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Variation (astronomy)
Encyclopedia
"Variation" is a generic word in astronomy as in other fields: many observable astronomical quantities can show variations of some kind. But the term variation also has a specific meaning in astronomy, referring to the variation of the Moon. This is the name given to one of the principal perturbation
s in the motion of the Moon.
The variation was first noted by Abū al-Wafā' Būzjānī. Independently rediscovering the phenomenon, Tycho Brahe
noticed that starting from a lunar eclipse
in December 1590, that at the times of syzygy
(new or full moon), the apparent velocity of motion of the Moon (along its orbit as seen against the background of stars) was faster than expected. On the other hand, at the times of first and last quarter, its velocity was correspondingly slower than expected. (Those expectations were based on the lunar tables widely used up to Tycho's time. They took some account of the two largest irregularities in the Moon's motion, i.e. those now known as the equation of the center
and the evection
, see also Lunar theory - History.)
The main visible effect (in longitude) of the variation of the Moon is that during the course of every month, at the octants of the Moon's phase that follow the syzygies (i.e. half way between the new or the full moon and the next-following quarter), the Moon is about two thirds of a degree farther ahead than would be expected on the basis of its mean motion (as modified by the equation of the centre and by the evection). But at the octants that precede the syzygies, it is about two thirds of a degree behind. At the syzygies and quarters themselves, the main effect is on the Moon's velocity rather than its position.
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In 1687 Newton published, in the 'Principia', his first steps in the gravitational analysis of the motion of three mutually-attracting bodies. This included a proof that the Variation is one of the results of the perturbation of the motion of the Moon caused by the action of the Sun, and that one of the effects is to distort the Moon's orbit in a practically elliptical manner (ignoring at this point the eccentricity of the Moon's orbit), with the centre of the ellipse occupied by the Earth, and the major axis perpendicular to a line drawn between the Earth and Sun.
The Variation has a period of half a synodic month and causes the Moon's ecliptic longitude to vary by nearly two-thirds of a degree
, more exactly by +2370"sin(2D) where D is the mean elongation of the Moon from the Sun.
The variational distortion of the Moon's orbit is a different effect from the eccentric elliptical motion of a body in an unperturbed orbit. The Variation effect would still occur if the undisturbed motion of the Moon had an eccentricity of zero (i.e. circular
). The eccentric Keplerian ellipse
is another and separate approximation for the Moon's orbit, different from the approximation represented by the (central) variational ellipse. The Moon's line of apses, i.e. the long axis of the Moon's orbit when approximated as an eccentric ellipse, rotates once in about nine years, so that it can be oriented at any angle whatever relative to the direction of the Sun at any season. (The angular difference between these two directions used to be referred to, in much older literature, as the "annual argument of the Moon's apogee".) Twice in every period of just over a year, the direction of the Sun coincides with the direction of the long axis of the eccentric elliptical approximation of the Moon's orbit (as projected on to the ecliptic).
Thus the (central) elliptical distortion of the Moon's orbit caused by the variation should not be confused with an undisturbed eccentric elliptical motion of an orbiting body. The variational effects due to the Sun would still occur even if the hypothetical undisturbed motion of the Moon had an eccentricity of zero (i.e. even if in the absence of the Sun it would be circular
).
Newton expressed an approximate recognition that the real orbit of the Moon is not exactly an eccentric Keplerian ellipse, nor exactly a central ellipse due to the variation, but "an oval of another kind". Newton did not give an explicit expression for the form of this "oval of another kind"; to an approximation, it combines the two effects of the central-elliptical variational orbit and the Keplerian eccentric ellipse. Their combination also continually changes its shape as the annual argument changes, and also as the evection shows itself in libratory changes in the eccentricity, and in the direction, of the long axis of the eccentric ellipse.
The Variation is the second-largest solar perturbation of the Moon's orbit after the Evection
, and the third-largest inequality in the motion of the Moon altogether; (the first and largest of the lunar inequalities is the equation of the centre, a result of the eccentricity – which is not an effect of solar perturbation).
Perturbation (astronomy)
Perturbation is a term used in astronomy in connection with descriptions of the complex motion of a massive body which is subject to appreciable gravitational effects from more than one other massive body....
s in the motion of the Moon.
The variation was first noted by Abū al-Wafā' Būzjānī. Independently rediscovering the phenomenon, Tycho Brahe
Tycho Brahe
Tycho Brahe , born Tyge Ottesen Brahe, was a Danish nobleman known for his accurate and comprehensive astronomical and planetary observations...
noticed that starting from a lunar eclipse
Lunar eclipse
A lunar eclipse occurs when the Moon passes behind the Earth so that the Earth blocks the Sun's rays from striking the Moon. This can occur only when the Sun, Earth, and Moon are aligned exactly, or very closely so, with the Earth in the middle. Hence, a lunar eclipse can only occur the night of a...
in December 1590, that at the times of syzygy
Syzygy (astronomy)
In astronomy, a syzygy is a straight line configuration of three celestial bodies in a gravitational system. The word is usually used in reference to the Sun, the Earth and either the Moon or a planet, where the latter is in conjunction or opposition. Solar and lunar eclipses occur at times of...
(new or full moon), the apparent velocity of motion of the Moon (along its orbit as seen against the background of stars) was faster than expected. On the other hand, at the times of first and last quarter, its velocity was correspondingly slower than expected. (Those expectations were based on the lunar tables widely used up to Tycho's time. They took some account of the two largest irregularities in the Moon's motion, i.e. those now known as the equation of the center
Equation of the center
For further closely related mathematical developments see also Two-body problem, also Gravitational two-body problem, also Kepler orbit, and Kepler problem...
and the evection
Evection
Evection , in astronomy, is the largest inequality produced by the action of the Sun in the monthly revolution of the Moon around the Earth. The evection, formerly called the moon's second anomaly, was approximately known in ancient times, and its discovery is attributed to Ptolemy...
, see also Lunar theory - History.)
The main visible effect (in longitude) of the variation of the Moon is that during the course of every month, at the octants of the Moon's phase that follow the syzygies (i.e. half way between the new or the full moon and the next-following quarter), the Moon is about two thirds of a degree farther ahead than would be expected on the basis of its mean motion (as modified by the equation of the centre and by the evection). But at the octants that precede the syzygies, it is about two thirds of a degree behind. At the syzygies and quarters themselves, the main effect is on the Moon's velocity rather than its position.
The Variation has a period of half a synodic month and causes the Moon's ecliptic longitude to vary by nearly two-thirds of a degree
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...
, more exactly by +2370"sin(2D) where D is the mean elongation of the Moon from the Sun.
The variational distortion of the Moon's orbit is a different effect from the eccentric elliptical motion of a body in an unperturbed orbit. The Variation effect would still occur if the undisturbed motion of the Moon had an eccentricity of zero (i.e. circular
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
). The eccentric Keplerian ellipse
Kepler orbit
In celestial mechanics, a Kepler orbit describes the motion of an orbiting body as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space...
is another and separate approximation for the Moon's orbit, different from the approximation represented by the (central) variational ellipse. The Moon's line of apses, i.e. the long axis of the Moon's orbit when approximated as an eccentric ellipse, rotates once in about nine years, so that it can be oriented at any angle whatever relative to the direction of the Sun at any season. (The angular difference between these two directions used to be referred to, in much older literature, as the "annual argument of the Moon's apogee".) Twice in every period of just over a year, the direction of the Sun coincides with the direction of the long axis of the eccentric elliptical approximation of the Moon's orbit (as projected on to the ecliptic).
Thus the (central) elliptical distortion of the Moon's orbit caused by the variation should not be confused with an undisturbed eccentric elliptical motion of an orbiting body. The variational effects due to the Sun would still occur even if the hypothetical undisturbed motion of the Moon had an eccentricity of zero (i.e. even if in the absence of the Sun it would be circular
Circular
Circular is a basic geometric shape such as a Circle.Circular may also refer to:-Documents:*Circular note, a document request by a bank to its foreign correspondents to pay a specified sum of money to a named person...
).
Newton expressed an approximate recognition that the real orbit of the Moon is not exactly an eccentric Keplerian ellipse, nor exactly a central ellipse due to the variation, but "an oval of another kind". Newton did not give an explicit expression for the form of this "oval of another kind"; to an approximation, it combines the two effects of the central-elliptical variational orbit and the Keplerian eccentric ellipse. Their combination also continually changes its shape as the annual argument changes, and also as the evection shows itself in libratory changes in the eccentricity, and in the direction, of the long axis of the eccentric ellipse.
The Variation is the second-largest solar perturbation of the Moon's orbit after the Evection
Evection
Evection , in astronomy, is the largest inequality produced by the action of the Sun in the monthly revolution of the Moon around the Earth. The evection, formerly called the moon's second anomaly, was approximately known in ancient times, and its discovery is attributed to Ptolemy...
, and the third-largest inequality in the motion of the Moon altogether; (the first and largest of the lunar inequalities is the equation of the centre, a result of the eccentricity – which is not an effect of solar perturbation).