453 Note_set_4a

453 Note_set_4a - MAE 453 – Intro to Space Flight Dr....

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Unformatted text preview: MAE 453 – Intro to Space Flight Dr. Scott Ferguson Note Set 4A Page 1 of 10 Note Set 4A – Lagrange Points and Orbital Position as a Function of Time The purpose of this note set is to two-fold. First, students will be introduced to the novel notion of Lagrange points. Lagrange points are the solution to a restricted three-body problem, and represent an equilibrium solution to the differential equations where the velocity and acceleration of a spacecraft are zero. Second, this note set explores the relationship of time using time past perigee as a reference in circular and elliptical orbits. Introduction • In Note Set 2A, we looked at the idea of the three-body problem ¡ We mentioned that it was impossible to solve by hand, and even computer simulations showed chaos ¡ A special formulation of the three-body problem will be presented that allows us to draw conclusions from the equations of motion ¡ We will identify five Lagrange points that represent equilibrium solutions • Next, we will bring back the notion of time in the analysis of an orbit ¡ Right now, we understand orbits as a function of true anomaly ¡ In this note set, we will explore the time-based equations for circular and elliptical orbits Lagrange Points • Before we move on, there is one special case that we can look at ¡ Consider m 1 and m 2 , moving only due to their own gravitational attraction, in a circular orbit about each other ¡ There is a co-moving, non-inertial reference frame xyz with the origin at the center of mass location • Now, let’s put a very small mass in the system ¡ It is going to have a significant impact? Why?? ¡ We call this the restricted three-body problem ¡ What we care about: motion of m (small mass) due to the gravitational fields of m 1 and m 2 ¡ Note: there is no closed-form solution for this problem! We can draw some conclusions, however • We can go through and set up the position, velocity, and acceleration equations ¡ The book does a full explanation...
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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_4a - MAE 453 – Intro to Space Flight Dr....

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