The Precession Dialogues--Part Two

Our coffee cups refilled, we resumed the dialogue.

CM: "I think it is a wonderful thread and a very good example of what can be done with a qualitative argument. We will now try to use this type of argument to deduce the direction of the precession of the equinoxes.

CM: "We start by noting that only the equatorial bulge matters. A perfectly spherical Earth with a spherically symmetric mass distribution would behave as if all the mass were concentrated at its center, that is, like a point-particle. So we will imagine the Earth as having a 'spare tire' of mass concentrated on the equator and treat it like a gyroscope.

CM: "As a further simplification we will consider how the Earth appears from the Sun, but the principles are the same for the Moon. It's just that with the Sun we can associate the various positions that I mention with the seasons and these can be pictured easily.

CM: "We consider ourselves to be in the plane of the ecliptic, looking out towards the Earth. Since this is being written just after the December solstice, we would see the North Pole turned away from us, the South Pole towards us, and the 'spare tire' above us on the side of Earth facing us and below us on the side of Earth facing away from us. Since the Earth rotates counter-clockwise in this coordinate system, points on the 'spare tire' are passing from left to right relative to us on the facing side. If we pull down on the facing side, this causes the motion to go from being left-to-right to being (slightly) left-up to right-down, and that causes the North Pole to shift to the right and the South Pole to the left."

DB: "But aren't we also pulling on the far side?"

CM: "Yes, we are, but remember on the far side the 'spare tire' is moving from right to left and we are pulling up, so the motion goes from being right-to-left to right-down to left-up and that also gives a rightward nudge to the North Pole and a leftward nudge to the South Pole. Now consider the June Solstice. Here the North Pole is turned towards us, the South Pole away from us, the 'spare tire' is now below us, so we are pulling up on it. This causes the North Pole to be nudged to the left and the South Pole to the right, and by the same argument as earlier, pulling down on the far side compounds this motion."

BH: "If the North Pole is first being pushed to the right and then to the left, how do we get a steady precession?"

CM: "Consider where the North Pole is during the solstices. During the December Solstice the North Pole is facing away from us and a rightward nudge causes it to move slightly in the clockwise sense about the vertical, which points to the ecliptic north pole. During the June Solstice the North Pole is facing towards us and a leftward nudge also causes it to move in a clockwise sense.

CM: "Finally, consider the equinoxes. At the equinoxes one pole is facing left and the other right, but both are at equal distance. Every point on the 'spare tire' that is above the plane is exactly matched by one below the plane at the exact same distance from us, so there is no turning of the axis. Thus the Earth's rotational axis is turned most strongly in the clockwise direction at the solstices and not at all at the equinoxes. But it all adds up and turns out to be about 50 seconds of arc per year."

BH: "Can you calculate this? I see that Nereid often asks the hard questions of many of the ATM proponents. Not all of them answer, some of them just blow right on by on their way to banishment. But I never see a mainstream proponent being asked to 'put up'."

CM: "Well, I could be arrogant and say we don't have to, that the mainstream calculations are there in the literature, from elementary treatments in the standard textbooks of celestial mechanics on up to the definitive treatments that our ephemerides are based on. But I think I could give a good elementary treatment here."

DB: "You've presented more difficult calculations on paper napkins at Couch Cafeteria!"

CM: "Yes, let me get a few paper napkins. It'll be just like old times!"

To be continued...

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