# NavList:

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

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Star-star distances for arc error
From: Frank Reed
Date: 2009 Jun 18, 22:21 -0700

```On a number of occasions, we've discussed using star-star distances as a means
of testing a sextant's arc error. The idea is that you measure the angle
between two stars and then compare it with known true angles. Any difference
is some combination of index error, arc error, shade error, personal error,
etc.

The true angles between stars are nearly fixed for many stars, especially
those at greater distances since they tend to have low proper motions, but
you do have to be careful to list those angles at monthly intervals since
aberration changes the angles during the year. A table of them can be
computed once and used for many years.

The observed distance between stars has to be cleared or corrected for
refraction. There is a straight-forward mathematical procedure for this which
is basically the same as clearing a lunar distance without the large
corrections for parallax. There are also simple graphical means for doing
this. But recently it occurred to me that the correction for refraction can
be handled very nearly with a trivially simple calculation for all measured
distances when the stars are above 45 degrees altitude.

Refraction shrinks all of the constellations under all observing conditions by
lifting the stars towards the zenith. But for altitudes above 45 degrees, the
"shrinking" is nearly uniform. That is, it is nearly proportional to the
zenith distance (since the refraction is proportional to tan(z.d.)). And
therefore all of the apparent distances between all of the stars are reduced
by the same proportional amount. Thus to clear the distances between stars
(when both stars are above 45 degrees altitude), all you have to do is
multiply the measured observed distance by 1.00034. By my calculations, which
somebody should check, the results never differ by more than 7 seconds of arc
and usually less than 3 seconds of arc from the exact calculation which is
pretty good for such a simple trick.

Another way of looking at this is to note that the apparent distances between
stars when they're both above 45 degrees does not change as they cross the
sky even though the refraction for each is changing continually. One could
publish a table listing a couple of dozen convenient pairs of bright stars at
various angular distances. The distances could be calculated with refraction
included for each month of the year (to deal with aberration), and there
would be no need for any further clearing. The observed distances could be
compared directly with the tabulated. For added utility, one could publish
three tables: one for standard pressure and temperature and one each for
pressure/temperature 10% above and 10% below the standard values (use factors
of 1+0.00034*P/T where P and T are given as ratios from their standard
values). That would cover nearly all observing conditions.

I suppose I should point out that for both stars to be above 45 degrees, the
measured distance would necessarily be less than 90 degrees so this trick
would only work with distances below that.

-FER

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