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    Re: Sun Moon Lunars to 155 degrees
    From: Frank Reed
    Date: 2010 Apr 3, 09:38 -0700

    George H, you wrote:
    "Then, from that modern ephemeris, at that GMT, we can predict the position,
    in GHA and dec., of both Sun and Moon. If we then use the observed altitudes of Sun and Moon, at that same moment, as the basis for a fix, there are only two places on Earth where those altitudes apply, at that GMT, and the other, false-fix, is thousands of miles away and we can forget about it. So we have found a true position in one go, without having to make prior assumptions about latitude, or anything else. And if we noted the chronometer reading at the same moment, we have also determined its error."

    That's right. You may have missed this procedure when we've talked about it before on NavList. It's simply a means of getting the most positional information out of a complete lunar sight. Luckily for us, although the altitudes are not required to high accuracy for clearing lunars, late 18th/early 19th century navigators seem to have been fairly careful about taking good altitudes for this purpose. Of course the process depends on an assumption of zero error in the lunar. If there was in fact error (and except by chance, there was), then the position will be shifted east or west just as would happen with any error in GMT.

    In many historical lunars, and even lunars today, the geometry is not favorable for determining a fix from the altitudes. This is because the most convenient lunars for shooting purposes are found when the Moon and Sun are on nearly opposite azimuths. And of course in that case the corresponding lines of position are nearly parallel usually rendering the latitdue uncertain. Last year or the year before that, Jeremy posted a lunar he had shot somewhere in the Caribbean which a couple of us wokred up on NavList and in that case the two LOPs crossed at a very shallow angle.

    For anyone else following along, this was NOT a historical method for working lunars, and it also doesn't have anything to do with the procedure I've described for getting a position fix from lunar observations under modern conditions at a known instant of GMT. This is primarily a method for re-analyzing historical lunars using the modern concepts of the two-body fix. In the 19th century, the technique for getting latitude as well as local time from the two altitudes taken with a lunar observation was known to nautical astronomers and mathematicians and even made it into some of the navigation manuals, including Bowditch, but it was an exotic trick, and I've never seen the slightest evidence that it was used at sea or even familiar conceptually to navigators at sea.

    Just as a reminder to all concerned, the required accuracy in the altitudes for a lunar observation are as follows:
    h_acc_moon = 6'*tan(LD)/cos(h_moon)
    h_acc_body = 6'*sin(LD)/cos(h_body)
    These express the approximate accuracy required such that the resulting error in the clearin process is less than 0.1 minutes of arc. As an example, suppose the lunar distance is 89 degrees, the altitude of the Moon is 30 degrees, and the altitude of the Sun is 45 degrees. Then you can have an error as large as +/-6.6 DEGREES in the Moon's altitude with no significant consequence in the clearing process. By contrast, you would need an accuracy of +/-8 minutes in the Sun's altitude. If the altitudes are the same while the LD is instead 20 degrees (to borrow from the subject line), then the required accuracy in the Moon's altitude is +/-2.5 minutes and the required accuracy in the Sun's altitude is +/-2.9 minutes of arc. Note that this is the required accuracy of independent errors in the altitudes. The process is much less affected by a common error in the altitudes. So if you get the dip wrong, no problem.

    -FER
    PS: It will take me a couple of days to work through the messages I didn't read while I was preparing for the class, teaching the class, and commuting back and forth from Chicago to Connecticut so I am answering current posts along with older posts.


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