Propellantless Stationkeeping at Enceladus via the Electromagnetic Lorentz Force

Abstract

An introduction to the dynamics of an electrostatically charged spacecraft in two-and three-body regimes is presented, with particular attention to a promising application at Enceladus. Equilibrium solutions to the equations of motion are found, and the stability of the orbits assessed. The perturbative Lorentz force, produced by interaction of the spacecraft with a planetary magnetic field, creates trajectories that are not possible with Newtonian gravity alone. A heuristic, analytical control law with single-variable scalar feedback allows a charged spacecraft to remain near the collinear Lagrange points on either side of Enceladus. The mission feasibility depends on the charge requirement, which is primarily affected by the navigational accuracy. For example, an insertion error into the cis-Enceladus Lagrange point of 10 km in position and 1 m=s in velocity requires a specific charge of approximately 0:05 C=kg. Other significant gravitating bodies in the Saturnian system do not have a substantive effect on the charge requirements or on the stability. Given a sufficiently accurate insertion, a charged spacecraft could maintain station near Enceladus without propellant. Nomenclature A= magnetic vector potential, T km A = tether cross-sectional area, mm 2 a = acceleration vector at Enceladus, m=s 2 a = acceleration at Enceladus, m=s 2 B = magnetic field vector, T B 0 = magnetic moment, T km 3 E = planetary electric field vector, km=s 2 C F g = specific gravitational force vector, km=s 2 F L = specific Lorentz force vector, km=s 2 I = tether current vector, C=s I = tether current, C=s L = Lagrangian, km 2 =s 2 l = length of tether, km M = generalized specific angular momentum, km 2 =s m tot = total mass of tether spacecraft, kg P = tether power, W q = specific charge, C=kg R = radius of reference circular orbit, km r = position vector, km r = radial unit vector r = radial coordinate, km r E = orbital radius of Enceladus, km U eff = effective potential, km 2 =s 2 u k = roots of the effective potential (k 1, 2, 3), km 1 V = tether voltage, V v = inertial velocity, km=s v rel = velocity relative to magnetic field, km=s Z = resistance of tether ribbon, = tuning parameter of control law v i = perturbation of velocity in the i-coordinate direction, km=ŝ = polar unit vector = polar coordinate, rad = gravitational parameter, km 3 =s 2 E = gravitational parameter of Enceladus, km 3 =s 2 S = gravitational parameter of Saturn, km 3 =s 2 = azimuthal unit vector = azimuthal coordinate, rad = resistivity of tether material, m = orbital angular velocity of Enceladus, rad=s ! = planetary rotational angular velocity, rad=s ! S = rotational angular velocity of Saturn, rad=s

Lagrange’s planetary equations for the motion of electrostatically charged spacecraft

“…For the sake of brevity, we do not present here the lengthy expression for the mean motion; most of the effects on the mean motion are small and have been observed in earlier analyses (Gangestad et al 2009;. Changes in the mean motion will primarily affect the time-variation in the position of the Lorentz spacecraft.…”

“…5. The trajectory was numerically integrated with the original equations of motion (Gangestad et al 2009) (i.e. not the Lagrange planetary equations) using initial conditions at Jupiter, where a 0 = 1.74, R J = 0.8R sync and e 0 = 0.4 (s 0 = 0.755R sync ).…”

“…Also, we assume that Lorentz spacecraft may be modeled as a charged point mass, may modulate charge at high bandwidth, may create both charge polarities, and that the rise time of spacecraft charging is on the order of milliseconds (Schaub et al 2004). We will test the validity of our analytical results by comparing them to numerically integrated trajectories that use equations of motion which have been derived elsewhere (Gangestad et al 2009), in lieu of comparing them to numerical integrations of …”

Fuel-optimal Lorentz-augmented Spacecraft Formations Using Novelty Minimum Sliding Mode Error Feedback Controller

“…It always works as a means of auxiliary propulsion to reduce the expenditure of the fuel onboard, which can extend the work life and reduce the life cycle costs for the most space missions [3][4]. Thus, the concept of Lorentz-augmented spacecraft has attracted the enough attentions and become the hot research spot, since it is proposed [5][6][7].…”

“…Various applications of the Lorentz force acting on the spacecraft have been studied and used to achieve Earth synchronous orbit [2], propellantless maneuver [5] and rendezvous [8][9], planetary capture [10], spacecraft hovering [4,6,11], and spacecraft formation flying (SFF) [3,[12][13][14] and so on. In recent decades, SFF has been identified as an enabling technology for many future space missions [15].…”