Automated, closed form integration of formulas in elliptic motion
Algorithm for automatic closed form integration of formulas in elliptic motion arising in perturbation theories
Jefferys, W. H.↗
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Integrals of motion in the elliptic restricted three-body problem.
Integrals of motion in plane elliptic restricted three-body problems for orbits with small eccentricity near primaries
JACOBI INTEGRAL↗
Automated, closed form integration of formulas in elliptic motion
Description of algorithm for performing automated closed form integration of formulas in elliptic motion
Jefferys, W. H.↗
Shape identification technique for a two-dimensional elliptic system by boundary integral equation method
The geometrical structure of the boundary shape for a two-dimensional boundary value problem is identified. The output least square identification method is considered for estimating partially unknown boundary shapes. A numerical parameter estimation technique using the spline collocation method is proposed.
Kojima, Fumio↗
Calculation of potential flow past airship bodies in yaw
An outline of Von Karman's method of computing the potential flow of airships in yaw by means of partially constant dipolar superposition on the axis of the body is followed by several considerations for beginning and end of the superposition. Then this method is improved by postulating a continuous, in part linearly variable dipolar superposition on the axis. The second main part of the report brings the calculation of the potential flow by means of sources and sinks, arranged on the surface of the airship body. The integral equation which must satisfy this surface superposition is posed, and the core reduced to functions developed from whole elliptical normal integrals. The functions are shown diagrammatically. The integration is resolvable by iteration. The consequence of the method is good. The formulas for computing the velocity on the surface and of the potential for any point conclude the report.
Lotz, I↗
Determination of integration constants in the generalized problem of two fixed centers
Integration constants determination in generalized problem of two fixed centers in gravitational earth field
FIXED POINT↗
Numerical integration of periodic orbits in the elliptic restricted three-body problem
The problem of finding periodic orbits in the circular restricted three-body problem has been very extensively studied in celestial mechanics. It is well known that continuous families of periodic orbits exist, for which the period varies in a continuous way. However, all the applications which are found in the solar system correspond to cases with non-zero eccentricities, and the elliptic restricted three-body problem is thus a better approximation than the circular one. For instance, for the motion of a satellite in the Earth-Moon system, as a first approximation, we may assume that the Moon moves around the Earth in circular motion; but as a much better approximation, we can also assume that the Moon moves in an elliptic orbit around the Earth.
NUMERICAL INTEGRATION↗
The elliptic anomaly
An independent variable different from the time for elliptic orbit integration is used. Such a time transformation provides an analytical step-size regulation along the orbit. An intermediate anomaly (an anomaly intermediate between the eccentric and the true anomaly) is suggested for optimum performances. A particular case of an intermediate anomaly (the elliptic anomaly) is defined, and its relation with the other anomalies is developed.
Janin, G.↗
The Space Transportation System summer environment on launch pad
This paper describes a 2D flow and thermalanalysis to determine the solar effect on the Space Shuttle launch components subsequent to the external tank (ET) loading operation in extremely hot conditions. An existing CFD code Parabolic Hyperbolic or Elliptical Numerical Integration Code Series was used in the study. The analysis was done for a 2D slice between planes perpendicular to the longitudinal axis of the STS and passing through the lower portions of the Redesigned Solid Rocket Motors (RSRMs), the ET, and the wing of the Orbiter. The results are presented as local and average values of the heat transfer coefficient, and the Nusselt number, and the surface temperature around the RSRMs and the ET. Solar heating effects increased the surface temperatures of the RSRMs by 9-11 F. Higher prelaunch surface temperatures measured on the east and west RSRMs (in the inboard region between the RSRMs and the ET) during 19 most recent launches of the STS are correlated as a function of the ambient temperature.
Ahmad, R. A.↗
A fuselage/tank structure study for actively cooled hypersonic cruise vehicles, summary
A detailed analytical study was made to investigate the effects of fuselage cross section (circular and elliptical) and the structural arrangement (integral and nonintegral tanks) on aircraft performance. The vehicle was a 200 passenger, liquid hydrogen fueled Mach 6 transport designed to meet a range goal of 9.26 Mn (5000 NM). A variety of trade studies were conducted in the area of configuration arrangement, structural design, and active cooling design in order to maximize the performance of each of three point design aircraft: (1) circular wing-body with nonintegral tanks, (2) circular wing-body with integral tanks and (3) elliptical blended wing-body with integral tanks. Aircraft range and weight were used as the basis for comparison. The resulting design and performance characteristics show that the blended body integral tank aircraft weights the least and has the greatest range capability, however, producibility and maintainability factors favor nonintegral tank concepts.
Pirrello, C. J.↗
Optimal trajectories based on linear equations
The Principal results of a recent theory of fuel optimal space trajectories for linear differential equations are presented. Both impulsive and bounded-thrust problems are treated. A new form of the Lawden Primer vector is found that is identical for both problems. For this reason, starting iteratives from the solution of the impulsive problem are highly effective in the solution of the two-point boundary-value problem associated with bounded thrust. These results were applied to the problem of fuel optimal maneuvers of a spacecraft near a satellite in circular orbit using the Clohessy-Wiltshire equations. For this case two-point boundary-value problems were solved using a microcomputer, and optimal trajectory shapes displayed. The results of this theory can also be applied if the satellite is in an arbitrary Keplerian orbit through the use of the Tschauner-Hempel equations. A new form of the solution of these equations has been found that is identical for elliptical, parabolic, and hyperbolic orbits except in the way that a certain integral is evaluated. For elliptical orbits this integral is evaluated through the use of the eccentric anomaly. An analogous evaluation is performed for hyperbolic orbits.
Carter, Thomas E.↗
ON THE ELLIPTIC CASE OF THE RESTRICTED PROBLEM OF THREE BODIES AND THE REMOTE HISTORY OF THE EARTH-MOON SYSTEM
Jacobi integral and the elliptic case of the restricted problem of three bodies, and study of the remote history of the earth-moon system
THREE-BODY PROBLEM↗
Amplitudes, supersymmetric black hole scattering at $\mathcal{O}\left({G}^5\right)$, and loop integration
We compute the potential-graviton contribution to the scattering amplitude, the radial action, and the scattering angle of two extremal black holes in $\mathcal{N}$ = 8 supergravity at the fifth post-Minkowskian order and to next-to-leading order in a large mass expansion (first self-force order). Properties of classical unitarity cuts allow us to focus on the integration-by-parts reduction of planar integrals, while nonplanar integrals at this order are obtained from the planar ones by straightforward manipulations. We present the solution to the differential equations for all master integrals necessary to evaluate the classical scattering amplitudes of massive scalar particles at this order in all gravitational theories, in particular in $\mathcal{N}$ = 8 supergravity, and in general relativity. Despite the appearance of higher-weight generalized polylogarithms and elliptic functions in the solution to the differential equation for master integrals, the final supergravity answer is remarkably simple and contains only (harmonic) polylogarithmic functions up to weight 2. The systematic analysis of elliptic integrals discussed here, as well as the particular organization of boundary integrals in $\mathcal{N}$ = 8 observables are independent of supersymmetry and may have wider applications, including to aspects of collider physics.
Black Holes↗
Integral Kernel Methods for Nonlinear Parabolic-Elliptic Systems
Nonlinear parabolic-elliptic systems arise in many physical, biological, and chemical phenomena such as chemotaxis, ion transport, self-gravitating particles, and Brownian vortices. Existing methods struggle with the strong coupling and high nonlinearity and nonlocality of some of these systems, especially the ill-conditioned, convection-dominated problems. To overcome numerical difficulties, current approaches rely on initial guesses, preconditioning, or iterative techniques with no convergence guarantees. They might suffer from poor scalability, large memory usage, and difficulty to parallelize. Inspired by the connection of parabolic-elliptic systems to stochastic processes, we introduce a novel meshless, monolithic, and fully explicit method that naturally encapsulates the elliptic and parabolic operators into a single step which updates each node deterministically with global information. By being fully quadrature-based, it avoids solving systems of discretized equations and does not utilize initial guesses or preconditioning, while requiring little memory and being easy to parallelize. We first derive the method in an integral kernel formulation with quadratic complexity in the number of integration nodes and then leverage kernel-independent fast multipole methods (FMM) to present a scalable algorithm with linear complexity. We provide numerical examples for the Poisson-Nernst-Planck equations in one, two, and three dimensions, together with the derivation of the integral kernel for each case. Furthermore, the examples demonstrate the fast convergence and scalability of the FMM-accelerated algorithm, as well as its suitability for convection-dominated problems, making it competitive against traditional PDE solvers.
PDE systems↗
[Elliptic Differential Equations]
Reflection laws of systems of elliptic equations, on-line programming to numerically solve nonlinear integral equations, and reduction of electrostriction problem
ON-LINE PROGRAMMING↗
The evolution of high-metallicity horizontal-branch stars and the origin of the ultraviolet light in elliptical galaxies
Evolutionary calculations of high-metallicity horizontal-branch stars show that for the relevant masses and helium abundances, post-HB evolution in the HR diagram does not proceed toward and along the AGB, but rather toward a 'slow blue phase' in the vicinity of the helium-burning main sequence, following the extinction of the hydrogen shell energy source. For solar and twice solar metallicity, the blue phase begins during the helium shell-burning phase (in agreement with the work of Brocato and Castellani and Tornambe); for 3 times solar metallicity, it begins earlier, during the helium core-burning phase. This behavior differs from what takes place at lower metallicities. The implications for high-metallicity old stellar populations in the Galactic bulge and for the integrated colors of elliptical galaxies are discussed.
Horch, E.↗
A calculation procedure for viscous flow in turbomachines, volume 1
A method for analyzing the nonadiabatic viscous flow through turbomachine rotors is presented. The field analysis is based upon the numerical integration of the full incompressible stream function vorticity form of the Navier-Stokes equations, together with the energy equation, over the rotor blade-to-blade stream channels. The numerical code used to solve the governing equations employs a nonorthogonal boundary fitted coordinate system that suits the most complicated blade geometries. A numerical scheme is used to carry out the necessary integration of the elliptic governing equations. The flow characteristics within the rotor of a radial inflow turbine are investigated over a wide range of operating conditions. The calculated results are compared to existing experimental data. The flow in a radial compressor is analyzed in order to study the behavior of viscous flow in diffusing cascades. The results are compared qualitatively to known experimental trends. The solution obtained provides insight into the flow phenomena in this type of turbomachine. It is concluded that the method of analysis is quite general and gives a good representation of the actual flow behavior within turbomachine passages.
Khalil, I.↗
Viscous flow analysis in mixed flow rotors
A method for analyzing the viscous flow through turbomachine rotors is presented. The field analysis is based upon the solution of the full Navier-Stokes equations over the rotor blade-to-blade stream channels. An Alternating-Direction-Implicit method is employed to carry out the necessary numerical integration of the elliptic governing equations. The flow analysis may be applied to various types of turbomachine rotors. Preliminarily, only the case of laminar flows are considered in this paper. The flow characteristics within the rotors of a mixed flow turbine and a radial bladed compressor are investigated over a wide range of operating conditions. Excellent results are obtained when compared with existing experimental data. The method of this analysis is quite general and can deal with a wide range of applications. Possible modification of the present study to deal with turbulent flow cases are also identified.
Khalil, I.↗