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Name: John
Status: other
Grade: other
Location: NV
Country: N/A
Date: N/A


Question:
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 publications.


Replies:
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.

R. Avakian


The problem with your conclusion is an assumption that "creeps" into the analysis. If you restrict yourself to three bodies -- Sun, Earth, Moon -- 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.

Vince Calder


John-

The moon's orbit is extremely slowly being pumped up by the earth's faster spin. 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,
-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 to spin, 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 altitude.

Sorry for the lack of grammar. Faster that way.

Jim Swenson


Dear John,

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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