Newton's derivation of Kepler's laws

In summary: 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.- try to integrate the equation to determine the time of arrival of a planet at a particular point in space.In summary, the student is trying to reconstruct Newton's derivation of Kepler's laws and is stuck on how to translate Newton's laws into kepler's first law. He needs hints, and
  • #1
inkyvoyd
2
0
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.
 
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  • #2
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.
 
  • #3
haruspex said:
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?
 
  • #4
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.
 
  • #5


First, it is important to understand that Kepler's laws were derived based on observational data and not on mathematical equations like Newton's laws. Kepler used Tycho Brahe's precise and extensive observations of planetary motion to formulate his laws. Therefore, it may be challenging to use Newton's laws to derive Kepler's laws, as they were not originally derived from these laws.

However, it is possible to show the equivalence between Kepler's laws and Newton's laws. For Kepler's first law, you can use Newton's second law to show that the elliptical orbit described by Kepler is consistent with the gravitational force acting on the planet. This can be done by considering the centripetal force required for circular motion and how it changes as the planet moves along its elliptical path. This will require some knowledge of parametric equations and polar coordinates, as you mentioned in your attempt at a solution.

For Kepler's second law, you can use polar integration to show that the area swept out by the planet in a given time is equal, regardless of where the planet is in its orbit. This is essentially a conservation of angular momentum, which is a consequence of Newton's second law.

As for Kepler's third law, you can use Newton's law of gravitation to show that the square of a planet's orbital period is proportional to the cube of its semi-major axis. This can be done by equating the gravitational force with the centripetal force and solving for the orbital period.

In terms of data, you can use the known values for the masses and distances of planets in our solar system to test the validity of Kepler's laws and their equivalence to Newton's laws. This will also help you understand the limitations and assumptions involved in deriving these laws.

In summary, you can use Newton's laws to show the equivalence of Kepler's laws, but it may be challenging to use them to derive Kepler's laws from scratch. It will require a good understanding of parametric equations, polar coordinates, and differential equations. Additionally, using observational data can help you test the validity and limitations of these laws.
 

What are Newton's laws of motion?

Newton's laws of motion are three fundamental laws that describe the behavior of objects in motion. The first law states that an object at rest will remain at rest, and an object in motion will remain in motion at a constant velocity unless acted upon by an external force. The second law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. The third law states that for every action, there is an equal and opposite reaction.

What are Kepler's laws of planetary motion?

Kepler's laws of planetary motion are three laws that describe the motion of planets around the sun. The first law, also known as the law of orbits, states that all planets move in elliptical orbits with the sun at one focus. The second law, also known as the law of equal areas, states that a line connecting a planet to the sun will sweep out equal areas in equal times. The third law, also known as the law of harmonies, states that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

How did Newton derive Kepler's laws?

Newton derived Kepler's laws using his laws of motion and law of universal gravitation. He used calculus to mathematically prove that a planet moving in an elliptical orbit around the sun would experience a force that varied inversely with the square of the distance between the planet and the sun. This force would result in the planet sweeping out equal areas in equal times and following Kepler's third law.

What was the significance of Newton's derivation of Kepler's laws?

Newton's derivation of Kepler's laws provided a unified explanation for the motion of objects in the solar system. It also demonstrated the power of his laws of motion and law of universal gravitation in predicting and explaining the behavior of objects in the universe. This paved the way for further advancements in the field of physics and our understanding of the natural world.

Are Newton's laws and Kepler's laws still relevant today?

Yes, Newton's laws and Kepler's laws are still relevant and widely used in modern physics and astronomy. They form the basis for our understanding of motion and gravity, and are essential in fields such as space exploration and satellite technology. While they have been refined and expanded upon over the years, their fundamental principles remain unchanged.

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