo are $\text{0.7≤}\text{ω}{}_0\text{(}\text{red}\text{) ≤0.92}$ and $\text{0.5≤}\text{ω}{}_0\text{(}\text{blue}\text{) ≤0.7}$, depending on the choice of phase function. No indication was found that the particle photometric properties might depend on the vertical distance from the central plane. Our results show that the ground-based photometry is entirely consistent with the classical, many-particle-thick ring model.     

Analysis of HI content in galaxies

We have collected data on 241 galaxies from 13 sources and made a statistical analysis after reduction to a uniform system. We found that the Hubble sequence is one of increasing $\text{〈}\text{M}{}_{\mathrm{<mtext>H</mtext>}}\text{M}{}_{\mathrm{<mtext>T</mtext>}}\text{〉}$ and $\text{〈}\text{M}{}_{\mathrm{<mtext>H</mtext>}}\text{L}{}_{\mathrm{<mtext>B</mtext>}}\text{〉}$, these mean values increasing monotonically from .0016 and .024 at E to .084 and .83 at Im, but the dispersion is large.The HI content in barred spiral is greater than that in ordinary spirals, and this is consistent with their statistics of angular momentum and colour.The HI content is related to colour; it is greater in bluer systems. The large dispersion suggests that it also depends on some other factors, but these are smoothed out when averaged over each type, resulting in a linear relation between $\text{〈log(}\text{M}{}_{\mathrm{<mtext>M</mtext>}}\text{M}{}_{\mathrm{<mtext>T</mtext>}}\text{〉}$ and 〈(B ? V^O_T)〉. Unlike the colour-colour diagram, the large dispersion on the $\text{log}\text{(}\text{M}{}_{\mathrm{<mtext>H</mtext>}}\text{L}{}_{\mathrm{<mtext>B</mtext>}}\text{) ? (B ? V}{}^0{}_{\mathrm{<mtext>T</mtext>}}\text{)}$ is not related to peculiar galaxies.     

Low-frequency upstream wave as a probable source of low-latitude Pc 3–4 magnetic pulsations

Daytime Pc 3–4 pulsation activities observed at globally coordinated low-latitude stations [SGC (L = 1.8,λ = 118.0°W), EWA(1.15,158.1°W), ONW(1.3,141.5°E)] are evidently controlled by the cone angle θ_XB of the IMF observed at ISEE 3. Moreover, the Pc 3–4 frequencies ($\text{?}$) at the low latitudes and high latitude (COL; L = 5.6 and λ = 147.9°W) on the ground and that of compressional waves at geosynchronous orbit (GOES 2; L = 6.67 and λ = 106.7°W) are also correlated with the IMFmagnitude(B_IMF).The correlation of $\text{?}$ of the compressional Pc 3–4 waves at GOES 2 against B_IMF is higher than those of the Pc 3–4 pulsations at the globally coordinated ground stations, i.e., γ = 0.70 at GOES 2, and (0.36,0.60,0.66,0.54) at (COL, SGC, EWA, ONW), respectively. The standard deviation ($\text{σ}{}_{\mathrm{n}}\text{= ± Δ?}\text{mHz}$) of the observed frequencies from the form $\text{?}$ (mHz) = 6.0 × B_IMF (nT) is larger at the ground stations than at GOES 2, i.e., $\text{Δ? = ± 6.6}\text{mHz at}$GOES 2, and ±(13.9, 9.1, 10.7, 12.1) mHz at (COL, SGC, EWA, ONW), respectively. The correlations between the IMF magnitude B_IMF and Pc 3–4 frequencies at the low latitudes are higher than that at the high latitude on the ground, which can be interpreted by a “filtering action” of the magnetosphere for daytime Pc 3–4 magnetic pulsations. The scatter plots of pulsation frequency $\text{?}$ against the IMF magnitude B_IMF for the compressional Pc 3–4 waves at GOES 2 are restricted within the forms $\text{? = 4.5 × B}{}_{\mathrm{<mtext>IMF</mtext>}}\text{and}\text{? = 7.5 × B}{}_{\mathrm{<mtext>IMF</mtext>}}$. The frequency distribution is in excellent agreement with the speculation ($\text{<ω}{}_{\mathrm{sc}}\text{Ω}{}_{\mathrm{i}}\text{= 0.3 ～ 0.5}$) of the spacecraft frame frequency of the magnetosonic right-hand waves excited by the anomalous ion cyclotron resonance with reflected ion beams with V₆ = 650 ～ 1150 km s^?1 in the solar wind frame observed by the ISEE satellite in the Earth's foreshock. These observational results suggest that the magnetosonic right-handed waves excited by the reflected ion beams in the Earth's foreshock are convected through the magnetosheath to the magnetopause, transmitted into the magnetosphere without significant changes in spectra, and then couple with various HM waves in the Pc 3–4 frequency range at various locations in the magnetosphere.     

Geopotential coefficients of order 29, from analysis of the orbit of satellite 1967-104B

The orbit of the satellite 1967-104B has been analysed as it passed through 29:2 resonance with the Earth's gravitational field between January 1977 and September 1978. From the changes in inclination and eccentricity the following lumped 29th-order geopotential harmonic coefficients were obtained: $\text{10}{}^9\text{C}\text{?}{}_{29}{}^{0.2}\text{= 4.1 ± 0.8}$, $\text{10}{}^9\text{S}\text{?}{}_{29}{}^{0.2}\text{= 10.3 ± 2.4}$, $\text{10}{}^9\text{C}\text{?}{}_{29}{}^{1.1}\text{= ? 160 ± 19}$, $\text{10}{}^9\text{S}\text{?}{}_{29}{}^{1.1}\text{= 79 ± 10}$, $\text{10}{}^9\text{C}\text{?}{}_{29}{}^{\mathrm{?1.3}}\text{= 38 ± 14}$, $\text{10}{}^9\text{S}\text{?}{}_{29}{}^{\mathrm{?1.3}}\text{= 19 ± 5}$. These values have been compared with existing comprehensive geopotential models: the best agreement is with the model of Rapp (1981).     

Analysis of the orbital inclination of Tansei 3 rocket 1977-12B

The orbit of Tansei 3rocket(1977-12B) has been determined at 47 epochs between 1 October 1977 and 19 March 1979 using over 1700 observations and the RAE orbit refinement program PROP6. The rate of change of the inclination was examined to evaluate values of the atmospheric rotation rate, Λ rev day^?1. Analysis yielded the value Λ = 1.1 ± 0.05 at height 315 ± 30 km, average conditions; or alternatively Λ = 1.1 ± 0.1 at height 347 ± 12 km, slight winter bias and Λ = 1.07 ± 0.1 at height 270 ± 18 km, average conditions, supplying further evidence of a decrease in rotation rates from the 1960s to the 1970s.Analysis of the inclination at 15th-order resonance yielded the lumped harmonic values

$\text{10}{}^9\text{C}{}^{\mathrm{0,1}}{}_{15}\text{= 13.4 ± 6.2, 10}{}^9\text{S}{}^{\mathrm{0,1}}{}_{15}\text{= 0.7 ± 13.3}$

for inclination 65.485°.     

Spherical harmonic representation of the gravity field from dynamic satellite data

Spherical harmonics are the natural parameters for the Earth's gravity field as sensed by orbiting satellites, but problems of resolution arise because the spectrum of effects is narrow and unique to each orbit. Comprehensive gravity models now contain many hundreds of thousands of observations from more than thirty different near-Earth artificial satellites. With refinements in tracking systems, newer data is capable of sensing the spherical harmonics of the field experienced by these satellites to very high degree and order. For example, altimeter, laser and satellite-tracking-satellite systems contain gravitational information well above present levels of satellite gravity field recovery (l = 20), but significant aliasing results because the orbital parameters are too restricted compared to the large number of spherical harmonics.It is shown however that the unique spectrum of information for each satellite contained within a comprehensive spherical harmonic model can be represented by simple gravitational constraint equations (lumped harmonics). All such constraints are harmonic in the argument of perigee (ω) with constants determinable directly from tracking data or reconstituted from the comprehensive solution:

$\text{(C}{}^1\text{, S}{}^1\text{) = (C}{}_{\mathrm{o}}\text{, S}{}_{\mathrm{o}}\text{) + Σ}{}_{\mathrm{i}}\text{= 1 (C}{}_{\mathrm{Ci}}\text{, S}{}_{\mathrm{Ci}}\text{) cos i ω + (C}{}_{\mathrm{Si}}\text{, S}{}_{\mathrm{Si}}\text{) sin i ω}$

. The constants are simple linear combinations of the geopotential harmonics. Through these lumped harmonics any satellite gravity field can be decomposed and then uniformly extended to any degree or tailored to a given orbit without reintegration of the trajectory and variational equations. They also make possible the inclusion of information into the field from special deep resonance passages, long arc zonal analyses, and satellites unique to other models. Numerous examples of the derivation, combination, extension and tailoring of the harmonics are presented. The importance of using data spanning an apsidal period is emphasized.     

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.