Equilibrium points in the photogravitational restricted four-body problem

We study numerically the photogravitational version of the problem of four bodies, where an infinitesimal particle is moving under the Newtonian gravitational attraction of three bodies which are finite, moving in circles around their center of mass fixed at the origin of the coordinate system, according to the solution of Lagrange where they are always at the vertices of an equilateral triangle. The fourth body does not affect the motion of the three bodies (primaries). We consider that the primary body m ₁ is dominant and is a source of radiation while the other two small primaries m ₂ and m ₃ are equal. In this case (photogravitational) we examine the linear stability of the Lagrange triangle solution. The allowed regions of motion as determined by the zero-velocity surface and corresponding equipotential curves, as well as the positions of the equilibrium points on the orbital plane are given. The existence and the number of the collinear and the non-collinear equilibrium points of the problem depends on the mass parameters of the primaries and the radiation factor q ₁. Critical masses m ₃ and radiation q ₁ associated with the existence and the number of the equilibrium points are given. The stability of the relative equilibrium solutions in all cases are also studied. In the last section we investigate the existence and location of the out of orbital plane equilibrium points of the problem. We found that such critical points exist. These points lie in the (x,z) plane in symmetrical positions with respect to (x,y) plane. The stability of these points are also examined.     

On the heat conduction in a high-temperature plasma in solar flares

We have developed three types of mathematical models to describe the mechanisms of plasma heating in the corona by intense heat fluxes from a super-hot (T _e ? 10⁸ K) reconnecting current layer in connection with the problem of energy transport in solar flares. We show that the heat fluxes calculated within the framework of self-similar solutions using Fourier’s classical law exceed considerably the real energy fluxes known from present-day multi-wavelength observations of flares. This is because the conditions for the applicability of ordinary heat conduction due to Coulomb collisions of thermal plasma electrons are violated. Introducing anomalous heat conduction due to the interaction of thermal runaway electrons with ion-acoustic turbulence does not give a simple solution of the problem, because it produces unstable temperature profiles. Themodels incorporating the effect of collisional heat flux relaxation describe better the heat transport in flares than Fourier’s law and anomalous heat conduction.     

Periodic orbits in the restricted four-body problem with two equal masses

The restricted (equilateral) four-body problem consists of three bodies of masses m ₁, m ₂ and m ₃ (called primaries) lying in a Lagrangian configuration of the three-body problem i.e., they remain fixed at the apices of an equilateral triangle in a rotating coordinate system. A massless fourth body moves under the Newtonian gravitation law due to the three primaries; as in the restricted three-body problem (R3BP), the fourth mass does not affect the motion of the three primaries. In this paper we explore symmetric periodic orbits of the restricted four-body problem (R4BP) for the case of two equal masses where they satisfy approximately the Routh’s critical value. We will classify them in nine families of periodic orbits. We offer an exhaustive study of each family and the stability of each of them.     

Radially accelerated optimal feedback orbits in central gravity field with linear drag

The solution of a feedback optimal control problem arising in orbital mechanics is addressed in this paper. The dynamics is that of a massless body moving in a central gravitational force field subject also to a drag and a radial modulated force. The drag is linearly proportional to the velocity and inversely proportional to the square of the distance from the center of attraction. The problem is tackled by exploiting the properties of a suitably devised linearizing map that transforms the nonlinear dynamics into an inhomogeneous linear system of differential equations supplemented by a quadratic objective function. The generating function method is then applied to this new system, and the solution is back transformed in the old variables. The proposed technique, in contrast to the classical optimal control problem, allows us to derive analytic closed-loop solutions without solving any two-point boundary value problem. Applications are discussed.     

Linear stability of equilibrium points in the generalized photogravitational Chermnykh’s problem

The equilibrium points and their linear stability has been discussed in the generalized photogravitational Chermnykh’s problem. The bigger primary is being considered as a source of radiation and small primary as an oblate spheroid. The effect of radiation pressure has been discussed numerically. The collinear points are linearly unstable and triangular points are stable in the sense of Lyapunov stability provided μ<μ _Routh =0.0385201. The effect of gravitational potential from the belt is also examined. The mathematical properties of this system are different from the classical restricted three body problem.     

Invariant shape solutions of the spinning three craft Coulomb tether problem

In this paper we study shape-preserving formations of three spacecraft, where the formation keeping forces arise from the electric charges deposed on each craft. Inspired by Lagrange’s 3-body problem, the general conditions that guarantee preservation of the geometric shape of the electrically charged formation are derived. While the classical collinear configuration is a solution to the problem, the equilateral triangle configuration is found to only occur with unbounded relative motion. The three collinear spacecraft problem is analyzed and the possible solutions are categorized based on the spacecraft mass–charge ratio. Precise statements on the number of solutions associated with each category are provided. Finally, a methodology is proposed to study boundedness of the collinear solution that is inspired by past understanding and results for the 3-body problem. Given the initial position and the velocity vectors of each craft along with the charges, analytical solutions are provided describing the resulting relative motion.     

Equilibrium solutions of the restricted problem of 2+2 bodies

Fourteen equilibrium solutions of the restricted problem of 2+2 bodies are shown to exist. Six of these solutions are located about the collinear Lagrangian points of the classical restricted problem of three bodies. Eight solutions are found in the neighborhood of the triangular Lagrangian points. Linear stability analysis reveals that all of the equilibrium solutions are unstable with the exception of four solutions; two in the vicinity of each of the triangular Lagrangian points. These four solutions are found to be stable provided the mass parameter of the primary masses is less than a critical value which depends also on the mass of the minor bodies.     

Application of high order expansions of two-point boundary value problems to astrodynamics

Two-point boundary value problems appear frequently in space trajectory design. A remarkable example is represented by the Lambert’s problem, where the conic arc linking two fixed positions in space in a given time is to be characterized in the frame of the two-body problem. Classical methods to numerically solve these problems rely on iterative procedures, which turn out to be computationally intensive in case of lack of good first guesses for the solution. An algorithm to obtain the high order expansion of the solution of a two-point boundary value problem is presented in this paper. The classical iterative procedures are applied to identify a reference solution. Then, differential algebra is used to expand the solution of the problem around the achieved one. Consequently, the computation of new solutions in a relatively large neighborhood of the reference one is reduced to the simple evaluation of polynomials. The performances of the method are assessed by addressing typical applications in the field of spacecraft dynamics, such as the identification of halo orbits and the design of aerocapture maneuvers.     

A second-order solution of the Ideal Resonance Problem by Lie series

A second-order libration solution of theIdeal Resonance Problem is construeted using a Lie-series perturbation technique. The Ideal Resonance Problem is characterized by the equations $$\begin{gathered} - F = B(x) + 2\mu ^2 A(x)sin^2 y, \hfill \\ \dot x = - Fy,\dot y = Fx, \hfill \\ \end{gathered} $$ together with the property thatB _x vanishes for some value ofx. Explicit expressions forx andy are given in terms of the mean elements; and it is shown how the initial-value problem is solved. The solution is primarily intended for the libration region, but it is shown how, by means of a substitution device, the solution can be extended to the deep circulation regime. The method does not, however, admit a solution very close to the separatrix. Formulae for the mean value ofx and the period of libration are furnished.     

A new approach to impulsive rendezvous near circular orbit

A new approach is presented for the problem of planar optimal impulsive rendezvous of a spacecraft in an inertial frame near a circular orbit in a Newtonian gravitational field. The total characteristic velocity to be minimized is replaced by a related characteristic-value function and this related optimization problem can be solved in closed form. The solution of this problem is shown to approach the solution of the original problem in the limit as the boundary conditions approach those of a circular orbit. Using a form of primer-vector theory the problem is formulated in a way that leads to relatively easy calculation of the optimal velocity increments. A certain vector that can easily be calculated from the boundary conditions determines the number of impulses required for solution of the optimization problem and also is useful in the computation of these velocity increments. Necessary and sufficient conditions for boundary conditions to require exactly three nonsingular non-degenerate impulses for solution of the related optimal rendezvous problem, and a means of calculating these velocity increments are presented. A simple example of a three-impulse rendezvous problem is solved and the resulting trajectory is depicted. Optimal non-degenerate nonsingular two-impulse rendezvous for the related problem is found to consist of four categories of solutions depending on the four ways the primer vector locus intersects the unit circle. Necessary and sufficient conditions for each category of solutions are presented. The region of the boundary values that admit each category of solutions of the related problem are found, and in each case a closed-form solution of the optimal velocity increments is presented. Similar results are presented for the simpler optimal rendezvous that require only one-impulse. For brevity degenerate and singular solutions are not discussed in detail, but should be presented in a following study. Although this approach is thought to provide simpler computations than existing methods, its main contribution may be in establishing a new approach to the more general problem.     

Families of Asymmetric Periodic Orbits in the Restricted Three-body Problem

This paper studies the asymmetric solutions of the restricted planar problem of three bodies, two of which are finite, moving in circular orbits around their center of masses, while the third is infinitesimal. We explore, numerically, the families of asymmetric simple-periodic orbits which bifurcate from the basic families of symmetric periodic solutions f, g, h, i, l and m, as well as the asymmetric ones associated with the families c, a and b which emanate from the collinear equilibrium points L ₁, L ₂ and L ₃ correspondingly. The evolution of these asymmetric families covering the entire range of the mass parameter of the problem is presented. We found that some symmetric families have only one bifurcating asymmetric family, others have infinity number of asymmetric families associated with them and others have not branching asymmetric families at all, as the mass parameter varies. The network of the symmetric families and the branching asymmetric families from them when the primaries are equal, when the left primary body is three times bigger than the right one and for the Earth–Moon case, is presented. Minimum and maximum values of the mass parameter of the series of critical symmetric periodic orbits are given. In order to avoid the singularity due to binary collisions between the third body and one of the primaries, we regularize the equations of motion of the problem using the Levi-Civita transformations.     

Symetries du Probleme desN Corps,Regularisation des Solutions Centrales Singulieres

The newtonian problem ofn mass points bodies is invariant by several changes of spatio-temporal variables. These symmetries correspond to arbitrary choices of the referential and they are related via Noether's theorem or by its generalization to conservative quantities of the motion. Forn=2 the author has defined two families of symmetriesS ₁ andS ₂ changing the eccentricity of a solution. The family of symmetries,S ₁, is associated to the arbitrary choice of thezero level of the potential and may related unbounded and bounded solutions. The family of symmetries,S ₂, is related to a possibleaffinity of the configurations space. Via a symmetry of theS ₂ family a zero angular momentum solution is equivalent to a non-zero angular momentum solution. Via a product of two symmetries of each family, denoted byS ₁.S ₂, any solution of the two-body problem is equivalent to a circular solution. In this paper it is shown that some of these transformations may be generalized to symmetries changing the quantityC ² H in then-body problem, whereC is the angular momentum andH is the energy. The extension is easily made to central solutions of then-body problem because involving several synchroneous two-body problems. We consider for exposition then=3 case. The principal results may be resumed by the following propositions:

1. The two families of symmetriesS ₁ andS ₂ are described by a spatial transformation product of an instantaneous homothethy and an instantaneous rotation completed by a change of temporal variable.
2. TheS ₁ family of symmetries may relate unbounded and bounded central solutions of the same type, i.e. unaligned or aligned.
3. TheS ₂ family of symmetries may regularize multiple collisions among central solutions of the same type.

Therefore any central solution, via a symmetryS ₁ orS ₂ orS ₁.S ₂, is equivalent to a central circular solution of the same type. That is a form of regularization.     

A multi-objective approach to the design of low thrust space trajectories using optimal control

In this article, we introduce a novel three-step approach for solving optimal control problems in space mission design. We demonstrate its potential by the example task of sending a group of spacecraft to a specific Earth L ₂ halo orbit. In each of the three steps we make use of recently developed optimization methods and the result of one step serves as input data for the subsequent one. Firstly, we perform a global and multi-objective optimization on a restricted class of control functions. The solutions of this problem are (Pareto-)optimal with respect to ΔV and flight time. Based on the solution set, a compromise trajectory can be chosen suited to the mission goals. In the second step, this selected trajectory serves as initial guess for a direct local optimization. We construct a trajectory using a more flexible control law and, hence, the obtained solutions are improved with respect to control effort. Finally, we consider the improved result as a reference trajectory for a formation flight task and compute trajectories for several spacecraft such that these arrive at the halo orbit in a prescribed relative configuration. The strong points of our three-step approach are that the challenging design of good initial guesses is handled numerically by the global optimization tool and afterwards, the last two steps only have to be performed for one reference trajectory.     

The vertical structure of atmospheric oscillations formulated by classical tidal theory

From the equations of classical tidal theory with Newtonian cooling (Chapman and Lindzen, Atmospheric Tides: thermal and gravitational, Reidel, 1970), formulae are obtained for wind, temperature and pressure oscillations generated by thermal, gravitational and lower-boundary excitations of given frequency. The analysis is an extension of that of Butler and Small (Proc. R. Soc. Lond.A274, 91, 1963) who formulated solutions of the vertical structure equation in terms of two independent solutions of the homogeneous equation and derived expressions for surface pressure oscillations. A comprehensive formulation is presented for wind, temperature and pressure oscillations as functions of height with the above-mentioned sources of excitation and an upper-boundary radiation condition. The formulae obtained are applied at the surface leading to evaluations of the surface oscillation weighting function W_p(z) which weights the thermal excitation at height z according to its differential contribution to the surface oscillation. The formulae are shown to simplify at heights above a region of excitation and evaluations are undertaken of the thermal response weighting function W_t(z) which weights the thermal excitation at height z according to its differential contribution to the oscillation at any height above the region of thermal excitation. Computational procedures are described for obtaining two independent solutions of the homogeneous equation and results are presented for an adopted profile of atmospheric scale height. The problem of deriving the surface pressure oscillation due to a tidal potential is briefly reviewed and results are presented as an example of the application of formulae that have been derived.     

K/S two-point-boundary-value problems

A method for developing the missing general K/S (Kustaanheimo/Stiefel) boundary conditions is presented, with use of the formalism of optimal control theory. As an illustrative example, the method is applied to the K/S Lambert problem to derive the missing terminal condition. The necessary equations are developed for a solution to this problem with the generalized eccentric anomaly,E, as the independent variable. This formulation, requiring the solution of only one nonlinear, well-behaved equation in one unknown,E, results in considerable simplification of the problem.Presented also at the AAS/AIAA Astrodynamics Specialist Conference, Nassau, Bahamas, July 1975 (AAS Paper No. 75-032).     

Families of periodic orbits in the planar three-body problem

A periodic orbit of the restricted circular three-body problem, selected arbitrarily, is used to generate a family of periodic motions in the general three-body problem in a rotating frame of reference, by varying the massm ₃ of the third body. This family is continued numerically up to a maximum value of the mass of the originally small body, which corresponds to a mass ratiom ₁:m ₂:m ₃?5:5:3. From that point on the family continues for decreasing massesm ₃ until this mass becomes again equal to zero. It turns out that this final orbit of the family is a periodic orbit of the elliptic restricted three body problem. These results indicate clearly that families of periodic motions of the three-body problem exist for fixed values of the three masses, since this continuation can be applied to all members of a family of periodic orbits of the restricted three-body problem. It is also indicated that the periodic orbits of the circular restricted problem can be linked with the periodic orbits of the elliptic three-body problem through periodic orbits of the general three-body problem.     

Galactic Sun's motion in the cold dark matter,MOdified Newtonian Dynamics and modified gravity scenarios

We numerically integrate the equations of motion of the Sun in Galactocentric Cartesian rectangular coordinates for –4.5 Gyr ≤ t ≤ 0 in Newtonian mechanics with two different models for the Cold Dark Matter (CDM) halo, in MOdified Newtonian Dynamics (MOND) and in MOdified Gravity (MOG) without resorting to CDM. The initial conditions used come from the latest kinematical determination of the 3D Sun's motion in the Milky Way (MW) by assuming for the rotation speed of the Local Standard of Rest (LSR) the recent value Θ₀ = 268 km s^–1 and the IAU recommended value Θ₀ = 220 km s^–1; the Sun is assumed located at 8.5 kpc from the Galactic Center (GC). For Θ₀ = 268 km s^–1 the birth of the Sun, 4.5 Gyr ago, would have occurred at large Galactocentric distances (12–27 kpc depending on the model used), while for Θ₀ = 220 km s^–1 it would have occurred at about 8.8–9.3 kpc for almost all the models used. The integrated trajectories are far from being circular, especially for Θ₀ = 268 km s^–1, and differ each other with the CDM models yielding the widest spatial extensions for the Sun's orbital path (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A dipole field model for axisymmetric alfvén waves with finite ionosphere conductivities

An axisymmetric model for approximate solution of the magnetospheric Alfvén wave problem at latitudes above the plasmapause is proposed, in which a realistic dipole geometry is combined with finite anisotropic ionosphere conductivities, thus bringing together various ideas of previous authors. It is confirmed that the axisymmetric toroidal and poloidal modes interact via the ionospheric Hall effect, and an approximate method of solution is suggested using previously derived closed solutions of the uncoupled wave equations.A solution for zero Hall conductivity is obtained, which consists of sets of independent shell oscillations, regardless of the magnitude of the Pedersen conductivity. One set reduces to the classical solutions for infinite Pedersen conductivity, while another predicts a new set of harmonics of a quarter-wave fundamental, with longer eigenperiods than the classical solutions for a given L-shell.