Classical Hyperbolic Trajectories: Importance of the Impact Angle and Requirements for Collinear Configurations

The mathematical analysis of the hyperbola is extended to distinguish and compare several angles which are important to physical applications, e.g., comets and Rutherford scattering. We quantify the impact angle, which plays a key role in angular momentum conservation, and distinguish it from the angle of deflection emphasized in historical analyses of gravitational bending of light. We elucidate requirements for collinear arrangement of points on opposite limbs with the focus, which configuration pertains to analysis of galaxy images for possible gravitational lensing effects. These geometrical relationships are obtained by transforming the curved trajectory of a mass propelled by Newtonian gravitational forces in polar (R, θ) or conventional Cartesian (x, y) Euclidean space into linear trajectories in Y, R or X, R space, where R, X, and Y are radial and Cartesian distances to the focus, now located at X = 0 and Y = 0. These coordinate transformations simplify geometric analysis of orbits and trajectories, as they allow any conic section to be recast into the compact linear form, R = eY + L, where e is the standard eccentricity and L is the length of the semi-latus rectum. Our formulae specify the limitations of approximations used in physical analyses of light bending, such as considering only asymptotic behavior and small angular deviations. Here we quantitatively describe the entire hyperbolic path, which should be useful in many applications.