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Let me try to summarise the current state-of-play, following Brad's initial 
question about predicting corrected star-star distances, for calibrating an 
instrument.

Unfortunately, Brad didn't make it clear whether he was considering the 
trivially-simple case; the angle between two stars, at a time chosen so that 
they have the same, or opposite,azimuths, as has recently been discussed on 
the list. Or the more general case of the angle between two stars wherever 
they happen to be in the sky.

Peter Hakel (I think wrongly) presumed the former, and provided a short and 
simple comment, which I wrongly took  to imply his committment to the 
procedure Brad had suggested.

I then pointed to flaws in Brad's procedure, and showed a more correct way 
of doing the job. Although, in principle, that was exactly right, 
difficulties can arise in practice, which I'll deal with later.

A string of postings followed from Douglas Denny, at least one of which he 
has retracted, identified as [12197]. Unfortunately, as my incoming messages 
still fail to show message numbers, I can't identify which, but suspect all 
were flawed, in various ways. I questioned his still-unexplained procedure, 
but in addition, "Just to make it clear in case of confusion", he added to 
the confusion by defining the way to correct observed altitude, for 
refraction, to get true; whereas in this case, the opposite is required.

A posting from Frank made some valid points, but was a victim of Frank's 
familiar attempts to take the trig out of navigation. There are certainly 
applications where, under some special circumstances, the trig can be 
simplified into plain arithmetic, and this can be one. But then, the user of 
any such tricks needs to know what the tricks are, the conditions under 
which they may be valid, the level of approximation that may be involved: 
and still remains ignorant of how to proceed in other situations in which 
that special rule-of-thumb isn't valid. Isn't it better to know a procedure 
which applies all the time, even if a bit of trig is involved?

Now, having dealt with defects in everyone else's postings, let me return to 
assess the defects in my own, when I wrote-

"First, obtain the predicted position of star1, in altitude and azimuth. Add
the appropriate refraction correction, to get the apparent altitude. Then do
the same for star 2. Now, using those two apparent positions, calculate the
angle between them using spherical trig. Then, the result will adjust itself
automatically for refraction, depending on  how the two azimuths differ.

There are other ways to make the same calculation, but that' the simplest,
conceptually."

And conceptually, it is. And it works perfectly well, as long as altitude 
and azimuth are calculated precisely, using a calculator or computer. But it 
calls for precision to a small fraction of an arc-minute, in both predicted 
altitude and azimuth, to get a sufficiently-accurate answer in the end. This 
is a much higher level of precision than is usually expected, for navigation 
purposes, particularly in azimuth. It's not that the actual azimuth and 
altitude of the two bodies needs to be known by the user, to that precision; 
it's just an important step in the calculation. Ordinary altitude-azimuth 
tables are not intended to provide that precision.

The known information we start off with is the dec and SHA of the two stars. 
First, the geometrical angle between them needs to be obtained, and clearly 
Brad knows how to do that; it's the first step in his calculation procedure. 
Here's the calculation for the true angular distance between the stars-

cos D = sin dec1 sin dec2 + cos dec1 cos dec2 cos (SHA1-SHA2)

Frank's point about taking star predictions for the appropriate time of year 
is a valid one if high precision is called for, to allow for aberration. The 
year itself doesn't matter, over wide limits, because star-star distances 
are unaffected by precession.

But now this angle D needs to be corrected, for the two refractions. To work 
out what the refractions are, we need the altitudes of the two stars, though 
(because it's only to make a small correction) high precision isn't needed. 
We could actually measure them, if the horizon is visible, but for star 
observations that's usually restricted to twilight. That would give us 
altitudes h1 and h2. We can apply the refraction correction to predict the 
true altitudes H1 and H2, which will be a bit less

Alternatively (but only if we know our position and the time) we can 
calculate the true altitudes H1, from dec1 and GHA1, and H2, from dec2 and 
GHA2. From those true altitudes, and the appropriate refraction, we can 
predict what the observed altitudes h1 and h2 would have been, if a true 
horizon had been visible to measure from . This is now making the refraction 
correction the opposite way round, so h1 and h2 will be a bit more than H1 
and H2.

Now we correct the true angular distance D for the effect of refraction, to 
predict the angle d that we will actually measure across the sky This is 
similar to the lunar-distance job for which many, many methods were devised 
in the days when log tables were essential to the navigator, except that in 
this case, we are working the opposite way, to get the observed distance 
from the true distance, rather than vice versa.

A useful formula for the calculator era is this -
cos d = (cos d +cos (H1 + H2)) cos h1 cos h2/(cosH1cosH2) -cos (h1+h2)

That provides the star spacing that is to be compared with the sextant 
reading.

That is completely rigorous, but alternatve procedures have been devised 
over the years, allowing a simpler correction to be added/subtracted to D. 
Especially so for the star-star case, in which the corrections are much 
smaller than for a lunar with its parallax. I suspect that Frank has 
something suitable in his locker.

George.

contact George Huxtable, at  george@hux.me.uk
or at +44 1865 820222 (from UK, 01865 820222)
or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

----- Original Message ----- 
From: "Brad Morris" 
To: 
Sent: Tuesday, March 09, 2010 9:48 PM
Subject: [NavList] Star - Star Observations


Gentlemen

I have been playing around with star-star observations to determine the 
accuracy of the sextant arc.  Calculating the star to star distance, without 
refraction is not a challenge, nor is correcting for refraction when both 
stars are on the same side as the zenith.  Derive GHA Aries for the 
observation instant, apply SHA objects, determine instantaneous altitude for 
both objects using spherical trig, compute refraction correction based on 
altitude for both objects, create delta refraction correction and finally, 
subtract from star to star distance to get the observable distance for my 
location.  It sounds like a lot of work, but I have set up a spreadsheet 
that uses the Celestron SkyScout as inputs.  I just point at two stars, 
enter some data from the SkyScout (in Right Ascension & Declination), and 
the observable distance from my location at a known time is the 
instantaneous result.  All the mindless tabular work is done by the 
spreadsheet.  I don’t really even need to know which stars they are, as long 
as the Celestron does!  Of course, I have checked my spreadsheet against 
some hand done calculations to check to see if it is working the way I 
expect it to…and it is.

Here is the dilemma.  When I get to larger angles, I need to go beyond my 
zenith.  For example, I have been looking at Polaris vs Sirius.  My latitude 
is about 40 degrees north.  So Sirius is to my south, Polaris, naturally, is 
to my north.  The nominal distance works out to about 106 degrees 20 odd 
minutes (forgive me, I don’t have the exact numbers in front of me). 
Because each object is on either side of my zenith, both objects will appear 
to be lower in the sky compared to the horizon, due to refraction.  Yet 
because they oppose each other in azimuth, the observable distance between 
them should be larger by the sum of the refraction corrections, not reduced 
by the difference of the refraction corrections.  That is, compute the true 
distance without refraction.  Since each object is lowered by refraction, 
but in opposite directions, shouldn’t we add the refraction corrections to 
the nominal distance to obtain the observable distance?

Best Regards
Brad

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