This preview shows pages 1–8. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: CDS140B: Computation of Halo Orbit Wang Sang Koon Control and Dynamical Systems, Caltech koon@cds.caltech.edu Importance of Halo Orbits: Genesis Discovery Mission I Genesis spacecraft will collect solar wind from a L 1 halo orbit for 2 1 2 years, return those samples to Earth in 2004 for analysis. I Will contribute to understanding of origin of Solar system. L&#13; 1&#13; L&#13; 2&#13; x&#13;(&#13;k&#13;m)&#13; y&#13;(&#13;k&#13;m)&#13;1E+06&#13; 0&#13; 1E+06&#13;1E+06&#13;500&#13;000&#13; 0&#13; 500&#13;000&#13; 1E+06&#13; L&#13; 1&#13; L&#13; 2&#13; x&#13;(&#13;k&#13;m)&#13; z&#13;(&#13;k&#13;m)&#13;1E+06&#13; 0&#13; 1E+06&#13;1E+06&#13;500&#13;000&#13; 0&#13; 500&#13;000&#13; 1E+06&#13; y&#13;(&#13;k&#13;m)&#13; z&#13;(&#13;k&#13;m)&#13;1E+06&#13; 0&#13; 1E+06&#13;1E+06&#13;500&#13;000&#13; 0&#13; 500&#13;000&#13; 1E+06&#13; 0&#13; 500&#13;000&#13; z&#13;1E+06&#13; 0&#13; 1E+06&#13; x &#13;500&#13;000&#13; 0&#13; 500&#13;000&#13; 1E+06&#13; y &#13; X&#13; Y&#13; Z&#13; 3D Equations of Motion I Recall equations of CR3BP: X 2 Y = X Y + 2 X = Y Z = Z where = ( X 2 + Y 2 ) / 2 + (1 ) d 1 1 + d 1 2 . &#13; 1&#13; x z&#13; y&#13; S&#13; S/C&#13; E&#13; 2&#13; d&#13; 1&#13; d&#13; 3D Equations of Motion I Equations for satellite moving in vicinity of L 1 can be obtained by translating the origin to the location of L 1 : x = ( X 1 + + ) /, y = Y/, z = Z/, where = d ( m 2 , L 1 ) I In new coordinate sytem, variables x, y, z are scale so that the distance between L 1 and small primary is 1.10.5 0.5 110.5 0.5 1 y (nondimensional units, rotating frame) m S = 1 &#13; m J = &#13; S J Jupiter's orbit L 2 L 4 L 5 L 3 L 1 comet 3D Equations of Motion I CR3BP equations can be developed using Legendre polynomial P n x 2 y (1 + 2 c 2 ) x = x X n 3 c n n P n ( x ) y + 2 x + ( c 2 1) y = y X n 3 c n n P n ( x ) z + c 2 z = z X n 3 c n n P n ( x ) where 2 = x 2 + y 2 + z 2 , and c n =  3 ( +( 1) n (1 )( 1 ) n +1 ). Useful if successive approximation solution procedure is carried to high order via algebraic manipulation software programs. Analytic and Numerical Methods: Overview I Lack of general solution motivated researchers to develop semianalytical method. I ISEE3 halo was designed in this way. See Farquhar and Kamel [1973], and Richardson [1980]. I Linear analysis suggested existence of periodic (and quasiperiodic) orbits near L 1 . I 3rd order approximation, using LindstedtPoincar e method, provided further insight about these orbits. I Differential corrector produced the desired orbit using 3rd order solution as initial guess . Periodic Solutions of Linearized Equations I Periodic nature of solution can be seen in linearized equations: x 2 y (1 + 2 c 2 ) x = 0 y + 2 x + ( c 2 1) y = 0 z + c 2 z = 0 I The zaxis solution is simple harmonic, does not depend on x...
View Full
Document
 Fall '10
 list

Click to edit the document details