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In "Astronomical Algorithms" (1st edition, chapter 45) Jean Meeus has a
method to compute the Moon's geocentric apparent place. He claims an
accuracy of 10 arc seconds in ecliptic longitude and 4 seconds in latitude.

I compared his algorithm to the Jet Propulsion Laboratory DE422
ephemeris at several hundred random times in the 21st century. The root
mean square angular separation between Moon positions calculated with
these methods was 3 arc seconds. His accuracy seems comparable to the
Nautical Almanac. If this performance is confirmed in more extensive
tests, I may add the Meeus routine to the next release of my SofaJpl
astronomy DLL. It has the self contained SOFA ephemerides for the Sun
and planets, but lacks anything for the Moon.

One annoyance is that the angles in the Meeus formulae are in degrees,
so you have to convert to radians if an angle is a trig function
parameter. I don't understand why he did that, especially since in the
first chapter Meeus says, "There is the added complication that most
computers can calculate only in radians, not in degrees. It is an
infernal nuisance having to convert degrees to radians all the time, but
on most computers this has to be done before calculating a trigonometric
function of an angle given in degrees."

Well, yes, if your fundamental angles are computed in degrees it is an
infernal nuisance! Initially I neglected to convert his additive angles
A1, A2, and A3 to radians.

I will say this for Meeus, he includes an example and gives the values
of several key quantities. Thus I was able to debug my code without
trouble. The discrepancy was only an arc minute or so, but I was sure
that was too much. After correcting my code, it matched his values
within a few millionths of a degree.

In the same set of tests I also compared geocentric positions of the
Moon in the 21st century as computed by the JPL long term ephemerides
DE406 and DE422. Root mean square difference in position was only .01
arc second. For years I have used the old (1997) DE406 as my main
reference, and based on this limited test I haven't missed much, at
least in epochs not far from the present.

   

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