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Kepler's 1st Law

"Now between the circle and the ellipse there is no other intermediary except a different ellipse. Therefore the path of the planet is an Ellipse ..." - Johannes Kepler ( 1571 - 1630 )
[ Source: "The New Astronomy": Astronomia nova ( Heidelberg, 1609 ) Chapter 58, 284 - 85, KGW 3 366, from School of Mathematics and Statistics, University of St Andrews, Scotland ]

kepler.png

Johannes Kepler ( 1571 – 1630 )
German mathematician, astronomer and astrologer
portrait circa 1610 - artist unknown


§ Kepler's 1st Law ( Planetary Law of Ellipses: Sun - centered model ):

All planetary orbits are ellipses with the Sun at one of the two foci.

An ellipse is defined as the locus of points, the sum of whose distance from two fixed points ( the foci ) is constant. That is, an ellipse is a special curve where the sum of the distances from every point on the curve to two other points is a fixed constant.

The ellipse equation is therefore

ellipse_equation

An ellipse is drawn by using two tacks into a piece of cardboard with a taut string and by moving a pencil held just inside the string.

Using this picture you can draw an ellipse as follows: 

draw_ellipse.jpg

The closer together which these points are, the more closely that the ellipse resembles the shape of a circle. In fact, a circle is the special case of an ellipse in which the two foci are at the same location. 

Where the two tacks ( foci ) are come closer, the ellipse will approach a circle. In fact, every circle is a special case ellipse where the two foci are identical. 

Hence,

e = eccentricity of a circle = ea/a = c/a = 0. See below for this definition.

0 < eccentricity of an ellipse < 1

eccentricity of a straight line = ∞, infinity [ ∞ = lemniscate, latin for ribbon ]

Kepler Ellipse2.jpg

 Sun = red circle, one of two foci; stationary yellow circle is an imaginary 2nd foci

 Planet = moving yellow circle

 Blue arrow = initial condition

 Red arrow = moving planet and is proportional to planet's velocity

 The Perigee: Closer to the sun, the faster the planet passes in its transit orbit

 The Apogee: Further from the sun, the slower the planet passes in its transit orbit

[ note: the last two observations hold because of Kepler's 2nd Law of Equal Areas where a planet sweeps out equal areas during equal intervals of time. ]

Sun at one of the two foci

Major Axis = Rp + Ra = 2a

Minor Axis = 2b

a = Semi-major axis of ellipse

Rp = perihelion radius

Ra = aphelion radius

Rav = a = 1/2(Ra + Rp) = average orbital radius

c =ea = 1/2(Ra - Rp) = interfocal radius

e = eccentricity of ellipse = ea/a = c/a.

source: http://dev.physicslab.org/Document.aspx?doctype=3&filename=OscillatoryMotion_KeplersLaws.xml

§ Equations for Planetary Orbital Eccentricity:


§ See Proof: Kepler's 1st Law




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