Chapter 2 - Part 1 The two body problem AERO 423 – Spring 2006

General Definitions Celestial mechanics or dynamical astronomy is the study of the motion of celestial bodies. Astrodynamics is the study of the motion of man-made objects in space, subject to both natural and artificially induced forces. Celestial bodies and man-made objects are subject of the same Kepler and Newton laws.

The Three Kepler’s Laws The planets orbit the Sun in elliptical orbits, with the Sun at one focus of the ellipse. The radius vector sweeps out equal areas in equal times. The period of the orbit squared is proportional to the semi-major axis cubed.

Kepler’s second law

The Three Newton’s Laws   Inertia: Every object in a state of uniform motion remains in that state of motion unless an external force is applied to it. Dynamics: The relationship between an object's mass m, its acceleration a, and the applied force F is F = ma. Action and reaction: For every action there is an equal and opposite reaction.

Center of mass moves at constant velocity. CM m1 m2 Center of mass moves at constant velocity.

Two body problem Fundamental equation (#1) Derive the following eqs.:

Energy – Escape velocity Fundamental equation (#2)

Angular momentum Fundamental equation (#3)

Eccentricity vector

Fundamental equation solution

Conic Section Geometry a r a(1+e) a(1-e) F b ae p 2a-p 

Circular orbit In general we have

Kepler’s 3rd Law which is the Kepler’s 3rd law!

LEO satellite period (circular)

Satellite period for higher circular orbits