lambert universal variable algorithm - Arabian Journal for Science ...

M.A. Sharaf, A.N. Saad, and M.I. Nouh
• Computational Sequence:
2 2 2 1 2
0 0 0
1. r0 = ( x + y + z ) .
2. r = (x 2 + y 2 + z 2 ) 1/2 .
3. γ = (x*x0 + y*y0 + z*z0)/(r*r0).
4. β = tm (1 – γ 2 ) 1/2 .
5. A = tm [r*r0*(1 + γ)] 1/2 .
6. If A = 0, we cannot calculate the orbit, then go to step 11.
7. For i: = 1 to M do
begin{i}
ψ = ψ0
Compute C2(≡ C2(ψ)) and C3(≡ C3(ψ)) using algorithm 2.
1
B = r + r+ A∗ ∗C −
{ ( ψ 1)
}
0 2
C2
If A > 0.0 and B < 0.0, then readjust ψL until B > 0.0
χ=
B
C
2
3 ( 3 )
1
∆ t = χ ∗ C + A∗ B
µ
If ∆t−∆ t< Tol, go to step 8
∆t ≤∆t set ψ =ψ , go to step 7.1
If L
Set ψu = ψ
7.1.
ψ = ( ψ +ψ )
1
1 2 u L
ψ0 = ψ1
End {i}
8. Compute F, G, and G from Equations (2.34), (2.35), and (2.36) respectively.
1 1 1
x ( x x F); y ( y y F); z( z z F)
G G G
9. 0 = − 0 0 = − 0 0 = − 0
1 1 1
( ); ( ); ( )
G G G
10. x = Gx− x0 y = Gy− y0 z = Gz−z0 11. End
94 The Arabian Journal for Science and Engineering, Volume 28, Number 1A. January 2003