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    Re: Lunars - Even Easier
    From: Frank Reed
    Date: 2008 Jul 02, 19:24 -0400

    George, you wrote:
    "I don't see the basis on which he decided to replace an observed zenith
    distance for the Moon of 38 degrees by one of 40 degrees. Where did that
    value of 40 come from?" 
    
    The LD was 85. The Sun's zenith distance was 45. So if we force the Moon's 
    zenith distance to be 40, then the sight is perfectly aligned through the 
    zenith because that's the only case where the sum of the zenith distances 
    would be equal to the angle between the objects. And when that happens, the 
    triangle is degenerate so we don't need any trig to clear the sight which is 
    of course the goal of all this. Sounds crazy, right? It works because the 
    altitude of the Moon really doesn't matter much at all under some 
    circumstances, so we can introduce an "error" with very little downside 
    which converts the problem into a simple case. 
    
    Let's do a realistic example. Let's take the lunar observation we've all 
    been talking about (Jeremy's observation on June 10) and move the observer 
    to another location. Instead of being at 15� 14'N DR latitude, we move him 
    to 25� 14'N. But all of the other setup conditions remain the same. We keep 
    the DR longitude, temperature/pressure, date and time of observation exactly 
    the same. That way we don't have to look up a lot of new almanac data. You 
    can see that the Moon and Sun from that location would no longer
    be as nicely aligned. In fact, at that time, their difference in azimuth 
    would amount to 155� --a good distance away from being aligned in opposite 
    azimuths, and clearly out of line even to a casual observer. 
    
     From the shifted DR, our observer takes these sights at 06:23:00 GMT:
     Sun LL  35� 38'
     Moon UL 56� 14'
     LD Near 85� 40.3'
    If you clear this lunar observation, you will find that it is exactly 
    correct for that time and location. I've set it up that way. Run it through 
    the lunar distance calculator at www.HistoricalAtlas.com/lunars, and you 
    will get error=0.0'. 
    
    Now let's see if we can adjust this observation and turn it into a simple 
    lunar with no trig required. We need the observed altitudes of the objects 
    centers above the true horizon, and we need the observed center-to-center 
    lunar distance (this is the normal "pre-clearing" step):
     Sun LL:  35�38' -10'+16'  = 35� 44'
     Moon UL: 56�14' -10'-16'  = 55� 48'
     LD Near: 85�40.3' +15.8' +15.7' = 86� 11.8'
    And now we add these up. The total is 177� 43.8'. It doesn't total 180� 
    because the objects are not aligned in opposite azimuths. And HERE is where 
    we apply the trick. If we raise the Moon's observed altitude by 2� 16.2' 
    then, of course, the total WOULD add up to 180�, and as far as the math is 
    concerned, this means they're now in opposite azimuths. So let's do that... 
    
    We work the same lunar observation again, but this time with a Moon UL 
    altitude of 58� 30.2'. If you do it by any of the standard spherical 
    triangle approaches, you will find that this modified observation has an 
    azimuth difference of very nearly 180�. And when we clear this modified 
    observation, the results are almost exactly the same. The error this time 
    around is 0.1'. But the important point is that we don't need to use any 
    spherical trig to solve a degenerate triangle. It reduces to a very simple 
    case of addition and/or subtraction. 
    
    There is one calculation we need to do. We need to make sure that it's 
    legitimate to shift the Moon's altitude by more than two degrees (legitimate 
    in the sense that the error introduced is within tolerable limits --the 
    exact limits of what is "tolerable" depends on the end-user). So we 
    calculate (6')*tan(LD)/cos(Moon_alt). In this case, this gives 173', nearly 
    three degrees, so modifying the Moon's altitude should not introduce an 
    error larger than a tenth of a minute of arc, and sure enough, that's what 
    we have already found. 
    
    Imagine if they had known about this 225 years ago. Back then, a somewhat 
    larger error in clearing might have been counted as "tolerable". A really 
    large number of lunar observations could have been reduced to simple cases 
    of addition or subtraction. The calculational work would have taken five 
    minutes at most... 
    
    Oh well. Can't change history! 
    
     -FER 
    
    
    
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