Kepler's laws

Kepler's 2nd law

Johannes Kepler (1571-1630) was an astronomer who continued the work of Tycho Brahe (1546-1601).

Brahe, himself a great astronomer, had measured the heavens collecting a mass of data that fell to Kepler to unravel.

What Kepler discovered is one of the great moments in science and shows just how fantastic and simplistic the universe can be.

In order to derive these laws Kepler had to make a massive intellectual step one that contradicted his former bosses view. One of the difficulties was that even for the orbits of earth and Mars seen from Earth appear really odd and would be difficult to try and model mathemtaically. If however, it is accepted that the sun is the center of the orbit, the one can start to see that the orbits may actually be almost circular. By doing this Kepler was then able to use the data to develop his laws.

Keplers 3 laws can be described as follows 

Kepler's elliptical orbit law
The planets orbit the sun in elliptical orbits with the sun at one focus.

Kepler's equal-area law
The line connecting a planet to the sun sweeps out equal areas in equal amounts of time.

Kepler's law of periods
The time required for a planet to orbit the sun, called its period, is proportional to the long axis of the ellipse raised to the 3/2 power. The constant of proportionality is the same for all the planets.

The Ellipse. An ellipse is like a circle that has been squashed slightly. (In fact it makes more sense to me to say that a circle is a special case of an ellipse where both foci (plural of focus)are the same.

For the Earth this means that the closest it gets to the sun is 91 million miles and the furthest distance is 95 million.

So the first law tells us something that these days everyone takes for granted, the planets orbit the sun in almost circular orbits.

The second and third laws are no so well known. Both of these I find to be amazing, If you draw a line from the center of the sun to the center of a planet, it sweeps out equal AREAs in EQUAL amounts of TIME! (see the diagram at the beginning of this post)

For a circular orbit this would be expected in a way. But in an elliptical orbit it means that when the planet is nearer the sun it is moving faster relative to the sun than when the planet is at its extreme distance from the sun.

The last law says that the time needed to complete and orbit squared is related to the long axis of the elliptical orbit cubed! Lets try a simple calculation to see what this means. The Earth is 150 million km from the sun (just to make the maths a little bit easier at this point). If we take the cube of 150,000,000 we get

150,000,000 x 150,000,000 x 150,000,000 = 3,375,000,000,000,000,000,000,000

now we know that the earth takes about 365 days to orbit the sun, each day has 24 hours, made up of 60 minutes, which are in turn made up of 60 seconds. This means that in 1 years we have

1 year = 365 x 24 (hours) x 60 (minutes) x 60 (seconds) = 31,536,000,

so Kepler tells us that we need to square this

so for earth this is

31,536,000 x 31,536,000 ~ 995,000,000,000,000

now according to Kepler we have

995,000,000,000,000 = k x 3,375,000,000,000,000,000,000,000

cancelling a few zeros from each side gives

9 = k x 33,750,000,000

so k = 9.95 / 33,750,000,000 with units of seconds squared / km cubed in this case.

k ~ 2.9 x 10^-10s²/km³, or 2.9 10x10^-19 s²/m³

What this also implies, given the second law that the time to orbit is proprtional to r 3/2, means that the planets further out must be moving more slowly than the inner planets.

The fact that the outer planets moved more slowly can be explained if the sun's force was some how dependent on distance, getting weaker as the distance increased and it was Newton who showed this in his law of gravity.

So, Kepler was able to show that by taking the Sun as the center of the solar system he could  demonstrate that the motion of the planets could be explained by very simple school boy level mathematics. I find this absolutely amazing. Here in the universe in which we exist these massive planets move in paths described by remarkable simple maths.

For me this raises a question. Is it the case that all of physics can be explained by equally simplistic methods? and the fact that we currently use some exceptionally difficult maths to explain certain phenomena is just misunderstanding and a lack of understanding on our part and we have not truly realised the real nature of the universe?

Note: Kepler's laws (and Newton's law of gravity) do not appear to hold in galaxies. This has led to the emergence of a number of competing theories. It is unclear at this time which, if any, is correct.