Inverse problem with two-parametric families of planar orbits

As a possible extension of recent work we study the following version of the inverse problem in dynamics: Given a two-parametric familyf(x, y, b)=c of plane curves, find an autonomous dynamical system for which these curves are orbits.We derive a new linear partial differential equation of the first order for the force componentsX(x, y) andY(x, y) corresponding to the given family. With the aid of this equation we find that, depending on the given functionf, the problem may or may not have a solution. Based on given criteria, we present a full classification of the various cases which may arise.     

Generalization of Szebehely's equation

Szebehely's partial differential equation for the force functionU=U(x,y) which gives rise to a given family of planar orbitsf(x,y)=Constant is generalized to account for velocity-dependent potentials V^*=V^*(x,y, ). The new partial differential equation is quasi-linear and of the first order. An example is given and a comparison is made of the two equations.     

Intrinsic stability of periodic orbits

Families of orbits of a conservative, two degree-of-freedom system are represented by an unsteady velocity field with componentsu(x, y, t) andv(x, y, t). Intrinsic stability properties depend on velocity field divergence and curl, whose dynamical evolution is determined by a matrix Riccati equation. Near equilibrium, divergence-free or irrotational fields are dynamically compatible with the conservative force field. It is shown that a necessary condition for stable periodic orbits is satisfied when the orbitaveraged divergence is zero, which results in bounded normal variations. A sufficient condition for stability is derived from the requirement that tangential variations do not exhibit secular growth.In a steady, divergence-free field, velocity component functionsu(x, y) andv(x, y) may be continuedanalytically from any initial condition, except when velocity is parallel to U or at equilibria. In an unsteady field, the orbit-averaged divergence is zero when the vorticity function is periodic. When such a field exists, initial conditions for stable periodic orbits (i.e., characteristic loci) may be determinedanalytically.     

Solution of the three-dimensional inverse problem

The three dimensional inverse problem for a material point of unit mass, moving in an autonomous conservative field, is solved. Given a two-parametric family of space curvesf(x, y, z)=c ₁,g(x, y, z)=c ₂, it is shown that, in general, no potentialU=U(x, y, z) exists which can give rise to this family. However, if the given functionsf(x, y, z) andg(x, y, z) satisfy certain conditions, the corresponding potentialU(x, y, z), as well as the total energyE=E(f, g) are determined uniquely, apart from a multiplicative and an additive constant.     

A new integration theory for conservative two degree-of-freedom systems

A two degree-of-freedom, conservative system is reduced to a single degree-of-freedom, kinematic system with Hamiltonian integral under the change of independent variable: $$dt = \zeta dt (\zeta = \upsilon _x - \upsilon _y )$$ where ζ is the curl (or vorticity) of the velocity field with cartesian inertial componentsu(x, y, t) andv(x, y, t). In the autonomous case whenu _t=v _t=0, orbits are globally represented by the level curves of an autonomous Hamiltonian functionH(x,y) satisfying a second-order quasilinear partial differential equation (Szebehely's Equation): $$2(H + U)\left( {H_{xx} H_y^2 - 2H_{xy} H_x H_y + H_{yy} H_x^2 } \right) + (H_x U_x + H_y U_y )\left( {H_x^2 + H_y^2 } \right) = 0$$ whereU(x, y) is the autonomous potential function. An inversion of dependent and independent variables reduces this equation to a second-order, ordinary differential equation for a function specifying the orbital curve. The true time variable is recovered by evaluating a quadrature. Fundamental differences exist between this approach and Hamilton-Jacobi theory.     

A direct numerical approach to the Chandrasekhar'sH-functions for arbitrary characteristic functions

The purpose of this paper is to present a numerical technique to directly compute the Chandrasekhar'sH ()-function for anisotropic scattering in terms of the roots of the characteristic equations as well as the quadrature points of a certain degreen employed to approximate the definite integral involved in the basic equation. The principal feature of the algorithm proposed here is a compact computer code to enumerate _n C _m combinations ofn distinct integers {1,...,n} takenm at a time. With these quantities available, the coefficients of the polynomial equation of the characteristics equation can be readily computed for any given characteristic function, so that a standard technique such as the Laguerre method can be applied to find all the roots.It is shown that the results obtained for some representativeH()-functions using the present technique with relatively low-order formula (e.g.,n=7) are sufficiently accurate for all practical purposes.     

A unified presentation of the Voigt functions

This paper aims at presenting a unified study of the Voigt functionsK(x,y) andL(x,y) which play a rather important role in several diverse fields of physics such as astrophysical spectroscopy and the theory of neutron reactions. Explicit expressions for these functions are given in terms of relatively more familiar special functions of one and two variables; indeed, each of these representations will naturally lead to various other needed properties of the Voigt functions.     

A unified presentation of the Voigt functions

In the present paper, we have given a generalization of a unified study of the Voigt functionsK(x, y) andL(x, y) obtained by Srivastava and Miller (1987; Vol. 135, pp. 111–118) which play an important role in several diverse fields of physics-such as astrophysical spectroscopy and the theory of neutron reactions. Explicit expressions for these functions are given in terms of relatively more familiar special functions of one and two variables; indeed, each of these representations will naturally lead to various other needed properties of the Voigt functions.     

Boundary curves for families of planar orbits

For monoparametric familiesf(x,y)=c of planar orbits, created by a planar potentialV(x,y), we introduce the notion of the family boundary curves (FBC). All members of the familyf(x,y)=c are traced in an allowable region of thexy plane, defined by the corresponding FBC, with total energyE=E(c) varying along the family. Family boundary curves are also found for two-parametric familiesf(x,y,b)=c. The relation of equilibrium points and asymptotic orbits, possibly possessed by the potentialV(x,y), to be FBC is studied.     

Solvable cases of Szebehely's equation

Szebehely’s equation is a first order partial differential equation relating a given family of orbits f (x, y) = q traced by a unit mass material point, the total energy E=E(f), and the unknown potential V=V (x, y) which produces the family. Although linear in V, this equation cannot generally be solved. In this paper we develop the reasoning for finding several cases for which Szebehely’s equation can be solved by quadratures. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the adelphic potentials compatible with a set of planar orbits

Given a planar potentialB=B(x, y), compatible with a monoparametric family of planar orbitsf(x, y)=c, we face the problem of producing potentialsA=A(x, y), adelphic toB(x, y), i.e. nontrivial potentials which have in common withB(x, y) the given set of orbits. We establish a linear, second order partial differential equation for a functionP(x, y) and we prove that, to any definite positive solution of this equation, there corresponds a potentialA(x, y) adelphic toB(x, y).     

Families of planar orbits generated by homogeneous potentials

The second order partial differential equation which relates the potentialV(x,y) to a family of planar orbitsf(x,y)=c generated by this potential is applied for the case of homogeneousV(x,y) of any degreem. It is shown that, if the functionf(x,y) is also homogeneous, there exists, for eachm, a monoparametric set of homogeneous potentials which are the solutions of an ordinary, linear differential equation of the second order. Iff(x,y) is not homogeneous, in general, there is not a homogeneous potential which can create the given family; only if =f _y /f _x satisfies two conditions, a homogeneous potential does exist and can be determined uniquely, apart from a multiplicative constant. Examples are offered for all cases.     

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.     

A simplification in the calculation of the characteristic exponents of the plane circular model of three bodies

It is shown that, to evaluate the two nontrivial characteristic exponents of the plane circular model of three bodies, we need only to integrate the variational equation fortwo displacements, rather thanfour, as is usually done. The two displacements should be chosen to lie in the constant Jacobi manifold and to be initially perpendicular to the given periodic motion. This simplification is applicable to both general orbits (integration over one whole period) and symmetric orbits (integration over one half period).     

On the integrability cases of the equation of motion for a satellite in an axially symmetric gravitational field

The projection of an axially symmetric satellite's orbit on a plane perpendicular to the rotation axis (z=const.) is given by the second-order differential equation. $$\frac{{y''}}{{1 + y'^2 }} = \bar \Psi _y - y'\bar \Psi _{x,}$$ where the prime denotes the derivative with respect tox and \(\bar \Psi (x,y)\) is a known function. Two integrability cases have been investigated and it has been shown that for these two cases the integration can be carried out either by quadratures or reduced to a first-order differential equation. Analytical and physical properties are expressed, and it is shown that the equation can be derived from the calssical plane eikonal equation of geometric optics.     

A measurement of the high energy secondary charged cosmic radiation

An experiment performed with a balloon-borne large plastic scintillator is described. It was launched from Reconquista, province of Santa Fe, on 24 February, 1978. The energy loss spectra of both atmospheric gamma radiation (forE > 4.15 MeV) and the charged component of the secondary cosmic radiation, were alternately measured at different altitudes, during the ascent of the balloon and at ceiling altitude. The atmospheric gamma radiation spectrum is analyzed in an earlier paper (Azcárateet al., 1992). The shape of the energy loss spectrum due to charged radiation is justified, in its more characteristic features, when the path length distribution in the detector of minimum ionization relativistic particles is taken into account. It is concluded that, at the ceiling altitude, the observed peak in the spectrum is due mainly to relativisticµ-mesons incident from the horizontal direction. The growth curve for the counting rate below the peak and the horizontal intensity of relativisticµ-mesons are also obtained.Member of the Carrera del Investigador Científico y Tecnológico, from the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) from ArgentinaMember of the Carrera del Investigador Científico y Tecnológico, from the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) from Argentina     

The solution of radiation transfer problem for a slab with space-dependent albedo and anisotropic scattering

The transport of thermal radiation has been considered within a finite slab which absorb and scatter anisotropically. The problem involves the space-dependent single-scattering albedow(x). Two approximations are taken forw(x). In the first it is represented in exponential form asw(x)=w ₀ exp(–x/s), wherew ₀ ands are given constants andx is the optical variable. The second approximation assumes the formw(x) = _r=0 ^R d _r ^* p _r (x/a), whered _r ^* are known expansion coefficients anda is the half optical thickness of the slab. Analytic expressions for the forward, backward radiation intensities and fluxes are given in each approximation. The solution of the linear transport equation is performed on the basis of integral Fourier transforms.     

A fourier transform solution of radiation transfer in plane-parallel medium with space-dependent albedo

A method is presented for solving radiation transfer problems involving space-dependent single-scattering albedoc(x) for a grey plane non-emitting medium with isotropic scattering. Expressions for the exit distributions and the reflection and transmission coefficients, relevant to a medium having a slab geometry, are given. The solution of the linear transport equation is performed on the basis of integral Fourier transforms. Numerical results are obtained in the case of exponentially varying single-scattering albedo and compared with previous results.     

An efficient method to calculate Ambarzumian-Chandrasekhar's and hopf's functions

An efficient method is proposed to calculate scalar Ambarzumian-Chandrasekhar's and Hopf's functions. This method is based on the approximation of Sobolev's resolvent function using exponent series, the coefficients of which are readily found from approximate characteristic equation and from a system of linear algebraic equations.The approximate expressions for the above functions are given. For checking purposes the calculations were carried out in single, double, and quadruple precision. For isotropic, Rocard, and Rayleigh scattering we present a sample of results in 14 significant figures.The Hopf function for isotropic and Rayleigh scattering is presented in 18 significant figures and the well-known Hopf constantq() is found in 59 significant figures.     

The gravitational field of a disk

This note gives the gravitational potential of the disk {(x, y, z):x ² +y ² p ² , z=0} and the gravitational field at the point (x, y, z). Formulas for a ring can be obtained as the difference of our results for two different values ofp. Results are obtained in terms of elliptic integrals and we indicate how these functions can be computed efficiently. Formulas necessary for the computation of partial derivatives are also given.This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract NAS7-100, sponsored by the National Aeronautics and Space Administration.