Apsidal Angle Research Articles

On the non-dissipative orbital evolution of a binary system comprising non-compact components with misaligned spin and orbital angular momenta

Abstract In this Paper we determine the non-dissipative tidal evolution of a close binary system with an arbitrary eccentricity in which the spin angular momenta of both components is misaligned with the orbital angular momentum. We focus on the situation where the orbital angular momentum dominates the spin angular momenta and so remains at small inclination to the conserved total angular momentum. Torques arising from rotational distortion and tidal distortion taking account of Coriolis forces are included. This extends the previous work of Ivanov & Papaloizou relaxing the limitation resulting from the assumption that one of the components is compact and has zero spin angular momentum. Unlike the above study, the evolution of spin-orbit inclination angles is driven by both types of torque. We develop a simple analytic theory describing the evolution of orbital angles and compare it with direct numerical simulations. We find that the tidal torque prevails near ’critical curves’ in parameter space where the time-averaged apsidal precession rate is close to zero. In the limit of small spin, these curves exist only for systems that have at least one component with retrograde rotation. As in our previous work, we find solutions close to these curves for which the apsidal angle librates. As noted there, this could result in oscillation between prograde and retrograde states. We consider the application of our approach to systems with parameters similar to those of the misaligned binary DI Herc.

An interesting class of exact solutions to Binet's equation, with references to Newton's and Einstein's theories of gravity

Abstract To adequately describe the motion of a particle in a central force field, one has to specify the apsidal angle (α). The particle's trajectory can only be truly periodic (closed), if α/π is a rational number. For example, F∼1/r^2 then α=π, and the trajectory is an ellipse. This is the simplest case when the trajectory equation (Binet's equation) is exactly solvable. In this article, bounded motion in a two-component central potential field V(r)=-a_1/r-a_2/r^2 will be examined and illustrated. This case is more complex, but the corresponding Binet's equation is still exactly solvable. Moreover, the perturbational solution to the relativistic trajectory equation is, in fact, the exact solution for a force field of the same type. It means that the relativistic force F∼1/r^4 is formally replaced with an effective F∼1/r^3 force. As will be shown, this differently interpretable solution gives a very accurate prediction to the anomalous shift of Mercury's perihelion.

Periodic perturbations of central force problems and an application to a restricted 3-body problem

We consider a perturbation of a central force problem of the formx¨=V′(|x|)x|x|+ε∇xU(t,x),x∈R2∖{0}, where ε∈R is a small parameter, V:(0,+∞)→R and U:R×(R2∖{0})→R are smooth functions, and U is τ-periodic in the first variable. Based on the introduction of suitable time-maps (the radial period and the apsidal angle) for the unperturbed problem (ε=0) and of an associated non-degeneracy condition, we apply an higher-dimensional version of the Poincaré–Birkhoff fixed point theorem to prove the existence of non-circular τ-periodic solutions bifurcating from invariant tori at ε=0. We then prove that this non-degeneracy condition is satisfied for some concrete examples of physical interest (including the homogeneous potential V(r)=κ/rα for α∈(−∞,2)∖{−2,0,1}). Finally, an application is given to a restricted 3-body problem with a non-Newtonian interaction.

On precession of Lagrange’s top

The article describes the results obtained for the upper and lower bounds (estimates) for the apsidal angle (precession angle) in the theory of the motion of the heavy symmetrical solid body about fixed point (Lagrange’s case) for arbitrary initial conditions and parameters of the body. All regions of initial conditions is divided into two sets. In the first set there is a direct precession of the top, in the second set there is a retrograde precession of the top.

On the non-dissipative tidal evolution of the misalignment between spin and orbital angular momenta

ABSTRACT We extend our previous work on the evolution of close binary systems with misaligned orbital and spin angular momenta resulting from non-dissipative tidal interaction to include all physical effects contributing to apsidal motion. In addition to tidal distortion of the primary by the compact secondary, these include relativistic Einstein precession and the rotational distortion of the primary. The influence of the precession of the line of nodes is included. The dependence of the tidal torque on the apsidal angle $\hat{\varpi }$ couples the apsidal motion to the rate of evolution of the misalignment angle β which is found to oscillate. We provide analytical estimates for the oscillation amplitude Δβ over a wide range of parameter space confirmed by numerical integrations. This is found to be more significant near critical curves on which ${\mathrm{d}}{\hat{\varpi }} /{\mathrm{d}}t=0$ for a specified β. We find that to obtain 0.1 < Δβ < ∼1, the mass ratio q > ∼1, the initial eccentricity should be modest, $\cos \beta \lt 1/\sqrt{5},$ with cos β < 0 corresponding to retrograde rotation, initially, and the primary rotation rate should be sufficiently large. The extended discussion of apsidal motion and its coupled evolution to the misalignment angle given here has potential applications to close binaries with anomalous apsidal motion as well as transiting exoplanets such as warm Jupiters.

The geometry of isochrone orbits: from Archimedes’ parabolae to Kepler’s third law

In classical mechanics, the Kepler potential and the Harmonic potential share the following remarkable property: in either of these potentials, a bound test particle orbits with a radial period that is independent of its angular momentum. For this reason, the Kepler and Harmonic potentials are called \it{isochrone}. In this paper, we solve the following general problem: are there any other isochrone potentials, and if so, what kind of orbits do they contain? To answer these questions, we adopt a geometrical point of view initiated by Michel H\'enon in 1959, in order to explore and classify exhaustively the set of isochrone potentials and isochrone orbits. In particular, we provide a geometric generalization of Kepler's third law, and give a similar law for the apsidal angle, for any isochrone orbit. We also relate the set of isochrone orbits to the set of parabolae in the plane under linear transformations, and use this to derive an analytical parameterization of any isochrone orbit. Along the way we compare our results to known ones, pinpoint some interesting details of this mathematical physics problem, and argue that our geometrical methods can be exported to more generic orbits in potential theory.

Geodesics and periodic orbits in Kehagias-Sfetsos black holes in deformed Hor̆ava-Lifshitz gravity

The motion of a massive test particle around a Kehagias-Sfetsos black hole in deformed Ho\u{r}ava-Lifshitz gravity is studied. Employing the effective potential, the marginally bound orbits and the innermost stable circular orbits are analyzed. For the marginally bound orbits, their radius and angular momentum decrease with the parameter $\omega$ of the gravity. For the innermost stable circular orbits, the energy and angular momentum also decrease with $\omega$. Based on these results, we investigate the periodic orbits in the Kehagias-Sfetsos black holes. It is found that the apsidal angle parameter increases with the particle energy, while decreases with the angular momentum. Moreover, compared to the Schwarzschild black hole, the periodic orbits in Kehagias-Sfetsos black holes always have lower energy. These results provide us a possible way to distinguish the Kehagias-Sfetsos black hole in deformed Ho\u{r}ava-Lifshitz gravity from the Schwarzschild black hole.

The Monotonicity of the Apsidal Angle Using the Theory of Potential Oscillators

In a central force system the angle between two successive passages of a body through pericenters is called the apsidal angle. In this paper we prove that for central forces of the form $$f(r)\sim \lambda r^{-\,(\alpha +1)}$$ with $$\alpha <2$$ the apsidal angle is a monotonous function of the energy, or equivalently of the orbital eccentricity.

The monotonicity of the apsidal angle in power-law potential systems

In a central force system the apsidal angle is the angle at the centre of force between two consecutive apsides and measures the precession rate of the line of apsis. The apsidal angle has applications in different fields and Newton's apsidal precession theorem has been extensively studied by astronomers, physicist and mathematicians. The perihelion precession of Mercury, the dynamics of galaxies, the vortex dynamics, the JWKB quantisation condition are some examples where the apsidal angle is of interest. In case of eccentric orbits and forces far from inverse square, numerical investigations provide evidence of the monotonicity of the apsidal angle with respect to the orbit parameters, such as the orbit eccentricity. However, no proof of this statement is available. In this paper central force systems with f(r)∼μr−(α+1) are considered. We prove that for any −2<α<1 the apsidal angle is a monotonic function of the orbital eccentricity, or equivalently of the angular momentum. As a corollary, the conjecture stating the absence of isolated cases of zero precession is proved.

A study of the apsidal angle and a proof of monotonicity in the logarithmic potential case

This paper concerns the behaviour of the apsidal angle for orbits of central force system with homogeneous potential of degree −2⩽α⩽1 and logarithmic potential. We derive a formula for the apsidal angle as a fixed end-points integral and we study the derivative of the apsidal angle with respect to the angular momentum ℓ. The monotonicity of the apsidal angle as function of ℓ is discussed and it is proved in the logarithmic potential case.

A study of the orbits of the logarithmic potential for galaxies

The logarithmic potential is of great interest and relevance in the study of the dynamics of galaxies. Some small corrections to the work of Contopoulos & Seimenis (1990) who used the method of Prendergast (1982) to find periodic orbits and bifurcations within such a potential are presented. The solution of the orbital radial equation for the purely radial logarithmic potential is then considered using the p-ellipse (precessing ellipse) method pioneered by Struck (2006). This differential orbital equation is a special case of the generalized Burgers equation. The apsidal angle is also determined, both numerically as well as analytically by means of the Lambert W and the Polylogarithm functions. The use of these functions in computing the gravitational lensing produced by logarithmic potentials is discussed.

On the habitability of the OGLE-2006-BLG-109L planetary system

We investigate the dynamics of putative Earth-mass planets in the habitable zone (HZ) of the extrasolar planetary system OGLE-2006-BLG-109L, a close analogue of the Solar system. Our work is inspired by the work of Malhotra & Minton. Using the linear Laplace–Lagrange theory, they identified a strong secular resonance that may excite large eccentricity of orbits in the HZ. However, due to uncertain or unconstrained orbital parameters, the subsystem of Jupiters may be found in a dynamically active region of the phase space spanned by low-order mean-motion resonances. To generalize this secular model, we construct a semi-analytical averaging method in terms of the restricted problem. The secular orbits of large planets are approximated by numerically averaged osculating elements. They are used to calculate the mean orbits of terrestrial planets by means of a high-order analytic secular theory developed in our previous works. We found regions in the parameter space of the problem in which stable, quasi-circular orbits in the HZ are permitted. The excitation of eccentricity in the HZ strongly depends on the apsidal angle of jovian orbits. For some combinations of that angle, eccentricities and semimajor axes consistent with the observations, a terrestrial planet may survive in low eccentric orbits. We also study the effect of post-Newtonian gravity correction on the innermost secular resonance.

An extension of Newton's apsidal precession theorem

Newton's apsidal precession theorem in Proposition 45 of Book I of the ‘Principia’ has great mathematical, physical, astronomical and historical interest. The lunar theory and the precession of the perihelion of the planet Mercury are but two examples of the applications of this theorem. We have examined the precession of orbits under varying force laws as measured by the apsidal angle θ(N, e), where N is the index for the centripetal force law, for varying eccentricity e. The paper derives a general function for the apsidal angle, dependent only on e and N as the potential is spherically symmetric. Further, we explore approximate ways of the solution of this equation, in the neighbourhood of N= 2 which happens to be the case of greatest historical interest. Exact solutions are derived where they are possible. The first derivatives ∂θ/∂N and ∂θ/∂h[where h(N, e) is the angular momentum] are analytically expressed in the neighbourhood of N= 2 (case of the inverse square law). The value of ∂θ/∂N is computed numerically as well for 1 ≤N < 3. The resulting integrals are interesting improper integrals with singularities at both limits. Some of the integrals, especially for N= 2, can be given in closed form in terms of generalized hypergeometric functions which are reducible in terms of algebraic and logarithmic functions. No evidence was found for isolated cases of zero precession as e was increased. The N= 1 case of the logarithmic potential is also briefly discussed in view of its interest for the dynamics of eccentric orbits and its relevance to realistic galaxy models. The possibility of apsidal precession was also examined for a few cases of high-eccentricity asteroids and extrasolar planets. We find that these systems may provide interesting new laboratories for studies of gravity.

The significance of the Mathieu-Hill differential equation for Newton's apsidal precession theorem

Newton's precession theorem in Proposition 45 of Book I of Principia relates a centripetal force of magnitude μrn-3 as a power of the distance from the center to the apsidal angle θ, where θ is the angle between the point of greatest distance and the point of least distance. The formula θ = π/[Formula: see text] is essentially restricted to orbits of small eccentricity. A study of the apsidal angle for appreciable orbital eccentricity leads to an analysis of the differential equation of the orbit. We show that a detailed perturbative approach leads to a Mathieu-Hill-type of inhomogeneous differential equation. The homogeneous and inhomogeneous differential equations of this type occur in many interesting problems across several disciplines. We find that the approximate solution of this equation is the same as an earlier one obtained by a bootstrap perturbative approach. A more thorough analysis of this inhomogeneous differential equation leads to a modified Hill determinant. We show that the roots of this determinant equation can be solved to obtain an accurate solution for the orbit. This approach may be useful even for cases where n deviates noticeably from 1. The derived analytic results were applied to the Moon, Mercury, the asteroid Icarus, and a hypothetical object. We show that the differential equation that occurs in a perturbative relativistic treatment of the perihelion precession of Mercury also leads to a simplified form of the Mathieu-Hill differential equation.PACS Nos.: 95.100.C, 95.10.E, 95.90, and 02.30.H

Bertrand spacetimes

The author determines all spherically symmetric and static Lorentzian spacetimes in which any bounded trajectory is periodic. The author calls such spacetimes 'Bertrand spacetimes', thereby referring to Bertrand's classical theorem which deals with the analogous situation in Newtonian mechanics. Any Bertrand spacetime is characterized by a rational number beta which gives the apsidal angle of all bounded trajectories in fractions of pi . Whereas Newtonian Bertrand potentials exist only for beta =1 (Kepler potential) and for beta =2 (harmonic oscillator potential), general relativity allows one to construct Bertrand spacetimes for any beta . An asymptotically Minkowskian Bertrand spacetime, however, must have beta =1, and a Bertrand spacetime with regular centre must have beta =2. Whereas all asymptotically Minkowskian Bertrand spacetimes violate the weak energy condition, some Bertrand spacetimes with regular centre admit an interpretation as charged perfect fluid solution of Einstein's field equation.

A Proof of the Puiseux - Halphen Inequality in the Theory of Spherical Pendulum Based on an Interesting Identity of S. Ramanujan

The classical problem of finding bounds for the apsidal angle in the case of the spherical pendulum has been considered by Halphen in his classical treatise on 'Fonctions Elliptiques'. His proof is based on certain inequalities among the Elliptic-function constant. We prove these inequalities of Halphen and Puiseux, in a simple way, using an interesting identity of S. Ramanujan; besides, we obtain positive-term series for the quantities involved which may enable one to improve such bounds.

Upper and lower bounds for the apsidal angle in the theory of the heavy symmetrical top

Abstract : A simple method is developed for obtaining upper and lower bounds for the apsidal angle which occurs in the theory of the heavy symmetrical top. This method is employed to give a quick derivation and an improvement of the upper and lower bounds obtained by W. Kohn. An advantage of the present method is the simplification which arises from eliminating the need for contour integration. The sharpness of the bounds is demonstrated, again without contour integration. The particular method in question is illustrated first in the theory of the spherical pendulum. (Author)

On a result of Hadamard concerning the sign of the precession of a heavy symmetrical top

Consider Lagrange's integrable case of the motion of a rigid body about a fixed point, when the center of gravity of the body lies on the polar axis of the spheroid of inertia of the body [l, case II, pp. 216— 249]. In [l, §110, pp. 233-235] one finds An interesting proof that the precession for a complete period has the same sign as uk (which) has been given by Hadamard (see [2]). Hadamard's proof employs the theory of residues of functions of a complex variable. Quite recently (see [3], which cites the recent literature), upper and lower bounds for the apsidal angle in the theory of the heavy symmetrical top have been obtained by an elementary method, not relying upon the theory of residues. This same direct method applies equally well in deducing Hadamard's result. This application will be carried out here, in the notation of [l], for the immediate comparison of the two methods. The argument requires the preliminary evaluation of two simple definite integrals:

Apsidal angles for symmetrical dynamical systems with two degrees of freedom

The bob of a spherical pendulum (or particle on a smooth sphere under gravity) oscillates between two levels, and the change of azimuth in passing from the lower to the higher level, or vice versa, is the same for all such passages during a single motion. Using the language of orbit theory, we shall refer to this change of azimuth as the apsidal angle. It is a function of the two constants of the motion, total energy and angular momentum, but the remarkable fact is that the apsidal angle a always satisfies the inequalities