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    Re: Lunar trouble, need help
    From: Frank Reed
    Date: 2008 Jun 26, 00:41 -0400

    So Jeremy mentioned in his initial account of his lunar on June 10 that the 
    Sun and the Moon were almost exactly on opposite bearings. That was a 
    nuisance for observing the Moon's altitude, but it has one really big 
    benefit. It dramatically simplifies the process of clearing this lunar. The 
    lunar arc in this case runs almost through the zenith and therefore the 
    corrections to the distance are just the ordinary altitude corrections. 
    
    The measured lunar distance was 85� 38.8 [converted from an "Inside Limb" 
    lunar to a normal "Near Limb" lunar]. The altitudes, adjusted to match the 
    time of the lunar distance at 06:23:00, were Moon UL: 61� 05' and Sun LL: 
    33� 07'. Historically, at this point we would use the 12/20 rule to correct 
    approximately for dip and semi-diameter (+12' for a LL sight and -20' for a 
    UL sight), but in this case, the observer's height of eye was 106 feet so we 
    need to subtract an extra 6' for dip making it a 6/26 rule. So the observed 
    altitude of the Sun's center above the true horizon is 33� 13'. The observed 
    altitude of the Moon's center above the true horizon is 60� 39'. 
    
    The observation of the altitudes and the clearing of them for dip and 
    semi-diameter does not need to be exact. But the next two steps, adjusting 
    the lunar distance itself for semi-diameters and then getting the altitude 
    corrections and applying them to the lunar distance needs be done with high 
    accuracy. 
    
    We need the measured center-to-center lunar distance (because that's what 
    tables of predicted lunar distances would give). To get this we add the 
    exact values of the semi-diameters of the Sun and Moon to the lunar 
    distance. The Sun's SD was 15.8'. The Moon's we get from 
    0.2724*HP+augmentation. Since the Moon's HP was 56.6' around the time of the 
    sight,the unaugmented SD for the Moon was 15.4'. The augmentation at 60 
    degrees altitude is 0.3' so the augmented SD for the Moon is 15.7'. We add 
    15.8 and 15.7 to the Near Limb LD of 85� 38.8' and get 86� 10.3. 
    
    Let's stop for a moment to see what these numbers imply. Notice here that 
    the observed altitude of the Moon's center above the true horizon added to 
    the observed angle of the Sun's center above the true horizon added to the 
    observed angle between the center of the Sun and the Moon gives 180� 02'. 
    This confirms that the objects are nearly aligned opposite each other in the 
    sky. In fact, technically, the total shouldn't exceed 180 but a 1' or 2' 
    error in either altitude is not unusual (and not critical either). Because 
    the difference in azimuth between them is very close to 180 degrees, we do 
    not need to solve any spherical triangles to clear the sight. Or in 
    different words, the spherical triangles that we would ordinarily solve are 
    degenerate in this case. 
    
    Next we need to correct the lunar distance for refraction and parallax. 
    Since the lunar arc passes through the zenith, the corrections to the 
    distance are simply the ordinary altitude corrections. Refraction pushes 
    both bodies towards the zenith (by about 1.5' for the Sun at its altitude 
    and about 0.5' for the Moon at its altitude). Parallax shifts both bodies 
    away from the zenith (by about half a degree for the Moon at that altitude 
    and by about 0.1' of arc for the Sun at its altitude). So we need to add to 
    the observed distance to correct for refraction and subtract to correct for 
    parallax. We could work these out from formulae, but we can get both of 
    these corrections combined from the standard tables in the Nautical Almanac. 
    The only catch is that we are interested in the corrections for the object's 
    center so we have to use the rules for bubble sextant observations. Namely, 
    we average the LL and UL corrections. For the Moon, the correction is 42.6 
    LL and 11.8 UL so the mean is 27.2'. For the Sun (use either column), the 
    correction is 14.5 LL and -17.3 UL so the mean is -1.4'. This is slightly 
    reduced in magnitude by the high temperature at the time of observation so 
    make it -1.3'. Now we have to think a bit to get the signs right. Refraction 
    pushes objects towards the zenith so a geocentric observer would see the Sun 
    at a slightly lower altitude. That means the lunar distance should be 
    increased so we add 1.3' for the Sun's correction. Parallax pushes objects 
    towards the horizon; a geocentric observer would see the Moon higher in the 
    sky. That means the lunar distance would be decreased so we subtract 27.2' 
    for the Moon's correction. Together, those two adjustments give us the 
    cleared lunar distance: it is 85d 44.4'. And that's it! We have cleared the 
    lunar distance without any long calculations and without using any tables 
    except the standard almanac altitude corrections. 
    
    Finally we compare this cleared distance with the expected distance based on 
    the assumed GMT at the time of the observation of 06:23:00. At 6h GMT, the 
    predicted LD was 85d 32.0. At 9h GMT, the predicted LD was 87d 01.1 [in a 
    practical case today, you would pre-calculate these or print them from any 
    of various sources including my web site; in a historical case you would 
    have taken them from the almanac]. We do this by simple, linear 
    interpolation. The rate of increase is 89.1' in three hours, just shy of 
    half a degree per hour. In 23 minutes, that implies an increase of 11.4' 
    (89.1*23/180) so the true geocentric LD at that time should be 85d 43.4'. 
    Hence the error implied for this lunar observation is 1.0 minutes of arc. 
    
    A more exact clearing calculation for this lunar observation gives an 
    implied error of 0.8 minutes of arc, rather than 1.0. I would guess that 
    0.1' of that difference is probably due to small rounding and calculation 
    differences in the altitude corrections while another 0.1' is due to the 
    slight change in geometry created by assuming the Sun and Moon were exactly 
    aligned on opposite azimuths. This is awfully close, no matter how you look 
    at it. 
    
    In the real world, in the tropics, the Sun and Moon will be "in distance" 
    (having a readily observable distance between them) and on opposite azimuths 
    quite often. If you have the time to take the observations at your leisure, 
    observing lunars when the Sun and Moon are opposite each other can make this 
    whole clearing business a lot simpler. Certainly for explaining or teaching 
    the principle of clearing a lunar, working one with this special property 
    could help a great deal. 
    
     -FER 
    
    
    
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