Hyperbolic Trajectories Research Articles

TheK-shell ionization induced by protons: Projectile scattering angle and electronic relativistic effects in the SCA calculations

The behaviour ofK-shell ionization probabilities for heavy charged projectiles scattered at large angles is investigated in the SCA with hyperbolic trajectories and both relativistic and nonrelativistic functions. It is found that the increase of ionization probabilities with increasing projectile scattering angle will be less pronounced for the heavy targets.

The existence of an infinite number of elliptic and hyperbolic periodic trajectories for a convex billiard

The existence of an infinite number of elliptic and hyperbolic periodic trajectories for a convex billiard

Gravitational Radiation from Relativistic Systems

Gravitational radiation is calculated for two selected systems, a pair of masses interacting through scalar or vector fields, to illustrate the behavior of gravitational radiation when the emitting bodies have arbitrarily large velocities. Explicit expressions are obtained for the radiation potentials and radiated power, which can be used for any orbit or trajectory. Although the radiation exhibits the same relativistic peaking as is found in electromagnetism, the amount of energy radiated is found to depend critically on the kind of stresses binding the system. Numerical calculations are done for the cases of circular orbits and extreme hyperbolic trajectories, in order to show the transition from the non-relativistic to extreme relativistic limits. The extreme hyperbolic case also allows comparison with previous results obtained for the gravitational deflection of a small mass in an unbound trajectory.

Optimal Nonplanar Escape from Circular Orbits

Minimum impulse transfer between circular orbit and hyperbolic trajectory velocity vector, assuming fixed transfer times

Vibrational and Rotational Excitations of HD+ by Collision with Positive Ions

Dipole-Type vibrational and rotational excitations of a molecular ion by positive ion impact are theoretically invesitigated. Semi-classical treatment with hyperbolic (Coulombic) trajectory is employed. Cross sections for the vibrational and rotational excitations of HD + are evaluated in the case of low-incident energies as a typical example. The mechanism of the process is considered to be the effect of the time-dependent electric field induced by the traverse of incident ions. The maximum cross section for the vibrational excitation from the ground to the first excited state is eatimated to be 11 a 0 2 (square of Bohr radius) at 40 eV proton impact. For the rotational excitation from the ground to the first excited state, the total cross section is estimated to exceed 200 a 0 2 for proton impact at energies more than 1.3 eV. The comparison with the straight line trajectory approximation is made and the validity of the semi-classical approximation is discussed.

Paraxial Solutions for Accelerated and Decelerated Axially Symmetric Space-Charge Flow

Particle trajectories in the presence of space charge are calculated for prescribed axial potential variations, including linear, parabolic, and sinusoidal variations, using the paraxial-ray equation including space charge for axial symmetry. Solutions are also presented for the potential as a function of distance corresponding to prescribed hyperbolic and parabolic trajectories. The off-axis equipotentials are calculated for some representative cases. These two-dimensional results are interpreted in the light of one-dimensional space-arge theory in which exact solutions can be obtained for the potential as a function of distance. Some of the limitations of the present theory in which either the potential or the trajectory is prescribed are pointed out.

Solution of the Free-Fall Guidance and Control Problem in N-Body Space

The solution of the free-fall guidance and control problem in an inverse-square central force field is treated as an introduction. Circular, elliptical, parabolic and hyperbolic trajectories are covered. Consideration of the control problem (steering and thrust cutoff signals) is applied to the case of a rocket where only the direction of the thrust vector and thrust cutoff are controllable. Consideration is given to the energy aspects, and to launch and atmospheric conditions as they affect guidance and control. These techniques are then extended to the solution of the free-fall guidance problem in N-body space. In this latter category, Bonnet's Theorem, as extended by Egorov, is covered to show its application to the solution of the N-body problem by superimposing solutions of the inverse-square central force field problem. The theorem is applied to treat the problem of finding possible conic trajectories in two-body space. The problem of finding possible conic and nonconic trajectories in three-body space is also covered and a solution to the guidance problem for these trajectories is included. The libration points of the N-body problem are treated as special cases of the solutions discussed. The equations discussed are valid for a vehicle of negligible mass acted upon by forces due only to N inverse-square central force fields. No attempt has been made to cover the effects of noninverse-square central force fields, although Bonnet's Theorem can be applied to constrained trajectories such as those that will be made possible with a continuous low thrust vehicle.

The motion of a sphere in a rotating liquid

This chapter illustrates a vector-based approach to the classical problem of determining the motion of two bodies due solely to their own mutual gravitational attraction. The path of one of the masses relative to the other is a conic section (circle, ellipse, parabola, or hyperbola) whose shape is determined by the eccentricity. Several fundamental properties of the different types of orbits are developed with the aid of the laws of conservation of angular momentum and energy. These properties include the period of elliptical orbits, the escape velocity associated with parabolic paths, and the characteristic energy of hyperbolic trajectories. Following the presentation of the four types of orbits, the perifocal frame is introduced, and this frame of reference is used to describe orbits in three dimensions. The chapter also presents a discussion of the restricted three-body problem to provide a basis for understanding the concepts of Lagrange points and the Jacobi constant.