[time-nuts] Calculating sidereal time

[Steve Allen]

On Sat 2019-01-19T11:33:28-0600 Didier Juges hath writ:
> The formula I am using uses Modified Julian Date, which has an offset of
> 51544.5 days with Julian Date, so I believe my formula is correct since the
> mjd2gmst() function adds that factor to the mjd value fed to it.

Most software packages find it expedient to break JD or MJD into
two separate doubles, one with the integral part and one with the
fractional part.  Otherwise precision gets lost.

The relationship is defined between GMST and UT1, not UTC.

The USNO site is almost certainly using the IAU2000/P03 models
which were refined in a series of papers through around 2009.
They are almost certainly not using the FK5 version.

The expressions change significantly because there are significant
changes to fundamental reference frames in which the catalog positions
are expressed.  Any catalog exists in a reference frame, and measurement
of earth rotation is done in that frame using that catalog.

In the old days a change of star catalog caused systematic shifts
of the stars and systematic shifts of the equinox.  The convention
has always been that earth rotation is a physical quantity which does
not jump merely because humans changed their catalogs.  The necessary
side effect is that whenever there was a change of catalogs there was
also a systematic shift of the longitude of observatories and thus
all geodetic positions.

FK5 was a relativistic improvement over non-relativistic FK4, but all
reference frames through those were built on an underlying assumption
that the earth is a rigid body without weather in its air, ocean, and
core fluids.  That simplification allowed for the notion of an equinox
at the intersetion of the equator and ecliptic.  That fictitious
intersection was the origin for measuring Universal Time.  But
strictly speaking even the FK5 formalisms had redefined GMST such that
it was no longer the hour angle of the equinox.

In reality the pole position wanders around South Pole Station
sub-diurnally by an amount that the folks there could walk around and
draw on the ice with a marker during the permanent daylight of summer.
(Not something that could be imagined doing in the winter.)  Also, the
pole on the reference frame of the sky moves around sub-diurnally.
That makes it very hard to define what is meant by equinox because
the intersection is moving back and forth during any day.
When VLBI made measuring those diurnal shifts possible it became
necessary to re-think the notions of celestial and terrestrial
coordinates and UT1 with models complex enough to exceed the
measurement precision.

By the late 1990s there were at least three different kinds of equinox
use for various purposes.  The new IAU 2000 model did not require the
existence of an ecliptic plane, and thus did not have an explicit
equinox.  It also did not have a self-consistent precession and
nutation model until the P03 model was approved in 2006.  The
expressions for those were refined in papers through 2008, and they
allowed for a self-consistent notion of ecliptic and equinox, and.  a
new formalism for GMST.  All of this removes the interdependence of
terrestrial coordinates and celestial coordinates so that future star
catalogs need not shift terrestrial coordinates.

The IAU Working Group for Nomenclature in Astronomy gave names and
explanations for the new models
https://syrte.obspm.fr/iauWGnfa/

The new expressions are buried in a series of papers and errata.
The right place to find them is in the current IERS Conventions
https://www.iers.org/IERS/EN/Publications/TechnicalNotes/tn36.html
but those are the full ugly of the model.

The most expedient place to find them are roughly pages B7 to B12 in a
current Astronomical Almanac.  See for example
https://babel.hathitrust.org/cgi/pt?id=uc1.31822038913307;view=1up;seq=116

--
Steve Allen                    <sla at ucolick.org>              WGS-84 (GPS)
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