## celestial mechanics

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## Celestial mechanics

# Celestial mechanics

Modern celestial mechanics began with the generalization by English physicist and mathematician Sir Isaac Newton (1642–1727) of Kepler’s laws published in his *Principia* in 1687. Newton used his three laws of motion and his law of universal gravitation to do this. The three generalized Kepler’s law are explained in more detail below.

First, the orbits of two bodies around their center of mass (barycenter) are conic sections (ellipses, circles, parabolas, or hyperbolas) with the center of mass at a focus of each conic sections.

Second, the line joining the center of the two bodies sweeps out equal areas in their orbits in equal time intervals. Newton showed that this is a consequence of conservation of angular momentum of an isolated two-body system unperturbed by other forces (Newton’s third law of motion).

Third, from his law of universal gravitation, which states that Bodies l and 2 of masses M_{1} and M_{2}, respectively, whose centers are separated by a distance (r), experience equal and opposite attractive gravitational forces F_{g} of magnitudes:

where G is the Newtonian gravitations factor, and from his second law of motion, Newton derived the following general form of Kepler’s third law for these bodies moving around the center of mass along elliptical or circular orbits:

where P is the sidereal period of revolution of the bodies around the center of mass, π is the ratio of the circumference of a circle to its diameter, X, M_{1} and M _{2} are the same as in Equation 1 and a is the semi-major axis of the *relative* orbit of the center of the less massive Body 2 around the center of the more massive Body l.

These three generalized Kepler’s law form the basis of the two-body problem of celestial mechanics. Astrometry is the branch of celestial mechanics that is concerned with making precise measurements of the positions of celestial bodies, then calculating precise orbits for them based on the observations. In theory, only three observations are needed to define the orbit of one celestial body relative to a second one. Actually, many observations are needed to obtain an accurate orbit.

However, for the most precise orbits and predictions, the vast majority of systems investigated are not strictly two-body systems but consist of many bodies (the solar system, planetary satellite systems, multiple star systems, star clusters, and galaxies).

## Planetary perturbations

To a first approximation, the solar system consists of the Sun and eight major planets, a system much more complicated than a two-body problem. However, use of Equation 2 with reasonable values for the astronomical unit (a convenient unit of length for the solar system) and for G showed that the Sun is far more massive than even the most massive planet Jupiter (whose mass is 0.000955 the Sun’s mass). This showed that the gravitational forces of the planets on each other are much weaker than the gravitational forces between the Sun and each of the planets. This concept enabled astronomers to consider the gravitational interactions of the planets as producing small changes with time perturbations) in the elliptical orbit of each planet around the center of mass of the solar system (which is always in or near the Sun). If the Sun and a planet (say the Earth or Jupiter) were alone in empty space, scientists would have an ideal two-body problem and they would expect the two-body problem as defined by the generalized Kepler’s law to exactly describe their orbits around the systems’ center of mass. Then, the seven orbital elements (of which a and y are two) of a planet’s orbit should remain constant forever.

However, the gravitational forces of the other planets on a particular planet cause its orbit to change slightly over time. These changes can be accurately allowed for over limited time intervals by calculating the perturbations of its orbital elements over time that are caused by the gravitational forces of the other planets.

Historically, perturbation theory has been more useful than merely providing accurate predictions of future planetary positions. Only six major planets were known when Newton published his *Principia.* German-born English astronomer William Herschel (1738–1822) fortuitously discovered Uranus, the seventh major planet from the Sun, in March 1781. The initial orbital elements calculated for Uranus did not accurately allow prediction of its future position even after inclusion of the perturbations caused by the six other major planets. Before 1821, Uranus was consistently observed to be ahead of its predicted position in its orbit; afterwards, it lagged behind its predicted positions.

John Couch Adams (1819–1892) in England and Urbain Leverier (1811–1877) in France, hypothesized that Uranus had passed an undiscovered massive planet further than it was from the Sun in the year 1821. They both made detailed calculations to locate the position of the undiscovered planet perturbing the motion of Uranus. Johann Galle (1812–1910) in Berlin, Germany, used Leverier’s calculations to discover the unknown planet in September 1846, which was then named Neptune.

Further unexplained perturbations of the orbits of Uranus and Neptune led American astronomer Percival Lowell (1855–1916) and several other astronomers to use them to calculate predicted positions for another undiscovered (trans-Neptunian) planet beyond Neptune’s orbit. Lowell searched for the trans-Neptunian planets he predicted from 1906 until his death in November 1916 without finding it. The search for a trans-Neptunian planet was resumed in 1929 at Lowell Observatory, where American astronomer Clyde Tombaugh (1906–1997) who discovered Pluto in February 1930.

Lowell had predicted that a planet more massive than Earth produced the unexplained perturbations. During the years following Pluto’s discovery, however, detailed studies of its perturbations of the orbits of Uranus and Neptune showed that Pluto is considerably less massive than Earth. The discovery of Pluto’s satellite, Charon, in 1978 allowed the determination of the total mass of Pluto and Charon from Equation 2 that is about 0.00237 Earth’s mass (about 0.2 the mass of Earth’s moon). There are two consequences of this discovery; Tombaugh’s discovery of Pluto may have been fortuitous, and one may make the case that Pluto is not a major planet. (In fact, as of August 24, 2006, Pluto had been demoted to a dwarf planet by the International Astronomical Union.)

The discrepancy in mass between the masses predicted by Lowell and others for the trans-Neptunian planet and the mass of the Pluto-Charon double planet has led to a renewed search for one or more additional trans-Neptunian planet(s) that still continues. The opinion also exists that the unexplained perturbations of the orbits of Uranus and Neptune are caused by systematic errors in some early measurements of their positions and that no trans-Neptunian planets with masses on the order of Earth’s mass exist.

## Resonance phenomena

Ceres, the first asteroid or minor planet, was discovered to orbit the Sun between the orbits of Mars and Jupiter in 1801. Thousands of other asteroids have been discovered in that part of interplanetary space, which is now called the Main Asteroid Belt.

Daniel Kirkwood (1815–1895) noticed in 1866 that the periods of revolution of the asteroids around the Sun did not form a continuous distribution over the Main Asteroid Belt but showed gaps (now known as Kirkwood’s gaps) at periods corresponding to one-half, one-third, and two-fifths of Jupiter’s period of revolution (11.86 sidereal years). This phenomenon can be explained by the fact that, if an asteroid is in one of Kirkwood’s gaps, then every second, third, or fifth revolution around the Sun, it will experience a perturbation by Jupiter of the same direction and magnitude; over the course of millions of years. These perturbations move asteroids out of the Kirkwood’s gaps. This is a resonance effect of planetary perturbations, and it is only one of several resonance phenomena found in the solar system.

Ratios between the periods of revolution of several planets around the Sun are another resonance phenomenon that is poorly understood. The periods of revolution of Venus, Earth, and Mars around the Sun are nearly in the ratio is 5:8:15. The periods of revolution of Jupiter and Saturn are nearly in a 2:5 ratio, and for Uranus, Neptune, and Pluto they are nearly in a 1:2:3 ratio. The 2:3 ratio between the periods of revolution of Neptune and Pluto makes Pluto’s orbit more stable. Due to the ellipticity of its orbit, near perihelion (the point on its orbit closest to the Sun) Pluto comes closer to the Sun than Neptune. Pluto last reached perihelion in September 1989; it has been closer to the Sun than Neptune since 1979 and will continue to be closer until 1998, when it will resume its usual place further from Neptune. However, Neptune was the Sun’s most distant known planet from 1995 to 1998! Recent calculations showed that, because of the 2:3 ratio of the orbital periods, the orientation of Pluto’s orbit, and of the positions of Neptune and Pluto in their orbits, Neptune and Pluto have never been closer than 2, 500,000,000 km (1.5 billion miles) in the last 10, 000, 000 years. Without the 2:3 ratio of their orbital periods, Pluto probably would have had a close encounter with Neptune, which could have ejected Pluto and Charon into separate orbits around the Sun that are drastically different from the system’s present orbit.

Jupiter’s inner three Gallean satellite, Io, Europa, and Ganymede, have orbital periods of revolution around Jupiter that are nearly in the ratio 1:2:4. Five of Saturn’s closest satellites, Pandora, Mimas, Enceladus, Tethys, and Dione, have orbital periods of revolution around Saturn that are nearly in the ratio 4:6:9:12:18. Resonance effects produced by some of these satellites, especially the 1:2 resonance with Mimas’ period of revolution around Saturn, seem to have produced the Cassini Division between Saturn’s A and B rings, which is analogous to the 1:2 Kirkwood’s gap in the Main Asteroid Belt. A 3:4 orbital period resonance seems to exist between Saturn’s largest satellite Titan and its next satellite out Hyperion.

Other resonances between the satellites and ring systems of Jupiter, Uranus, and Neptune are not clear because these ring systems are far less developed than that of Saturn.

## Tidal effects

A tidal effect is produced when the gravitational pull of one body on a second one is appreciably greater on the nearer part of the second one than on its center, and in turn, the first body’s pull on the second one’s center is greater than its pull on the second one’s most distant part. Unlike the gravitational force (F_{g}), which varies as the inverse square of the distance (r) between the centers of the two bodies (1/r_{2}); see Equation 1, the tidal effect varies as the inverse cube (1/r^{3}) of the distance between their centers. Both the Moon and the Sun raise tides in Earth’s oceans, atmosphere, and solid body. The lag of the tides raised in the oceans behind the Moon’s crossings of the celestial meridian causes a gravitational interaction between Earth and Moon that slows the Earth’s rotation and moves the Moon’s orbit further from Earth.

Tides raised in the Moon’s solid body by Earth have slowed its rotation until it has become tidally locked to Earth (the Moon keeps the same hemisphere turned towards Earth, and its periods of rotation and revolution around Earth are the same, 27.32 mean solar days). Eventually Earth’s rotation will be slowed to where Earth will be tidally locked to the Moon, and the durations of the sidereal day and sidereal month will both equal about 47 present mean solar days.

Tidal evolution has forced most planetary satellites to become tidally locked to their planets. This includes all of Jupiter’s Galilean satellites and its four small satellites closer to Jupiter than Io, probably most of Saturn’s satellites out to Iasetus (Titan, Saturn’s largest satellite, is probably tidally locked to Saturn), the satellites of Uranus and Neptune, and the Pluto-Charon double dwarf planet. Tidal action in Io’s interior produced by Jupiter (and to a lesser degree by its next satellite out Europa) powers volcanism on Io, making it the most volcanically active body in the solar system. Tidal effects also may have powered volcanic activity on Europa and Ganymede, Saturn’s satellite Enceladus, and Uranus satellites Miranda and Ariel; all of them show some evidence of resurfacing. Some of Earth’s internal heat may have been produced by the Moon’s tidal action.

The Sun’s tidal action on Mercury at perihelion has tidally locked Mercury’s rotation to its angular velocity near perihelion, which is 1.5 times Mercury’s average orbital angular velocity; Mercury’s rotation period is 58.6 days, two-thirds of its 87.9 day period of revolution around the Sun.

When two bodies are very close together, tidal forces tending to disrupt a body can equal or exceed the attractive gravitational forces holding it together. If the tidal stresses exceed the yield limits of the body’s material, the body will gradually disintegrate into many smaller bodies. French mathematician Edouard Albert Roche (1820–1883) studied the limiting separation of two bodies where the tidal and gravitational forces are equal; it usually between two to three times the radius of the more massive body and depends on the relative densities of the bodies and their state of motion. If two bodies approach closer than this Roche limit, one (usually the smaller, less massive body) or both bodies will begin to disintegrate. The rings of some of the Jovian planets may have formed from the tidal disintegration of one or more of their close satellites. Theory predicts that after the Earth’s and Moon’s rotations become tidally locked (see above), the Sun’s tides raised on Earth will cause the Moon to approach Earth. If this effect lasts long enough, the Moon may get closer to Earth than its Roche limit, be disintegrated by Earth’s tidal forces, and form a ring of small bodies that orbit Earth.

Tidal effects act on close double stars, distorting their shapes, changing their orbits, and sometimes tidally locking their rotations. In some cases, tidal effects cause streams of gas to flow in a double star system and can transfer matter from one star to the other or allow it to escape into interstellar space. Tidal effects even seem to act between galaxies, with one galaxy distorting the form of its neighbor.

## Precession

Rapidly rotating planets and satellites have appreciable equatorial bulges as a consequence of Newton’s first law of motion. If the rotation axis of such a body is not perpendicular to its orbit, other bodies in the system will exert stronger gravitational attractions on the near part of the bulge than its far part. The effect of this difference is to tend to turn the body’s rotation axis perpendicular to the plane of its orbit. Because the body is rotating rapidly, however, this does not happen, and, like the rotation axis of a spinning top, the body’s rotation axis describes a cone in space whose axis is the perpendicular to the body’s orbit (in a two-body system). This phenomenon is called precession, and it is important for Earth, Mars, and the Jovian planets. For Earth, precession causes its celestial poles to describe small circles of 23°.5 arc radii around its ecliptic poles and the equinoxes to move westward on the ecliptic. They require 25,800 years to make one 360° circuit around the ecliptic poles and the ecliptic. For Mars, the estimated period of precession is about 175,000 years.

## Non-gravitational effects

Physicists in the twentieth century found that photons of light possess momentum that, when they are absorbed or reflected by material bodies, transfers momentum to the bodies, producing a light pressure effect. The interaction of photon velocity of light with the orbital velocities of bodies orbiting the Sun produces a retarding effect on their orbits known as the Poynting-Robertson effect. These effects are insignificant for large solar system bodies, but are important for bodies smaller than 0.394 in (1 cm) in diameter. The Poynting-Robertson effect causes such small interplanetary particles to spiral inwards towards the Sun and to eventually be vaporized by heating from its radiation. Much smaller (micron-sized) particles will be pushed out away from the Sun by light pressure that, along with electromagnetic forces, are the dominant mechanisms for the formation of comet tails.

## The three-body problem

No closed general solution has been found for the problem of systems of three or more bodies whose motions are controlled by their mutual gravitational attractions in a form analogous to the generalized Kepler’s laws for the two-body problem.

However, in 1772 Joseph Lagrange (1736–1813) found a special stable solution known as the restricted three-body problem. If the second body in the three-body system has a mass M_{2} less than 0.04M _{1} where M_{1} is the mass of the most massive Body 1, then there are five stability points in the orbital plane of Bodies 1 and 2. Three of these points, L_{1}, L_{2}, and L _{3} lie on the line joining Bodies 1 and 2. The stability of particles placed at these points is minimal; slight perturbations will cause them to move away from these points indefinitely. The points L_{4} and L_{5}, respectively 60° ahead of and 60° behind Body 2 in its orbit around the system’s center of mass, are more stable; particles placed there will, if slightly perturbed, go into orbits around these points.

Lagrange’s solution became relevant to the solar system in 1906 when Max Wolf (1863–1952) discovered the asteroid Achilles in Jupiter’s orbit but about 60° ahead of it (near the L_{4} point of the solution). Several hundred such asteroids are now known; they are called the Trojan asteroids, since they are named for heroes of the Trojan War. Following the three-body problem, the Sun is Body 1, Jupiter is Body 2, and the asteroids Achilles, Agamemnon, Ajax, Diomedes, Odysseus, and other asteroids named after Greek heroes cluster around the L_{4} point of Jupiter’s orbit, forming the Greek camp. The asteroids Anchises, Patroclus, Priam, Aneas, which are named for Trojan heroes, cluster around the L_{5} point (60° behind Jupiter in its orbit), forming the Trojan camp. The L_{4} and L_{5} points of the orbits of Earth, Mars, and Saturn around the Sun and of the Moon’s orbits around the Earth have been searched for the presence of small bodies ranging in size from asteroids to interplanetary dust without confirmed success. In Saturn’s satellite system, with Saturn as Body 1 and its satellite Dione as Body 2, Saturn’s small satellite Helene orbits Saturn in Dione’s orbit near the L_{4} point. With Saturn’s satellite Tethys as Body 2, Saturn’s satellite Telesto orbits in Tethys’ orbit close to the leading L_{4} point and Calypso orbits close to the-following L_{5} point.

## The n-body problem

For systems of n gravitationally interacting bodies where n=3 to thousands (that is, multiple stars and star clusters where the member stars are of comparable mass), the Virial theorem—by working with a system’s gravitational potential energy and the kinetic energies of the member stars—can give some insight into the system’s stability and evolution. However, the theorem gives mainly information of a statistical nature about the system. It cannot define the space trajectory of a specific star in the system over an extended time interval. Therefore, it cannot predict close encounters of it with other stars nor whether or not this specific star will remain part of the system or will be ejected from it.

## Recent developments

In the last 35 years, high performance computers have been used to study the n-body problem (n=3 to n=10 or more) by stepwise integration of the orbits of the gravitationally interacting bodies. Earlier computers were incapable of performing such calculations over sufficiently long time intervals. The study of the stability of Pluto’s orbits over the last 10,000,000 years, as mentioned above, was made for n=5 (the Sun and the Jovian planets) perturbing Pluto’s orbit. Some other studies have treated the solar system as a n=9 system (the Sun and the eight major solar system bodies) over time intervals of several million years.

However, the finite increments of space and time used in stepwise integrations introduce small uncertainties in the predicted positions of solar system objects. These uncertainties increase as the time interval covered by the calculations increases. This has led to the application to celestial mechanics of a new concept in science, chaos, which started to develop in the 1970s. Chaos studies indicate that, due to increasing inaccuracy of prediction from integration calculations and, also, due to incompleteness of the mathematical models integrated, meaningful predictions about the state or position of a system cannot be made beyond some finite time. One result is that Pluto’s orbit is chaotic over times of about 800 million years, so that its orbit and position in the early solar system or billions of years from now cannot be specified. In addition, the rotation of Saturn’s satellite Hyperion appears to be chaotic. Chaos is now being applied to studies of the stability of the solar system, a problem which celestial mechanics has considered for centuries without finding a definite answer.

Chaos has also been able to show how certain orbits of main belt asteroids can, over billions of years, evolve into orbits that cross the orbits of Mars and Earth, producing near-Earth asteroids (NEAs), of which over 100 are now known. Computer predictions of NEA orbits are now being made to identify NEAs that may collide with Earth in the future. Such collisions would threaten the very existence of human civilization. The prediction of such Earth-impacting asteroids may allow them to be dejected past Earth or to be destroyed. The space technology to do this may be available soon.

High performance computers and the concept of chaos are now also being used to study the satellite systems of the Jovian planets. They have also been used to study the orbits of stars in multiple star systems and the trajectories of stars in star clusters and galaxies.

The search for planets around other stars is also a recent development. It uses the theory of the two-body problem, starting from earlier work on astrometric double stars. Such stars have proper motions in the sky that are not straight lines as is the case for single stars. Instead, they are wavelike curves with periods of some years. This indicates that they are actually double stars with the visible star moving around the system’s center of mass (which has straight-line proper motion) with an unseen companion. The stellar companions of Airius A (Gliese 244A), Procyon A (Gliese 280A), Ross 614 A (Gliese 234A), and Mu Cassiopeiae (Gliese 53A) were first detected as astrometric double stars before being observed optically. Small departures of the proper motions of stars from straight lines have been used since 1940 to predict the presence of companions of substellar mass (less than 0.07 solar mass) around nearby stars.

Action of a star around a double star system’s center of mass produces periodic variations of the Doppler shift of the star’s spectral lines as the star first approaches Earth, then recedes from it as seen from the system’s center of mass. Since 1980, very precise spectroscopic observations have allowed searches for companions of substellar mass of visible stars to be made at several observatories.

These methods have allowed many companions of substellar mass (so-called brown dwarfs and bodies of Jovian planet mass) to be discovered near stars other than the Sun. In fact, since 1995, hundreds of brown dwarfs have been discovered and it appears that they may be very numerous. In addition, as of October 2006 according to the *Extrasolar Planets Encyclopedia,* 208 planetary systems have been discovered—154 in single-planet systems, 48 in 20 multiple-planet systems, and six orbiting pulsars. The search for planets orbiting normal stars continues; this is closely associated with the Search for Extra-Terrestrial Intelligence (the SETI Project).

Since 1957, the Space Age has accelerated the development of the branch of celestial mechanics called astro-dynamics, which is becoming increasingly important. In addition to the traditional gravitational interactions between celestial bodies, astrodynamics must also consider (rocket) propulsion effects that are necessary for inserting artificial satellites and other spacecraft into their necessary orbits and trajectories. Aerodynamic effects must sometimes be considered for planets and satellites with appreciable atmospheres (Venus, Earth, Mars, the Jovian planets, Io, Titan, Neptune’s satellite Britons, and Pluto). Trajectory building is a new part of astrodynamics; it consists of combining different conic section orbits and propulsion segments along with planet and planetary satellite flybys to increase spacecraft payload on missions requiring very large propellant expenditures. The spacecraft *Voyager 1* and *Voyager 2, Magellan, Galileo,* and Cassini/Huygens have all used trajectory building to reach their destinations. Minor perturbations due to light pressure, the Poynting-Robertson Effect, and electromagnetic effects sometimes must also be considered.

The solar sail is now being studied in spacecraft design as a way of using the light pressure from starlight on solar sails to maneuver spacecraft and propel them through interplanetary space. Finally, the development of astrodynamics has increased the importance of hyperbolic orbits, since so far, most flybys of planets and planetary satellites by spacecraft have occurred along hyperbolic orbits. The spacecraft *Pioneers 10* and *11* and *Voyagers 1* and *2* are leaving the solar system along hyperbolic orbits with respect to the Sun that will take them into interstellar trajectories around the center of the Milky Way galaxy. Their hyperbolic orbits are being checked by intermittent radio signals from their transmitters as they leave the solar system for perturbations that could be produced by the gravitational attractions of undiscovered trans-Neptunian planets.

*See also* Brown dwarf; Celestial coordinates; Gravity and gravitation; Precession of the equinoxes; Stellar evolution.

## Resources

### BOOKS

Dyorak, R, R. Freistetter, and J. Kurths, eds. *Chaos and Stability in Planetary Systems.* Berlin, Germany, and New York: Springer, 2005.

Morbidelli, Alessandro. *Modern Celestial Mechanics: Aspects of Solar System Dynamics.* London and New York: Taylor and Francis, 2002.

Roy, Archie E. *Orbital Motion.* Philadelphia, PA: Institute of Physics, 2005.

### OTHER

*Astronomical Virtual Observatory, Paris Observatory.* “The

Extrasolar Planets Encyclopaedia.” <http://vo.obspm.fr/exoplanetes/encyclo/encycl.html> (accessed October 4, 2006).

## Celestial Mechanics

# Celestial mechanics

Modern celestial mechanics began with Isaac New ton's generalization of **Kepler's laws** published in his *Principia* in 1687. Newton used his three **laws of motion** and his law of universal gravitation to do this. The three generalized Kepler's law are:

1) The orbits of two bodies around their center of **mass** (barycenter) are **conic sections** (ellipses, circles, parabolas, or hyperbolas) with the center of mass at a focus of each conic sections; 2) The line joining the center of the two bodies sweeps out equal areas in their orbits in equal **time** intervals. Newton showed that this is a consequence of conservation of angular **momentum** of an isolated two-body system unperturbed by other forces (Newton's third law of motion); 3) From his law of universal gravitation, which states that Bodies l and 2 of masses M1 and M2 whose centers are separated by a **distance** r experience equal and opposite attractive gravitational forces Fg of magnitudes

where G is the Newtonian gravitations factor, and from his second law of motion, Newton derived the following general form of Kepler's third law for these bodies moving around the center of mass along elliptical or circular orbits:

where P is the sidereal period of revolution of the bodies around the center of mass, π is the **ratio** of the circumference of a **circle** to its diameter, X, m1 and m2 are the same as in Equation 1 and a is the semi-major axis of the *relative* orbit of the center of the less massive Body 2 around the center of the more massive Body l.

These three generalized Kepler's Law form the basis of the two-body problem of celestial mechanics. **Astrometry** is the branch of celestial mechanics which is concerned with making precise measurements of the positions of celestial bodies, then calculating precise orbits for them based on the observations. In theory, only three observations are needed to define the **orbit** of one celestial body relative to a second one. Actually, many observations are needed to obtain an accurate orbit.

However, for the most precise orbits and predictions, the vast majority of systems investigated are not strictly two-body systems but consist of many bodies (the **solar system** , planetary **satellite** systems, multiple **star** systems, star clusters, and galaxies).

## Planetary perturbations

To a first **approximation** , the solar system consists of the **Sun** and eight major planets, a system much more complicated than a two-body problem. However, use of Equation 2 with reasonable values for the **astronomical unit** (a convenient unit of length for the solar system) and for G showed that the Sun is far more massive than even the most massive **planet** Jupiter (whose mass is 0.000955 the Sun's mass). This showed that the gravitational forces of the planets on each other are much weaker than the gravitational forces between the Sun and each of the planets, which enabled astronomers to consider the gravitational interactions of the planets as producing small changes with time perturbations) in the elliptical orbit of each planet around the center of mass of the solar system (which is always in or near the Sun). If the Sun and a planet (say the **Earth** or **Jupiter** ) were alone in empty **space** , we would have an ideal two-body problem and we would expect the two-body problem as defined by the generalized Kepler's law to exactly describe their orbits around the systems center of mass. Then the seven orbital elements (of which a and y are two) of a planet's orbit should remain constant forever.

However, the gravitational forces of the other planets on a planet cause its orbit to change slightly over time; these changes can be accurately allowed for over limited time intervals by calculating the perturbations of its orbital elements over time that are caused by the gravitational forces of the other planets.

Historically, perturbation theory has been more useful than merely providing accurate predictions of future planetary positions. Only six major planets were known when Newton published his *Principia*. William Herschel (1738-1822) fortuitously discovered **Uranus** , the seventh major planet from the Sun, in March 1781. The initial orbital elements calculated for Uranus did not accurately allow prediction of its future position even after inclusion of the perturbations caused by the six other major planets. Before 1821, Uranus was consistently observed to be ahead of its predicted position in its orbit; afterwards, it lagged behind its predicted positions.

John Couch Adams (1819-1892) in England and Urbain Leverier (1811-1877) in France, hypothesized that Uranus had passed an undiscovered massive planet further than it was from the Sun in the year 1821. They both made detailed calculations to locate the position of the undiscovered planet perturbing the **motion** of Uranus. Johann Galle (1812-1910) in Berlin, Germany used Leverier's calculations to discover the unknown planet in September 1846, which was then named **Neptune** .

Further unexplained perturbations of the orbits of Uranus and Neptune led Percival Lowell (1855-1916) and several other astronomers to use them to calculate predicted positions for another undiscovered (trans-Neptunian) planet beyond Neptune's orbit. Lowell searched for the trans-Neptunian planets he predicted from 1906 until his death in November 1916 without finding it. The search for a trans-Neptunian planet was resumed in 1929 at Lowell Observatory, where Clyde Tombaugh (1905–1997) who discovered **Pluto** in February 1930.

Lowell had predicted that a planet more massive than Earth produced the unexplained perturbations. During the years following Pluto's discovery, however, detailed studies of its perturbations of the orbits of Uranus and Neptune showed that Pluto is considerably less massive than Earth. The discovery of Pluto's satellite, Charon, in 1978 allowed the determination of the total mass of Pluto and Charon from Equation 2 which is about 0.00237 Earth's mass (about 0.2 the mass of Earth's **moon** ). There are two consequences of this discovery; Tombaugh's discovery of Pluto may have been fortuitous, and one may make the case that Pluto is not a major planet.

The discrepancy in mass between the masses predicted by Lowell and others for the trans-Neptunian planet and the mass of the Pluto-Charon double planet has led to a renewed search for one or more additional trans-Neptunian planet(s) that still continues. The opinion also exists that the unexplained perturbations of the orbits of Uranus and Neptune are caused by systematic errors in some early measurements of their positions and that no trans-Neptunian planets with masses on the order of Earth's mass exist.

## Resonance phenomena

Ceres, the first asteroid or **minor planets** , was discovered to orbit the Sun between the orbits of liars and Jupiter in 1801. Thousands of other asteroids have been discovered in that part of interplanetary space, which is now called the Main asteroid Belt.

Daniel Kirkwood (1815-1895) noticed in 1866 that the periods of revolution of the asteroids around the Sun did not form a continuous distribution over the Main Asteroid Belt but showed gaps (now known as Kirkwood's gaps) at periods corresponding to 1/2, 1/3, and 2/5 Jupiter's period of revolution (11.86 sidereal years). This phenomenon can be explained by the fact that, if an asteroid is in one of Kirkwood's gaps, then every second, third, or fifth revolution around the Sun, it will experience a perturbation by Jupiter of the same direction and magnitude; over the course of millions of years, these perturbations move asteroids out of the Kirkwood's gaps. This is a **resonance** effect of planetary perturbations, and it is only one of several resonance phenomena found in the solar system.

Ratios between the periods of revolution of several planets around the Sun are another resonance phenomenon that is poorly understood. The periods of revolution of **Venus** , Earth, and **Mars** around the Sun are nearly in the ratio is 5:8:15. The periods of revolution of Jupiter and **Saturn** are nearly in a 2:5 ratio, and for Uranus, Neptune, and Pluto they are nearly in a 1:2:3 ratio. The 2:3 ratio between the periods of revolution of Neptune and Pluto makes Pluto's orbit more stable. Due to the ellipticity of its orbit, near perihelion (the point on its orbit closest to the Sun) Pluto comes closer to the Sun than Neptune. Pluto last reached perihelion in September 1989; it has been closer to the Sun than Neptune since 1979 and will continue to be closer until 1998, when it will resume its usual place as the Sun's most distant known planet. However, Neptune will be the Sun's most distant known planet from 1995 to 1998! Recent calculations showed that, because of the 2:3 ratio of the orbital periods, the orientation of Pluto's orbit, and of the positions of Neptune and Pluto in their orbits, Neptune and Pluto have never been closer than 2,500,000,000 km in the last 10,000,000 years. Without the 2:3 ratio of their orbital periods, Pluto probably would have had a close encounter with Neptune which could have ejected Pluto and Charon into separate orbits around the Sun that are drastically different from the systems present orbit.

Jupiter's inner three Gallean satellite, Io, Europa, and Ganymede, have orbital periods of revolution around Jupiter that are nearly in the ratio 1:2:4. Five of Saturn's closest satellites, Pandora, Mimas, Enceladus, Tethys, and Dione, have orbital periods of revolution around Saturn that are nearly in the ratio 4:6:9:12:18. Resonance effects produced by some of these satellites, especially the 1.2 resonance with Mimas' period of revolution around Saturn, seem to have produced the Cassini Division between Saturn's A and B rings, which is analogous to the 1:2 Kirkwood's gap in the Main Asteroid Belt. A 3:4 orbital period resonance seems to exist between Saturn's largest satellite Titan and its next satellite out Hyperion.

Other resonances between the satellites and ring systems of Jupiter, Uranus, and Neptune are not clear because these ring systems are far less developed than that of Saturn.

## Tidal effects

A tidal effect is produced when the gravitational pull of one body on a second one is appreciably greater on the nearer part of the second one than on its center, and in turn, the first body's pull on the second one's center is greater than its pull on the second one's most distant part. Unlike the gravitational **force** Fg, which varies as the inverse square of the distance r between the centers of the two bodies (1/r2); see Equation 1, the tidal effect varies as the inverse cube (1/r3) of the distance between their centers. Both the Moon and the Sun raise **tides** in Earth's oceans, atmosphere, and solid body. The lag of the tides raised in the oceans behind the Moon's crossings of the celestial meridian causes a gravitational interaction between Earth and Moon which slows the **earth's rotation** and moves the Moon's orbit further from Earth.

Tides raised in the Moon's solid body by Earth have slowed its **rotation** until it has become tidally locked to Earth (the Moon keeps the same hemisphere turned towards Earth, and its periods of rotation and revolution around Earth are the same, 27.32 mean solar days). Eventually Earth's rotation will be slowed to where Earth will be tidally locked to the Moon, and the durations of the sidereal day and sidereal month will both equal about 47 present mean solar days.

Tidal evolution has forced most planetary satellites to become tidally locked to their planets. This includes all of Jupiter's Galilean satellites and its four small satellites closer to Jupiter than Io, probably most of Saturn's satellites out to Iasetus (Titan, Saturn's largest satellite, is probably tidally locked to Saturn), the satellites of Uranus and Neptune, and the Pluto-Charon double planet. Tidal action in Io's interior produced by Jupiter (and to a lesser degree by its next satellite out Europa) powers volcanism on Io, making it the most volcanically active body in the solar systems. Tidal effects also may have powered volcanic activity on Europa and Ganymede, Saturn's satellite Enceladus, and Uranus satellites Miranda and Ariel; all of them show some evidence of resurfacing. Some of Earth's internal **heat** may have been produced by the Moon's tidal action.

The Sun's tidal action on Mercury at perihelion has tidally locked Mercury's rotation to its angular **velocity** near perihelion, which is 1.5 times Mercury's average orbital angular velocity; Mercury's rotation period is 58.6 days, 2/3 of its 87.9 day period of revolution around the Sun.

When two bodies are very close together, tidal forces tending to disrupt a body can equal or exceed the attractive gravitational forces holding it together. If the tidal stresses exceed the yield limits of the body's material, the body will gradually disintegrate into many smaller bodies. The mathematician E. Roche (1820-1885) studied the limiting separation of two bodies where the tidal and gravitational forces are equal; it usually between 2-3 times the radius of the more massive body and depends on the relative densities of the bodies and their state of motion. If two bodies approach closer than this Roche limit, one (usually the smaller, less massive body) or both bodies will begin to disintegrate. The rings of some of the Jovian planets may have formed from the tidal disintegration of one or more of their close satellites. Theory predicts that after Earth's and Moon's rotations become tidally locked (see above), the Sun's tides raised on Earth will cause the Moon to approach Earth. If this effect lasts long enough, the Moon may get closer to Earth than its Roche limit, be disintegrated by Earth's tidal forces, and form a ring of small bodies which orbit Earth.

Tidal effects act on close double stars, distorting their shapes, changing their orbits, and sometimes tidally locking their rotations. In some cases, tidal effects cause streams of gas to flow in a double star system and can transfer **matter** from one star to the other or allow it to escape into interstellar space. Tidal effects even seem to act between galaxies, with one **galaxy** distorting the form of its neighbor.

## Precession

Rapidly rotating planets and satellites have appreciable equatorial bulges as a consequence of Newton's First Law of Motion. If the rotation axis of such a body is not **perpendicular** to its orbit, other bodies in the system will exert stronger gravitational attractions on the near part of the bulge than its far part. The effect of this difference is to tend to turn the body's rotation axis perpendicular to the **plane** of its orbit. Because the body is rotating rapidly, however, this does not happen, and, like the rotation axis of a spinning top, the body's rotation axis describes a cone in space whose axis is the perpendicular to the body's orbit (in a two-body system). This phenomenon is called precession, and it is important for Earth, Mars, and the Jovian planets. For Earth, precession causes its celestial poles to describe small circles of 23.°5 **arc** radii around its ecliptic poles and the equinoxes to move westward on the ecliptic. They require 25,800 years to make one 360 circuit around the ecliptic poles and the ecliptic. For liars, the estimated period of precession is about 175,000 years.

## Non-gravitational effects

Twentieth century **physics** have found that photons of **light** possess momentum which, when they are absorbed or reflected by material bodies, transfers momentum to the bodies, producing a light **pressure** effect. The interaction of **photon** velocity of light with the orbital velocities of bodies orbiting the Sun produces a retarding effect on their orbits known as the Poynting-Robertson Effect. These effects are insignificant for large solar system bodies, but are important for bodies smaller than 0.394 in (1 cm) in diameter. The Poynting-Robertson Effect causes such small interplanetary particles to **spiral** inwards towards the Sun and to eventually be vaporized by heating from its **radiation** . Much smaller (micron-sized) particles will be pushed out away from the Sun by light pressure which, along with electromagnetic forces, are the dominant mechanisms for the formation of comet tails.

## The three-body problem

No closed general solution has been found for the problem of systems of three or more bodies whose motions are controlled by their mutual gravitational attractions in a form analogous to the generalized Kepler's Laws for the two-body problem.

However, in 1772 Joseph Lagrange (1736-1813) found a special stable solution known as the Restricted Three-Body Problem. If the second body in the three-body system has a mass M2 less than 0.04M1 where M1 is the mass of the most massive Body 1, then there are five stability points in the orbital plane of Bodies 1 and 2. Three of these points, L1, L2, and L3 lie on the line joining Bodies 1 and 2. The stability of particles placed at these points is minimal; slight perturbations will cause them to move away from these points indefinitely. The points L4 and L5, respectively 60° ahead of and 60° behind Body 2 in its orbit around the system's center of mass, are more stable; particles placed there will, if slightly perturbed, go into orbits around these points.

Lagrange's solution became relevant to the solar system in 1906 when Max Wolf (1863–1952) discovered the asteroid Achilles in Jupiter's orbit but about 60° ahead of it (near the L4 point of the solution). Several hundred such asteroids are now known; they are called the Trojan asteroids, since they are named for heroes of the Trojan War. Following the three-body problem, the Sun is Body 1, Jupiter is Body 2, and the asteroids Achilles, Agamemnon, Ajax, Diomedes, Odysseus, and other asteroids named after Greek heroes cluster around the L4 point of Jupiter's orbit, forming the "Greek camp." The asteroids Anchises, Patroclus, Priam, Aneas, that are named for Trojan heroes cluster around the L5 point (60° behind Jupiter in its orbit), forming the "Trojan camp." The L4 and L5 points of the orbits of Earth, Mars, and Saturn around the Sun and of the Moon's orbits around Earth have been searched for the presence of small bodies ranging in size from asteroids to interplanetary dust without confirmed success. In Saturn's satellite system, with Saturn as Body 1 and its satellite Dione as Body 2, Saturn's small satellite Helene orbits Saturn in Dione's orbit near the L4 point; with Saturn's satellite Tethys as Body 2, Saturn's satellite Telesto orbits in Tethys' orbit close to the leading L4 point and Calypso orbits close to the-following L5 point.

## The n-body problem

For systems of n gravitationally interacting bodies where n = 3 to thousands, that is, multiple stars and star clusters where the member stars are of comparable mass, the Virial **Theorem** , by working with a systems gravitational potential **energy** and the kinetic energies of the member stars, can give some insight into the system's stability and evolution. However, the theorem gives mainly information of a statistical nature about the system; it cannot define the space trajectory of a specific star in the system over an extended time interval, and therefore, it cannot predict close encounters of it with other stars nor whether or not this specific star will remain part of the system or will be ejected from it.

## Recent developments

In the last 30 years high performance computers have been used to study the n-body problem (n = 3 to n = 10 or more) by stepwise integration of the orbits of the gravitationally interacting bodies. Earlier computers were incapable of performing such calculations over sufficiently long time intervals. The study of the stability of Pluto's orbits over the last 10,000,000 years mentioned above was made for n = 5 (the Sun and the Jovian planets) perturbing Pluto's orbit. Some other studies have treated the solar system as a n = 9 system (the Sun and the eight major planets) over time intervals of several million years.

However, the finite increments of space and time used in stepwise integrations introduce small uncertainties in the predicted positions of solar system objects; these uncertainties increase as the time interval covered by the calculations increases. This has led to the application to celestial mechanics of a new concept in science, **chaos** , which started to develop in the 1970s. Chaos studies indicate that, due to increasing inaccuracy of prediction from integration calculations and also due to incompleteness of the mathematical models integrated, meaningful predictions about the state or position of a system cannot be made beyond some finite time. One result is that Pluto's orbit is chaotic over times of about 800 million years, so that its orbit and position in the early solar system or billions of years from now cannot be specified. Also the rotation of Saturn's satellite Hyperion appears to be chaotic. Chaos is now being applied to studies of the stability of the solar system, a problem which celestial mechanics has considered for centuries without finding a definite answer.

Chaos has also been able to show how certain orbits of main belt asteroids can, over billions of years, evolve into orbits which cross the orbits of Mars and Earth, producing near-Earth asteroids (NEA), of which about 100 are now known. Computer predictions of NEA orbits are now being made to identify NEA which may collide with Earth in the future; such collisions would threaten the very existence of our civilization. The prediction of such Earth-impacting asteroids may allow them to be dejected past Earth or to be destroyed; the space technology to do this may be available soon.

High performance computers and the concept of chaos are now also being used to study the satellite systems of the Jovian planets. They have also been used to study the orbits of stars in multiple star systems and the trajectories of stars in star clusters and galaxies.

The search for planets around other stars is also a recent development. It uses the theory of the two-body problem, starting from earlier work on astrometric double stars. These are stars whose proper motions on the sky are not straight lines as are the case for single stars, but are wavelike curves with periods of some years. This indicates that they are actually double stars with the visible star moving around the system's center of mass (which has straight-line proper motion) with an unseen companion. The stellar companions of Airius A (Gliese 244A), Procyon A (Gliese 280A), Ross 614 A (Gliese 234A), and Mu Cassiopeiae (Gliese 53A) were first detected as astrometric double stars before being observed optically. Small departures of the proper motions of stars from straight lines have been used since 1940 to predict the presence of companions of substellar mass (less than 0.07 solar mass) around nearby stars.

Action of a star around a double star system's center of mass produces periodic variations of the Doppler shift of the star's **spectral lines** as the star first approaches Earth, then recedes from it as seen from the system's center of mass. Since 1980, very precise spectroscopic observations have allowed searches for companions of substellar mass of visible stars to be made at several observatories.

These methods have allowed several dozen companions of substellar mass (so-called "brown dwarfs" and bodies of Jovian planet mass) to be suspected near stars other than the Sun. Unfortunately, as of late 1994 none of the suspected bodies of planetary mass associated with other stars has been confirmed by consistent observations at two or more observatories. Surprisingly, the two or three most reliably established planets have been detected orbiting a **pulsar** , which is a **neutron star** , a star that has used up its nuclear energy sources and has almost completed its evolution. The planets have been detected by apparent periodic variations in the period of the **radio** pulses from this neutronstar pulsar PSR 1257 + 12, and moreover, they seem to have masses on the order of Earth's mass or less. The search for planets orbiting normal stars continues; this is closely associated with the Search for Extra-Terrestrial Intelligence (the **SETI** Project).

Since 1957, the Space Age has accelerated the development of the branch of celestial mechanics called astrodynamics, which is becoming increasingly important. In addition to the traditional gravitational interactions between celestial bodies, astrodynamics must also consider (rocket) propulsion effects that are necessary for inserting artificial satellites and other spacecraft into their necessary orbits and trajectories. Aerodynamic effects must sometimes be considered for planets and satellites with appreciable atmospheres (Venus, Earth, Mars, the Jovian planets, Io, Titan, Neptune's satellite Britons and Pluto). Trajectory building is a new part of astrodynamics; it consists of combining different conic section orbits and propulsion segments along with planet and planetary satellite flybys to increase spacecraft payload on missions requiring very large propellant expenditures. The spacecraft *Voyagers 1* and *2*, *Magellan*, and *Galileo* have all used trajectory building, and future spacecraft such as the Cassini/Huygens mission to Saturn and Titan plan to use it to reach their destinations. Minor perturbations due to light pressure, the Poynting-Robertson Effect, and electromagnetic effects sometimes must also be considered. The solar sail is now being studied in spacecraft design as a way of using the light pressure from sunlight on solar sails to maneuver spacecraft and propel them through interplanetary space. Finally, the development of astrodynamics has increased the importance of hyperbolic orbits, since so far all flybys of planets and planetary satellites by spacecraft have occurred along hyperbolic orbits. The spacecraft Pioneers 10 and 11 Voyagers 1 and 2 are leaving the solar system along hyperbolic orbits with respect to the Sun that will take them into interstellar trajectories around the center of our **Milky Way** galaxy. Their hyperbolic orbits are being checked by intermittent radio signals from their transmitters as they leave the solar system for perturbations that could be produced by the gravitational attractions of undiscovered trans-Neptunian planets.

See also Brown dwarf; Celestial coordinates; Gravity and gravitation; Precession of the equinoxes; Solar system; Stellar evolution.

## Resources

### books

Glelek, Jame. *Chaos: Making a New Science.* New York: Viking Penguin, Inc. 1988.

Motz, Lloyd, and Anneta Duveen. *Essentials of Astronomy.* Belmont, CA: Wadsworth, 1966.

### periodicals

"Pulsar's Planets Confirmed." *Sky and Telescope."* 87 (1994).

## Celestial Mechanics

# Celestial mechanics

Celestial mechanics is a branch of astronomy that studies the movement of bodies in outer space. Using a mathematical theory, it explains the observed motion of the planets and allows us to predict their future movements. It also comes into play when we launch a satellite into space and expect to direct its flight.

## Early Greeks

Until English scientist and mathematician Isaac Newton (1642–1727) founded the science of celestial mechanics over 300 years ago, the movement of the planets regularly baffled astronomers or anyone who studied the heavens. This is because those bodies called planets, a word which comes from the Greek word for "wanderer," literally "wandered" about the sky in a seemingly unpredictable manner. To the early astronomers, the stars were fixed in the heavens and the Sun seemed to make the same regular journey every year, but the planets followed no such pattern. Their unpredictable behavior was especially frustrating to the ancients, and it was not until around a.d. 140 that Greek astronomer Claudius Ptolemaeus or Ptolemy provided some kind of order to this chaotic situation.

## Words to Know

**Copernican system:** Theory proposing that the Sun is at the center of the solar system and all planets, including Earth, revolve around it.

**Epicycle:** A circle on which a planet moves and which has a center that is itself carried around at the same time on the rim of a larger circle.

**Gravity:** Force of attraction between objects, the strength of which depends on the mass of each object and the distance between them. Also, the special acceleration of 9.81 meters per second per second exerted by the attraction of the mass of Earth on nearby objects.

**Orbit:** The path followed by a body (such as a planet) in its travel around another body (such as the Sun).

**Ptolemaic system:** Theory proposing that Earth is at the center of the solar system and the Sun, the Moon, and all the planets revolve around it.

In what came to be known as the Ptolemaic (pronounced tahl-uh-MAY-ik) system, he placed Earth at the center of the universe and had the Sun revolving around it, along with all the other known planets. Ptolemy's system of predicting which planet would be where and when was described as the epicyclic theory because it was based on the notion of epicycles (an epicycle is a small circle whose center is on the rim of a larger circle). Since Ptolemy and most Greeks of his time believed that all planetary motion must be circular (which it is not), they had to keep adding more and more epicycles to their calculations to make their system work. Despite the fact that it was very complicated and difficult to use, his system was able generally to tell where the planets would be with some degree of accuracy. This is even more amazing when we realize that it was based on an entirely incorrect notion of the solar system (since it had the Earth and not the Sun at the center). Yet its ingenious use of the off-center epicycles, which were regularly adjusted, permitted the Ptolemaic system to approximate the irregular movements of the planets. This, in turn, mostly accounts for the fact that his incorrect system survived and was used by every astronomer and astrologer for 1,400 years.

## Copernican revolution

During the Middle Ages (period in European history usually dated from about 500 to 1450), the Ptolemaic system was dominant, and all educated people whether in Europe or in the Arabian world used it to explain the movement of the planets. By then, the system had the seven known planets riding more than 240 different epicycles. It was this level of complexity that led Spanish King Alfonso X (1221–1284), who was called "the Wise," to state that had he been present at the creation of the world, he would have suggested that God make a simpler planetary system. The clutter of all of this complexity was eventually done away with (although by no means immediately) when Polish astronomer Nicolaus Copernicus (1473–1543) offered what is called his heliocentric (pronounced hee-lee-oh-SEN-trik) theory in 1543.

Copernicus placed the Sun at the center of the solar system and made the planets (including Earth) orbit the Sun on eccentric circles (which are more egg-shaped than perfectly circular or round ones). The Copernican system took a long time to be adopted, mainly because it was actively condemned for over a century by the Catholic Church. The Church objected to the fact that his system took Earth out of its stationary center position and made it revolve instead around the Sun with all the other planets. Although Copernicus could explain certain phenomena—for example he correctly stated that the farther a planet lies from the Sun the slower it moves—his system still did not have a mathematical formula that could be used to explain and predict planetary movement.

## Kepler's laws

By the time the Church was condemning the work of Italian astronomer and physicist Galileo Galilei (1564–1642), who defended the Copernican model of the solar system, German astronomer Johannes Kepler (1571–1630) had already published his three laws of planetary motion, which would lay the groundwork for all of modern astronomy. His first two laws were contained in his *Astronomia nova* (*The New Astronomy* ), published in 1609, and his third was stated in his book *Harmonices mundi* (*Harmony of the World* ), published in 1619. Basically, the laws state that the orbits of planets can be drawn as ellipses (elongated egg shapes) with the Sun always at one of their central points; that a planet moves faster the closer it is to the Sun (and slower the farther away it is); and, lastly, that it is possible to calculate a planet's relative distance from the Sun knowing its period of revolution. Kepler's laws about the planets and the Sun laid the groundwork for English physicist Isaac Newton to be able to go further and generalize about what might be called the physics of the universe—in other words, the mechanics of the heavens or celestial mechanics (celestial being another word for "the heavens").

## Newtonian mechanics

Celestial mechanics is, therefore, Newtonian mechanics. Newton's greatness was in his ability to seek out and find a generalization or a single big idea that would explain the behavior of bodies in motion. Newton was able to do this with what is called his law of universal gravitation and his three laws of motion. The amazing thing about his achievement is that he discovered certain general principles that unified the heavens and Earth. He showed that all aspects of the natural world, near and far, were subject to the same laws of motion and gravitation, and that they could be demonstrated in mathematical terms within a single theory.

In 1687, Newton published his epic work, *Philosophiae naturalis principia mathematica* (*Mathematical Principles of Natural Philosophy* ). In the first part of the book, Newton offers his three laws of motion. The first law is the principle of inertia, which says that a body stays at rest (or in motion) until an outside force acts upon it. His second law defines force as the product of how fast something is moving and how much matter (called mass) is in it. His third law says that for every action there is an equal and opposite reaction. It was from these laws that Newton arrived at his law of universal gravitation, which can be said to have founded the science of celestial mechanics.

## Gravity as a universal force

The law of universal gravitation states that every particle of matter attracts every other particle with a force that is directly proportional (an equal ratio such as 1:1) to the product of the masses of the particles, and is inversely proportional (the opposite ratio) to the square of the distance between them. Although this may sound complicated, it actually simplified things because celestial mechanics now had an actual set of equations that could be used with the laws of motion to figure out how two bodies in space influenced and affected each other.

It was Newton's great achievement that he discovered gravity to be the force that holds the universe together. Gravity is a mutual attraction or a two-way street between bodies. That is, a stone falls to the ground mainly because Earth's gravity pulls it downward (since Earth's mass is much greater than that of the stone). But the stone also exerts its influence on Earth, although it is so tiny it has no effect. However, if the two bodies were closer in size, this two-way attraction would be more noticeable. We see this with Earth and the Moon. Earth's gravity holds the Moon in orbit around it, but just as Earth exerts a force on the Moon, so the Moon pulls upon Earth. We can demonstrate this by seeing how the free-flowing water of the oceans gets pulled toward the side of Earth that is facing the Moon (what we call high tide). The opposite side of Earth also experiences this same thing at the same time, as the ocean on that side also bulges away from Earth since the Moon's gravity pulls the solid body of Earth away from the water on Earth's distant side.

When the principles discovered by Newton are applied only to the movements of bodies in outer space, it is called celestial mechanics instead of just mechanics. Therefore, using Newton's laws, we can analyze the orbital movements of planets, comets, asteroids, and human-made

satellites and spacecraft, as well as the motions of stars and even galaxies. However, Newton's solution works best (and easiest) when there are only two bodies (like Earth and the Moon) involved. The situation becomes incredibly complicated when there are three or more separate forces acting on each other at once, and all these bodies are also moving at the same time. This means that each body is subject to small changes that are known as perturbations (pronounced pur-tur-BAY-shunz). These perturbations or small deviations do not change things very much in a short period of time, but over a very long period they may add up and make a considerable difference.

That is why today's celestial mechanics of complicated systems are really only very good approximations. However, computer advances have made quite a difference in the degree of accuracy achieved. Finally, with the beginning of the space age in 1957 when the first artificial satellite was launched, a new branch of celestial mechanics called astrodynamics was founded that considers the effects of rocket propulsion in putting an object into the proper orbit or extended flight path. Although our space activity has presented us with new and complicated problems of predicting the motion of bodies in space, it is still all based on the celestial mechanics laid out by Isaac Newton over three centuries ago.

[*See also* **Gravity and gravitation; Laws of motion; Orbit; Tides** ]

## celestial mechanics

celestial mechanics, the study of the motions of astronomical bodies as they move under the influence of their mutual gravitation. Celestial mechanics analyzes the orbital motions of planets, dwarf planets, comets, asteroids, and natural and artificial satellites within the solar system as well as the motions of stars and galaxies. Newton's laws of motion and his theory of universal gravitation are the basis for celestial mechanics; for some objects, general relativity is also important. Calculating the motions of astronomical bodies is a complicated procedure because many separate forces are acting at once, and all the bodies are simultaneously in motion. The only problem that can be solved exactly is that of two bodies moving under the influence of their mutual gravitational attraction (see ephemeris). Since the sun is the dominant influence in the solar system, an application of the two-body problem leads to the simple elliptical orbits as described by Kepler's laws; these laws give a close approximation of planetary motion. More exact solutions, which consider the effects of the planets on each other, cannot be found in a straightforward way. However, methods accounting for these other influences, or perturbations, have been devised; they allow successive refinements of an approximate solution to be made to almost any degree of precision. In computing the motions of stars and the rotations of galaxies, statistical methods are often used. Columbia astronomer Wallace Eckert was the first to use a computer for orbit calculations; now computers are used for this work almost exclusively.

## celestial mechanics

**celestial mechanics** Branch of astronomy concerned with the relative motions of stars and planets that are associated in systems (such as the Solar System or a binary star system) by gravitational fields. Introduced by Isaac Newton in the 17th century, celestial mechanics, rather than general relativity, is usually sufficient to calculate the various factors determining the motion of planets, satellites, comets, stars, and galaxies around a centre of gravitational attraction.