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Frank Reed has written-

>Regarding the calculation of the lunar distance predictions in the almanac,
>George H wrote:
>"Then these 3-hour Moon positions were converted to declination and right
>ascension"
>
>Although they did that anyway for other tables in the almanac, it was not
>strictly necessary for the lunars tables and really would have made the problem
>more difficult. Consider especially the case of Sun-Moon lunars tables. If you
>stay in ecliptic coordinates, the calculation is very short since the Sun's
>ecliptic latitude is zero.

=============

Response from George-

I described the work undertaken by the human "computers" for Maskelyne as
follows.

"Lunar distances were computed in four steps, as I understand it.

First the position of the Moon was calculated, in terms of ecliptic
latitude and ecliptic longitude, at noon and midnight, using Mayer's
predictions. These were VERY complex calculations, so the largest possible
time-interval was chosen, of 12 hours.

Then interpolations were made between those positions, to get the Moon's
position at 3-hour intervals. Because the moon's lat and long vary so
wildly, linear interpolation wasn't sufficient. Instead, a second-order
interpolation was done, which took the curvature into account, by looking
at four successive positions of the Moon.

Then these 3-hour Moon positions were converted to declination and right
ascension,

Then appropriate bodies (Sun or stars) were chosen, East and West of the
Moon, near the Moon's path and with an appropriate angular distance from
the Moon for easy measurement with a sextant. Then the angle between the
Moon and the star was calculated from the decs and RAs."

==============

Frank is quite correct.

When I wrote that the predicted Moon positions (in ecliptic coordinates)
would be converted to dec. and RA, that was an assumption that I made; I
didn't KNOW it to be true. And Frank's sensible comments, together with
some evidence from the first edition of Maskelyne's "British Mariner's
Guide" (1763), persuade me that I was indeed quite wrong. The step of
converting the Moon positions to dec and RA was NOT required. Instead, the
calculation of the angle between Moon and star would be made in ecliptic
coordinates, from the ecliptic latitudes and longitudes of the two bodies.
The evidence from Maskelyne's work was his table giving the ecliptic
coordinates of 12 zodiacal stars (with corrections for the slow changes
over the years).

So the third step in my account of the computation process should be
omitted, and the final step altered to-

"Then appropriate bodies (Sun or stars) were chosen, East and West of the
Moon, near the Moon's path and with an appropriate angular distance from
the Moon for easy measurement with a sextant. Then the angle between the
Moon and the body was calculated from their ecliptic lats and longs.

===============

I'm less convinced by Frank's follow-up, however, in which he said-

>The lunar distance, LD, is calculated from the
>difference in ecliptic longitude and latitude using the simple rule
>  cos(LD) = cos(diff_longitude)*cos(diff_latitude)
>With the stars and planets, the ecliptic latitudes are low enough that
>certain approximations can be applied which also reduce the task of
>computation.
>Rotating to RA and Dec would lengthen the calculations quite a bit.

===============

Using the "simple rule" quoted by Frank, the lunar distance between the
Moon at a celestial lat of S 5 deg and a zodiacal star at N 10deg, 90 deg
apart in celestial longitude, would be exactly 90 deg. Calculated properly,
however, I make that lunar distance to be 90.867 deg. Far too large a
discrepancy to use for calculating lunar distances, needed to a small
fraction of an arc-minute.

That precise calculation can be made using the familiar solution to the
analogous problem of zenith angle, from -

cos ZA = sin lat sin dec + cos lat cos dec cos hour-angle

in which the celestial lats of Moon and body are used instead of lat and
dec (using + for N, and - for S), and the difference in celestial long
substitutes for hour-angle.

George.

================================================================
contact George Huxtable by email at george@huxtable.u-net.com, by phone at
01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
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