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Chapter 4: Gravity and Orbits Newton’s Law of Universal Gravitation 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 their centers The Force of Gravity depends on both distance and mass Newton’s Laws can be used st to derive Kepler’s 1 Law Newton’s equations give several possible orbits. The planets follow elliptical orbits. Some comets have parabolic or hyperbolic orbits. Circular orbits are only possible if there are only two bodies: a star and a single planet with nothing else in the system. Newton demonstrated that an object in orbit is actually falling Play Newton’s Cannon applet Newton’s Laws can be used to derive Kepler’s 3rd Law Kepler’s 3rd Law was P kA 2 3 where k is a constant Newton showed that, starting with his universal law of gravitation, a little algebra would give 2 4π P A3 GM 2 G is the universal gravitation constant and M is the total mass. In the case of the solar system, M is the mass of the Sun. For multi-star systems, M is the combined mass of the stars in the system. Newton’s form of Kepler’s 3rd Law allows us to determine the mass of the Sun, stars and even galaxies 2 4π 3 P A GM 2 Rearranges to give 2 3 4π A M 2 G P Where M is the total mass in the system Newton’s Universal Law of Gravitation sure seems simple enough m1m2 F G 2 r The gravitational force between any two objects is proportional to the product of their masses and the inverse square of the distance between them. What if there are three objects? How about 4? How do you handle a trillion objects? Let’s take a closer look at gravity…self gravity It works like all the mass is at a point. Once again we have two object, you and the Earth. What if you aren’t on the surface but inside somewhere? If you were at the very center there would be no gravity Now let’s look at three objects at different distances from the Moon mM moon F G ma 2 r All three have the same mass so the closest experiences the largest acceleration and the farthest the smallest Now imagine those three masses are parts of the Earth The Moon’s tidal force is stretching Earth If we look all around Earth we find a tidal force everywhere The solid ground can’t move much but the water can Earth’s rotation drags the tidal bulge around with it The result is high tides occur a little after the Moon is directly overhead The Sun also causes tides but not as strong as the Moon’s Solar tides are less than half the strength of lunar tides The tides are strongly influenced by the shape of the coast and sea floor The tides of Earth on the Moon are much stronger The Moon’s tidal bulge is locked in place. It caused the crust and mantle to be much thinner on the Near Side than the Far Side Earth’s pull on the Moon’s tidal bulge caused it to lock on us Shortly after formation, the tides on the Moon were much stronger. The extreme friction from those tides caused the Moon’s rotation to slow until it its orbital period matched its rotational period. The Moon is spiraling away from us and that is causing our rotation to slow The rate has been carefully measured since 1969 The Moon is receding away from Earth at 3.8 cm/year. Our rotation is slowing at 0.014 sec/century Tidal forces can be strong enough to disrupt bodies Comet Shoemaker-Levy 9 was fractured by tidal forces from Jupiter. It later smashed into Jupiter Watch YouTube video of the impact at http://www.youtube.com/watch?v=DgOTcIfU75Y&NR=1 &feature=fvwp Newton’s Dirty Little Secret After writing his Universal Law of Gravity, Newton immediately saw that adding a third body could make the orbit of an object impossible to calculate. We now call the problem “Chaos” and it means that the orbits of smaller bodies like asteroids and small moons can only be calculated for a few decades into the future. Beyond that, any small difference in the initial conditions make the final result wildly different.