Download - Chapter 4: Gravity and Orbits
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Chapter 4: Gravity and Orbits
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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
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The Force of Gravity depends on both distance and mass
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Newton’s Laws can be used to derive Kepler’s 1st
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.
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Newton demonstrated that an object in orbit is actually
falling
Play Newton’s Cannon applet
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Newton’s Laws can be used to derive Kepler’s 3rd
Law
32
2 AGM4πP
Kepler’s 3rd Law was where k is aconstant
2 3P kANewton showed that, starting with his universal law of gravitation, a little algebra would give
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.
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Newton’s form of Kepler’s 3rd Law allows us to determine the mass of the Sun, stars
and even galaxies3
22 AGM4πP
2
32
PA
G4πM Rearranges to give
Where M is the total mass in the system
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Newton’s Universal Law of Gravitation sure seems
simple enough
221
rmmGF
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?
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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.
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What if you aren’t on the surface but inside
somewhere?
If you were at the very center there would be no gravity
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Now let’s look at three
objects at different
distances from the Moon
mar
mMGF moon 2
All three have the same mass so the closest experiences the largest acceleration and the farthest the smallest
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Now imagine those three masses are parts of the
EarthThe Moon’s tidal force is stretching Earth
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If we look all around Earth we find a tidal force everywhere
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The solid ground can’t move much but the water can
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Earth’s rotation drags the tidal bulge around with it
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The result is high tides occur a little after the Moon is
directly overhead
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The Sun also causes tides but not as strong as the Moon’s
Solar tides are less than half the strength of lunar tides
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The tides are strongly influenced by the shape of
the coast and sea floor
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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
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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.
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The Moon is spiraling away from us and that is causing our rotation to
slow
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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
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Tidal forces can be strong enough to disrupt bodies
Comet Shoemaker-Levy 9 was fractured by tidal forces from Jupiter. It later smashed into JupiterWatch YouTube video of the impact at http://www.youtube.com/watch?v=DgOTcIfU75Y&NR=1&feature=fvwp
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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.