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Dear George and all,

I very much liked your sketch of an LD algorithm. You give a very clear explanation of
a method that is different from the 'classical' one.


Instead of reducing the apparent distance to the true distance, you work it out the
other way around. You introduce the assumed position (lat + Long + time) of the
observer early into the calculation. That makes it possible, after application of the
(reversed) corrections for refraction and parallax, to compute just the apparent lunar
distance and not the true LD. That is how you circumvent the 'clearing of the
distance'.

Still, I believe that this procedure is computationally just about as expensive as the
'classical' way. Instead of clearing the distance you have to do a transformation from
equatorial coordinates (Dec,GHA) of the bodies to horizon coordinates (Az,Alt). In
terms of amount of work, that is quite comparable to the clearing process (I didn't
make a detailed count).

Early introduction of the observer's position prohibits the use of precomputed lunar
distance tables. That is one of the reasons why your method is more suitable for
computerisation than for hand work. (Actually not using distance tables makes your
process more efficient than any process based on such tables; lunars are so seldom
used nowadays that any table will remain largely unused, and say 99% of the work that
would be put into them would be in vain.)

I wondered for a while if it is at all allowable to introduce the observer's estimate
of lat, long and time in this way. I think it is OK. As long as the estimate is not
too far off, i.e. as long as the computed altitudes are within a few minutes of the
altitudes that would have been observed.  Think of it this way: the distance between
moon and other body does not depend on the coordinate system that you use to compute
it; it only depends on the places of these bodies relative to the earth.

It is much more important, as you rightly remark, to get the refraction and parallax
corrections right, because those cause the observed distance to differ from the true
distance. It is not difficult to adjust the formulas underlying the correction tables
so as to make them yield the reverse corrections. As an alternative, one could use the
ordinary tables, look up the corrections with the true altitudes, and check if the
apparent altitudes that you find by applying them, give rise to the same corrections.
Correction for parallax is rather complicated and should make allowance for the
oblateness of the earth. When altitude of sun or moon is below 15 degrees include a
correction for oblique semidiameter: this is caused by refraction being unequal at the
top and bottom side of the body. Don't correct for dip, the visible horizon is not
used!

As an alternative to the iteration for GMT, it would be possible to interpolate. Do
the whole calculation for two time instances of an hour apart. Linear interpolation is
then almost always sufficient.

Finally a note on accuracy. With a good sextant it is possible to take the lunar
distance (well, any angle) to approx. 0.1' arc. This corresponds to a time interval of
12 seconds, and to a longitude difference of 3'. I like to keep that (ie the 0.1') as
the working accuracy. The calculations and corrections should then be carried out to
the same precision. I would like to increase the precision of the calculation of the
celestial position of the bodies to 6".

Would be fun to implement this process and see if it holds in practice...

Bye,
_Steven.

   

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