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I modified my Monte Carlo test program to compare the low accuracy Sun
ephemeris algorithms from Umland ("Short Guide to Celestial
Navigation"), Meeus ("Astronomical Algorithms"), and the USNO
(http://aa.usno.navy.mil/faq/docs/GAST.php and
http://aa.usno.navy.mil/faq/docs/SunApprox.php).

In celestial navigation we are really interested Greenwich hour angle,
not right ascension, so my error statistics include GHA, declination,
and semidiameter. The "dir" column is the total error in direction due
to the GHA and declination error. I also calculated error in Greenwich
hour angle of Aries, since that's necessary to reduce star observations.

The "gold standards" are the JPL DE431 ephemeris and IAU 2006/00B
precession / nutation / sidereal time routines, implemented in my
SofaJpl fundamental astronomy DLL.

All values in the table are the root mean squared error in minutes of
arc after 10,000 tests at random times in the 21st century.

GHA  dec  dir   SD   GHAA
.22  .13  .33  .006  .19   Umland
.29  .12  .30  .002  .00   USNO 1
.29  .12  .30  .002  .01   USNO 2
.29  .12  .30  .002  .01   USNO 3
.22  .06  .22  .001  .01   Meeus 1
.22  .06  .22  .001  .00   Meeus 2

Umland's formulas are the simplest. His GHAA expression does not correct
for Δψ (nutation in longitude), whereas the other routines apply the
correction. Curiously, his Sun GHA is as good as any method despite the
greater error in GHAA.

The USNO says the squared term in their GMST expression "can be omitted
in most applications." That's #2, and #3 is the result with the still
simpler "alternative formula" in which GMST is a linear function of UT1.

My Meeus implementation is from the "low accuracy" section of his "Solar
Coordinates" chapter. It is more elaborate than the other routines. In
#2 I used more precise expressions from his Nutation chapter for the
angles Δψ and Δε. The difference is hardly noticeable.

   

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