Department of
Science & Technology Studies
University College London

Nicholas Kollerstrom's
Newton's 1702 Lunar Theory

Modern Equations: A Comparison

Nowadays, about one and a half thousand lunar equations are likely to be contained in a modern program, giving a few arcseconds accuracy in historical time. I found that Newton's theory, with its seven steps, was equivalent in accuracy to the first thirteen of fourteen of the modern terms, as if each step of equation was twice as effective as the modern ones.

The main differences between the Newtonian and modern terms are:

Newton invented six new equations, four lunar (2nd, 3rd, 6th and 7th) and two additional 'annual equations' for the apse and node. It took some decades before astronomers started to prepare the new tables for these extra equations, partly because it was such a new idea. Surprisingly, I found that all four of the new lunar equations were valid, in that the program worked better with them than without. I tested each equation separately in this respect. All four of them have equivalent and similar modern terms, of comparable amplitude.

The modern equations for lunar longitude are normally cast in terms of just four terms: solar anomaly (M), lunar anomaly (M'), lunar elongation (D) (angular distance from the Sun) and mean lunar distance from ascending node (F). These are used because they turn up most often in the hundreds of terms comprising the theory. We may add an asterisk to the modern solar anomaly term, as M*, to avoid confusion with the TMM symbol. Before comparing these with the TMM program, we should recall that the modern definitions of anomaly, with respect to perigee and perihelion, are 180° out of phase with the old. We then transform them using the symbols M (Moon), S (Sun), N (node), A (Apogee), and H (Aphelion); thus,

For example, the eleventh of the modern terms in longitude, +0.041sin(M'-M*), becomes 2'28" sin [(M-A) - (S-H)] or 2'28" sin(M-S+H-A), which we can recognise as the sixth equation. The first fourteen modern equations in order of diminishing amplitude are as follows:




1 +6.2888sinM' 6°17'24" sin(M-A) ellipse function
2 +1.274sin(2D-M') 1°16'26" sin(M+A-2S-180) evection
3 +0.658sin2D 39'29" sin2(M-S) 35'32" sin2(M-S) ['5th eqn']
4 +0.213sin2M' 12'49" sin2(M-A) Horrock's theory
5 -0.185sinM* -11' 8" sin(S-H) 11'49" sin(H-S) ['1st eqn']
6 -0.114sin2F -6'51" sin2(M-N) 6'57" sin2(N-M) [reduction]
7 +0.058sin(2D-2M') 3'32" sin2(A-S) 3'45" sin2(A-S) ['2nd eqn']
8 +0.057sin(2D-M*-M') 3'26" sin(M-3S+A+H)
9 +0.053sin(2D+M') -3'12" sin(3M-A-2S)
10 +0.046sin(2D-M*) -2'44" sin(2M-3S+H)
11 +0.041sin(M'-M*) 2'28" sin(M-S+H-A) -2'25" sin(S-M+A-H) ['6th eqn' 1713]
12 -0.034sinD -2' 5" sin(M-S) 2'20" sin(S-M) ['7th eqn']
13 -0.030sin(M*+M') -1'49" sin(M-A+S-H)
14 +0.015sin(2D-2F) 55" sin2(N-S) 47" sin2(N-S) ['3rd eqn']

Table 10.2
The first fourteen modern terms for lunar longitude are given on the left, then restated in the adjacent column using TMM-PC symbols. The TMM equations, not in sequence, are given in the third column, where the first equation refers to the annual equation and the fifth to the Variation.
Here one may be surprised to notice, that the four new equations (2,3,6 and 7) have close to their optimal amplitudes, ie they correspond well to the modern values.

The contents of this page remain the copyrighted, intellectual property of Nicholas Kollerstrom.  Details. rev: May 1998