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    Refraction & star separation angles
    From: Paul Hirose
    Date: 2000 Sep 24, 10:54 PM

    Since the effect of refraction on the apparent angular separation
    between stars has come up, I did some research this afternoon. First,
    I assumed two stars of known and identical altitude. From a given
    altitude and a desired angular separation, I calculated the required
    difference in azimuth. Then, holding that angle constant (refraction
    doesn't affect azimuth), I increased both star altitudes by the
    almanac refraction value. Finally, I computed the new angular
    separation, and tabulated the change from the starting value. The
    following table shows the decrease in angular separation caused by
    refraction (altitude and separation are in degrees).
        |    15      30      45      60      75   <-alt
    sep |----------------------------------------------
     15 |   .25'    .26'    .26'    .27'    .29'
     30 |   .51'    .53'    .54'    .56'    .60'
     45 |   .80'    .81'    .83'    .86'     --
     60 |  1.12'   1.13'   1.15'   1.20'     --
     75 |  1.48'   1.51'   1.53'     --      --
    ( -- = not possible)
    The refraction table in the almanac is only to the nearest tenth
    minute, but I've carried the computations to one more decimal
    place to show the effect of changing the geometry. This table
    shows the "refraction effect" is mainly a function of angular
    separation when the stars have the same altitude, and is little
    affected by changing altitude.
    Next, let's see what happens when the stars are at the same azimuth,
    differing only in altitude. Blessedly, the computations are much
    simpler in this case, requiring little more than inspection of a
    refraction table. As before, refraction always decreases the apparent
    separation. (In the following table, "altitude" refers to the lower
    sep |    15      30      45      60      75   <-alt
     15 |   1.9'     .7'     .4'     .3'     .3'
     30 |   2.6'    1.1'     .7'     .6'     .6'*
     45 |   3.0'    1.4'    1.0'     .9'*    --
     60 |   3.3'    1.7'    1.3'*   1.2'*    --
     75 |   3.6'    2.0'*   1.6'*    --      --
    (* = backsight)  (-- = not practical)
    Someone said this year's almanac would have to be used to compute star
    separations. Strictly speaking, this is true, but in practical terms
    it makes little difference. If you look at the star table in the back
    of the almanac, you'll see an oscillation in a star's coordinates
    amounting to a few tenths of minutes, with the star returning to
    nearly the same spot after one year. Superimposed on this is a
    smaller, slow drift which is practically constant in direction and
    rate year after year.
    The short-term fluctuation is largely something called stellar
    aberration, caused by the velocity of Earth in its orbit and the
    finite speed of light. On the other hand, the slow change from year to
    year is largely precession, the slow wobble of Earth's axis.
    Aberration does change the APPARENT relative positions of stars and in
    theory is enough to be measurable with a sextant, so for work of the
    highest accuracy you'd want to allow for it. Using the correct month
    in the almanac ought to be quite enough. On the other hand, precession
    doesn't affect relative star positions, so an almanac several years
    out of date won't matter.
    There is something called "proper motion" which is a change the ACTUAL
    relative positions of the stars. Perhaps you've seen diagrams showing
    how a familiar constellation will radically change shape far in the
    future. That's proper motion. It's negligible for our purposes unless
    you're using a very old almanac, since the motion is so slow except
    for a few dim stars. E.g., proper motion of Acrux = .05" per year,
    Deneb .003"/yr, Castor .20"/yr, Regulus .25"/yr, Vega .35"/yr,
    Betelgeuse .03"/yr.
    One more digression and I'll hang up. For accurate latitude
    determination on land, surveyors in years past used a zenith
    telescope. This was an instrument optimized for measuring
    altitudes near 90 degrees. A balanced selection of stars north
    and south of the zenith would be observed to cancel refraction.
    It was this device that Mason and Dixon used to establish the
    starting point for their famous boundary line. These paid experts
    had to be brought from England to settle the dispute between
    Pennsylvania and Maryland, since no one in 1760s America had the
    know-how or equipment.

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