Nutation of the Earth's axis

Nutation of the Earth's axis[edit]

Causes[edit]

Precession and nutation are caused principally by the gravitational forces of the Moon and Sun acting upon the non-spherical figure of the Earth. Precession can be thought of as the effect of these forces when averaged over a very long period of time, and has a timescale of about 26,000 years. Nutation occurs because the forces are not constant, and vary as the Earth revolves around the Sun, and the Moon revolves around the Earth.

The largest term in nutation is caused by the fact that the orbit of the Moon around the Earth is inclined at slightly over 5° to the plane of the ecliptic, and the orientation of this orbital plane varies over a period of about 18.6 years. Because the Earth's equator is itself inclined at an angle of about 23.5° to the ecliptic (the obliquity of the ecliptic, ${\displaystyle \epsilon }$), these effects combine to mean that the inclination of the Moon's orbit to the equator varies between about 18.4° and 28.6° over the 18.6 year period. This causes the orientation of the Earth's axis to vary over the same period, with the true position of the celestial poles describing a small ellipse around their mean position. The maximum radius of this ellipse is approximately 9.2 arcseconds, a figure which is known as the constant of nutation.

Smaller terms in nutation must also be added to this principal term. These are caused by the monthly motion of the Moon around the Earth and the fact that its orbit is eccentric, and similar terms caused by the annual motion of the Earth around the Sun.

Effect on position of astronomical objects[edit]

Because nutation causes a change to the frame of reference, rather than a change in position of an observed object itself, it applies equally to all objects. Its magnitude at any point in time is usually expressed in terms of ecliptic coordinates, as nutation in longitude (${\displaystyle \Delta \psi }$) and nutation in obliquity (${\displaystyle \Delta \epsilon }$). The largest term in nutation is expressed numerically (in arcseconds) as follows:

${\displaystyle \Delta \psi =-17.2\sin \Omega }$

${\displaystyle \Delta \epsilon =9.2\cos \Omega }$

where ${\displaystyle \Omega }$ is the ecliptic longitude of the ascending node of the Moon's orbit.

Spherical trigonometry can then be used on any given object to convert these quantities into an adjustment in the object's right ascension and declination. For objects that are not close to a celestial pole, nutation in right ascension (${\displaystyle \Delta \alpha }$) and declination (${\displaystyle \Delta \delta }$) can be calculated approximately as follows:^[2]

${\displaystyle \Delta \alpha =(\cos \epsilon +\sin \epsilon \sin \alpha \tan \delta )\Delta \psi -\cos \alpha \tan \delta \Delta \epsilon }$

${\displaystyle \Delta \delta =\cos \alpha \sin \epsilon \Delta \psi +\sin \alpha \Delta \epsilon }$

History[edit]

Nutation was discovered by James Bradley from a series of observations of stars conducted between 1727 and 1747. These observations were originally intended to demonstrate conclusively the existence of the annual aberration of light, a phenomenon that Bradley had unexpectedly discovered in 1725-6. However, there were some residual discrepancies in the stars' positions that were not explained by aberration, and Bradley suspected that they were caused by nutation taking place over the 18.6 year period of the revolution of the nodes of the Moon's orbit. This was confirmed by his 20-year series of observations, in which he discovered that the celestial pole moved in a slightly flattened ellipse of 18 by 16 arcseconds about its mean position.^[3]

Although Bradley's observations proved the existence of nutation and he intuitively understood that it was caused by the action of the Moon on the rotating Earth, it was left to later mathematicians, d'Alembert and Euler, to develop a more detailed theoretical explanation of the phenomenon.^[4]