# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Lunars: Altitude Accuracy and Margetts**

**From:**Frank Reed CT

**Date:**2006 May 5, 23:02 -0500

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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