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Re: Star-star distances for arc error
From: Frank Reed
Date: 2009 Jun 19, 19:38 -0700

```Douglas you wrote:
"I've done this. It provides a method of checking a sextant but is not easy
with accuracy unless you have some clamping system to hold the sextant still
and be able to rotate the sextant into alignment."

With a little practice, it's not hard, but I agree that you can find yourself
in some awkward positions. It's good practice for lunars, too. Not that
lunars are for everybody, but some people get a kick out of shooting them.

And then you wrote:
"If you use horizontal angles with stars approximately of the same declination
the refraction element is negligable if they are above say 30 degrees."

No, that just isn't right. First, you say 'same declination'. I assume that
this was just a "typo" and you meant altitudes. No problem. Even so, then
your claim would be that refraction is negligible for stars at the same
altitude (above 30 degrees). This is a "navigation urban legend". It isn't
true, but it has even been published in some recent editions of Bowditch and
other places where the authors really should know better. The angular
distances between stars at the same altitude are indeed affected by
refraction.

Consider for example, the case of two stars at 45 degrees altitude on opposite
sides of the sky. Let's assume that the angle between them is exactly 90
degrees in the absence of refraction. Now, refraction lifts BOTH stars by
about 1'. They're pushed towards each other, so the observed angular distance
would be just about 89d 58' which is a difference that would be measurable
even with a fairly bad sextant.

Now consider a case of two pairs of stars: two are at 45 degrees altitude
separated by 20 degrees more or less horizontally. Another pair is located
with a more or less vertical separation, one at 45 degrees altitude, the
other at 65 degrees altitude. What I was saying my previous message is that
you can calculate (clear) the effect of refraction in these cases, and indeed
in any case where both objects are above 45 degrees, simply by multiplying
the observed distance by 1.00034. So in these pairs, one where the stars are
horizontally placed and the other in which they're vertically placed, the
effect of refraction is the same. Horizontal relative orientation of the two
stars makes no difference.

The idea that horizontal placement of the stars eliminates the effect of
reftraction is most likely due to a mistaken extrapolation from one extreme
case (which is never observable in practice but worth understanding in
principle). Imagine two stars right on the horizon (can't see stars that low,
but let's ignore that for the moment). The effect of refraction is to lift
the stars towards the zenith. There is no component in azimuth. Now near the
horizon the lines of constant azimuth show no convergence so if stars very
close to the horizon are shifted towards the zenith, the angle between them
will hardly change at all. So here we have a case where the distances between
stars at the same altitude (zero) are unaffected by refraction. It seems
likely that this case was mistakenly extrapolated decades ago leading to the
mistaken notion that the angles between stars at ANY altitude are unaffected
if the stars are at the same altitude. And again, this just isn't true.

"Most sextants only read with accuracy to about a half of a minute of arc anyway."

Accuracy in sextants depends on many factors, but I have seen plenty of folks
get significantly better accuracy than that with a decent metal sextant (so
long as it's equipped with a medium power telescope, e.g. 7x). And I myself
generally get 0.25' accuracy in lunars on individual sights or 0.1' when four
are averaged. But none of this will work unless you've checked for arc error
(and that's the motivation for measuring star-star distances for some
people). Arc error is generally a fixed error and therefore it is just as
easy to correct for it as it is to correct for index error.

And you concluded:
"Star distances can be calculated from declinations/ RA (or hour angles) with simple spherical trig."

Of course. And I probably should have outlined the exact calculation for those
who like setting things up in spreadsheets and in their own software, so if
anybody is uncertain on those details, let me know. But neat little rules
like the one I described earlier have another interesting use. Knowing that
the distances between all stars change by the same percentage amount when
they're above 45 degrees gives us useful planning information. As long as
you're sure the altitudes above 45, you don't have to record any other
information. Local time doesn't matter. The actual altitudes of the stars
don't matter. You can just measure the distance at your leisure and the
clearing process will not care about the details. Just multiply by 1.00034
and you're done.

-FER

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