CONTROL OF LAGRANGE POINT ORBITS USING SOLAR SAIL PROPULSION John Bookless*, Colin McInnes+ [email protected] *Dept. of Aerospace Engineering, University of Glasgow, Glasgow G12 8QQ, Scotland, UK +Dept. of Mechanical Engineering, University of Strathclyde, Glasgow G1 1XJ, Scotland, UK Abstract Several missions have utilised halo orbits around the L1 and L2 Lagrange points of the Earth-Sun system. Due to the instability of these orbits, station-keeping techniques are required to prevent escape after orbit insertion. This paper considers using solar sail propulsion to provide station-keeping at quasi-periodic orbits around L1 and L2. Stable manifolds will be identified which provide near-Earth insertion to a quasi- periodic trajectory around the libration point. The possible control techniques investigated include solar sail area variation and solar sail pitch and yaw angle variation. Hill’s equations are used to model the dynamics of the problem and optimal control laws are developed to minimise the control requirements. The constant thrust available using solar sails can be used to generate artificial libration points sunwards of L1 or Earthwards of L2. A possible mission to position a science payload sunward of L1 will be investigated. After insertion to a halo orbit at L1, gradual solar sail deployment can be performed to spiral sunwards along the Sun-Earth axis. Insertion ΔV requirements and area variation control requirements will be examined. This mission could provide advance warning of Earthbound CME (Coronal Mass Ejections) responsible for magnetic storms. Introduction conditions which lead to periodic solutions of the linearised three-body equations. Due to the Solar sailing is an emerging form of propulsion instability of these orbits, station-keeping which utilises solar radiation pressure to provide techniques must be applied to prevent escape a useful thrust. The sail is a thin reflective film after orbit insertion. The solar sail can be made with a thickness of order 2 μm, with a large to track the nominal trajectory by applying trims reflective area in order to intercept a flux of to the sail reflective area or sail pitch angle.11 photons. A key advantage of solar sails over Optimal control laws are applied to minimise the conventional propulsion systems is that missions variation of sail area or pitch angle required to are not constrained by the ΔV available from provide orbit control. stored reaction mass. This enables many new and exciting high-energy mission concepts. The Geostorm mission, developed at JPL, aims 1,2 3 Several authors including Leipold , Macdonald to position a science payload Sunwards of L1. 4 and Hughes have recently investigated the After insertion to a halo orbit around L1, the solar potential of solar sails for planetary missions. sail is slowly deployed resulting in a spiraling motion Sunwards along the Sun-Earth line. This 5-7 8 9 10 McInnes , Farquhar , Forward and Morrow would enable continuous monitoring of the solar have demonstrated that the constant wind to detect CMEs. The increase in charge acceleration from a solar sail can be used to density of the solar wind can result in magnetic generate artificial libration points in the Earth- storms being experienced in the vicinity of the Sun three-body problem. This is achieved by Earth. This space weather phenomenon poses directing the thrust due to solar radiation a considerable risk to geostationary satellites pressure in the anti-Sun direction adding to the which can be bombarded by high energy, 12 centripetal force in the rotating Earth-Sun frame. charged particles. A continuum of libration points can be produced Earthwards of L2 or sunwards of L1. Solar Sails Halo orbits can be generated around these libration points by selecting suitable initial 2 Solar sails consist of large area gossamer pressure = κ xi +κ y j+κ zk . These equations structures with a reflective coating which are non-dimensionalised using 1RE (Earth intercept the solar photon flux. Photons incident Radii) as the characteristic length and on the sail impart momentum and the reflection 3 results in a reaction force, thus providing double characteristic time τ = L μE , where μE is the the force which would be imparted to an Earth gravitational parameter.15 A schematic of absorbing surface. Possible sail substrate the three-body model is provided in Fig 1. candidates include the plastics Mylar, Kapton or CP-1 polyimide film developed at NASA- 13 It is assumed that the Earth is located at the Langley. origin and the separation distance between the The solar sail acceleration is dependent on the Sun and Earth, R, is constant. The sail is areal density, also known as the sail loading. assumed to have negligible mass relative to the The sail acceleration, κ, is defined by larger bodies. βμs 2 2 = η 2 cos α cos φn (1) Rs where η the sail reflectivity, Rs is the distance from the Sun, α is the pitch angle of the sail normal vector n to the Sun-line, φ is the yaw angle of the sail normal vector to the Sun-line and β is the solar sail lightness parameter defined as the ratio of solar radiation pressure to gravitational attraction L β = s (2) 2 cμπ sσ Fig 1: Schematic of Sun-Earth Model with 26 artificial libration point La where solar luminosity Ls = 3.86x10 W, c is the speed of light, μs is the solar gravitational parameter and σ is the ratio of solar sail mass to Jacobi Integral surface area known as the loading parameter. Decreasing sail loading increases the lightness The Jacobi integral can be extracted from Hill’s number thus increasing the characteristic equations with the form 14 acceleration. 2 v2 − − 3Ω x22 Ω+ z 22 − .2 r = C (4) Hill’s Equations r Hill’s approximation of the circular restricted where v = x& 2 + y& 2 + z& 2 is the velocity three-body problem can be adapted to include the dynamical effects of solar radiation pressure magnitude and r = xi + yj+ zk . as x 2 The constant of integration, C, is analogous to x&& − 2Ωy& −= + 3Ω x + κ x (3.1) r3 the total orbit energy E, where C = 2E . The y Jacobi constant can be evaluated for a set of y&& + 2Ωx& −= + κ (3.2) 3 y initial conditions. The obtained value of C can r be used to generate a surface of zero velocity. z 2 This surface of constant energy, also referred to z&& −= Ω− z + κ z (3.3) r3 as Hill’s surface, bounds the orbital motion as it 2 2 2 is forbidden for the trajectory to intersect the where r = x + y + z is the separation surface. between the solar sail and the Earth, Ω is the angular velocity of the Earth orbiting the Sun and the acceleration due to solar radiation Libration Points 3 The L1 and L2 Lagrange points of Hill’s problem Hill’s equations are linearised about the libration are symmetric about the origin on the x-axis. point, new coordinates ξ = x − xo , η = y − yo and The position can be calculated from Eqn (3) by ζ = z − z , where (x , y , z ). The linearised setting the acceleration and velocity components o o o o x& = x&& = y& = y&& = z& = z&& = 0 . Solving Eqn (3.1) for x, equation can be represented by the state equation x& = Ax , where the state vector 2 − 31 Τ gives the Lagrange point location x ±= (3Ω ) . x = [ξ η ς ξ& η& ς&] , The linear coefficient Substituting into Eqn (4), the Jacobi constant matrix is defined as evaluated at the Lagrange points has the form 32 15 C −= ()9Ω for the ballistic case, κ = 0 . ⎡ 0 0 0 1 0 0⎤ ⎢ 0 0 0 0 1 0⎥ An artificial libration point, located at x , can be ⎢ ⎥ o ⎢ 0 0 0 0 0 1⎥ generated using the solar sail acceleration to A = ⎢ ⎥ (6) cancel part of the gravitational forces exerted by ⎢ f xx 0 0 0 2Ω 0⎥ ⎢ ⎥ the Earth and the Sun. The required 0 f yy 0 − 2Ω 0 0 acceleration is given by ⎢ ⎥ ⎣⎢ 0 0 f zz 0 0 0⎦⎥ x κ = o − 3Ω2 x (5) o 3 o r nd o where fxx, fyy and fzz are the 2 order derivatives where =κ κoi in the case of a Sun-pointing sail. of the pseudo-potential Table 1 provides the values of Jacobi Constant 1 Ω2 zyxf ),,( = + 3( x2 − z 2 ) +κx (7) for the contours shown in Fig 2. These contours r 2 represent constant energy surfaces of zero- velocity. The values of C, evaluated at the The partial derivatives are evaluated at the libration points represent a critical value for a libration point. closed surface. As orbit energy is increased, a gap opens in the surface around the libration In the x-y plane the linear solution produces two point. real eigenvalues and two imaginary eigenvalues. Using the method outlined by Szebehely16, the No. xo Jacobi Constant Acceleration terms containing real eigenvalues are RE C κ suppressed which obtains the oscillatory 1 234.46 -0.012795 0 solution -6 2 210 -0.015626 6.383x10 3 190 -0.0182518 1.296 x10-5 ξ = yvA sin(λxyτ ) (8.1) 4 170 -0.021287 2.141 x10-5 -5 A 5 150 -0.024921 3.281 x10 η = y cos(λxyτ ) (8.2) Table 1: Jacobi Constant for libration points ς = Ay cos(λzyτ ) (8.3) where Ay is the y-axis orbit amplitude. The in-plane eigenvalue λxy is defined as ⎛ ⎞ 1 ⎜ 2 2 2 ⎟ λxy = ()4Ω− + fxx + f yy − ()4Ω − fxx − f yy − 4()xx ff yy 2 ⎜ ⎟ ⎝ ⎠ (9) and the out of plane eigenvalue λz = f zz .