A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Antoine Couëtte
Date: 2016 Jun 20, 08:44 -0700
Well, I will have to make it short because I have only limited time to-day if I want send you a quick reply. Also this subject has most probably been addressed here earlier, and extensively by Frank if I remember correctly.
Quick and dirty then : in case of a "near grazing" occultation, you will understand that when an occulted body gets nearer to the Moon, its distance to the Moon Limb decreases, stays constant (equal to zero, or very close to it) for some time and then increases. Therefore it cannot be adequately represented by a linear curve as a function of elapsed time. This fact is further compounded by some of the effects you have mentioned, and also by the reason that the sought after distance is not a "point to point" one, but a distance from "moving point relaticeley to apparent circle". The non-linearity of this phenomenon can be appreciable (at the 0.1' level) when the Body to Moon limb distances decrease under about 5°. For this reason, it is generally not recommended (with some exceptions though) to take short distance lunars, and in addition Frank has published excellent good sense rules as to where the second body should stay relatively to the Moon, and more specifically where it should stay relatively to the Moon horns.
No time for me to retrieve real world examples in my archives. However ....
... going through the "Astronomical Tables of the Sun, Moon and Planets" by Jean Meeus published by Willman-Bell inc. 1983 Edition, I can find that on Jul 30th, 2000 at 12:00 UT there was a daylight occultation of Mars by the Moon visible from South America, (south) Atlantic Ocean, South Africa and Madagascar. So if we assume that if someone observed this occultation from sufficiently far away northern Latitudes, it should have been seen as "more or less" a grazing occultation (no actual occultation). Therefore, if you use Frank's on-line Calculator (OLC) for Lunars as he just explained to us ("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.") you should be able to plot the Mars-Moon Limb distances vs. times as seen from some adequately chosen spot.
As a starting point to run this example (30 Jul 2000 between Lady Moon and Warrier Mars) assume a steady observer at 30°N and 10°W (near Agadir Morocco). At 11h59m45s assume that the Lunar Distance is equal to 0°30'8 . You are right on, since the OLC indicates to you "Error in Lunar 0.0' , Error in Longitude 0.0' ". Since both bodies - the Moon and Mars - at almost on top of your head, you can now start playing with the OLC and from the same spot compute by trial and error (as advocated and described by Frank) the Mars to Moon Limb distances at regular intervals (say every 30 minutes) in the 06:00 - 18:00 time frame. You should observe that such distances do not vary linearly with time. If this example does not work, I will dig out for you a real world OK example to support this point when I have more time.
Sorry, I can't work it out any further to-day, but this Mars Lunar example here-above should work ....
Enjoy !!! :-) Good luck !
Best Lunarian Regards,