• Nutation

    The Nutation, found by the British astronomer James Bradley (1693 - 1762), it's a periodic oscillation of the Earth's rotation axis around his "mean" position . Due to the nutation, the instantaneous pole of the rotation of the Earth oscillate around the "mean" pole that spins around, due to the precession, the ecliptic pole.

Fig. 01: Scheme with the Precession and Nutation effects.

    The nutation is associated mainly to the action of the Moon and can be described as a sum of many terms. The more important term have a period of about 6798.4 days (18.6 years), but some of the other ones have a very shorten period (less than 10 days).

    The nutation are distributed in two components, one parallel and the other perpendicular to the ecliptic. The parallel component is denoted by Dy and it calls Nutation in Longitude and affects the celestial longitude of all the celestial bodies, the other one, the perpendicular, it's detonated by De and it calls Nutation in Obliquity, since that affects the obliquity of the Equator regarding to the ecliptic. However it's important to remark that the nutation doesn't have any effect in the latitude of the celestial bodies.

    The components Dy and De can be determinate using the following procedure:

        T = (JDE - 2451545)/36525

        D = 297.85036 + 445267.111480T - 0.0019142T² + T³/189474

        M = 357.52772 + 35999.050340T - 0.0001603T² - T³/300000

        M' = 134.96298 + 477198.867398T + 0.0086972T² + T³/327270

        F = 93.27191+483202.017538T - 0.0036825T² + T³/327270

        W = 125.04452 - 1934.136261T + 0.0020708T² + T³/450000

    These quantities Dy and De are obtain through the sum of many terms givens in a table in the “1980 IAU Theory of Nutation” (Ibid page S23).

    It's obvious that if not higher accuracy is needed it's possible to use only the terms with the largest coefficients. If an accuracy of 0’’.5 in Dy and 0’’.1 in De is enough then we can ignore the terms in T² and in T³ in the expression of W. So if we made that we will obtain the following expressions:

Dy = -17''.20sin(W) - 1''.32sin(2L) - 0''.23sin(2L') + 0''.21sin(2W)

De = 9''.20cos(W) + 0''.57cos(2L) + 0''.10cos(2L') - 0''.09cos(2W)

    Where L and L’ represents the mean longitudes of the Sun and of the Moon respectively.

L=280º.4665+36000º.7698T

L’=218º.3165+481267º.8813T

    • Obliquity of the Ecliptic:

    The obliquity of the ecliptic is the angle between the Equator and the Ecliptic. We need to make the distinction between mean and true obliquity, being the angles between the ecliptic and the mean and the true equator, respectively. In other words, the adjective mean means that the nutation effect is not taken into account.

    The mean obliquity of the ecliptic is given by the following formula:

e₀ = 23º26'21''.448 - 46''.8150T - 0''.00059T² + 0''.001813T³

(formula adopted by the International Astronomical Union)

    The accuracy of the previous formula is not enough for a long period of time, since that after 2000 years the error is about 10’’. So, we need to adopt an improved formula (formula founded by J. Laskar, "Astronomy and Astrophysics", Vol. 157, page 68 (1986)).

    Take U=T/100, then:

e₀ = 23º26'21''.448 - 4680.93U - 1.55U² + 1999.25U³ - 51.38U⁴

- 249.67U⁵ - 39.05U⁶ + 7.12U⁷ + 27.87U⁸ + 5.79U⁹ + 2.45U¹⁰

    After 1000 years the accuracy of the previous formula is about 0.’’01 and only a few seconds after 10000 years. However this formula it's only correct for a period of about 10000 years for each side of the initial epoch (J2000.0).

    As a curiosity, we show a graphic with the variation of e₀ since 10000 years before and 10000 years after J2000.0.

Fig. 02: Graphic with the variation of the Mean Obliquity of the Ecliptic during 200 centuries.

    The true obliquity of the ecliptic is e ₌ e₀ + De, where De is the nutation in obliquity.

    • Corrections to made in the position of the Star:

    The easiest way is to apply the nutation effect to the mean position and add Dy to the ecliptic longitude of the object. This procedure can be used in the calculation of the position of the planets, but for the stars it's preferable to determinate the corrections in right ascension and declination.

    The first order corrections to the right ascension a and to the declination d of a star due to the nutation are:

Da₁ = (cos(e) + sin(e)sin(a)tan(d))Dy - (cos(a)tan(d))De

Dd₁ = (sin(e)cos(a))Dy + sin(a)De

    These expressions are invalids for some stars near the celestial poles, in this case is preferable to work in ecliptic coordinates and simply add Dy to the longitude.