Geometric Analysis of Free-Return Trajectories Following a Gravity-Assisted Flyby

Abstract

The purpose of this study is to systematically identify all feasible trajectories following a gravity-assisted flyby that immediately return to the flyby body with no intermediate maneuvers. Every class of possible transfer angles is considered including even-nπ, odd-nπ, and generic return orbits. Lambert's problem is solved for a desired timeof-flight range allowing the possibility for multiple spacecraft and celestial body revolutions. The solutions are expressed geometrically, and the resulting velocity diagram is a mission-planning tool with potential applications that include cycler trajectories and planetary moon tours. The generalized free-return solutions can be used to construct loitering orbits about one celestial body or transfers between multiple bodies. Several previously documented cycler trajectories are improved using the discussed solutions. Nomenclature a = semimajor axis E = eccentric anomaly F = focus of a transfer ellipse M = number of complete revolutions made by the celestial body N = number of complete revolutions made by the spacecraft r = position vector T = period t = time v = velocity vector α, β, S = intermediate variables for Lambert's equation γ = flight-path angle θ, δ, r 1 , r 2 , c = input geometry for Lambert's equation; see Fig. 2a µ = gravitational parameter of the primary ψ = referencing angle for generic returns; see Fig. 12 Subscripts B = celestial body F = full-revolution return f = final H = half-revolution return r = radial component v = vacant focus θ = transverse component 0 = initial ∞ = relative to celestial body before or after flyby

Global Search for Idealized Free-Return Earth-Mars Cyclers

“…Figures 1 and 2 illustrate these solutions for an example v ∞ using the methods described in Ref. 11. These are three-dimensional velocity diagrams showing the common v ∞ sphere associated with gravity-assisted flybys.…”

“…Because of symmetry, when the Earth is in a circular orbit v ∞ at the end of a leg is identical to v ∞ at the beginning. 10,11 Figures 1 and 2 show the outbound v ∞ solutions that lead to all three types of free returns for a given v ∞ . Table 1 summarizes how to obtain the x, y, and z components of the inbound v ∞ based on the x, y, and z components of the outbound v ∞ …”

“…All generic returns with times of flight less than 0.876 years are simply the Earth's orbit. 11 Limiting the search to p = 1 → 3, or cyclers with repeat times of one, two, or three synodic periods, is very reasonable because only one Mars-Earth or Earth-Mars transit is guaranteed during each cycler period. Thus a three-synodic-period cycler typically requires six vehicles to maintain an Earth-Mars transit and a Mars-Earth transit every synodic period.…”

“…To improve the accuracy of the solutions, the current study takes the linearly interpolated solutions obtained from the solution method in Ref. 11 and uses them as initial guesses in a two-dimensional solver that fixes v ∞ and iterates on TOF and ψ in a Keplerian integration until the trajectory reencounters Earth to a desired tolerance. For v ∞ = 5.5 km/s, from Fig.…”

“…11, previously documented cyclers use generic-return transfers for the Earth-Mars transit legs. The current study includes both generic returns and full-revolution returns as transit legs.…”