1. Oct. 2, 2007 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 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 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…

2. 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 inertial mass –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

3. 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

4. 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/R 2 = 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 = V 2 /R so GMm/R 2 = m V 2 /R and thus V 2 = GM/R

5. Oct. 2, 2007 5 Newton’s form of Kepler’s 3 rd 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 P 2 = (4π 2 /GM) R 3 Or, P 2 = R 3 /M

6. 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