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Kepler's laws of planetary motion
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Kepler's laws of planetary motion

     
Johannes Kepler's primary contribution to astronomy/astrophysics were the three laws of planetary motion. Kepler derived these laws, in part, by studying the observations of Brahe. Isaac Newton would later verify these laws with his laws of motion and universal gravity. The generic term for an orbiting object is "satellite".

Table of contents
1 Kepler's laws of planetary motion
2 Kepler's first law
3 Kepler's second law
4 Kepler's third law (harmonic law)
5 Applicability
6 Kepler's understanding of the laws
7 See also

Kepler's laws of planetary motion

Kepler's first law

The
orbit of a planet about a star is an ellipse with the star at one focus.

There is no object at the other focus of a planet's orbit. The semimajor axis, a, is the average distance between the planet and its star.

Proof of Kepler's first law:

Newton proposed that "every object in the universe attracts every other object along a line of the centers of the objects proportinal to each objects mass, and inversely proportional to the square of the distance between the objects"

We begin with Newton's law F=ma.

Here we express F as the product of its magnitude and its direction. Recall that in polar coordinates:

In component form we have:

Now consider the angular momentum.

So

Where l = L/m is the angular momentum per unit mass. Now we substitute. Let

The equation of motion in the direction becomes:

Newton's law of gravitation states that the central force is inversely proportional to the square of the distance so we have:

where k is our proportionality constant.

This differential equation has the general solution:

Replacing u with r and letting θ0=0,

.

This is indeed the equation of a conic section with the origin at one focus. Q.E.D.

Kepler's second law

A line joining a planet and its star sweeps out equal areas during equal intervals of time.

This is also known as the law of equal areas. Suppose a planet takes 1 day to travel from points A to B. During this time, an imaginary line, from the Sun to the planet, will sweep out a roughly triangular area. This same amount of area will be swept every day.

As a planet travels in its elliptical orbit, its distance from the Sun will vary. As an equal area is swept during any period of time and since the distance from a planet to its orbiting star varies, one can conclude that in order for the area being swept to remain constant, a planet must vary in velocity. Planets move most rapidly when at perihelion and more slowly when at aphelion.

This law was developed, in part, from the observations of Brahe that indicated that the velocity of planets was not constant.

This law is the explicit expression of the angular momentum conservation law in given situation.

Proof of Kepler's second law:

By definition, the angular momentum of a point mass with mass and velocity is :

.

where is the position vector of the particle.

Since , we have:

taking the time derivative of both sides:

since the cross product of parallel vectors is 0. We can now say that is constant.

The area swept out by the line joining the planet and the sun, is half the area of the parallelogram formed by and .

Since is constant, the area swept out by is also constant. Q.E.D.

Kepler's third law (harmonic law)

The square of the sidereal period of an orbiting planet is directly proportional to the cube of the orbit's semimajor axis.

P2 ~ a3
P = object's sidereal period in years
a = object's semimajor axis, in AU

The larger the distance (between a planet and its sun), a, the longer the sidereal period. By understanding this and the second law, one can determine that the larger an orbit is the slower the average velocity of an orbiting object will be, (as the satellite will be consistently farther from the object being orbited).

Newton would modify this third law, noting that the period is also affected by the orbiting body's mass, however typically the central body is so much more massive that the orbiting body's mass may be ignored. (See below.)

Applicability

The laws are applicable whenever a comparatively light object revolves around a much heavier one because of gravitational attraction. It is assumed that the gravitational effect of the lighter object on the heavier one is negligible. An example is the case of a satellite revolving around Earth.

Kepler's understanding of the laws

Kepler did not understand why his laws were correct; it was Isaac Newton who discovered the answer to this more than fifty years later. Newton, understanding that his third law of motion was related to Kepler's third law of planetary motion, devised the following:

where: Astronomers doing celestial mechanics often use units of years, AU, G=1, and solar masses, and with m2<<m1, this reduces to Kepler's form. SI units may also be used directly in this formula.

See also

circular motion.