AA 4362 Astrodynamics

1
AA 4362Astrodynamics
Week3 Vallado Chapter 2 Sections 2.1, 2.2, 2.3
2
Kepler's Second Law
3
Area Swept out by an Elliptical Arc
4
Area Swept out by an Elliptical Arc (concluded)
5
Graphical Solution ofThe Area Integral
6
True and Eccentric Anomaly
7
Relationship of EccentricAnomaly to the Elliptic
Area Integral
8
Relationship of EccentricAnomaly to the Elliptic
Area Integral (cont'd)
9
Relationship of EccentricAnomaly to Elliptic
Area Integral (Concluded)
10
AND FINALLY ... (WHEW!) KEPLER'S EQUATION
11
Using Mean Anomaly toPropagate the Orbital
Position
12
Using Mean Anomaly toPropagate the Orbital
Position(concluded)
13
The Propagation Algorithm
14
Propagation Algorithm(continued)
15
Propagation Algorithm(continued)
16
Propagation Algorithm(continued)
17
Propagation Algorithm(concluded)
18
Numerical Solution of Kepler's Equation
19
Numerical Solution of Kepler's Equation (cont'd)
20
Numerical Solution of Kepler's Equation (cont'd)
21
Numerical Solution of Kepler's Equation (cont'd)
22
Steepest-Descent Solution of Kelper's Equation
(Newton's Method)
M 2.0, e 0.5
23
Steepest-Descent Solution of Kelper's Equation
(Iteration 1)
M 2.0, e 0.5
24
Steepest-Descent Solution of Kelper's Equation,
Iteration 1 (contd))
25
Steepest-Descent Solution of Kelper's Equation
(Iteration 2)
M 2.0, e 0.5
26
Steepest-Descent Solution of Kelper's Equation
(Iteration 2) (contd)
27
Steepest-Descent Solution of Kelper's Equation
(Iteration j) (contd)
28
The Solver Algorithm
29
Anatomy of the Solver Algorithm
30
Convergence Criterion
31
Starting Value
M1.818
32
Starting Value (contd)
M1.136
33
Starting Value (contd)
34
Starting Value (contd)
35
Starting Value (contd)
36
Starting Value (contd)
37
Starting Value (concluded)
Since we are solving Keplers Equation for
each and every Time point in our propagation of
the orbit convergence speed becomes critically
important Clearly, since convergence near
the Perigee of an eccentric orbit can be a bit Of
a problem Its clear that we need . a
better way to start each iteration Next time
Startup Algorithms
38
Homework 6