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