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    Re: Lunars - Even Easier
    From: Dan Allen
    Date: 2008 Jul 6, 22:44 -0600
    Frank, I am impressed.  Great advances simplify things and this is quite a simplification.

    -- Dan

    On Wed, Jul 2, 2008 at 5:24 PM, <frankreed@historicalatlas.net> wrote:

    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 15º 14'N DR latitude, we move him
    to 25º 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 155º --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  35º 38'
     Moon UL 56º 14'
     LD Near 85º 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:  35º38' -10'+16'  = 35º 44'
     Moon UL: 56º14' -10'-16'  = 55º 48'
     LD Near: 85º40.3' +15.8' +15.7' = 86º 11.8'
    And now we add these up. The total is 177º 43.8'. It doesn't total 180º
    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 2º 16.2'
    then, of course, the total WOULD add up to 180º, 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 58º 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 180º. 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!


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