453 Note_set_3b

453 Note_set_3b - MAE 453 Intro to Space Flight Dr Scott...

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MAE 453 – Intro to Space Flight Dr. Scott Ferguson Note Set 3B Page 1 of 11 Note Set 3B –Parabolic Trajectories, Hyperbolic Trajectories and the Perifocal Frame The purpose of this note set is to introduce the fundamental principles of parabolic and hyperbolic trajectories. Students will be able to calculate the velocity needed to escape the orbit of a body and the velocity needed for interplanetary missions. Also, students will be introduced to the perifocal frame, which will be used when we explore orbits in three dimensions. Introduction In the last note set, we looked at circular and elliptical orbits Cases where 0 < e < 1 Here, we will look at values of eccentricity equal to one for parabolic trajectories Allows us to determine the escape velocity It’s a trajectory, not an orbit We will also look at values of eccentricity greater to one Allows us to determine the hyperbolic excess velocity – needed for interplanetary missions Finally, we will discuss the perifocal frame, which is needed for when we look at three dimensional orbits Parabolic Trajectories (e = 1) If the eccentricity of the orbit is 1, the orbit equation becomes:  cos 1 1 2 h r ( 3 - 1 9 ) What happens when the true anomaly approaches 180 0 ?? Recalling back to Equation (2-31)  2 2 2 1 2 1 e h ( 2 - 3 1 ) o With the eccentricity equal to 1, this equation reduces to: 0 Which means the conservation of energy equation (Equation (2-30)) becomes: 0 2 2 r v So the speed anywhere on the parabolic path is:

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MAE 453 – Intro to Space Flight Dr. Scott Ferguson Note Set 3B Page 2 of 11 r v 2 ( 3 - 2 1 ) Not only does Equation (3-21) allow us to calculate velocity, but we can also learn some important things from it As m 2 moves radially away from m 1 , what happens to the relative velocity?? But is this really what happens?
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453 Note_set_3b - MAE 453 Intro to Space Flight Dr Scott...

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