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Unformatted text preview: J. Peraire, S. Widnall 16.07 Dynamics Fall 2008 Version 1.1 Lecture L16- Central Force Motion: Orbits In lecture L12, we derived three basic relationships embodying Keplers laws: Equation for the orbit trajectory, r = h 2 / = a (1 e 2 ) . (1) 1 + e cos 1 + e cos elliptical orbits Conservation of angular momentum, h = r 2 = | r v | . (2) Relationship between the major semi-axis and the period of an elliptical orbit, 2 2 = a 3 . (3) Time of Flight (TOF) expressions for elliptical orbits, TOF = t B t A = 2 ( M B M A ) (4) M = u e sin u (5) e + cos cos u = . (6) 1 + e cos In this lecture, we will first derive an additional useful relationship expressing conservation of energy, and then examine different types of trajectories. Energy Integral Since there are no dissipative mechanisms and the only force acting on m can be derived from a gravitational potential, the total energy for the orbit will be conserved. Recall that the gravitational potential per unit mass is given by /r . That is, F /m = ( /r ) = ( /r 2 ) e r . Note that the origin (zero potential) for the gravitational potential is taken to be at infinity. Therefore, for finite values of r , the potential is negative. The kinetic energy per unit mass is v 2 / 2. Therefore, 1 v 2 = E constant . 2 r 1 The total specific energy, E , can be related to the parameters defining the trajectory by evaluating the total energy at the orbits periapsis ( = 0). From equation 1, r = ( h 2 / ) / (1 + e ), and, from equation 2, v 2 = h 2 /r 2 = (1 + e ) /r , since r and v are orthogonal at the periapsis. Thus, E = 2 1 v 2 r = 2 h 2 2 ( e 2 1) . (7) We see that the value of the eccentricity determines the sign of E . In particular, for e < 1 the trajectory is closed (ellipse), and E < 0, e = 1 the trajectory is open (parabola), and E = 0, e > 1 the trajectory is open (hyperbola), and E > 0. Equations 1, 2, and 3, together with the energy integral 7, provide most of relationships necessary to solve basic engineering problems in orbital mechanics....
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