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    Re: Raw data for bubble
    From: Gary LaPook
    Date: 2007 Mar 11, 15:16 -0700

    Gary LaPook wrote:
    
    Congratulations, you guys have managed to reinvent the wheel!
    
    Every volume of H.O. 249 contains two tables for correction for the
    motion of the body (MOB) as well as tables for correction for motion
    of the observer. The  MOB tables show the change in altitude both for
    a one minute interval and for a four minute interval as well as an
    interpolation table for other time intervals based on the observer's
    latitude and the azimuth of the body. I will try to post them
    tomorrow.
    
    Your formula will also compute this rate. The basis for the formula is
    that the earth turns 15 minutes of arc in one minute of time which is
    equal to 15 NM at the equator and at a  slower rate at other latitudes
    based on the cosine of the latitude, e.g. at 40� lat the earth is
    turning only 11.5 NM per minute. This, then, is the rate of altitude
    change per minute, the slope, for a body on azimuth 90� or 270�. Then
    by multiplying by the sine of the azimuth you find how much of this
    maximum change will affect the altitude of a body on a different
    azimuth.
    
    There is also a table for the motion of the observer (MOO) which is
    used to adjust the Hc to allow for the motion of the observer between
    shots and is the equivalent to advancing the LOP to obtain a running
    fix. All fixes in the air are "running" since the plane moves a
    significant distance between the first an last shot, about 60 NM at
    450 knots. Even though a boat is moving between shots its small amount
    of movement can be disregarded.
    
    The reason that these tables are provided in H.O 249 has to do with
    the very different way that celestial is done in the air compared to
    on the surface.
    
     First, since a bubble sextant is used you can shoot stars anytime you
    want to during the night and are not restricted to the limited time
    around twilight. On a boat you wait until twilight and shoot the stars
    and record the times of the observation, which are random, for your
    computations. In the air, you decide what time you want a fix and then
    schedule the times you want to take the sights and then take the
    sights at the pre planned times.
    
    Second, you must come up with a fix rapidly. On a boat you can take
    the sights and only then go below to start the computations and you
    could wait until the next day, if you wanted to, to compute the fix.
    Since a plane is moving so quickly, a ten minute delay in plotting the
    fix will mean the plane could be 100 NM  from the fix by the time it
    is plotted so procedures are used to minimize the time between taking
    the shots and finishing the plot. This includes doing all the
    computations before taking any sights and this is what these MOB and
    MOO tables are used for.
    
    Third, the level of accuracy achievable and the level of accuracy
    needed are much less than for marine navigation so is is perfectly
    acceptable to do the calculations to a lower order of precision, more
    quickly, and the fixes obtained will be within the achievable level of
    accuracy. As we say in the artillery, "it is a waste of time to polish
    the cannon ball."
    
    So here is an example of how it is done. First, you decide what time
    you want a fix, which is usually on the hour. The Air Almanac gives
    the data for every ten minutes  (I will post a page from it also) so
    by choosing one of the listed times (usually on the hour) you don't
    need to do any interpolation of the data. You assume a longitude so
    that LHA Aries is a whole number and then go to H.O.249 Volume 1 for
    selected stars and choose which stars you want to shoot which are well
    spaced in azimuth. Since you are usually above the clouds you can
    shoot in any direction. You take the values of altitude and azimuth
    from H.O.249 without any interpolation. These would be the Hc's if all
    the shots were taken at the planned fix time, which is not possible.
    
    You usually plan to space the shots by four minutes since each shot
    takes two minutes for the use of the averager and this allows two
    minutes then between shooting to write down the measured altitude
    (maybe actually plot the LOP) and reset the sextant to get ready for
    the next star. A common shooting schedule would be to start the first
    shot at 51 after the hour. You set up the sextant, using the expected
    altitude and azimuth, and start tracking the body and then you check
    your watch and trigger the averager at 51:00. You usually shoot the
    first star near the wing tip since advancing its LOP to the fix time
    will have little effect on its accuracy. You continue shooting until
    the shutter closes on the sextant, blocking the view, which tells you
    that two minutes have elapsed, you have, therefore, shot until 53 so
    that the mid time is 52 , which is 8 minutes before the fix time.
    
    You use the next two minutes to reset the sextant and start tracking
    the second star and start the averager at 55:00 so the mid time of the
    second shot is 56:00, 4 minutes prior to fix time. You start the last
    shot at 59 and continue shooting until 01 after the hour, so the mid
    time of the third star is on the hour.
    
    In order to be able to plot the fix as quickly as possible after
    shooting the last star you pre compute the expected altitudes so you
    can compare them immediately with the SEXTANT altitudes (Hs) to
    determine the intercepts. So, using the MOO and MOB tables you adjust
    the Hc from H.O 249 to allow for the two shots taken 4 and 8 minutes
    before fix time. No correction is need for the shot centered on 00.
    You look at the MOO table and take out the correction for 4 minutes
    (this is the reason for the 4 minute table) without any interpolation,
    and add to it the 4 minute correction from the MOO table. This will be
    the correction for these "motions" for the star shot at 56. You do the
    same for the first star but you multiply the sum by 2 for the total
    "motions" for the 52 shot. You add these motions to the Hcs obtained
    from  H.O. 249. You also ADD the refraction correction (that' right,
    ADD) and add the index error (if any) so as to arrive at Hp, pre
    computed altitude. Since you have allowed for index error and
    refraction (no need for dip when using the bubble sextant) in
    computing the Hp you do not have to apply them to the Hs so you can
    compare Hs directly with Hp to determine intercept. It is obvious that
    the this procedure allows for the determination of intercept much more
    rapidly after the shot than in marine practice.
    
    As part of the pre computation process you have plotted the A.P. on
    the chart  (only one is needed with H.O. 249 vol. 1 ) after applying
    the correction for coriolis, precession and nutation, and the azimuths
    so you can quickly plot the LOPs on the chart (or plotting board). You
    have completed the fix in one or two minutes after the last shot
    depending if you had time to plot the first LOPs between shots. So you
    have a fix at 02 or 03 after the hour and can compute the winds
    encountered over the last hour and compute a new heading to
    destination. So by 06 you can give the pilot a new heading and a new
    ETA.
    
    If anybody is interested I can post an example of how this is done.
    
    On Mar 10, 5:29 pm, "Peter Fogg"  wrote:
    > Bill asked:
    >
    > > Does this look like the method you use to calculate slope?
    >
    > > Delta H (rate of change in arc minutes) per minute of time  (divide by 60
    > > for degrees)
    >
    > > Delta H = 15 * cosine latitude * sine azimuth (or its supplement)
    >
    > I used a graphical method to indicate slope which gave +32 minutes of
    > arc over 5 minutes.
    >
    > This formula gives 32.02; so looks good.
    >
    > Thanks Bill
    
    
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