Astronomical Angle Computation

The harmonic prediction equation requires knowing the values of the six fundamental variables (\(\tau, s, h, p, N', p_1\)) at any given time. These angles are computed from polynomial expressions in the Julian century \(T\) measured from a reference epoch. PyFES provides four sets of formulae, selectable via the Formulae enumeration.

This section documents the mathematical expressions implemented in each formula set and the auxiliary angles derived from them.

General Form

All formula sets express each fundamental variable as a polynomial in the Julian century \(T\):

\[\theta(T) = a_0 + a_1 T + a_2 T^2 + a_3 T^3 + \ldots\]

where the coefficients \(a_i\) are given in arc-degrees (or, for the IERS formulae, in arc-seconds with subsequent conversion). The Julian century is defined as:

\[T = \frac{t - t_{\text{epoch}}}{36\,525 \times 86\,400}\]

where \(t\) is in seconds and \(t_{\text{epoch}}\) is the reference epoch of the formula set.

Schureman Order 1

Enumeration value:

Formulae.SCHUREMAN_ORDER_1

Reference epoch:

J1900.0 (January 0.5, 1900 Greenwich Civil Time)

Time scale:

UTC

Polynomial order:

1 (linear)

Source:

Schureman (1940), Table 1, p. 162

This is the simplest formula set, using first-order polynomials that give the mean rate of change of each element over a century:

\[\begin{split}\begin{aligned} N &= 259°\,10'\,57.12'' - (5 \times 360° + 482\,912.63'') \, T \\ h &= 279°\,41'\,48.04'' + 129\,602\,768.13'' \, T \\ s &= 270°\,26'\,14.72'' + (1336 \times 360° + 1\,108\,411.20'') \, T \\ p_1 &= 281°\,13'\,15.0'' + 6\,189.03'' \, T \\ p &= 334°\,19'\,40.87'' + (11 \times 360° + 392\,515.94'') \, T \end{aligned}\end{split}\]

The multiples of \(360°\) represent complete revolutions per century that are removed modulo \(360°\) in practice.

Schureman Order 3

Enumeration value:

Formulae.SCHUREMAN_ORDER_3

Reference epoch:

J1900.0

Time scale:

UTC

Polynomial order:

3 (cubic)

Source:

Schureman (1940), Table 1, p. 162

This extends the Order 1 formulae to cubic polynomials, providing improved accuracy over multi-century intervals. The additional quadratic and cubic coefficients capture slow secular variations in the orbital elements. For example:

\[N = 259°\,10'\,57.12'' - (5 \times 360° + 482\,912.63'') \, T + 7.58'' \, T^2 + 0.008'' \, T^3\]

The other variables follow the same pattern with their respective higher-order coefficients.

Meeus

Enumeration value:

Formulae.MEEUS

Reference epoch:

J2000.0 (January 1.5, 2000 TDT)

Time scale:

Terrestrial Dynamical Time (TDT)

Polynomial order:

4 (quartic)

Source:

Meeus (1998), Astronomical Algorithms, Ch. 47

The Meeus formulae use the J2000.0 epoch and Terrestrial Dynamical Time rather than UTC. The fundamental variables are computed via intermediate quantities:

\[\begin{split}\begin{aligned} s &= 218.3164477° + 481\,267.88123421° \, T - 0.0015786° \, T^2 + \ldots \\ d &= 297.8501921° + 445\,267.1114034° \, T - 0.0018819° \, T^2 + \ldots \\ h &= s - d \\ g &= 357.5291092° + 35\,999.0502909° \, T - 0.0001536° \, T^2 + \ldots \\ p_1 &= h - g \\ N &= 125.0445479° - 1\,934.1362891° \, T + 0.0020754° \, T^2 + \ldots \\ p &= 83.3532465° + 4\,069.0137287° \, T - 0.0103200° \, T^2 + \ldots \end{aligned}\end{split}\]

Here, \(d\) is the mean elongation of the Moon from the Sun and \(g\) is the mean anomaly of the Sun. The higher-order terms provide improved precision for modern epochs.

Note

When using the Meeus or IERS formulae, PyFES automatically converts UTC to TDT using the utc_2_tdt() function, applying the appropriate \(\Delta T\) correction.

IERS

Enumeration value:

Formulae.IERS

Reference epoch:

J2000.0

Time scale:

Terrestrial Dynamical Time (TDT)

Polynomial order:

4 (quartic)

Source:

IERS Conventions (2010), based on Simon et al. (1994)

The IERS formulae use the five lunisolar fundamental arguments of the nutation theory:

\[\begin{split}\begin{aligned} l &= 485\,868.249036'' + 1\,717\,915\,923.2178'' \, T + 31.8792'' \, T^2 + \ldots \quad \text{(mean anomaly of the Moon)} \\ l' &= 1\,287\,104.79305'' + 129\,596\,581.0481'' \, T - 0.5532'' \, T^2 + \ldots \quad \text{(mean anomaly of the Sun)} \\ F &= 335\,779.526232'' + 1\,739\,527\,262.8478'' \, T - 12.7512'' \, T^2 + \ldots \quad \text{(argument of latitude)} \\ D &= 1\,072\,260.70369'' + 1\,602\,961\,601.2090'' \, T - 6.3706'' \, T^2 + \ldots \quad \text{(mean elongation)} \\ \Omega &= 450\,160.398036'' - 6\,962\,890.5431'' \, T + 7.4722'' \, T^2 + \ldots \quad \text{(longitude of ascending node)} \end{aligned}\end{split}\]

These are then converted to the traditional Doodson fundamental variables:

\[\begin{split}\begin{aligned} s &= F + \Omega & \text{(mean longitude of the Moon)} \\ h &= F + \Omega - D & \text{(mean longitude of the Sun)} \\ p &= F + \Omega - l & \text{(longitude of lunar perigee)} \\ p_1 &= -l' + F - D + \Omega & \text{(longitude of solar perihelion)} \\ N &= \Omega & \text{(longitude of the ascending node)} \end{aligned}\end{split}\]

Comparison of Formula Sets

Property	Schureman O1	Schureman O3	Meeus	IERS
Epoch	J1900.0	J1900.0	J2000.0	J2000.0
Time scale	UTC	UTC	TDT	TDT
Poly. order	1	3	4	4
Default engine	Darwin	Darwin	Either	PERTH

Derived Auxiliary Angles

After computing the six fundamental variables, PyFES derives several auxiliary angles required for the nodal corrections. These are computed once per time step in the update() method.

Obliquity of the Lunar Orbit (\(I\))

The inclination \(I\) of the lunar orbit with respect to the celestial equator varies between approximately 18.3° and 28.6° over the 18.61-year nodal cycle. From Schureman (p. 156):

\[\cos I = \cos i \cos \omega - \sin i \sin \omega \cos N\]

where \(i = 5.145°\) is the inclination of the lunar orbit to the ecliptic and \(\omega = 23.452°\) is the obliquity of the ecliptic. The numerical constants are:

\[\cos I = 0.9137 - 0.0357 \cos N\]

Longitude of the Lunar Intersection (\(\xi\))

The longitude in the Moon’s orbit of the intersection with the celestial equator is:

\[\xi = -\arctan\!\left(1.0188 \, \tan \tfrac{N}{2}\right) -\arctan\!\left(0.6441 \, \tan \tfrac{N}{2}\right) + N\]

Right Ascension of the Lunar Intersection (\(\nu\))

\[\nu = \arctan\!\left(1.0188 \, \tan \tfrac{N}{2}\right) -\arctan\!\left(0.6441 \, \tan \tfrac{N}{2}\right)\]

These two auxiliary angles appear in the phase corrections \(u\) of most tidal constituents (see Nodal Corrections: Amplitude and Phase Modulations).

Factors for K₁ and K₂

The lunisolar constituents \(K_1\) and \(K_2\) combine lunar and solar terms, requiring dedicated correction angles:

\[\nu' = \arctan\!\left( \frac{\sin 2I \sin \nu}{\sin 2I \cos \nu + 0.3347} \right)\]

\[\nu'' = \frac{1}{2}\arctan\!\left( \frac{\sin^2 I \sin 2\nu}{\sin^2 I \cos 2\nu + 0.0727} \right)\]

where the numerical constants 0.3347 and 0.0727 are the ratios of the solar to lunar coefficients (Schureman formulas 224 and 232).

Factors for L₂

The constituent \(L_2\) requires two additional quantities:

\[x_{1ra} = \sqrt{1 + 36\,\tan^2\!\left(\tfrac{I}{2}\right)^2 - 12\,\tan^2\!\left(\tfrac{I}{2}\right) \cos 2(p - \xi)}\]

\[R = \arctan\!\left( \frac{\sin 2(p - \xi)}{(6\,\tan^2(I/2))^{-1} - \cos 2(p - \xi)} \right)\]

These are derived from Schureman formulas 213–214 (p. 44).

Connection to the AstronomicAngle Class

The AstronomicAngle Python class exposes all the angles described above:

Property	Description
t	Mean solar angle relative to Greenwich (hour angle of the mean Sun)
s	Mean longitude of the Moon
h	Mean longitude of the Sun
p	Longitude of the lunar perigee
p1	Longitude of the solar perihelion
n	Longitude of the Moon’s ascending node
i	Obliquity of the lunar orbit (\(I\))
xi	Longitude in Moon’s orbit of the lunar intersection (\(\xi\))
nu	Right ascension of the lunar intersection (\(\nu\))
nuprim	Correction angle for \(K_1\) (\(\nu'\))
nusec	Correction angle for \(K_2\) (\(\nu''\))
x1ra	Amplitude factor for \(L_2\) (\(x_{1ra}\))
r	Phase factor for \(L_2\) (\(R\))

References

• Schureman, P. (1940). Manual of Harmonic Analysis and Prediction of Tides, SP 98, Table 1 (p. 162), p. 156, formulas 213–214, 224, 232.

• Meeus, J. (1998). Astronomical Algorithms, 2nd ed., Ch. 47.

• Petit, G. & Luzum, B. (2010). IERS Conventions (2010), IERS Technical Note 36.

• Simon, B. (2013). Marées Océaniques et Côtières (943-MOC), Appendix A.