This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 twofold. First, students will be introduced to the novel notion of Lagrange points. Lagrange points are the solution to a restricted threebody 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 threebody problem ¡ We mentioned that it was impossible to solve by hand, and even computer simulations showed chaos ¡ A special formulation of the threebody 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 timebased 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 comoving, noninertial 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 threebody 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 closedform 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...
View
Full
Document
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.
 Spring '08
 Mazzoleni,A

Click to edit the document details