Lambert problem solution in the hill model of motion

The goal of this paper is obtaining a solution of the Lambert problem in the restricted three-body problem described by the Hill equations. This solution is based on the use of pre determinate reference orbits of different types giving the first guess and defining the sought-for transfer type. A mathematical procedure giving the Lambert problem solution is described. This procedure provides step-by-step transformation of the reference orbit to the sought-for transfer orbit. Numerical examples of the procedure application to the transfers in the Sun–Earth system are considered. These examples include transfer between two specified positions in a given time, a periodic orbit design, a halo orbit design, halo-to-halo transfers, LEO-to-halo transfer, analysis of a family of the halo-to-halo transfer orbits. The proposed method of the Lambert problem solution can be used for the two-point boundary value problem solution in any model of motion if a set of typical reference orbits can be found.     

An analytical solution for Kepler's problem

In this paper, we present a framework which provides an analytical (i.e. infinitely differentiable) transformation between spatial coordinates and orbital elements for the solution of the gravitational two-body problem. The formalism omits all singular variables which otherwise would yield discontinuities. This method is based on two simple real functions for which the derivative rules are only required to be known, all other applications – e.g. calculating the orbital velocities, obtaining the partial derivatives of radial velocity curves with respect to the orbital elements – are thereafter straightforward. As it is shown, the presented formalism can be applied to find optimal instants for radial velocity measurements in transiting explanatory systems to constrain the orbital eccentricity as well as to detect secular variations in the eccentricity or in the longitude of periastron.     

On an analytical solution of the optimum trajectory problem in a gravitational field

Mayer's variational problem for a point with a limited mass flow rate is described by differential equations of the fourteenth order, allowing for a few first integrals. By reducing the equations to closed canonical form, these integrals are analyzed from the viewpoint of finding a possible solution to the problem via quadratures on zero, intermediate, and maximum thrust sections. In addition to confirming well-known cases of total integrability, this approach enabled us to establish that the essential difficulty of the solution of the space problem with intermediate thrust is reduced to finding one integral, and the solution of the problem with maximum thrust requires two integrals in involution. It is shown that these integrals can be applied to find particular solutions.     

Periodic solutions of the singly averaged hill problem

Based on the ideas of Lyapunov’s method, we construct a family of symmetric periodic solutions of the Hill problem averaged over the motion of a zero-mass point (a satellite). The low eccentricity of the satellite orbit and the sine of its inclination to the plane of motion of the perturbing body are parameters of the family. We compare the analytical solution with numerical solutions of the averaged evolutionary system and the rigorous (nonaveraged) equations of the restricted circular three-body problem.     

Constructive-analytical solution of the problem of the secular evolution of polar satellite orbits

The well-known twice-averaged Hill problem is considered by taking into account the oblateness of the central body. This problem has several integrable cases that have been studied qualitatively by many scientists, beginning with M.L. Lidov and Y. Kozai. However, no rigorous analytical solution can be obtained in these cases due to the complexity of the integrals. This paper is devoted to studying the case where the equatorial plane of the central body coincides with the plane of its orbital motion relative to the perturbing body, while the satellite itself moves in a polar orbit. A more detailed qualitative study is performed, and an approximate constructive-analytical solution of the evolution system in the form of explicit time dependences of the eccentricity and pericenter argument of the satellite orbit is proposed. The methodical accuracy for the polar orbits of lunar satellites has been estimated by comparison with the numerical solution of the system.     

Towards solution of the pulsar problem

The 24-year-old pulsar problem is reconsidered. New results are obtained by replacing the assumption of steady-state discharges near the polar caps by oscillatory discharges, and by creating the neutral-excess pair plasma via inverse-Compton collisions rather than via curvature radiation. As a result, the electrons and positrons which compose the pulsar wind have different bulk velocities and an oscillating space density, and (strong) coherent curvature radiation is implied (without invoking the excitation of instabilities, and contrary to existing proofs of its impossibility). The magnetospheres of young pulsars are likely to have considerable higher-order multipole components, in particular octupole. Radiation transfer through the pulsar magnetosphere results in fan beams whose polarization is dictated by the bottom of the radiation zone, hence, looks like curvature radiation from dipole-like polar caps.Wind generation depends mainly on the quantityB² which takes similar values for the ms pulsars; the latter compensate for (somewhat) weaker fields by wider polar caps and smaller curvature radii.     

Chemical evolution of the Magellanic Clouds: analytical models

We have extended our analytical chemical evolution modelling ideas for the Galaxy to the Magellanic Clouds. Unlike previous authors (Russell &38; Dopita, Tsujimoto et al. and Pilyugin), we assume neither a steepened initial mass function nor selective galactic winds, since among the α-particle elements only oxygen shows a large deficit relative to iron and a similar deficit is also found in Galactic supergiants. Thus we assume yields and time delays identical to those that we previously assumed for the solar neighbourhood. We include inflow and non-selective galactic winds and consider both smooth and bursting star formation rates, the latter giving a better fit to the age–metallicity relations. We predict essentially solar abundance ratios for primary elements and these seem to fit most of the data within their substantial scatter. Our model for the Large Magellanic Cloud also gives a remarkably good fit to the anomalous Galactic halo stars discovered by Nissen &38; Schuster.   Our models predict current ratios of Type Ia supernova to core-collapse supernova rates enhanced by 50 and 25 per cent respectively relative to the solar neighbourhood, in fair agreement with ratios found by Cappellaro et al. for Sdm–Im relative to Sbc galaxies, but these ratios are sensitive to detailed assumptions about the bursts and a still higher enhancement in the Large Magellanic Cloud has been deduced from X-ray studies of remnants by Hughes et al. The corresponding ratios integrated over time up to the present are slightly below 1, but they exceed 1 if one compares the Magellanic Clouds with the Galaxy at times when it had the corresponding metallicities.     

Theory of the isotropic scattering of radiation in a plane layer: Feasibility of obtaining a complete analytical solution of the problem

Analytical investigations of the method of linear nonsingular integral equations, originally proposed by É. Kh. Danielyan [Astrofizika 36,225 (1993)] for the solution of problems in the theory of radiative transport in a medium of finite optical thickness with isotropic scattering, are continued in the present article. It is shown that the solution of problems of the stated class reduce to the determination of only the functions u ^± (, ) in the general case with true absorption. Explicit expressions are obtained for these functions at =0. The feasibility of a complete analytical solution of the problem is newly formulated as the solution of a Fredholm integral equation on the semiaxis with a kernel that admits representation by a superposition of exponential functions [Eq. (25)]. The choice of an efficient procedure for determining the Ambartsumyan -function for a semiinfinite medium is discussed. In particular, a new equation is given for this function.Translated from Astrofizika, Vol. 37, No. 1, pp. 129–145, January–March, 1994.     

An exact analytical solution of Kepler's equation

Complex-variable analysis is used to develop an exact solution to Kepler's equation, for both elliptic and hyperbolic orbits. The method is based on basic properties of canonical solutions to appropriately posed Riemann problems, and the final results are expressed in terms of elementary quadratures.     

A uniform solution of the Lambert problem

A uniform method (with respect to the type of the conic section) for solving the Lambert boundary value problem is presented. It is derived by using the KS-transformation and consists in solving one nonlinear equation. It can be used either in the ordinaryx-frame (physical space) as well as in the KS-coordinates (regularized space).     

Global solution of the ideal resonance problem

If a dynamical problem ofN degress of freedom is reduced to the Ideal Resonance Problem, the Hamiltonian takes the form 1 $$\begin{array}{*{20}c} {F = B(y) + 2\mu ^2 A(y)\sin ^2 x_1 ,} & {\mu \ll 1.} \\ \end{array} $$ Herey is the momentum-vectory _k withk=1,2?N, x ₁ is thecritical argument, andx _k fork>1 are theignorable co-ordinates, which have been eliminated from the Hamiltonian. The purpose of this Note is to summarize the first-order solution of the problem defined by (1) as described in a sequence of five recent papers by the author. A basic is the resonance parameter α, defined by 1 $$\alpha \equiv - B'/\left| {4AB''} \right|^{1/2} \mu .$$ The solution isglobal in the sense that it is valid for all values of α² in the range 1 $$0 \leqslant \alpha ^2 \leqslant \infty ,$$ which embrances thelibration and thecirculation regimes of the co-ordinatex ₁, associated with α² < 1 and α² > 1, respectively. The solution includes asymptotically the limit α² → ∞, which corresponds to theclassical solution of the problem, expanded in powers of ε ≡ μ², and carrying α as a divisor. The classical singularity at α=0, corresponding to an exact commensurability of two frequencies of the motion, has been removed from the global solution by means of the Bohlin expansion in powers of μ = ε^1/2. The singularities that commonly arise within the libration region α² < 1 and on the separatrix α² = 1 of the phase-plane have been suppressed by means of aregularizing function 1 $$\begin{array}{*{20}c} {\phi \equiv \tfrac{1}{2}(1 + \operatorname{sgn} z)\exp ( - z^{ - 3} ),} & {z \equiv \alpha ^2 } \\ \end{array} - 1,$$ introduced into the new Hamiltonian. The global solution is subject to thenormality condition, which boundsAB″ away from zero indeep resonance, α² < 1/μ, where the classical solution fails, and which boundsB′ away from zero inshallow resonance, α² > 1/μ, where the classical solution is valid. Thedemarcation point 1 $$\alpha _ * ^2 \equiv {1 \mathord{\left/ {\vphantom {1 \mu }} \right. \kern-\nulldelimiterspace} \mu }$$ conventionally separates the deep and the shallow resonance regions. The solution appears in parametric form 1 $$\begin{array}{*{20}c} {x_\kappa = x_\kappa (u)} \\ {y_1 = y_1 (u)} \\ {\begin{array}{*{20}c} {y_\kappa = conts,} & {k > 1,} \\ \end{array} } \\ {u = u(t).} \\ \end{array} $$ It involves the standard elliptic integralsu andE((u) of the first and the second kinds, respectively, the Jacobian elliptic functionssn, cn, dn, am, and the Zeta functionZ (u).     

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.     

On the global solution of the N-body problem

In connection with the publication (Wang Qiu-Dong, 1991) the Poincaré type methods of obtaining the maximal solution of differential equations are discussed. In particular, it is shown that the Wang Qiu-Dong'sglobal solution of the N-body problem has been obtained in Babadzanjanz (1979). First the more general results on differential equations have been published in Babadzanjanz (1978).     

Canonical solution of the two critical argument problem

The solution by Sessin and Ferraz-Mello (Celes. Mech. 32, 307–332) of the Hori auxiliary system for the motion of two planets with periods nearly commensurate in the ratio 21 is considerably simplified by the introduction of canonical variables. An analogous canonical transformation simplifies the elliptic restricted problem.     

A fourth-order solution of the ideal resonance problem

The second-order solution of the Ideal Resonance Problem, obtained by Henrard and Wauthier (1988), is developed further to fourth order applying the same method. The solutions for the critical argument and the momentum are expressed in terms of elementary functions depending on the time variable of the pendulum as independent variable. This variable is related to the original time variable through a Kepler-equation. An explicit solution is given for this equation in terms of elliptic integrals and functions. The fourth-order formal solution is compared with numerical solutions obtained from direct numerical integrations of the equations of motion for two specific Hamiltonians.     

A new simple method for the analytical solution of Kepler's equation

A new simple method for the closed-form solution of nonlinear algebraic and transcendental equations through integral formulae is proposed. This method is applied to the solution of the famous Kepler equation in the two-body problem for elliptic orbits. The resulting formulae are quite elementary and, beyond their analytical interest, they can also provide quite accurate numerical results by using Gausstype quadrature rules.     

An exact analytical solution in radiation gas dynamics

The Brinkley-Kirkwood theory, as modified by Bhatnagar and Kushwaha for the inclusion of radiation pressure, is applied to obtain an exact analytical solution for radiation pressure, shock velocity, etc., when a strong explosion takes place in a cold undisturbed gas obeying an exponential density distribution. Cases involving spherical symmetry, axial symmetry or spheroidal symmetry are also considered.