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    Re: Correcting for the movement of an observer: a plausible explanation?
    From: Frank Reed
    Date: 2019 Dec 22, 19:11 -0800

    Picture the celestial sphere as an actual sphere a thousand miles across with the stars created by little lights on the inner surface of the sphere. You're sailing under this sphere at a speed of some knots, and you take some star sights over a period of, let's say, fifteen or twenty minutes. You would like to synchronize those sights -- to find the equivalent sights all taken at one instant of time. The value here is that the sights are much easier to plot when synchronized --if wwe're using fast lookup tables like Pub.249. These tables require integral values for LHA Aries, which means that you have to pick your AP longitude such that the minutes either cancel the minutes of GHA Aries (in West longitude) or complement the minutes (in East longitude). If you shoot four stars at four random times, you would need to select four unique AP longitudes leading to a messy plot. And that's not even worrying about the motion of the vessel which complicates the plot even further. It would all be so much nicer if we could use one AP position, and eliminate the process of advancing LOPs for vessel motion.

    Enter the MOO and MOB tables... The MOO table adjusts a set of sights for "Motion Of the Observer", synchronizing them all to one time, backing out vessel motion. The MOB table does the same for the "Motion Of the Body" (celestial body, in other words, the normal motion of a star due to the rotation of the Earth). The MOB table, combined with the MOO table, removes the changes in altitude that occur over relatively short periods of time.

    To picture the effect of the MOO table, imagine a vessel crossing the Earth's surface under that celestial sphere. Stars off the bow climb. They rise at a rate of one minute of arc for every nautical miles that we travel. Stars astern sink at the same rate. What happens to the rest of the celestial sphere? If you imagine an axis running abeam, from exactly port to exactly starboard, the celestial sphere rotates about that axis. Stars to port and starboard rotate around the fixed points on the horizon which are exactly perpendicular to our track over the ocean. The changes in altitude, Δh ("delta h"), are given by

       Δh = d · cos(C – Zn)

    where d is the distance made good which may be computed from the elapsed time in hours multiplied by the speed in knots (adjusted for currents if possible). The difference between C, the true course of the vessel, and Zn, the true azimuth of the star, gives the angle off the bow in degrees. You don't need to use the table if you don't have access to a copy. You can work this up easily on a calculator. You do have to think a little about signs, but it's pretty obvious in practice. Suppose I'm in latitude 41.6° N, and I observe the altitude of the star Vega when its azimuth is 310° (which I get from the main table while planning the round of sights) and my course is 090°. I'm travelling at 10 knots. Five minutes and 45 seconds later I observe the altitude of Pollux. Rather than plotting and advancing the Vega LOP, I can bring my altitude of Vega forward. How much? Distance made good is 0.96 nautical miles, and the cosine of course minus star's azimuth is -0.77 so the change in altitude is just about -0.7 minutes of arc. Our vessel is moving (mostly) away from Vega, so the sign is right as it stands. To synchronize the earlier Vega sight with the later Pollux sight, we subtract 0.7 minutes from the observed altitude of Vega. Not a big correction, but easy to do. But we're not done. Time for MOB.

    During the 5.75 minutes between the Pollux and the Vega sights, the Earth was turning, so we also need to adjust the altitude of Vega for the daily rotation of the celestial sphere. The sphere is turning "under its own power", if you like, on an axis that is parallel to the Earth's axis. It's like a spinning umbrella above our heads with the handle pointed nearly at Polaris. It makes one turn in a sidereal day (23h56m). The rate of change of change of altitude is directly proportional to the elapsed time in minutes, T, the cosine of our latitude, and the sine of the true azimuth of the star. The maximum rising rate for stars for observers on the equator with azimuths due East or due West (Zn 90° or 270°) is 15.04’ per minute of time. The rate of change of altitude, in full, is:

      Δh = 15.04ʹ · T · cos(Lat) · sin(Zn)

    This is tabulated in MOB tables, and it's also easy to work up on a calculator. For the Vega sight above, T is 5.75 minutes (as before) and Lat is 41.6° while Zn is 310°. Putting it all together, the change in altitude is -49.5'. If we were at rest, we would synchronize our Vega sight to the time of our Pollux sight, by lowering it, reducing the observed altitude, by 49.5 minutes of arc. Combining this with the MOO correction, the net amount is a reduction in altitude of 50.2 minutes of arc. The Vega and Pollux sights are not synchronized. They have the same GHA Aries and thus the same LHA Aries, and we do not need to advance the earlier LOP. Where the two synchronized lines of position cross, we mark the fix and that's it. By the way, it's normal for surface navigation cases that the MOO correction will be something like two orders of magnitude smaller than MOB. That's because the speed of rotation of points on the Earth's equator (relative to points on the axis of rotation) is 900 knots. Surface vessel speeds will usually be 50 to 100 times smaller. If you're flying in a supersonic aircraft, you can make MOO and MOB cancel out. The stars will stand still.

    Calculations and tables like these should not be used for time intervals greater than 15 to 30 minutes though naturally it all depends on how much accuracy you want in the end.

    It's worth mentioning/reminding here that there's another way of dealing with MOB, without using a table (or the calculation above), and that's to use carefully timed sight intervals. Use a "stop watch" or equivalent and shoot sights at exactly four minute intervals (3m 59.2s, if you want to worry). This increases GHA Aries by exactly one degree between each pair of sights and implies the same AP for each and every sight in the series. On smaller vessels in heavy seas, this sort of timing is not possible, but on a large, stable vessel, it's a valuable trick.

    Frank Reed

    PS: Tony, I realize I'm not directly addressing your question, but it seemed like a good context to make this post.

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