Accuracy of satellite positions

From: Mike Waterman (Mike.Waterman@marconi.com)
Date: Mon Sep 04 2000 - 07:11:37 PDT

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    There have been a few postings recently about observing accuracy, so
    here is my contribution.
    
    Many (probably most) observers determine satellite positions as a 
    fraction between 2 stars. The error in position is partly due to the
    accuracy of the star positions, partly due to the error in determining
    the fraction. 
    
    In the pre-computer days star positions were measured by ruler on star 
    maps (error maybe 0.02deg with a good map); if you still do this then 
    check the accuracy by measuring a few stars whose true positions are known.
    These days star catalogs are commonly used, so the error in star positions
    is very small.
    
    The major error is due to the fraction.
    
    On 1968 October 16 an experiment was performed in which a simulated 
    satellite passed among a projected star field, and about 30 satellite 
    observers estimated the fraction (which was later measured accurately). 
    Here is a summary of the results collected by Gordon Taylor, all in %;
    see below for column f2.
      Test   True      Mean    Standard   f2
            fraction   error   deviation
       1       55       -1         4       5
       2       34        1         6       4
       3      157      -10        11      18
       4       27       -1         4       4
       5       53       -3         5       5
       6      203       -4        20      42
       7       85        0         4       3
       8      -21        3         4       5
       9       27       -2         4       4
      10       73        1         5       4
      11      -59        8        11      19
      12       20        0         4       3
    Note that tests 3 6 8 11 extrapolate beyond the stars, the others
    are interpolations. As one might expect, the extrapolations are 
    generally less accurate. Unfortunately none of the tests went close
    to either star. I believe that close to a star, the error would be
    approximately proportional to the distance from that star.
    
    In 1985 I started interpolating by computer, and I wanted a formula 
    to give me accuracy, depending on fraction and star separation. I chose:
         error = e+(star separation)*abs(f*(1-f))*k
    where e and k are constants.
    The formula is symmetric about f=0.5 (=50%), and gives the desired 
    behaviour near either star (f near 0 or 1), and larger errors for 
    extrapolations (f<0 or f>1).
    
    Column f2 above is this function with constant e=0 and k=0.2 (=20%), 
    which is a reasonable fit to the standard deviations except for the 3
    biggest extrapolations.
     
    Constant e allows for the resolution of your binoculars (plus star map
    errors if used), and should be small, but not zero. Currently I use 
    a peculiar e=0.02+0.005*(star separation) deg and k=1/12 which gives 
    3.6% for separation 2deg and f=0.5, (similar to my personal results at 
    the meeting). 
    I probably should use e=0.01deg and k=0.12 which would usually give 
    similar results. Those with higher magnification than my 11* should use
    smaller e.
    Note that e is much larger than the nominal resolution of the binoculars,
    since we are watching moving objects, often variable or faint.
    
    Sometimes the accuracy from the formula has to be degraded, for example
    if the visibility is bad, the satellite very faint, or only briefly
    visible, or a cat has just jumped onto your lap.
    
    You can get some idea of your accuracy by finding 3 stars in a line,
    quickly estimating the fraction, and later calculating the exact 
    fraction from the star positions. Or estimate the ratio (separation
    stars A & B)/(separation stars B & C). Or write a PC simulation of the
    1968 experiment (I have a QBASIC program for this).
    
    Mike Waterman      mike.waterman@marconi.com
    Site Yateley = COSPAR 2115 =  51.3286N  0.7950W  75m.
    
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