wings of fire-2

7/31/2019 Wings of Fire-2

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BY:Pranav Kanuparthi and TusharTrehon

P r o j e c t

D o c u

m e n

t a t i o n

Keplers laws of planetary motion


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Johannes Kepler published his first two laws in 1609, having found them by analyzing the astronomical observations of Tycho Brahe. Kepler discovered his third law many years later, and it was published in 1619.At the time, Kepler's laws were radical claims; the prevailing belief (particularly in epicycle -based 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 long-accepted 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).

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 .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. Voltaire' s Elments de la philosophie de
http://en.wikipedia.org/wiki/Johannes_Keplerhttp://en.wikipedia.org/wiki/Tycho_Brahehttp://en.wikipedia.org/wiki/Epicyclehttp://en.wikipedia.org/wiki/Sunhttp://en.wikipedia.org/wiki/Epicyclehttp://en.wikipedia.org/wiki/Geocentric_modelhttp://en.wikipedia.org/wiki/Aristotlehttp://en.wikipedia.org/wiki/Ptolemyhttp://en.wikipedia.org/wiki/Copernican_heliocentrismhttp://en.wikipedia.org/wiki/Nicolaus_Copernicushttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Newton%27s_laws_of_motionhttp://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitationhttp://en.wikipedia.org/wiki/Perturbation_(astronomy)http://en.wikipedia.org/wiki/Voltairehttp://en.wikipedia.org/wiki/Voltairehttp://en.wikipedia.org/wiki/Perturbation_(astronomy)http://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitationhttp://en.wikipedia.org/wiki/Newton%27s_laws_of_motionhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Nicolaus_Copernicushttp://en.wikipedia.org/wiki/Copernican_heliocentrismhttp://en.wikipedia.org/wiki/Ptolemyhttp://en.wikipedia.org/wiki/Aristotlehttp://en.wikipedia.org/wiki/Geocentric_modelhttp://en.wikipedia.org/wiki/Epicyclehttp://en.wikipedia.org/wiki/Sunhttp://en.wikipedia.org/wiki/Epicyclehttp://en.wikipedia.org/wiki/Tycho_Brahehttp://en.wikipedia.org/wiki/Johannes_Kepler


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Newton ( Elements of Newton's Philosophy ) was in 1738 the first publication to call Kepler's Laws "laws". Together with Newton's mathematical theories, they are part of the foundation of modern astronomy and physics .
http://en.wikipedia.org/wiki/Astronomyhttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Astronomy


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The orbit of every planet is an ellipse with the Sun at one of

the two foci .

An ellipse is a particular class of mathematical shapes that resemble a stretched out circle. 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 neither of which is in the center of the ellipse (except for the one special case of the ellipse being a circle). 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 varies from 0 (a simple circle)to 1 (an ellipse that is so stretched out that it is a straight line back and forth between the two focal points). The eccentricities of the planets known to Kepler varies from 0.007 ( Venus ) to 0.2( Mercury ).

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.
http://en.wikipedia.org/wiki/Orbithttp://en.wikipedia.org/wiki/Planethttp://en.wikipedia.org/wiki/Ellipsehttp://en.wikipedia.org/wiki/Focus_(geometry)http://en.wikipedia.org/wiki/Orbital_eccentricityhttp://en.wikipedia.org/wiki/Venushttp://en.wikipedia.org/wiki/Mercury_(planet)http://en.wikipedia.org/wiki/Comethttp://en.wikipedia.org/wiki/Asteroidhttp://en.wikipedia.org/wiki/Dwarf_planethttp://en.wikipedia.org/wiki/Dwarf_planethttp://en.wikipedia.org/wiki/Plutohttp://en.wikipedia.org/wiki/Parabolic_trajectoryhttp://en.wikipedia.org/wiki/Hyperbolic_trajectoryhttp://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitationhttp://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitationhttp://en.wikipedia.org/wiki/Hyperbolic_trajectoryhttp://en.wikipedia.org/wiki/Parabolic_trajectoryhttp://en.wikipedia.org/wiki/Plutohttp://en.wikipedia.org/wiki/Dwarf_planethttp://en.wikipedia.org/wiki/Dwarf_planethttp://en.wikipedia.org/wiki/Asteroidhttp://en.wikipedia.org/wiki/Comethttp://en.wikipedia.org/wiki/Mercury_(planet)http://en.wikipedia.org/wiki/Venushttp://en.wikipedia.org/wiki/Orbital_eccentricityhttp://en.wikipedia.org/wiki/Focus_(geometry)http://en.wikipedia.org/wiki/Ellipsehttp://en.wikipedia.org/wiki/Planethttp://en.wikipedia.org/wiki/Orbit


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Figure : Heliocentric coordinate system (r, ) for ellipse. Also shown are: semi-major axis a , semi-minor axis b and semi-latus rectum p ; center of ellipse and its two foci marked by large dots.For = 0, r = r min and for = 180, r = r max .

Symbolically an ellipse can be represented in polar coordinates as:

where ( r , ) are the polar coordinates (from the focus) for the ellipse, p is the semi-latus rectum , and is the eccentricity of the ellipse. For a planet orbiting the Sun then r is the distance from the Sun to the planet and is the angle with its vertex at the Sun from the location where the planet is closest to the Sun.

At = 0, perihelion , the distance is minimum

At = 90 and at = 270, the distance is

At = 180, aphelion , the distance is maximum
http://en.wikipedia.org/wiki/Polar_coordinateshttp://en.wikipedia.org/wiki/Semi-latus_rectumhttp://en.wikipedia.org/wiki/Eccentricity_(mathematics)http://en.wikipedia.org/wiki/Perihelionhttp://en.wikipedia.org/wiki/Aphelionhttp://en.wikipedia.org/wiki/File:Ellipse_latus_rectum.PNGhttp://en.wikipedia.org/wiki/File:Ellipse_latus_rectum.PNGhttp://en.wikipedia.org/wiki/File:Ellipse_latus_rectum.PNGhttp://en.wikipedia.org/wiki/File:Ellipse_latus_rectum.PNGhttp://en.wikipedia.org/wiki/File:Ellipse_latus_rectum.PNGhttp://en.wikipedia.org/wiki/Aphelionhttp://en.wikipedia.org/wiki/Perihelionhttp://en.wikipedia.org/wiki/Eccentricity_(mathematics)http://en.wikipedia.org/wiki/Semi-latus_rectumhttp://en.wikipedia.org/wiki/Polar_coordinates


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The semi-major axis a is the arithmetic mean between r min and r max :

so

The semi-minor axis b is the geometric mean between r min and r max :

so

The semi-latus rectum p is the harmonic mean between r min and r max :

so

The eccentricity is the coefficient of variation between r min and r max :

The area of the ellipse is
http://en.wikipedia.org/wiki/Semi-major_axishttp://en.wikipedia.org/wiki/Arithmetic_meanhttp://en.wikipedia.org/wiki/Semi-minor_axishttp://en.wikipedia.org/wiki/Geometric_meanhttp://en.wikipedia.org/wiki/Semi-latus_rectumhttp://en.wikipedia.org/wiki/Harmonic_meanhttp://en.wikipedia.org/wiki/Eccentricity_(mathematics)http://en.wikipedia.org/wiki/Coefficient_of_variationhttp://en.wikipedia.org/wiki/Areahttp://en.wikipedia.org/wiki/Areahttp://en.wikipedia.org/wiki/Coefficient_of_variationhttp://en.wikipedia.org/wiki/Eccentricity_(mathematics)http://en.wikipedia.org/wiki/Harmonic_meanhttp://en.wikipedia.org/wiki/Semi-latus_rectumhttp://en.wikipedia.org/wiki/Geometric_meanhttp://en.wikipedia.org/wiki/Semi-minor_axishttp://en.wikipedia.org/wiki/Arithmetic_meanhttp://en.wikipedia.org/wiki/Semi-major_axis


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The special case of a circle is = 0, resulting in

r = p = r min = r max = a = b and A = r 2 .


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A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.

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
http://en.wikipedia.org/wiki/Line_(geometry)http://en.wikipedia.org/wiki/Areal_velocityhttp://en.wikipedia.org/wiki/Areal_velocityhttp://en.wikipedia.org/wiki/Line_(geometry)


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.

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.
http://en.wikipedia.org/wiki/Angular_velocityhttp://en.wikipedia.org/wiki/Newton_notation_for_differentiationhttp://en.wikipedia.org/wiki/Mean_motionhttp://en.wikipedia.org/wiki/Mean_motionhttp://en.wikipedia.org/wiki/Newton_notation_for_differentiationhttp://en.wikipedia.org/wiki/Angular_velocity


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The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

The third law, published by Kepler in 1619 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 4times 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 42=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 ,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.

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.

Symbolically:
http://en.wikipedia.org/wiki/Square_(algebra)http://en.wikipedia.org/wiki/Orbital_periodhttp://en.wikipedia.org/wiki/Proportionality_(mathematics)http://en.wikipedia.org/wiki/Cube_(arithmetic)http://en.wikipedia.org/wiki/Semi-major_axishttp://en.wikipedia.org/wiki/Centripetal_forcehttp://en.wikipedia.org/wiki/Centripetal_forcehttp://en.wikipedia.org/wiki/Musica_universalishttp://en.wikipedia.org/wiki/Exoplanethttp://en.wikipedia.org/wiki/Starhttp://en.wikipedia.org/wiki/Habitable_zonehttp://en.wikipedia.org/wiki/Habitable_zonehttp://en.wikipedia.org/wiki/Starhttp://en.wikipedia.org/wiki/Exoplanethttp://en.wikipedia.org/wiki/Musica_universalishttp://en.wikipedia.org/wiki/Centripetal_forcehttp://en.wikipedia.org/wiki/Centripetal_forcehttp://en.wikipedia.org/wiki/Semi-major_axishttp://en.wikipedia.org/wiki/Cube_(arithmetic)http://en.wikipedia.org/wiki/Proportionality_(mathematics)http://en.wikipedia.org/wiki/Orbital_periodhttp://en.wikipedia.org/wiki/Square_(algebra)


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where P is the orbital period of the planet and a is the semi-major axis of the orbit.

Interestingly, the constant is theoretically same for both circular and elliptical orbits.

The proportionality constant is the same for any planet around the Sun.

So the constant is 1 ( sidereal year ) 2( astronomical unit ) 3 or 2.9747250510 19 s 2m 3 . See the actual figures: attributes of major planets.
http://en.wikipedia.org/wiki/Proportionality_constanthttp://en.wikipedia.org/wiki/Sidereal_yearhttp://en.wikipedia.org/wiki/Astronomical_unithttp://en.wikipedia.org/wiki/List_of_gravitationally_rounded_objects_of_the_Solar_Systemhttp://en.wikipedia.org/wiki/List_of_gravitationally_rounded_objects_of_the_Solar_Systemhttp://en.wikipedia.org/wiki/List_of_gravitationally_rounded_objects_of_the_Solar_Systemhttp://en.wikipedia.org/wiki/List_of_gravitationally_rounded_objects_of_the_Solar_Systemhttp://en.wikipedia.org/wiki/Astronomical_unithttp://en.wikipedia.org/wiki/Sidereal_yearhttp://en.wikipedia.org/wiki/Proportionality_constant


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The project has scope of improvement in terms of

presentation(using drop based menu)

Background music could have been added

Interactive tools could have used(background audio

explanation)


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www.wikipedia.com

www.google.com

www.drennon.com
http://www.wikipedia.com/http://www.google.com/http://www.drennon.com/http://www.drennon.com/http://www.drennon.com/http://www.google.com/http://www.wikipedia.com/

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