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 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.

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

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.

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

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?

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

mar

mMGF moon 2

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

EarthThe 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 JupiterWatch 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.