Equations of Motions First order perturbation about natural circular Keplerian orbit Modified Hill's Equations 0r=-152-2)nx-2(nc)y (velocity vector) a,=j+2(nc)i a =z+k where 3E[1+3cs(2 √1+s re 3nJR- k=nv1+s+ Possible trajectory for Techsat 21: 200+-----i----i-Zx1 Ellipse i cOS Eliptical Trajectory x=A cos(ntv1-s) 0 2√1+s R-200 A sin(nt S 400 ------ Projected Ci 400 2√1+s A coS(kt -200 S Cross axis 400400 Velocity Vector Space Systems Laboratory Massachusetts Institute of TechnologySpace Systems Laboratory Massachusetts Institute of Technology Equations of Motions Equations of Motions • First order perturbation about natural circular Keplerian orbit • Modified Hill’s Equations: ( ) ( ) ( ) 2 2 2 5 2 2 2 x y z a x c n x nc y a y nc x a z k z = − − − = + = + && & && & && cos( 1 ) 2 1 sin( 1 ) 1 2 1 cos( ) 1 o o o x A nt s s y A nt s s s z A kt s = − + = − − − + = − − • Possible trajectory for Techsat 21: -400 -200 0 200 -400 400 -200 0 200 400 -400 -200 0 200 400 Velocity Vector Elliptical Trajectory Projected Circle 2x1 Ellipse Cross Axis Z eni t h-Nad i r ( ) 2 2 2 3 1 3cos 2 8 e ref ref J R s i r = + ⎡ ⎤ ⎣ ⎦ where c s = 1+ ( ) 2 2 2 2 3 1 cos 2 e ref ref nJ R k n s i r = + + ⎡ ⎤ ⎣ ⎦