Methods of Celestial Mechanics
- ... Generally , the truncation is not explicit because there is no explicit expansion in power of the eccentricity. But in practice, Fourier series of the mean anomaly which converge slowly must be truncated and this relies mainly on the D'Alembert rule (see Brouwer and Clemence, 1961) which guarantees an accelerated convergence as long as the eccentricity is small. Because this is indeed the case of numerous natural bodies or artificial satellites, these expansions of the disturbing function are well suited in many situations. ...... with µ ′ the third-body gravitational constant, r and r ′ respectively the geocentric position vector of the artificial satellite and the disturbing body, and r and r ′ their associated radial distances. Since we are interested in the orbits such as r ′ /r > 1, R ′ can be expressed in power series of r ′ /r as (Plummer, 1960; Brouwer and Clemence, 1961): ...... where X n,k q (e) are the well known Hansen coefficients (Hansen, 1853; Tisserand, 1889; Brouwer and Clemence, 1961). In the general case, the series (2.19) always converge as Fourier series, but can converge rather slowly (see e.g. ...(Section 4.1 required a little correction... asap...) Traditional analytical theories of celestial mechanics are not well-adapted when dealing with highly elliptical orbits. On the one hand, analytical solutions are quite generally expanded into power series of the eccentricity and so limited to quasi-circular orbits. On the other hand, the time-dependency due to the motion of the third body (e.g. Moon and Sun) is almost always neglected. We propose several tools to overcome these limitations. Firstly, we have expanded the third-body disturbing function into a finite polynomial using Fourier series in multiple of the satellite's eccentric anomaly (instead of the mean anomaly) and involving Hansen-like coefficients. Next, by combining the classical Brouwer-von Zeipel procedure and the time-dependent Lie-Deprit transforms, we have performed a normalization of the expanded Hamiltonian in order to eliminate all the periodic terms. One of the benefits is that the original Brouwer solution for J2 is not modified. The main difficulty lies in the fact that the generating functions of the transformation must be computed by solving a partial differential equation, involving derivatives with respect to the mean anomaly, which appears implicitly in the perturbation. We present a method to solve this equation by means of an iterative process. Finally we have obtained an analytical tool useful for the mission analysis, allowing to propagate the osculating motion of objects on highly elliptical orbits (e>0.6) over long periods efficiently with very high accuracy, or to determine initial elements or mean elements. Comparisons between the complete solution and the numerical simulations will be presented.
- ... In the case of satellite gradiometry, since we directly measure the second gradients of the gravitational potential, the observational equations of the gravitational tensors are mathematically straightforward ( [93,95] ). Linear perturbation methods are to find an approximate solution to nonlinear Lagrange's planetary equations and mathematically rigorous, which have been well developed and documented (see e.g., [11,12,34,49,52,60,102,122] ). Soon after the launches of first artificial satellites in 1950s and 60s, with camera and Doppler tracking measurements, linear perturbation solutions and simplified variants for small and/or zero divisors were used to compute the flattening and/or the eccentricity of the Earth (see e.g., [13,45,54,75,77] ), lumped and/or resonance-derived harmonic coefficients (see e.g., [1,[16][17][18][55][56][57]119] ), and gravitational models with low degrees and orders (see e.g., [17,18,30,31,34,46,49,52,61,62] ). ...... One may equivalently rewrite Lagrange's planetary Eqs. (1) in six other orbital elements (see e.g., [12,96,102] ). Without loss of generality, we denote the general solution to Lagrange's planetary Eqs. ...... On the other hand, given the general solution K ( t , p , c k ) and a sufficient number of measurements on K ( t , p , c k ), one can then estimate the (unknown) force parameters p from the measurements. Unfortunately, the general analytical solution K ( t , p , c k ) can only be obtained for the idealized two-body problem in which a particle of negligible mass is attracted by another point mass (see e.g., [12,52,83,102] ). In general, no analytical solution to (1) can be possible, for two reasons: (i) Lagrange's planetary Eqs. ...The numerical integration method has been routinely used by major institutions worldwide, for example, NASA Goddard Space Flight Center and German Research Center for Geosciences (GFZ), to produce global gravitational models from satellite tracking measurements of CHAMP and/or GRACE types. Such Earth's gravitational products have found widest possible multidisciplinary applications in Earth Sciences. The method is essentially implemented by solving the differential equations of the partial derivatives of the orbit of a satellite with respect to the unknown harmonic coefficients under the conditions of zero initial values. From the mathematical and statistical point of view, satellite gravimetry from satellite tracking is essentially the problem of estimating unknown parameters in the Newton's nonlinear differential equations from satellite tracking measurements. We prove that zero initial values for the partial derivatives are incorrect mathematically and not permitted physically. The numerical integration method, as currently implemented and used in mathematics and statistics, chemistry and physics, and satellite gravimetry, is groundless, mathematically and physically. Given the Newton's nonlinear governing differential equations of satellite motion with unknown equation parameters and unknown initial conditions, we develop three methods to derive new local solutions around a nominal reference orbit, which are linked to measurements to estimate the unknown corrections to approximate values of the unknown parameters and the unknown initial conditions. Bearing in mind that satellite orbits can now be tracked almost continuously at unprecedented accuracy, we propose the measurement-based perturbation theory and derive global uniformly convergent solutions to the Newton's nonlinear governing differential equations of satellite motion for the next generation of global gravitational models. Since the solutions are global uniformly convergent, theoretically speaking, they are able to extract smallest possible gravitational signals from modern and future satellite tracking measurements, leading to the production of global high-precision, high-resolution gravitational models. By directly turning the nonlinear differential equations of satellite motion into the nonlinear integral equations, and recognizing the fact that satellite orbits are measured with random errors, we further reformulate the links between satellite tracking measurements and the global uniformly convergent solutions to the Newton's governing differential equations as a condition adjustment model with unknown parameters, or equivalently, the weighted least squares estimation of unknown differential equation parameters with equality constraints, for the reconstruction of global high-precision, high-resolution gravitational models from modern (and future) satellite tracking measurements.
- ... In a subsequent work [6] he tacked the problem of secular motion in a 2 nd degree and order-gravity field with no rotation qualitatively. In this respect the main problem of artificial satellite theory is very useful; the major contributors to this subject were [7][8][9][10] formed the Hamiltonian of the motion of an A.S. about a planet with an inhomogeneous gravitational field including the leading zonal and tesseral harmonics. He used the Whit- taker variables and then he normalized the Hamiltonian us- ing the method of elimination of the parallax developed by [11]. ...The motion of a satellite about a rotating triaxial body will be investigated, stressing on the case of slow rotation. The Hamiltonian of the problem will be formed including the zonal harmonic J 2 and the leading tesseral harmonics C 22 and S 22. The small parameter of the problem is the spin rate () of the primary. The solution proceeds through three canonical transformations to eliminate in succession; the short, intermediate and long-period terms. Thus secular and periodic terms are to be retained up to orders four and two respectively.
- ... If we focus on the resonances affecting the eccentricity evolution, Musen (1960) andCook (1962)were the first to highlight the existence of six SRP resonances, and to show their location in the (i, a) plane.Cook (1962), in particular, noted that, contrary to lunisolar gravitational resonances, the SRP resonances are able to give a variation in eccentricity also in the case of circular orbits.Musen (1960)labelled Ω + ω − n S as the 'most interesting resonance', as done by all the following authors. In the case of the gravitational perturbation, this resonance is also known as evection resonance 1 (e.g.,Brouwer & Clemence 1961;Touma & Widsom 1998;Frouard et al. 2010). Breiter (1999, basing his analysis on the work done byCook (1962), noted that 'resonant solar perturbations can be much stronger than the lunar ones'. ...The aim of this work is to highlight the crucial role that orbital resonances associated with solar radiation pressure can have in Low Earth Orbit. We review the corresponding literature, and provide an analytical tool to estimate the maximum eccentricity which can be achieved for well-defined initial conditions. We then compare the results obtained with the simplified model with the results obtained with a more comprehensive dynamical model. The analysis has important implications both from a theoretical point of view, because it shows that the role of some resonances was underestimated in the past, but also from a practical point of view in the perspective of passive deorbiting solutions for satellites at the end-of-life.
- ... Hence, for the purpose of this paper we proceed with a different way of obtaining the secular Hamiltonian. Following Lee & Peale (2003), we make use of the following variables, which is a set of canonical ones (Brouwer & Clemence 1961; Murray & Dermott 1999) and for a coplanar three body system they are defined as: ...We study the long term orbital evolution of a terrestrial planet under the gravitational perturbations of a giant planet. In particular, we are interested in situations where the two planets are in the same plane and are relatively close. We examine both possible configurations: the giant planet orbit being either outside or inside the orbit of the smaller planet. The perturbing potential is expanded to high orders and an analytical solution of the terrestrial planetary orbit is derived. The analytical estimates are then compared against results from the numerical integration of the full equations of motion and we find that the analytical solution works reasonably well. An interesting finding is that the new analytical estimates improve greatly the predictions for the timescales of the orbital evolution of the terrestrial planet compared to an octupole order expansion. Finally, we briefly discuss possible applications of the analytical estimates in astrophysical problems.
- ... The corrections to the orbit elements tend to be of two classes: periodic corrections, which vary on an orbital timescale, and secular corrections, which vary on a longer timescale, depending upon the nature of the perturbations. There are many approaches to separating secular from periodic effects (see [22] , for example), but we will adopt a specific approach, known as " multiple-scale analysis " , that is used in many problems in physics [23]. One advantage of this method is that it can be carried out systematically to higher orders in perturbation theory. ...We derive the secular evolution of the orbital elements of a stellar-mass object orbiting a spinning massive black hole. We use the post-Newtonian approximation in harmonic coordinates, with test-body equations of motion for the conservative dynamics that are valid through 3PN order, including spin-orbit, quadrupole and (spin)$^2$ effects, and with radiation-reaction contributions linear in the mass of the body that are valid through 4.5PN order, including the 4PN damping effects of spin-orbit coupling. The evolution equations for the osculating orbit elements are iterated to high PN orders using a two-timescale approach and averaging over orbital timescales. We derive a criterion for terminating the orbit when its Carter constant drops below a critical value, whereupon the body plunges across the event horizon at the next closest approach. The results are valid for arbitrary eccentricities and arbitrary inclinations. We then analyze numerically the orbits of objects injected into high-eccentricity orbits via interactions within a surrounding star cluster, obtaining the number of orbits and the elapsed time between injection and plunge, and the residual orbital eccentricity at plunge as a function of inclination. We derive an analytic approximation for the time to plunge in terms of initial orbital variables. We show that, if the black hole is spinning rapidly, the flux of gravitational radiation during the final orbit before plunge may be suppressed by as much as three orders of magnitude if the orbit is retrograde on the equatorial plane compared to its prograde counterpart.
- ... Following Brouwer and Clemenc [14] ...The effect of resonance on the motion of two cylindrical rigid bodies has been studied in the light of Bhatnagar [1] [2] [3] and under some defined axiomatic restrictions. Here we have calculated variation in Eulerian angles due to resonance in terms of orbital elements and unperturbed Eulerian angles.
- ... the Lagrange-Laplace planetary equations up to order O(e 2 1 ) that determine the satellite's orbit evolution read (Brouwer & Clemence 1961): ...We present a semi-analytical correction to the seminal solution for the secular motion of a planet's orbit under gravitational influence of an external perturber derived by Heppenheimer (1978). A comparison between analytical predictions and numerical simulations allows us to determine corrective factors for the secular frequency and forced eccentricity in the co-planar restricted three-body problem. The correction is given in the form of a polynomial function of the system's parameters that can be applied to first-order forced eccentricity and secular frequency estimates. The resulting secular equations are simple, straight forward to use and improve the fidelity of Heppenheimer's solution well beyond higher-order models. The quality and convergence of the corrected secular equations are tested for a wide range of parameters and limits of its applicability are given.
- ... – 8 – It is possible to express the true anomaly as a function of time through the mean anomaly according toBrouwer & Clemence (1961,p. 77) ...One of the directly measured quantities which are used in monitoring the orbital motions of many of the S stars revolving around the Supermassive Black Hole (SMBH) in the Galactic Center (GC) is their radial velocity (RV) $V$ obtained with near-infrared spectroscopy. Here, we devise a general approach to calculate both the instantaneous variations $\Delta V\left(t\right)$ and the net shifts per revolution $\left\langle\Delta V\right\rangle$ induced on such an observable by some post-Keplerian (pK) accelerations. In particular, we look at the general relativistic Schwarzschild (gravitoelectric) and Lense-Thirring (gravitomagnetic frame-dragging) effects, and the mass quadrupole. It turns out that we may be on the verge of measuring the Schwarzschild-type 1pN static component of the SMBH's field with the S2 star for which RV measurements accurate to about $\simeq 30-50~\textrm{km s}^{-1}$ dating back to $t_0 = 2003.271$ are currently available, and whose orbital period amounts to $P_\textrm{b} = 16$ yr. Indeed, while its expected general relativistic RV net shift per orbit amounts to just $\left\langle\Delta V^\textrm{GE}\right\rangle = -11.6~\textrm{km s}^{-1}$, it should reach a peak value as large as $\Delta V_\textrm{max}^\textrm{GE}\left(t_\textrm{max}\right) = 551~\textrm{km s}^{-1}$ at $t_\textrm{max} = 2018.35$. The periastron shift $\Delta\omega^\textrm{GE}$ of S2 over the same time span will not be larger than $0.2$ deg, while the current accuracy in estimating such an orbital element for this star is of the order of $0.6$ deg. The frame-dragging and quadrupole-induced RV shifts are far smaller for S2, amounting to, at most, $0.19~\textrm{km s}^{-1},0.0039~\textrm{km s}^{-1}$, respectively. Further studies should be dedicated to the impact on the RV of possible diffused mass distribution in the GC and of other individual stars inside and outside the orbit of S2.
- ... Depending on the specific perturbing acceleration A at hand, the calculation of Equation (8) can be, in general, rather unwieldy. A time-dependent time series for ∆F can be conveniently obtained by expressing the true anomaly f as a function of time t through the mean anomaly according toBrouwer & Clemence (1961,p. 77) ...Several kinds of observable quantities $\mathcal{O}$ are more or less directly measured in studying gravitationally bound binary systems like, e.g., exoplanets, ordinary binary stars, binaries hosting one or more compact objects: radial velocities, timing, astrometric observations in the plane of the sky, transit durations, temporal intervals between primary and secondary eclipses, periods of the primary eclipse, etc. We develop a general approach to analytically calculate the instantaneous changes $\Delta F$ experienced by the mathematical model $F$ of $\mathcal{O}$ due to arbitrary post-Keplerian dynamical accelerations without restrictions on the binary's orbital geometry. We successfully test our method with the well-known post-Newtonian Schwarzschild-type acceleration and the S2 main sequence star orbiting the Supermassive Black Hole in the center of the Milky Way. Our approach allows for a closer connection between theoretical investigations, aiming, e.g., to design dedicated tests of fundamental physics, and what is actually measured by astronomers. Furthermore, it is able to provide for more accurate sensitivity analyses and re-analyses of existing data.
- ... Let the satellite of the mass M have the maximal moment of inertia C , and let the planet of the mass M * have the maximal moment of inertia C * . For a negligibly small obliquity, and with θ and θ * denoting the rotation angles of the moon and the planet, the total mechanical energy of this system in the inertial frame is (Brouwer & Clemence 1961 ...In the preceding paper (Efroimsky 2017), we demonstrated that under weak libration in longitude most dissipation is due to the gravitational tides (including the additional tides generated by libration). The other three sources of dissipation$~-~$which are the alternating parts of the centripetal, toroidal and purely radial deformations$~-~$are less important when libration is weak. Whether this is so for large-magnitude libration requires a separate study. In Ibid. it was also shown that in some situations the forced libration in longitude can provide a considerable and even leading input into the tidal heating: 52% in Phobos, 33% in Mimas, 12% in Enceladus, and 96% in Epimetheus. Equating our expression for the tidal dissipation rate (with the libration-generated input included) to the outgoing energy flux due to the vapour plumes, we estimate the mean tidal viscosity of Enceladus, under the assumption that the Enceladean mantle behaves as a Maxwell body. This method yields a value of $\,0.4\times 10^{14}$ Pa s for the mean tidal viscosity, which is remarkably close to the viscosity of ice near the melting point. We then demonstrate that, with such a value of the tidal viscosity, the tidal dissipation in Enceladus is too low to influence its orbital evolution. Thus the orbital evolution of Enceladus is defined to a much greater extent by the tidal friction in Saturn.
- ... The mutual gravitational potential of rigid bodies is a classical problem. The basic form of the potential is a double volume integration of reciprocal distance between mutual mass elements over both bodies (Brouwer and Clemence 1961). The usual approach is expanding the potential in asymptotic series in terms of ratio of body sizes over the distance. ...Binary systems are quite common within the populations of near-Earth asteroids, main-belt asteroids, and Kuiper belt asteroids. The dynamics of binary systems, which can be modeled as the full two-body problem, is a fundamental problem for their evolution and the design of relevant space missions. This paper proposes a new shape-based model for the mutual gravitational potential of binary asteroids, differing from prior approaches such as inertia integrals, spherical harmonics, or symmetric trace-free tensors. One asteroid is modeled as a homogeneous polyhedron, while the other is modeled as an extended rigid body with arbitrary mass distribution. Since the potential of the polyhedron is precisely described in a closed form, the mutual gravitational potential can be formulated as a volume integral over the extended body. By using Taylor expansion, the mutual potential is then derived in terms of inertia integrals of the extended body, derivatives of the polyhedron’s potential, and the relative location and orientation between the two bodies. The gravitational forces and torques acting on the two bodies described in the body-fixed frame of the polyhedron are derived in the form of a second-order expansion. The gravitational model is then used to simulate the evolution of the binary asteroid (66391) 1999 KW4, and compared with previous results in the literature.
- ... Their computation can be done directly from the equations of motion (1.101), (1.102) and (1.103). As in[13], we adapt the expression of the potential U , in order to facilitate the computation of the derivatives involved. Considering the definitions of the distances δ 0 and δ 1 in (1.95) and (1.96), we have ...The main subject of this work is the study of the problem of the Trojan orbits from a perturbative Hamiltonian perspective. We face this problem by introducing first a novel Hamiltonian formulation, exploiting the well-differentiated temporal scales of the Trojan motion. The resulting Hamiltonian allows to separate the secular (very slow) component of the motion from the librating and fast degrees of freedom. This decompositon provides the foundation of a so-called Basic Hamiltonian model (Hb), i.e. the part of the Hamiltonian for Trojan orbits independent of all secular angles. Our study shows that, up to some extent, the model Hb successfully represents the features of the motion under more complete models, in a range of physical parameters relevant for dynamics in the Solar System or in extrasolar planetary systems. (continues...)
- ... (β) and their derivatives that can be calculated numerically ( Brouwer & Clemence 1961) using the definitions given in Murray & Dermott (1999, Eq. 6.67). In the limit m >> 1, m/(m-1) ≈ 1, β → 1 and A m,0 ≈ 0.8m for a first order resonance (k=0), and A m,1 ≈ 0.48m 2 for a second order resonance (k=1). ...The viscous spreading of planetary rings is believed to be counteracted by satellite torques, through either an individual resonance or overlapping resonances. For the A ring of Saturn, it has been commonly believed that the satellite Janus alone can prevent the ring from spreading, via its 7:6 Lindblad resonance. We discuss this common misconception and show that, in reality, the A ring is confined by the contributions from the group of satellites Pan, Atlas, Prometheus, Pandora, Janus, Epimetheus, and Mimas, whose cumulative torques from various resonances gradually decrease the angular momentum flux transported outward through the ring via density and bending waves. We further argue that this decrease in angular momentum flux occurs through "flux reversal." Furthermore, we use the magnitude of the satellites' resonance torques to estimate the effective viscosity profile across the A ring, showing that it decreases with radius from ~50 cm² s⁻¹ to less than ~10 cm² s⁻¹. The gradual estimated decrease of the angular momentum flux and effective viscosity are roughly consistent with results obtained by balancing the shepherding torques from Pan and Daphnis with the viscous torque at the edges of the Encke and Keeler gaps, as well as the edge of the A ring. On the other hand, the Mimas 2:1 Lindblad resonance alone seems to be capable of confining the edge of the B ring, and contrary to the situation in the A ring, we show that the effective viscosity across the B ring is relatively constant at ~24–30 cm² s⁻¹.
- ... The periodic orbits found in the 3DHP, besides mimicking closely the real lunar orbit, are also very similar to those found in Valsecchi et al. (1993) in the CR3D3BP, as also shown by the closeness of the values of the geocentric orbital elements reported by Tables 3, 4. In fact, the principal lunar short-period terms (Brouwer and Clemence, 1961) are contained in the periodic orbits in both problems: ...In the framework of the restricted, circular, 3-dimensional 3-body problem Sun-Earth-Moon, Valsecchi et al. (1993) found a set of 8 periodic orbits, with duration equal to that of the Saros cycle, and differing only for the initial phases, in which the motion of the massless Moon follows closely that of the real Moon. Of these, only 4 are actually independent, the other 4 being obtainable by symmetry about the plane of the ecliptic. In this paper the problem is treated in the framework of the 3-dimensional Hill's problem. It is shown that also in this problem there are 8 periodic orbits of duration equal to that of the Saros cycle, and that in these periodic orbits the motion of the Moon is very close to that of the real Moon. Moreover, as a consequence of the additional symmetry of Hill's problem about the y-axis, only 2 of the 8 periodic orbits are independent, the other ones being obtainable by exploiting the symmetries of the problem.
- This paper is extension to the classical Sitnikov problem, when the four primaries of equal masses lie at the vertices of a square for all time and moving in elliptic orbits around their center of mass of the system, the distances between the primaries vary with time but always in such a way that their mutual distances remain in the same ratio. First we have established averaged equation of motion of the Sitnikov five-body problem in the light of Jalali and Pourtakdoust (1997), by applying the Van der Pol transformation and averaging technique of Guckenheimer and Holmes (Nonlinear oscillations, dynamical system bifurcations of vector fields, Springer, Berlin, 1983). Next the Hamiltonian equation of motion has been solved with the help of action angle variables . Finally the periodicity and stability of the Sitnikov five-body problem have been examined with the help of Poincare surfaces of section (PSS). It is shown that chaotic region emerging from the destroyed islands, can easily be seen by increasing the eccentricity of the primaries.
- The role of different sets of variables in analytical or semianalytical solutions to the artificial satellite theory is reviewed. In particular, it is shown that using polar-nodal variables or non-singular variables based on them is clearly advantageous in the evaluation of the transformation from mean to osculating elements.
- Semianalytical theories provide tools for propagating perturbed satellite orbits. The power of these theories lies in the ability to efficiently calculate, and thereby comprehend, the orbital dynamics of satellites. In many cases of practical interest, as we will see in Chap. 14, satellite orbit control laws also utilize mean orbital elements. The mean elements are most commonly defined as doubly-averaged elements, obtained from a doubly-averaged Hamiltonian. However, the mean elements in the current chapter will be singly-averaged elements, obtained by removing the short-periodic oscillations only.
- This section is meant as an introduction to notation and procedures involved in numerical work that occurs in celestial mechanics and astrodynamics . No theoretical developments will be presented; for theory the reader might consult a work on finite differences. The notation to be used and reference to more information is available from Nautical Almanac Office (1956).
- We have seen the complexity of the problem when more than two gravitating masses are involved. We have seen two methods of determining the orbits, Cowell’s and Encke’s methods . Now, let us look at the basic mathematical description of the perturbation problem.
- We use a new method of solving equations in variations for perturbed central force fields to obtain first order perturbations of Cartesian coordinates of a small body moving in a perturbed central force field. In the case of Newtonian field, we derive formulae for first order perturbations of coordinates of a small body moving under piecewise polynomial (in true anomaly) low-thrust. We suppose that the small body under consideration is initially moving in an elliptic orbit.
- In the last years considerable efforts have been made to construct analytical theories of the major planets and the moon. But I do not know of any new attempt to construct a theory of an asteroid since Duncombe’s numerical integration (1969) of the orbits of Ceres, Pallas, Juno, and. Vesta.
- A number of applications exist in astronomical research for planetary and lunar ephemerides covering an extended length of time. This paper discusses such a set of ephemerides, DE102/LE51, produced at JPL, covering the time 1411 B.C. to 3001 A.D. The ephemerides are dynamically self-consistent, in that the equations of motion were integrated simultaneously. They also represent the most accurately known positions covering such a time span. They have already been used by a number of different users in a variety of different applications.
- Celestial mechanics is the topic of this chapter. The 2-body solution is given, the restricted 3-and n-body solutions discussed, and the effects of perturbations on the orbital elements are treated in detail. Tidal friction and its effects in the Earth-Moon system, spin-orbit and orbit-orbit resonances are discussed.
- In this chapter we deal with the terminology and basic well known results, which are necessary to the development of the subsequent chapters. It is not the scope of this chapter to describe Hamiltonian Systems and their general properties. They are found in several books and monographs, among which we wish to mention the classics of Birkhoff (1927), Siegel (1956), Wintner (1947), Abraham (1966), Moser (1968). We avoid any and every sophistication in arriving at intrinsic representations and definitions of Hamiltonian systems on manifolds, not because they are not important, but because they are of no essential necessity in what has to follow.
- In this section we give a more precise characterization of trajectories which are solutions of an integrable system. We start from Liouville’s result. Given a Hamiltonian System $$\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{{{\dot{y}}}_{k}} = + {{H}_{{{{x}_{k}}}}},} \\ {{{{\dot{x}}}_{k}} = - {{H}_{{{{y}_{k}}}}},} \\ \end{array} } & {(k = 1,2,. .,n)} \\ \end{array}$$ (3.1.1) let H = H(y;x) be analytic in a given domain D of the phase space. If n uniform integrals F1, F2,...,Fn, in involution, are known, in a domain D’ ⊂ D, then in D’ the system is integrable, i.e., reducible to quadratures. Let $$ {F_i}(y;x) = {C_i} = {\text{const}} $$ (3.1.2) for i = 1,2,...,n. In general one verifies that the closed manifolds generated by Eqs. (3.1.2.) are tori and, on these, the motion is quasiperiodic. One can actually show that this is so, in general, for a Liouville system. More precisely, the following theorem (Arnol’d, 1963) can be stated. “Let the system (3.1.1.), with n degrees of freedom, have n first uniform integrals F1,...,Fn in involution. The equations Fi = Cidefine a compact manifold M = MCin very point of which the vectors grad Fi (i = 1,2,...,n) are linearly independent in the phase space of dimensions 2n. Then M is a torus of dimension n and the point (y(t); x(t)), solution of (3.1.1) in D’, has a quasiperiodic motion on M.” This theorem justi fies the fact that we always consider intergrable systems as given by a Hamiltonian H = HO(x), a function of the momenta only.
- This chapter is devoted to two main goals. First introduce the reader to known methods of canonical perturbations, describe them in a heuristic way and give examples so as to motivate the theorems presented in Chapters III and IV. Second, present some basic results about iterative procedures of fundamental importance on methods of averaging. Major contributors to this area are Lindstedt (1884), Poincaré (1893), Whittaker (1916), Siegel (1941), Krylov (1947), Bogoliubov (1945), Kolmogorov (1953), Arnol’d (1963), Diliberto (1961), Pliss (1966), Kyner (1961), Moser (1962), Hale (1961) with several overlappings in results. Many of these results have been unified and consolidated in celebrated books by Siegel (1956), Wintner (1947), Newytskii-Stepanov (1960), Cesari (1963), Hale (1969), Abraham (1967), Birkhoff (1927), Bogoliubov-Mitropolskii (1961), Lefschetz (1959), Minorsky (1962), Sansone-Conti (1964), Sternberg (1970).
- This paper introduces a simple analytical approximation to three-dimensional heliocentric solar sail orbits where the only forces considered are solar gravity and solar radiation. The approximation is based upon the previously studied hodograph transformation and provides a description of the inclination, longitude of ascending node and true latitude for a specific set of initial conditions. It is shown that the rotational symmetry of a heliocentric orbit allows this specific solution to be mapped onto a solution with arbitrary initial conditions. The approximation is then compared to the numerical results for a solar sail on an Earth escape trajectory with an area to mass ratio up to twice as high as current technology allows.
- Satellites are attracted not only by the central force of the earth, but also by the non-central force, the attraction forces of the sun and the moon, and the drag force of the atmosphere. They are also affected by solar radiation pressure, earth and ocean tides, general relativity effects (cf. Chap. 5), and coordinate perturbations. Equations of satellite motion must be represented by perturbed equations. In this chapter, after discussions of the perturbed equations of motion and the attraction forces, for convenience of the earth tide and ocean loading tide computations, the ephemerides of the sun and the moon are described. Orbit correction is discussed based on an analysis solution of the \( \overline{C}_{20} \) perturbation. Emphasis is given to the precise orbit determination, which includes the principle of orbit determination , algebraic solution of the variation equation, numerical integration, and interpolation algorithms, as well as the related partial derivatives .
- Studies of the perturbations of Vesta can be divided into three phases in the early nineteenth century. Burckhardt in France was at the forefront in this research, writing a paper on it the very same year Vesta was discovered. The following decade, Pierre Daussy (also in France) published two major studies, and in 1810 Santini (in Italy) “gave Vesta’s secular variations and formulae for her periodic inequalities to the first order of small quantities, on two hypotheses of the value of the major axis.” (Airy 1835:158) A review of a two-volume work by Santini (1819) says it “contains the most noted theorems in spherics, and the formulas generally used in calculations of astronomy, particularly, a detailed account of the methods of Olbers and Gauss for computing the orbits of comets or planets, with Burckhardt’s tables of motion for a parabola, and Gauss’ tables for an ellipse or hyperbola.” (Bowditch 1825:52) Volume 1 of his work treated the asteroids.
- This paper considers the spatial elliptic restricted three body problem in the case that the particle with negligible mass is revolving around one of the primaries. The system is modelled in an inertial frame through a Hamiltonian function representing a non-autonomous dynamical system with three degrees of freedom that depends periodically on time. Three Lie transformations are applied at first order to eliminate successively the mean anomaly of the infinitesimal particle’s motion, the time dependence of the system and the argument of the node of the particle with negligible mass. All the transformations are implemented in a compact way, that is, carrying out the computations by means of infinite series. This approach can be very useful to deal with certain artificial satellite problems or, in general, with systems such that the ratio between the distance of the infinitesimal particle to the body around it orbits and the distance between the two primaries is smaller than one but close to it. In this case the Legendre expansion of the potential converges slowly and many terms of the series must be taken into consideration.
- The potential of the Earth is developed in ellipsoidal harmonics, and the mathematical tools required, which are the generation of Lame’s functions and the relationship between rectangular and ellipsoidal coordinates, are compiled. Brief reference is made to a procedure utilized for the determination of the gravity coefficients in the expansion ot the geopotential in ellipsoidal harmonics when precise satellite tracking data is available. With the aim of carrying out a numerical integration of the Lagrangian equations of planetary motion, the functional dependence of the disturbing earth potential on the orbital elements for elliptic motion is given. In particular, formulae for the partial derivatives of the disturbing potential with respect to the orbital elements are derived, thus making possible the numerical calculation of these partial derivatives from orbital elements.
- Prolegomena – Classical Mechanics in a NutshellThe N-body ProblemThe Reduced Two-Body ProblemA Perturbed Two-Body ProblemRe-examining the ObviousEpilogue to the ChapterReferences
- Solar sails are a type of propulsion that uses solar radiation pressure to generate acceleration. The fundamental goal for any solar sail design is to provide a large and flat reflective film which requires a minimum of structural support mass. This research takes into account the non-sphericity of the central body, the perturbation of the third body and the solar radiation pressure to analyze the behavior of the orbit of a spacecraft when it has a solar sail around Mercury. We present an approach where we plot maps to analyze frozen orbits with longer lifetimes around Mercury. A set of initial conditions, which may contribute with the scientific missions planned to visit the planet Mercury in the next few years, are presented. Frozen orbits were found, i.e., orbits with smaller variation of the orbital elements. An approach is also presented to analyze the effect of the non-sphericity of Mercury on the motion of the spacecraft. In addition, the J2 and J3 zonal terms are also considered, as well as the C22 sectorial term.
- This paper deals with a new formulation of the creep tide theory (Ferraz-Mello in Celest Mech Dyn Astron 116:109, 2013—Paper I) and with the tidal dissipation predicted by the theory in the case of stiff bodies whose rotation is not synchronous but is oscillating around the synchronous state with a period equal to the orbital period. We show that the tidally forced libration influences the amount of energy dissipated in the body and the average perturbation of the orbital elements. This influence depends on the libration amplitude and is generally neglected in the study of planetary satellites. However, they may be responsible for a 27% increase in the dissipation of Enceladus. The relaxation factor necessary to explain the observed dissipation of Enceladus (\(\gamma =1.2{-}3.8\times 10^{-7}\ \mathrm{s}^{-1}\)) has the expected order of magnitude for planetary satellites and corresponds to the viscosity \(0.6{-}1.9 \times 10^{14}\) Pa s, which is in reasonable agreement with the value recently estimated by Efroimsky (Icarus 300:223, 2018) (\(0.24 \times 10^{14}\) Pa s) and with the value adopted by Roberts and Nimmo (Icarus 194:675, 2008) for the viscosity of the ice shell (\(10^{13}{-}10^{14}\) Pa s). For comparison purposes, the results are extended also to the case of Mimas and are consistent with the negligible dissipation and the absence of observed tectonic activity. The corrections of some mistakes and typos of paper II (Ferraz-Mello in Celest Mech Dyn Astron 122:359, 2015) are included at the end of the paper.
- Dissipation in a tidally perturbed librating body differs from that in a steadily spinning rotator. First, libration changes the spectral distribution of tidal damping across the tidal modes, as compared to the steady-spin case. This adds both to the dissipation rate and the tidal torque. Second, while a non-librating rotator experiences alternating deformation only due to the potential force exerted on it by the perturber, a librating body is also subject to a toroidal force proportional to the angular acceleration. Third, while the centrifugal force in a steadily spinning body renders only a permanent deformation, in a librating body this force contains two alternating components $-$ one purely radial, another a degree-2 potential force. Both contribute to heating, as well as to the tidal torque and potential (and thereby to the orbital evolution). We build a formalism to describe dissipation of small-amplitude libration in longitude. This formalism incorporates a linear rheological law defining the response of the rotator's material to forcing. While the developed formalism can work with an arbitrary linear rheology, we consider a simple example of a Maxwell material. We find that, independent of the rheology, the forced libration in longitude can provide a considerable and even leading input in the tidal heating. Based on the observed parameters, this input amounts to 52% in Phobos, 33% in Mimas, 12% in Enceladus, and 96% in Epimetheus. This supports the hypothesis by Makarov & Efroimsky (2014) that the additional tidal damping due to the forced libration may have participated in the early heating of some of the large moons. E.g., a moon could have been chipped by collisions $-$ so it acquired a higher triaxiality and a higher forced-libration magnitude and, consequently, a higher heating rate. After the moon warms up, its triaxiality reduces, and so does the tidal heating
- This paper investigates the motion of two infinitesimal masses on the location and stability of the equilibrium points in Robe’s restricted problem of 2 + 2 bodies with the bigger primary a Roche ellipsoid and the smaller a triaxial body. We suppose the bigger primary of mass m1 to be filled with a homogeneous incompressible fluid of density ρ1. The third and the fourth bodies (of mass m3 and m4 respectively) are small solid spheres of density ρ3 and ρ4 respectively inside the ellipsoid, with the assumption that the mass and the radius of the third and the fourth body are infinitesimal. We assume that m2 is describing a circle around m1. The masses m3 and m4 mutually attract each other, do not influence the motion of m1 and m2 but are influenced by them. We have taken into consideration all the three components of the pressure field in deriving the expression for the buoyancy force viz (i) due to the own gravitational field of the fluid (ii) that originating in the attraction of m2 (iii) that arising from the centrifugal force. In this paper, equilibrium solutions of m3 and m4 and their linear stability are analyzed.
- An implementation of the ideal frame formulation of perturbed Keplerian motion is presented which only requires the integration of a differential system of dimension 7, contrary to the 8 variables traditionally integrated with this approach. The new formulation is based on the integration of a scaled version of the Eulerian set of redundant parameters, and slightly improves runtime performance with respect to the 8 dimensional case while retaining comparable accuracy.
- In extending the analysis of the four secular resonances between close orbits in Li and Christou (Celest Mech Dyn Astron 125:133-160, 2016) (Paper I), we generalise the semianalytical model so that it applies to both prograde and retrograde orbits with a one to one map between the resonances in the two regimes. We propose the general form of the critical angle to be a linear combination of apsidal and nodal differences between the two orbits $ b_1 \Delta \varpi + b_2 \Delta \Omega $, forming a collection of secular resonances in which the ones studied in Paper I are among the strongest. Test of the model in the orbital vicinity of massive satellites with physical and orbital parameters similar to those of the irregular satellites Himalia at Jupiter and Phoebe at Saturn shows that $> 20\%$ and $> 40\%$ of phase space is affected by these resonances, respectively. The survivability of the resonances is confirmed using numerical integration of the full Newtonian equations of motion. We observe that the lowest order resonances with $b_1+|b_2|\le 3$ persist, while even higher order resonances, up to $b_1+|b_2|\ge 7$, survive. Depending on the mass, between $10\%-60\%$ of the integrated test particles are captured in these secular resonances, in agreement with the phase space analysis in the semianalytical model.
- Hamiltonian formalisms provide powerful tools for the computation of approximate analytic solutions of the Einstein field equations. The post-Newtonian computations of the explicit analytic dynamics and motion of compact binaries are discussed within the most often applied Arnowitt–Deser–Misner formalism. The obtention of autonomous Hamiltonians is achieved by the transition to Routhians. Order reduction of higher derivative Hamiltonians results in standard Hamiltonians. Tetrad representation of general relativity is introduced for the tackling of compact binaries with spinning components. Configurations are treated where the absolute values of the spin vectors can be considered constant. Compact objects are modeled by use of Dirac delta functions and their derivatives. Consistency is achieved through transition to d-dimensional space and application of dimensional regularization. At the fourth post-Newtonian level, tail contributions to the binding energy show up. The conservative spin-dependent dynamics finds explicit presentation in Hamiltonian form through next-to-next-to-leading-order spin–orbit and spin1–spin2 couplings and to leading-order in the cubic and quartic in spin interactions. The radiation reaction dynamics is presented explicitly through the third-and-half post-Newtonian order for spinless objects, and, for spinning bodies, to leading-order in the spin–orbit and spin1–spin2 couplings. The most important historical issues get pointed out.
- The desire to understand the orbits of the planets has a history as long as that of mankind. How and why the planets orbit around the sun are questions in two categories. One focuses on geometry and the other on physics. However, without knowing the answer to why, the how may not be answered theoretically, with the exception made by astronomical genius Kepler. After Newton’s second law, all three Kepler laws may be derived theoretically.
- This paper deals with the existence and stability of the libration points in the restricted-three body problem when the smaller primary is a finite straight segment and bigger primary is an oblate spheroid. We have observed that five libration points do exist in this problem, out of which three are collinear and two are non-collinear with the primaries. The collinear libration points are unstable for all values of mass parameter /u and the non-collinear libration points are stable if jU < /Jc, where jU is a critical value of mass parameter /u. The range of stability depends length of the finite straight segment and oblateness factor.
- Low Earth orbit (LEO) constellations have potentialities to augment global navigation satellite systems for better service performance. The prerequisite is to provide the broadcast ephemerides that meet the accuracy requirement for navigation and positioning. In this study, the Kepler ephemeris model is chosen as the basis of LEO broadcast ephemeris design for backward compatibility and simplicity. To eliminate the singularity caused by the smaller eccentricity of LEO satellites compared to MEO satellites, non-singular elements are introduced for curve fitting of parameters and then transformed to Kepler elements to assure the algorithm of ephemeris computation remains unchanged for the user. We analyze the variation characteristics of LEO orbital elements and establish suitable broadcast ephemeris models considering fit accuracy, number of parameters, fit interval, and orbital altitude. The results of the fit accuracy for different fit intervals and orbital altitudes suggest that the optimal parameter selections are \((Crs3,Crc3)\), \((Crs3,Crc3, \, \dot{a},\dot{n})\) and \(\left( {Crs3,Crc3, \, \dot{a},\dot{n}, \, \ddot{i},\ddot{a}} \right)\), i.e., adding two, four or six parameters to the GPS 16-parameter ephemeris. When adding four parameters, the fit accuracy can be improved by about one order of magnitude compared to the GPS 16-parameter ephemeris model, and fit errors of less than 10 cm can be achieved with 20-min fit interval for a 400–1400 km orbital altitude. In addition, the effects of the number of parameters, fit interval, and orbit altitude on fit accuracy are discussed in detail. The validation with four LEO satellites in orbit also confirms the effectiveness of proposed models.
- Following Giacaglia (1967), in Sect. 2 we have developed equation of motion of the Sitnikov restricted N+1-body problem in elliptic case. We assumed that the primaries are at the vertices of a regular N-gon so the distances of the primaries from center of mass are time depending. In Sect. 3 we have linearized the equation of motion to obtain the Hill’s type equation and then find the approximate solution. In Sects. 4 and 5 the series solutions of the Sitnikov restricted N+1-body problem have been developed by the method of Lindstedt-Poincaré and iteration of Green’s function respectively. In Sect. 6 the two series solutions have been compared graphically by putting N=2, 3 and 4 for different eccentricity.
- The planets have been targets of exploration since the launching of Sputnik in 1957. Space probes have been sent to each of the planets known in antiquity and to all but one of the known planets. In this exercise, we explore first the properties of planetary orbits, then how to calculate their coordinates on the sky from their orbital elements, and, finally, how to design the orbit of a spacecraft to travel to a target planet.
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