Expansion of the planetary mutual distance raised to any negative power by the method of symbolic differential operators

We present a literal approach to evaluate ^–s necessary for the construction of high order planetary theories. This approach is valid to be applied on very large scale digital computers with standard Poisson series programs, for high order and high degree planetary theories. We apply the method of symbolic differential operators for single variable functions, and the binomial theorem expansions, for the evaluation of ^–s . We utilize Laplace coefficients and its derivatives to carry out the development, without dealing with Newcomb operators or Hansen's coefficients.     

On the solving of the Hill's differential equation through the method of operational calculus

Using a method previously applied to the treatment of the Mathieu differential equation, we solve the Hill's differential equation of lunar theory through the way of operational calculus, which avoids the cumbersome infinite determinants of the classical procedure. The one-sided Laplace transformation changes it into a difference equation with an infinite number of terms and variable coefficients. When its first member is divided by a suitable factor, this difference equation is the image of an integral equation of the Volterra type which is equivalent to the initial Hill's differential equation. Solution of this Volterra integral equation is unique and it is the general solution of the Hill's differential equation. This solution is a series in the powers of a small dimensionless parameter ² which appears as a factor in the second member of the Hill's differential equation. We reduce it to the sum of its terms of degree 12 with respect to which is the precision usually required in a lunar theory and we write down effectively the coefficients of the terms in ², ( ²)² and the coefficient of the term in ( ²)³ which depends upon the initial valuey(0) of the Hill's differential equation.     

The Impact of the Static Part of the Earth's Gravity Field on Some Tests of General Relativity with Satellite Laser Ranging

In this paper we calculate explicitly the classical secular precessions of the node and the perigee of an Earth artificial satellite induced by the even zonal harmonics of the static part of the geopotential up to degree l = 20. Subsequently, their systematic errors induced by the mismodelling in the even zonal spherical harmonics coefficients J _l are compared to the general relativistic secular gravitomagnetic and gravitoelectric precessions of the node and the perigee of the existing laser-ranged geodetic satellites and of the proposed LARES. The impact of the future terrestrial gravity models from CHAMP and GRACE missions is discussed as well. Preliminary estimates with the recently released EIGEN-1S gravity model including the first CHAMP data are presented.     

Formation of short-period orbits of minor bodies of the Mars family

It is shown that the orbits of minor bodies of the Mars family can be formed as a result of single close encounters with the inner planets (Venus -Mars) of minor bodies moving along orbits with the aphelion distanceq' 4.26 AU. The Laplace method of the unperturbed two-body problem was applied.     

On a new form for the differential equations of relative motion of the three-body problem

A new method for the development of the disturbing function of the three-body problem is outlined in this paper. A special process is devised to get the distance between two planetsP ₁ andP ₂ in terms of their heliocentric distances. It is then shown that the differential equations of relative motion of this problem can be brought in an homogeneous set of differential equations.     

Generalized Laplace coefficients and Newcomb derivatives

It is shown how the generalized Laplace coefficients can be employed to deduce explicit formulas for ordinary and Newcomb derivatives of the Laplace coefficients.      

Unsteady magnetohydrodynamic flows in a rotating elastico-viscous fluid

This paper points out the errors in the solutions of a research work by N. Nanousis under the same title published in this journal, volume 199, 1993. The correct solutions of the problem for the velocity field and the drag on the plate, by the Laplace transform technique, are presented. The results are discussed for two cases of an arbitrary time-dependent forcing effect. It is shown that the viscoelastic parameterk > 0 influences the velocity and introduces reverse flow. For a suddenly accelerated plate,k > 0 increases the velocity forz < and decreases it forz > . In the case of the ramp-type boundary condition,k > 0 tends to decrease the velocity.     

The global solution of the N-body problem

The problem of finding a global solution for systems in celestial mechanics was proposed by Weierstrass during the last century. More precisely, the goal is to find a solution of the n-body problem in series expansion which is valid for all time. Sundman solved this problem for the case of n = 3 with non-zero angular momentum a long time ago. Unfortunately, it is impossible to directly generalize this beautiful theory to the case of n > 3 or to n = 3 with zero-angular momentum.A new blowing up transformation, which is a modification of McGehee's transformation, is introduced in this paper. By means of this transformation, a complete answer is given for the global solution problem in the case of n > 3 and n = 3 with zero angular momentum.The main result in this paper has appeared in Chinese in Acta Astro. Sinica. 26 (4), 313–322. In this version some mistakes have been rectified and the problems we solved are now expressed in a much clearer fashion.     

The main problem of lunar theory solved by the method of Brown

Brown's method for solving the main problem of lunar theory has been adapted for the computation by machine. The computations are carried out with the help of an algebraic processor called POLYPAK, which can manipulate power series in several real or complex variables. Brown's result have been recovered and refined first and the solution, in Cartesian coordinates to the sixth order, has been compared to the work of Eckert (see Gutzwiller, 1979). The solution has then been expanded to include most terms through order nine. This order is necessary to get an accuracy of 0.00001 for the terms in longitude and latitude and of 0.000001 for the terms in the sine parallax. A preliminary comparison with the theories of Chapront and Henrard (1980) indicates that the solution has an accuracy which is close to the one desired. For details see Schmidt (1980).The next step in developing a complete lunar theory requires the computation of the partial derivatives of the solution with respect to the primary parameters. Since Brown's method gives a semianalytical solution only the derivative with respect ton, the mean motion of the Moon, is difficult to compute. It is possible to find this derivative with one quadrature from the other derivatives if one takes into account that the Jacobian has to be a symplectic matrix when a canonical set of primary parameters is used.The mean motions of the perigee and node often exhibit the largest discrepancies among the different theories. Therefore it is not too surprising that also their derivatives show significant differences. It is hoped by providing another independent computation of the derivatives by a different method their accuracy can be improved.Proceedings of the Conference on Analytical Methods and Ephemerides: Theory and Observations of the Moon and Planets. Facultés universitaires Notre Dame de la Paix, Namur, Belgium, 28–31 July, 1980.     

A New Method of Numerical Solution of Nonsteady Problems in the Theory of Radiative Transfer

A new method is proposed for the numerical solution of nonsteady problems in the theory of radiative transfer. In this method, if the solution at some time t (such as the initial time) is known, then by representing the radiation intensity and all time-dependent quantities (level populations, kinetic temperature, etc.) in the form of Taylor series expansions in the vicinity of t, one can, from the transfer equation and the equations accompanying it (population equations, energy-balance equation, etc.), find all derivatives of that solution at the given time from certain recursive equations. From the Taylor series one can then calculate the solution at some later time t + t, and so forth. The method enables one to analyze nonsteady tradiative transfer both in stationary media and in media with characteristics that vary with time in a given way. This method can also be used to solve nonlinear problems, i.e., those in which the radiation field significantly affects the characteristics of the medium. No iterations are used for this: everything comes down to calculations based on recursive equations. Several problems, both linear and nonlinear, are solved as examples.     

The flow of viscoelastic non-Newtonian fluid in the Ekman layer

In this paper the unsteady flow in the Ekman layer of a visco-elastic non-Newtonian fluid near a flat plate is discussed. Laplace transform technique has been employed to show the basic differential equations. Expressions for velocity profile, the skin friction have been calculated. It is shown that the time to attain the steady state increases with the elastic parameter. It is shown that normally the ultimate steady state is reached through a decay of inertial oscillations whose frequency decreases with increase in the elastic parameter. In the present study we examine the following unsteady problem in non-Newtonian fluid. Consider an infinite plate coinciding with the platez=0 and rotating in unison with elasticoviscous liquid occupying the regionz>0 with a uniform angular velocity about thez-axis for timet<-0. At timet>0, the plate starts moving with a uniform velocityU _o along thex-axis relative to the rotating frame of reference. The horizontal homogeneity of the problem demands that conditions depend onz andt only. The equation of continuity together with the no slip condition at the plate then shows that thez-component of the velocity vanishes everywhere.     

Expansion theory for the elliptic motion of arbitrary eccentricity and semi-major axis

In this paper of the series, literal analytical expressions for the coefficients of the Fourier series representation ofF will be established for anyx _i ; withn, N positive integers 1 and | _i | fori=1, 2,...n. Moreover, the recurrence formulae satisfied by these coefficients will also be established. Illustrative analytical examples and a full recursive computational algorithm, with its numerical results, are included. The applications of the recurrence formulae are also illustrated by their stencils. As a by-product of the analyses is an integral which we may call a complete elliptic integral of thenth kind, in which the known complete elliptic integrals (1st, 2nd and 3rd kinds) are special cases of it.     

Relationships of quasi-stationary solar wind flows with their sources on the Sun

In this paper, by considering an example of four Carrington rotations (1671,1672,1681, and 1682), it is shown that there generally exists an exhaustive correspondence between quasi-stationary flows of fast and slow solar wind (SW), on the one hand, and their sources on the Sun: coronal holes (CHs) and the heliospheric current sheet (HCS), on the other. It is also shown that by knowing characteristics such as the coordinate of the center of gravity of CHs on the Sun, their areas S and the positions of the neutral line (NL) and of the HCS without the NL on the Sun, it becomes possible to calculate the time of appearance and the amplitude of three points on the SW velocity profile at the Earth's orbit, namely time, t _F ,and velocity amplitude, V _F ,corresponding to the mean point of the forward front of the SW flow velocity profile, the value of V = V _m in the central part of the flow, and angular width ⁺ of the flow at level V = V _F .Calculated values agree with those observed.     

Restricted quantum-mechanical three-body problems

Eigenvalues and normalized wave functions of an electron are derived in the field of a generalized dipole with chargesZ ₁ andZ ₂ (Z ₁+Z ₂ 0) in the asymptotic region up to the third order (inr ₂) where the distance,r ₂, between the two charges is small. These asymptotic wave functions render it possible to calculate the asymptotic expansion for the coefficients of a coupled infinite system of second order differential equations arising from a perturbative (analytic) solution to the Schrödinger equation of helium-like ions if the nuclear charge is not less than 2.     

Partial separability of the Laplace equation in Roche coordinates

The Laplace equation in the coordinatesu, v, w is calledu-separable if there are solutions of the formF(u)G(v, w). If the surfacesv = constant andw = constant are orthogonal tou = constant the coordinate system is called semi-orthogonal. The Laplace equation is notu-separable for the rotation problem semi-orthogonal Roche coordinate system (n0, q=0) or the general problem (n0, q0) ifv andw are analytic functions ofn andq and the coordinate system is proper in some region of then, q plane including the origin,n=q=0 (u is the Roche potential).     

An exact solution of the equation of transfer with three-term scattering indicatrix in an exponential atmosphere

The general equation for radiative transfer in the Milne-Eddington model is considered here. The scattering function is assumed to be quadratically anisotropic in the cosine of the scattering angle and Planck's intensity function is assumed for thermal emission. Here we have taken Planck's function as a nonlinear function of optical depth, viz.,B _v(T)=b _o+b ₁ e ^–. The exact solution for emergent intensity from the bounding face is obtained by the method of the Laplace transform in combination with the Wiener-Hopf technique.     

Solar center-to-limb infrared intensity from the Halogen Occultation Experiment

Data from the Halogen Occultation Experiment (HALOE) provide the first opportunity to examine solar center-to-limb relative intensity measured exoatmospherically at wavelengths from 2.4 to 10 m. The data were obtained from limb-to-limb scans across the solar equator on days of very low activity in May 1994. Coefficients for a function describing limb darkening are obtained at eight infrared wavelengths using a nonlinear least-squares fitting technique. Relative intensities produced by the limb-darkening functions are precise to 0.1% (2). From the limb-darkening coefficients, it is possible to calculate temperature information about the photosphere. At each of the eight HALOE wavelengths, the brightness temperature from the flux, T _b ^disk(), and the temperature as a function of monochromatic optical depth, T(), are determined and normalized using Kondratyev et al. (1965) and calibrated Pierce (1954) central intensity measurements. The two temperature quantities are compared with the predictions of Vernazza, Avrett, and Loeser's (1976) model M, and in general there is good agreement. The largest differences occur between 2.4 and 3 m and suggest that the central intensities used in this spectral region are low.     

Expansions of (r/a)m cos jv and (r/a)m sin jv to high eccentricities

Expansions of the functions (r/a)^cos jv and (r/a)m ^sin jv of the elliptic motion are extended to highly eccentric orbits, 0.6627 ... <e<1. The new expansions are developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients of these expansions are expressed in terms of the derivatives of Hansen's coefficients with respect to the eccentricity. The new expansions are convergent for values of the eccentricity such that |e–e*|<(e*), where the radius of convergence (e*) is the same of the extended solution of Kepler's equation. The new expansions are intrinsically related to Lagrange's series.     

Representation of the Earth's gravitational potential

The paper represents the Earth's gravitational potentialV, outside a sphere bounding the Earth, by means of its difference V from the author's spheroidal potential. The difference V is in turn represented as arising from a surface density on the sphere bounding the Earth. Because of the slow decrease with ordern of the normalized coefficients in the spherical harmonic expansion ofV, the density anomalies from which the higher coefficients arise must occur in regions close to the Earth's surface. The surface density is thus an idealization of the product of the density anomaly and the crustal thicknessb. Values of are computed from potential coefficients obtained from two sources, Rapp and the Smithsonian Astrophysical Observatory. The two sources give qualitative agreement for the values of and for its contour map. The numerical values obtained for are compatible with the idea that the responsible density anomalies are reasonably small, i.e., less than 0.05 g/cm³, and occur in the crust alone.This paper was prepared under the sponsorship of the Electronics Research Center of the National Aeronautics and Space Administration through NASA Grant NGR 22-009-311.     

Empirical NLTE analyses of solar spectral lines

A method is suggested for empirical NLTE analyses of solar spectral lines. Special depthdependent departure coefficients (x) are introduced and the formulas are given for further application. From test calculations it is shown that the separation of the departure coefficients for the upper and the lower level from each other and from the uncertainties in several input parameters (oscillator strength times abundance, turbulent velocities and damping constant) is greatly facilitated when spectral lines are analysed on the disk, around the limb, as well as in the flash spectrum. Therefore it is necesssary that all accessible line data are used, from accurate line profiles to equivalent widths or integrated intensities. To reduce the number of independent variables the analysis should include many multiplets between a few spectroscopic terms.We discuss the advantages of translating the departure coefficients into a set of temperatures. Some published LTE and NLTE analyses are compared in terms of these temperatures. The results do not show inexplicable contradictions. For several atomic spectra the departures from LTE seem large enough to be established quantitatively by the detailed method suggested in this paper.