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A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Re: Lunars - Even Easier
From: Frank Reed
Date: 2008 Jul 02, 19:24 -0400
From: Frank Reed
Date: 2008 Jul 02, 19:24 -0400
George, you wrote: "I don't see the basis on which he decided to replace an observed zenith distance for the Moon of 38 degrees by one of 40 degrees. Where did that value of 40 come from?" The LD was 85. The Sun's zenith distance was 45. So if we force the Moon's zenith distance to be 40, then the sight is perfectly aligned through the zenith because that's the only case where the sum of the zenith distances would be equal to the angle between the objects. And when that happens, the triangle is degenerate so we don't need any trig to clear the sight which is of course the goal of all this. Sounds crazy, right? It works because the altitude of the Moon really doesn't matter much at all under some circumstances, so we can introduce an "error" with very little downside which converts the problem into a simple case. Let's do a realistic example. Let's take the lunar observation we've all been talking about (Jeremy's observation on June 10) and move the observer to another location. Instead of being at 15� 14'N DR latitude, we move him to 25� 14'N. But all of the other setup conditions remain the same. We keep the DR longitude, temperature/pressure, date and time of observation exactly the same. That way we don't have to look up a lot of new almanac data. You can see that the Moon and Sun from that location would no longer be as nicely aligned. In fact, at that time, their difference in azimuth would amount to 155� --a good distance away from being aligned in opposite azimuths, and clearly out of line even to a casual observer. From the shifted DR, our observer takes these sights at 06:23:00 GMT: Sun LL 35� 38' Moon UL 56� 14' LD Near 85� 40.3' If you clear this lunar observation, you will find that it is exactly correct for that time and location. I've set it up that way. Run it through the lunar distance calculator at www.HistoricalAtlas.com/lunars, and you will get error=0.0'. Now let's see if we can adjust this observation and turn it into a simple lunar with no trig required. We need the observed altitudes of the objects centers above the true horizon, and we need the observed center-to-center lunar distance (this is the normal "pre-clearing" step): Sun LL: 35�38' -10'+16' = 35� 44' Moon UL: 56�14' -10'-16' = 55� 48' LD Near: 85�40.3' +15.8' +15.7' = 86� 11.8' And now we add these up. The total is 177� 43.8'. It doesn't total 180� because the objects are not aligned in opposite azimuths. And HERE is where we apply the trick. If we raise the Moon's observed altitude by 2� 16.2' then, of course, the total WOULD add up to 180�, and as far as the math is concerned, this means they're now in opposite azimuths. So let's do that... We work the same lunar observation again, but this time with a Moon UL altitude of 58� 30.2'. If you do it by any of the standard spherical triangle approaches, you will find that this modified observation has an azimuth difference of very nearly 180�. And when we clear this modified observation, the results are almost exactly the same. The error this time around is 0.1'. But the important point is that we don't need to use any spherical trig to solve a degenerate triangle. It reduces to a very simple case of addition and/or subtraction. There is one calculation we need to do. We need to make sure that it's legitimate to shift the Moon's altitude by more than two degrees (legitimate in the sense that the error introduced is within tolerable limits --the exact limits of what is "tolerable" depends on the end-user). So we calculate (6')*tan(LD)/cos(Moon_alt). In this case, this gives 173', nearly three degrees, so modifying the Moon's altitude should not introduce an error larger than a tenth of a minute of arc, and sure enough, that's what we have already found. Imagine if they had known about this 225 years ago. Back then, a somewhat larger error in clearing might have been counted as "tolerable". A really large number of lunar observations could have been reduced to simple cases of addition or subtraction. The calculational work would have taken five minutes at most... Oh well. Can't change history! -FER --~--~---------~--~----~------------~-------~--~----~ Navigation List archive: www.fer3.com/arc To post, email NavList@fer3.com To , email NavList-@fer3.com -~----------~----~----~----~------~----~------~--~---