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I had written, about working a lunar-

Next we have to "clear" the observed lunar distance d of the effects of
parallax and refraction, to arrive at the true lunar distance D.

s and m are the altitudes of Sun and Moon corrected for everything except
parallax and refraction.
S and M are the altitudes of Sun and Moon corrected for everything including
parallax and refraction.

I use the formula-
D = arc cos[(cos d - sin s sin m) cos S cos M /(cos s cos m) +Sin S sin M]
and I get D= 85.7384�. This is the corrected lunar distance, that has to be
compared with prediction.

Jeremy replied:

"I think this is the first time I've seen a LD formula without using
Haversines, and am copying it so I can do these with my calculator. "

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

From George, again-

It was never used in the days of logarithms, because of the awkwardness of 
the subtraction and addition within it, but it works fine with a calculator.

It's quite easy to see how it comes about. First, mark on a diagram of the 
celestial sphere surrounding the observer, the apparent positions of the Sun 
and Moon, (corrected for everything except parallax and refraction) and join 
them to his zenith point. The (slanting) great circle distance between those 
positions is the lunar distance, as observed. This is a standard 
navigational triangle, and the included angle, at the zenith, is the 
difference in azimuth between Sun and Moon, which we can call Z. We could 
work out Z from the standard cosine formula, which is -

cos Z = (cos d - cos s cos m) / sin s sin m,

but there's no need to do so, as you will see.

Now apply the corrections, for parallax and refraction, to both bodies, to 
provide new positions M and S. Because the Moon's parallax is so huge, it 
always overrides Moon's refraction, which works in the opposite direction, 
so the net effect is always to decrease the Moon's altitude. So M, the 
corrected altitude, is always greater than m, the correction always pushing 
the Moon upwards, towards the zenith, without making any significant change 
to the Moon's azimuth. For the Sun, parallax is tiny (and for a star, zero) 
, so the corrected Sun altitude, S, is always less than s, but again, the 
correction doesn't alter the Sun's azimuth.

We can draw a different spherical triangle now, joining M, S, and the 
zenith. Because the azimuths are unchanged, the lines radiating from the 
zenith, at angle Z, are just the same as before, but the lengths are 
changed, and the great-circle between M and S represents the true lunar 
distance D, the quantity we are after. That is, the angle between Moon and 
Sun that would have been measured from the centre of a transparent Earth, 
which is what the Almanac predictions apply to.

Now, as before, we can apply the standard cosine formula to that triangle, 
as follows-

cos Z = (cos D - cos S cos M) / sin S sin M

Because, in our two equations, the left-hand side is the same in both cases, 
then the right sides are also equal, so we can write-

(cos D - cos S cos M) / sin S sin M = (cos d - cos s cos m) / sin s sin m

and simply rearranging that leads to-

D = arc cos[(cos d - sin s sin m) cos S cos M /(cos s cos m) +sin S sin M]

which is what we were after.

George.

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







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