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    Re: Angular Distance Between Stars By Camera and Sextant
    From: Marcel Tschudin
    Date: 2012 Sep 19, 15:18 +0300
    You are right Paul. I misinterpreted the results obtained from the different calculations as refracted, whereas they relate to unrefracted. I started to notice this while doing last night a similar relative accuracy comparison as you kindly provided but using only the unrefracted distances. Thank you Paul for noticing and informing on it. May I ask the list members to please accept my apologies for having mislead or confused you. This confirms again that one should not use results from programs before one fully understands them. In order to avoid others to make the same mistakes, I mention here my pitfalls:

    Andrés' program:
    Here I misinterpreted the results shown under the heading "Star-star distance: Calculated and Observed" . I presumed the program would compare here the directly measured (i.e. the apparent) star-star distance with the calculated apparent one, thus that the calculated one would be the refracted. This is not true. These results are used to calculates the sextant error relative to the true, unrefracted distance. The program does not show the refracted distance, it has to be calculated separately from the intermediate results. But how did you, Paul, obtain for the Alioth-Alkaid refracted distance the 10.455432°? I was not able to reproduce this value. Transferring e.g. the Dec, GHA and Hc values from Andrés' program into Bill's Excel sheet I obtain Alioth-Alkaid refracted distance of 10.457595°, or when using Frank's refraction approximation 10.457409°.

    For the location and time of observation CalSky shows the following results:

    Right Ascension:   12h 54m 33.114s   Apparent coordinates
    Declination:       +  55° 53' 32.72" Apparent coordinates
    Right Ascension:   12h 54m 01.633s   J2000
    Declination:       +  55° 57' 35.43" J2000

    Right Ascension:   13h 48m 01.509s   Apparent coordinates
    Declination:       +  49° 15' 10.34" Apparent coordinates
    Right Ascension:   13h 47m 32.546s   J2000
    Declination:       +  49° 18' 47.89" J2000

    Here I misinterpreted "Apparent" to relate to refracted whereas they "possibly" relate to "true" or air-free coordinates. Their J2000 values show some differences to the Simbad FK5 values:

    Right Ascension:   12h 54m 01.750s   FK5(J2000)
    Declination:       +  55° 57' 35.36" FK5(J2000)

    Right Ascension:   13h 47m 32.438s   FK5(J2000)
    Declination:       +  49° 18' 47.76" FK5(J2000)

    This means that the CalSky data would require further verification to what they relate to.

    The unrefracted Alioth-Alkaid distance is with CalSky data also considerably longer, by about 0.06 moa, compared to the others (all values in moa):
    CalSky: 627.717803
    MICA:   627.655466
    Paul:   627.655103
    Andrés: 627.653756
    USNO: 627.580582 (Online Almanac)

    How can now those of you proceed who would like to use observed star-star distances for calibration purposes?

    If an accuracy of several seconds of arc is sufficient, then the USNO Almanac data can be used and the GHA, Dec and Hc values transferred into Bill's Excel sheet for calculating the refracted distance. Since the inaccuracy results from rounding the values in the Almanac to 1/10th moa the error will reduce with a larger nuber of observations.

    For higher accuracy:

    Windows users can obtain Andrés program and calculate with it GHA, Dec and Hc which then can be transferred into Bill's Excel sheet for calculating the refracted distance. May be Andrés provides in a future version of it also this result.

    Of course also Paul's routines can be used. They can unfortunately not be made running "straight forward" and the data input in its present form is cumbersome.

    I do not know the capabilities of MICA. May be someone who is familiar using it could explain up to which point it could help calculating refracted star-star distances. MICA appears to have the advantage of being available to Windows and Mac users.


    On Wed, Sep 19, 2012 at 8:02 AM, Paul Hirose <cfuhb-acdgw@earthlink.net> wrote:
    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

    03:05:00 UT1

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

    Geocentric equatorial coordinates
    GHA = 208.084158 º = 208º  5.0'
    Dec = 55.892403 º =  55º 53.5'
    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

    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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