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    Re: 2016, June 14th Mars Lunars
    From: Frank Reed
    Date: 2016 Jun 19, 14:20 -0700

    Sean, you wrote:
    "I did notice that there was also a slight shift in RA. I suspected the problem was more complex than my current understanding of the relationship between the observer's position and the apparent position of the body."

    We may want to back up a minute. There may well be a substantial shift in RA. Remember, the altitude corrections are simple only when we look at azimuth and altitude since refraction is entirely an altitude correction, and parallax is 99.7% an altitude correction. When you rotate to RA and Dec, then all that simplicity disappears. But in any case, you don't need to worry about RA and Dec when thinking about lunars.

    Note that the small correction that Antoine has brought up is the oblateness correction (the correction for the polar flattening of the earth). It's a very minor matter. The lunars clearing web app on my web site includes it as it has for over a decade, of course, and for complete accuracy you should add it if you're working lunars by some other technique, but it's right at the very limit of real practical concerns in lunars. The correction for oblateness is normally either 0.0 or 0.1 minutes of arc, only rarely as much as 0.2 minute of arc.

    You conctinued:
    "I tried working the parallax and other corrections into the calculated geocentric distances in order to obtain an approximation of the topocentric distances. (Or, more accurately, an approximation of the slope of the topocentric change in distance.) I was pleased with the result."

    Yes, this is an interesting puzzle, and the approach you're taking is definitely valid and up to the task. There's a spectrum of methods by which we can calculate the topocentric slope. At one end of the spectrum, we can calculate the perfect topocentric distances using something like my web app. You do this by entering the coordinates for your position (as near as you know it) and then you enter an observed lunar distance and by trial and error you find the distance that gives a perfect result. Do this for a time near the start of your observations and another time near the end, and assuming the interval isn't more than about thirty minutes (frequently a much longer interval would be fine) you can draw a straight line through those points. That line represents the topocentric rate of change of the apparent lunar distance, which is the quantity that we observe.

    At the opposite extreme, we can try to come up with some short calculation method suited for paper calculations or at least a quick calculator check. You wrote:
    "The method I used to calculate the corrections was the same one that Frank gives on his 'easy lunars' page. The only real difference is that I subtracted the inverse "DH" values (multiplied by the "corner cosines") from the calculated distance. I then added in the augmented semi-diameter."

    That's essentially how I would go about it, too, but we can simplify somewhat. First, notice that you didn't need to add in the SD since you were only interested in the slope, and the SD does not change in any reasonable period of time for a run of sights. Next, we can get excellent results by dropping all the minor bits. This is quite analogous to the methodology that leads to Thomson's method of clearing lunars, later adopted as Bowditch's Second method. If you ignore the refractions for the current analysis as well as the quadratic correction and the parallax of the other body, assuming that these are all small, slowly-varying details, then the correction to the lunar distance becomes rather simple and short:

    d' = d + HP · cos h1 · cos α

    and given the usual equation for cos α, this reduces to 

    d = d' - [(HP · sin h2 / sin d') - (HP · sin h1 / tan d')].

    Here h1 is the altitude of the Moon, h2 is the altitude of the Sun, d is the topocentric distance, and d' is the cleared distance. Over the course of half an hour, the HP barely changes so you can treat that as a constant value. Most of the variation in the topocentric distance comes from the changing altitudes. Does this work? Does it pick up most of the variation as required? Incidentally, this is already rather short, but it may be able to simplify even further. It's worth some numerical experimentation. 

    You wrote:
    "So far, this seems like a viable way to weed out outliers and select a suitable observation for the actual clearing process."

    Yes, this is definitely how you would extend the standard "error banding" procedure to lunars. We work out the correct slope that the observations should follow. Then we look at a band with that slope and a width of two s.d.'s based on prior experience. Slide the band up and down until it catches the majority of sights. Kill off the others (which should be few in number!).

    You concluded:
    "I'll have to go back and try it on some of my old lunar sets to be absolutely sure."

    Absoluteness you will never find in the art of knocking off outliers. There are methods that do work given the unique statistics of manual celestial observations, but you may well find that they only work modestly better than common averaging. And don't ignore the "median" method proposed in that ON article. It's worth some experimentation.

    Frank Reed

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