The Precession Dialogues--Part Six
BH: "I'd like to ask a few questions about the derivation you just presented."
CM: "OK, fire away!"
BH: "Well, can replacing the equatorial bulge, which is actually spread over all latitudes, with a ring at the equator actually be justified?"
CM: "As is the case in most gravitational problems, the exact distribution of the matter in a body is of no consequence at the distances involved. As a zeroth approximation we may pretend that the Sun and the planets are point-particles. In fact we can count the Earth and Moon as a single mass concentrated at the Earth-Moon barycenter as far as the rest of the Solar System is concerned. And this approximation also suffices for calculating the motion of the Earth-Moon system about the Sun; the corrections due to Earth and Moon being separate bodies are small and can be added in after the other perturbations have been calculated.
CM: "But back to the problem at hand: if the Earth were perfectly spherical and the mass distributed in a perfectly spherical manner the gravitational potential U would simply be M/R, where M is the total mass and R the distance from the center of the Earth. All information about the exact distribution of mass is lost, only the total mass matters. (Pardon the pun!
)
CM: "Now the Earth is not exactly spherical, and we can measure the departure using moments. The total mass represents the zeroth moment, the first moment is identically zero because we choose the center of mass of the Earth to be at the origin, so we are left with the second moment as our first indicator of departure from sphericity. It is a symmetric tensor with nine components (six independent), but because of the symmetry of the system only the three diagonal components are non-zero. The three diagonal components, conventionally called A, B, and C are subject to one more condition on account of the symmetry of the Earth: B=A, so we really only have two independent quantities here, A and C. Since we are not considering the higher moments, M, A, and C are the only quantities relevant to calculating the potential and any forces derived from it. Any distribution of mass that gives these three quantities will suffice."
BH: "What about the quantity you called X?"
CM: "As you will recall, there were two equations in two unknowns, m representing the mass of the equatorial bulge and X representing the moment of inertia for the Earth minus its equatorial bulge. We could have solved for X if we wished; it is 2*A-C. I did not solve for it because no further use is made of it; we are only interested in the bulge and its movement."
DB: "Was it really necessary to use integration?"
CM: "The integration is necessary, there is no avoiding it. The usual presentation involves such things as Euler's equations and McCullagh's formula, where the integrations are performed 'offstage' so to speak. It was a pedagogical decision on my part; I wanted to emphasize the role of the bulge, and to use simple definitions such as that of torque. This required me to perform the integration after calculating the torques on the individual bits of mass in the ring. But the integration is over the 'longitude' angle of the ring, it involves only trigonometric functions and is easy, almost trivial. Why, even
Zanket could do it!"
BH: "Can the eccentricities and inclinations really be neglected?"
CM: "In this case they can be neglected because they will only enter at the second power. The eccentricity of the Moon is about 0.055 so its square is about 0.003025. The eccentricity of the Earth is about 0.0167, so its square is even less. The inclination of the Moon is about 5.15 degrees, with a sine of .08976, the square of which is 0.0081. This last one might account for my one percent error earlier."
DB: "But why don't the first powers matter?"
CM: "Because they will be associated with angular arguments that will generally average to zero. Now if we were interested in the periodic terms we would have to consider these terms, but in calculating a non-periodic term such as the precession we must ignore these terms. By the way, if you'll indulge me I will present a calculation of one of those periodic terms, the principal term in the nutation of the longitude and the obliquity in a future dialogue."
BH: "Is it legitimate to subtract the force at the center?"
CM: "This question is often asked when considering tides and tidal effects such as this. The thing to remember is that we are interested in the force on the ring, relative to the rest of the Earth, and the torque that it causes on the ring. The Earth as a whole moves with an acceleration given by the inverse square law as applied at the center of the Earth. We are not interested in the translational motion of the rings that it shares with the rest of the Earth; we want to know what forces and hence what torques are acting on each section of the ring, relative to the Earth as a whole."
BH: "Why did you project the angular momentum vector down to the ecliptic plane?"
CM: "I had just shown that the constant part of the torque was perpendicular to the Earth's angular momentum, so the length of that vector remains constant. What I did not show, and perhaps I was a bit sloppy about it, is that there is no torque in the 'latitudinal' direction, so that the angular momentum vector remains pointed at the same angle with respect to the vertical, that is, the obliquity does not change. This means that the effect of the lunisolar torque is to cause the angular momentum vector to precess and sweep out a cone. Now the torque gives us the velocity in rectangular coordinates but that is not the velocity in angular coordinates. Since the angular momentum vector of the Earth is at co-latitude epsilon, it describes a circle of radius C*n*sin(epsilon) but there are still 360 degrees of 'longitude'."
BH: "One last question. Is the obliquity a constant?"
CM: "Actually, no. The obliquity oscillates over a period of about 40,000 years with an amplitude of 3/4 of a degree. It just so happens that we are at about the mean value of the obliquity, so we can probably use it to estimate the period of the precession of the equinoxes as we have done; but we must realize that it is only an estimate and represents an average period for the equinox to precess through 360 degrees."
To be continued...