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    Re: Time of meridian passage accuracy
    From: Andrés Ruiz
    Date: 2009 Oct 8, 08:16 +0200

    Antoine is right in saying: “Only the presence of higher order terms (in t**4 and maybe t**6) prevented Andres to do the perfect pick of the exact Culmination time.”

     

    I have simulated two theoretical cases with very high speed, and the curve obtained for the altitude near meridian transit is like this:

     

    R=180º

    V=3000 kt

     

    R=135º

    V=3000 kt

     

    In this case the least squares fit gives for Hmax:

    yLS = a0+a1*x+a2*x2+a3*x3+...+anxn

    a0 = -51168.479530

    a1 = 4885.116188

    a2 = 118.025783

    a3 = -20.281600

    a4 = 0.107351

    a5 = 0.025145

    a6 = -0.000455

    x

    y

    yLS

    y - yLS

    ErrorRel_y

    ErrorRel_yLS

    17.508889

    53.34053

    53.178178

    0.162352

    0.30437

    0.305299

     

    And the solution is:

    R

    V

     

    UT1

    B

    L

    GHA

    Dec

    LHA

    H

    Z

     

    UT1

    tTransit-tMax

    135

    3000

    Transit:

    17.0775

    21.4216279

    -76.4695651

    76.471408

    -23.434998

    0.00184292

    45.1433741

    180.002397

     

    17:04:39

     

    135

    3000

    Hmax:

    17.5088889

    6.16972746

    -60.8005176

    82.9400037

    -23.4348594

    22.1394861

    53.3405304

    215.389641

     

    17:30:32

    0:25:53

     

     

    Andrés Ruiz

    Navigational Algorithms

    http://sites.google.com/site/navigationalalgorithms/

     

     

     

     

    Asunto: [NavList 10061] Re: Time of meridian passage accuracy

    ...

    I am quite surprised by these results because for culminations close to zenit (say above 85d) the effect of a high N/S speed component and/or change in declination significantly skews the heights vs times curve. Given the almost perfect symmetry of this Altitude curve, the algorithm used by Andres - apparently a second degree best fit as seems to be indicated on his site - was very powerful to pick up almost the exact the Culmination time. Only the presence of higher order terms (in t**4 and maybe t**6) prevented Andres to do the perfect pick of the exact Culmination time.

     

    ...

    Antoine M. "Kermit" Couette


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