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8/28/06. Problem of Two Bodies. XYZ is nonrotating, with zero acceleration; an inertial reference frame.
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1. 8/28/06
2. 8/28/06 Problem of Two Bodies
3. 8/28/06 Equations of Motion in the Orbit Plane
4. 8/28/06 Solution of ur Equations of Motion The solution of the ur equation is (as function of ? instead of t):
5. 8/28/06 Types of Orbital Motion
6. 8/28/06 The Orbit and Time If angle f is known, r can be determined from conic equation
Time is preferred independent variable instead of f
Introduce E, “eccentric anomaly” related to time t by Kepler’s Equation:
E – e sin E = M = n (t – tp)
where M is “mean anomaly”
7. 8/28/06 Properties of Hyperbolic Orbits
8. n-Body Problem Assume that we have a s/c denoted as m2 whose motion relative to m1 we wish to describe. Assume that the motion of m2 is perturbed by mj, other masses (planets) where
9. n-Body Problem
10. n-Body Problem
11. n-Body Problem
Where
Recall that , but since there are no external forces, is a constant.
12. n-Body Problem The total energy of the system is given by
The total work done by the external forces is equal to the change in total energy of the system. Because there are no external forces, total energy is conserved.
13. Julian Day Calculation of the Julian Day No. at 0h UT Ref. (Curtis p214). Julian day is a continuous count from the year 4713 BC and begins at 12h UT, i.e., Greenwich mean noon.
Where Y, M, D lie in the following range:
14. Julian Day At any other UT the Julian Day is given by
where
Example: What is the Julian Day Number for Oct. 4, 1957 UT 19:26:24 (The launch date of Sputnick I)?
15. Julian Day
16. Patch Conics The Patch Conic divides the planetary mission into three phases:
The Departure Phase – Bodies are the Earth and S/C, the trajectory is a departure hyperbola with the Earth at the focus. Influence of the Sun and Target Planet are neglected.
Cruise Phase – Bodies are the Sun and S/C and the trajectory is a transfer ellipse with the Sun as the focus. Planets are neglected.
Arrival Phase – Bodies are target planet and S/C. The trajectory is an arrival hyperbola with the Planet at the focus. Sun and Earth are neglected.
17. Patch Conics Phases begin and end at the sphere of influence (SOI). In reality, the SOI often is neglected since the effect is small. For the Earth, the true anomaly on the transfer ellipse should be reduced by 0.15° to account for the Earth’s SOI.
18. Patch Conics Example: Consider an Interplanetary mission from Earth to Venus.
Assumptions:
1) Planetary orbits are circular and coplaner.
2) Transfer ellipse is tangent to Earth & Venus orbit i.e. and transfer is a simple Hohmann ellipse
19. Patch Conics The transfer orbit elements are easily calculated:
20. Patch Conics
21. Patch Conics
22. Patch Conics
23. Patch Conics
