Celestial mechanics

Encyclopedia

**Celestial mechanics**is the branch of astronomy

Astronomy

Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

that deals with the motion

Motion (physics)

In physics, motion is a change in position of an object with respect to time. Change in action is the result of an unbalanced force. Motion is typically described in terms of velocity, acceleration, displacement and time . An object's velocity cannot change unless it is acted upon by a force, as...

s of celestial objects. The field applies principles of physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, historically classical mechanics

Classical mechanics

In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

, to astronomical objects such as star

Star

A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...

s and planet

Planet

A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

s to produce ephemeris

Ephemeris

An ephemeris is a table of values that gives the positions of astronomical objects in the sky at a given time or times. Different kinds of ephemerides are used for astronomy and astrology...

data. Orbital mechanics (astrodynamics) is a subfield which focuses on the orbit

Orbit

In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...

s of artificial satellites. Lunar theory

Lunar theory

Lunar theory attempts to account for the motions of the Moon. There are many irregularities in the Moon's motion, and many attempts have been made over a long history to account for them. After centuries of being heavily problematic, the lunar motions are nowadays modelled to a very high degree...

is another subfield focusing on the orbit of the Moon.

## History of celestial mechanics

Modern analytic celestial mechanics started over 300 years ago with Isaac NewtonIsaac Newton

Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

's Principia

Principia

Principia could refer to:*Philosophiae Naturalis Principia Mathematica, Isaac Newton's three-volume work containing explanations of his laws of motion and his law of universal gravitation*Principia , a stem-group coralline alga...

of 1687. The name "celestial mechanics" is more recent than that. Newton wrote that the field should be called "rational mechanics." The term "dynamics" came in a little later with Gottfried Leibniz

Gottfried Leibniz

Gottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....

, and over a century after Newton, Pierre-Simon Laplace

Pierre-Simon Laplace

Pierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste...

introduced the term "celestial mechanics." Nevertheless, prior studies addressing the problem of planetary positions are known going back perhaps 3,000 or more years, as early as the Babylonian astronomers.

### Ancient Greece

Classical Greek writers speculated widely regarding celestial motions, and presented many geometrical mechanisms to model the motions of the planets. Their models employed combinations of uniform circular motion and were centered on the earth. An independent philosophical tradition was concerned with the physical causes of such circular motions. An extraordinary figure among the ancient Greek astronomers is Aristarchus of SamosAristarchus of Samos

Aristarchus, or more correctly Aristarchos , was a Greek astronomer and mathematician, born on the island of Samos, in Greece. He presented the first known heliocentric model of the solar system, placing the Sun, not the Earth, at the center of the known universe...

(310 BCE–c.

Circa

Circa , usually abbreviated c. or ca. , means "approximately" in the English language, usually referring to a date...

230 BCE), who suggested a heliocentric model

Heliocentrism

Heliocentrism, or heliocentricism, is the astronomical model in which the Earth and planets revolve around a stationary Sun at the center of the universe. The word comes from the Greek . Historically, heliocentrism was opposed to geocentrism, which placed the Earth at the center...

of the universe and attempted to measure Earth's distance from the Sun.

The only known supporter of Aristarchus was Seleucus of Seleucia

Seleucus of Seleucia

Seleucus of Seleucia was a Hellenistic astronomer and philosopher. Coming from Seleucia on the Tigris, the capital of the Seleucid empire, or, alternatively, Seleukia on the Red Sea, he is best known as a proponent of heliocentrism and for his theory of the origin of tides.- Heliocentric theory...

, a Babylonian astronomer who is said to have proved heliocentrism through reasoning in the 2nd century BCE. This may have involved the phenomenon of tide

Tide

Tides are the rise and fall of sea levels caused by the combined effects of the gravitational forces exerted by the moon and the sun and the rotation of the Earth....

s, which he correctly theorized to be caused by attraction to the Moon

Moon

The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

and notes that the height of the tides depends on the Moon's position relative to the Sun. Alternatively, he may have determined the constants of a geometric

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

model for the heliocentric theory and developed methods to compute planetary positions using this model, possibly using early trigonometric

Trigonometry

Trigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...

methods that were available in his time, much like Copernicus.

Also in the second century, the Antikythera mechanism

Antikythera mechanism

The Antikythera mechanism is an ancient mechanical computer designed to calculate astronomical positions. It was recovered in 1900–1901 from the Antikythera wreck. Its significance and complexity were not understood until decades later. Its time of construction is now estimated between 150 and 100...

was constructed. This device mechanically computes the positions of celestial bodies "with reference to the observer's position on the surface of the earth."

### Claudius Ptolemy

Claudius Ptolemy (c.Circa

Circa , usually abbreviated c. or ca. , means "approximately" in the English language, usually referring to a date...

120 CE

CE

CE, Ce or ce may refer to:* Common Era , secular alternative to Anno Domini * Cerium, chemical element with symbol Ce- Titles :* Chief Executive, administrative head of some regions...

) was an ancient astronomer and astrologer in early Imperial Roman

Roman Empire

The Roman Empire was the post-Republican period of the ancient Roman civilization, characterised by an autocratic form of government and large territorial holdings in Europe and around the Mediterranean....

times who wrote several books on astronomy. The most significant of these was the

*Almagest*

, which remained the most important book on predictive geometrical astronomy for some 1400 years. Ptolemy selected the best of the astronomical principles of his Greek predecessors, especially Hipparchus

Almagest

The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...

Hipparchus

Hipparchus, the common Latinization of the Greek Hipparkhos, can mean:* Hipparchus, the ancient Greek astronomer** Hipparchic cycle, an astronomical cycle he created** Hipparchus , a lunar crater named in his honour...

, and appears to have combined them either directly or indirectly with data and parameters obtained from the Babylonians. Although Ptolemy relied mainly on the work of Hipparchus, he introduced at least one idea, the equant

Equant

Equant is a mathematical concept developed by Claudius Ptolemy in the 2nd century AD to account for the observed motion of heavenly bodies....

, which appears to be his own, and which greatly improved the accuracy of the predicted positions of the planets. Although his model was extremely accurate, it relied solely on geometrical constructions rather than on physical causes; Ptolemy did not use celestial mechanics.

### Early Middle Ages

B. L. van der WaerdenBartel Leendert van der Waerden

Bartel Leendert van der Waerden was a Dutch mathematician and historian of mathematics....

has interpreted the planetary models developed by Aryabhata

Aryabhata

Aryabhata was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy...

(476–550 CE

CE

CE, Ce or ce may refer to:* Common Era , secular alternative to Anno Domini * Cerium, chemical element with symbol Ce- Titles :* Chief Executive, administrative head of some regions...

), an Indian astronomer, and Albumasar

Ja'far ibn Muhammad Abu Ma'shar al-Balkhi

Abū Maʿshar, Jaʿfar ibn Muḥammad al-Balkhī , was a Persian astrologer, astronomer, and Islamic philosopher, thought to be the greatest astrologer of the Abbasid court in Baghdad...

(787–886 CE

CE

CE, Ce or ce may refer to:* Common Era , secular alternative to Anno Domini * Cerium, chemical element with symbol Ce- Titles :* Chief Executive, administrative head of some regions...

), a Persian astronomer

Islamic astronomy

Islamic astronomy or Arabic astronomy comprises the astronomical developments made in the Islamic world, particularly during the Islamic Golden Age , and mostly written in the Arabic language. These developments mostly took place in the Middle East, Central Asia, Al-Andalus, and North Africa, and...

, to be heliocentric models

Heliocentrism

Heliocentrism, or heliocentricism, is the astronomical model in which the Earth and planets revolve around a stationary Sun at the center of the universe. The word comes from the Greek . Historically, heliocentrism was opposed to geocentrism, which placed the Earth at the center...

but this view has been strongly disputed by others. In the 9th century CE

CE

, the Persian physicist

Islamic physics

Physics in medieval Islam is the development of physics in the medieval Islamic world in the history of physics. In the course of the expansion of the Islamic world, Muslim scholars encountered the science, mathematics, and medicine of antiquity through the works of Aristotle, Archimedes, Galen,...

and astronomer

Islamic astronomy

Islamic astronomy or Arabic astronomy comprises the astronomical developments made in the Islamic world, particularly during the Islamic Golden Age , and mostly written in the Arabic language. These developments mostly took place in the Middle East, Central Asia, Al-Andalus, and North Africa, and...

, Ja'far Muhammad ibn Mūsā ibn Shākir, hypothesized that the heavenly bodies and celestial spheres

Celestial spheres

The celestial spheres, or celestial orbs, were the fundamental entities of the cosmological models developed by Plato, Eudoxus, Aristotle, Ptolemy, Copernicus and others...

are subject to the same laws of physics

Physical law

A physical law or scientific law is "a theoretical principle deduced from particular facts, applicable to a defined group or class of phenomena, and expressible by the statement that a particular phenomenon always occurs if certain conditions be present." Physical laws are typically conclusions...

as Earth

Earth

Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

, unlike the ancients who believed that the celestial spheres followed their own set of physical laws different from that of Earth.

### Ibn al-Haytham

In the early 11th century CECE

, Ibn al-Haytham presented a development of Ptolemy's geocentric epicyclic models in terms of nested celestial spheres

Celestial spheres

The celestial spheres, or celestial orbs, were the fundamental entities of the cosmological models developed by Plato, Eudoxus, Aristotle, Ptolemy, Copernicus and others...

. In chapters 15–16 of his

*Book of Optics*

, he also discovered that the celestial spheres do not consist of solid

Book of Optics

The Book of Optics ; ; Latin: De Aspectibus or Opticae Thesaurus: Alhazeni Arabis; Italian: Deli Aspecti) is a seven-volume treatise on optics and other fields of study composed by the medieval Muslim scholar Alhazen .-See also:* Science in medieval Islam...

Solid

Solid is one of the three classical states of matter . It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a...

matter.

### Late Middle Ages

There was much debate on the dynamics of the celestial spheresCelestial spheres

The celestial spheres, or celestial orbs, were the fundamental entities of the cosmological models developed by Plato, Eudoxus, Aristotle, Ptolemy, Copernicus and others...

during the late Middle Ages. Averroes

Averroes

' , better known just as Ibn Rushd , and in European literature as Averroes , was a Muslim polymath; a master of Aristotelian philosophy, Islamic philosophy, Islamic theology, Maliki law and jurisprudence, logic, psychology, politics, Arabic music theory, and the sciences of medicine, astronomy,...

(Ibn Rushd), Ibn Bajjah

Ibn Bajjah

Abū-Bakr Muhammad ibn Yahya ibn al-Sāyigh , known as Ibn Bājjah , was an Andalusian polymath: an astronomer, logician, musician, philosopher, physician, physicist, psychologist, botanist, poet and scientist. He was known in the West by his Latinized name, Avempace...

(Avempace) and Thomas Aquinas

Thomas Aquinas

Thomas Aquinas, O.P. , also Thomas of Aquin or Aquino, was an Italian Dominican priest of the Catholic Church, and an immensely influential philosopher and theologian in the tradition of scholasticism, known as Doctor Angelicus, Doctor Communis, or Doctor Universalis...

developed the theory of inertia

Inertia

Inertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...

in the celestial spheres, while Avicenna

Avicenna

Abū ʿAlī al-Ḥusayn ibn ʿAbd Allāh ibn Sīnā , commonly known as Ibn Sīnā or by his Latinized name Avicenna, was a Persian polymath, who wrote almost 450 treatises on a wide range of subjects, of which around 240 have survived...

(Ibn Sina) and Jean Buridan

Jean Buridan

Jean Buridan was a French priest who sowed the seeds of the Copernican revolution in Europe. Although he was one of the most famous and influential philosophers of the late Middle Ages, he is today among the least well known...

developed the theory of impetus

Theory of impetus

The theory of impetus was an auxiliary or secondary theory of Aristotelian dynamics, put forth initially to explain projectile motion against gravity...

in the celestial spheres.

In the 14th century, Ibn al-Shatir

Ibn al-Shatir

Ala Al-Din Abu'l-Hasan Ali Ibn Ibrahim Ibn al-Shatir was an Arab Muslim astronomer, mathematician, engineer and inventor who worked as muwaqqit at the Umayyad Mosque in Damascus, Syria.-Astronomy:...

produced the first model of lunar

Moon

The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

motion which matched physical observations, and which was later used by Copernicus. In the 13th–15th centuries, Tusi

Nasir al-Din al-Tusi

Khawaja Muḥammad ibn Muḥammad ibn Ḥasan Ṭūsī , better known as Naṣīr al-Dīn al-Ṭūsī , was a Persian polymath and prolific writer: an astronomer, biologist, chemist, mathematician, philosopher, physician, physicist, scientist, theologian and Marja Taqleed...

and Ali Kuşçu

Ali Kusçu

Ala al-Dīn Ali ibn Muhammed , known as Ali Qushji , was a Turkic or Persian astronomer, mathematician and physicist originally from Samarkand, who settled in the Ottoman Empire some time before 1472...

provided the earliest empirical evidence

Empirical research

Empirical research is a way of gaining knowledge by means of direct and indirect observation or experience. Empirical evidence can be analyzed quantitatively or qualitatively...

for the Earth's rotation, using the phenomena of comet

Comet

A comet is an icy small Solar System body that, when close enough to the Sun, displays a visible coma and sometimes also a tail. These phenomena are both due to the effects of solar radiation and the solar wind upon the nucleus of the comet...

s to refute Ptolemy's claim that a stationary Earth can be determined through observation. Kuşçu further rejected Aristotelian physics

Aristotelian physics

Aristotelian Physics the natural sciences, are described in the works of the Greek philosopher Aristotle . In the Physics, Aristotle established general principles of change that govern all natural bodies; both living and inanimate, celestial and terrestrial—including all motion, change in respect...

and natural philosophy

Natural philosophy

Natural philosophy or the philosophy of nature , is a term applied to the study of nature and the physical universe that was dominant before the development of modern science...

, allowing astronomy and physics to become empirical and mathematical instead of philosophical. In the early 16th century, the debate on the Earth's motion was continued by Al-Birjandi

Al-Birjandi

Abd Ali ibn Muhammad ibn Husayn Birjandi was a prominent 16th century Persian astronomer, mathematician and physicist who lived in Birjand, Iran.- His works :...

(d. 1528), who in his analysis of what might occur if the Earth were rotating, develops a hypothesis similar to Galileo Galilei

Galileo Galilei

Galileo Galilei , was an Italian physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations and support for Copernicanism...

's notion of "circular inertia

Inertia

Inertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...

", which he described in the following observational test:

### Johannes Kepler

Johannes KeplerJohannes Kepler

Johannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...

(27 December 1571–15 November 1630) was the first to closely integrate the predictive geometrical astronomy, which had been dominant from Ptolemy to Copernicus, with physical concepts to produce a

*New Astronomy, Based upon Causes, or Celestial Physics...*

Astronomia nova

The Astronomia nova is a book, published in 1609, that contains the results of the astronomer Johannes Kepler's ten-year long investigation of the motion of Mars...

. His work led to the modern laws of planetary orbits

Kepler's laws of planetary motion

In astronomy, Kepler's laws give a description of the motion of planets around the Sun.Kepler's laws are:#The orbit of every planet is an ellipse with the Sun at one of the two foci....

, which he developed using his physical principles and the planet

Planet

A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

ary observations made by Tycho Brahe

Tycho Brahe

Tycho Brahe , born Tyge Ottesen Brahe, was a Danish nobleman known for his accurate and comprehensive astronomical and planetary observations...

. Kepler's model greatly improved the accuracy of predictions of planetary motion, years before Isaac Newton

Isaac Newton

Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

developed his law of gravitation.

See Kepler's laws of planetary motion

Kepler's laws of planetary motion

In astronomy, Kepler's laws give a description of the motion of planets around the Sun.Kepler's laws are:#The orbit of every planet is an ellipse with the Sun at one of the two foci....

and the Keplerian problem for a detailed treatment of how his laws of planetary motion can be used.

### Isaac Newton

Isaac NewtonIsaac Newton

Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

(4 January 1643–31 March 1727) is credited with introducing the idea that the motion of objects in the heavens, such as planet

Planet

A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

s, the Sun

Sun

The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

, and the Moon

Moon

The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...

, and the motion of objects on the ground, like cannon

Cannon

A cannon is any piece of artillery that uses gunpowder or other usually explosive-based propellents to launch a projectile. Cannon vary in caliber, range, mobility, rate of fire, angle of fire, and firepower; different forms of cannon combine and balance these attributes in varying degrees,...

balls and falling apples, could be described by the same set of physical law

Physical law

A physical law or scientific law is "a theoretical principle deduced from particular facts, applicable to a defined group or class of phenomena, and expressible by the statement that a particular phenomenon always occurs if certain conditions be present." Physical laws are typically conclusions...

s. In this sense he unified

*celestial*and

*terrestrial*dynamics. Using Newton's law of universal gravitation

Newton's law of universal gravitation

Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them...

, proving Kepler's Laws for the case of a circular orbit is simple. Elliptical orbits involve more complex calculations, which Newton included in his Principia

Philosophiae Naturalis Principia Mathematica

Philosophiæ Naturalis Principia Mathematica, Latin for "Mathematical Principles of Natural Philosophy", often referred to as simply the Principia, is a work in three books by Sir Isaac Newton, first published 5 July 1687. Newton also published two further editions, in 1713 and 1726...

.

### Joseph-Louis Lagrange

After Newton, Lagrange (25 January 1736–10 April 1813) attempted to solve the three-body problem, analyzed the stability of planetary orbits, and discovered the existence of the Lagrangian points. Lagrange also reformulated the principles of classical mechanicsClassical mechanics

In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

, emphasizing energy more than force and developing a method

Lagrangian mechanics

Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788....

to use a single polar coordinate equation to describe any orbit, even those that are parabolic and hyperbolic. This is useful for calculating the behaviour of planets and comet

Comet

A comet is an icy small Solar System body that, when close enough to the Sun, displays a visible coma and sometimes also a tail. These phenomena are both due to the effects of solar radiation and the solar wind upon the nucleus of the comet...

s and such. More recently, it has also become useful to calculate spacecraft

Spacecraft

A spacecraft or spaceship is a craft or machine designed for spaceflight. Spacecraft are used for a variety of purposes, including communications, earth observation, meteorology, navigation, planetary exploration and transportation of humans and cargo....

trajectories

Trajectory

A trajectory is the path that a moving object follows through space as a function of time. The object might be a projectile or a satellite, for example. It thus includes the meaning of orbit—the path of a planet, an asteroid or a comet as it travels around a central mass...

.

### Simon Newcomb

Simon NewcombSimon Newcomb

Simon Newcomb was a Canadian-American astronomer and mathematician. Though he had little conventional schooling, he made important contributions to timekeeping as well as writing on economics and statistics and authoring a science fiction novel.-Early life:Simon Newcomb was born in the town of...

(12 March 1835–11 July 1909) was a Canadian-American astronomer who revised Peter Andreas Hansen

Peter Andreas Hansen

Peter Andreas Hansen was a Danish astronomer, was born at Tønder, Schleswig.-Biography:The son of a goldsmith, Hansen learned the trade of a watchmaker at Flensburg, and exercised it at Berlin and Tønder, 1818–1820...

's table of lunar positions. In 1877, assisted by George William Hill

George William Hill

George William Hill , was an American astronomer and mathematician.Hill was born in New York City, New York to painter and engraver John William Hill. and Catherine Smith Hill. He moved to West Nyack with his family when he was eight years old. After attending high school, Hill graduated from...

, he recalculated all the major astronomical constants. After 1884, he conceived with A. M. W. Downing a plan to resolve much international confusion on the subject. By the time he attended a standardisation conference in Paris

Paris

Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...

, France in May 1886, the international consensus was that all ephemerides should be based on Newcomb's calculations. A further conference as late as 1950 confirmed Newcomb's constants as the international standard.

### Albert Einstein

Albert EinsteinAlbert Einstein

Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

(14 March 1879–18 April 1955) explained the anomalous precession of Mercury's perihelion in his 1916 paper

*The Foundation of the General Theory of Relativity*. This led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy. Binary pulsar

Binary pulsar

A binary pulsar is a pulsar with a binary companion, often a white dwarf or neutron star. Binary pulsars are one of the few objects which allow physicists to test general relativity in the case of a strong gravitational field...

s have been observed, the first in 1974, whose orbits not only require the use of General Relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

for their explanation, but whose evolution proves the existence of gravitational radiation, a discovery that led to the 1993 Nobel Physics Prize.

## Examples of problems

Celestial motion without additional forces such as thrustThrust

Thrust is a reaction force described quantitatively by Newton's second and third laws. When a system expels or accelerates mass in one direction the accelerated mass will cause a force of equal magnitude but opposite direction on that system....

of a rocket

Rocket

A rocket is a missile, spacecraft, aircraft or other vehicle which obtains thrust from a rocket engine. In all rockets, the exhaust is formed entirely from propellants carried within the rocket before use. Rocket engines work by action and reaction...

, is governed by gravitational acceleration of masses due to other masses. A simplification is the

*n*-body problem

N-body problem

The n-body problem is the problem of predicting the motion of a group of celestial objects that interact with each other gravitationally. Solving this problem has been motivated by the need to understand the motion of the Sun, planets and the visible stars...

, where the problem assumes some number

*n*of spherically symmetric masses. In that case, the integration of the accelerations can be well approximated by relatively simple summations.

- Examples:
- 4-body problem: spaceflight to Mars (for parts of the flight the influence of one or two bodies is very small, so that there we have a 2- or 3-body problem; see also the patched conic approximation)
- 3-body problem:
- Quasi-satelliteQuasi-satelliteA quasi-satellite is an object in a 1:1 orbital resonance with its planet that stays close to the planet over many orbital periods.A quasi-satellite's orbit around the Sun takes exactly the same time as the planet's, but has a different eccentricity , as shown in the diagram on the right...
- Spaceflight to, and stay at a Lagrangian pointLagrangian pointThe Lagrangian points are the five positions in an orbital configuration where a small object affected only by gravity can theoretically be stationary relative to two larger objects...

- Quasi-satellite

In the case that

*n*=2 (two-body problem

Two-body problem

In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. Common examples include a satellite orbiting a planet, a planet orbiting a star, two stars orbiting each other , and a classical electron orbiting an atomic nucleus In...

), the situation is much simpler than for larger

*n*. Various explicit formulas apply, where in the more general case typically only numerical solutions are possible. It is a useful simplification that is often approximately valid.

- Examples:
- A binary starBinary starA binary star is a star system consisting of two stars orbiting around their common center of mass. The brighter star is called the primary and the other is its companion star, comes, or secondary...

, e.g., Alpha CentauriAlpha CentauriAlpha Centauri is the brightest star in the southern constellation of Centaurus...

(approx. the same mass) - A binary asteroidBinary asteroidA binary asteroid is a system of two asteroids orbiting their common center of mass, in analogy with binary stars. 243 Ida was the first binary asteroid to be identified when the Galileo spacecraft did a flyby in 1993...

, e.g., 90 Antiope90 AntiopeThe most remarkable feature of Antiope is that it consists of two components of almost equal size , making it a truly "double" asteroid. Its binary nature was discovered on 10 August 2000 by a group of astronomers using adaptive optics at the Keck Telescope on Mauna Kea. The "secondary" is...

(approx. the same mass)

- A binary star

A further simplification is based on the "standard assumptions in astrodynamics", which include that one body, the orbiting body

Orbiting body

In astrodynamics, an orbiting body is a body that orbits a primary body .The orbiting body is properly referred to as the secondary body.It is less massive than the primary body ....

, is much smaller than the other, the central body

Central body

In astrodynamics a central body is a body that is being orbited by an secondary body, or satellite .The central body is properly referred to as the primary body.Under standard assumptions in astrodynamics:...

. This is also often approximately valid.

- Examples:
- Solar systemSolar SystemThe Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...

orbiting the center of the Milky WayMilky WayThe Milky Way is the galaxy that contains the Solar System. This name derives from its appearance as a dim un-resolved "milky" glowing band arching across the night sky... - A planet orbiting the Sun
- A moon orbiting a planet
- A spacecraft orbiting Earth, a moon, or a planet (in the latter cases the approximation only applies after arrival at that orbit)

- Solar system

Either instead of, or on top of the previous simplification, we may assume circular orbit

Circular orbit

A circular orbit is the orbit at a fixed distance around any point by an object rotating around a fixed axis.Below we consider a circular orbit in astrodynamics or celestial mechanics under standard assumptions...

s, making distance and orbital speed

Orbital speed

The orbital speed of a body, generally a planet, a natural satellite, an artificial satellite, or a multiple star, is the speed at which it orbits around the barycenter of a system, usually around a more massive body...

s, and potential and kinetic energies constant in time. This assumption sacrifices accuracy for simplicity, especially for high eccentricity

Orbital eccentricity

The orbital eccentricity of an astronomical body is the amount by which its orbit deviates from a perfect circle, where 0 is perfectly circular, and 1.0 is a parabola, and no longer a closed orbit...

orbits which are by definition non-circular.

- Examples:
- The orbit of the dwarf planetDwarf planetA dwarf planet, as defined by the International Astronomical Union , is a celestial body orbiting the Sun that is massive enough to be spherical as a result of its own gravity but has not cleared its neighboring region of planetesimals and is not a satellite...

PlutoPlutoPluto, formal designation 134340 Pluto, is the second-most-massive known dwarf planet in the Solar System and the tenth-most-massive body observed directly orbiting the Sun...

, ecc. = 0.2488 - The orbit of MercuryMercury (planet)Mercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 Earth days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three rotations about its axis for every two orbits...

, ecc. = 0.2056 - Hohmann transfer orbitHohmann transfer orbitIn orbital mechanics, the Hohmann transfer orbit is an elliptical orbit used to transfer between two circular orbits, typically both in the same plane....
- Gemini 11Gemini 11Gemini 11 was the ninth manned spaceflight mission of NASA's Project Gemini, which flew from September 12 to 15, 1966. It was the 17th manned American flight and the 25th spaceflight to that time . Astronauts Charles "Pete" Conrad, Jr. and Richard F. Gordon, Jr...

flight - Suborbital flights

- The orbit of the dwarf planet

## Perturbation theory

Perturbation theoryPerturbation theory

Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...

comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly. (It is closely related to methods used in numerical analysis

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

, which are ancient.) The earliest use of perturbation theory

Perturbation theory

Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...

was to deal with the otherwise unsolveable mathematical problems of celestial mechanics: Newton

Isaac Newton

's solution for the orbit of the Moon

Moon

, which moves noticeably differently from a simple Keplerian ellipse

Kepler's laws of planetary motion

In astronomy, Kepler's laws give a description of the motion of planets around the Sun.Kepler's laws are:#The orbit of every planet is an ellipse with the Sun at one of the two foci....

because of the competing gravitation of the Earth

Earth

Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

and the Sun

Sun

The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

.

Perturbation methods

Perturbation theory

Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...

start with a simplified form of the original problem, which is carefully chosen to be exactly solvable. In celestial mechanics, this is usually a Keplerian ellipse

Kepler's laws of planetary motion

, which is correct when there are only two gravitating bodies (say, the Earth

Earth

Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

and the Moon

Moon

), or a circular orbit, which is only correct in special cases of two-body motion, but is often close enough for practical use. The solved, but simplified problem is then

*"perturbed"*to make its starting conditions closer to the real problem, such as including the gravitational attraction of a third body (the Sun

Sun

The Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...

). The slight changes that result, which themselves may have been simplified yet again, are used as corrections. Because of simplifications introduced along every step of the way, the corrections are never perfect, but even one cycle of corrections often provides a remarkably better approximate solution to the real problem.

There is no requirement to stop at only one cycle of corrections. A partially corrected solution can be re-used as the new starting point for yet another cycle of perturbations and corrections. The common difficulty with the method is that usually the corrections progressively make the new solutions very much more complicated, so each cycle is much more difficult to manage than the previous cycle of corrections. Newton

Isaac Newton

is reported to have said, regarding the problem of the Moon

Moon

's orbit

*"It causeth my head to ache."*

This general procedure – starting with a simplified problem and gradually adding corrections that make the starting point of the corrected problem closer to the real situation – is a widely used mathematical tool in advanced sciences and engineering. It is the natural extension of the "guess, check, and fix" method used anciently with numbers.

## See also

- AstrometryAstrometryAstrometry is the branch of astronomy that involves precise measurements of the positions and movements of stars and other celestial bodies. The information obtained by astrometric measurements provides information on the kinematics and physical origin of our Solar System and our Galaxy, the Milky...

is a part of astronomy that deals with measuring the positions of stars and other celestial bodies, their distances and movements. - AstrodynamicsAstrodynamicsOrbital 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 Newton's law of universal gravitation. It...

is the study and creation of orbits, especially those of artificial satellites. - Celestial navigationCelestial navigationCelestial navigation, also known as astronavigation, is a position fixing technique that has evolved over several thousand years to help sailors cross oceans without having to rely on estimated calculations, or dead reckoning, to know their position...

is a position fixing technique that was the first system devised to help sailors locate themselves on a featureless ocean. - GravitationGravitationGravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
- Numerical analysisNumerical analysisNumerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

is a branch of mathematics, pioneered by celestial mechanicians, for calculating approximate numerical answers (such as the position of a planetPlanet

in the sky) which are too difficult to solve down to a general, exact formula. - Creating a numerical model of the solar systemNumerical model of solar systemA numerical model of the Solar System is a set of mathematical equations, which, when solved, give the approximate positions of the planets as a function of time. Attempts to create such a model established the more general field of celestial mechanics. The results of this simulation can be...

was the original goal of celestial mechanics, and has only been imperfectly achieved. It continues to motivate research. - An
*orbit*is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity.OrbitIn physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System... - Orbital elementsOrbital elementsOrbital 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...

are the parameters needed to specify a Newtonian two-body orbit uniquely. - Osculating orbitOsculating orbitIn astronomy, and in particular in astrodynamics, the osculating orbit of an object in space is the gravitational Kepler orbit In astronomy, and in particular in astrodynamics, the osculating orbit of an object in space (at a given moment of time) is the gravitational Kepler orbit In astronomy,...

is the temporary Keplerian orbit about a central body that an object would continue on, if other perturbations were not present. - Retrograde motionRetrograde motionRetrograde motion is motion in the direction opposite to the movement of something else, and is the contrary of direct or prograde motion. This motion can be the orbit of one body about another body or about some other point, or the rotation of a single body about its axis, or other phenomena such...
- SatelliteSatelliteIn the context of spaceflight, a satellite is an object which has been placed into orbit by human endeavour. Such objects are sometimes called artificial satellites to distinguish them from natural satellites such as the Moon....

is an object that orbits another object (known as its primary). The term is often used to describe an artificial satellite (as opposed to natural satellites, or moons). The common noun moon (not capitalized) is used to mean any natural satelliteNatural satelliteA natural satellite or moon is a celestial body that orbits a planet or smaller body, which is called its primary. The two terms are used synonymously for non-artificial satellites of planets, of dwarf planets, and of minor planets....

of the other planets. - Tidal forceTidal forceThe tidal force is a secondary effect of the force of gravity and is responsible for the tides. It arises because the gravitational force per unit mass exerted on one body by a second body is not constant across its diameter, the side nearest to the second being more attracted by it than the side...
- The Jet Propulsion Laboratory Developmental EphemerisJet Propulsion Laboratory Developmental EphemerisThe name Jet Propulsion Laboratory Development Ephemeris , or its abbreviation JPL DE , designates one of a series of numbered versions of the astronomical ephemerides produced at the Jet Propulsion Laboratory in Pasadena, California, primarily for purposes of space navigation and astronomy...

(JPL DE) is a widely used model of the solar system, which combines celestial mechanics with numerical analysisNumerical analysisNumerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

and astronomical and spacecraft data. - Two solutions, called VSOP82 and VSOP87Secular variations of the planetary orbitsThe secular variations of the planetary orbits 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...

are versions one mathematical theory for the orbits and positions of the major planets, which seeks to provide accurate positions over an extended period of time. - Lunar theoryLunar theoryLunar theory attempts to account for the motions of the Moon. There are many irregularities in the Moon's motion, and many attempts have been made over a long history to account for them. After centuries of being heavily problematic, the lunar motions are nowadays modelled to a very high degree...

attempts to account for the motions of the Moon.

## External links

- Astronomy of the Earth's Motion in Space, high-school level educational web site by David P. Stern

**Research**

**Artwork**

**Course notes**

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