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Kepler's first law

An imperfect theory

It was convenient to assume that the planets orbited around the sun in perfect circles when early astronometers first began modeling the solar system. It was an idea that goes back to Plato and remained commonplace up until the 17th Century.
Heliocentric model printed in the book “On the Revolutions of the Heavenly Spheres.” by Nicolaus Copernicus, first printed 1543
The heliocentric model assumes that the Earth revolves around the sun in a perfect circle. Yet there is a problem with this model when you observe the motion of planets closely. Here is a sequence of images of the sun (taken from the earth) over the course of a year. Pay close attention to the size:
Image Credit: Big Bear Solar Observatory
Notice the size of the sun is gradually changing? This is not a result of the sun growing and shrinking. This is phenomena is a result of the changing distance between the earth and sun.

Elliptical orbits

Johannes Kepler (1571 – 1630) was a german astronomer who realized that circular orbits wouldn’t work while investigating the orbital motion of Mars in close detail. Kepler writes about his discovery to a fellow astronomer (David Fabricius) on October 11th 1605:
“So, Fabricius, I already have this: that the most true path of the planet [Mars] is an ellipse, which Dürer also calls an oval, or certainly so close to an ellipse that the difference is insensible.”
An ellipse can have different values for its width and height. This means the radius will change depending on the angle through the full orbit. One simple way to think about an ellipse is the addition of two different sized circles which defines an x and y coordinate respectively. In the example below the x-coordinate comes from the larger circle and the y-coordinate comes from the smaller circle. Convince yourself of this:
Image Credit: Peter Collingridge
It's very important to notice that a circle is a set of points which are a fixed distance from the center. However an ellipse is a set of points a certain distance from two points (called the foci). These are two points on the major axis such that the sum of the distance between any point on the ellipse and both foci is constant.
Below is an interactive illustration. You can click and drag the foci to change the shape. Notice that the green and blue lines always add up to the same distance:
Which leads to Kepler’s first law:
The orbit of every planet is an ellipse with the sun at one of the two foci.

Preparing to animate

To draw an elliptical orbit, we define the x-axis radius (a) and the y-axis radius (b). The major axis is the larger of the two, the minor axis, the smaller. Notice that if a=b, then the equations are the same as the ones we used for a perfect circle.
x = a x cos(θ)
y = b x sin(θ)
We can now define an ellipse with three properties: its centre, its major axis and its minor axis.
Next let's review the equations of an ellipse.

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