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Celestial Navigation

Celestial Navigation. Mr Finn 841 Physics I March, 2006. Trip to Mars. How does NASA send a spacecraft from Earth to Mars? What orbit is used to transfer the satellite from Earth orbit to Mars orbit? Orbits = ellipse described by length of semi-major axis and eccentricity

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Celestial Navigation

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  1. Celestial Navigation Mr Finn 841 Physics I March, 2006

  2. Trip to Mars • How does NASA send a spacecraft from Earth to Mars? • What orbit is used to transfer the satellite from Earth orbit to Mars orbit? • Orbits = ellipse described by length of semi-major axis and eccentricity • How much time does take to make the trip? • Period of this orbit • How frequently can spacecraft be sent from Earth to Mars • How often do “opportunities” arise?

  3. HohmannTransfer Orbit Ellipse a = (REarth+RMars)/2  = Ra = RMars Aphelion Perihelion Rp = REarth

  4. Period of Transfer Orbit • a = (1.0 AU + 1.52 AU)/2 = 1.26 AU •  = 0.206 • Kepler 3rd Law: T  a3/2 • PEarth = 1 yr • aEarth = 1 AU • T = a3/2 with T in “Earth” yr and a in AU • Transfer orbit • T = (1.26)3/2 = 1.42 yr = 518 d • Transit time = T/2 = 259 d or  8.5 mo Time for trip between Mars-Earth

  5. “Opportunity” Position of Mars at Launch Position of Mars at Arrival Position of Earth at Launch

  6. Orientation at Launch (259 d)(360/687 d) or 136 Find angle  between Earth-Mars at launch Mars 259 d later 180 - 136 or 44

  7. How frequently are Earth-Mars 44 apart? How frequently are they aligned? Synodic period = time between alignments of planets True for any orientation: aligned or 44 off! Same problem as: How often do the min and hour hands of a clock align? Time between “Opportunities” Topp 780 d

  8. Summary • Transfer orbit • Semi-major axis = 1.26 AU • Eccentricity = 0.206 • Transit time = 259 d • Opportunity • Orientation at 44 • Occurs every 780 d

  9. Kepler’s Laws • Kepler’s 1st Law on Ellipses • Transfer orbit = ellipse • Provided the geometry • Kepler’s2nd Law on Equal Areas • Gives the launch velocity when combined with conservation of energy • Kepler’s3rd Law on P a3/2 • Period of transfer orbit from geometry

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