Analytical Study of Tangent Orbit and Conditions for Its Solution Existence

Abstract

This paper presents an analytical study of the two-body tangent orbit technique by providing solution-existence conditions. The flight-direction angle is used to describe and solve this problem. Closed-form solutions are obtained for three classic problems: specified arrival flight-direction angle, specified departure flight-direction angle, and cotangent transfers. Not all of the problems admit solutions; thus, closed-form conditions for solution existence are provided by imposing a positive semilatus rectum constraint and a negative transfer-orbit energy (elliptic orbit transfer) constraint. The final solution-existence condition is then provided in terms of the true anomaly range for initial or final orbit. The singularity problem of 180 deg orbit transfer is also analyzed. Several examples are provided to verify the proposed analytical methods.

Tangent Orbit Technique in Three Dimensions

“…Similar to the 2D case [12], for the tangent to initial orbit problem and the cotangent transfer problem, the conditions of solution existence can be expressed in the range of true anomaly for point P 2 ; for the tangent to final orbit problem, the conditions can be expressed in the range of true anomaly for point P 1 .…”

“…For the cotangent transfer problem, the point P 2 is given, then the following expression should be satisfied for point P 1 [12]:…”

“…Moreover, based on the orbital hodograph theory, Thompson et al [11] solved the tangent transfer orbit problem, but a numerical iterative technique was implemented for the cotangent transfer. Furthermore, Zhang et al [12] provided closed-form solutions for the tangent transfer orbit and its solution-existence conditions. In addition, Zhang et al [13] studied a tangent orbit rendezvous problem which requires the same flight time and the same direction of terminal velocities.…”

Two-impulse transfer between coplanar elliptic orbits using along-track thrust

“…These results were mainly used in orbit escape (Benney 1958;Mengali and Quarta 2009), orbit raising (Kechichian 1998), and asteroid deflection (Bombardelli and Baù 2012) problems. Besides continuous constant thrust, impulse thrust and continuous variable thrust in these three special directions were also studied by many researchers (Petropoulos and Longuski 2004;Quarta and Mengali 2010;Wall and Conway 2009;Zhang et al 2012).…”

“…For impulse tangential thrust, the two-impulse cotangent transfer and rendezvous problems between coplanar noncoaxial elliptic orbits were recently solved (Zhang et al 2012(Zhang et al , 2013. With the assumption of continuous tangential thrust, the coplanar motion equations in polar coordinates can be simplified so that the shape-based method is a widely used approach for low-thrust trajectory approximation.…”