Kepler's Laws and Pareto distribution: Difference between pages

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[[File:Kepler laws diagram.svg|thumb|300px|Figure 1: Illustration of [[Johannes Kepler|Kepler's]] three laws with two planetary orbits.<br> (1) The orbits are ellipses, with focal points ''&fnof;''<sub>1</sub> and ''&fnof;''<sub>2</sub> for the first planet and ''&fnof;''<sub>1</sub> and ''&fnof;''<sub>3</sub> for the second planet. The Sun is placed in focal point ''&fnof;''<sub>1</sub>. <br><br> (2) The two shaded sectors ''A''<sub>1</sub> and ''A''<sub>2</sub> have the same surface area and the time for planet 1 to cover segment ''A''<sub>1</sub> is equal to the time to cover segment ''A''<sub>2</sub>. <br><br> (3) The total orbit times for planet 1 and planet 2 have a ratio ''a''<sub>1</sub><sup>3/2</sup>&nbsp;:&nbsp;''a''<sub>2</sub><sup>3/2</sup>.]]
The '''Pareto distribution''' is a [[probability distribution]] named after [[Vilfredo Pareto]]. It is [[Continuous function|continous]] and follows a [[power law]]. It is a [[heavy tailed distribution]]. The Pareto distribution was first used to show how [[income]] is distributed among households (which is commonly called [[Income distribution]]). Pareto distributions are often used in the cases when many different small independent factors contribute to a result. Pareto wrote a book called ''Cours d'économie politique'', where he tried to show that the number of people who have an income greater than x is proportional to <math>1/x^k</math>. The parameter <math>k</math> is constant, and about 1.5. This is all that is needed to describe the Pareto distribution. [[Insurance]] companies often use Pareto distributions to model damage.
'''Kepler's laws of planetary motion''' are three laws that describe the motion  of [[Planet|planets]] around the [[sun]]:
{{stub}}
#Planets move around the sun in [[Ellipse|elliptic]] [[Orbit|orbits]]. The sun is in one of the two foci of the orbit.
#A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
#The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.
[[Johannes Kepler]] found these laws, between 1609 and 1619.  


==Comparison to Copernicus==
[[Category:Probability distributions]]
Kepler's laws improve the [[Scientific model|model]] of Copernicus. If the eccentricities of the planetary [[orbit]]s are taken as zero, then Kepler basically agrees with Copernicus:
#The planetary orbit is a [[circle]]
#The Sun at the center of the orbit
#The speed of the planet in the orbit is constant
 
The eccentricities of the orbits of those planets known to Copernicus and Kepler are small, so the rules above give good approximations of planetary motion; but Kepler's laws fit the observations better than Copernicus's.
 
Kepler's corrections 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'' of the elliptical orbit.
#Neither the linear speed nor the angular speed of the planet in the orbit is constant, but the ''area speed'' is constant.
 
The 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. A diameter would cut the orbit into equal parts, but the plane through the sun parallel to the [[equator]] of the earth cuts the orbit into two parts with areas in a 186 to 179 ratio, so the eccentricity of the orbit of the Earth is approximately
:<math>\varepsilon\approx\frac \pi 4 \frac {186-179}{186+179}\approx 0.015,</math>
which is close to the correct value (0.016710219) (see [[Earth's orbit]]).
The calculation is correct when [[perihelion]], the date the Earth is closest to the Sun, falls on a [[solstice]]. The current perihelion, near January 4, is fairly close to the solstice of December 21 or 22.
 
[[Category:Astronomy]]

Latest revision as of 21:23, 15 January 2016

The Pareto distribution is a probability distribution named after Vilfredo Pareto. It is continous and follows a power law. It is a heavy tailed distribution. The Pareto distribution was first used to show how income is distributed among households (which is commonly called Income distribution). Pareto distributions are often used in the cases when many different small independent factors contribute to a result. Pareto wrote a book called Cours d'économie politique, where he tried to show that the number of people who have an income greater than x is proportional to [math]\displaystyle{ 1/x^k }[/math]. The parameter [math]\displaystyle{ k }[/math] is constant, and about 1.5. This is all that is needed to describe the Pareto distribution. Insurance companies often use Pareto distributions to model damage. Template:Stub

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