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

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Re: Star-star distances for arc error
From: Frank Reed
Date: 2009 Jun 26, 21:05 -0700

```George H, you wrote:
"but one of its serious weaknesses is in that star-star distance table, appendix G."

Well, correct me if I'm wrong, but couldn't those weaknesses in his appendix G
be addressed very simply? You're quite correct to point out the errors, of
course. But calling them "serious weaknesses" when there are simple solutions
strikes me as a bit of an exaggeration. The two issues are:

Problem #1: "the star-star distances would only be correct during
the rising part of path of star 1, to the Eastwards".

OK. So can't he just include a line in the instructions (and perhaps at the
bottom of each page) saying exactly that?

Problem #2: "No account has been taken of annual aberration, which varies
cyclically over the year, and in the worst case can amount to errors in
star-star angle amounting to 40 arc-seconds".

OK, but the error should be no more than 20 arcseconds if the mean distance
has been used (I haven't checked to see whether he did that or not). In any
case, to correct for that, all he needs is a short table, maybe on the right
margin, giving the change in each distance for each month of the year.
Something like: Jan -0.3, Feb -0.2, Mar -0.2, Apr -0.1, etc.

Those are simple solutions to relatively minor problems. It's a shame that
they weren't caught in proofreading but it would take no more than an index
card sized "errata" sheet to cover those particular items (I do understand
that there are some other small errors, as in any book published in a finite
amount of time).

And you wrote:
"So, anyone wishing to make high-precision checks on a sextant using
star-star distances would be well advised to calculate, and correct for, the
refractions from first principles"

Yes, of course, that's one solution. Or they could just write in by hand the
extremely brief monthly correction table (for each pair) as noted above for
Problem #2 and take note of the special instructions for dealing with Problem
#1.

And George, you concluded:
"rather than follow Frank's recommendation of Table G in Karl's book. I wonder
if Frank has ever tried it?"

I'm sorry my post confused you, George. If you re-read it, I did not issue a
"recommendation" to USE those tables. I merely pointed out there existence in
a popular new book. Douglas Denny had asserted repeatedly that using
star-star distances to measure arc error was not possible. You read those
posts, too. I suggested that Douglas might want to "take a peek" at those
tables in John Karl's book. My point was to persuade him, just maybe, that he
needed to re-think his certainty that such observations were worthless. Do
you understand now??

So how should you clear star-star distances? One way would be to make those
small corrections to the tables in John Karl's book as noted above. But
probably the easiest way for a modern observer is with a spreadsheet or a
purpose-built piece of software. You calculate refraction for each star as
exactly as possible based on its altitude (since this is a "backyard"
observation the altitudes would presumably be calculated from the observer's
position and the GMT and date of the sight) and then you do the usual
spherical trig to get the correct apparent distance. Compare that with the
observed distance and you're done. No problem. BUT here's a fun trick (that
started this thread): for stars above 45 degrees altitude (about one-third of
the usable distances at any moment), the refraction in the star-star
distances does not change. The refraction is directly proportional to the
distance. All of the constellations above 45 degrees shrink proportionately
by a factor of just about 1.00034. Even if you don't do any calculations this
way, this refractional morsel is also great for planning a round of sights
because you know in advance that you don't have to record the time since the
refracted distances won't change by more than about 0.1 minutes of arc (until
one of the stars is below 45 degrees).

-FER

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