On the inclination functions and a rapid stable procedure for their evaluation together with derivatives

The authors’ individual work on the inclination functions over a period of more than 30 years has led to the need for a joint paper. Intervening papers by other authors have demonstrated misunderstandings needing to be corrected, in particular concerning the key recurrence relation published by the present first author in 1971. This relation is remarkably stable, though this has not always been recognized. The real source of error with the specific functions that are involved in the recurrence relation arises from the possibilities for underflow (as well as overflow) in the computation. The problem exists even with normalized versions of the functions, and is carefully addressed. Very important, for both academic and practical reasons, is a general invariance relation that had been found earlier by the second author, for which a proof is given here for the first time. Some numerical results from our new (and highly efficient) procedure for computing the inclination functions are tabulated, and comparisons made with the results of other authors. Finally, Fortran code for an optimized implementation of this procedure is in supplementary material. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

An exercise in symbolic programming: Computation of general normalized inclination functions

Celestial Mechanics and Dynamical Astronomy - Computation of the general normalized inclination functionsA k lm(i) defined by R. H. Gooding is studies as an example of a representative problem in...     

The critical inclination problem: A simple treatment

A short analysis is presented in the hope of clarifying the situation.     

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.     

Outcomes of tidal evolution for orbits with arbitrary inclination

Tides raised by a satellite on a rotating planet dissipate energy and result in an exchange of angular momentum between the orbit and the spin. A set of diagrams is constructed which shows the evolution of the angular momentum vectors. The results are applied to possible histories of the Uranus system.     

A possible method for the determination of the magnitude and inclination of a sunspot magnetic-field

A method is presented in this paper for the determination of both the H and the γ of a sunspot magnetic field by means of two polarizers with transmission axes inclined at ± 45° to the fast axis of a quarter-wave plate. Previously, this arrangement was used to determine H only.It is shown that the profiles of the Stokes parameters are sensitive to the sunspot atmosphere model used whereas the results given by the present method are not. It is shown, more ever, that the formula for determining H from the wave length shift, Δλ = 4.67 × 10^?5gHλ², cannot be used in many cases.     

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.     

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.     

Sun-synchronous orbits near critical inclination

The long period dynamics of Sun-synchronous orbits near the critical inclination 116.6° are investigated. It is known that, at the critical inclination, the average perigee location is unchanged by Earth oblateness. For certain values of semimajor axis and eccentricity, orbit plane precession caused by Earth oblateness is synchronous with the mean orbital motion of the apparent Sun (a Sun-synchronism). Sun-synchronous orbits have been used extensively in meteorological and remote sensing satellite missions. Gravitational perturbations arising from an aspherical Earth, the Moon, and the Sun cause long period fluctuations in the mean argument of perigee, eccentricity, inclination, and ascending node. Double resonance occurs because slow oscillations in the perigee and Sun-referenced ascending node are coupled through the solar gravity gradient. It is shown that the total number and infinitesimal stability of equilibrium solutions can change abruptly over the Sun-synchronous range of semimajor axis values (1.54 to 1.70 Earth radii). The effect of direct solar radiation pressure upon certain stable equilibria is investigated.     

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.     

A new method of determining the inclination angle in interacting binaries

We describe a method of determining the system parameters in non-eclipsing interacting binaries. We find that the extent to which an observer sees the shape of the Roche lobe of the secondary star governs the amount of distortion of the absorption-line profiles. The width and degree of asymmetry of the phase-resolved absorption-line profiles show a characteristic shape, which depends primarily on the binary inclination and gravity darkening exponent. We show that, in principle, by obtaining high spectral and time resolution spectra of quiescent cataclysmic variables or low-mass X-ray binaries in which the mass-losing star is visible, fitting the shape of absorption-line profiles will allow one to determine not only the mass function of the binary, but also the binary inclination and hence the mass of the binary components.     

A dispersion relation for open spiral galaxies

The Lin-Shu dispersion relation is applicable in the (asymptotic) case of tight spirals (large wave numberk _R). Here we reconsider the various steps leading to the Lin-Shu dispersion relation in higher approximation, under the assumption that the wave numberk _R is not large [(k _Rr) =O(1)], and derive a new dispersion relation. This is valid for open spiral waves and bars. We prove that this dispersion relation is the appropriate limit of the nonlinear self-consistency condition in the case where the linear theory is applicable.     

A PRC relation for delta Scuti stars

The absolute radii of 78 Scuti variables in the Catalogue of the Geneva Observatory have been computed. A PRC relation among the period, radius and color has been obtained.     

A short review of the pulsar magnetic inclination angles (II)

The pulsar magnetic inclination angle is a key parameter for pulsar physics. It influences the observable properties of pulsars, such as the pulse beam width, braking index, polarization, and emission geometry. In this study, we give a brief overview of the current state of knowledge and research on this parameter and its implications for the internal physics of pulsars. We use the observed pulsar data of magnetic inclination angle and braking index to constrain the star's number of precession cycles, $\xi$, which reflects the interaction between superfluid neutrons and other particles inside a neutron star (NS). We apply the method proposed by Cheng et al. (Cheng, Q., Zhang, S. N., Zheng, X. P., & Fan, X. L., 2019, Phys. Rev. D, 99, 083011) to analyze the data of PSR J2013 + 3845 and obtain the constraints for $\xi$ ranging from $2.393\times 10^5$ to $1.268\times 10^6$. And further analysis suggests that the internal magnetic field structure of PSR J2013 + 3845 is likely dominated by toroidal components. This study may help us understand the process of internal viscous dissipation and the related evolution of the inclination angles of pulsars, and may have important implications for the study of continuous gravitational wave emissions from NS.     

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.     

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.     

Determination of the optimal conditions for inclination maneuvers using a Swing-by

The search for methods to reduce the fuel consumption in orbital transfers is something relevant and always current in astrodynamics. Therefore, the maneuvers assisted by the gravity, also called Swing-by maneuvers, can be an advantageous option to save fuel. The proposal of the present research is to explore the influence of some parameters in a Swing-by of an artificial satellite orbiting a planet with one of the moons of this mother planet, with the goal of changing the inclination of the artificial satellite around the main body of the system. The fuel consumption of this maneuver is compared with the required consumption to perform the same change of inclination using the classical approach of impulsive maneuvers.     

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.