Roughly speaking, the Earth's rotation can be considered as consisting
of two components, the tidally driven component with precisely known
frequencies and the component driven by an exchange of the angular momentum
between the solid Earth and geophysical fluids. The latter component is not
predictable in principle. The atmosphere contributes to the UT1 at a level
of 10^{-6} rad, more than three orders of magnitude higher than the
accuracy of observations. The first component is also affected by the
atmosphere and can be predicted only at a level of 10^{-9} rad.
Therefore, the Earth's rotation has to be continuously measured with
modern space geodesy techniques.

The Earth rotation is mathematically expressed as a transformation of
a vector in the rotating terrestrial coordinate system **r _{t}**
to the inertial celestial coordinate

**r _{c}** = M̂

where, E_{1}(t), E_{2}(t), E_{3}(t) are Euler angles
with respect to axes 1,2,3 and M̂_{x}(E_{x}) is a rotation matrix
with respect to axis x:

1 0 0 M1(E1) = 0 cos E1 sin E1 0 -sin E1 cos E1

cos E2 0 -sin E2 M2(E2) = 0 1 0 sin E2 0 cos E2

cos E3 sin E3 0 M3(E3) = -sin E3 0 cos E3 0 1 0However, accordint to the adopted so-called Newcomb-Andoyer formalism, the rotation matrix is decomposed as a product of 12 elementary rotations:

M̂(t) = M̂_{3}(ζt)) ·
M̂_{2}(-θ(t)) ·
M̂_{3}(z(t)) ·
M̂_{1}(-ε_{0}(t)) ·
M̂_{3}(Δψ(t)) ·
M̂_{1}(ε_{0}(t) + Δε(t)) ·
M̂_{3}(-S_{1}(t) + E_{3}(t)) ·
M̂_{2}(E_{2}(t)) ·
M̂_{1}(E_{1}(t)) ·
M̂_{3}(H_{3}(t)) ·
M̂_{2}(H_{2}(t)) ·
M̂_{1}(H_{1}(t))

where

- ζ(t) — the first angle of the precession in right ascension. It is expressed as a lower degree polynomial with respect to TDB argument. TDB (Time Dynamic Barycentric) is a function of TAI.
- θ(t) — precession declination ascension. It is expressed as a lower degree polynomial with respect to TDB argument.
- z(t) — the second argument of precession in right ascension. It is expressed as a lower degree polynomial with respect to TDB argument.
- ε
_{0}(t) — the mean inclination of the ecliptic to the equator. It is expressed as a lower degree polynomial with respect to TDB argument. - Δψ(t) — nutation in longitude. It is expressed in a quasi-harmonic
expansion

∑ (a_{s}(i) + b_{s}(i) t)*sin ( p(i) + q(i) t + 1/2 r(i) t^{2}) + (a_{c}(i) + b_{c}(i) t)*cos ( p(i) + q(i) t + 1/2 r(i) t^{2}) + Ψ_{0}+ Ψ̇ t

where t is the TDB argument. - Δε(t) — nutation in obliquity. It is expressed in a quasi-harmonic
expansion

∑ (a_{c}(i) + b_{c}(i) t)*cos ( p(i) + q(i) t + 1/2 r(i) t^{2}) + (a_{s}(i) + b_{s}(i) t)*sin ( p(i) + q(i) t + 1/2 r(i) t^{2}) + Ε_{0}+ Ε̇ t

where t is the TDB argument. - S
_{1}(t) — modified stellar argument. It is a function of low degree polynomials, ζ(t), θ(t), z(t); ε_{0}(t), Δψ(t), Δε(t) - E
_{3}(t) — Euler angle around axis 3 (i.e. axial rotation). It is related to a commonly used argument UT1(t) or UT1MTAI(t) (UT1 minus Tai):

E_{3}(t) = - κ UT1(t) = - κ (t - UT1MTAI(t)), where κ = 1.00273781191135448*2π/86400.0. Units for E_{3}(t), units for UT1(t) or UT1MTAI(t) is seconds. E_{3}(t) is a slowly variating function of time and is determined from observations. - E
_{2}(t) — Euler angle around axis 2 (i.e. axial rotation). It is related to a commonly used argument X pole coordinate. E_{2}(t) is a slowly variating function of time and is determined from observations. - E
_{1}(t) — Euler angle around axis 1 (i.e. axial rotation). It is related to a commonly used argument Y pole coordinate. E_{1}(t) is a slowly variating function of time and is determined from observations. - H
_{3}(t) — Euler angle around axis 3 that holds harmonic variations. It is expressed in a form of quasi-harmonic expansion

H_{3}(t) = ∑ a_{c}(i) * cos ( p(i) + q(i) t + 1/2 r(i) t^{2}) + a_{s}(i) * sin ( p(i) + q(i) t + 1/2 r(i) t^{2}) +

Coefficients of the expansion are determined from analysis of space geodesy observations. - H
_{2}(t) — Euler angle around axis 2 that holds harmonic variations. It is expressed in a form of quasi-harmonic expansion

H_{2}(t) = ∑ ( a_{c}(i) + ȧ_{c}) * sin ( p(i) + q(i) t + 1/2 r(i) t^{2}) - ( a_{s}(i) + ȧ_{s}) * cos ( p(i) + q(i) t + 1/2 r(i) t^{2})

Coefficients of the expansion are determined from analysis of space geodesy observations. - H
_{1}(t) — Euler angle around axis 1 that holds harmonic variations. It is expressed in a form of quasi-harmonic expansion

H_{1}(t) = ∑ ( a_{c}(i) + ȧ_{c}) * cos ( p(i) + q(i) t + 1/2 r(i) t^{2}) + ( a_{s}(i) + ȧ_{s}) * sin ( p(i) + q(i) t + 1/2 r(i) t^{2})

Coefficients of the expansion are determined from analysis of space geodesy observations.

*NERS* library keeps numerical coefficients of expansion
ζ(t), θ(t), z(t); ε_{0}(t), Δψ(t),
Δε(t), and S_{1}(t) and has the code that computes them
on the specified moment of time. Empirical functions E(t) and H(t) are taken
from the *NERS* server EOP message. Function E(t) comes
as a table of values on specified, in general non-equidistant epochs.
The tables are updated several times a day. Function H(t) comes in a form of
a table of expansion coefficients determined from analysis of observations.
It is updated 4–6 times a year. See NERS how
for explanation how *NERS server generates the EOP message.
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NERS client automatically downloads the EOP message and
extracts from there E(t) and H(t) functions relevant to the request, and computes
the Earth's rotation matrix.
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