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    Re: Venus, Jupiter, and the Moon
    From: Dave Walden
    Date: 2008 Dec 22, 06:16 -0800

    Here's another solution.  As a reminder, this 'no altitude' method relies on 
    the effect of the moon's parallax, not its motion relative to the stars.  
    Previous posts have described solutions of the intersecting cones and sphere 
    problem by various methods.  This post will describe the "Nothing But Frank" 
    method, with no close initial position required.
    
    As Frank points out, we know the ground point of the moon, 74W, 22S.  Since we 
    have observed lunar distances, the moon must be above the horizon, so we must 
    be in the hemisphere centered on the moon's GP.  (Since in our case, the 
    other two bodies are so close, not much is to be gained by consideration of 
    the hemispheres where they are visible.)  Let's take the GP as our starting 
    point.  We'll set up Frank's calculator with the GP as the lat and long, and 
    the observed Jupiter distance.  We first fix the long and iterate to look for 
    the lat were the LD error is zero.  If we're really lucky, there will be such 
    a point at the moon's GP long.  But this is unlikely.  We'll use a "hill 
    climbing" algorithm.
    
    Imagine while hiking at night you approach a hill and want to go to the summit 
    which you can't see. Your plan is to approach the hill along a north-south 
    line and continue as long as you ascend.  When you reach a crest and are 
    about to descend, you stop, turn 90deg right and take a step.  If you're 
    going up, proceed, it you're going down, stop turn back, then turn left and 
    proceed on the east-west line.  When you reach another crest, repeat.  When 
    right and left both lead you down, you're at the summit.  Works for hills of 
    "reasonable" shape and no "local" maxima.
    
    We proceed, using Frank's calculator, moving along the 74W north-south line, 
    find a local minima (ok, we're using a "valley descending" algorithm.)  We 
    find a minimum at 60S.  We then proceed east 'til we find another minimum at 
    20E.  We go north to 43S and call this close enough.  The first figure plots 
    LD error vs. lat on the 74W line.  The second shows the LD error on a 
    lat-long grid with circles proportional to the LD error.
    
    Once we've gotten to the right neighborhood, we can move along lat or long 
    lines finding the LD error equal zero points.  Plotting 4-5 of these gives up 
    the shape of the Jupiter curve.  We proceed in the same manner for Venus (not 
    forgetting to reset the observed LD).   Where these curves intersect, we have 
    a good approximate position and proceed as Frank describes.
    
    The last figure shows a view of the earth from the moon.  The GP of the moon 
    is at the center of the earth's disk.  The curved lines result from plotting 
    the LD error zero points we found above.  The points at the ends of the lines 
    from the earth's disk are the points in space (at a distance of 60 earth 
    radii from the moon) were the center to center LD would be zero.  I.e. the 
    planet would lie directly behind the moon.  Circles around the point 
    represent lines on which the observed lunar distance would be constant.  The 
    arc segments we have drawn are constant LD lines on the earth's surface (they 
    are the intersections of the cones of constant LD with the surface of the 
    earth.)
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