Canonical transformations depending on a small parameter

The concept of a Lie series is enlarged to encompass the cases where the generating function itself depends explicity on the small parameter. Lie transforms define naturally a class of canonical mappings in the form of power series in the small parameter. The formalism generates nonconservative as well as conservative transformations. Perturbation theories based on it offer three substantial advantages: they yield the transformation of state variables in an explicit form; in a function of the original variables, substitution of the new variables consists simply of an iterative procedure involving only explicit chains of Poisson brackets; the inverse transformation can be built the same way.     

A programming system for analytical series expansions on a computer

A programming system is described for the manipulation of Poisson series on a computer. The general structure is described together with the most important individual subroutines. The system exists in two versions, one with 3 angular and 3 polynomial variables and the other with twice as many variables. The programming has been done first in FORTRAN IV, but the most crucial subroutines have been rewritten in machine language for more efficiency. The storage mechanism is such that obsolete series can be erased from the main memory in order to make room for new series to be created.This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. NAS 7-100, sponsored by the National Aeronautics and Space Administration.     

The boundary-layer flow past a semi-infinite heated porous plate for a two-component plasma

The flow in the boundary layer past a semi-infinite vertical hot porous plate is investigated for a two-component plasma model when the suction at the plate is large. Employing asymptotic expansion procedure the velocity and temperature fields are computed in the presence of free convection currents and viscous dissipation heat. The results are compared with the case when the flow is flow ionized. It is observed that the assumption of a fully ionized situation overestimates the basic flow variables.     

A linear solution of the equations of motion of an earth-orbiting satellite based on a Lie-series

Using Hill's variables, an analytical solution of a canonical system of six differential equations describing the motion of a satellite in the gravitational field of the earth is derived. The gravity field, expanded into spherical harmonics, has to be expressed as a function of the Hill variables. The intermediary is chosen to include the main secular terms. The first order solution retains the highly practical formal structure of Kaula's linear solution, but is valid for circular orbits and provides of course a spectral decomposition of radius vector and radial velocity. The resulting eccentricity functions are much simpler than the Hansen functions, since a series evaluation of the Kepler equation is avoided. The present solution may be extended to higher order solutions by Hori's perturbation method.     

Rotational evolution of a symmetric gyrostat with visco-elastic bars around the center of mass in a circular orbit

The motion of a gyrostat in a circular orbit in a Newtonian field of force is considered. The gyrostat has four homogeneous viscoelastic bars attached to it. Rotation of the symmetric rotor inside the rigid body is statically and dynamically balanced. Bending deformations of the bars, accompanied by dissipation of energy, are the cause of the evolution of the system's rotational motion. Approximate equations describing this evolution are derived, together with averaged equations in Andoyer variables.     

Moment-based ellipticity measurement as a statistical parameter estimation problem

We show that galaxy ellipticity estimation for weak gravitational lensing with unweighted image moments reduces to the problem of measuring a combination of the means of three independent normal random variables. Under very general assumptions, the intrinsic image moments of sources can be recovered from observations including effects such as the point-spread function and pixellation. Gaussian pixel noise turns these into three jointly normal random variables, the means of which are algebraically related to the ellipticity. We show that the random variables are approximately independent with known variances, and provide an algorithm for making them exactly independent. Once the framework is developed, we derive general properties of the ellipticity estimation problem, such as the signal-to-noise ratio, a generic form of an ellipticity estimator, and Cramér-Rao lower bounds for an unbiased estimator. We then derive the unbiased ellipticity estimator using unweighted image moments. We find that this unbiased estimator has a poorly behaved distribution and does not converge in practical applications, but demonstrates how to derive and understand the behaviour of new moment-based ellipticity estimators.     

The construction of a fifth-order Jupiter-Saturn motion: Analytical theory

We construct a fifth-order with respect to masses Jupiter-Saturn secular theory by Hori-Lie canonical technique. The J-S Hamiltonian includes both parts of the perturbing function. The influence of the 2:5 critical terms is taken into consideration. The Jacobi-Radau system of origins is adopted and the theory is expressed in terms of the Poincaré canonical variables.     

The elimination of short-periodic terms in a Uranus-Neptune general planetary theory by von Zeipel's method

We eliminate by the method of von Zeipel the short-period terms in a first order-with respect to planetary masses—general planetary Uranus-Neptune theory. We exclude in the expansion terms of eccentricities and sines of inclinations higher than the third power.Our variables are the Poincaré canonical variables. We use the Jacobi-Radau set of origins, and we refer the planes of the osculating ellipses to a common fixed plane, the longitudes to a common origin. The short-periodic terms arising from the indirect and principal parts of the disturbing functions, are eliminated separately. The Fourier series of the principal part of the disturbing function, is reduced to the sum of only the first three terms.     

RR Lyrae variables in M5 as a test of pulsational theory

We present B and V CCD photometry for variables in the cluster central region, adding new data for 32 variables and giving suitable light curves, mean magnitudes and corrected colours for 17 RR Lyrae variables. Adding the data given in this paper to similar data that have already appeared in the literature, we discuss a sample of 42 variables, as given by 22 RR_ab and 20 RR_c, in the light of recent predictions from pulsational theories. We find that the observational evidence concerning M5 pulsators appears in marginal disagreement with predictions concerning the colour of the first overtone blue edge (FOBE), whereas a clear disagreement appears between the zero-age horizontal branch (ZAHB) luminosities predicted through evolutionary and pulsational theories.     

Radiation transfer in a sphere with internal source

From the presence of superhumps in its outburst light curve, the dwarf nova BR Lup is shown to be a member of the SU UMa sub-class of cataclysmic variables. Its orbital period is estimated to be 0.0789 d.     

A general model of the planetary radiation pressure on a satellite with a complex shape

The purpose of this paper is to present a general model for the acceleration exerted on a spacecraft by the radiation coming from a planet. Both the solar radiation reflected by the planet and the thermal emission associated with its temperature are considered. The planet albedo and the planet emissive power are expanded in spherical harmonics with respect to an equatorial reference frame attached to the planet. The satellite external surface is assumed to consist of a juxtaposition of planar surfaces. A particular choice of variables allows to reduce the surface integrals over the lit portion of the planet visible to the satellite to one-dimension integrals.     

The construction of a seventh-order Jupiter-Saturn analytical theory

We extend the construction of the Jupiter-Saturn theory to include all the terms up to the seventh order in the masses. The Hori-Lie transformation technique is employed. The Jacobian coordinates are adopted and the theory is expressed in terms of the canonical non-singular variables of H. Poincaré.     

Magnetic field reconnection in a compressible plasma

An extension of Sonnerup's model for the magnetic field-line reconnection for a compressible plasma is given. The plasma is considered to be only slightly compressible so that the leading wave in Sonnerup's model can still be taken to be a thin discontinuity. The flow is assumed to occur under adiabatic conditions, and de Hoffmann-Teller jump conditions are used to connect the state variables across the shocks. The compressibility effects are found to increase the reconnection rate. The signaling problem is finally considered to study the evolution of MHD waves in a compressible, dissipative plasma so as to investigate the conditions under which the assumption of MHD waves in a compressible plasma to be thin discontinuities is valid.     

Long nonlinear waves in a compressible magnetically structured atmosphere

The propagation of nonlinear slow sausage small-amplitude waves in a magnetic slab in a magnetic environment is considered. The equation for surface waves that is allied form of the Benjamin-Ono equation and the equation for body waves that is allied form of the equation for body waves in the slab are derived with the use of the method of stretching variables. The solutions of the equation for surface waves in the form of solitary waves are obtained numerically.     

Exact analytical solutions basic to a class of two-body orbits

The theory of complex variables is used to establish exact analytical solutions to a class of two-body problems. In view of Lambert's theorem, two points on the conic, the chord-distance between the two points, and the time interval are considered given, and subsequently the solutions for the semi-major axis required to define the orbit are developed and expressed ultimately in terms of elementary quadratures.     

Self-similar MHD shock waves for a rotating atmosphere under the force of gravitation

A study has been made of self-similar magnetohydrodynamic spherical shock waves for a rotating atmosphere taking into account the effect of self-gravitation. The energy is assumed to vary with some power of time. A study has been made to investigate the effects of magnetic field in the presence and absence of gravitation. The variation of flow variables is shown in tables for several different cases of physical interest.     

On the elimination of the critical term in a general first-order Uranus-Neptune theory

A solution of the Uranus-Neptune planetary canonical equations of motion through the Von Zeipel technique is presented. A unique determinging function which depends upon mixed canonical variables, reduces the 12 critical terms of the Hamiltonian to the set of its secular terms. The Poincaré canonical variables are used. We refer to a common fixed plane, and apply the Jacobi-Radau set of origins. In our expansion we neglected terms of power higher than the fourth with respect to the eccentricities and sines of the inclinations.     

The Lissajous transformation I. Basics

A new canonical transformation is proposed to handle elliptic oscillators, that is, Hamiltonian systems made of two harmonic oscillators in a 1-1 resonance. Lissajous elements pertain to the ellipse drawn with a light pen whose coordinates oscillate at the same frequency, hence their name. They consist of two pairs of angle-action variables of which the actions and one angle refer to basic integrals admitted by an elliptic oscillator, namely, its energy, its angular momentum and its Runge-Lenz vector. The Lissajous transformation is defined in two ways: explicitly in terms of Cartesian variables, and implicitly by resolution of a partial differential equation separable in polar variables. Relations between the Lissajous variables, the common harmonic variables, and other sets of variables are discussed in detail.     

A new treatment of the albedo radiation pressure in the case of a uniform albedo and of a spherical satellite

The purpose of this paper is to present a model for the radiation pressure acceleration of a spherical satellite, due to the radiation reflected by a planet with a uniform albedo. A particular choice of variables allows one to reduce the surface integrals over the lit portion of the planet visible to the satellite to one-dimensional integrals. Exact analytical expressions are found for the integrals corresponding to the case where the spacecraft does not "see" the terminator. The other integrals can be computed either numerically, or analytically in an approximate form. The results are compared with those of Lochry (1966). The model is applied to Magellan, a spacecraft orbiting Venus.