+ Kepler's laws of planetary motion +
+ + +In astronomy, Kepler's laws give a description of the motion of planets around the Sun.
+Kepler's laws are:
+
+
 The orbit of every planet is an ellipse with the Sun at one of the two foci. +
 A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.^{[1]} +
 The square of the orbital period of a planet is directly proportional to the cube of the semimajor axis of its orbit. +
+
+
+
+Contents+ 
[edit] History
+Johannes Kepler published his first two laws in 1609, having found them by analyzing the astronomical observations of Tycho Brahe.^{[2]} Kepler discovered his third law many years later, and it was published in 1619.^{[2]} At the time, Kepler's laws were radical claims; the prevailing belief (particularly in epicyclebased theories) was that orbits should be based on perfect circles. Most of the planetary orbits can be rather closely approximated as circles, so it is not immediately evident that the orbits are ellipses. Detailed calculations for the orbit of the planet Mars first indicated to Kepler its elliptical shape, and he inferred that other heavenly bodies, including those farther away from the Sun, have elliptical orbits too. Kepler's laws and his analysis of the observations on which they were based, the assertion that the Earth orbited the Sun, proof that the planets' speeds varied, and use of elliptical orbits rather than circular orbits with epicycles—challenged the longaccepted geocentric models of Aristotle and Ptolemy, and generally supported the heliocentric theory of Nicolaus Copernicus (although Kepler's ellipses likewise did away with Copernicus's circular orbits and epicycles).^{[2]}
+Some eight decades later, Isaac Newton proved that relationships like Kepler's would apply exactly under certain ideal conditions that are to a good approximation fulfilled in the solar system, as consequences of Newton's own laws of motion and law of universal gravitation.^{[3]}^{[4]} Because of the nonzero planetary masses and resulting perturbations, Kepler's laws apply only approximately and not exactly to the motions in the solar system.^{[3]}^{[5]} Voltaire's Eléments de la philosophie de Newton (Elements of Newton's Philosophy) was in 1738 the first publication to call Kepler's Laws "laws".^{[6]} Together with Newton's mathematical theories, they are part of the foundation of modern astronomy and physics.^{[3]}
+[edit] First Law
+An ellipse is a particular class of mathematical shapes that resemble a stretched out circle (see the figure to the right). Note as well that the Sun is not at the center of the ellipse but is at one of the focal points. The other focal point is marked with a lighter dot but is a point that has no physical significance for the orbit. Ellipses have two focal points and the center of the ellipse is the midpoint of the line segment joining them. Circles are a special case of an ellipse that are not stretched out and in which both focal points coincide at the center.
+How stretched out that ellipse is from a perfect circle is known as its eccentricity; a parameter that can take any value greater than or equal to 0 (a simple circle) and smaller than 1 (when the eccentricity tends to 1, the ellipse tends to a parabola). The eccentricities of the planets known to Kepler varies from 0.007 (Venus) to 0.2 (Mercury). (See List of planetary objects in the Solar System for more detail.)
+After Kepler, though, bodies with highly eccentric orbits have been identified, among them many comets and asteroids. The dwarf planet Pluto was discovered as late as 1929, the delay mostly due to its small size, far distance, and optical faintness. Heavenly bodies such as comets with parabolic or even hyperbolic orbits are possible under the Newtonian theory and have been observed.^{[7]}
+ +Symbolically an ellipse can be represented in polar coordinates as:
+
+
+
where (r, θ) are the polar coordinates (from the focus) for the ellipse, p is the semilatus rectum, and ε is the eccentricity of the ellipse. For a planet orbiting the Sun then r is the distance from the Sun to the planet and θ is the angle with its vertex at the Sun from the location where the planet is closest to the Sun.
+At θ = 0°, perihelion, the distance is minimum
+
+
+
At θ = 90° and at θ = 270°, the distance is
+At θ = 180°, aphelion, the distance is maximum
+
+
+
The semimajor axis a is the arithmetic mean between r_{min} and r_{max}:
+
+
+
so
+
+
+
The semiminor axis b is the geometric mean between r_{min} and r_{max}:
+
+
+
so
+
+
+
The semilatus rectum p is the harmonic mean between r_{min} and r_{max}:
+
+
+
so
+
+
+
The eccentricity ε is the coefficient of variation between r_{min} and r_{max}:
+
+
+
The area of the ellipse is
+
+
+
The special case of a circle is ε = 0, resulting in r = p = r_{min} = r_{max} = a = b and A = π r^{2}.
+[edit] Second law
+ +
+
 "A line joining a planet and the Sun sweeps out equal areas during equal intervals of time."^{[1]} +
In a small time
+
+
+
the planet sweeps out a small triangle having base line
+
+
+
and height
+
+
+
The area of this triangle is
+
+
+
and so the constant areal velocity is
+
+
+
Now as the first law states that the planet follows an ellipse, the planet is at different distances from the Sun at different parts in its orbit. So the planet has to move faster when it is closer to the Sun so that it sweeps equal areas in equal times.
+The total area enclosed by the elliptical orbit is
+
+
 . +
Therefore the period
+
+
+
satisfies
+
+
+
or
+
+
+
where
+
+
+
is the angular velocity, (using Newton notation for differentiation), and
+
+
+
is the mean motion of the planet around the Sun.
+[edit] Third law
+
+
 "The square of the orbital period of a planet is directly proportional to the cube of the semimajor axis of its orbit." +
The third law, published by Kepler in 1619 [1] captures the relationship between the distance of planets from the Sun, and their orbital periods. For example, suppose planet A is 4 times as far from the Sun as planet B. Then planet A must traverse 4 times the distance of Planet B each orbit, and moreover it turns out that planet A travels at half the speed of planet B, in order to maintain equilibrium with the reduced gravitational centripetal force due to being 4 times further from the Sun. In total it takes 4×2=8 times as long for planet A to travel an orbit, in agreement with the law (8^{2}=4^{3}).
+This third law used to be known as the harmonic law,^{[8]} because Kepler enunciated it in a laborious attempt to determine what he viewed as the "music of the spheres" according to precise laws, and express it in terms of musical notation.^{[9]}
+This third law currently receives additional attention as it can be used to estimate the distance from an exoplanet to its central star, and help to decide if this distance is inside the habitable zone of that star.^{[10]}
+Symbolically:
+
+
+
where is the orbital period of the planet and is the semimajor axis of the orbit.
+Interestingly, the constant of proportion is theoretically same for both circular and elliptical orbits, and the constant is essentially the same for all planets (and other objects) orbiting the Sun.
+
+
+
So the constant is 1 (sidereal year)^{2}(astronomical unit)^{−3} or 2.97472505×10^{−19} s^{2}m^{−3}. See the actual figures: attributes of major planets.
+[edit] Generality
+Godefroy Wendelin, in 1643, noted that Kepler's third law applies to the four brightest moons of Jupiter. These laws approximately describe the motion of any two bodies in orbit around each other. (The statement in the first law about the focus becomes closer to exactitude as one of the masses becomes closer to zero mass. Where there are more than two masses, all of the statements in the laws become closer to exactitude as all except one of the masses become closer to zero mass and as the perturbations then also tend towards zero).^{[4]} The masses of the two bodies can be nearly equal, e.g. Charon—Pluto (~1:10), in a small proportion, e.g. Moon—Earth (~1:100), or in a great proportion, e.g. Mercury—Sun (~1:10,000,000).
+In all cases of twobody motion, rotation is about the barycenter of the two bodies, with neither one having its center of mass exactly at one focus of an ellipse. However, both orbits are ellipses with one focus at the barycenter. When the ratio of masses is large, the barycenter may be deep within the larger object, close to its center of mass. In such a case it may require sophisticated precision measurements to detect the separation of the barycenter from the center of mass of the larger object. But in the case of the planets orbiting the Sun, the largest of them are in mass as much as 1/1047.3486 (Jupiter) and 1/3497.898 (Saturn) of the solar mass,^{[11]} and so it has long been known that the solar system barycenter can sometimes be outside the body of the Sun, up to about a solar diameter from its center.^{[12]} Thus Kepler's first law, though not far off as an approximation, does not quite accurately describe the orbits of the planets around the Sun under classical physics.
+[edit] Zero eccentricity
+Kepler's laws refine the model of Copernicus. If the eccentricity of a planetary orbit is zero, then Kepler's laws state:
+
+
 The planetary orbit is a circle +
 The Sun is in the center +
 The speed of the planet in the orbit is constant +
 The square of the sidereal period is proportionate to the cube of the distance from the Sun. +
Actually the eccentricities of the orbits of the six planets known to Copernicus and Kepler are quite small, so this gives excellent approximations to the planetary motions, but Kepler's laws give even better fit to the observations.
+Kepler's corrections to the Copernican model are not at all obvious:
+
+
 The planetary orbit is not a circle, but an ellipse +
 The Sun is not at the center but at a focal point +
 Neither the linear speed nor the angular speed of the planet in the orbit is constant, but the area speed is constant. +
 The square of the sidereal period is proportionate to the cube of the mean between the maximum and minimum distances from the Sun. +
The nonzero eccentricity of the orbit of the earth makes the time from the March equinox to the September equinox, around 186 days, unequal to the time from the September equinox to the March equinox, around 179 days. The equator cuts the orbit into two parts having areas in the proportion 186 to 179, while a diameter cuts the orbit into equal parts. So the eccentricity of the orbit of the Earth is approximately
+
+
+
close to the correct value (0.016710219). (See Earth's orbit). The calculation is correct when the perihelion, the date that the Earth is closest to the Sun, is on a solstice. The current perihelion, near January 4, is fairly close to the solstice on December 21 or 22.
+[edit] Relation to Newton's laws
+Isaac Newton computed in his Philosophiæ Naturalis Principia Mathematica the acceleration of a planet moving according to Kepler's first and second law.
+
+
 The direction of the acceleration is towards the Sun. +
 The magnitude of the acceleration is in inverse proportion to the square of the distance from the Sun. +
This suggests that the Sun may be the physical cause of the acceleration of planets.
+Newton defined the force on a planet to be the product of its mass and the acceleration. (See Newton's laws of motion). So:
+
+
 Every planet is attracted towards the Sun. +
 The force on a planet is in direct proportion to the mass of the planet and in inverse proportion to the square of the distance from the Sun. +
Here the Sun plays an unsymmetrical part which is unjustified. So he assumed Newton's law of universal gravitation:
+
+
 All bodies in the solar system attract one another. +
 The force between two bodies is in direct proportion to the product of their masses and in inverse proportion to the square of the distance between them. +
As the planets have small masses compared to that of the Sun, the orbits conform to Kepler's laws approximately. Newton's model improves Kepler's model and gives better fit to the observations. See twobody problem.
+A deviation of the motion of a planet from Kepler's laws due to attraction from other planets is called a perturbation.
+[edit] Computing position as a function of time
+Kepler used his two first laws for computing the position of a planet as a function of time. His method involves the solution of a transcendental equation called Kepler's equation.
+The procedure for calculating the heliocentric polar coordinates (r,θ) to a planetary position as a function of the time t since perihelion, and the orbital period P, is the following four steps.
+
+
 1. Compute the mean anomaly M from the formula
+

+
+
+  2. Compute the eccentric anomaly E by solving Kepler's equation:
+

+
+
+  3. Compute the true anomaly θ by the equation:
+

+
+
+  4. Compute the heliocentric distance r from the first law:
+

+
+
+
The important special case of circular orbit, ε = 0, gives simply θ = E = M. Because the uniform circular motion was considered to be normal, a deviation from this motion was considered an anomaly.
+The proof of this procedure is shown below.
+[edit] Mean anomaly, M
+ +The Keplerian problem assumes an elliptical orbit and the four points:
+
+
 s the Sun (at one focus of ellipse); +
 z the perihelion +
 c the center of the ellipse +
 p the planet +
and
+
+
 distance between center and perihelion, the semimajor axis, +
 the eccentricity, +
 the semiminor axis, +
 the distance between Sun and planet. +
 the direction to the planet as seen from the Sun, the true anomaly. +
The problem is to compute the polar coordinates (r,θ) of the planet from the time since perihelion, t.
+It is solved in steps. Kepler considered the circle with the major axis as a diameter, and
+
+
 the projection of the planet to the auxiliary circle +
 the point on the circle such that the sector areas zcy and zsx are equal, +
 the mean anomaly. +
The sector areas are related by
+The circular sector area
+The area swept since perihelion,
+
+
+
is by Kepler's second law proportional to time since perihelion. So the mean anomaly, M, is proportional to time since perihelion, t.
+
+
+
where P is the orbital period.
+[edit] Eccentric anomaly, E
+When the mean anomaly M is computed, the goal is to compute the true anomaly θ. The function θ=f(M) is, however, not elementary. [2]. Kepler's solution is to use
+
+
 , x as seen from the centre, the eccentric anomaly +
as an intermediate variable, and first compute E as a function of M by solving Kepler's equation below, and then compute the true anomaly θ from the eccentric anomaly E. Here are the details.
+
+
+

+
+
Division by a^{2}/2 gives Kepler's equation
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+
+
This equation gives M as a function of E. Determining E for a given M is the inverse problem. Iterative numerical algorithms are commonly used.
+Having computed the eccentric anomaly E, the next step is to calculate the true anomaly θ.
+[edit] True anomaly, θ
+Note from the figure that
+
+
+
so that
+
+
+
Dividing by and inserting from Kepler's first law
+
+
+
to get
+
+
 +
The result is a usable relationship between the eccentric anomaly E and the true anomaly θ.
+A computationally more convenient form follows by substituting into the trigonometric identity:
+
+
+
Get
+
+
 +
Multiplying by (1+ε)/(1−ε) and taking the square root gives the result
+
+
+
We have now completed the third step in the connection between time and position in the orbit.
+[edit] Distance, r
+The fourth step is to compute the heliocentric distance r from the true anomaly θ by Kepler's first law:
+
+
+
[edit] Computing the planetary acceleration
+In his Principia Mathematica Philosophiae Naturalis, Newton showed that Kepler's laws imply that the acceleration of the planets are directed towards the sun and depend on the distance from the sun by the inverse square law. However, the geometrical method used by Newton to prove the result is quite complicated. The demonstration below is based on calculus.^{[13]}
+[edit] Acceleration vector
+ +From the heliocentric point of view consider the vector to the planet where is the distance to the planet and the direction is a unit vector. When the planet moves the direction vector changes:
+
+
+
where is the unit vector orthogonal to and pointing in the direction of rotation, and is the polar angle, and where a dot on top of the variable signifies differentiation with respect to time.
+So differentiating the position vector twice to obtain the velocity and the acceleration vectors:
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+
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+
So
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+
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where the radial acceleration is
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and the tangential acceleration is
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[edit] The inverse square law
+Kepler's second law implies that the areal velocity is a constant of motion. The tangential acceleration is zero by Kepler's second law:
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+
+
So the acceleration of a planet obeying Kepler's second law is directed exactly towards the sun.
+Kepler's first law implies that the area enclosed by the orbit is , where is the semimajor axis and is the semiminor axis of the ellipse. Therefore the period satisfies or
+
+
+
where
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+
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is the mean motion of the planet around the sun.
+The radial acceleration is
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+
Kepler's first law states that the orbit is described by the equation:
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Differentiating with respect to time
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or
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Differentiating once more
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The radial acceleration satisfies
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Substituting the equation of the ellipse gives
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The relation gives the simple final result
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This means that the acceleration vector of any planet obeying Kepler's first and second law satisfies the inverse square law
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where
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is a constant, and is the unit vector pointing from the Sun towards the planet, and is the distance between the planet and the Sun.
+According to Kepler's third law, has the same value for all the planets. So the inverse square law for planetary accelerations applies throughout the entire solar system.
+The inverse square law is a differential equation. The solutions to this differential equation includes the Keplerian motions, as shown, but they also include motions where the orbit is a hyperbola or parabola or a straight line. See kepler orbit.
+[edit] Newton's law of gravitation
+By Newton's second law, the gravitational force that acts on the planet is:
+
+
+
where only depends on the property of the Sun. According to Newton's third Law, the Sun is also attracted by the planet with a force of the same magnitude. Now that the force is proportional to the mass of the planet, under the symmetric consideration, it should also be proportional to the mass of the Sun. So the form of the gravitational force should be
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+
+
where is a universal constant. This is Newton's law of universal gravitation.
+The acceleration of solar system body no i is, according to Newton's laws:
+
+
+
where is the mass of body no j, and is the distance between body i and body j, and is the unit vector from body i pointing towards body j, and the vector summation is over all bodies in the world, besides no i itself. In the special case where there are only two bodies in the world, Planet and Sun, the acceleration becomes
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+
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which is the acceleration of the Kepler motion.
+[edit] See also
+
+
 Kepler orbit +
 Kepler problem +
 Kepler's equation +
 Circular motion +
 Gravity +
 Twobody problem +
 Freefall time +
 Laplace–Runge–Lenz vector +
[edit] Notes
+
+
 ^ ^{a} ^{b} Bryant, Jeff; Pavlyk, Oleksandr. "Kepler's Second Law", Wolfram Demonstrations Project. Retrieved December 27, 2009. +
 ^ ^{a} ^{b} ^{c} Holton, Gerald James; Brush, Stephen G. (2001). Physics, the Human Adventure: From Copernicus to Einstein and Beyond (3rd paperback ed.). Piscataway, NJ: Rutgers University Press. pp. 40–41. ISBN 0813529085. http://books.google.com/?id=czaGZzR0XOUC&pg=PA40. Retrieved December 27, 2009. +
 ^ ^{a} ^{b} ^{c} See also G E Smith, "Newton's Philosophiae Naturalis Principia Mathematica", especially the section Historical context ... in The Stanford Encyclopedia of Philosophy (Winter 2008 Edition), Edward N. Zalta (ed.). +
 ^ ^{a} ^{b} Newton's showing, in the 'Principia', that the twobody problem with centripetal forces results in motion in one of the conic sections, is concluded at Book 1, Proposition 13, Corollary 1. His consideration of the effects of perturbations in a multibody situation starts at Book 1, Proposition 65, including a limit argument that the error in the (Keplerian) approximation of ellipses and equal areas would tend to zero if the relevant planetary masses would tend to zero and with them the planetary mutual perturbations (Proposition 65, Case 1). He discusses the extent of the perturbations in the real solar system in Book 3, Proposition 13. +
 ^ Kepler "for the first time revealed" a "real approximation to the true kinematical relations [motions] of the solar system", see page 1 in H C Plummer (1918), An introductory treatise on dynamical astronomy, Cambridge, 1918. +
 ^ Wilson, Curtis (May 1994). "Kepler's Laws, SoCalled". HAD News (Washington, DC: Historical Astronomy Division, American Astronomical Society) (31): 1–2. http://had.aas.org/hadnews/HADN31.pdf. Retrieved December 27, 2009. +
 ^ Dunbar, Brian (2008). SECCHI Makes a Fantastic Recovery!. NASA. http://erc.ivv.nasa.gov/mission_pages/stereo/news/SECCHI_P2003.html +
 ^ Gerald James Holton, Stephen G. Brush (2001). Physics, the Human Adventure. Rutgers University Press. p. 45. ISBN 0813529085. http://books.google.com/?id=czaGZzR0XOUC&pg=PA45&dq=Kepler+%22harmonic+law%22. +
 ^ Burtt, Edwin. The Metaphysical Foundations of Modern Physical Science. p. 52. +
 ^ http://www.astro.lsa.umich.edu/undergrad/Labs/extrasolar_planets/pn_intro.html +
 ^ Astronomical Almanac for 2008, page K7. +
 ^ The fact was already stated by Newton ('Principia', Book 3, Proposition 12). +
 ^ HaiChau Chang, WuYi Hsiang. "The epic journey from Kepler's laws to Newton's law of universal gravitation revisited". arXiv:0801.0308. +
[edit] References
+
+
 Kepler's life is summarized on pages 523–627 and Book Five of his magnum opus, Harmonice Mundi (harmonies of the world), is reprinted on pages 635–732 of On the Shoulders of Giants: The Great Works of Physics and Astronomy (works by Copernicus, Kepler, Galileo, Newton, and Einstein). Stephen Hawking, ed. 2002 ISBN 0762413484 +

+
 A derivation of Kepler's third law of planetary motion is a standard topic in engineering mechanics classes. See, for example, pages 161–164 of Meriam, J. L. (1966, 1971). Dynamics, 2nd ed. New York: John Wiley. ISBN 0471596019. +

+
 Murray and Dermott, Solar System Dynamics, Cambridge University Press 1999, ISBN 0521575974 +
 V.I. Arnold, Mathematical Methods of Classical Mechanics, Chapter 2. Springer 1989, ISBN 0387968903 +
[edit] External links
+
+
 B.Surendranath Reddy; animation of Kepler's laws: applet +
 Crowell, Benjamin, Conservation Laws, http://www.lightandmatter.com/area1book2.html, an online book that gives a proof of the first law without the use of calculus. (see section 5.2, p. 112) +
 David McNamara and Gianfranco Vidali, Kepler's Second Law  Java Interactive Tutorial, http://www.phy.syr.edu/courses/java/mc_html/kepler.html, an interactive Java applet that aids in the understanding of Kepler's Second Law. +
 Audio  Cain/Gay (2010) Astronomy Cast Johannes Kepler and His Laws of Planetary Motion +
 University of Tennessee's Dept. Physics & Astronomy: Astronomy 161 page on Johannes Kepler: The Laws of Planetary Motion [3] +
 Equant compared to Kepler: interactive model [4] +
 Kepler's Third Law:interactive model [5] +
 Solar System Simulator (Interactive Applet) +
 Kepler and His Laws, educational web pages by David P. Stern +
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