Degrading Orbits and Lunar Orbit Increase
Unless I propel it further, a satellite's orbit will
degrade, and an object in orbit will fall inward towards Earth. So,
why, is it that every year the Moon is obtaining a larger orbit, or
increasing in distance from Earth? Any impulse outside of the
mutual orbit of Earth and moon would produce an elliptic orbit, but
NOT an outward spiral as described to me in several texts and
Satellites near earth encounter a small but real friction with widely
spaced gasses of earth's upper atmosphere and their orbits slowly
degrade from this.
The moon's orbital velocity is linked to earth's tides. Not only does
the moon pull on the water, the water pulls back on the moon. Right
now, the moon orbits slower that the earth so the tides have the effect
of pulling the moon along. Energy is transferring from earth to moon
and is speeding the moon up. This means the moon will orbit further out
a small bit every day.
At a time in the far future, the moon will orbit just as fast as the
earth turns. It will hang over the same place on earth. Half the
people on earth will not see the moon in the sky. For those who can,
the moon will go through its phases every day. At this point, the moon
will begin to lose energy back to the earth and will orbit closer and
closer to earth with predictable results.
The problem with your conclusion is an assumption that "creeps" into
the analysis. If you restrict yourself to three bodies -- Sun, Earth,
you can, in principle (even though the actual computation could be
very messy), the trajectory of the Earth and Moon with respect to
the Sun. However, there is a hidden assumption that no other planets,
comets, stars, etc. play no part in exerting their gravitational
influence on each of these bodies. However, that is not a valid
assumption. For example, Jupiter, Venus, Mars, and Saturn all exert
a gravitational effect, and a different gravitational effect, on
each of these bodies. As a result the Keplerian elliptical orbits
are only approximate, a good approximation -- but an approximation
nonetheless. The solar system is fundamentally "unstable" to these
small effects, so in principle, if the planets, comets, stars, etc.
were to happen to align in just the right arrangement the Moon
could go "shooting off" in an unpredicted direction. The book
"Mathematics and the Unexpected" by Ivar Ekeland gives a good
qualitative description of the effects involved that are too long
to go into here. Chapter 2 of that small book ( less than 150 pages)
is especially enlightening about how presumably "negligible" effects
can accumulate to de-stabilize events that we take for granted.
The moon's orbit is extremely slowly being pumped up by the earth's
The moon-orbit's angular kinetic energy and momentum go up,
simultaneously Earth's rotational energy and rate go down.
So slowly we will not care for millions of years.
And some energy is also lost to heating of Earth's mantle.
All this because tidal distortion of Earth can cause
weak coupling between Earth's spin and moon's orbit.
Earth spins faster than the moon orbits (1 day vs 27 days),
and that sets the direction of energy flow from earth to moon.
If earth had zero spin, the opposite flow would happen.
Sequence of explanation-points is this:
-gravity gradient (1/R^2)
-tidal distortion (earth like egg-shaped drop of liquid)
-drag in fluid body: viscosity and partial rigidity of earth
- egg-distortion travels around earth-body like a sea-wave,
travelling backwards with respect to day/night revolution.
- due to viscosity, long axis of egg lags the earth-moon line with respect
but leads with respect to moon orbit.
- end of egg nearer to moon then has diagonal vector to moon,
tangential component of which pulls moon "forwards",
accelerating it in it's orbit
- gradual acceleration in line of orbit is always converted to orbit
Sorry for the lack of grammar. Faster that way.
You are right that a satellite will gradually fall into the earth
as it collides with molecules, dust particles, and more massive
objects. As it falls towards the earth, its velocity and so
kinetic energy increases, but its potential energy decreases
by a larger amount. For an object in a circular orbit, U = -2K
where U is the potential energy and K is the kinetic energy.
The total energy of the object is E = U/2 = -GMm/R. Here G is
the gravitational constant, M is the mass of the earth, m is the
mass of the object and R is the radius of the circular orbit.
Now to your question: The radius of the moon's orbit is
increasing by 3.8 cm/year as has been measured with great
accuracy using the laser reflectors installed on the moon by
Apollo 11. If you used energy considerations as I did above,
the calculation would say that the moon would spiral into the
earth. However, tidal forces are important and, since the tidal
motions of the earth and moon are not conservative (due to
friction as the water and solid material move due to the tidal
forces), you cannot use those simple energy considerations.
However, angular momentum is conserved. Since the bulge in the
earth due to the moon's tidal force is a little ahead of lining
up exactly with the moon's position due to the earth's rotation,
it increases the angular momentum of the moon in its orbit around
the earth. This takes angular momentum from the earth's rotation
and transfers it to the moon's orbit. This will continue for about
2 billion years at which time the earth and moon will always face
each other with the same side. Since the angular momentum is given
by L = mvR, and v is decreased, R is increased, explaining the
increase in the radius of the orbit.
I hope this is helpful to you.
Best, Dick Plano, Professor of Physics emeritus, Rutgers University.
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Update: June 2012