On Pfaff's equations of motion in dynamics; Applications to satellite theory

In this article we study a form of equations of motion which is different from Lagrange's and Hamilton's equations: Pfaff's equations of motion. Pfaff's equations of motion were published in 1815 and are remarkably elegant as well as general, but still they are much less well known. Pfaff's equations can also be considered as the Euler-Lagrange equations derived from the linear Lagrangian rather than the usual Lagrangian which is quadratic in the velocity components. The article first treats the theory of changes of variables in Pfaff's equations and the connections with canonical equations as well as canonical transformations. Then the applications to the perturbed two-body problem are treated in detail. Finally, the Pfaffians are given in Hill variables and Scheifele variables. With these two sets of variables, the use of the true anomaly as independent variable is also considered.     

The modified Lagrangian equations of general-relativistic hydrodynamics for a nonideal fluid

The general-relativistic equations of hydrodynamics for a nonideal fluid are derived in the modified Lagrangian form. Together with the zeroth and first moment equations of radiative transfer derived by Morita and Kaneko (1986), the equations provide a complete set of the modified Lagrangian equations of radiation hydrodynamics. The equations of hydrodynamics are specialized for a thermally conducting, Newtonian viscous fluid in the modified Lagrangian form, which are the generalization of the specialrelativistic equations of hydrodynamics derived by Greenberg (1975).     

Equations of radiation gas dynamics in tetrad form for astrophysical applications

This paper deals with the representation of relativistic equations of gas dynamics with due regard to the general relativity theory effects in the form accepted and widely applied in the special relativity theory. With this purpose, a strict formal definition of a non-inertial co-moving reference frame without rotation is carried out on the basis of a tetrad formalism by use of the Fermi—Walker rules of transport of 4-frame. The equations of physical kinetics, relativistic collapse, Einstein's equations, equations of relatiivistic radiation gas dynamics for ideal and dissipative gases, Taub's equations for a shock wave, which allow for radiation and electron-positron pairs, are obtained in this reference frame. On the basis of the local Lorentz transformation and the Ricci rotation coefficients, these equations are written in the laboratory reference frame, in order to illustrate the fact that the general relativity effects can be simply taken into account in the equations having a form accepted in the special relativity theory.     

The nth-order stellar hydrodynamic equation: transfer of comoving moments and pressures

The exact mathematical expression for an arbitrary n th-order stellar hydrodynamic equation is explicitly obtained depending on the central moments of the velocity distribution. In such a form the equations are physically meaningful, since they can be compared with the ordinary hydrodynamic equations of compressible, viscous fluids. The equations are deduced without any particular assumptions about symmetries, steadiness or particular kinematic behaviours, so that they can be used in their complete form, and for any order, in future works with improved observational data. Also, in order to work with a finite number of equations and unknowns, which would provide a dynamic model for the stellar system, the n th-order equation is needed to investigate in a more general way the closure conditions, which may be expressed in terms of velocity distribution statistics. A case example for a Schwarzschild distribution shows how the infinite hierarchy of hydrodynamic equations is reduced to the equations of orders   n = 0, 1, 2, 3  , owing to the recurrent form of the central moments and to the equations of order   n = 2  and 3, which become closure conditions for higher even- and odd-order equations, respectively. The closure example is generalized to a quadratic function in the peculiar velocities, so that the equivalence between moment equations and the system of equations that Chandrasekhar had obtained working from the collisionless Boltzmann equation is borne out.     

Time-varying transformations for Hill–Clohessy–Wiltshire solutions in elliptic orbits

The relative motion of chief and deputy satellites in close proximity with orbits of arbitrary eccentricity can be approximated by linearized time-periodic equations of motion. The linear time-invariant Hill–Clohessy–Wiltshire equations are typically derived from these equations by assuming the chief satellite is in a circular orbit. Two Lyapunov–Floquet transformations and an integral-preserving transformation are here presented which relate the linearized time-varying equations of relative motion to the Hill–Clohessy–Wiltshire equations in a one-to-one manner through time-varying coordinate transformations. These transformations allow the Hill–Clohessy–Wiltshire equations to describe the linearized relative motion for elliptic chief satellites.     

Regularization and the computation of optimal trajectories

A completely regular form for the differential equations governing the three-dimensional motion of a continuously thrusting space vehicle is obtained by using the Kustaanheimo-Stiefel regularization. The differential equations for the thrusting rocket are transformed using the K-S transformation and an optimal trajectory problem is posed in the transformed space. The canonical equations for the optimal motion in the transformed space are regularized by a suitable change of the independent variable. The transformed equations are regular in the sense that the differential equations do not possess terms with zero divisors when the motion encounters a gravitational force center. The resulting equations possess symmetry in form and the coefficients of the dependent variables are slowly varying quantities for a low-thrust space vehicle.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969.     

二阶后牛顿光线方程

近来相继提出一系列的空间天体测量计划，要求考虑在多参考系中二阶后牛顿部分对光线传播的贡献，也就是说，必须讨论在最近完成的扩展的DSX体系下的二阶后牛顿(2PN)光线方程．DSX体系是在20世纪90年代初建立的，用来讨论对N个任意形状和组成、自转可变形物体的一套完整的一阶后牛顿(1PN)天体力学理论．在此建议采用迭代的方法来推导2PN光线方程．从度规和Christoffel记号出发推导太阳系中的2PN光线方程．当忽略掉更高阶的项时，2PN光线方程将回到在很多教科书中广泛出现的1PN光线方程．利用这套方程就可以计算太阳系的光线传播．     

Dynamical equations of a superfluid in curved space-time and Cattaneo’s projection method

The absolute tensorial equations describing the dynamics of a superfluid in general relativity have been brought into the same form as the corresponding classical equations written in 3-vector forms. The expressions of the various forces acting on an element of a superfluid are explicitly displayed. In the Newtonian limit, these equations give the classical equations of motion of a superfluid in Galilean spacetime.     

Code Development of Three-Dimensional General Relativistic Hydrodynamics with AMR (Adaptive-Mesh Refinement) and Results from Special and General Relativistic Hydrodynamics

In this paper, the general procedure to solve the general relativistic hydrodynamical (GRH) equations with adaptive-mesh refinement (AMR) is presented. In order to achieve, the GRH equations are written in the conservation form to exploit their hyperbolic character. The numerical solutions of GRH equations are obtained by high resolution shock Capturing schemes (HRSC), specifically designed to solve nonlinear hyperbolic systems of conservation laws. These schemes depend on the characteristic information of the system. The Marquina fluxes with MUSCL left and right states are used to solve GRH equations. First, different test problems with uniform and AMR grids on the special relativistic hydrodynamics equations are carried out to verify the second-order convergence of the code in one, two and three dimensions. Results from uniform and AMR grid are compared. It is found that adaptive grid does a better job when the number of resolution is increased. Second, the GRH equations are tested using two different test problems which are Geodesic flow and Circular motion of particle In order to do this, the flux part of GRH equations is coupled with source part using Strang splitting. The coupling of the GRH equations is carried out in a treatment which gives second order accurate solutions in space and time. This revised version was published online in July 2006 with corrections to the Cover Date.     

The comoving-frame equation of radiative transfer in relativistic flows

The comoving-frame equations of radiative transfer and moment equations to accurate terms of all orders inv/c are derived in the modified Lagrangian form. The equations exactly describe the interaction of radiation with matter in a relativistically moving medium in flat or curved spacetime. Two specialized sets of equations are presented: (1) the equation of radiative transfer and moment equations accurate to terms of second order (v ²/c ²), and (2) the transfer equation and moment equations for a radial flow in curved spacetime with the Schwarzschild-type metric.     

Solutions to bi-Maxwellian transport equations for SAR-arc conditions

We have compared solutions obtained from the bi-Maxwellian based 16-moment transport equations with those obtained from the Maxwellian based 13-moment transport equations for conditions leading to the steady state, subsonic flow of a fully-ionized electron-proton plasma along geomagnetic field lines in the vicinity of the plasmapause. The bi-Maxwellian based equations can account for large temperature anisotropies and the flow of both parallel and perpendicular thermal energy, while the Maxwellian based equations account for small temperature anisotropies and only the total heat flow. Our comparison indicates that for Stable Auroral Red arc (SAR-arc) conditions leading to strong field-aligned heat flows (temperatures of 8000 K and temperature gradients of4K. km⁻¹ at 1500 km), the bi-Maxwellian based equations predict a different thermal structure in the topside ionosphere than the less rigorous Maxwellian based equations. In particular, the bi-Maxwellian based equations predict proton and electron temperature anisotropies with T > T, while the Maxwellian based equations predict the opposite behavior for the same boundary conditions. This difference is related to the way in which the temperature anisotropies and heat flows are treated in the two formulations. For the bi-Maxwellian based equations, the inclusion of separate heat flows for parallel and perpendicular thermal energy allows for the development of a pronounced tail in both the electron and proton distribution functions, which leads to temperature anisotropies with T > T. For the Maxwellian based equations, on the other hand, the tail development is restricted because only the total heat flow is considered. Consequently, as the heat flows down, the presence of an increasing magnetic field acts to produce an anisotropy with T > T, and this process dominates tail formation for the Maxwellian based equations.     

Matrix perturbation methods using regularized coordinates

Matrix methods for computing perturbations of non-linear perturbed systems, as formulated by Alexeev, involve an expression for the full solution of the first variational equations of the system evaluated about a reference orbit. These cannot be immediately applied to a regularized system of equations where perturbations about Keplerian motion are considered since the solution of the variational equations of regularized Keplerian motion does not in general correspond to the solution of the variational equations of the unregularized equations. But, as Kustaanheimo and Stiefel have pointed out, the regularized equations of Keplerian motion should be excellent for the initiation of a perturbation theory since they are linear in form. This paper describes a method for applying Alexeev's theorem to a regularized system where full advantage is taken of the basic linear form of the unperturbed equations.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969.     

Studies in the application of recurrence relations to special perturbation methods

A set of differential equations is derived that has a number of advantages in special perturbation work. In particular, the equations remain valid for all values of the orbital eccentricity and inclination including zero. They are therefore applicable to parabolic- and hyperbolic-type orbits as well as elliptic-type; a scheme for use when the orbit is rectilinear or nearly so is provided. The equations are also much simpler in form than the Lagrange planetary equations and the transformations of the osculating elements to and from the rectangular coordinates are straightforward.     

Slowly rotating superdense configurations in Rosen's bimetric theory of gravitation

In the limits of the Bimetric Theory of Gravitation the equations of axial-symmetric gravitational field, describing the field of rotating configurations, are obtained and examined. The analytical solutions of these equations out of configurations are found. The equations are solved numerically in the first approximation in the angular velocity for the configurations consisting of the real baryon gas (cf. Sahakian, 1972). The moments of inertia of these configurations in BTG are calculated. It is shown that the equations and the appropriate solution, obtained by Falik and Opher (1979), are incorrect.     

Exact solution of a basic equation in finite atmosphere by the method of Laplace transform and linear singular operators

A finite atmosphere having distribution of intensity at both surfaces with definite form of scattering function and source function is considered here. The basic integro-differential equation for the intensity distribution at any optical depth is subjected to the finite Laplace transform to have linear integral equations for the surface quantities under interest. These linear integral equations are transformed into linear singular integral equations by use of the Plemelj's formulae. The solution of these linear singular integral equations are obtained in terms of theX-Y equations of Chandrasekhar by use of the theory of linear singular operators which is applied in Das (1978a).     

A Simplified Kinetic Element Formulation for the Rotation of a Perturbed Mass-Asymmetric Rigid Body

Euler's equations, describing the rotation of an arbitrarily torqued mass asymmetric rigid body, are scaled using linear transformations that lead to a simplified set of first order ordinary differential equations without the explicit appearance of the principal moments of inertia. These scaled differential equations provide trivial access to an analytical solution and two constants of integration for the case of torque-free motion. Two additional representations for the third constant of integration are chosen to complete two new kinetic element sets that describe an osculating solution using the variation of parameters. The elements' physical representations are amplitudes and either angular displacement or initial time constant in the torque-free solution. These new kinetic elements lead to a considerably simplified variation of parameters solution to Euler's equations. The resulting variational equations are quite compact. To investigate error propagation behaviour of these new variational formulations in computer simulations, they are compared to the unmodified equations without kinematic coupling but under the influence of simulated gravity-gradient torques.     

Linear fREDHOLM integral equations for radiative transfer problems in finite plane parallel media. I. Imbedding in an infinite medium

Linear FREDHOLM integral equations are derived for the STOKES vector of the radiation emerging from a scattering plane parallel medium of finite optical thickness. The integral equations are obtained by means of imbedding the slab in an infinite medium. They are formulated in terms of GREEN 's function matrices and renormalized for the asymptotic eigenmode. Explicitly, linear integral equations are given for the reflection and transmission matrices. The reciprocity principle is employed to obtain integral equations also for the mean intensity of the inner radiation field in the case of the slab albedo problem.     

Electron distribution function and ion concentrations in the earth's lower ionosphere from Boltzmann-Fokker-Planck theory

Using the Boltzmann-Fokker-Planck method and the local approximation we derive coupled non-linear equations for the electron and ion concentrations and the energy-dependent electron distribution function in the Earth's lower ionosphere. These equations are new and give the appropriate generalization of the standard electron-ion continuity equations in the local approximation when electron-neutral particle impact ionization is treated rigorously. We report stable, numerical solutions to these equations and compare our calculated electron concentration to the experimentally determined result for a rocket experiment where the electron concentration and solar EUV spectral flux were measured simultaneously.     

Equations for a plasma consisting of matter and antimatter

A set of fluid type equations is derived to describe the macroscopic behaviour of a plasma consisting of a mixture of matter and antimatter. The equations are written in a form which displays the full symmetry of the medium with respect to particle charge and mass, a symmetry absent in normal plasmas. This symmetry of the equations facilitates their manipulation and solution, and by way of illustration the equations are used to analyze the propagation of electromagnetic and acoustic waves through a matter-antimatter plasma. Some differences from the propagation of such waves in a normal plasma are noted.     

Hill stability of the planar three-body problem: General and restricted cases

The equations of motion of the planar three-body problem split into two parts, called an external part and an internal part. When the third mass approaches zero, the first part tends to the equations of the Kepler motion of the primaries and the second part to the equations of motion of the restricted problem.We discuss the Hill stability from these equations of motion and the energy integral. In particular, the Jacobi integral for the circular restricted problem is seen as an infinitesimal-mass-order term of the Sundman function in this context.