Chapter 4: Gravity and Orbits

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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.
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