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Review from Last Lecture

Review from Last Lecture. Newton’s Law of Universal Gravitation Spherically symmetric masses act as points Gravitational Field Reduced Mass Two masses replaced by one: Central Force Angular Momentum: Conservation of Energy: Conservative force, define potential energy

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Review from Last Lecture

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  1. Review from Last Lecture • Newton’s Law of Universal Gravitation • Spherically symmetric masses act as points • Gravitational Field • Reduced Mass • Two masses replaced by one: • Central Force • Angular Momentum: • Conservation of Energy: • Conservative force, define potential energy • Remove rotational part using centrifugal potential

  2. Kepler’s Laws • Tycho Brahe (1546-1601) made extensive observations of the planets and the brightest stars (using a sextant, not a telescope!) • Johannes Kepler (1571-1630) used this data to deduce laws about the motion of planets • Kepler’s Laws: • Planets move in elliptical orbits with the sun at one focus. • The radial vectors of planets sweep out equal areas in equal times. • The square of the planet’s orbital period is proportional to the cube of the semimajor axis.

  3. Kepler’s 1st Law • Planets move in elliptical orbits with the sun at one focus. • Major axis: 2a • Minor axis: 2b • The eccentricity e describes how flattened the ellipse is • Ellipse from inverse square nature of gravity

  4. Kepler’s 1st Law • Planets move in elliptical orbits with the sun at one focus. • Point closest to sun: perihelion (perigee for earth) • Point farthest: aphelion (apogee) Aphelion Perihelion

  5. Kepler’s 2nd Law • The radial vectors of planets sweep out equal areas in equal times. • Direct result of any central force and conservation of angular momentum

  6. Kepler’s 3rd Law • The square of the planet’s orbital period is proportional to the cube of the semimajor axis. • Do the circular case

  7. Gravitational Potential Energy • Central Forces are conservative • Approximate path as radial and circular sections • Only radial section do work • Define Potential Energy (w.r.t infinity)

  8. Gravitational Potential Energy • Potential energies from different sources just add • Consider potential in the Earth-moon system (neglecting moon’s orbital velocity) • Red line is potential energy due to Earth • Blue adds in Moon’s PE • Zoom into where force is zero • Lagrange point • Not stable

  9. Total Energy for Orbits • What’s the total energy for an object in orbit? • What’s the velocity? • So

  10. Example: Insertion into Geosynchronous Orbit • Begin in low earth orbit • Find r for geosynchronous orbit

  11. Example: Insertion into Geosynchronous Orbit • First go into elliptical transfer orbit • Then boost into final orbit

  12. Escape Velocity • Total energy tells one the height a projectile can reach • What value of vi lets rmax go to infinity?

  13. Black Holes • What if escape velocity is greater than speed of light? • Then no light can get out of that object! • Call this a Black Hole • The radius at which the escape velocity is c is the Schwarzchild radius

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