Variations In Orbital Inclination Research Articles

Forced obliquity variations of Mercury

The spin pole of Mercury is very nearly, but not quite, aligned with its orbit pole. Tidal dissipation has driven the free obliquity to very small values, and the high rate of spin pole precession allows the forced obliquity variations to remain small despite significant variations in orbital inclination and eccentricity. We present calculations of the obliquity for a 10 million year time span, centered on the present. The obliquity remains small, with typical values of 2–4 minutes of arc. The dominant period of obliquity oscillations is 895 kyr, which is also the main period at which the orbital inclination varies. If the orbit pole precession rate were uniform, dissipation would have driven Mercury into a Cassini state, in which the spin pole and orbit pole remain coplanar with the invariable pole, as the spin pole precesses about the moving orbit pole. However, due to the nonuniform orbit precession rate, this simple coplanar configuration is not maintained, except on a mode‐by‐mode basis. That is, when the orbit pole motion is represented as a sum of normal modes of the coupled oscillations of the planetary system, the spin pole coprecesses with the orbit pole at each modal frequency. This is a generalization of Cassini's second and third laws of lunar rotation to the case of nonuniform orbit precession. We compare results of a linearized obliquity model with a numerical integration of the equations of motion. The two solutions agree at the level of a few seconds of arc.

Pacemaking the ice ages by frequency modulation of Earth's orbital eccentricity

Evidence from power spectra of deep-sea oxygen isotope time series suggests that the climate system of Earth responds nonlinearly to astronomical forcing by frequency modulating eccentricity-related variations in insolation. With the help of a simple model, it is shown that frequency modulation of the approximate 100,000-year eccentricity cycles by the 413,000-year component accounts for the variable duration of the ice ages, the multiple-peak character of the time series spectra, and the notorious absence of significant spectral amplitude at the 413,000-year period. The observed spectra are consistent with the classic Milankovitch theories of insolation, so that climate forcing by 100,000-year variations in orbital inclination that cause periodic dust accretion appear unnecessary.

Chaos in the Kepler system

The long-term dynamical evolution of a Keplerian binary orbit due to the emission and absorption of gravitational radiation is investigated. This work extends our previous results on transient chaos in the planar case to the three-dimensional Kepler system. Specifically, we consider the nonlinear evolution of the relative orbit due to gravitational radiation damping as well as external gravitational radiation that is obliquely incident on the initial orbital plane. The variation of orbital inclination, especially during resonance capture, turns out to be very sensitive to the initial conditions. Moreover, we discuss the novel phenomenon of chaotic transition.

Analysis of the orbit of intercosmos 10 rocket (1973-82B) in its last 15 days

Intercosmos 10 rocket (1973-82B) was launched on 30 October 1973 into an orbit of eccentricity 0.08 inclined at 74° to the equator, and decayed on 8 October 1977. Orbits have been determined daily in the last 15 days of its life from NORAD and U.S. Navy observations; and an orbital accuracy of about 55 m radially and 70 m cross-track was obtained. Analysis of the decrease in perigee distance has provided two values of atmospheric density scale height at heights of 190 and 200 km. The variations in orbital inclination due to atmospheric rotation and 16thorder resonance with the geopotential have been analysed, and give a morning value for the atmospheric rotation rate of 0.90 ± 0.06 rev day −1 at a height of approximately 200 km and two values for lumped 16th-order harmonic coefficients with accuracies equivalent to 1.5 and 5.0 cm in geoid height.

The determination and analysis of the orbit of NIMBUS 1 rocket, 1964-52B: Variations in orbital inclination

The influence of aerodynamic drag and the geopotential on the motion of the satellite 1964-52B is considered. A model of the atmosphere is adopted that allows for oblateness, and in which the density behaviour approximates to the observed diurnal variation. A differential equation governing the variation of the orbital inclination combining the effects of air drag with those of the Earth's gravitational field is given. The 310 observed values of inclination are modified by the removal of perturbations due to luni-solar attraction, solid Earth and ocean tides, solar radiation pressure, low-order long-periodic tesseral harmonic perturbations and changes due to precession. The method of removal of these effects is given in some detail. The variations in inclination due to drag are analysed to give four values of the average atmospheric rotation rate at heights of 296–476 km at latitude 0–54°. These values are as expected from previous analyses. The analysis of the change in inclination due to solar radiation pressure shows that this rapidly tumbling cylindrical satellite may be considered as equivalent to a spherical satellite of a given area-to-mass ratio. Analysis of the inclination near 15:1 resonance with the geopotential yields values of lumped geopotential harmonics of order 15 and 30, namely, 10 9 C ̄ 0.1 15 = −31.2 ± 2.3 10 9 S ̄ 0.1 15 = −4.4 ± 3.2 10 9 C ̄ 0.2 30 = 39.0 ± 10.7 10 9 S ̄ 0.2 30 = 51.8 ± 10.0

Upper-atmosphere features revealed by the orbit of 1980-43A

The satellite NOAA-B (1980-43A) was launched in May 1980 into an orbit with perigee height near 260 km and apogee height 1440 km, at an inclination of 92.2°.The lifetime was 11 months. The orbit has been determined at 40 epochs between October 1980 and May 1981 from about 3000 radar and optical observations. The average orbital accuracy, radial and cross-track, was about 100 m, with rather better accuracy in the final 14 days. The variation of orbital inclination has been analysed to determine two good values of atmospheric rotation rate, namely 1.10 ± 0.10 rev day −1 at 300 km (average local time) and 1.15 ± 0.06 rev day −1 at 225 km (evening). The effect of atmospheric rotation on the precession of the orbital plane of an actual satellite has never previously been detected; it is clearly apparent for 1980-43A in its last days and conforms to the expected theoretical change. The variation of perigee height has been analysed to determine ten values of atmospheric density scale height, for heights of 280–370 km. These values, accurate to about 3%, exceed by 15% the values indicated by the COSPAR International Reference Atmosphere. Solar activity was higher in the years 1980–1981 than at any time since early 1958 and it appears that the CIRA model underestimates the density and density scale height at high levels of solar activity.

The orbit of proton 4 redetermined, with geophysical implications

The orbit of Proton 4, 1968-103A, has been redetermined, in greater detail and with better accuracy, in order to clarify previously puzzling features in the variation of orbital inclination. Orbital parameters have been determined at 25 epochs between December 1968 and July 1969, using about 1600 optical and radar observations with the RAE orbit refinement program PROP 6. During January 1969 the orbit passed through 31:2 resonance—when the ground track over the Earth repeats every two days after 31 revolutions of the satellite. A simultaneous least-squares fitting of theoretical curves to the values of inclination and eccentricity between 14 December 1968 and 6 March 1969 has yielded values for two pairs of lumped 31st-order geopotential coefficients, appropriate to an inclination of 51.5°. This is the first specific evaluation of 31st-order coefficients. The 15 values of inclination after the resonance, from March to near decay in July 1969, have been used to determine mean, morning and afternoon-evening values for the rotation rate of the atmosphere at a height near 260 km; the values of rotation rate, namely 1.1, 0.9 and 1.3 rev/day respectively, confirm the trends already established from analysis of other satellite orbits.

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 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.