Variation (astronomy)

Last updated June 13, 2019

In astronomy, the variation of the Moon is one of the principal perturbations in the motion of the Moon.

Astronomy is a natural science that studies celestial objects and phenomena. It applies mathematics, physics, and chemistry in an effort to explain the origin of those objects and phenomena and their evolution. Objects of interest include planets, moons, stars, nebulae, galaxies, and comets; the phenomena also includes supernova explosions, gamma ray bursts, quasars, blazars, pulsars, and cosmic microwave background radiation. More generally, all phenomena that originate outside Earth's atmosphere are within the purview of astronomy. A branch of astronomy called cosmology is the study of the Universe as a whole.

In astronomy, perturbation is the complex motion of a massive body subject to forces other than the gravitational attraction of a single other massive body. The other forces can include a third body, resistance, as from an atmosphere, and the off-center attraction of an oblate or otherwise misshapen body.

Discovery

The variation was discovered by Tycho Brahe, who noticed that, starting from a lunar eclipse in December 1590, 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.)^[1]

Tycho Brahe was a Danish nobleman, astronomer, and writer known for his accurate and comprehensive astronomical and planetary observations. He was born in the then Danish peninsula of Scania. Well known in his lifetime as an astronomer, astrologer and alchemist, he has been described as "the first competent mind in modern astronomy to feel ardently the passion for exact empirical facts." His observations were some five times more accurate than the best available observations at the time.

A lunar eclipse occurs when the Moon passes directly behind Earth and into its shadow. This can occur only when the Sun, Earth, and Moon are exactly or very closely aligned, with Earth between the other two. A lunar eclipse can occur only on the night of a full moon. The type and length of a lunar eclipse depend on the Moon's proximity to either node of its orbit.

In astronomy, a syzygy is a (roughly) straight-line configuration of three or more celestial bodies in a gravitational system.

Variation

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. halfway 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.

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.

<i>Philosophiæ Naturalis Principia Mathematica</i> tract by Isaac Newton

Philosophiæ Naturalis Principia Mathematica, often referred to as simply the Principia, is a work in three books by Isaac Newton, in Latin, first published 5 July 1687. After annotating and correcting his personal copy of the first edition, Newton published two further editions, in 1713 and 1726. The Principia states Newton's laws of motion, forming the foundation of classical mechanics; Newton's law of universal gravitation; and a derivation of Kepler's laws of planetary motion.

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.^[2]

A degree, usually denoted by °, is a measurement of a plane angle, defined so that a full rotation is 360 degrees.

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).

Circle simple curve of Euclidean geometry

A circle is a simple closed shape. It is the set of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any of the points and the centre is called the radius. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.

In celestial mechanics, a Kepler orbit is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, solar radiation pressure, a non-spherical central body, and so on. It is thus said to be a solution of a special case of the two-body problem, known as the Kepler problem. As a theory in classical mechanics, it also does not take into account the effects of general relativity. Keplerian orbits can be parametrized into six orbital elements in various ways.

Elliptical distortion

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 the orbit would be circular in the absence of the Sun).

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".^[3] 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).

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In astronomy, Kepler's laws of planetary motion are three scientific laws describing the motion of planets around the Sun.

In physics, an orbit is the gravitationally curved trajectory of an object, such as the trajectory of a planet around a star or a natural satellite around a planet. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the central mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.

Eclipses may occur repeatedly, separated by certain intervals of time: these intervals are called eclipse cycles. The series of eclipses separated by a repeat of one of these intervals is called an eclipse series.

Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are generally considered in classical two-body systems, where a Kepler orbit is used. There are many different ways to mathematically describe the same orbit, but certain schemes, each consisting of a set of six parameters, are commonly used in astronomy and orbital mechanics.

Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics to astronomical objects, such as stars and planets, to produce ephemeris data.

A lunar day is the period of time for Earth's Moon to complete one rotation on its axis with respect to the Sun. Due to tidal locking, it is also the time the Moon takes to complete one orbit around Earth and return to the same phase. A lunar month is the period between two new moons. A lunar month lasts about 29.5 solar days on earth.

Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and law of universal gravitation. It is a core discipline within space-mission design and control.

Precession is the change in orientation of a rotational axis with respect to a reference plane. The orbit of the Moon undergoes two important types of precessional motion: apsidal and nodal. The axis of the Moon also experiences precession.

Elliptic orbit Kepler orbit with an eccentricity of less than 1

In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1. In a wider sense, it is a Kepler orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1.

In astronomy, evection 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. The current name itself dates much more recently, from the 17th century: it was coined by Bullialdus in connection with his own theory of the Moon's motion.

Lunar theory attempts to account for the motions of the Moon. There are many small variations in the Moon's motion, and many attempts have been made to account for them. After centuries of being problematic, lunar motion is now modeled to a very high degree of accuracy.

The Moon orbits Earth in the prograde direction and completes one revolution relative to the stars in about 27.32 days and one revolution relative to the Sun in about 29.53 days. Earth and the Moon orbit about their barycenter, which lies about 4,600 km (2,900 mi) from Earth's center. On average, the distance to the Moon is about 385,000 km (239,000 mi) from Earth's center, which corresponds to about 60 Earth radii.

The semi-analytic planetary theory VSOP is a concept describing long-term changes in the orbits of the planets Mercury to Neptune. If one ignores the gravitational attraction between the planets and only models the attraction between the Sun and the planets, then with some further idealisations, the resulting orbits would be Keplerian ellipses. In this idealised model, the shape and orientation of these ellipses would be constant in time. In reality, while the planets are at all times roughly in Keplerian orbits, the shape and orientation of these ellipses do change slowly over time. Over the centuries increasingly complex models have been made of the deviations from simple Keplerian orbits. In addition to the models, efficient and accurate numerical approximation methods have also been developed.

In celestial mechanics, apsidal precession is the precession of the line connecting the apsides of an astronomical body's orbit. The apsides are the orbital points closest (periapsis) and farthest (apoapsis) from its primary body. The apsidal precession is the first derivative of the argument of periapsis, one of the six main orbital elements of an orbit. Apsidal precession is considered positive when the orbit's axis rotates in the same direction as the orbital motion. An apsidal period is the time interval required for an orbit to precess through 360°.

In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the widest points of the perimeter.

In lunar calendars, a lunar month is the time between two successive syzygies. The precise definition varies, especially for the beginning of the month.

References

↑ V E Thoren, "Tycho and Kepler on the Lunar theory", Publications of the Astronomical Society of the Pacific, vol.79 (1967), pp. 482-489, especially at p.485. The discovery of the variation was made accidentally, after Tycho planned to observe the lunar eclipse of December 1590, and checked the Moon's position a day or two beforehand, to determine more exactly, as he thought, the starting time. But when he went out to observe just before the expected time, he found that he had missed the start of the eclipse. The Moon was earlier by about an hour, the eclipse already in progress. Tycho's subsequent investigations pinned down the cause, a previously unrecognized inequality, since then called the variation.
↑ See Lunar theory - Delaunay arguments.
↑ D T Whiteside (ed.) (1973), The Mathematical papers of Isaac Newton, Volume VI: 1684-1691, Cambridge University Press, at page 533.

Bibliography

Brown, E.W. An Introductory Treatise on the Lunar Theory. Cambridge University Press, 1896 (republished by Dover, 1960).

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[1] V E Thoren, "Tycho and Kepler on the Lunar theory", Publications of the Astronomical Society of the Pacific, vol.79 (1967), pp. 482-489, especially at p.485. The discovery of the variation was made accidentally, after Tycho planned to observe the lunar eclipse of December 1590, and checked the Moon's position a day or two beforehand, to determine more exactly, as he thought, the starting time. But when he went out to observe just before the expected time, he found that he had missed the start of the eclipse. The Moon was earlier by about an hour, the eclipse already in progress. Tycho's subsequent investigations pinned down the cause, a previously unrecognized inequality, since then called the variation.

[2] See Lunar theory - Delaunay arguments.

[3] D T Whiteside (ed.) (1973), The Mathematical papers of Isaac Newton, Volume VI: 1684-1691, Cambridge University Press, at page 533.