Re: Reason for Procession of the Equinox?

From: Brian Tung (brian_at_isi.edu)
Date: 11/30/04 

Date: Tue, 30 Nov 2004 20:56:18 +0000 (UTC)

Richard DeLuca wrote:
> I'm not a complete stranger to celestial mechanics, being familiar and
> comfortable with such concepts as nutation and cyclical changes in the
> obliquity of the Ecliptic. However, I've never really understood the
> precise *cause* of the Earth's precession, and why its period is so
> long. My Google searches have not been particularly productive, and
> some explanations I've found seem rather farfetched, even bizarre.
>
> I have other questions about some peculiar little-known effects of
> precession, but they should wait until I have a better understanding of
> precession itself. Any help is much appreciated.

Hey, this is one of my upcoming Astronomical Games essays. Here's a
summary.

As others have pointed out, precession occurs because of the gravitational
pull of the Moon on the Earth's equatorial bulge. I've never been
satisfied with that explanation myself, because it's just too brief. For
instance, let me open with a couple of questions.

    1. Precession is an example of symmetry breaking. Why does the
        Earth's axis precess in the direction that it does, and not the
        other?

    2. The precession of the Earth's axis has been likened to that of
        a spinning top. Anyone who's played with a top knows that it
        precesses (although they usually don't know the name for it).
        But observe: If you spin a top clockwise, as seen from above,
        the top's axis also precesses clockwise.

        Now consider the Earth's axis. The Earth rotates on that axis
        from west to east--that is, counter-clockwise as seen from above
        the north pole. But the path of the celestial pole in the skies
        as seen from the Earth is counter-clockwise, meaning that as
        seen from above the north pole, the axis precesses *clockwise*,
        or the direction opposite from the spin. Why should they be
        different, if the precession is like that of a top?

To answer these questions, consider a simple top, which consists of a
flat disc impaled by the spin axis. If you lean the top over, so that
it's not standing straight up, and then let go, it falls over in the
direction that it was leaning; that's simple gravity. But if you spin
it first, it precesses. Why?

The common answer is that it now has angular momentum. You'll excuse
me if I find that a non-answer. Let's suppose the top is rotating
clockwise as seen from above, and that we have it leaning to the left.
If it weren't for gravity, then the lowest point of the top, at the
left, would be moving away from you, and the highest point of the top,
at the right, would be moving toward you.

But gravity adds a tipping force to the top. The leftmost point of the
top therefore moves away from you *and a little downward*, and the
rightmost point moves toward you *and a little upward*. But that's just
the same as a top that is tilted to the left *and a little away from
you*. That is a slight clockwise motion of the top's axis.

What's more, if you now shift your way around the top so that the new
axis is pointing directly to your left, gravity will now act in the same
way to tip the axis again to the left *and a little bit away from you*.
This continues to happen no matter how far around the top you revolve
to maintain the same perspective on the top. In other words, the axis
of the top precesses.

Note that the top precesses clockwise, as seen from the top. This is
now seen not simply as a consequence of the spin of the top, but the way
that spin interacts with the direction of gravity's pull.

Now, over to the Earth and the Moon. The Moon revolves around the Earth
in more or less the ecliptic plane. (We'll ignore the inclination of
the orbit for the time being.) If the Earth were perfectly spherical,
there would be no "handle" for the Moon to pull on, and no precession.

But the Earth does bulge, as a result of its relatively fast rotation.
The Moon pulls differently on different parts of that bulge. That part
of the bulge which lies above the ecliptic plane, the Moon pulls down,
toward the plane. That part of it which lies below the plane, the Moon
pulls up, again toward the plane. The net force is to tip the Earth's
"upward," so that it is perpendicular to the ecliptic. Observe that
this tipping is opposite to the tip that gravity exerts on the top; there
gravity tended to pull the axis down so the top falls. And this makes
all the difference in the world! (Ahem.)

Now, when it's northern summer, say, the northern axis is tilted toward
the Sun. If you view the Sun and Earth so that the Sun is on the left,
the axis is also tilted leftward, and the leftmost portion of the bulge
is rotating toward you. The rightmost portion of the bulge is rotating
away from you. But the tipping force exerted by the Moon means that
that leftmost part (which is below the ecliptic) is also moving just a
tiny bit upward, and the rightmost part (which is above the ecliptic)
just a tiny bit downward. That's the same as saying the axis is now
tilted to the left *plus a little bit away from you*. And that's the
clockwise precession of the axis. Note that it goes opposite from the
Earth's spin, because the Moon's pull does not tend to tip the axis over;
it tends to tip it back up, so to speak.

The Moon's pull is so weak, and the Earth's bulge so small (it amounts to
only about a third of a percent of the Earth's average diameter) that the
precession is very slow indeed--about 26,000 years, as measured directly.
Amazingly, this precession was first measured, by comparing star maps of
different ages, by Hipparchus around the second century B.C.!

Brian Tung <brian@isi.edu>
The Astronomy Corner at http://astro.isi.edu/
  Unofficial C5+ Home Page at http://astro.isi.edu/c5plus/
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  My Own Personal FAQ (SAA) at http://astro.isi.edu/reference/faq.txt

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