### Re: Why isn't earth's moon farther away?

Date: Mon May 21 18:42:35 2001
Posted By: Erika Gibb, Grad student, Physics & Astronomy/Origins of Life, RPI
Area of science: Astronomy
ID: 989954309.As
Message:
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Hi Jeremy,

The Moon is indeed moving farther away from the Earth at a rate of about
3.8 cm per year.  At the same time, the Earth's rotation is being slowed by
1.5 milliseconds per century.  This is due to the tidal interaction between
the two bodies.  Tides are due to the difference in force between the side
of Earth facing the Moon and the side facing away.  The force due to
gravity between 2 bodies is F = GMm/R^2 where G is the gravitational
constant, M and m are the masses of the 2 bodies involved (Earth and the
Moon in this case), and R is the distance between the two bodies.  The
difference in force from the side of Earth facing the Moon and the side of
Earth facing away from the Moon (called the differential tidal force) is
proportional to 1/R^3.  This means that as objects get closer together, the
gravitational force between them increases as 1/R^2 and the differential
tidal force increases as 1/R^3.  As an example, if the Moon were half as
far from Earth as it is now, the gravitational force would be 4 times
stronger and the differential tidal forces would be 8 times stronger.

Here's a diagram to help visualize this concept.  The arrows represent the
strength of the force due to gravity from the Moon on Earth (note, the
sizes aren't exact, they are just meant to illustrate the concept) The +
represents the Moon.  The side of Earth closest to the Moon experiences the
greatest force.

-->  --->  ---->                                                       +

From the point of view of the center of Earth, these forces look like

<-  .  ->                                                          +

which shows that the Earth gets "stretched" by the Moon.

Now, let's see the same diagram if the Moon is half as far away.

--------> ------------> ---------------->     +

so from the center of Earth, we see

<-------- . -------->                        +

The Moon's gravity pulls a tidal bulge on Earth.  Earth is rotating faster
than the moon is revolving around Earth, so the bulge gets a little ahead
of the Moon.  The Moon's gravity pulls back on this bulge creating a torque
on Earth.  (The Earth also creates a bulge on the Moon, but since the Moon
orbits Earth in the same amount of time it takes to rotate on its axis, the
bulge is always facing Earth directly and this torque action does not
occur.)  Likewise, the gravitational attraction due to Earth's bulge
accelerates the Moon slightly, causing it to move away from the Earth.  In
the distant past, the Moon was closer to Earth than it is now.  That means
the gravitational force and the differential tidal forces would have been
greater.  The tides would have been much higher than they are now.

Models of the formation of the Moon estimate that it formed at a distance
of about 12,500 to 20,000 miles from Earth.  The Moon is currently about
240,000 miles from Earth.  That means Early in the Moon's history, it was
about 12-20 times closer to Earth than it is now.  The tidal forces would
therefore have been more than a thousand times greater--resulting in tides
of about a mile high or so.  Early in Earth's history, most of the surface
would have been covered by water every high tide.  By the time land animals
evolved, however, the Moon had moved far enough away that the tides were
much lower.  Eventually, far in the future, the Earth's rotation will be
slowed until a day is 47 days long.  At this time, a month will also be 47 days
long.  Earth will always have the same side facing the Moon and the Earth and
Moon will no longer move apart.

For general information about the Moon, visit
http://seds.lpl.arizona.edu/nineplanets/nineplanets/luna.html
therein.

Erika

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