Stabilization of Kepler's problem

A regularization of Kepler's problem due to Moser is used to ‘stabilize’ the equations of motion, that is, imbed a particular solution of Kepler's problem in a Lyapounov stable system.     

On the history of some explicit formulae for solving Kepler's equation

In this paper we examine, in their historical context, some approximate solutions for Kepler's equation. These explicit formulae, obtained by Trembley, Pacassi, Fergola, and Horrebow, had not a great diffusion and were thus often rediscovered by other astronomers. We will prove that the formulae are equivalent and, moreover, we will give an evaluation of the error. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Indian record for Kepler's supernova: Evidence from Kashmir Valley

We revisit a strange representation of the Sagittarius constellation painted on a door arch of a tomb in Kashmir. We show that it is a very strong case of a representation of Kepler's supernova (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Kepler,Galileo, the telescope and its consequences

In the beginning Copernicus' system of the world did not have empirical confirmation. In this situation, Kepler's research, as well as the astronomical observations with the telescope, invented in 1608, played a decisive role. Under the assumption of the central position of the Sun, Kepler discovered the elliptical orbital motion of the planets as a base of the computation of noticeably improved ephemerides. The first telescopic observations – Jupiter's moons, phases of Venus, sunspots, surface features of the moon, gave important arguments for Copernicus' system. Galilei was one of the first who used the telescope for astronomical research (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The solution for seasonal variations in the Earth's rate of rotation

New physical principles for an explanation of seasonal variations in the Earth's rate of rotation are proposed. It is thought that the variations are caused by a variation of the total energy of the Earth's atmosphere in the course of the planet's revolution about the Sun in elliptic orbit. Jacobi's virial equation for the Earth's atmosphere is derived from the Eulerian equations. The virial theorem is obtained. The existence of the relationship between Jacobi's function and potential energy of the atmosphere is confirmed. In the framework of this relationship, Jacobi's equation is reduced to the equation of unperturbed virial oscillations. The solution of the above-mentioned equation expresses the periodic virial oscillations of Jacobi's function (moment of inertia) of the Earth's atmosphere with time. The solution of the perturbed virial oscillation problem of the atmosphere-solid Earth system is obtained. The perturbation term in Jacobi's virial equation regards, in explicit form, the energy changes occurring in the atmosphere in the course of the planet's revolution about the Sun in elliptic orbit. The annual and semi-annual periodic variations in the Earth's rate of rotation can be considered as an astrometrical result following from the obtained solution. A satisfactory accord of the theoretical results with experimental data is shown.     

Un formulaire pour le calcul des perturbations d'ordres élevés dans les problèmes planétaires

The authors present formulas in compact form for constructing high order planetary perturbations with respect to the disturbing masses. They have been built by an iterative process and give the variations of osculating elements. Singularities due to vanishing eccentricities and inclinations are not present in the differential equations. All elementary operations are manipulations of Fourier series with numerical coefficients, and great care has been taken to economize algebraic operations. Results are presented in three forms:

1. vectorial form, with real components which may be useful in numerical integrations;
2. complex form, to put in evidence the symmetries of the system of variables;
3. scalar form, which is the most elaborate. This last form has been used for constructing the first order perturbations for any pair of planets. Two illustrations are given (Jupiter and Saturn, Venus and Earth). Further remarks are made about the practical manipulation of Fourier series, resolution of Kepler's equation in complex form and construction by iteration of the inverse of the distance between two bodies.

     

Solution of the equation of transfer for coherent scattering in an exponential atmosphere by Eddington's method

An approximate solution of the transfer equation for coherent scattering in stellar atmospheres with Planck's function as a nonlinear function of optical depth, viz., $$B_v \left( T \right) = b_0 + b_1 e^{ - \beta \tau } $$ is obtained by Eddington's method. is obtained by Eddington's method.     

Thread safe astronomy

Observational astronomy is the beneficiary of an ancient chain of apprenticeship. Kepler's laws required Tycho's data. As the pace of discoveries has increased over the centuries, so has the cadence of tutelage (literally, “watching over”). Naked eye astronomy is thousands of years old, the telescope hundreds, digital imaging a few decades, but today's undergraduates will use instrumentation yet unbuilt – and thus, unfamiliar to their professors – to complete their doctoral dissertations. Not only has the quickening cadence of astronomical data‐taking overrun the apprehension of the science within, but the contingent pace of experimental design threatens our capacity to learn new techniques and apply them productively. Virtual technologies are necessary to accelerate our human processes of perception and comprehension to keep up with astronomical instrumentation and pipelined dataflows. Necessary, but not sufficient. Computers can confuse us as efficiently as they illuminate. Rather, as with neural pathways evolved to meet competitive ecological challenges, astronomical software and data must become organized into ever more coherent ‘threads’ of execution. These are the same threaded constructs as understood by computer science. No datum is an island. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Die Hadamard-Konstruktion Greenscher Funktionen auf gekrümmter Raum-Zeit: Physikalische Zusammenhänge und explizite strenge Resultate

First, in connection with their construction due to HADAMARD, the mathematical and physical meaning of covariant Green's functions in relativistic gravitational fields - according to EINSTEIN: on curved space-time - is discussed. Then, in the case of a general static spherically symmetric space-time the construction equations for a scalar Green's function are cast into symmetry-adapted form providing a convenient starting point for an explicit calculation of the Hadámard building elements. In applying the obtained basic scheme to a special one-parameter family of model metrics one succeeds in advancing to the explicit exact calculation of tail-term coefficients of a massless Green's function which are simultaneously coefficients in the Schwinger-De Witt expansion of the Feynman propagator for the corresponding massive Klein-Gordon equation on curved space-time.     

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.     

One of the solutions for the Laplace equation and its physical interpretation

In this paper it is confirmed once more that there exists the general solution of Laplace's equation in ellipsoidal coordinates which satisfies the Stäckel theorem and which was derived earlier by M. Jarov-Jarovoi and S. J. Madden. The author interprets physically the general solution in real space as potentials of layers of charge and double layers in which the distribution of densities is defined by Green's formula.     

On solving Kepler's equation

Intrigued by the recent advances in research on solving Kepler's equation, we have attacked the problem too. Our contributions emphasize the unified derivation of all known bounds and several starting values, a proof of the optimality of these bounds, a very thorough numerical exploration of a large variety of starting values and solution techniques in both mean anomaly/eccentricity space and eccentric anomaly/eccentricity space, and finally the best and simplest starting value/solution algorithm: M + e and Wegstein's secant modification of the method of successive substitutions. The very close second is Broucke's bounds coupled with Newton's second-order scheme.This work was sponsored by the Department of the Air Force under Contract F19628-85-C-0002. The views are those of the authors and do not reflect the official policy or position of the U.S. Government.Now at Space Telescope Science Institute operated by AURA, Inc. for NASA.     

Green's functions of the induction equation on regions with boundary. I. General properties and a construction principle

The evolution of a magnetic field is considered which pervades an electrically conducting fluid and its non-conducting surroundings under the influence of electromotive forces due to internal motion and other causes. The governing equations -- among which the induction equation of magnetohydrodynamics is the most prominent -- pose an initial value problem for the magnetic flux density. Properties of this initial value problem and of the solving Green's function are reviewed and a general construction principle for the Green's function is given. Detailed treatment of cases where the fluid occupies a sphere or an evenly bounded half-space are presented in two subsequent papers     

Solution of the equation of transfer for coherent scattering in an exponential atmosphere by the method of discrete ordinates

A solution of the transfer equation for coherent scattering in stellar atmosphere with Planck's function as a nonlinear function of optical depth, viz., $$B_v (T) = b_0 + b_1 {\text{ }}e^{ - \beta \tau } $$ is obtained by the method of discrete ordinates originally due to Chandrasekhar.     

Viscous damping of nonlinear magneto-acoustic waves

The motivation for the present work is discussed, and in Section 2 the nonlinear Burger's equation is derived for wave propagation in a compressible unstratified viscous fluid permeated by a magnetic field (initially constant), using the reductive perturbation method of Taniuti and Wei (1968). An analytic solution of the equation is given (after Sakai, 1972) and the angular behaviour is shown for certain parameters describing the nonlinearity and damping of the system, for both fast- and slow-mode disturbances.     

On coronal streamers with T-type neutral points

The gas-magnetic field interaction of an isothermal axisymmetric corona is considered. A method is suggested for solving the MHD equations in the case when a uniform gas pressure and the radial component of the magnetic field (as in a dipole) are specified at the Sun's surface. The flux of open field lines (φ) can be given arbitrarily, and no reconnection or opening of field lines can take place. If configurations in hydrostatic equilibrium between the regions of open and closed field lines can be found, then the method of solution converges. The equation of hydrostatic equilibrium at the neutral point (assumed to be of the T-type) is written in a simple form, and it is shown that if φ is smaller than a certain φ_min, this equation cannot be satisfied. Configurations in hydrostatic equilibrium between the regions of open and closed field lines are expected to exist for any value of φ larger than φ_min.     

An exact solution of the equation of transfer for coherent scattering in an exponential atmosphere

An exact solution of the transfer equation for coherent scattering in stellar atmospheres with Planck's function as a nonlinear function of optical depth, of the form $$B_v (T) = b_0 + b_1 {\text{ }}e^{ - \beta \tau } $$ is obtained by the method of the Laplace transform and Wiener-Hopf technique.     

Solution of the equation of transfer for coherent scattering in an exponential atmosphere by Busbridge's method

A solution of the transfer equation for coherent scattering in stellar atmosphere with Planck's function as a nonlinear function of optical depth, viz. $$B{\text{ }}_v (T) = b_0 + b_1 {\text{ }}e^{ - \beta \tau } $$ is obtained by the method developed by Busbridge (1953).     

The Fragmentation of Asteroids – A Synopsis of Analytical Methods (Mitteilungen der Universitäts-Sternwarte Jena Nr. 119)

Basing on the author's work a review of the possibilities as well as the limits of treating the problem of the collisional history of the asteroids by analytical methods is given. Using empirical data on rock fragmentation and general principles like symmetry and mass conservation the distribution function of the fragments arising from a single collision is analytically formulated. The size distribution of asteroids adjusting when crushing collisions have taken place a sufficiently long time can be obtained as the solution of an integrodifferential equation with partial derivatives (equation of fragmentation). Quasi-stationary solutions of the equation of fragmentation are discussed for particular cases. The problem of the steady state is reduced to the solution of a transcendental equation. The results obtained show that analytical methods already offer a good theoretical understanding of the observed size distribution of the asteroids. They should be, therefore, a useful basis of carrying out numerical calculations.