NavList:

Recently Ken Muldrew wrote about the great  practical utility of Margetts's
graphical tables for clearing lunar distances.  And they are wonderful! They
also can help clarify some theoretical  issues.

About a year and a half ago, I investigated the required accuracy  of the
altitudes of the Sun and Moon that are required for clearing lunars. Most  of the
period navigation manuals and even the theoretical works say simply that  the
altitudes do not need to be measured with great accuracy. But they give no 
numbers. Some resources available on the Internet have suggested that the
Moon's  altitude should be measured with great accuracy. Others suggest that +/-5 
minutes of arc is a good rule of thumb for both altitudes. As I wrote in
October  of 2004, it turns out that there are fairly simple expressions that define
the  required altitude accuracy at least when the objects are high enough
that  refraction is not large --in practice when the altitudes are above about 10
to  15 degrees. Those expressions are:
MoonAltError = +/-6' *  tan(dist)/cos(MoonAlt)
BodyAltError = +/-6' *  sin(dist)/cos(BodyAlt)
This is the allowable error in the Moon's altitude and  the other body's
altitude. As long as the measured altitudes are taken at this  level of accuracy,
the cleared lunar distance will not be subject to an error  larger than a
tenth of a minute of arc or so. What do these equations imply?  First if either
altitude is high, the required accuracy decreases. For example,  if the distance
is 45 degrees, then MoonAltError (the required accuracy in the  Moon's
altitude) varies from just about +/-6' down at low altitude to +/-12' at  60 degrees
altitude (since cos(60)=0.5) to +/-34' at 80 degrees altitude.  Second, for
short distances, the required accuracy is basically proportional to  the
distance since sin(x) and tan(x) are both about equal to x for small angles.  That
means that the altitudes have to be measured to about +/-1' accuracy at 10 
degrees distance and +/-2' accuracy at 20 degrees distance. Third, while the 
required accuracy of the other body's distance is stable at +/-6' for distances 
around 90 degrees, the required accuracy for the Moon's altitude changes 
dramatically. If the distance is close to 90 degrees, through a surprising 
cancellation, the Moon's measured altitude can be wrong by many degrees and it  will
make no difference in the clearing process. By the way, as far as I have 
been able to determine, these results have never been published  anywhere.

And now to Margetts. His graphs let the navigator read off the  correction to
the lunar distance directly. Each graph shows the correction for  one degree
of lunar distance. The Moon's altitude is found along the top of the  graphs
--each vertical line represents one degree of the Moon's altitude. The 
correction of the distance is along the left-hand margin --each horizontal line 
represents one minute of the correction to the distance. And the other object's 
altitude is picked off from a series of curves generally trending from the 
lower-left to the upper-right. Each curve represents one degree in the other 
object's altitude.

From the curves in the graphs, we can directly read  off the effect of a 1
degree error in the altitude of the Moon or the other  body. When the curves
have a steep slope, this implies that a small error in the  Moon's altitude leads
to a large error in the distance correction. And sure  enough, the curves are
steep for the lower distances and progressively become  more horizontal. At
90 degrees, the curves are almost exactly horizontal  implying that any value
for the Moon's altitude will yield almost the same  correction for the
distance, just as described above. In addition, on all the  graphs, the curves flatten
out when the Moon's altitude approaches 90 degrees  (in the left-hand
corner). That's the influence of cos(MoonAlt) in the  denominator of the equations
above. The spacing between the curves indicates the  influence of the other
body's altitude. Where the curves are closely spaced, a  small error in the other
body's altitude has a reduced influence on the cleared  distance. And as
expected, the curves are very close together when the other  body's altitude is
close to 90 degrees.

A navigator using Margetts's  graphs would have had an almost instinctive
understanding of the required  accuracy in the altitudes used in a lunar
observation. For us today, the graphs  provide a nice graphical way of seeing those
nice short equations (above) in  action. And in addition, we can see clearly
where those approximate equations  break down at low altitudes and observe the
changes in those rules at low  altitudes. For example, there is a little "local
maximum" in the curves for  distance=45 when the Moon's altitude is around 7
degrees and the other object's  altitude is between about 35 an 50 degrees.
Since the curves are horizontal  there, an error in the Moon's altitude is
relatively insignificant in those  cases.

Pretty slick, huh?

-FER
42.0N 87.7W, or 41.4N  72.1W.
www.HistoricalAtlas.com/lunars 


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