11 - Gravitation -- Kepler and Newton

As discussed in class, Kepler's laws were based on Tycho Brahe's massive amount of data. The laws can be summarized as follows:

1. Planetary orbits are elliptical with the Sun at one focus.

2. Planets sweep out equal areas in equal amounts of time.

3. The square of the period of orbit is proportional to the cube of the semi-major axis. If the units are years and AUs, this is an equality:

T^2 = a^3

e.g, Consider an asteroid with a 4 AU semi-major axis of orbit. How long does it take to orbit once?

Answer: 8 years

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Several decades after Kepler, Newton derived (from geometry) his law of universal gravitation. The derivation is prohibitive to discuss here, but it can be found in Principia (prop. 71). I don't recommend that you read it - many key points are omitted by Newton. In modern language:

F = G m1 m2 / r^2

That is, the force of gravitational attraction is equal to a constant (6.67 x 10^-11 Nm^2/kg^2) times the product of the masses, divided by the distance between the masses squared.

Setting this equal to the local force of gravity (weight) yields a simple expression for local gravitation:

g = G m(planet) / r^2

Finally, we saw in class how Newton's law of gravitation, along with the expression for centripetal acceleration (v^2 / r) can yield Kepler's third law. That is, Newton's law was powerful enough to predict anything known before it, as well as make predictions about the future.