On the artificial satellite theory

Some of the basic ideas of an analytical orbiter theory which is being developed by Hubert Claes in Namur are presented.The theory is based on the Lie transform technique and will be expressed in a closed form up to second order. The inclusion of additional terms of the third order (expanded in power series of the eccentricity) will be considered.Special attention is being given to the choice of the elements and to the final form of the theory. Three main criteria are used. The removal of the virtual singularities of small inclination and eccentricity. The simplicity of the final form of the theory once the elements have been given their numerical values. The numerical stability of the evaluation of the theory.     

On one classical problem in the radial orbit instability theory

Antonov’s classical problem of stability of a collisionless sphere with a purely radial motion of stars is considered as a limit of the problem in which stars move in nearly radial orbits. We provide the proper limiting equations that take into account the singularity in the density distribution at the sphere center and give their solutions. We show that there is instability for even and odd spherical harmonics, with all unstable modes being not slow. The growth rates of aperiodic even modes increase indefinitely when approaching purely radial models. The physics of the radial orbit instability is discussed.     

On the numerical transformation of variables in perturbation theory

Once the generating function of a Lie-type transformation is known, canonical variables can be transformed numerically by application of a Runge-Kutta type integration method or any other appropriate numerical integration algorithm. The proposed approach avails itself of the fact, that the transformation is defined by a system of differential equations with a small parameter as the independent variable. The integration of such systems arising in the perturbation theories of Hori and Deprit is discussed. The method allows to compute numerical values of periodic perturbations without deriving explicitly the perturbation series. This saving of an algebraic work is achieved at the expense of multiple evaluations of the generator's derivatives.     

Virtual singularities in the artificial satellite theory

When the coordinate system used in perturbation theory presents a geometrical singularity and when the perturbation technique fails to take account of this, the solution developed may present singularities which are no longer easily explained by purely geometrical means. These singularities have been calledvirtual singularities by Deprit and Rom (1970). We propose to demonstrate that virtual singularities can in general be avoided by the use of Lie transforms. In general, it is sufficient to recognize that the original Hamiltonian function presents the d'Alembert characteristic with respect to pairs of action-angle variables and that the averaging operations preserve this characteristic. We then apply this criterion to the artificial satellite theory (for small to moderate eccentricity) showing that all of three possible virtual singularities can be avoided at the same time. A new set of elliptic elements, well suited to the problem at hand, is introduced.     

The abundance and radial distribution of satellite galaxies

Using detailed mock galaxy redshift surveys (MGRSs) we investigate the abundance and radial distribution of satellite galaxies. The mock surveys are constructed using large numerical simulations and the conditional luminosity function (CLF), and are compared against data from the Two Degree Field Galaxy Redshift Survey (2dFGRS). We use Monte Carlo Markov chains to explore the full posterior distribution of the CLF parameter space, and show that the average relation between light and mass is tightly constrained and in excellent agreement with our previous models and with that of Vale & Ostriker. The radial number density distribution of satellite galaxies in the 2dFGRS reveals a pronounced absence of satellites at small projected separations from their host galaxies. This is (at least partly) owing to the overlap and merging of galaxy images in the 2dFGRS parent catalogue. Owing to the resulting close-pair incompleteness we are unfortunately unable to put meaningful constraints on the radial distribution of satellite galaxies; the data are consistent with a radial number density distribution that follows that of the dark matter particles, but we cannot rule out alternatives with a constant number density core. Marginalizing over the full CLF parameter space, we show that in a ΛCDM concordance cosmology the observed abundances of host and satellite galaxies in the 2dFGRS indicate a power spectrum normalization of  σ₈≃ 0.7  . The same cosmology but with  σ₈= 0.9  is unable to match simultaneously the abundances of host and satellite galaxies. This confirms our previous conclusions based on the pairwise peculiar velocity dispersions and the group multiplicity function.     

On a transformation of the differential equations of the lunar theory

This work contains a transformation of Hill-Brown differential equations for the coordinates of the satellite to a type which can be integrated in a literal form using an analytical programming language. The differential equation for the parallax of the satellite is also established. Its use facilitates the computation of Hill's periodic intermediary orbit of the satellite and provides a good check for the expansion of the coordinates and frequencies. The knowledge of the expansion of the parallax facilitates the formation of differential equations for terms with a given characteristic. These differential equations are put into a form which favors the solution by means of iteration on the computer. As in the classical theory we obtain the expansions of the coordinates and of the parallax in the form of trigonometric series in four arguments and in powers of the constants of integration. We expand the differential operators into series in squares of the constants of integration. Only the terms of order zero in these expansions are employed in the integration of the differential equations. The remaining terms are responsible for producing the cross-effects between the perturbations of different order. By applying the averaging operator to the right sides of the differential equations we deduce the expansion of the frequencies in powers of squares of the constants of integration.Basic Notations f the gravitational constant - E the mass of the planet - M the mass of the satellite - t dynamical time - x, y, z planetocentric coordinates of the satellite - u x+y–1 - s x–y–1 - the planetocentric distance of the satellite - w 1/ - ₀ the variational part of - w ₀ the variational part ofw, - n the mean daily sidereal motion of the satellite - a the mean semi-major axis of the satellite defined by means of the Kepler relation:a ³ n ²=f(E+M) - a the mean semi-major axis defined as the constant factor attached to the variational solution - e the constant of the eccentricity of the satellite - the sine of one half the orbital inclination of the satellite relative to the orbit of the sun - c(n–n) the anomalistic frequency of the satellite - c ₀ the part ofc independent frome,e, and - g(n–n) the draconitic frequency of the satellite, - g ₀ the part ofg independent frome,e, and - exp (n–n)t–1 - D d/d - e the eccentricity of the solar planetocentric orbit - a the semi-major axis of the solar orbit - n the mean daily motion of the sun in its orbit around the planet - m n/(n–n) - a/a-the parallactic factor - the disturbing function     

On Pfaff's equations of motion in dynamics; Applications to satellite theory

In this article we study a form of equations of motion which is different from Lagrange's and Hamilton's equations: Pfaff's equations of motion. Pfaff's equations of motion were published in 1815 and are remarkably elegant as well as general, but still they are much less well known. Pfaff's equations can also be considered as the Euler-Lagrange equations derived from the linear Lagrangian rather than the usual Lagrangian which is quadratic in the velocity components. The article first treats the theory of changes of variables in Pfaff's equations and the connections with canonical equations as well as canonical transformations. Then the applications to the perturbed two-body problem are treated in detail. Finally, the Pfaffians are given in Hill variables and Scheifele variables. With these two sets of variables, the use of the true anomaly as independent variable is also considered.     

On the choice of reference orbit,canonical variables,and perturbation method in satellite theory

The author's second-order artificial satellite theory (Aksnes, 1970) is reviewed and compared with that of Kozai (1962). These theories differ in that the former makes use of: (1) an intermediate orbit, being a rotating ellipse instead of a fixed ellipse, (2) Hill variables instead of Delaunay variables, and (3) Hori's perturbation method in Lie series rather than Von Zeipel's method in Taylor series.It is demonstrated that because of these differences, the former theory enjoys a greater simplicity and compactness, it is non-singular at zero eccentricity, and the process of deriving the perturbations is considerably simplified (Aksnes, 1972). For example, the number of second-order short-period terms due to the planet's oblateness (J ₂) is reduced by a factor of about three (Hori, 1970). The intermediate orbit and Hori's perturbation method contribute about equally to this reduction.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.     

The elimination of the parallax in satellite theory

When the perturbation affecting a Keplerian motion is proportional tor ^–n (n3), a canonical transformation of Lie type will convert the system into one in which the perturbation is proportional tor ^–2. Because it removes parallactic factors, the transformation is called the elimination of the parallax.In the main problem for the theory of artificial satellites, the elimination of the parallax has been conducted by computer to order 4. The first order in the reduced system may now be integrated in closed form, thereby revealing the fundamental property of the first-order intermediary orbits in line with Newton's Propositio XLIV.Extension beyond order 1 leads to identify a new class of intermediaries for the main problem in nodal coordinates, namely the radial intermediaries.The technique of smoothing a perturbation prior to normalizing the perturbed Keplerian system, of which the elimination of the parallax is an instance, is applied to derive the intermediaries in nodal coordinates proposed by Sterne, Garfinkel, Cid-Palacios and Aksnes, and to find the canonical diffeomorphisms which relate them to one another and to the radial intermediaries.     

Stablization and time transformation

We suggest a simple stabilization technique to reduce the along-track error in the numerical Integration of the Lagrange's equations of motion. We also investigate the equations of motion of the two-body problem after applying the Sundman transformation dt = r^αds, both with and without introducing the energy integral. In both cases, we show how the stability of the equations varies with α and in the case with the energy integral, we show that every solution is a quasi-periodic function of s with two frequencies.     

The critical inclination in artificial satellite theory

Certain it is that the critical inclination in the main problem of artificial satellite theory is an intrinsic singularity. Its significance stems from two geometric events in the reduced phase space on the manifolds of constant polar angular momentum and constant Delaunay action. In the neighborhood of the critical inclination, along the family of circular orbits, there appear two Hopf bifurcations, to each of which there converge two families of orbits with stationary perigees. On the stretch between the bifurcations, the circular orbits in the planes at critical inclinmation are unstable. A global analysis of the double forking is made possible by the realization that the reduced phase space consists of bundles of two-dimensional spheres. Extensive numerical integrations illustrate the transitions in the phase flow on the spheres as the system passes through the bifurcations.A delicacy so very susceptible of offence...—Hester Lynch PIOZZI,Observations and Reflections made in the Course of a Journey through France, Italy and Germany (1789)NAS/NRC Postgraduate Research Associate in 1984–1985.     

Classes of orbits in the main problem of satellite theory

We consider the main problem in satellite theory restricted to the polar plane. For suitable values of the energy the system has two unstable periodic orbits. We classify the trajectories in terms of their ultimate behavior with respect these periodic orbits in: oscillating, asymptotic and capture orbits. We study the energy level set and the existence and properties of the mentioned types of motion.     

Exact linearization of non-planar intermediary orbits in the satellite theory

In this paper we consider the reduction of the equations of motion for non-planar perturbed two body problems into linear form. It is seen that this can be easily accomplished for any element of the class of radial intermediaries to the satellite problem proposed by Deprit in 1981, since they have a functional dependence suitable for linearization. The transformation is worked out by using an adequate set of redundant variables. Four harmonic oscillators are obtained, of which two are coupled through gyroscopic terms. Their constant frequencies contain the secular contribution of the main problem of artificial satellite theory up to the order of the considered intermediary. Therefore, this result may well be interesting in relation to the study and prediction of accurate long-term solutions to satellite problems.     

The gravitational perturbation spectrum in linear satellite theory

A new method for calculating the perturbation spectrum in the framework of Kaula's linear satellite theory (LST) is introduced. The novelty of this approach consists in using recent results on the spectral decomposition of the perturbation frequencies in LST to provide a closed formulation for the amplitude and the phase of each line in the perturbation spectrum. The theory presented here can be applied to perturbations in the elements or in the radial and transverse directions due to the geopotential or to the tides. Separate algorithms are developed for application to orbits with circulating or frozen perigee.     

Relativistic effects in the critical inclination problem in artificial satellite theory

It is well known that in artifical satellite theory special techniques must be employed to construct a formal solution whenever the orbital inclination is sufficiently close to the critical value cos^–1 (1/5). In this article the authors investigate the consequences of introducing certain relativistic effects into the motion of a satellite about an oblate primary. Particular attention is paid to the critical inclination(s), and for such critical motions an appropriate method of solution is formulated.     

A note on Hansen's coefficients in satellite theory

Recurrence relations are derived for the Eccentricity FunctionsG andH and their derivatives, as they appear in the evaluation of geopotential and third body perturbations of an artificial satellite.     

Galactic satellite systems: radial distribution and environment dependence of galaxy morphology

We have studied the radial distribution of the early (E/S0) and late (S/Irr) types of satellites around bright host galaxies. We made a volume-limited sample of 4986 satellites brighter than   M _r=−18.0  associated with 2254 hosts brighter than   M _r=−19.0  from the Sloan Digital Sky Survey Data Release 5 sample. The morphology of satellites is determined by an automated morphology classifier, but the host galaxies are visually classified. We found segregation of satellite morphology as a function of the projected distance from the host galaxy. The amplitude and shape of the early-type satellite fraction profile are found to depend on the host luminosity. This is the morphology–radius/density relation at the galactic scale. There is a strong tendency for morphology conformity between the host galaxy and its satellites. The early-type fraction of satellites hosted by early-type galaxies is systematically larger than that of late-type hosts, and is a strong function of the distance from the host galaxies. Fainter satellites are more vulnerable to the morphology transformation effects of hosts. Dependence of satellite morphology on the large-scale background density was detected. The fraction of early-type satellites increases in high-density regions for both early- and late-type hosts. It is argued that the conformity in morphology of galactic satellite system is mainly originated by the hydrodynamical and radiative effects of hosts on satellites.     

Second-order theory of the rotation of an artificial satellite

This work presents the expansion of the second-order of an analytical theory of the attitude evolution of an artificial satellite perturbed by given torques. The first-order of the theory has already been presented by the author in Celestial Mechanics39 (1986) 309–327. It is a theory that is valid under very general conditions including slow rotation and inequal axes of inertia. The present theory is suitable for any internal or external disturbing forces producing the torques. A formal solution is expanded in the second-order according to powers of a small parameter characteristic of the order of magnitude of the disturbing torques. These torques are expanded in Fourier series and the theory applies whatever is the length of these series. The coefficients of the solution are given by an iterative formation law. The comparison of the results with a numerical integration based upon a HIPPARCOS model shows that the second order has brought an improvement to the theory by at least one order of magnitude over the results of the first order.     

Unstable non-radial modes in radial pulsators: theory and an example

We study the possibility of the excitation of non-radial oscillations in classical pulsating stars. The stability of an RR Lyrae model is examined through non-adiabatic non-radial calculations. We also explore stability in the presence of non-linear coupling between radial and non-radial modes of nearly identical frequency.   In our model, a large number of unstable low-degree (ℓ = 1,2) modes have frequencies in the vicinity of unstable radial mode frequencies. The growth rates of such modes, however, are considerably smaller than those of the radial modes. We also recover an earlier result that at higher degrees (ℓ = 6–12) there are modes trapped in the envelope with growth rates similar to those of radial modes.   Subsequently, monomode radial pulsation of this model is considered. The destabilizing effect of the 1:1 resonance between the radial mode and nearby non-radial modes of low degrees is studied, with the assumption that the excited radial mode saturates the linear instability of all other modes. The instability depends on the radial mode amplitude, the frequency difference, the damping rate of the non-radial mode, and the strength of the non-linear coupling between the modes considered. At the pulsation amplitudes typical for RR Lyrae stars, the instability of the monomode radial pulsation and the concomitant resonant excitation of some non-radial oscillation modes is found to be very likely.