Johannes Kepler Johannes Kepler (December 27, 1571 – November 15, 1630), a key figure in the scientific revolution, was a German astronomer, mathematician and astrologer. He is best known for his laws of planetary motion. He is sometimes referred to as the first theoretical astrophysicist, although Carl Sagan also refers...
Johannes Kepler's primary contributions to Astronomy, which etymologically means law of the stars, (from Greek: αστρονομία = άστρον + νόμος) is a science involving the observation and explanation of events occurring outside Earth and its atmosphere. It studies the...
astronomy/ Spiral Galaxy ESO 26957 Astrophysics is the branch of astronomy that deals with the physics of the universe, including the physical properties ( luminosity, density, temperature and chemical composition) of astronomical objects such as stars, galaxies, and the interstellar medium, as well as their interactions. The study of cosmology is...
astrophysics were the three laws of planetary motion. Kepler derived these laws, in part, by studying the Observation basically means watching something and taking note of anything it does. For instance, you might observe a bird flying by watching it closely. The sciences of biology and astronomy have their historical basis in observations by amateurs. There is pleasure in observation, which explains the participation of hobbyists. The...
observations of Tycho Brahe (December 14, 1546 Knudstrup, Denmark – October 24, 1601 Prague, Bohemia (now Czech Republic)) was a Danish nobleman, well known as an astronomer/astrologer (the two were not yet distinct) and alchemist. He had Uraniborg built, which became an early research institute. For purposes of publication, he owned...
Brahe. Sir Isaac Newton in Knellers portrait of 1689. Sir Isaac Newton (25 December 1642 – 20 March 1727 by the Julian calendar in use in England at the time; or 4 January 1643 – 31 March 1727 by the Gregorian calendar) was an English physicist, mathematician, astronomer, philosopher, and...
Isaac Newton would later design his 1...
laws of motion and The law of universal gravitation states that gravitational force between masses decreases with the distance between them, according to an inversesquare law. In addition, the theory notes that the greater an objects mass, the greater its gravitational force on another mass. Newton published his argument in Philosophiae Naturalis...
universal gravitation and verify that Kepler's laws could be derived from them. The generic term for an orbiting object is " For other uses, please see Satellite (disambiguation) A satellite 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). Because all objects exert gravity, the motion of the primary object is also...
satellite". Kepler's laws of planetary motion
 Kepler's first law (1609): The For other meanings of the term orbit, see orbit (disambiguation) In physics, 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. History Orbits were first analysed mathematically by Kepler who formulated his results in...
orbit of a A planet (from the Greek πλανήτης, planetes or wanderers) is a body of considerable mass that orbits a star and that produces very little or no energy through nuclear fusion. Prior to the 1990s only nine were known (all of them in our...
planet about a For alternate meanings see star (disambiguation) Hundreds of stars are visible in this image taken by the Hubble Space Telescope of the Sagittarius Star Cloud in the Milky Way Galaxy. A star (Greek astron) is any massive gaseous celestial body in outer space. Stars appear as shining points in the...
star is an In mathematics, an ellipse (from the Greek for absence) is a curve where the sum of the distances from any point on the curve to two fixed points is constant. The two fixed points are called foci (plural of focus). Thus an ellipse can be drawn with two pins, a...
ellipse with the star at one The word focus (pl. foci) has several meanings: focus (geometry) focus (optics) See also: autofocus, focal length focus (linguistics) Focus (band) focus (computing) Focus (encyclopedia) Focus software Ford Focus (automobile) Focus (DIY) Ltd a UK chain of DIY stores. See also: focal point (another disambiguation page) This is a disambiguation...
focus.
 Kepler's second law (1609): A A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i.e. a curve that is long and straight. Given two points, in Euclidean geometry, one can always find exactly one line that passes through the two points; the line provides the shortest connection...
line joining a planet and its star sweeps out equal This article explains the meaning of area as a physical quantity. The article area (geometry) is more mathematical. Area is a quantity expressing the size of a region of space. Surface area refers to the summation of the areas of the exposed sides of an object. Units Units for measuring...
areas during equal intervals of (Clockwise from upper left) Notable Time magazine covers from May 7, 1945; July 25, 1969; December 31, 1999; September 14, 2001; and April 21, 2003. Note that on the September 14, 2001 edition, the usual red border was colored black due to the Sept. 11 attacks Time (officially capitalized TIME...
time.
 Kepler's third law (1618): The In algebra, the square of x is written x2 and is defined as the product of x with itself: x × x. Taking the square is exponentiation with exponent two. If x is a positive real number, the value of x2 is equal to the area of a (geometric) square of...
square of the The orbital period is the time it takes a planet (or another object) to make one full orbit. There are several kinds of orbital periods for objects around the Sun: The sidereal period is the time that it takes the object to make one full orbit around the Sun, relative...
sidereal period of an orbiting planet is directly This article is about proportionality, the mathematical relation. For other uses of the term proportionality, see proportionality (disambiguation). In mathematics, two related quantities x and y are called proportional (or directly proportional) if there exists a functional relationship with a constant, nonzero number k such that . In this case...
proportional to the In arithmetic and algebra, the cube of a number n is its third power — the result of multiplying it by itself two times: n3 = n × n × n. This is also the volume formula for a geometric cube of side length n, giving rise to the name. The term cube...
cube of the orbit's In mathematics, an ellipse (from the Greek for absence) is a curve where the sum of the distances from any point on the curve to two fixed points is constant. The two fixed points are called foci (plural of focus). Thus an ellipse can be drawn with two pins, a...
semimajor axis.
Kepler's first law The For other meanings of the term orbit, see orbit (disambiguation) In physics, 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. History Orbits were first analysed mathematically by Kepler who formulated his results in...
orbit of a A planet (from the Greek πλανήτης, planetes or wanderers) is a body of considerable mass that orbits a star and that produces very little or no energy through nuclear fusion. Prior to the 1990s only nine were known (all of them in our...
planet about a For alternate meanings see star (disambiguation) Hundreds of stars are visible in this image taken by the Hubble Space Telescope of the Sagittarius Star Cloud in the Milky Way Galaxy. A star (Greek astron) is any massive gaseous celestial body in outer space. Stars appear as shining points in the...
star is an ellipse with the star at one The word focus (pl. foci) has several meanings: focus (geometry) focus (optics) See also: autofocus, focal length focus (linguistics) Focus (band) focus (computing) Focus (encyclopedia) Focus software Ford Focus (automobile) Focus (DIY) Ltd a UK chain of DIY stores. See also: focal point (another disambiguation page) This is a disambiguation...
focus. There is no object at the other focus of a planet's orbit. The semimajor axis, a, is half the major axis of the ellipse. In some sense it can be regarded as the average distance between the planet and its star, but it is not the time average in a strict sense, as more time is spent near This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. In architecture, apsis is a synonym for apse; Apogee is also the name of a video game publisher. elements of an orbit In astronomy, an apsis (plural...
apocentre than near This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. In architecture, apsis is a synonym for apse; Apogee is also the name of a video game publisher. elements of an orbit In astronomy, an apsis (plural...
pericentre.
Connection with Newton's laws Newton proposed that "every object in the universe attracts every other object along a line of the centers of the objects proportional to each objects mass, and inversely proportional to the square of the distance between the objects." This section proves that Kepler's first law is consistent with 1...
Newton's laws of motion. We begin with Newton's law F=ma: Here we express F as the product of its magnitude and its direction. Recall that in This article describes some of the common coordinate systems that appear in elementary mathematics. For advanced topics, please refer to coordinate system. For more background, see Cartesian coordinate system. The coordinates of a point are the components of a tuple of numbers used to represent the location of the point...
polar coordinates: 

In component form we have: Now consider the In physics, angular momentum intuitively measures how much the linear momentum is directed around a certain point called the origin; the moment of momentum. Since angular momentum depends upon the origin of choice, one must be careful when discussing angular momentum to specify the origin and not to combine angular...
angular momentum: 
So: where is the In astrodynamics specific relative angular momentum () of orbiting body () relative to central body () is the relative angular momentum of per unit mass. Specific relative angular momentum plays a pivotal role in definition of orbit equations. Specific relative angular momentum ()is defined as cross product of position vector and velocity vector...
angular momentum per unit mass. Now we substitute. Let: 

The equation of motion in the direction becomes: 
Newton's law of gravitation states that the central force is inversely proportional to the square of the distance so we have: where k is our proportionality constant. This differential equation has the general solution: Replacing u with r and letting θ_{0}=0:  .
This is indeed the equation of a In mathematics, a conic section (or just conic) is a curved locus of points, formed by intersecting a cone with a plane. The conic sections were named and studied as long ago as 200 BC, when Apollonius of Perga undertook a systematic study of their properties. Types of conics Two...
conic section with the origin at one focus. Q.E.D.
Kepler's second law A A huge number of links to this disambiguation page should point to one of the pages listed below, or perhaps to some page that should be listed below but is not yet. Please help fix these. The word line apparently derives from the Latin linum, meaning flax plant from which...
line joining a planet and its star sweeps out equal This article explains the meaning of area as a physical quantity. The article area (geometry) is more mathematical. Area is a quantity expressing the size of a region of space. Surface area refers to the summation of the areas of the exposed sides of an object. Units Units for measuring...
areas during equal intervals of (Clockwise from upper left) Notable Time magazine covers from May 7, 1945; July 25, 1969; December 31, 1999; September 14, 2001; and April 21, 2003. Note that on the September 14, 2001 edition, the usual red border was colored black due to the Sept. 11 attacks Time (officially capitalized TIME...
time. This is also known as the law of equal areas. Suppose a planet takes 1 A day is any of several different units of time. The word refers either to the period of light when the Sun is above the local horizon or to the full day covering a dark and a light period. Introduction Different definitions of the day are based on the apparent...
day to travel from The word point can refer to: a location in physical space a unit of angular measurement; see navigation point is a typographic unit of measure in typography equal inch or sometimes approximated as inch; on computer displays it should be equal to point in typography if the correct display resolution...
points A to B. During this (Clockwise from upper left) Notable Time magazine covers from May 7, 1945; July 25, 1969; December 31, 1999; September 14, 2001; and April 21, 2003. Note that on the September 14, 2001 edition, the usual red border was colored black due to the Sept. 11 attacks Time (officially capitalized TIME...
time, an imaginary line, from the Sun to the planet, will sweep out a roughly For alternate meanings, such as the musical instrument, see triangle (disambiguation). A triangle is one of the basic shapes of geometry: a twodimensional figure with three vertices and three sides which are straight line segments. Types of triangles Triangles can be classified according to the lengths of their sides...
triangular This article explains the meaning of area as a physical quantity. The article area (geometry) is more mathematical. Area is a quantity expressing the size of a region of space. Surface area refers to the summation of the areas of the exposed sides of an object. Units Units for measuring...
area. This same amount of area will be swept every day. As a planet travels in its elliptical orbit, its For distance between people, see proxemics. Distance between two points The distance between two points is the length of a straight line between them. In the case of two locations on Earth, usually the distance along the surface is meant: either as the crow flies (along a great circle) or...
distance from the Sun will vary. As an equal area is swept during any period of time and since the distance from a planet to its orbiting star varies, one can conclude that in order for the area being swept to remain In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. This is in contrast to a variable, which is not fixed. Constant number The most widely mentioned sort of constant is a fixed, but possibly unspecified, number. Usually the term constant is used in connection...
constant, a planet must vary in This article is about velocity in physics. For the use of the term velocity in economics, see velocity of money Velocity (symbol: v) is a vector measurement of the rate and direction of motion. The scalar absolute value (magnitude) of velocity is speed. Velocity can also be defined as rate...
velocity. Planets move fastest when at This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. In architecture, apsis is a synonym for apse; Apogee is also the name of a video game publisher. elements of an orbit In astronomy, an apsis (plural...
perihelion and slowest when at This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. In architecture, apsis is a synonym for apse; Apogee is also the name of a video game publisher. elements of an orbit In astronomy, an apsis (plural...
aphelion. This law was developed, in part, from the observations of Tycho Brahe (December 14, 1546 Knudstrup, Denmark – October 24, 1601 Prague, Bohemia (now Czech Republic)) was a Danish nobleman, well known as an astronomer/astrologer (the two were not yet distinct) and alchemist. He had Uraniborg built, which became an early research institute. For purposes of publication, he owned...
Brahe that indicated that the velocity of planets was not constant. This law corresponds to the In physics, angular momentum intuitively measures how much the linear momentum is directed around a certain point called the origin; the moment of momentum. Since angular momentum depends upon the origin of choice, one must be careful when discussing angular momentum to specify the origin and not to combine angular...
angular momentum In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. The following is a partial listing of conservation laws that have never been shown to be inexact. (Actually, in General relativity energy, momentum and angular momentum are...
conservation law in the given situation.
Proof of Kepler's second law: Assuming Newton's laws of motion, we can show that Kepler's second law is consistent. By definition, the In physics, angular momentum intuitively measures how much the linear momentum is directed around a certain point called the origin; the moment of momentum. Since angular momentum depends upon the origin of choice, one must be careful when discussing angular momentum to specify the origin and not to combine angular...
angular momentum of a point mass with mass m and velocity is :  .
where is the position vector of the particle. Since , we have: taking the time derivative of both sides: since the In mathematics, the cross product is a binary operation on vectors in three dimensions. It is also known as the vector product or outer product. It differs from the dot product in that it results in a vector rather than in a scalar. Its main use lies in the fact...
cross product of parallel A vector in physics and engineering typically refers to a quantity that has close relationship to the spatial coordinates, informally described as an object with a magnitude and a direction. The word vector is also now used for more general concepts (see also vector and generalizations below), but in this...
vectors is 0. We can now say that is constant. The area swept out by the line joining the A planet (from the Greek πλανήτης, planetes or wanderers) is a body of considerable mass that orbits a star and that produces very little or no energy through nuclear fusion. Prior to the 1990s only nine were known (all of them in our...
planet and the The Sun (occasionally referred to as Sol) is the star at the centre of our solar system. Planet Earth orbits the Sun, as do innumerable other bodies including other planets, asteroids, meteoroids, comets and dust. In common usage, the primary stellar body around which an object orbits is called its...
sun, is half the This article explains the meaning of area as a physical quantity. The article area (geometry) is more mathematical. Area is a quantity expressing the size of a region of space. Surface area refers to the summation of the areas of the exposed sides of an object. Units Units for measuring...
area of the A parallelogram is a foursided plane figure that has two sets of opposite parallel sides. Every parallelogram is a polygon, and more specifically a quadrilateral. Special cases of a parallelogram are the rhombus, in which all four sides are of equal length, the rectangle, in which the two sets...
parallelogram formed by and . 
Since is constant, the area swept out by is also constant. Q.E.D.
Kepler's third law (harmonic law) The In algebra, the square of x is written x2 and is defined as the product of x with itself: x × x. Taking the square is exponentiation with exponent two. If x is a positive real number, the value of x2 is equal to the area of a (geometric) square of...
square of the The orbital period is the time it takes a planet (or another object) to make one full orbit. There are several kinds of orbital periods for objects around the Sun: The sidereal period is the time that it takes the object to make one full orbit around the Sun, relative...
sidereal period of an orbiting planet is directly The word proportionality may have one of a number of meanings: In mathematics, proportionality is a mathematical relation between two quantities. In law and politics, proportionality is a maxim in some theories of governance and a principle underpinning the constitution of the European Union. In politics, proportional representation is a...
proportional to the In arithmetic and algebra, the cube of a number n is its third power — the result of multiplying it by itself two times: n3 = n × n × n. This is also the volume formula for a geometric cube of side length n, giving rise to the name. The term cube...
cube of the orbit's In mathematics, an ellipse (from the Greek for absence) is a curve where the sum of the distances from any point on the curve to two fixed points is constant. The two fixed points are called foci (plural of focus). Thus an ellipse can be drawn with two pins, a...
semimajor axis.  P^{2} ~ a^{3}
 P = object's sidereal period in years
 a = object's semimajor axis, in The astronomical unit (AU or au or a.u. or rarely ua) is a unit of distance, approximately equal to the mean distance between Earth and Sun. The currently accepted value of the AU is 149,597,870,691 ± 30 metres (about 150 million kilometres or 93 million miles...
AU
Thus, not only does the length of the orbit increase with distance, also the 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. It can be used to refer to either the mean orbital speed...
orbital speed decreases, so that the increase of the sidereal period is more than proportional. See the actual figures: Mosaic of the planets of the solar system, excluding Pluto, and including Earths Moon. Note: planets are not portrayed in the same scale. The Solar System consists of the Sun and all the objects that orbit around it, including asteroids, comets, moons, and planets). The Earth is the third...
attributes of major planets. Newton would modify this third law, noting that the period is also affected by the orbiting body's Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. It is a central concept of classical mechanics and related subjects. Strictly speaking, there are two different quantities called mass: Inertial mass is a measure of an objects inertia: its resistance to...
mass, however typically the central body is so much more massive that the orbiting body's mass may be ignored. (See below.)
Applicability The laws are applicable whenever a comparatively light object revolves around a much heavier one because of gravitational attraction. It is assumed that the gravitational effect of the lighter object on the heavier one is negligible. An example is the case of a For other uses, please see Satellite (disambiguation) A satellite 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). Because all objects exert gravity, the motion of the primary object is also...
satellite revolving around Earth.
Application Assume an orbit with semimajor axis a, semiminor axis b, and (This page refers to eccitricity in astrodynamics. For other uses, see the disambiguation page eccentricity.) In astrodynamics, under standard assumptions any orbit must be of conic section shape. The eccentricity of this conic section, the orbits eccentricity, is an important parameter of the orbit that defines its absolute shape...
eccentricity ε. To convert the laws into predictions, Kepler began by adding the orbit's auxiliary circle (that with the major axis as a diameter) and defined these points:  c center of auxiliary circle and ellipse
 s sun (at one focus of ellipse);
 p the planet
 z perihelion
 x is the projection of the planet to the auxiliary circle; then
 y is a point on the circle such that
and three angles measured from perihelion:  In astronomy, the true anomaly (, also written ) is the angle between the direction zs of periapsis and the current position p of an object on its orbit, measured at the focus s of the ellipse (the point around which the object orbits). In the diagram below, true anomaly is...
true anomaly , the planet as seen from the sun
 The eccentric anomaly is the angle between the direction of periapsis and the current position of an object on its orbit, projected onto the ellipses circumscribing circle perpendicularly to the major axis, measured at the centre of the ellipse. In the diagram below, it is E (the angle zcx...
eccentric anomaly , x as seen from the centre
 In the study of orbital dynamics the mean anomaly is a measure of time, specific to the orbiting body p, which is a multiple of 2π radians at and only at periapsis. It is the fraction of the orbital period that has elapsed since the last passage at periapsis...
mean anomaly , y as seen from the centre
Then  area cxz = area cxs + area sxz = area cxs + area cyz
giving Kepler's equation  .
To connect E and T, assume r = length sp then  and rsinT = bsinE

which is ambiguous but useable. A better form follows by some trickery with In mathematics, trigonometric identities (or trig identities for short) are equations involving trigonometric functions that are true for all values of the occurring variables. These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of nontrigonometric functions: a common trick...
trigonometric identities: 
(So far only laws of geometry have been used.) Note that area spz is the area swept since perihelion; by the second law, that is proportional to time since perihelion. But we defined and so M is also proportional to time since perihelion—this is why it was introduced. We now have a connection between time and position in the orbit. The catch is that Kepler's equation cannot be rearranged to isolate E; going in the timetoposition direction requires an iteration (such as In numerical analysis, Newtons method (or the NewtonRaphson method) is an efficient algorithm for finding approximations to the zeros (or roots) of a realvalued function. As such, it is an example of a rootfinding algorithm. It can also be used to find a minimum or maximum...
Newton's method) or an approximate expression, such as 
via the This page is about Lagrange reversion. For inversion, see Lagrange inversion theorem. In mathematics, the Lagrange reversion theorem gives series or formal power series expansions of certain implicitly defined functions; indeed, of compositions with such functions. Let z be a function of x and y in terms of another function...
Lagrange reversion theorem. For the small ε typical of the planets (except Atmospheric characteristics Atmospheric pressure 0.150.30 Pascal Composition Nitrogen,Methane Pluto is the ninth planet from the Sun in our solar system. Because Pluto is also the smallest planet in our solar system and has a highly eccentric orbit (which takes it inside the orbit of Neptune) there...
Pluto) such series are quite accurate with only a few terms; one could even develop a series computing T directly from M. [1] (http://info.ifpan.edu.pl/firststep/awworks/fsII/mul/mueller.html)
Kepler's understanding of the laws Kepler did not understand why his laws were correct; it was Sir Isaac Newton in Knellers portrait of 1689. Sir Isaac Newton (25 December 1642 – 20 March 1727 by the Julian calendar in use in England at the time; or 4 January 1643 – 31 March 1727 by the Gregorian calendar) was an English physicist, mathematician, astronomer, philosopher, and...
Isaac Newton who discovered the answer to this more than fifty years later. Sir Isaac Newton in Knellers portrait of 1689. Sir Isaac Newton (25 December 1642 – 20 March 1727 by the Julian calendar in use in England at the time; or 4 January 1643 – 31 March 1727 by the Gregorian calendar) was an English physicist, mathematician, astronomer, philosopher, and...
Newton, understanding that his 1...
third law of motion was related to Kepler's third law of planetary motion, devised the following: where:  P = object's The orbital period is the time it takes a planet (or another object) to make one full orbit. There are several kinds of orbital periods for objects around the Sun: The sidereal period is the time that it takes the object to make one full orbit around the Sun, relative...
sidereal period
 a = object's In mathematics, an ellipse (from the Greek for absence) is a curve where the sum of the distances from any point on the curve to two fixed points is constant. The two fixed points are called foci (plural of focus). Thus an ellipse can be drawn with two pins, a...
semimajor axis
 G = 6.67 × 10^{−11} N · m^{2}/kg^{2} = the According to the law of universal gravitation, the attractive force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. The constant of proportionality is called , the gravitational constant, the universal gravitational constant, or Newtons constant. The...
gravitational constant
 m_{1} = Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. It is a central concept of classical mechanics and related subjects. Strictly speaking, there are two different quantities called mass: Inertial mass is a measure of an objects inertia: its resistance to...
mass of object 1
 m_{2} = Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. It is a central concept of classical mechanics and related subjects. Strictly speaking, there are two different quantities called mass: Inertial mass is a measure of an objects inertia: its resistance to...
mass of object 2
 π = mathematical constant The title given to this article is incorrect due to technical limitations. The correct title is π. The minuscule, or lowercase, pi The mathematical constant π (written as pi when the Greek letter is not available) is ubiquitous in mathematics and physics. In Euclidean plane geometry, π may be...
pi
Astronomers doing celestial mechanics often use units of years, AU, G=1, and solar masses, and with m_{2}<<m_{1}, this reduces to Kepler's form. SI (disambiguation). The International System of Units (abbreviated SI from the French phrase, Système International dUnités) is the most widely used system of units. It is used for everyday commerce in virtually every country of the world except the United States, Liberia and Myanmar, and...
SI units may also be used directly in this formula.
See also  In physics, circular motion is movement with constant speed around in a circle: a circular path or a circular orbit. It is one of the simplest cases of accelerated motion. Circular motion involves acceleration of the moving object by a centripetal force which pulls the moving object towards the center...
Circular motion
 This article covers the physics of gravitation. See also gravity (disambiguation). Gravitation is the tendency of masses to move toward each other. The first mathematical formulation of the theory of gravitation was made by Sir Isaac Newton and proved astonishingly accurate. He postulated the force of universal gravitational attraction. Newton...
Gravity
 In mechanics, the twobody problem is a special case of the nbody problem that admits a closed form solution. The most commonly encountered version of the problem, involving an inverse square law force, is encountered in celestial mechanics and the Bohr model of the hydrogen atom. This problem...
Twobody problem
