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Kepler's Law of Planetary Motion

Johannes Kepler was a German astronomer and mathematician of the late sixteenth and early seventeenth centuries. Kepler was born in Wurttemberg, Germany in 1571. His parents were poor, but he was sent to the University of Tubingen on a scholarship in recognition of his abilities in mathematics.

History:

Unlike Brahe, Kepler believed firmly in the Copernican system. In retrospect, the reason that the orbit of Mars was particularly difficult was that Copernicus had correctly placed the Sun at the center of the Solar System, but had erred in assuming the orbits of the planets to be circles. Thus, in the Copernican theory epicycles were still required to explain the details of planetary motion.

Kepler's laws give a description of the motion of planets around the Sun.

Astronomy

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

The radius vector of a planet sweeps over equal areas in equal intervals of time.

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

Kepler's laws

Isaac Newton solidified Kepler's laws by showing that they were a natural consequence of his inverse square law of gravity with the limits set in the previous paragraph. Further, Newton extended Kepler's laws in a number of important ways such as allowing the calculation of orbits around other celestial bodies.

The solutions to the two-body problem, where there are only two particles involved, say, the sun and one planet, can be expressed analytically. These solutions include the elliptical Kepler orbits, but motions along other conic section (parabolas, hyperbolas and straight lines) also satisfy Newton's differential equations.

Derivation from Newton's laws

The focus of the conic section is at the center of mass of the two bodies, rather than at the center of the Sun itself.

The period of the orbit depends a little on the mass of the planet.

The solutions deviate from Kepler's laws in that:

"The orbit of every planet is an ellipse with the Sun at one of the two foci.“

First Law

An ellipse is a particular class of mathematical shapes that resemble a stretched out circle. Ellipses have two focal points neither of which are 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.

Ellipse

Eccentricity of an ellipse is defined as the ratio of distance between foci to length of major axis. In the figure below, if eccentricity is denoted as “e”, the distance between the foci as “c” and the lengths of the major axis as “a”, the formula is:

e = c a

Kepler's First Law is illustrated in the image shown above. The Sun is not at the center of the ellipse, but is instead at one focus (generally there is nothing at the other focus of the ellipse). The planet then follows the ellipse in its orbit, which means that the Earth-Sun distance is constantly changing as the planet goes around its orbit. For purpose of illustration we have shown the orbit as rather eccentric; remember that the actual orbits are much less eccentric than this.

When c is nearly to a, the ellipse appears flattened.

When c is smaller than a, the ellipse appears rounded.

When c is zero, the ellipse becomes a circle.

Therefore:

The distance between the foci, c, the major axis, a, and the minor axis, b, can be solved using the Pythagorean Theorem:

c2 = a2 - b2

Moreover:

“The radius vector of a planet sweeps over equal areas in equal intervals of time.”

Second Law

Radius Vector is an imaginary line connecting the sun and the planet.

Note:

Kepler's second law is illustrated in the preceding figure. The line joining the Sun and planet sweeps out equal areas in equal times, so the planet moves faster when it is nearer the Sun. Thus, a planet executes elliptical motion with constantly changing angular speed as it moves about its orbit. The point of nearest approach of the planet to the Sun is termed perihelion; the point of greatest separation is termed aphelion. Hence, by Kepler's second law, the planet moves fastest when it is near perihelion and slowest when it is near aphelion.

Kepler's second law is equivalent to the fact that the force perpendicular to the radius vector is zero. The "areal velocity" is proportional to angular momentum, and so for the same reasons, Kepler's second law is also in effect a statement of the conservation of angular momentum.

About January 4th, by about 1.5%, not enough to make the Sun look different.

When are we closest to the Sun?

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

Third Law

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.

Harmonic law

P2planet P2

earth

A3

planet A3

earth

Symbolically:

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