USNO Circular 179 - U.S. Naval Observatory

16 TIME SCALES
method and the time span considered. Differences in rate of some tens of microseconds per century
are possible (Fairhead & Bretagnon 1990). These rate ambiguities are probably unimportant
(amounting to fractional errors of only ∼10 −14 ) for retrieving positions and velocities from the
ephemeris but do affect pulsar timing that is reduced to the barycentric time scale.
2.6.2 Formulas for Time Scales Based on the Rotation of the Earth
For the time scales that are based on the rotation of the Earth, we again start with UTC.
UT1 = UTC + (UT1–UTC) (2.7)
≈ UTC + DUT1 (2.8)
where DUT1 is a broadcast approximation to UT1–UTC (precision ±0. s 1). We also have
UT1 = TT − ∆T (2.9)
where ∆T = 32. s 184 + ∆AT – (UT1–UTC). The most recent values of UT1–UTC and ∆AT are
listed in IERS Bulletin A (URL 5). Values of ∆T are listed in The Astronomical Almanac on page
K9.
The Earth Rotation Angle, θ, is
θ = 0.7790572732640 + 1.00273781191135448 DU
(2.10)
where DU is the number of UT1 days from 2000 January 1, 12 h UT1: DU = JD(UT1) – 2451545.0.
The angle θ is given in terms of rotations (units of 2π radians or 360 ◦ ). The above rate coefficient
gives an Earth rotation period of 86164.0989036903511 seconds of UT1. If we consider this to be
the adopted average rotation period of the Earth in SI seconds, it is consistent with the nominal
mean angular velocity of Earth rotation, ω = 7.292115×10 −5 radian s −1 , used by the International
Association of Geodesy. The above expression is taken directly from note 3 to res. B1.8 of 2000. An
equivalent form of this expression (if the integral number of rotations is neglected) that is usually
more numerically precise is
θ = 0.7790572732640 + 0.00273781191135448 DU + frac(JD(UT1)) (2.11)
where frac(JD(UT1)) is the fractional part of the UT1 Julian date, i.e., JD(UT1) modulus 1.0.
Then Greenwich mean sidereal time in seconds is
GMST = 86400 · θ + (0.014506 + 4612.156534 T + 1.3915817 T 2
−0.00000044 T 3 − 0.000029956 T 4 − 0.0000000368 T 5 )/15 (2.12)
where T is the number of centuries of TDB (equivalently for this purpose, TT) from J2000.0: T
= (JD(TDB) – 2451545.0)/36525. The polynomial in parentheses is the accumulated precession of
the equinox in right ascension, in arcseconds, as given for the P03 solution (eq. 42) in Capitaine
et al. (2003). Note that two time scales are now required to compute sidereal time: in the “fast
term”, θ is a function of UT1, while in the remaining terms, T is expressed in TDB (or TT).
To obtain Greenwich apparent sidereal time in seconds, we must add the equation of the
equinoxes:
GAST = GMST + EΥ/15 (2.13)