Elementary Calculation of the Nodal Precession (Part 3--Dance Mix)

In our previous two posts we calculated a first approximation to the nodal precession as:

dOmega/dt = -3/4 * N^2 / n,

where N and n are the mean motions of the Sun and Moon. According to Chapront-Touzé and Chapront [1] the mean motions of arguments L and D for one year (Julian year of 365.25 days) are:

Mean motion of L (n): 4812.6788°/yr,
Mean motion of D (n-N): 4452.6711°/yr,

so the mean motion of L' (N) is 360.0077°/yr.

Thus dOmega/dt = -0.75 * (360.0077)^2 / 4812.6788 °/yr = -20.1975 °/yr.

The node takes 360/20.1975 = 17.8 years to complete a revolution.

Of course this is only the first approximation, but still this is about 95 percent of the observed value. Not bad for a first approximation!

If we include the next term, +9/32 * N^3/n^2 [2], we find dOmega/dt = -20.1975 + 0.5666 = -19.6309 °/yr, which gives 360/19.6309 = 18.34 years. This is about 98 percent of the result.

Also from [1], the mean motion of F (n-dOmega/dt) is 4832.0202°/yr, so that dOmega = 4812.6788 - 4832.0202 = - 19.3414&deg/yr, which gives 360/19.3414 = 18.6129 years.

As claimed in an earlier post the Sun (not the Earth's equatorial bulge!) is the main source of perturbations in the Moon's motion, accounting for all but the tiniest bit of the precession of the nodes.

The astronomy books do not "lie".

References:
[1] Chapront-Touzé , Michelle and Chapront, Jean, 1991. Lunar Tables and Programs from 4000 BC to AD 8000, p. 12, Willmann-Bell, Inc.
[2] Peterson, Ivars, 1993. Newton's Clock, Chaos in the Solar System, p. 137, W.H. Freeman and Company, New York.

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