453 Note_set_4b

453 Note_set_4b - MAE 453 Intro to Space Flight Dr Scott...

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MAE 453 – Intro to Space Flight Dr. Scott Ferguson Note Set 4B Page 1 of 8 Note Set 4B – Continuing Orbital Position as a Function of Time The purpose of this note set is to two-fold. First, we will finish exploring the development of orbital position as a function of time for parabolic and hyperbolic situations. Finally, we will also introduce the idea of the universal anomaly, and how it can be used to develop a single representation of Kepler’s Equation. Introduction In Note Set 4A, we looked at orbital position as a function of time for circular and elliptical orbits We saw that we needed to use integral tables for the elliptical case We are going to build upon the approach for parabolic and hyperbolic cases Finally, we will also introduce the idea of the universal anomaly It will allow us to come up with a single form of our equations To do this, we are going to use Stumpff functions Parabolic Trajectories When eccentricity has a value of one, Equation (4-1) becomes: 0 2 3 2 ) cos 1 ( e d t h 2 tan 6 1 2 tan 2 1 3 ( 4 - 1 4 ) What are the units of the right-hand side of this equation?? We can define the right-hand side by the variable M p 2 tan 6 1 2 tan 2 1 3 p M ( 4 - 1 5 ) t h M p 3 2 o This is called Barker’s Equation o We can also think of it as the ‘mean anomaly’ So from Equation (4-15), there are two ways the problem can be posed: If the true anomaly is given:
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This note was uploaded on 05/19/2010 for the course MAE 453 taught by Professor Mazzoleni,a during the Spring '08 term at N.C. State.

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453 Note_set_4b - MAE 453 Intro to Space Flight Dr Scott...

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