Equations of Motions assumption First order perturbation about natural circular Keplerian orbit Hills Equations x-3n'x-2ny = y+2nx n Total Spacecraft Velocity Increment △V m212, atat a Example: AV required to hold a spacecraft stationary at (x, y, z) Spacecraft instantaneous acceleration △ V required n X a.=n2 △V=nTmv9x2+z Space Systems Laboratory Massachusetts Institute of TechnologySpace Systems Laboratory Massachusetts Institute of Technology Equations of Motions Equations of Motions Assumption : First order perturbation about natural circular Keplerian orbit (cross-range) z x N y (zenith-nadir) (velocity vector) S a z n z a y nx a x n x ny z y x 2 2 2 3 2 = + = + = − − && && & && & a a a dt Tlife ∆ = ∫ x + y + z 0 2 2 2 V Hill’s Equations : Total Spacecraft Velocity Increment : Example : ∆V required to hold a spacecraft stationary at (x,y,z) 2 2 2 V n T 9x z ∆ = life + Spacecraft instantaneous acceleration : ∆V required : ay = 0 a n z z 2 ax n x = 2 = −3