The Minute Waltz

Chopin’s masterpiece for the piano, written in 1847, takes between 1.5 to 2.5 minutes to play depending on the pianist.

The period of rotation of the Earth

The true siderial day is the period of rotation of the Earth relative to the distant (fixed) stars and is 86164.09053 seconds.

The lunar month

Although the Moon appears relatively stable in its orbit about the Earth, its motion is complex and there are several different definitions of lunar months. The siderial month is approximately 27.32166 days, the time for it to return to a fixed point relative to the distant stars. The tropical lunar month is the time for the Moon to return to the same point on the ecliptic (the imaginary point on the celestial equator intersected by the plane of the Earth’s orbit, discussed below). Because of the precession of the equinoxes, the tropical lunar month is approximately 27.32158 days, slightly shorter than the siderial lunar month.

The year

Whilst Newtonian orbit theory predicts that the Earth’s orbit is a perfect ellipse with a definite period known as a year, a full description of the motion of the Earth around the Sun is complicated by many factors. There are, consequently, several different definitions of an Earth ‘year’. The sidereal year is the time required for the Earth to complete an orbit of the Sun relative to the distant stars and is 365 days, 6 hours, 9 minutes, and 9.5 seconds. This would be the length of the year if there were no complicating factors.

The ecliptic plane, the plane of the Earth’s orbit, is inclined by just over 23 degrees relative to the equatorial plane, the plane perpendicular to the axis of the Earth’s daily rotation. Over a year, observers on the Earth see the Sun moving slowly relative to the distant stars on a path known as the ecliptic. This path intersects the equatorial plane at two points known as the equinoxes. One of the equinoxes takes place approximately on 20 March (the vernal equinox) at a point on the ecliptic known as the First point of Aries. The other equinox is on 22 September (the autumnal equinox).

A significant complicating factor is that the Earth is not a perfect sphere and on that account there is a ‘wobble’ or precession of its axis of rotation. One of the triumphs of Newtonian mechanics is that it can account for such wobbles in the motion of a rotating mass. It was noticed in Antiquity that the equinoxes move gradually eastwards, a phenomenon known as the precession of the equinoxes. Every year, the First Point of Aries crosses the equatorial plane just over 20 minutes earlier than the previous year. The astronomer Hipparchus estimated that the period of the Earth’s ‘wobble’ was about 36 thousand years. Modern measurements give a figure of 25,722 years. A tropical or solar year is the time for the Sun to return to the same position, relative to the cycle of seasons, as seen from Earth. The mean tropical year on 1 January 2000 was about 20 minutes and 25 seconds shorter than the sidereal year.

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