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    Re: Lunars with Young's formula: always off by several minutes
    From: William Porter
    Date: 2018 Nov 2, 12:36 -0700

    As promised, here is a worked example of my first draft of how to clear lunars by haversines using only the most basic external inputs I can get down to. The idea is to use only sextant and paper. At this stage, for obvious reasons, I am using known UT, Lat and Long for proof of concept. Again, thank you everyone on here for your help in finding the RA error: see below. I know the GST/RA method is cumbersome, and thanks for the suggestions in improving it. When I said “GST” in earlier posts, I meant the value interpolated between the two 0UT values, so really GST-UT. Sorry for the imprecision. And I use decimal degrees and minutes randomly. Sorry.

    Moon-Aries 19Oct18 18:46:47U Obs distance (near): 41deg 16.8’

    Lat: 51.432 deg Long -0.219 deg

    Ephemera:

    Moon (from USNO polynomials)

    RA: 332.155 deg Dec: -13.891 deg HP:3274” AD:1784”

    Aries

    RA J2000: 19:50:47=297.696deg

    PLUS TO A FIRST ORDER APPROX, 18.8years’ precession at 360/25771 deg year=0.263deg=297.959deg

    Thank you Navlist for helping me find this error

    Dec, ignoring precession for now: 8.868 deg

    GST-UT: 1:52:47 (from astropixels.com)

    1) Preparation

    GST (GHA Aries) = 20:39:34 = 309.890 deg

    GHA moon = 337.735 deg

    GHA Altair = 12.194 deg

    d”, observed distance = 41.528 deg after adding semi-diameter

    2) Calculate m and M (obs and corrected lunar elevations)

    Log Cos Lat -0.2052066199

    + log Cos Dec -0.0128907218

    + Log Hav LHA -1.4201185859

    = Log term -1.6382159277

    term 0.0230029784

    + Hav (L-D) 0.2912516608

    = Hav CZD 0.3142546392

    CZD Degrees 68.192

    M 21.818

    + Refraction 0.03

    = 1st guess 21.838

    - Parallax 0.844 (HP* cos guess)

    = “m” 20.993

    3) Calculate s and S (obs and corrected star elevations)

    Log Cos Lat -0.2052066199

    + log Cos Dec -0.0052231779

    + Log Hav LHA -1.9633314089

    = Log term -2.1737612067

    term 0.0067025304

    + Hav (L-D) 0.131739016

    = Hav CZD 0.1384415465

    CZD Degrees 43.688

    “S” 46.312

    + refraction 0.01

    = “s” 46.322

    4) Calculate D, true LD

    Log Cos Dec moon 0.0128907218

    + log Cos Dec Altair -0.0052231779

    + Log Hav diff LHA -1.0623346349

    = Log term -1.0804485347

    term 0.0830905178

    + Hav (Dec moon – Dec Altair) 0.0389310211

    = Hav CZD 0.1220215388

    CZD Degrees 40.891

    5) Apply Young’s formula in Haversine form: the fun part. I am working off the description in "Astronomical and Nautical Tables" by James Andrew. 1805

    "Take the log-sines of half (the sum and difference) of ((the apparent distance) and (the difference of the apparent altitudes)), the log-cosines of the true altitudes , and the log-secants of the apparent altitudes; and having added these six logarithms, rejecting the tens from the index, find the natural number corresponding to their sum, to which natural number add S.S.C. difference of true altitudes, and the sum will be the S.S.C. true distance required."

    m 21.003deg log sec 0.0298583243

    s 46.344 log sec 0.1609426257

    M 21.808 log cos -0.0322477164

    S 46.344 log cos -0.1609426257

    m~s 25.340

    d 41.528

    half the sum 33.434 log sin -0.258867254

    Half the diff 8.094 log sin -0.8514168363

    sum -1.1126734823

    natural 0.0771483279

    +hav(M-S)=hav(-24.536) 0.0451500994

    =hav D 0.1222984273

    D 40.939 deg

    This is very close to what I get with Frank’s amazing online tool (40deg56.1) but annoyingly far from my computed value of 40.89 deg. So, all further help welcome.

       
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