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Marcel Tschudin wrote:
> In the mean time Andrés was so kind to provide me with his Navigation
> calculator.  For the purpose of sextant calibration it allows also to
> calculate star-star distances. Not calibrating a sextant the corresponding
> values were left 0.0.
> 
> For Greg's Alioth-Alkaid-observation his program provides the following
> output:
> 
> 16/09/2012
> 03:05:00 UT1

My original computation used UTC, not UT1. However, in this message I 
will use UT1.

> Geocentric equatorial coordinates
> Alioth
> GHA = 208.084158 º = 208º  5.0'
> Dec = 55.892403 º =  55º 53.5'
> Alkaid
> GHA = 194.718161 º = 194º 43.1'
> Dec = 49.252840 º =  49º 15.2'

 From SIMBAD (http://simbad.u-strasbg.fr/simbad/) I got the catalog 
coordinates (barycentric ICRS at J2000.0), then converted them to 
geocentric apparent place. I don't know if the program by Andrés 
corrects for polar motion. At this precision it significantly affects 
GHS/dec and az/el, but not the refracted separation angle. I did the 
computation both ways. With no polar motion correction:

208.084157° +55.892369° Alioth GHA, dec
194.718145° +49.252780° Alkaid

With polar motion correction:

208.084064° +55.892458° Alioth
194.718050° +49.252852° Alkaid

Great circle error in position, compared to Andrés.
.000034° Alioth, no polar motion
.000076° Alioth, with polar motion
.000061° Alkaid, no polar motion
.000073° Alkaid, with polar motion

The errors are .12" to .27", which is not great accuracy, but OK for our 
purposes.

At .000001° precision the computation should include polar motion, so I 
will.

> Sextant Error by a Star-Star Distance
> 
> Input data:
> Star 1: Dec = 55.892403 GHA = 208.084158
> Star 2: Dec = 49.252840 GHA = 194.718161
> Star-star distance - sextant: DSSs = 0.000000 =   0º  0.0'
> Position of the observer:
> B = 34.173333 =  34º 10.4'
> L = -119.230000 = -119º 13.8'
> hEye = 1.830000
> Atmospheric parameters:
> P = 1015.600000
> T = 22.200000
> 
> Calculated altitudes:
> Hc1 = 28.317414
> Z1 = 320.442970
> Hc2 = 34.115333
> Z2 = 310.248963

My topocentric apparent unrefracted angles:
320.443072° +28.317505° Alioth az el
310.249013° +34.115437° Alkaid

Total (great circle) error compared to Andrés:
.000128° Alioth
.000112° Alkaid

> Refraction:
> R1 = 0.028946
> R2 = 0.023032
> Apparent altitudes:
> Ha1 = 28.347441
> Ha2 = 34.139225

Refraction is probably the weakest part of my computation. I use the 
formulas in The Astronomical Almanac, Section B. The one for altitudes 
above 15° simply assumes refraction is proportional to air density, 
times the tangent of apparent zenith distance. The Almanac says it's 
usually accurate to .1' - not an enthusiastic endorsement!

Furthermore, its implementation in the SofaJpl DLL is flawed. It assumes 
the formula is a function of unrefracted altitude, so if refracted 
altitude is the known quantity, the solution proceeds by iteration. In 
reality, the opposite is true. The error is not serious, though - about 
.9" at worst.

So, from my not great but not bad refraction model, here are the 
refractions and refracted altitudes:

.028846° +28.346351° Alioth
.022945° +34.138382° Alkaid

Altitude error compared to Andrés.
-.001090° Alioth
-.000843° Alkaid

My refracted distance, Alioth to Alkaid, is 10.455680°.

> Star-star distance: Calculated and Observed
> DSSc = 10.460896 =  10º 27.7'

That angle is consistent with the *unrefracted* coordinates from Andrés' 
program. What we need is the refracted angle. I don't see it in Marcel's 
message. But from the refracted coordinates he quoted, I can calculate 
it: 10.455432°. That's .000248°, or .89", from my value.


Marcel has mentioned CalSky. Unfortunately, he did not give us the 
coordinates from CalSky, and I don't feel like doing all those steps. In 
the past I've tested this site, and was not impressed.

Instead, let us try the USNO MICA program. At 2012 Sep 16 03:05 UT1, 
topocentric apparent azimuth and altitude at Greg's location:

320.442972 28.317417 Alioth (unref)
310.248944 34.115361 Alkaid (unref)

MICA does not compute refraction, so I will use the Nautical Almanac 
formula, including nonstandard temperature (72 F) and pressure (29.99). 
It is completely different from the formula I normally use. Refracted 
coordinates:

320.442972 28.347035 Alioth
310.248944 34.138979 Alkaid

Now compute the refracted distance, and compare:
10.455432° Alioth to Alkaid, refracted, Andrés
10.455561° MICA
10.455680° me

The difference between smallest and largest is only .9 arc seconds. But 
note that the angle from "Andrés" is really one I computed, based on the 
refracted coordinates from his program, as quoted by Marcel. The 
distance Marcel quoted is clearly UNREFRACTED.

-- 

   

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