Problem: • The time equation for elliptical orbits has the form M = φ - E sin φ ; where M is known as the mean anomaly, φ is known as the eccentric anomaly and E is the eccentricity (between 0 and 1) of the elliptic orbit. Obtain a solution that satisfies the εra < 0.001. Use Simple Fixed Point Iteration if E = 1.
Solution: • φ - E sin φ - M = 0 • εra < 0.001 • • • •
E=1 M = mean anomaly φ = eccentric anomaly E is the eccentricity (between 0 and 1) of the elliptic orbit.
• f(φ) = φ - E sin φ -M • M = (0+1)/2 = 0.5 • f(φ) = φ - 1 sin φ - 0.5
• f(φ) = φ - sin φ - 0.5
E=1
SFPI:
f(φ) = φ - sin φ - 0.5
• φ = sin φ + 0.5
---- 1
• φ =sin -1 ( φ - 0.5 )
---- 2
φ = sin φ + 0.5 φi
F(φi+1)
φi
F(φi+1)
0.0000 0.5000 0.9794 1.3302 1.4712 1.4950 1.4971 1.4973
0.5000 0.9794 1.3302 1.4712 1.4950 1.4971 1.4973 1.4973
1 1.3415 1.4738 1.4953 1.4972 1.4973
1.3415 1.4738 1.4953 1.4972 1.4973 1.4973
NRM:
f(φ) = φ - sin φ - 0.5
• F’(φ) = 1 - cos φ if φ = 0 φi F(φi) F’(φi) φi+1 0.0000 -0.5000 0.0000 #DIV/0! #DIV/0! #DIV/0! #DIV/0! #DIV/0!
f(φ) = φ - sin φ - 0.5
NRM:
• F’(φ) = 1 - cos φ if φ = 1.0 φi
F(φi)
F’(φi)
φi+1
1 1.7428 1.5229 1.4976 1.4973
-0.3415 0.2576 0.0240 0.0003 0.0000
0.4597 1.1712 0.9521 0.9269 0.9266
1.7428 1.5229 1.4976 1.4973 1.4973