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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
star.)

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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