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    Re: Angular Distance Between Stars By Camera and Sextant
    From: Paul Hirose
    Date: 2012 Sep 18, 22:02 -0700

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