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Re: Real accuracy of the method of lunar distances
From: Frank Reed CT
Date: 2004 Jan 12, 19:08 EST
From: Frank Reed CT
Date: 2004 Jan 12, 19:08 EST
Jared Sherman wrote:
"The "discovery" that there are in fact two parallax corrections rather than one needed to clear a lunar would indicate that all prior lunars taken have been wrong, and that the diagrams in Arthur's PDF file must also therefore be wrong, since they only mention one parallax correction. "
No, there aren't two corrections, and (naturally) all those earlier analyses are not wrong. George is talking about something else that I think most folks without the math background will find pretty obscure. It's an interesting issue on its own merits, but don't worry if you don't follow it. And you shouldn't get the impression that George is suggesting a complete re-write of the methods of clearing lunars.
Jared Sherman asked:
"Have I missed something while my eyes were glazing over, or has there been any correction made for the low altitude refraction that must also be affecting the parallax correction for a low-altitude moon? And wouldn't that then need to be made based on air temperature and density, the same way that a conventional sun sight takes these factors into account?"
In fact, yes (to both questions), but perhaps surprisingly the difference is VERY small and can be ignored in practical cases. Chauvenet takes this small correction into account. The idea is that the parallax correction should be based on the Moon's (or other body's) altitude AFTER correcting for refraction. Parallax, after all, is a purely geometric correction so it has to be based on the altitude after refraction is taken out. Mathematically, that means that the corrected altitude is given by
H = H0 - R + P*cos(H0 - R)
where H0 is the measured altitude, P is the horizontal parallax (at altitude zero), and R is the refraction. Most accounts of altitude corrections use
H = H0 - R + P*cos(H0)
where you don't necessarily bother correcting for refraction before calculating parallax. But try some numbers. The difference between these two equations is very small. For high altitudes, R is small. At low altitudes when R can grow large, the whole argument of the cosine is nearly zero, so the value of the cosine is very close to 1 no matter whether use the first equation or the second. I think the maximum value for this difference is around one or two seconds of arc even in the worst cases (the Moon at 10 degrees altitude or so), but that's from memory. It never amounts to a practical difference.
Frank E. Reed
[X] Mystic, Connecticut
[ ] Chicago, Illinois
Frank E. Reed
[X] Mystic, Connecticut
[ ] Chicago, Illinois
"The "discovery" that there are in fact two parallax corrections rather than one needed to clear a lunar would indicate that all prior lunars taken have been wrong, and that the diagrams in Arthur's PDF file must also therefore be wrong, since they only mention one parallax correction. "
No, there aren't two corrections, and (naturally) all those earlier analyses are not wrong. George is talking about something else that I think most folks without the math background will find pretty obscure. It's an interesting issue on its own merits, but don't worry if you don't follow it. And you shouldn't get the impression that George is suggesting a complete re-write of the methods of clearing lunars.
Jared Sherman asked:
"Have I missed something while my eyes were glazing over, or has there been any correction made for the low altitude refraction that must also be affecting the parallax correction for a low-altitude moon? And wouldn't that then need to be made based on air temperature and density, the same way that a conventional sun sight takes these factors into account?"
In fact, yes (to both questions), but perhaps surprisingly the difference is VERY small and can be ignored in practical cases. Chauvenet takes this small correction into account. The idea is that the parallax correction should be based on the Moon's (or other body's) altitude AFTER correcting for refraction. Parallax, after all, is a purely geometric correction so it has to be based on the altitude after refraction is taken out. Mathematically, that means that the corrected altitude is given by
H = H0 - R + P*cos(H0 - R)
where H0 is the measured altitude, P is the horizontal parallax (at altitude zero), and R is the refraction. Most accounts of altitude corrections use
H = H0 - R + P*cos(H0)
where you don't necessarily bother correcting for refraction before calculating parallax. But try some numbers. The difference between these two equations is very small. For high altitudes, R is small. At low altitudes when R can grow large, the whole argument of the cosine is nearly zero, so the value of the cosine is very close to 1 no matter whether use the first equation or the second. I think the maximum value for this difference is around one or two seconds of arc even in the worst cases (the Moon at 10 degrees altitude or so), but that's from memory. It never amounts to a practical difference.
Frank E. Reed
[X] Mystic, Connecticut
[ ] Chicago, Illinois
Frank E. Reed
[X] Mystic, Connecticut
[ ] Chicago, Illinois
