Newton's derivation of Kepler's laws

[inkyvoyd]

Introduction
This is not a homework or coursework question (if it were it would be of the project type), and I am looking for hints not spoilers.
Hi,
I recently passed by kepler's laws again in a science class (this time Earth science), and am concurrently taking calculus in my math class.

I realized that my current knowledge of calculus should let me be able to re-find kepler's laws (or show equivalence to Newton's laws) - for kepler's first law, I should be able to prove (with Newton's second law) that the elliptical orbit described is consistent. For his second law, I should be able to use polar integration to complete the consistency proof. As for the third, I haven't had any ideas, but my problems are really with where to start.

Homework Statement

I contacted my math teacher with this question, and we had a short discussion, with my teacher suggesting I get data of planetary locations over time. I searched for these, with no avail (I am not looking for conclusions - which are all I could seemingly find). I'm trying to understand kepler's first law and how it relates to Newton's laws - but I do realize some problems. Since initial velocity (and position) must be known in order to determine the elliptical path, one must have these accounted for - and I have no idea how to do that.

I need hints, and if possible, data.
I do not want a result, or work and a result, because I want to in a sense "repioneer" this - the thinking that is involved with creating an idea previously unknown to one differs from that of learning about an idea.

tl;dr:I am trying to reconstruct Newton's derivation of Kepler's laws, and am stuck on how to translate Newton's laws into kepler's first law. I need hints, and if possible, data.

Homework Equations

Newton's second law: f=ma
Newton's gravitational law:f=g(m_1m_2)/(r^2)
kepler's 3 laws

The Attempt at a Solution

I'm not sure where to start.
I know that once I get an idea I will break motion into x and y-axis (put in parametric form), and try to convert to polar form as well. I am guessing I will encounter simple differential equations.

 

[haruspex]

The main problem relating Kepler's first law (elliptical orbits) to Newton's equations is that Kepler's law says nothing about time.
You could try:
- obtain a differential equation r, theta, t from Newton's laws;
- assume the relationship between r and theta implied by Kepler I, and on the basis of that obtain expressions for r and its time derivatives in terms of theta and its time derivatives;
- substitute in your ODE to eliminate references to r and show that appropriate assignments of the constants satisfy the equation.

 

[inkyvoyd]

The main problem relating Kepler's first law (elliptical orbits) to Newton's equations is that Kepler's law says nothing about time.
You could try:
- obtain a differential equation r, theta, t from Newton's laws;
- assume the relationship between r and theta implied by Kepler I, and on the basis of that obtain expressions for r and its time derivatives in terms of theta and its time derivatives;
- substitute in your ODE to eliminate references to r and show that appropriate assignments of the constants satisfy the equation.

I see - but since Kepler's second and third laws are about orbital time and velocity, I'm guessing I would have to incorporate all three simultaneously?

 

[johns123]

F = GMm/r^2 = mv^2/r ... T = 2 x Pi x r / v and therefor v = 2 x Pi x r / T

sub for v in mv^2/r ( centrepital force ) and simplify to get T^2/r^3 = Constant

which holds for all the planets around the sun. That is Kepler's 3rd law.