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Lunars: A 90 Degree Miracle
From: Frank Reed CT
Date: 2004 Oct 31, 22:59 EST
From: Frank Reed CT
Date: 2004 Oct 31, 22:59 EST
Some days a singularity is a useful thing...
In historical logbooks, lunars were frequently observed close to first or last quarter or when the distance from a star was roughly 90 degrees. Why? There are some practical observational reasons that you can imagine for yourself. But there's a "miracle" involved, too. In another post to the list, I recently wrote that the required accuracy for the Moon's altitude is (with slight approximation) given by:
AccuracyMoon = 6' * tan(Distance) / cos(MoonAltitude)
(details in that other post).
OK... so what happens in this expression when the lunar distance is 90 degrees? The required accuracy is infinite. Which means that you can tolerate HUGE errors in the Moon's altitude when the distance is close to 90 degrees. You could measure the Moon's altitude with a simple cross-staff, for example, and have it wrong by 3 degrees, and there would be no problem. You could take a wild guess at the Moon's altitude, and it still wouldn't matter. As long as the observed center-to-center distance is close to 90 degrees, the Moon's altitude makes almost no difference at all. The altitude of the other body should still be measured to six minutes of arc or better, quite a bit better if it's going to be used for a matching time sight.
In addition to this convenience, lunars close to 90 degrees also have an advantage in the clearing process. In series methods, it's only necessary to calculate the two linear terms (see "Easy Lunars" on my web site). The quadratic term is negligible when the distance is near 90. All told, lunar observations for distances close to 90 degrees are significantly easier than those for other distances, both shorter and greater.
By the way, the "series expansion" for clearing lunars is very helpful here for seeing why the errors should depend on distance in this fashion and also for deriving the expression above. Details upon request.
If you would like to experiment with this issue, you can go to my web site and try the "Clear a Lunar" tool. Experiment with dates and objects until you find one where the distance is just about 90 degrees (+/- a degree or two). On the first pass, let the software calculate the objects' altitudes. If they're negative, move your DR position until they're in the sky (not hard to estimate from the GHAs and Decs). Then take the calculated altitudes and enter them as if they had been observed (for the Moon you'll have to undo the semi-diameter by hand until the tool says that you have no error in altitude. The clearing results should be the same since you're feeding in altitudes that are the same as the previously calculated altitudes. Now the fun starts.. Try entering a different altitude for the Moon. You'll soon see that the altitude can be way off before there is any serious impact on the cleared distance. Think it's a fluke? Pick another DR. Move the objects around in the sky as much as you like (but keeping them above the thick air and high refraction below 10 degrees). As long as the distance is close to 90 degrees, the accuracy of the Moon's altitude is not a critical factor in clearing a lunar.
I think I've got that all right... More later.
www.HistoricalAtlas.com/lunars
Frank R
[ ] Mystic, Connecticut
[X] Chicago, Illinois
In historical logbooks, lunars were frequently observed close to first or last quarter or when the distance from a star was roughly 90 degrees. Why? There are some practical observational reasons that you can imagine for yourself. But there's a "miracle" involved, too. In another post to the list, I recently wrote that the required accuracy for the Moon's altitude is (with slight approximation) given by:
AccuracyMoon = 6' * tan(Distance) / cos(MoonAltitude)
(details in that other post).
OK... so what happens in this expression when the lunar distance is 90 degrees? The required accuracy is infinite. Which means that you can tolerate HUGE errors in the Moon's altitude when the distance is close to 90 degrees. You could measure the Moon's altitude with a simple cross-staff, for example, and have it wrong by 3 degrees, and there would be no problem. You could take a wild guess at the Moon's altitude, and it still wouldn't matter. As long as the observed center-to-center distance is close to 90 degrees, the Moon's altitude makes almost no difference at all. The altitude of the other body should still be measured to six minutes of arc or better, quite a bit better if it's going to be used for a matching time sight.
In addition to this convenience, lunars close to 90 degrees also have an advantage in the clearing process. In series methods, it's only necessary to calculate the two linear terms (see "Easy Lunars" on my web site). The quadratic term is negligible when the distance is near 90. All told, lunar observations for distances close to 90 degrees are significantly easier than those for other distances, both shorter and greater.
By the way, the "series expansion" for clearing lunars is very helpful here for seeing why the errors should depend on distance in this fashion and also for deriving the expression above. Details upon request.
If you would like to experiment with this issue, you can go to my web site and try the "Clear a Lunar" tool. Experiment with dates and objects until you find one where the distance is just about 90 degrees (+/- a degree or two). On the first pass, let the software calculate the objects' altitudes. If they're negative, move your DR position until they're in the sky (not hard to estimate from the GHAs and Decs). Then take the calculated altitudes and enter them as if they had been observed (for the Moon you'll have to undo the semi-diameter by hand until the tool says that you have no error in altitude. The clearing results should be the same since you're feeding in altitudes that are the same as the previously calculated altitudes. Now the fun starts.. Try entering a different altitude for the Moon. You'll soon see that the altitude can be way off before there is any serious impact on the cleared distance. Think it's a fluke? Pick another DR. Move the objects around in the sky as much as you like (but keeping them above the thick air and high refraction below 10 degrees). As long as the distance is close to 90 degrees, the accuracy of the Moon's altitude is not a critical factor in clearing a lunar.
I think I've got that all right... More later.
www.HistoricalAtlas.com/lunars
Frank R
[ ] Mystic, Connecticut
[X] Chicago, Illinois
