lecture_halo_2004

lecture_halo_2004 - CDS140B: Computation of Halo Orbit Wang...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the 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
 1
 L
 2
 x
(
k
m)
 y
(
k
m)
-1E+06
 0
 1E+06
-1E+06
-500
000
 0
 500
000
 1E+06
 L
 1
 L
 2
 x
(
k
m)
 z
(
k
m)
-1E+06
 0
 1E+06
-1E+06
-500
000
 0
 500
000
 1E+06
 y
(
k
m)
 z
(
k
m)
-1E+06
 0
 1E+06
-1E+06
-500
000
 0
 500
000
 1E+06
 0
 500
000
 z
-1E+06
 0
 1E+06
 x 
-500
000
 0
 500
000
 1E+06
 y 
 X
 Y
 Z
 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 . 
 1-
 x z
 y
 S
 S/C
 E
 2
 d
 1
 d
 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.-1-0.5 0.5 1-1-0.5 0.5 1 y (nondimensional units, rotating frame) m S = 1- 
 m J = 
 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 semi-analytical method. I ISEE-3 halo was designed in this way. See Farquhar and Kamel [1973], and Richardson [1980]. I Linear analysis suggested existence of periodic (and quasi-periodic) orbits near L 1 . I 3rd order approximation, using Lindstedt-Poincar 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 z-axis solution is simple harmonic, does not depend on x...
View Full Document

Page1 / 32

lecture_halo_2004 - CDS140B: Computation of Halo Orbit Wang...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online