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Encyclopedia > Kepler's Laws of Planetary Motion
Illustration of Kepler's three laws with two planetary orbits. (1) The orbits are ellipses, with focal points f1 and f2 for the first planet and f1 and f3 for the second planet. The sun is placed in focal point f1. (2) The two shaded sectors A1 and A2 have the same surface area and the time for planet 1 to cover segment A1 is equal to the time to cover segment A2. (3) The total orbit times for planet 1 and planet 2 have a ratio a13 / 2:a23 / 2.
Illustration of Kepler's three laws with two planetary orbits. (1) The orbits are ellipses, with focal points f1 and f2 for the first planet and f1 and f3 for the second planet. The sun is placed in focal point f1. (2) The two shaded sectors A1 and A2 have the same surface area and the time for planet 1 to cover segment A1 is equal to the time to cover segment A2. (3) The total orbit times for planet 1 and planet 2 have a ratio a13 / 2:a23 / 2.

In astronomy, Kepler's Laws of Planetary Motion are three mathematical laws that describe the motion of planets in the Solar System. German mathematician and astronomer Johannes Kepler (15711630) discovered them. Image File history File links Question_book-3. ... For other uses, see Astronomy (disambiguation). ... This article is about the astronomical term. ... This article is about the Solar System. ... Leonhard Euler, considered one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is the field of mathematics. ... Galileo is often referred to as the Father of Modern Astronomy. ... Kepler redirects here. ... Events January 11 - Austrian nobility is granted Freedom of religion. ... Events February 22 - Native American Quadequine introduces Popcorn to English colonists. ...


Kepler studied the observations of the legendarily precise Danish astronomer Tycho Brahe. Around 1605, Kepler found that Brahe's observations of the planets' positions followed three relatively simple mathematical laws. For other uses, see Observation (disambiguation). ... This article is about the astronomer. ...


Kepler's laws challenged Aristotelean and Ptolemaic astronomy and physics. His assertion that the Earth moved, his use of ellipses rather than epicycles, and his proof that the planets' speeds varied, changed astronomy and physics. Nevertheless, the physical explanation of the planets' behavior came almost a century later, when Isaac Newton was able to deduce Kepler's laws from Newton's own laws of motion and his law of universal gravitation, using his invention of calculus. Other models of gravitation would give empirically false results. In the Ptolemaic system of astronomy, the epicycle (literally: on the cycle in Greek) was a geometric model to explain the variations in speed and direction of the apparent motion of the Moon, Sun, and planets. ... For other uses, see Astronomy (disambiguation). ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... Isaac Newtons theory of universal gravitation (part of classical mechanics) states the following: Every single point mass attracts every other point mass by a force pointing along the line combining the two. ... For other uses, see Calculus (disambiguation). ...


Kepler's three laws are:

  1. The orbit of every planet is an ellipse with the sun at one of the foci. An ellipse is characterized by its two focal points; see illustration. Thus, Kepler rejected the ancient Aristotelean, Ptolemaic,and Copernican belief in circular motion.
  2. A line joining a planet and the sun sweeps out equal areas during equal intervals of time as the planet travels along its orbit. This means that the planet travels faster while close to the sun and slows down when it is farther from the sun. With his law, Kepler destroyed the Aristotelean astronomical theory that planets have uniform velocity.
  3. The squares of the orbital periods of planets are directly proportional to the cubes of the semi-major axes (the "half-length" of the ellipse) of their orbits. This means not only that larger orbits have longer periods, but also that the speed of a planet in a larger orbit is lower than in a smaller orbit.

Kepler's laws are formulated below, and are derived from Newton's laws, using heliocentric polar coordinates  (r,theta). However, Kepler's laws can alternatively be formulated and derived using Cartesian coordinates.[1] Two bodies with a slight difference in mass orbiting around a common barycenter. ... This article is about the astronomical term. ... For other uses, see Ellipse (disambiguation). ... Line redirects here. ... This article is about velocity in physics. ... In algebra, the square of a number is that number multiplied by itself. ... The orbital period is the time it takes a planet (or another object) to make one full orbit. ... This article is about proportionality, the mathematical relation. ... 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. ... In geometry, the semi-major axis (also semimajor axis) a applies to ellipses and hyperbolas. ... Heliocentric Solar System Heliocentrism (lower panel) in comparison to the geocentric model (upper panel) In astronomy, heliocentrism is the theory that the sun is at the center of the Universe and/or the Solar System. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

Contents

Mathematical description

First law

Kepler's first law
Kepler's first law

The first law says: "The orbit of every planet is an ellipse with the sun at one of the foci." Image File history File links Kepler-first-law. ... Image File history File links Kepler-first-law. ... Two bodies with a slight difference in mass orbiting around a common barycenter. ... This article is about the astronomical term. ... For other uses, see Ellipse (disambiguation). ... In geometry, the focus (pl. ...


The mathematics of the ellipse is as follows.


The equation is

r=frac{p}{1+epsiloncdotcostheta}

where (r,θ) are heliocentric polar coordinates for the planet, p is the semi-latus rectum, and ε is the eccentricity, which is greater than or equal to zero, and less than one. In mathematics, the latus rectum of a conic section is the chord parallel to the directrix and passing through the single focus, or one of the two foci. ... (This page refers to eccentricity in mathematics. ...


For θ=0 the planet is at the perihelion at minimum distance: This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ...

r_mathrm{min}=frac{p}{1+epsilon}

for θ=90°: r=p, and for θ=180° the planet is at the aphelion at maximum distance: This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ...

r_mathrm{max}=frac{p}{1-epsilon}

The semi-major axis is the arithmetic mean between rmin and rmax: The semi-major axis of an ellipse In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ...

a=frac{p}{1-epsilon^2}

The semi-minor axis is the geometric mean between rmin and rmax: In geometry, the semi-minor axis (also semiminor axis) applies to ellipses and hyperbolas. ... The geometric mean of a collection of positive data is defined as the nth root of the product of all the members of the data set, where n is the number of members. ...

b=frac p{sqrt{1-epsilon^2}}

and it is also the geometric mean between the semimajor axis and the semi latus rectum: The geometric mean of a collection of positive data is defined as the nth root of the product of all the members of the data set, where n is the number of members. ...

frac a b=frac b p

Second law

Illustration of Kepler's second law.
Illustration of Kepler's second law.

The second law: "A line joining a planet and the sun sweeps out equal areas during equal intervals of time."[2] Line redirects here. ...


This is also known as the law of equal areas. It is a direct consequence of the law of conservation of angular momentum; see the derivation below. This gyroscope remains upright while spinning due to its angular momentum. ...


Suppose a planet takes one day to travel from point A to B. The lines from the Sun to A and B, together with the planet orbit, will define a (roughly triangular) area. This same amount of area will be formed every day regardless of where in its orbit the planet is. This means that the planet moves faster when it is closer to the sun. A spatial point is an entity with a location in space but no extent (volume, area or length). ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...


This is because the sun's gravity accelerates the planet as it falls toward the sun, and decelerates it on the way back out, but Kepler did not know that reason.


The two laws permitted Kepler to calculate the position, (r,θ), of the planet, based on the time since perihelion, t, and the orbital period, P. The calculation is done in four steps. This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms. ...

1. Compute the mean anomaly M from the formula
M=frac{2pi t}{P}
2. Compute the eccentric anomaly E by numerically solving Kepler's equation:
 M=E-epsiloncdotsin E
3. Compute the true anomaly θ by the equation:
tanfrac theta 2 = sqrt{frac{1+epsilon}{1-epsilon}}cdottanfrac E 2
4. Compute the heliocentric distance r from the first law:
r=frac p {1+epsiloncdotcostheta}

The proof of this procedure is shown below. 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 astronomy, the true anomaly (, also written ) is the angle between the direction z-s 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). ...


Third law

The third law : "The squares of the orbital periods of planets are directly proportional to the cubes of the semi-major axis of the orbits." Thus, not only does the length of the orbit increase with distance, the orbital speed decreases, so that the increase of the orbital period is more than proportional. In algebra, the square of a number is that number multiplied by itself. ... The orbital period is the time it takes a planet (or another object) to make one full orbit. ... This article is about proportionality, the mathematical relation. ... 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. ... The semi-major axis of an ellipse In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae. ... 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. ... The orbital period is the time it takes a planet (or another object) to make one full orbit. ...

P^2 propto a^3
P = orbital period of planet
a = semimajor axis of orbit

So the expression P2·a–3 has the same value for all planets in the Solar System as it has for Earth. When certain units are chosen, namely P is measured in sidereal years and a in astronomical units, P2·a–3 has the value 1 for all planets in the Solar System. This article is about the Solar System. ... This article is about Earth as a planet. ... The sidereal year is the time for the Sun to return to the same position in respect to the stars of the celestial sphere. ... The astronomical unit (AU or au or a. ...


In SI units: frac{P^{2}}{a^{3}} = 3.00times 10^{-19} frac{s^{2}}{m^{3}} pm  0.7%, . The International System of Units (symbol: SI) (for the French phrase Système International dUnités) is the most widely used system of units. ...


The law, when applied to circular orbits where the acceleration is proportional to a·P−2, shows that the acceleration is proportional to a·a−3 = a−2, in accordance with Newton's law of gravitation. In astrodynamics or celestial mechanics a circular orbit is an elliptic orbit with the eccentricity equal to 0. ... Acceleration is the time rate of change of velocity and/or direction, and at any point on a velocity-time graph, it is given by the slope of the tangent to the curve at that point. ... The law of universal gravitation states that gravitational force between masses decreases with the distance between them, according to an inverse-square law. ...


The general equation, which Kepler did not know, is

left({frac{P}{2pi}}right)^2 = {a^3 over G (M+m)},

where G is the gravitational constant, M is the mass of the sun, and m is the mass of planet. The latter appears in the equation since the equation of motion involves the reduced mass. Note that P is time per orbit and P/2π is time per radian. Reduced mass is an algebraic term of the form that simplifies an equation of the form The reduced mass is typically used as a relationship between two system elements in parallel, such as resistors; whether these be in the electrical, thermal, hydraulic, or mechanical domains. ... Two bodies with a slight difference in mass orbiting around a common barycenter. ... For the musical group, see Radian (band). ...


See the actual figures: attributes of major planets. These tables list the atributes of the solar systems largest objects (excluding the Sun). ...


This law is also known as the harmonic law.


Position as a function of time

The Keplerian problem assumes an elliptical orbit and the four points: Two bodies with similar mass orbiting around a common barycenter with elliptic orbits. ...

  • s the sun (at one focus of ellipse);
  • z the perihelion
  • c the center of the ellipse
  • p the planet

and

 a=|cz|, distance from center to perihelion, the semimajor axis,
 varepsilon={|cs|over a}, the eccentricity,
 b=asqrt{1-varepsilon^2}, the semiminor axis,
 r=|sp| , the distance from sun to planet.

and the angle

nu=angle zsp, the planet as seen from the sun, the true anomaly.

The problem is to compute the polar coordinates (r,ν) of the planet from the time since perihelion, t. In astronomy, the true anomaly (, also written ) is the angle between the direction z-s 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). ... This article describes some of the common coordinate systems that appear in elementary mathematics. ...


It is solved in steps. Kepler began by adding the orbit's auxiliary circle (that with the major axis as a diameter) and defined these points:

  • x is the projection of the planet to the auxiliary circle; then the area |zsx|=frac a b cdot|zsp|
  • y is a point on the auxiliary circle such that the area  |zcy|=|zsx|

and

M=angle zcy, y as seen from the centre, the mean anomaly.

The area of the circular sector  |zcy| = frac{a^2 M}2, and the area swept since perihelion, 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. ... A circular sector or circle sector also known as a pie piece is the portion of a circle enclosed by two radii and an arc. ...

|zsp|=frac b a cdot|zsx|=frac b a cdot|zcy|=frac b acdotfrac{a^2 M}2 = frac {a b M}{2} ,

is by Kepler's second law proportional to time since perihelion. So the mean anomaly, M, is proportional to time since perihelion, t.

M={2 pi t over T},

where T is the orbital period. The orbital period is the time it takes a planet (or another object) to make one full orbit. ...


The mean anomaly M is first computed. The goal is to compute the true anomaly ν. The function ν=f(M) is, however, not elementary. Kepler's solution is to use

E=angle zcx, 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 ν from the eccentric anomaly E. Here are the details. 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. ...

 |zcy|=|zsx|=|zcx|-|scx|
frac{a^2 M}2=frac{a^2 E}2-frac {avarepsiloncdot asin E}2

Division by a²/2 gives Kepler's equation

M=E-varepsiloncdotsin E.

The catch is that Kepler's equation cannot be rearranged to isolate E. The function E=f(M) is not an elementary formula. Kepler's equation is solved either iteratively by a root-finding algorithm or, as derived in the article on eccentric anomaly, by an infinite series A root-finding algorithm is a numerical method, or algorithm, for finding a value x such that f(x) = 0, for a given function f. ... 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 mathematics, a series is a sum of a sequence of terms. ...

Eapprox M+left(varepsilon-frac18varepsilon^3right)sin M+frac12varepsilon^2sin 2M+frac38varepsilon^3sin 3M+ cdots

For the small ε typical of the planets (except Pluto), such series are quite accurate with only a few terms. Atmospheric characteristics Atmospheric pressure 0. ...


Having computed the eccentric anomaly E from Kepler's equation, the next step is to calculate the true anomaly ν from the eccentric anomaly E.


Note from the geometry of the problem that

acdotcos E=acdotvarepsilon+rcdotcos nu.

Dividing by a and inserting from Kepler's first law

 frac r a =frac{1-varepsilon^2}{1+varepsiloncdotcos nu}

to get

cos E =varepsilon+frac{1-varepsilon^2}{1+varepsiloncdotcos nu}cdotcos nu =frac{varepsiloncdot(1+varepsiloncdotcos nu)+(1-varepsilon^2)cdotcos nu}{1+varepsiloncdotcos nu} =frac{varepsilon +cos nu}{1+varepsiloncdotcos nu}.

The result is a usable relationship between the eccentric anomaly E and the true anomaly ν.


A computationally more convenient form follows by substituting into the trigonometric identity: In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ...

tan^2frac{x}{2}=frac{1-cos x}{1+cos x}.

Get

tan^2frac{E}{2} =frac{1-cos E}{1+cos E} =frac{1-frac{varepsilon+cos nu}{1+varepsiloncdotcos nu}}{1+frac{varepsilon+cos nu}{1+varepsiloncdotcos nu}} =frac{(1+varepsiloncdotcos nu)-(varepsilon+cos nu)}{(1+varepsiloncdotcos nu)+(varepsilon+cos nu)} =frac{1-varepsilon}{1+varepsilon}cdotfrac{1-cos nu}{1+cos nu}=frac{1-varepsilon}{1+varepsilon}cdottan^2frac{nu}{2}.

Multiplying by (1+ε)/(1−ε) and taking the square root gives the result

tanfrac nu2=sqrtfrac{1+varepsilon}{1-varepsilon}cdottanfrac E2.

We have now completed the third step in the connection between time and position in the orbit.


One could even develop a series computing ν directly from M. [1]


The fourth step is to compute the heliocentric distance r from the true anomaly ν by Kepler's first law:

 r=acdotfrac{1-varepsilon^2}{1+varepsiloncdotcos nu}.

Derivation from Newton's laws

Kepler's laws are about the motion of the planets around the sun, while Newton's laws more generally are about the motion of point particles attracting each other by the force of gravitation. In the special case where there are only two particles, and one of them is much lighter than the other, and the distance between the particles remains limited, then the lighter particle moves around the heavy particle as a planet around the sun according to Kepler's laws, as shown below. Newton's laws however also admit other solutions, where the trajectory of the lighter particle is a parabola or a hyperbola. These solutions show that there is a limitation to the applicability of Kepler's first law, which states that the trajectory will always be an ellipse. In the case where one particle is not much lighter than the other, it turns out that each particle moves around their common center of mass, so that the general two body problem is reduced to the special case where one particle is much lighter than the other. While Kepler's laws are expressed either in geometrical language or as equations connecting the coordinates of the planet and the time variable with the orbital elements, Newton's second law is a differential equation. So the derivations below involve the art of solving differential equations. The second law is derived first, as the derivation of the first law depends on the derivation of the second law. Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ... Gravity redirects here. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: παραβολή) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... In mathematics, a hyperbola (Greek literally overshooting or excess) is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves of the cone. ... In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ... In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. ... The elements of an orbit are the parameters needed to specify that orbit uniquely, given a model of two ideal masses obeying the Newtonian laws of motion and the inverse-square law of gravitational attraction. ... Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ...


Deriving Kepler's second law

Newton's law of gravitation says that "every object in the universe attracts every other object along a line of the centers of the objects, proportional to each object's mass, and inversely proportional to the square of the distance between the objects," and his second law of motion says that "the mass times the acceleration is equal to the force." So the mass of the planet times the acceleration vector of the planet equals the mass of the sun times the mass of the planet, divided by the square of the distance, times minus the radial unit vector, times a constant of proportionality. This is written: In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ...

 mcdotddotmathbf{r} = frac{Mcdot m}{r^2}cdot(-hat{mathbf{r}})cdot G

where a dot on top of the variable signifies differentiation with respect to time, and the second dot indicates the second derivative.


Assume that the planet is so much lighter than the sun that the acceleration of the sun can be neglected.

 dothat{mathbf{r}} = dottheta hat{boldsymboltheta}

where  hat{boldsymboltheta} is the tangential unit vector, and

 dothat{boldsymboltheta} = -dottheta hat{mathbf{r}}.

So the position vector

mathbf{r} = r hat{mathbf{r}}

is differentiated twice to give the velocity vector and the acceleration vector

dotmathbf{r} =dot r hatmathbf{r} + r dothatmathbf{r} =dot r hat{mathbf{r}} + r dottheta hat{boldsymboltheta},
ddotmathbf{r} = (ddot r hat{mathbf{r}} +dot r dothat{mathbf{r}} ) + (dot rdottheta hat{boldsymboltheta} + rddottheta hat{boldsymboltheta} + rdottheta dothat{boldsymboltheta}) = (ddot r - rdottheta^2) hat{mathbf{r}} + (rddottheta + 2dot r dottheta) hat{boldsymboltheta}.

Note that for constant distance,  r, the planet is subject to the centripetal acceleration, rdottheta^2, and for constant angular speed, dottheta, the planet is subject to the coriolis acceleration, 2dot r dottheta. A centripetal force is a force pulling an object toward the center of a circular path as the object goes around the circle. ... In the inertial frame of reference (upper part of the picture), the black object moves in a straight line. ...


Inserting the acceleration vector into Newton's laws, and dividing by m, gives the vector equation of motion It has been suggested that SUVAT equations be merged into this article or section. ...

 (ddot r - rdottheta^2) hat{mathbf{r}} + (rddottheta + 2dot r dottheta) hat{boldsymboltheta}= -GMr^{-2}hat{mathbf{r}}

Equating component, we get the two ordinary differential equations of motion, one for the radial acceleration and one for the tangential acceleration: In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ...

ddot r - rdottheta^2 = -GMr^{-2},
rddottheta + 2dot rdottheta = 0.

In order to derive Kepler's second law only the tangential acceleration equation is needed. Divide it by  r dottheta:

frac{ddottheta}{dottheta} +2frac{dot r}{r}=0

and integrate:

logdottheta +2log r = logell,

where logell is a constant of integration, and exponentiate: In calculus, the indefinite integral of a given function (i. ...

 r^2dot theta =ell .

This says that the specific angular momentum  r^2 dot theta is a constant of motion, even if both the distance  r and the angular speed dottheta vary. In astrodynamics specific relative angular momentum () of orbiting body () relative to central body () is the relative angular momentum of per unit mass. ... In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. ... Angular frequency is a measure of how fast an object is rotating In physics (specifically mechanics and electrical engineering), angular frequency ω (also called angular speed) is a scalar measure of rotation rate. ...


The area swept out from time t1 to time t2,

 int_{t_1}^{t_2}frac 1 2 cdot basecdot heightcdot dt = int_{t_1}^{t_2}frac 1 2 cdot rcdot rdot thetacdot dt=frac 1 2 cdotell cdot(t_2-t_1)

depends only on the duration t2t1. This is Kepler's second law.


Deriving Kepler's first law

The expression

p=ell ^2 G^{-1}M^{-1}

has the dimension of length and is used to make the equations of motion dimensionless. We define

 u =pr^{-1}

and get

-GMr^{-2}=-ell^2 p^{-3}u^{2}

and

 dot theta =ell r^{-2}=ell p^{-2}u^2.

Differentiation with respect to time is transformed into differentiation with respect to angle:

 dot X=frac {dX}{d theta}cdot dottheta=frac {dX}{d theta}cdotell p^{-2}u^2.

Differentiate

 r =pu^{-1}

twice:

dot r = frac{d(pu^{-1})}{dtheta}cdotell p^{-2}u^{2} = -pu^{-2}frac{du}{dtheta}cdotell p^{-2}u^{2}= -ell p^{-1}frac{du}{dtheta}
ddot r = frac{ddot r}{dtheta}cdotell p^{-2}u^{2}= frac{d}{dtheta}(-ell p^{-1}frac{du}{dtheta})cdotell p^{-2}u^{2}= -ell^2 p^{-3}u^{2}frac{d^2 u}{dtheta^2}

Substitute into the radial equation of motion

ddot r - rdottheta^2 = -GMr^{-2}

and get

(-ell^2 p^{-3}u^2frac{d^2u}{dtheta^2}) - (pu^{-1})(ell p^{-2}u^2)^2 = -ell ^2 p^{-3} u^2

Divide by -ell^2 p^{-3}u^2 to get a simple non-homogeneous linear differential equation for the orbit of the planet: In mathematics, a linear differential equation is a differential equation of the form Ly = f, where the differential operator L is a linear operator, y is the unknown function, and the right hand side f is a given function. ...

frac{d^2u}{dtheta^2} + u = 1 .

An obvious solution to this equation is the circular orbit

 u = 1.

Other solutions are obtained by adding solutions to the homogeneous linear differential equation with constant coefficients In mathematics, a linear differential equation is a differential equation of the form Ly = f, where the differential operator L is a linear operator, y is the unknown function, and the right hand side f is a given function. ...

frac{d^2u}{dtheta^2} + u = 0

These solutions are

 u = epsiloncdotcos(theta-A)

where  epsilon and  A are arbitrary constants of integration. So the result is

 u = 1+ epsiloncdotcos(theta-A)

Choosing the axis of the coordinate system such that  A=0, and inserting  u=pr^{-1}, gives: In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ...

 pr^{-1 } = 1+ epsiloncdotcostheta .

If  epsilon<1 , this is Kepler's first law.


Kepler's third law

Newton used the third law as one of the pieces of evidence used to build the conceptual and mathematical framework of his Law of Gravitation. If we take Newton's laws of motion as given, and consider a hypothetical planet that happens to be in a perfectly circular orbit of radius r, then we have F = mv2 / r for the sun's force on the planet. The velocity is proportional to r/T, which by Kepler's third law varies as one over the square root of r. Substituting this into the equation for the force, we find that the gravitational force is proportional to one over r squared. Newton's actual historical chain of reasoning is not known with certainty, because in his writing he tended to erase any traces of how he had reached his conclusions. Reversing the direction of reasoning, we can consider this as a proof of Kepler's third law based on Newton's law of gravity, and taking care of the proportionality factors that were neglected in the argument above, we have: An artists rendering of a hypothetical exoplanet. ...

T^2 = frac{4pi^2}{GM} cdot r^3

where:

The same arguments can be applied to any object orbiting any other object. This discussion implicitly assumed that the planet orbits around the stationary sun, although in reality both the planet and the sun revolve around their common center of mass. Newton recognized this, and modified this third law, noting that the period is also affected by the orbiting body's mass. However typically the central body is so much more massive that the orbiting body's mass may be ignored. Newton also proved that in the case of an elliptical orbit, the semimajor axis could be substituted for the radius. The most general result is: The orbital period is the time it takes a planet (or another object) to make one full orbit. ... 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. ... For other uses, see Mass (disambiguation). ... For other uses, see Mass (disambiguation). ... For other uses, see Ellipse (disambiguation). ...

T^2 = frac{4pi^2}{G(M + m)} cdot a^3

where:

For objects orbiting the sun, it can be convenient to use units of years, AU, and solar masses, so that G, 4π² and the various conversion factors cancel out. Also with m<<M we can set m+M = M, so we have simply T2 = a3. Note that the values of G and planetary masses are not known with good accuracy; however, the products GM (the Keplerian attraction) are known to extremely high precision. The orbital period is the time it takes a planet (or another object) to make one full orbit. ... For other uses, see Ellipse (disambiguation). ... 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. ... For other uses, see Mass (disambiguation). ... For other uses, see Mass (disambiguation). ... In astronomy, the solar mass is a unit of mass used to express the mass of stars and larger objects such as galaxies. ... Conversion of units refers to conversion factors between different units of measurement for the same quantity. ...


Define point A to be the periapsis, and point B as the apoapsis of the planet when orbiting the sun. A diagram of Keplerian orbital elements. ... A diagram of Keplerian orbital elements. ...


Kepler's second law states that the orbiting body will sweep out equal areas in equal quantities of time. If we now look at a very small periods of time at the moments when the planet is at points A and B, then we can approximate the area swept out as a triangle with an altitude equal to the distance between the planet and the sun, and the base equal to the time times the speed of the planet. In astrodynamics, an orbiting body () is a body that orbits central body (). Under standard assumptions in astrodynamics: it is orders of magnitude lighter than central body (i. ...

begin{matrix}frac{1}{2}end{matrix} cdot(1-epsilon)acdot V_A,dt= begin{matrix}frac{1}{2}end{matrix} cdot(1+epsilon)acdot V_B,dt
(1-epsilon)cdot V_A=(1+epsilon)cdot V_B
V_A=V_Bcdotfrac{1+epsilon}{1-epsilon}

Using the law of conservation of energy for the total energy of the planet at points A and B, Look up conservation of energy in Wiktionary, the free dictionary. ... In classical physics, the total energy of an object is the sum of its potential energy and its kinetic energy. ...

frac{mV_A^2}{2}-frac{GmM}{(1-epsilon)a} =frac{mV_B^2}{2}-frac{GmM}{(1+epsilon)a}
frac{V_A^2}{2}-frac{V_B^2}{2} =frac{GM}{(1-epsilon)a}-frac{GM}{(1+epsilon)a}
frac{V_A^2-V_B^2}{2}=frac{GM}{a}cdot left ( frac{1}{(1-epsilon)}-frac{1}{(1+epsilon)} right )
frac{left ( V_Bcdotfrac{1+epsilon}{1-epsilon}right ) ^2-V_B^2}{2}=frac{GM}{a}cdot left ( frac{1+epsilon-1+epsilon}{(1-epsilon)(1+epsilon)} right )
V_B^2 cdot left ( frac{1+epsilon}{1-epsilon}right ) ^2-V_B^2=frac{2GM}{a}cdot left ( frac{2epsilon}{(1-epsilon)(1+epsilon)} right )
V_B^2 cdot left ( frac{(1+epsilon)^2-(1-epsilon)^2}{(1-epsilon)^2}right )=frac{4GMepsilon}{acdot(1-epsilon)(1+epsilon)}
V_B^2 cdot left ( frac{1+2epsilon+epsilon^2-1+2epsilon-epsilon^2}{(1-epsilon)^2} right) =frac{4GMepsilon}{acdot(1-epsilon)(1+epsilon)}
V_B^2 cdot 4epsilon =frac{4GMepsiloncdot (1-epsilon)^2}{acdot(1-epsilon)(1+epsilon)}

Now that we have VB, we can find the rate at which the planet is sweeping out area in the ellipse. This rate remains constant, so we can derive it from any point we want, specifically from point B.

However, the total area of the ellipse is equal to . (That's the same as πab, because ). The time the planet take out to sweep out the entire area of the ellipse equals the ellipse's area, so,

However, if the mass m is not negligible in relation to M, then the planet will orbit the sun with the exact same velocity and position as a very small body orbiting an object of mass M + m (see reduced mass). To integrate that in the above formula, M must be replaced with M + m, to give Reduced mass is an algebraic term of the form that simplifies an equation of the form The reduced mass is typically used as a relationship between two system elements in parallel, such as resistors; whether these be in the electrical, thermal, hydraulic, or mechanical domains. ...

Q.E.D. This article is about Latin phrase Q.E.D., as used in proofs. ...


References

  1. ^ Hyman, Andrew. "A Simple Cartesian Treatment of Planetary Motion", European Journal of Physics, Vol. 14, pp. 145-147 (1993).
  2. ^ "Kepler's Second Law" by Jeff Bryant with Oleksandr Pavlyk, The Wolfram Demonstrations Project.

See also

In mathematics, the Kepler conjecture is a conjecture about sphere packing in three dimensional Euclidean space. ... In physics, circular motion is rotation along a circle: a circular path or a circular orbit. ... Gravity is a force of attraction that acts between bodies that have mass. ... In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. ... The free-fall time is the characteristic time it would take a body to collapse under its own gravitational attraction, if no other forces existed to oppose the collapse. ...

External links

  • 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.
  • University of Tennessee's Dept. Physics & Astronomy: Astronomy 161 page on Johannes Kepler: The Laws of Planetary Motion [2]
  • Equant compared to Kepler: interactive model [3]
  • Kepler's Third Law:interactive model[4]

In stellar dynamics a box orbit refers to a particular type of orbit which can be seen in triaxial systems, that is, systems which do not possess a symmetry around any of its axes. ... In astrodynamics or celestial mechanics a circular orbit is an elliptic orbit with the eccentricity equal to 0. ... A non-inclined orbit is an orbit which is contained in the plane of reference. ... Two bodies with similar mass orbiting around a common barycenter with elliptic orbits. ... Highly Elliptical Orbit (HEO) is an elliptic orbit characterized by a relatively low-altitude perigee and an extremely high-altitude apogee. ... A graveyard orbit is an orbit where spacecraft are intentionally placed at the end of their operational life. ... In astrodynamics or celestial mechanics a hyperbolic trajectory is an orbit with the eccentricity greater than 1. ... A geostationary orbit occurs when an object (satellite) is placed 37,000 km (22,300 miles) above the Earths equator with the characteristic that, from a fixed observation point on the Earths surface, it appears motionless. ... In Astronomy, and in particular in Astrodynamics, the osculating orbit of an object in space is the gravitational Keplerian orbit about a central body which best approximates the (more complex) motion of the object at a given instant in time. ... In astrodynamics or celestial mechanics a parabolic trajectory is an orbit with the eccentricity equal to 1. ... A capture orbit is the high-energy parabolic orbit that allows the capture other than crashing directly to the central bodys surface (or atmospheric re_entry). ... An escape orbit (also known as C3 = 0 orbit) is the high-energy parabolic orbit around the central body. ... Semi-Synchronous Orbit (SSO): An orbit with approximately a 12-hour period. ... Unsurprisingly similar to synchronous orbit, this orbit is at a slightly different distance from the Earth, resulting in the satellite drifting slowly eastward. ... A synchronous orbit is an orbit in which an orbiting body (usually a satellite) has a period equal to the average rotational period of the body being orbited (usually a planet), and in the same direction of rotation as that body. ... A parking orbit is a temporary orbit used during the launch of a satellite or other space probe. ... Geocentric orbit refers to the orbit of any object orbiting the Earth, such as the Moon or artificial satellites. ... It has been suggested that this article or section be merged with geostationary orbit. ... Geostationary orbit A geostationary orbit (GEO) is a geosynchronous orbit directly above the Earths equator (0° latitude), with orbital eccentricity of zero. ... By analogy with the geosynchronous orbit, a heliosynchronous orbit is a heliocentric orbit of radius 24. ... A low Earth orbit (LEO) is an orbit in which objects such as satellites are below intermediate circular orbit (ICO) and far below geostationary orbit, but typically around 350 - 1400 km above the Earths surface. ... Intermediate circular orbit (ICO), also called medium earth orbit (MEO), is used by satellites between the altitudes of low earth orbit (up to 1400 km) and geostationary orbit (ca. ... Molniya orbit is a class of a highly elliptic orbit with inclination of +/-63. ... A near equatorial orbit is an orbit that lies close to the equatorial plane of the object orbited. ... The orbit of the Moon around the Earth is completed in approximately 27. ... A polar orbit is an orbit in which a satellite passes above or nearly above both poles of the planet orbiting on each revolution. ... Tundra orbit is a class of a highly elliptic orbit with inclination of 63. ... Areosynchronous orbits are class of synchronous orbits for artificial satellites around the planet Mars. ... An areostationary orbit (abbreviated ASO) is a circular areo­synchronous orbit in the Martian equatorial plane 11,000 km above the surface, any point on which revolves about Mars in the same direction and with the same period as the Martian surface. ... A halo orbit is an orbit around a Lagrange point between two larger bodies. ... In orbital mechanics, a Lissajous orbit is a quasi-periodic orbital trajectory an object can follow around a colinear libration point of a two-body system without requiring any propulsion. ... In astronomy, lunar orbit refers just to the orbit of the Moon around the Earth. ... A heliocentric orbit is an orbit around the sun. ... By analogy with the geosynchronous orbit, a heliosynchronous orbit is a heliocentric orbit of radius 24. ... An orbital maneuver is a change from one orbit to another, accomplished by applying thrust. ... In astronautics and aerospace engineering, the Bi-elliptic transfer is an orbital maneuver that moves a spacecraft from one orbit to another and may, in certain situations require less delta-v than a Hohmann transfer. ... A geostationary transfer orbit (GTO) is a Hohmann transfer orbit around the Earth between a low Earth orbit (LEO) and a geostationary orbit (GEO). ... In orbital mechanics and aerospace engineering, a gravitational slingshot or gravity assist is the use of the gravity of a planet or other celestial body to alter the path and speed of a spacecraft. ... In astronautics and aerospace engineering, the Hohmann transfer orbit is an orbital maneuver that, under standard assumption, moves a spacecraft from one circular orbit to another using two engine impulses. ... Orbital inclination change is a orbital maneuver aimed at changing inclination of orbiting bodys orbit. ... In astrodynamics orbital phasing is the adjustment of the time-position of spacecraft along its orbit, usually described as adjusting the orbiting spacecrafts true anomaly. ... A space rendezvous between two spacecraft, often between a spacecraft and a space station, is an orbital maneuver where the two arrive at the same orbit, make the orbital velocities the same, and bring them together (an approach maneuver, taxiing maneuver); it may or may not include docking. ... A diagram of Keplerian orbital elements. ... In astronomy, a celestial coordinate system is a coordinate system for mapping positions in the sky. ... Delta-v budget (or velocity change budget) is a term used in astrodynamics and aerospace industry for velocity change (or delta-v) requirements for the various propulsive tasks and orbital maneuvers over phases of the space mission. ... In astronomy, an epoch is a moment in time for which celestial coordinates or orbital elements are specified. ... An ephemeris (plural: ephemerides) (from the Greek word ephemeros = daily) is a device giving the positions of astronomical objects in the sky. ... The equatorial coordinate system is probably the most widely used celestial coordinate system, whose equatorial coordinates are: declination () right ascension () -also RA-, or hour angle () -also HA- It is the most closely related to the geographic coordinate system, because they use the same fundamental plane, and the same poles. ... A gravity turn is a maneuver used in launching spacecraft into, and descending from, orbits around a celestial body such as a planet or a moon. ... Ground track of the International Space Station for approximately two periods. ... Artists concept of the Interplanetary Transport Network. ... A contour plot of the effective potential (the Hills Surfaces) of a two-body system (the Sun and Earth here), showing the five Lagrange points. ... This article is about the problem in classical mechanics. ... The Oberth effect is a feature of astronautics where using a rocket engine close to a gravitational body gives a higher final speed than the same burn executed further from the body. ... In astrodynamics an orbit equation defines the path of orbiting body around central body relative to , without specifying position as a function of time. ... In astrodynamics or celestial dynamics orbital state vectors (sometimes State Vectors) are vectors of position () and velocity () that together with their time (epoch) () uniquely determine the state of an orbiting body. ... Perturbation is a term used in astronomy to describe alterations to an objects orbit caused by gravitational interactions with other bodies. ... Direct motion is the motion of a planetary body in a direction similar to that of other bodies within its system, and is sometimes called prograde motion. ... In astrodynamics the specific orbital energy (or vis-viva energy) of an orbiting body traveling through space under standard assumptions is the sum of its potential energy () and kinetic energy () per unit mass. ... In astrodynamics specific relative angular momentum () of orbiting body () relative to central body () is the relative angular momentum of per unit mass. ... The following is a list of types of orbits: // Box orbit Circular orbit Ecliptic orbit Elliptic orbit Highly Elliptical Orbit Graveyard orbit Hohmann transfer orbit Hyperbolic trajectory Inclined orbit Osculating orbit Parabolic trajectory Capture orbit Escape orbit Semi-synchronous orbit Subsynchronous orbit Synchronous orbit Geocentric orbit Geosynchronous orbit Geostationary orbit...


  Results from FactBites:
 
The Galileo Project | Science | Johannes Kepler (1275 words)
Johannes Kepler was born in Weil der Stadt in Swabia, in southwest Germany.
Kepler remained in Graz until 1600, when all Protestants were forced to convert to Catholicism or leave the province, as part of Counter Reformation measures.
Kepler served as Tycho Brahe's assistant until the latter's death in 1601 and was then appointed Tycho's successor as Imperial Mathematician, the most prestigious appointment in mathematics in Europe.
Johannes Kepler: The Laws of Planetary Motion (1409 words)
Kepler's Third Law implies that the period for a planet to orbit the Sun increases rapidly with the radius of its orbit.
As an example of using Kepler's 3rd Law, let's calculate the "radius" of the orbit of Mars (that is, the length of the semimajor axis of the orbit) from the orbital period.
Kepler's Laws Calculator that allows you to make simple calculations for periods, separations, and masses for Keplers' laws as modified by Newton (see subsequent section) to include the effect of the center of mass.
  More results at FactBites »

 
 

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