Tables for the trigonometric series representations of the orbital inclination function

In this paper, tables for the trigonometric series representations of the orbital inclination functionF _lmp (i) in multiples of cosines or sines will be represented forl=2(1)10;m=0(1)l;p=0(1)l.Astrophysics and Space Science Review Article.     

Analysis of the orbital inclination of 1963-27A

The Agena B upper-stage rocket 1963-27A was launched into near-circular orbit, inclined at 82.3° to the Equator, on 29 June 1963. Its orbital elements were available at 52 epochs over the 16 month interval prior to its decay on 26 October 1969. During this period the satellite passed through 31:2 resonance and the variation in orbital inclination near this event was analysed to obtain two lumped geopotential harmonics of order 31. Since the resonance was a weak feature in the data, the resulting values are poorly defined.Either side of the resonance period, the inclination was used to estimate the mean atmospheric rotation rate Λ rev day^?1. The values obtained were Λ = 0.85 ± 0.18 at a height of 440 km for the period June 1968 to February 1969 and Λ = 1.13 ± 0.10 at 338 km for the period June to October 1969.     

Recurrence relations of the inclination function

In this paper, the inclination function is expressed in terms of the hypergeometric function and this has resulted in some simple recurrence relations involving no more than three neighbouring functions.     

Derivative of Kaula's inclination function

The derivative of Kaula's inclination function is expressed as a linear combination of two inclination functions themselves.On leave from Institute of Applied Astronomy, 197042 St.-Petersburg, Russia.     

Upper-atmosphere rotation rate from analysis of the orbital inclination of explorer 1

Explorer 1, 1958α, ths first U.S. artificial satellite, was launched on 1 February 1958 and remained in orbit for 12 years. In this paper theoretical curves have been fitted to the values of inclination, giving three values of the average atmospheric rotation rate at heights of 350–400 km, and latitudes 0–20°:

     

6.

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

P. Moore 《Planetary and Space Science》1984,32(9):1101-1109

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°.     

7.

Lunar cratering asymmetries with high lunar orbital obliquity and inclination of the Moon     

《天文和天体物理学研究(英文版)》2021,(6)

Accurate estimation of cratering asymmetry on the Moon is crucial for understanding Moon evolution history. Early studies of cratering asymmetry have omitted the contributions of high lunar obliquity and inclination. Here, we include lunar obliquity and inclination as new controlling variables to derive the cratering rate spatial variation as a function of longitude and latitude. With examining the influence of lunar obliquity and inclination on the asteroids population encountered by the Moon, we then have derived general formulas of the cratering rate spatial variation based on the crater scaling law. Our formulas with addition of lunar obliquity and inclination can reproduce the lunar cratering rate asymmetry at the current Earth-Moon distance and predict the apex/ant-apex ratio and the pole/equator ratio of this lunar cratering rate to be 1.36 and 0.87, respectively. The apex/ant-apex ratio is decreasing as the obliquity and inclination increasing. Combining with the evolution of lunar obliquity and inclination, our model shows that the apex/ant-apex ratio does not monotonically decrease with Earth-Moon distance and hence the influences of obliquity and inclination are not negligible on evolution of apex/ant-apex ratio. This model is generalizable to other planets and moons, especially for different spin-orbit resonances.     

8.

Optical spectroscopy and photometry of main-belt asteroids with a high orbital inclination     

《天文和天体物理学研究(英文版)》2017,(2)

We carried out low-resolution optical spectroscopy of 51 main-belt asteroids, most of which have highly-inclined orbits. They are selected from D-type candidates in the SDSS-MOC 4 catalog.Using the University of Hawaii 2.2 m telescope and the Inter-University Centre for Astronomy and Astrophysics 2 m telescope in India, we determined the spectral types of 38 asteroids. Among them,eight asteroids were classified as D-type asteroids. Fractions of D-type asteroids are 3.0 ± 1.1% for low orbital inclination main-belt asteroids and 7.3 ± 2.0% for high orbital inclination main-belt asteroids.The results of our study indicate that some D-type asteroids were formed within the ecliptic region between the main belt and Jupiter, and were then perturbed by Jupiter.     

9.

Dynamics of Trojan asteroids with large orbital inclination: A preliminary report     

Reinhold Bien 《Earth, Moon, and Planets》1980,22(2):163-166

Some Trojans have a remarkably large inclination. This fact raises the question whether or not these asteroids are dynamically different from Trojans with an inclination, say smaller than 10°. In this paper the author investigates the dynamics of selected Trojans. Common properties and differences between highly inclined orbits and practically planar ones are discussed.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratorio di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.     

10.

The inclination function in terms of nonsingular elements     

Edwin Wnuk  Iwona Wytrzyszczak 《Celestial Mechanics and Dynamical Astronomy》1987,42(1-4):251-261

Different formulas for computation of the inclination function in terms of nonsingular elements have been presented and compared. Among the ways to compute the inclination function presented below, the best one is based on the recurrence relations (eqs. 16) derived in this paper.     

11.

Analytical development of the lunisolar disturbing function and the critical inclination secular resonance     

Alessandra Celletti  Cătălin Galeş  Giuseppe Pucacco  Aaron J. Rosengren 《Celestial Mechanics and Dynamical Astronomy》2017,127(3):259-283

We provide a detailed derivation of the analytical expansion of the lunar and solar disturbing functions. Although there exist several papers on this topic, many derivations contain mistakes in the final expansion or rather (just) in the proof, thereby necessitating a recasting and correction of the original derivation. In this work, we provide a self-consistent and definite form of the lunisolar expansion. We start with Kaula’s expansion of the disturbing function in terms of the equatorial elements of both the perturbed and perturbing bodies. Then we give a detailed proof of Lane’s expansion, in which the elements of the Moon are referred to the ecliptic plane. Using this approach the inclination of the Moon becomes nearly constant, while the argument of perihelion, the longitude of the ascending node, and the mean anomaly vary linearly with time. We make a comparison between the different expansions and we profit from such discussion to point out some mistakes in the existing literature, which might compromise the correctness of the results. As an application, we analyze the long-term motion of the highly elliptical and critically-inclined Molniya orbits subject to quadrupolar gravitational interactions. The analytical expansions presented herein are very powerful with respect to dynamical studies based on Cartesian equations, because they quickly allow for a more holistic and intuitively understandable picture of the dynamics.     

12.

Analytic characterization of measurement uncertainty and initial orbit determination on orbital element representations     

R. M. Weisman  M. Majji  K. T. Alfriend 《Celestial Mechanics and Dynamical Astronomy》2014,118(2):165-195

This paper presents an approach to characterize the uncertainty associated with the state vector obtained from the Herrick-Gibbs orbit determination approach using transformation of variables. The approach is applied to estimate the state vector and its probability density function for objects in low Earth orbit using sparse observations. The state vector and associated uncertainty estimates are computed in Cartesian coordinates and Keplerian elements. The approach is then extended to accommodate the $J_2$ perturbation where the state vector is written in terms of mean orbital elements. The results obtained from the analytical approach presented in this paper are validated using Monte Carlo simulations and compared with the often utilized similarity transformation for Kepler, mean, and nonsingular elements. The measurement uncertainty characterization obtained is used to initialize conventional nonlinear filters as well as operate a Bayesian approach for orbit determination and object tracking.     

13.

The change in satellite orbital inclination caused by a rotating atmosphere with day-to-night density variation     

D. G. King-Hele  Doreen M. C. Walker 《Celestial Mechanics and Dynamical Astronomy》1972,5(1):41-54

The theory specifying the change i in a satellite's orbital inclination due to atmospheric rotation, in terms of the decrease in orbital period T, has been extended to an atmosphere with sinusoidal variation of density between day and night. It is found that with certain special sets of values for the orbital parameters, the day-to-night variation in the Earth's atmosphere can alter the equation for i/T by as much as 25% though only for a few days. Appreciable changes in i/T persisting for several months can only occur for certain resonant orbits: the maximum change is then about 8%. Near-resonance is very unlikely, but the resonance conditions are derived so that orbits can be recognised and avoided.     

14.

Extension of the critical inclination     

Xiaodong Liu  Hexi Baoyin  Xingrui Ma 《Astrophysics and Space Science》2011,334(1):115-124

The critical inclination is of special interest in artificial satellite theory. The critical inclination can maintain minimal deviations of eccentricity and argument of pericentre from the initial values, and orbits at this inclination have been applied to some space missions. Most previous researches about the critical inclination were made under the assumption that the oblateness term J ₂ is dominant among the harmonic coefficients. This paper investigates the extension of the critical inclination where the concept of the critical inclination is different from that of the traditional sense. First, the study takes the case of Venus for instance, and provides some preliminary results. Then for general cases, given the values of argument of pericentre and eccentricity, the relationship between the multiplicity of the solutions for the critical inclination and the values of J ₂ and J ₄ is analyzed. Besides, when given certain values of J ₂ and J ₄, the relationship between the multiplicity of the solutions for the critical inclination and the values of semimajor axis and eccentricity is studied. The results show that for some cases, the value of the critical inclination is far away from that of the traditional sense or even has multiple solutions. The analysis in this paper could be used as starters of correction methods in the full gravity field of celestial bodies.     

15.

On the classical expansion of the perturbing function in individual orbital elements     

Irina Tupikova  Michael Soffel  Sergei Klioner 《Celestial Mechanics and Dynamical Astronomy》1999,74(3):147-152

Starting from the classical expansion of the perturbing function in the three‐body problem, the transformation to individual orbital elements is performed in principle up to any degree in small parameters. Some corrections to the results presented in the well‐known article by Yuasa on secular perturbations of asteroids are given. Consequences for the expansion of the indirect part of the perturbing function are discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

16.

Dynamical modelling of stars and gas in NGC 2974: determination of mass-to-light ratio, inclination and orbital structure using the Schwarzschild method   总被引：1，自引：0，他引：1  

Davor Krajnovi&#;  Michele Cappellari  Eric Emsellem  Richard M. McDermid  P. Tim de Zeeuw 《Monthly notices of the Royal Astronomical Society》2005,357(4):1113-1133

     

17.

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

W.J. Boulton 《Planetary and Space Science》1985,33(8):965-982

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, $\text{10}{}^9\text{C}\text{?}{}^{0.1}{}_{15}\text{= ?31.2 ± 2.3 10}{}^9\text{S}\text{?}{}^{0.1}{}_{15}\text{= ?4.4 ± 3.2 10}{}^9\text{C}\text{?}{}^{0.2}{}_{30}\text{= 39.0 ± 10.7 10}{}^9\text{S}\text{?}{}^{0.2}{}_{30}\text{= 51.8 ± 10.0}$     

18.

On the stability of high inclination Trojans     

Seppo Mikkola  Kimmo Innanen 《Earth, Moon, and Planets》1995,71(3):195-198

Numerical integrations have been performed for orbits of Venus', Earth's and Mars' Trojan-asteroid test particles in a self consistent model of the solar system to study the stability of inclined Trojan orbits. In the case of Mars low inclination orbits tend to be unstable while the contrary seems to apply to Venus and Earth, although the stability of some very high inclination orbits may not be excluded on the basis of these computations.     

19.

The problem of the critical inclination revisited     

Alan H. Jupp 《Celestial Mechanics and Dynamical Astronomy》1975,11(3):361-378

The behaviour of the argument of the pericentre is investigated for the orbit of an artificial satellite which is moving under the potential when the inclination of the orbit is close to thecritical value tan^?1 2. The theory is developed to first order and it is applicable to all values of the eccentricity, with the exception of those in the neighbourhood of zero and unity. Four principal types of behaviour are noted and these are illustrated in appropriate phase-plane diagrams. It is shown that the two types which exhibit double libration in the argument of the pericentre are restricted to a relatively small domain in the (a, e)-plane of possible motions. Moreover, it is demonstrated that for double libration to occur it is necessary, but not sufficient, that \(e > \sqrt 6/13\) . The ranges of values of the inclination for which libration of the pericentre is a possibility are given for the more important cases. The general results are applied to the specific case of artificial Earth satellites whose orbits are inclined to the equator at angles close to the value of the critical inclination.     

20.

Critical inclination of Trojan asteroids     

Bálint Érdi 《Celestial Mechanics and Dynamical Astronomy》1984,34(1-4):435-441

The author's earlier solution for Trojan asteroids is developed further. It is shown that depending on the amplitude of libration around the Lagrangian point L₄, there is a critical inclination which determines the sign of the variation of the ascending node. If the orbital inclination of a Trojan is smaller than the critical one, then the ascending node decreases and otherwise it increases. The variation of the eccentricity and of the longitude of the perihelion has also a dependence on the critical inclination.     

Feb 1958 to mid 1960	1.5 rev/day
Mid 1960 to Dec 1967	1.2 rev/day
Jan 1968 to Mar 1970	1.3 rev/day

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°.     

Lunar cratering asymmetries with high lunar orbital obliquity and inclination of the Moon

Accurate estimation of cratering asymmetry on the Moon is crucial for understanding Moon evolution history. Early studies of cratering asymmetry have omitted the contributions of high lunar obliquity and inclination. Here, we include lunar obliquity and inclination as new controlling variables to derive the cratering rate spatial variation as a function of longitude and latitude. With examining the influence of lunar obliquity and inclination on the asteroids population encountered by the Moon, we then have derived general formulas of the cratering rate spatial variation based on the crater scaling law. Our formulas with addition of lunar obliquity and inclination can reproduce the lunar cratering rate asymmetry at the current Earth-Moon distance and predict the apex/ant-apex ratio and the pole/equator ratio of this lunar cratering rate to be 1.36 and 0.87, respectively. The apex/ant-apex ratio is decreasing as the obliquity and inclination increasing. Combining with the evolution of lunar obliquity and inclination, our model shows that the apex/ant-apex ratio does not monotonically decrease with Earth-Moon distance and hence the influences of obliquity and inclination are not negligible on evolution of apex/ant-apex ratio. This model is generalizable to other planets and moons, especially for different spin-orbit resonances.     

Optical spectroscopy and photometry of main-belt asteroids with a high orbital inclination

We carried out low-resolution optical spectroscopy of 51 main-belt asteroids, most of which have highly-inclined orbits. They are selected from D-type candidates in the SDSS-MOC 4 catalog.Using the University of Hawaii 2.2 m telescope and the Inter-University Centre for Astronomy and Astrophysics 2 m telescope in India, we determined the spectral types of 38 asteroids. Among them,eight asteroids were classified as D-type asteroids. Fractions of D-type asteroids are 3.0 ± 1.1% for low orbital inclination main-belt asteroids and 7.3 ± 2.0% for high orbital inclination main-belt asteroids.The results of our study indicate that some D-type asteroids were formed within the ecliptic region between the main belt and Jupiter, and were then perturbed by Jupiter.     

Dynamics of Trojan asteroids with large orbital inclination: A preliminary report

Some Trojans have a remarkably large inclination. This fact raises the question whether or not these asteroids are dynamically different from Trojans with an inclination, say smaller than 10°. In this paper the author investigates the dynamics of selected Trojans. Common properties and differences between highly inclined orbits and practically planar ones are discussed.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratorio di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.     

The inclination function in terms of nonsingular elements

Different formulas for computation of the inclination function in terms of nonsingular elements have been presented and compared. Among the ways to compute the inclination function presented below, the best one is based on the recurrence relations (eqs. 16) derived in this paper.     

Analytical development of the lunisolar disturbing function and the critical inclination secular resonance

We provide a detailed derivation of the analytical expansion of the lunar and solar disturbing functions. Although there exist several papers on this topic, many derivations contain mistakes in the final expansion or rather (just) in the proof, thereby necessitating a recasting and correction of the original derivation. In this work, we provide a self-consistent and definite form of the lunisolar expansion. We start with Kaula’s expansion of the disturbing function in terms of the equatorial elements of both the perturbed and perturbing bodies. Then we give a detailed proof of Lane’s expansion, in which the elements of the Moon are referred to the ecliptic plane. Using this approach the inclination of the Moon becomes nearly constant, while the argument of perihelion, the longitude of the ascending node, and the mean anomaly vary linearly with time. We make a comparison between the different expansions and we profit from such discussion to point out some mistakes in the existing literature, which might compromise the correctness of the results. As an application, we analyze the long-term motion of the highly elliptical and critically-inclined Molniya orbits subject to quadrupolar gravitational interactions. The analytical expansions presented herein are very powerful with respect to dynamical studies based on Cartesian equations, because they quickly allow for a more holistic and intuitively understandable picture of the dynamics.     

Analytic characterization of measurement uncertainty and initial orbit determination on orbital element representations

This paper presents an approach to characterize the uncertainty associated with the state vector obtained from the Herrick-Gibbs orbit determination approach using transformation of variables. The approach is applied to estimate the state vector and its probability density function for objects in low Earth orbit using sparse observations. The state vector and associated uncertainty estimates are computed in Cartesian coordinates and Keplerian elements. The approach is then extended to accommodate the $J_2$ perturbation where the state vector is written in terms of mean orbital elements. The results obtained from the analytical approach presented in this paper are validated using Monte Carlo simulations and compared with the often utilized similarity transformation for Kepler, mean, and nonsingular elements. The measurement uncertainty characterization obtained is used to initialize conventional nonlinear filters as well as operate a Bayesian approach for orbit determination and object tracking.     

The change in satellite orbital inclination caused by a rotating atmosphere with day-to-night density variation

The theory specifying the change i in a satellite's orbital inclination due to atmospheric rotation, in terms of the decrease in orbital period T, has been extended to an atmosphere with sinusoidal variation of density between day and night. It is found that with certain special sets of values for the orbital parameters, the day-to-night variation in the Earth's atmosphere can alter the equation for i/T by as much as 25% though only for a few days. Appreciable changes in i/T persisting for several months can only occur for certain resonant orbits: the maximum change is then about 8%. Near-resonance is very unlikely, but the resonance conditions are derived so that orbits can be recognised and avoided.     

Extension of the critical inclination

The critical inclination is of special interest in artificial satellite theory. The critical inclination can maintain minimal deviations of eccentricity and argument of pericentre from the initial values, and orbits at this inclination have been applied to some space missions. Most previous researches about the critical inclination were made under the assumption that the oblateness term J ₂ is dominant among the harmonic coefficients. This paper investigates the extension of the critical inclination where the concept of the critical inclination is different from that of the traditional sense. First, the study takes the case of Venus for instance, and provides some preliminary results. Then for general cases, given the values of argument of pericentre and eccentricity, the relationship between the multiplicity of the solutions for the critical inclination and the values of J ₂ and J ₄ is analyzed. Besides, when given certain values of J ₂ and J ₄, the relationship between the multiplicity of the solutions for the critical inclination and the values of semimajor axis and eccentricity is studied. The results show that for some cases, the value of the critical inclination is far away from that of the traditional sense or even has multiple solutions. The analysis in this paper could be used as starters of correction methods in the full gravity field of celestial bodies.     

On the classical expansion of the perturbing function in individual orbital elements

Starting from the classical expansion of the perturbing function in the three‐body problem, the transformation to individual orbital elements is performed in principle up to any degree in small parameters. Some corrections to the results presented in the well‐known article by Yuasa on secular perturbations of asteroids are given. Consequences for the expansion of the indirect part of the perturbing function are discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

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, $\text{10}{}^9\text{C}\text{?}{}^{0.1}{}_{15}\text{= ?31.2 ± 2.3 10}{}^9\text{S}\text{?}{}^{0.1}{}_{15}\text{= ?4.4 ± 3.2 10}{}^9\text{C}\text{?}{}^{0.2}{}_{30}\text{= 39.0 ± 10.7 10}{}^9\text{S}\text{?}{}^{0.2}{}_{30}\text{= 51.8 ± 10.0}$     

On the stability of high inclination Trojans

Numerical integrations have been performed for orbits of Venus', Earth's and Mars' Trojan-asteroid test particles in a self consistent model of the solar system to study the stability of inclined Trojan orbits. In the case of Mars low inclination orbits tend to be unstable while the contrary seems to apply to Venus and Earth, although the stability of some very high inclination orbits may not be excluded on the basis of these computations.     

The problem of the critical inclination revisited

The behaviour of the argument of the pericentre is investigated for the orbit of an artificial satellite which is moving under the potential when the inclination of the orbit is close to thecritical value tan^?1 2. The theory is developed to first order and it is applicable to all values of the eccentricity, with the exception of those in the neighbourhood of zero and unity. Four principal types of behaviour are noted and these are illustrated in appropriate phase-plane diagrams. It is shown that the two types which exhibit double libration in the argument of the pericentre are restricted to a relatively small domain in the (a, e)-plane of possible motions. Moreover, it is demonstrated that for double libration to occur it is necessary, but not sufficient, that \(e > \sqrt 6/13\) . The ranges of values of the inclination for which libration of the pericentre is a possibility are given for the more important cases. The general results are applied to the specific case of artificial Earth satellites whose orbits are inclined to the equator at angles close to the value of the critical inclination.     

Critical inclination of Trojan asteroids

The author's earlier solution for Trojan asteroids is developed further. It is shown that depending on the amplitude of libration around the Lagrangian point L₄, there is a critical inclination which determines the sign of the variation of the ascending node. If the orbital inclination of a Trojan is smaller than the critical one, then the ascending node decreases and otherwise it increases. The variation of the eccentricity and of the longitude of the perihelion has also a dependence on the critical inclination.