**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 ]

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

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:

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 ]

• 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 the Chapter on the Proof for Kepler's 1st Law.**