History. Geocentric model (Ptolemy) Heliocentric model (Aristarchus of Samos)

Transcription

1 Orbital Mechanics

2 History Geocentric model (Ptolemy) Heliocentric model (Aristarchus of Samos)

3 Nicholas Copernicus ( ) In De Revolutionibus Orbium Coelestium ("On the Revolutions of the Celestial Orbs"), which was published in Nuremberg in 1543, the year of his death, stated that the Sun was the center of the universe and that the Earth orbited around this center. His theory gave a simple and elegant explanation of the retrograde motions of the planets (the annual motion of the Earth necessarily projected onto the motions of the planets in geocentric astronomy) and settled the order of the planets definitively.

4 Copernican Universe

5 Tycho Brahe ( ) Tycho designed and built new instruments, calibrated them, and instituted nightly observations. Changed observational practice profoundly: earlier astronomers observed the positions of planets and the Moon at certain important points of their orbits (e.g., opposition, quadrature, station), Tycho observed these bodies throughout their orbits. As a result, a number of orbital anomalies never before noticed were made explicit by Tycho. Without these complete series of observations of unprecedented accuracy, Kepler could not have discovered that planets move in elliptical orbits.

6 Johannes Kepler ( ) Using the precise data that Tycho had collected, Kepler discovered that the orbit of Mars was an ellipse. In 1609 he published Astronomia Nova, delineating his discoveries, which are now called Kepler's first two laws of planetary motion. In 1619 he published Harmonices Mundi, in which he describes his "third law." Kepler published the seven-volume Epitome Astronomiae in This was his most influential work and discussed all of heliocentric astronomy in a systematic way. He was a sustainer of the copernican system.

7 Isaac Newton Derived three laws of motion Derived the Law of Universal Gravitation Explained why Kepler s laws worked

8 2-Body Problem = =

9 Not Solving a Problem Can Get You a Prize! The 3-Body Problem remained a nagging problem until..in 1887, the King of Sweden offered a prize for the answer to the question: Is the solar system stable? Poincaré showed the impossibility of solution

10 AERO 660 Nonlinear Flight Dynamics Instructor: Dr. T

11 Nonlinear Dynamical Systems Dynamic changes with time Nonlinear not linear If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living Henri Poincaré

12 We Can Use This Stuff! Space Flight Fluid Mixing

13 Kepler s 1st Law: Law of Ellipses The orbits of the planets are ellipses with the sun at one focus

14 Ellipses Period (T) Semi-Minor Axis (b) FOCI Semi-Major Axis (a)

15 Kepler s 2nd Law: Law of Equal Areas The line joining the planet to the center of the sun sweeps out equal areas in equal times t2 Area 2 t3 Area 1 t1 t0 t1-t0 = t3-t2 Area 1 = Area 2

16 Kepler s 3rd Law: Law of Harmonics In Chinese: The squares of the periods of two planets orbits are proportional to each other as the cubes of their semimajor axes: T1 2 /T2 2 = a1 3 /a2 3 In English: Orbits with the same semimajor axis will have the same period

17 Newton s Laws Law of Inertia: Every body continues in a state of uniform motion unless it is compelled to change that state by a force imposed upon it Law of Momentum: Change in momentum is proportional to the applied force Action Reaction: For every action, there is an equal and opposite reaction Universal Gravitation: Between any two objects there exists a force of attraction that is proportional to the product of their masses and inversely proportional to the square of the distance between them

18 ORBIT CLASSIFICATION Location (equatorial, polar) Shape (circular, elliptical, parabolic, hyperbolic) Size/Period

19 ORBIT CLASSIFICATION Size/Period a Low Earth Orbit (LEO) High Earth Orbit (HEO) Semi-synchronous Orbit Geo-synchronous Orbit LEOs are elliptical/circular orbits at a height of less than 2,000 km above the surface HEOs typically have a perigee at about 500 km above the surface of the earth and an apogee as high as 50,000 km.

20 ORBIT CLASSIFICATION Shape (Conic Sections) Apollonius of Perga ~BC Hypatia of Alexandria ~AD

21 ORBIT CLASSIFICATIONS Characteristics Constant speed Circular Orbits Nearly constant altitude Typical Missions Reconnaissance/Weather (DMSP) Manned Navigational (GPS) Geo-synchronous (Comm sats)

22 ORBIT CLASSIFICATIONS Elliptical Orbits Characteristics Varying speed Varying altitude Asymmetric Ground Track Typical Missions Deep space surveillance (Pioneer) Communications Ballistic Missiles

23 ORBIT CLASSIFICATIONS Parabolic/Hyperbolic Trajectories Characteristics Escaped Earth s gravitational influence Heliocentric Typical Missions Interplanetary exploration (Galileo, Phobos, Magellan)

24 ORBIT GEOMETRY Eccentricity = c/a a Perigee Apogee c c

25 ORBIT CLASSIFICATIONS Eccentricity e = 0 e = 1 0 < e < 1 e > 1

26 ORBIT CLASSIFICATIONS Eccentricity Eccentricity = c/a c = 0 c a a e = 0 0 < e < 1

27 ORBIT CLASSIFICATIONS Eccentricity e = 0.75 e =.45 e = 0 Eccentricity = c/a

28 F g m r r 2 0 m 2r r?

29 Angular momentum H C/S r C/S mv re r m r e r r e mr 2 k d dt mr2 m 2rr r 2 rm 2r r 0 r 2 h p p Gm S h 2