A new approach to the librational solution in the Ideal Resonance Problem

The librational motion of the Ideal Resonance Problem (Garfinkel, 1966, Jupp, 1969) is treated through an initialnon-canonical transformation which, however, leaves the equations of motion in a quasi-canonical form, with Hamiltonian expressed in standard trigonometric functions amenable to traditional averaging techniques. The perturbed solutions, similarly expressed intrigonometric near-identity transformations, and their frequencies can be found to arbitrary order, with the elliptic integrals expected of the system introduced only in a final explicit quadrature for a Kepler-type equation in the angular variable. The specific transformations and resulting equations of motion are introduced, and explicit solutions for the original variables are found to second order, with mean motion accurate to fifth. Limitation of the present solution to the librational region, the extension of that solution to higher order, and observations on the form of the associated Hamiltonian are also discussed.     

Regularization in the Ideal Resonance Problem

The Ideal Resonance Problem, defined by the HamiltonianF=B(y)+2A (y) sin² x, 1, has been solved in Garfinkelet al. (1971). There the solution has beenregularized by means of a special function _j , introduced into the new HamiltonianF, under the tacit assumption thatA anB¨' are of order unity.This assumption, violated in some applications of the theory, is replaced here by the weaker assumption ofnormality, which admits zeros ofA andB inshallow resonance. It is shown here that these zeros generate singularities, which can be suppressed if _j is suitably redefined.With the modified _j , and with the assumption of normality, the solution is regularized for all values ofB, B¨', andA. As in the previous paper, the solution isglobal, including asymptotically the classical limit withB as a divisor of O(1).A regularized first-order aloorithm is constructed here as an illustration and a check.Presented at the XXII International Congress of I.A.F., Brussels, Belgium, Sept. 20, 1971.     

An extended Ideal Resonance Problem

An Extended Resonance Problem is defined by the Hamiltonian, $$F = B(y) + 2\mu ^2 A(y)[\sin x + \lambda (y)]^2 \mu<< 1,\lambda = O(\mu ).$$ It is noted here that the phase-plane trajectories exhibit adouble libration, enclosing two centers, for the initial conditions of motion satisfying the inequality $$1 - |\lambda |< |\alpha |< 1 + |\lambda |,$$ where α is the usualresonance parameter. A first order solution for the case of double libration is constructed here by a generalization of the procedure previously used in solving the Ideal Resonance Problem with λ=0. The solution furnishes a reference orbit for a Perturbed Ideal Problem if a double libration occurs as a result of perturbations.     

On The Ideal Resonance Problem

The Ideal Resonance Problem in its normal form is defined by the Hamiltonian (1) $$F = A (y) + 2B (y) sin^2 x$$ with (2) $$A = 0(1),B = 0(\varepsilon )$$ where ? is a small parameter, andx andy a pair of canonically conjugate variables. A solution to 0(?^1/2) has been obtained by Garfinkel (1966) and Jupp (1969). An extension of the solution to 0(?) is now in progress in two papers ([Garfinkel and Williams] and [Hori and Garfinkel]), using the von Zeipel and the Hori-Lie perturbation methods, respectively. In the latter method, the unperturbed motion is that of a simple pendulum. The character of the motion depends on the value of theresonance parameter α, defined by (3) $$\alpha = - A\prime /|4A\prime \prime B\prime |^{1/2} $$ forx=0. We are concerned here withdeep resonance, (4) $$\alpha< \varepsilon ^{ - 1/4} ,$$ where the classical solution with a critical divisor is not admissible. The solution of the perturbed problem would provide a theoretical framework for an attack on a problem of resonance in celestial mechanics, if the latter is reducible to the Ideal form: The process of reduction involves the following steps: (1) the ration ₁/n₂ of the natural frequencies of the motion generates a sequence. (5) $$n_1 /n_2 \sim \left\{ {Pi/qi} \right\},i = 1, 2 ...$$ of theconvergents of the correspondingcontinued fraction, (2) for a giveni, the class ofresonant terms is defined, and all non-resonant periodic terms are eliminated from the Hamiltonian by a canonical transformation, (3) thedominant resonant term and itscritical argument are calculated, (4) the number of degrees of freedom is reduced by unity by means of a canonical transformation that converts the critical argument into an angular variable of the new Hamiltonian, (5) the resonance parameter α (i) corresponding to the dominant term is then calculated, (6) a search for deep resonant terms is carried out by testing the condition (4) for the function α(i), (7) if there is only one deep resonant term, and if it strongly dominates the remaining periodic terms of the Hamiltonian, the problem is reducible to the Ideal form.     

Ideal Resonance Problem: the Post-Post-Pendulum Approximation

Explicit construction of the solutions of the Hamiltonian system given by H = H ₀(J) – A(J) cos (ideal resonance problem), two orders of approximation beyond the well-known pendulum approximation. The given solutions are valid for libration amplitudes of order . The procedure used is extended to allow the construction of the solutions of Hamiltonians with perturbations involving two degrees of freedom; the post-pendulum solution of an example of this kind is constructed.     

A second-order solution of the Ideal Resonance Problem by Lie series

A second-order libration solution of theIdeal Resonance Problem is construeted using a Lie-series perturbation technique. The Ideal Resonance Problem is characterized by the equations $$\begin{gathered} - F = B(x) + 2\mu ^2 A(x)sin^2 y, \hfill \\ \dot x = - Fy,\dot y = Fx, \hfill \\ \end{gathered} $$ together with the property thatB _x vanishes for some value ofx. Explicit expressions forx andy are given in terms of the mean elements; and it is shown how the initial-value problem is solved. The solution is primarily intended for the libration region, but it is shown how, by means of a substitution device, the solution can be extended to the deep circulation regime. The method does not, however, admit a solution very close to the separatrix. Formulae for the mean value ofx and the period of libration are furnished.     

Comparison of the classical and the global solutions of the Ideal Resonance Problem

The Ideal Resonance Problem is defined by the Hamiltonian $$F = B(y) + 2\varepsilon A(y) \sin ^2 x,\varepsilon \ll 1.$$ The classical solution of the Problem, expanded in powers of ε, carries the derivativeB′ as a divisor and is, therefore, singular at the zero ofB′, associated with resonance. With α denoting theresonance parameter, defined by $$\alpha \equiv - B'/|4AB''|^{1/2} \mu ,\mu = \varepsilon ^{1/2} ,$$ it is shown here that the classical solution is valid only for $$\alpha ^2 \geqslant 0(1/\mu ).$$ In contrast, the global solution (Garfinkelet al., 1971), expanded in powers ofμ=ε^1/2, removes the classical singularity atB′=0, and is valid for all α. It is also shown here that the classical solution is an asymptotic approximation, for largeα ², of the global solution expanded in powers ofα ^?2. This result leads to simplified expressions for resonancewidth and resonantamplification. The two solutions are compared with regard to their general behavior and their accuracy. It is noted that the global solution represents a perturbed simple pendulum, while the classical solution is the limiting case of a pendulum in a state offast circulation.     

The Ideal Resonance Problem: A comparison of the solutions expressed in terms of mean elements and in terms of initial conditions

In an earlier publication (Jupp, 1972), a solution of the Ideal Resonance Problem is exhibited explicitly in terms of the mean elements; to second order in the small parameter in the case of libration, and to first order in the case of deep circulation. Both representations possess a singularity when the mean modulus of the Jacobi elliptic functions is unity; this corresponds to the separatrix of the phase plane of the dynamical system.It is shown here that, provided particular coefficients associated with the problem satisfy specific relations, the singularity is removed, and the resulting solution is applicable throughout the deep resonance region.The solution is then expressed in terms of general initial conditions. Again, in general, the solution has a singularity associated closely with the limiting motion, and the circulation part of the solution is restricted to deep circulation. It is shown that when the previously-mentioned coefficients satisfy particular constraints, the singularity is removed. In addition, with the same constraints, the deep-circulation solution is applicable throughout the circulation region. It is of interest that these constraints are quite different from those associated with the mean, element formulation.     

A geometric method to determine the distances of galaxies

Es wird die Möglichkeit diskutiert, die Größenverteilungsfunktion von Kugelsternhaufen für die Entfernungsbestimmung von Galaxien zu nutzen.     

Instability of Libration Points and Resonance Phenomena in the Photogravitational Elliptic Restricted Three-Body Problem

The stability of collinear and triangular libration points is investigated in the photogravitational elliptic restricted three-body problem, in which two primary bodies emit light energy simultaneously. The conditions of stability of the collinear and triangular libration points are obtained based on a linearized set of equations of perturbed motion for various values of the eccentricity of the Keplerian orbits and the mass ratio of the primary bodies. The maximal numerical value is defined for the eccentricity at which a stable libration point can still exist. It is demonstrated how the parametric resonance causes an instability of collinear and triangular libration points; the evolution of the origination of the instability zones is traced. The minimal eccentricity value is found at which zones of instability of triangular libration points arise.     

A geometric property of Hill's curves

The following is proved in this note: If we construct a circle passing through a given primary in the planar circular restricted three-body problem, center of which is the remaining primary, then the minor arc of this circle with endpoints represented by the triangular libration points represents-when the given primary is excluded-locus of all the points on the Hill's curves that are the least distant points from the given primary.     

A new solution to the Main Problem of Lunar Theory

A new solution for the Main Problem on Lunar Theory is given. This solution maintains the advantages of an analytical solution and should be more accurate than previous analytical or numerical solutions. It contains the effects of the mass ratios which are often neglected in the definition of the Main Problem.     

A simple approach to the high-redshift sub-millimeter galaxies

We aim at understanding the statistical properties of luminous sub-millimeter (submm) galaxies (SMGs) in the context of cosmological structure formation. By utilizing a cosmological N-body simulation to calculate the distribution of dark halos in the Universe, we consider the dust enrichment in individual halos by Type II supernovae (SNe II). The SN II rate is estimated under a star formation activity which is assumed to occur on a dynamical timescale in the dark matter potential. Our simple framework successfully explains the luminosity function, the typical star formation rate, and the typical dust mass of an observational SMG sample at z～3. We also examine the clustering properties of SMGs, since a positive cross correlation between SMGs and Lyα emitters (LAEs) is indeed observed by a recent observation. In the simulation, we select SMGs by FIR dust luminosity >10¹² L _⊙, while LAEs are chosen such that the age and the virial mass are consistent with the observed LAE properties. The SMGs and LAEs selected in this way show a spatial cross correlation whose strength is consistent with the observation. This confirms that the SMGs really trace the most clustered regions at z～3 and that their luminosities can be explained by the dust accumulation as a result of their star formation activities. We extend our prediction to higher redshifts, finding that a statistical sample of submm galaxies at z≥6 can be obtained by ALMA with a 100 arcmin² survey. With the same survey, a few submm galaxies at z～10 may be detected.     

A new analytic approach to the Sitnikov problem

A new analytic approach to the solution of the Sitnikov Problem is introduced. It is valid for bounded small amplitude solutions (z _max = 0.20) (in dimensionless variables) and eccentricities of the primary bodies in the interval (–0.4 < e < 0.4). First solutions are searched for the limiting case of very small amplitudes for which it is possible to linearize the problem. The solution for this linear equation with a time dependent periodic coefficient is written up to the third order in the primaries eccentricity. After that the lowest order nonlinear amplitude contribution (being of order z ³) is dealt with as perturbation to the linear solution. We first introduce a transformation which reduces the linear part to a harmonic oscillator type equation. Then two near integrals for the nonlinear problem are derived in action angle notation and an analytic expression for the solution z(t) is derived from them. The so found analytic solution is compared to results obtained from numeric integration of the exact equation of motion and is found to be in very good agreement. CERN SL/AP     

A Multi-Angle Averaging Theorem Applied to the J2 Problem

A particular multi-angle averaging theorem for systems admitting a finite Fourier expansion of the field is presented, together with its application to the problem of motion around an oblate planet (the J₂ problem), in harmonic oscillator formulation. This method of approximate integration has the advantage of working with (close on) directly measurable elements. This revised version was published online in July 2006 with corrections to the Cover Date.     

A sheet-current approach to coronal-interplanetary modeling

The most pertinent effect of the currents in the coronal-interplanetary space is their alteration of the magnetic topology to form configurations of open field lines. The important currents seem to be those in the neighborhoods of the interfaces between closed and open field lines or between oppositely directed open field lines in the coronal helmet-streamer structures. Thus, the coronal-interplanetary space may be regarded as being partitioned by current-sheets into several piecewise current-free regions. These current sheets overlie the photospheric neutral lines, where the vertical component of the magnetic field reverses its polarity on the solar surface. But, their locations and strengths are determined by force balance between the magnetic field and the gas pressure in the coronal-interplanetary space. Since the pressure depends on the flow velocity of the solar wind and the solar wind channels along magnetic flux tubes, there is a strong magnetohydrodynamic coupling between the magnetic field and the solar wind. The sheetcurrent approach presented in this paper seems to be a reasonable way to account for this complicated interaction.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

A realistic approach to magnetic evolution

An improved scheme for measuring and analyzing the magnetic field of flares is presented in general outline form. The techniques employed during the last solar cycle are reviewed very briefly. The point is made that those techniques were inadequate due to fundamental limitations in the data. A realistic scheme for acquiring the necessary data is then proposed. The scheme is realistic in that it makes only those assumptions which appear to be indeed valid, and it presupposes only instrumental techniques which are presently available. It is suggested that, with this improved data, the structure and evolution of the pre-flare magnetic field can be calculated with sufficient accuracy to form the basis of theoretical models.This research supported by Company funds of The Aerospace Corporation.     

Polygonal terrains on Mars: A contribution to their geometric and topological characterization

Detailed examination of large extensions of polygonal terrains on the surface of Mars and extraction of some characteristic geometric and topological parameters is made possible by the application of image analysis methods to scenes of the Martian surface acquired from orbit. This is illustrated by the analysis of a set of diverse Martian networks, clearly visible in MOC/MGS images with high spatial resolution. It is shown that these networks present, in average, a hexagonal habit, and that they verify two classic laws relative to 2D random networks (those of Lewis and Aboav-Weaire). This research can, through the quantified analysis of the differences and similarities between networks, lead into a much better understanding of the origin and dynamics of this type of features.     

Approximate Analytic Solutions of the Averaged Three-Body Problem at First-Order Resonance with Large Oblateness of the Central Planet

For the general spatial planetary three-body problem at first-order mean motion resonance under the large oblateness of the central planet, the analytic solutions of the averaged motion are obtained with the help of the Weierstrass functions accurate to the third-degree terms in the satellites' eccentricities and inclinations. The behavior of solutions is investigated on the phase plane.This revised version was published online in October 2005 with corrections to the Cover Date.     

A rigorous Hamiltonian approach to the rotation of elastic bodies

When the problem of the rotation of a non-rigid body is studied, the usual procedure consists of adding perturbations to the Hamiltonian of the rigid solid. In some cases, as occurs with the centrifugal deformation, the new perturbations contains potentials which depend on the velocity, but usually one alter neither the definition of the canonical variables nor the method for obtaining the Hamiltonian. Although this procedure gives good estimates and its formulation is simpler, it is incorrect from a theoretical point of view.In this paper we rigorously develop a Hamiltonian formulation of the problem, considering potentials that depend on the velocity. Thus the differences between the two procedures are clearly shown, giving special emphasis to the case of the elastic Earth, for which we show that the differences obtained cannot be ignored within the accuracy limits at present required.