MAIN CONCEPTS
The orbit of each planet is an ellipse with the sun at one focus. To show Kepler's first law, we aim to get an equation of the orbit in polar form, that is .
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We start by first examining Newton’s inverse square law of universal gravitation. We know that  is a central attractive force between m and M and is given by

Knowing that G, the gravitational constant, and M, the mass of the Sun, are constant, we simplify the algebra by writing

where , and our second equation of motion becomes,

So how do we proceed with this complicated differential equation? Here’s the plan. Now remember that we seek to find an equation of the orbit, possibly one written in polar form. Here are a few goals to look at:

1. We wish to somehow get , the equation of the orbit.
2. We need to eliminate .
3. Most probably  will be a dependent variable and  an independent one.

Now let’s get started. Remember the small equation we had from the previous section, namely

We will substitute this into our differential equation to get

The presence of  to a negative power suggest that it might be temporarily convenient to introduce a new dependent variable like . Now let’s see. To eliminate t, it makes sense to express  in terms of  through some hard work of differentiation.

Using our small equation again and differentiating a second time,

We make this substitution into our original differential equation to yield

and after simplifying

With t gone, the equation looks more manageable. We’re moving progress indeed. We further notice that, except the constant term on the right, this is a differential equation of simple harmonic motion where the acceleration is proportional to the displacement in the opposite direction. We simply put

With  and so

giving us the general solution of this equation as

and so

We find a particular solution by using the following reasoning. With reference to the diagram at the bottom, we shift the direction of the polar axis in such a way that r is minimal implying m is closest to the origin. This occurs at . So by our equation , this means z to be a maximum in this direction, so

 and

when . Differentiating the equation in z once and twice through and letting , we equate the coefficients A and B to get

 and

Replacing z and r, we finally get the equation we intended.

All left to be done is to recognize that this is an equation of an ellipse. We put  giving us the equation of the orbit as

where e is a positive constant.

What do we know about conic sections? The above represents the polar equation of a conic section with focus at the origin and that this conic section is an ellipse, a parabola, or a hyperbola when , , or . By logical reasoning, since the planets remain in the solar system and do not move infinitely far away from the sun, the ellipse is the only possibility.

While the actual calculation e still needs to be done by taking an example of the earth’s orbit, our deduction is sufficient to prove Kepler’s first law, namely the orbit of each planet is an ellipse with the sun at one focus.

ERRATA: In the video, I made the wrong substitution of  forgetting about the  term. Please accept my apologies. The equation here is the correct one. Nevertheless, the equations of conic sections still apply for both.

The same diagram has been place here for easy reference.