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Newton: Understanding Kepler’s Laws & Orbits

Newton: Understanding Kepler’s Laws & Orbits. Review: orbits as revealed by Kepler’s Laws for motion of the planets – a simulation (click on link below when in slide show): http://astro.unl.edu/naap/pos/animations/kepler.html

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Newton: Understanding Kepler’s Laws & Orbits

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  1. Newton: Understanding Kepler’s Laws & Orbits • Review: orbits as revealed by Kepler’s Laws for motion of the planets – a simulation (click on link below when in slide show): http://astro.unl.edu/naap/pos/animations/kepler.html • Experiments: how does period P depend on semi-major axis a? on e? • This explains (and was motivated by) retrograde motion: http://mars.jpl.nasa.gov/allabout/nightsky/nightsky04-2003animation.html • Experiments: how is synodic period, S, of conjunction related to sidereal period, P? If Mars were closer to Earth, would synodic period be longer or shorter? • But how to understand why and how orbits work? Need to introduce concepts of force, mass (and inertia) and acceleration… Oct. 2, 2007

  2. Deconstructing force, mass&acceleration • Start with mass: the quantity of matter (which is proportional to its weight, but not equal – your mass does not change when you are weightless in Shuttle!) • Mass is what “causes” inertia. Push on a car vs. a bike, and your ability to move it with fixed strength (force…) is a measure of its inertialmass • Inertial mass is same in space shuttle on orbit as on ground. Mass is a direct measure of number of atoms (or molecules, or ions, etc.) in a given object; not dependent on location • Mass is what will govern the fate of the stars Oct. 2, 2007

  3. And now for Force and Acceleration • Force was just defined: it’s what must be applied to overcome inertia and move a mass • To “move a mass” by “applying a force”, we must accelerate the mass: change its velocity, from 0 (rest) • The force needed to achieve a given acceleration, a, is directly proportional to the mass, M: F = M a Oct. 2, 2007

  4. Connection to Orbits? • The force needed to continuously change the direction of an orbiting object with mass m is due to Gravity between it and mass M, F = GMm/R2 = m a where G is Newton’s constant and R is the distance between M and m (and is the semi-major axis, a, of an orbit) • The acceleration of mass m moving about M at velocity V can be shown to be a = V2/R so GMm/R2 = m V2/R and thus V2 = GM/R Oct. 2, 2007

  5. Newton’s form of Kepler’s 3rd Law • But orbital velocity V around an orbit with “radius” or semi-major axis R is just V = 2π R/P = circumference/period So substituting this for V we have P2 = (4π2/GM) R3 Or, P2 = R3/M Oct. 2, 2007

  6. Discussion of Errors…. • Concept of uncertainties in our labs • Examples (on blackboard) of measurements and their “scatter” about a mean • Conversion of this scatter into estimate of overall uncertainty: • Mean error • “Root Mean Square” (rms) error Oct. 2, 2007

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