Lecture 8. Kepler's IInd: Angular Momentum

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1 Lecture 8 Gravity and Orbits Angular Momentum Deducing the Law of Gravity Escape Orbits Orbits: common misconceptions Feb 3, 2006 Astro 100 Lecture 8 1 Kepler's IInd: Angular Momentum Kepler's IInd: (motion sweeps out equal areas/time) According to Newton, this is due to the fact that the direction of the force of gravity is along the line between the objects ("down", not "across"). In such a case, mechanics says angular momentum is conserved (angular momentum = mass of orbiting body x distance from center x sideways speed). When orbiting bodies are closer, sideways motion becomes faster. (principle used in figure skating to create rapid spin by pulling in arms). URL for orbit applet: Feb 3, 2006 Astro 100 Lecture 8 2

2 The Universal Force of Gravity Newton suspected that there was one simple force which explained falling objects on the Earth as well as the motions described by Kepler: On the Earth, the acceleration due to gravity does not depend on the mass of the falling body. This fact (unique to gravity) means that the force must be proportional to mass, to compensate for the extra force needed to accelerate more massive bodies. The acceleration due to gravity on the surface of the Earth is: 32 (ft/sec)/sec, towards center of Earth Feb 3, 2006 Astro 100 Lecture 8 3 Moon's acceleration Orbiting bodies must be accelerated, too, since the direction of their velocity is constantly changing. Newton assumed that the Moon was also being accelerated by Earth's gravity. The Moon is 63 times farther than Earth's surface is from the center of the Earth. The acceleration is 3900x smaller than that at the Earth's surface, which is just 63 2! Feb 3, 2006 Astro 100 Lecture 8 4

3 Gravity So, Newton suggests the following law works for everything: Gravity: Two bodies attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. For instance, Newton showed Kepler's Laws follow from his 3 Laws of Mechanics plus the force of gravity operating between sun and planets: F = G m sun m planet / r(sun,planet) 2. Feb 3, 2006 Astro 100 Lecture 8 5 Extending Kepler Assuming this works between any two masses, can make predictions which extend Kepler F = G m 1 m 2 / r(1,2) 2 "G", "Gravitational Constant", is a constant which defines the strength of gravity anywhere in our Universe). Feb 3, 2006 Astro 100 Lecture 8 6

4 Kepler's First and Escape Kepler's Ist: (orbits are ellipses, sun at one focus) According to Newton, possible orbits for this kind of gravity are "conic sections": circle/ellipse: planets around sun, satellites around planets. Closed orbits parabola: not closed. One of the two focuses is "at infinity". Visualize throwing rock horizontally: at circular orbit speed, get circle; faster, get ellipse; at 1.41 x circular speed (escape speed) rock leaves forever in parabolic orbit. eg: Orbits of "long-period" comets. hyperbola: not closed. Throw at speed > escape speed. eg: "Fly-by" orbits of space probes Next: Using this to weigh astronomical objects Feb 3, 2006 Astro 100 Lecture 8 7 Odd notes on orbits Orbits do not work in an atmosphere because friction introduces sideways force. (Shuttle gets out of orbit by firing engine backwards along orbit to reduce its orbital angular momentum; then falls low enough that atmosphere drags it out of orbit). If there were no atmosphere, every thrown object would be in a Keplerian orbit (until it hits the ground). Feb 3, 2006 Astro 100 Lecture 8 8

5 "weightlessness" and Astrogation "weightlessness": Objects in orbit are not really "weightless", they are "supportless": In orbit there is no force opposing gravity => constantly accelerating. Better to say orbiting objects are in "free-fall". within the shuttle, all objects are accelerated at same rate : simulates gravity-free environment. (can do in airplane, too) Astrogation with orbits: very counter-intuitive: To get to higher orbit, thrust to increase velocity in direction of orbit, not up. If you escape from orbit around Earth, you are still in orbit around sun. It is very difficult to visit the sun, since you must get rid of all your angular momentum Feb 3, 2006 Astro 100 Lecture 8 9 The "Vomit Comet" Feb 3, 2006 Astro 100 Lecture 8 10

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