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    Sumner Line of position
    From: Andrés Ruiz
    Date: 2009 Nov 17, 09:31 +0100

    [NavList 10640] Re: AP terminology, WAS: 2-Body Fix -- take three

    ----------------------------

    So we need to get at least 2 Sumner points, and preferably three to expose plotting and/or math errors. Sounds like at least as much work, if not more, than St. Hilaire.  I don't know, since I've never plotted Sumner lines.  As an aside, we use the same equations with different names in our Great Circle sailings.

    -----------------------------------

     

     

    Jeremy, an example of a FIX by Sumner line.

     

     

    TWO Circles of Position

    19/02/2004   20:00:00 Fix

    UT1

    date

    body

    GHA

    Dec

    Ho

    19/02/2004

    20:00:00

    Procyon

    334.23

    5.22

    28.5

    19/02/2004

    20:00:00

    Sirius

    347.78

    -16.72

    19.55

    DR at tFix: 42N 30.5W

     

    Choose a latitude near your DR position

    B1 = Be - e

    B2 = Be + e

    Obtain the LHA for these two latitudes:

    LHA = ACOS( ( SIN( Ho )-SIN( B )*SIN( Dec ) )/ COS( B )/COS( Dec ) )

    LHAprima = -LHA

    Do: 0 <= LHA <= 360º

    Obtain the longitude for this two points of the LoP:

    L = LHA-GHA

    Do: -180ºW <= L <= +180ºE

    Choose L near your DR longitude

     

    Plot: on plotting sheet or white chart for celnav. No protractor is needed.

     

    For two simultaneous LoP fix: Chauvenet  Vol I, pg 428

     

    Restrictions: the intrinsic for a time sight

    • Body near the transit. Line N/S
    • Line W/E
    • High altitudes and declination 1/COS( B )/COS( Dec ) term

     

    I have the felling, subjectively, that when I use Sumner it is simpler than the Saint-Hilaire. Doing the calculations with a standard calculator and obtaining the fix with a RADAR plotting sheet.

     

    The results are, (see Sumner double altitudes.pdf):

    L = 81.583253

    Lprima = -30.043253

    L = 81.499574

    Lprima = -29.959574

    SumnerLoP 1:   41.916667         -30.043253        42.083333         -29.959574

     

    L = 54.553448

    Lprima = -30.113448

    L = 54.313510

    Lprima = -29.873510

    SumnerLoP 2:   41.916667         -30.113448        42.083333         -29.873510

     

    Sumner2CoPFix:           41.991536         -30.005998

     

     

     

     

    For comparison:

    .

    TWO Circles of Position

    CoP1 = 334.230000   5.220000     28.500000

    CoP2 = 347.780000   -16.720000   19.550000

     

    Vector Solution for the Intersection of two Circles of Equal Altitude

    THE JOURNAL OF NAVIGATION (2008), 61, 355-365. The Royal Institute of Navigation

    Andrés Ruiz González - Navigational Algorithms - http://sites.google.com/site/navigationalalgorithms/

    iter   Err    Be     Le     B1     L1     B2     L2     GHA1f  dec1f  GHA2f  dec2f

    0      22.047719943373227  41.991775    -30.005655   -11.987157   85.157807    41.991775    -30.005655       334.230000   5.220000     347.780000   -16.720000

    I1: -11.987157      85.157807

    I2: 41.991775 -30.005655

     

     

    An analytical solution of the two star sight problem of celestial navigation

    James A. Van Allen. NAVIGATION Vol. 28, No. 1, 1981

    iter   Err    Be     Le     B1     L1     B2     L2     GHA1f  dec1f  GHA2f  dec2f

    0      22.047719943364704  41.991775    -30.005655   41.991775    -30.005655   -11.987157   85.157807       334.230000   5.220000     347.780000   -16.720000

    I1: 41.991775 -30.005655

    I2: -11.987157      85.157807

     

    Marq Saint Hilaire: ( 41º 59.5' -30º  0.3')

     

    Andrés Ruiz

    Navigational Algorithms

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

     

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