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    Re: Lunars.
    From: Steven Wepster
    Date: 2001 Jul 06, 1:01 PM

    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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