Kepler's 2nd Law of Planetary Motion

"Moving about the perimeter of its orbital ellipse, a planet sweeps-out from the focal point wherein the sun resides, equal areas in equal times."

Establishing that planets orbit the sun following an elliptical and not a circular path was a critical breakthrough for Kepler, for it led on to the realisation that the specific geometry of this shape held the solution to an understanding of the non-uniform motion of the planets, as observed from the Earth. This solution is what is encapsulated in his second law of planetary motion. Whereas the first law simply defines the character of the path taken by the planets as they orbit the sun, Kepler's second law uniquely accounts for the continuous variation in their speed throughout their journey, by linking this to the actual time required by each planet to complete one full orbit.

The specific realisation that Kepler had in this regard was that the total area of a planet’s orbital ellipse could be equated to the total time taken for the planet to complete one single orbit. As a consequence of this thinking, were a planet to complete but a small arc of movement about the sun upon its elliptical perimeter, a line continuously connecting the planet to the focal point wherein the sun resides, would ‘sweep out’ or cover a portion of its total area. The area covered by the line thus amounts to a fraction of the total area of the ellipse shape, and therefore a fraction of the total time for one complete orbit.

In this way, a real connection may be perceived between the passage of time and the distance travelled by a planet as it moves through space. One simple consequence of this is that planets must increase their speed of movement within an elliptical orbit as they near the sun, and slow down as they move further away. This is detailed in the following diagram:

Diagram: Orbital configuration detailing how a planet travels about the sun in two different sections of its orbit in an equal amount of time. As pictured, from point ‘Y’, the planet moves a set distance along the perimeter of the ellipse as shown by the arrow. A line from the sun to the planet is swept out within the ellipse as the planet moves, covering an area as shown in black. This area equates to a given amount of time that passed whilst this movement took place. (The total area of the ellipse is equal to the total orbital time.)

At another stage in the planet’s orbit, a line from position ‘X’ to the closest approach to the sun is swept out. The area covered in this second example is equal to the area swept out from point ‘Y’, earlier. Kepler’s law states that equal areas are swept out in equal times. Thus, the same amount of time passed as both sections of the orbit were completed.

In one instance though, the planet covered only a short distance in its elliptical orbit, when it was farthest from the sun; but when nearer to the sun it covered a far greater arc distance. The planet was therefore travelling at a significantly greater speed when it was closest to the sun, for it moved a greater arc length in the same amount of time that was taken to travel a far lesser arc length when it was at its farthest point from the sun. Such indeed is typical of elliptical orbits in general.

The reality of Kepler’s second law being operative within the solar system thus allows for a true understanding of the precise nature of the non-uniform motion of the planets. Under its direction, each planet in orbit about the sun continuously changes its speed as it moves through space; accelerating as it moves towards the central solar body, or decelerating as it moves away from it.

At its closest point of approach (perihelion) it travels at its fastest speed through space, and at its farthest point (aphelion) it moves at its slowest orbital speed. Consequently, in elliptical orbits, planets never move at a constant speed at any point during their path about the sun. Their state of motion is irreducibly one of continuous change. A most careful consideration of this point allows one to understand the true importance of the second law, and how it marked out Kepler as distinct from his contemporaries.

The law of equal areas swept out in equal times is an idea that is knowable but not observable. Thus, what Kepler had discovered was a principle of intention underlying orbital motion; one that is in essence embedded within the solar system as a whole [1].


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References

[1] http://larouchein2004.net/pages/writings/2002/020125 enddelusionch1.htm
Internet Article: Economics: At The End Of A Delusion by Lyndon H. LaRouche, Jr. (January 12, 2002)