Showing that planets follow elliptical orbits

[kent davidge]

Is it difficult to show that planets follow elliptic orbits around the sun, using Newton's theory?

I have seen the equations showing it, but from General Relativity, considering the Newtonian limit.

How to arrive at them using only Newton?

 

[fresh_42]

Wikipedia has it: https://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion#Formulary
I find the German version a bit more detailed: https://de.wikipedia.org/wiki/Keplersche_Gesetze#Herleitung_und_moderne_Darstellung
Either you just read the formulas, or use the translation tool of chrome.

 

[kent davidge]

Wikipedia has it

The part of the Wiki page you linked to shows the three Keplers laws, one of which, namely the first law, states that the planets follow elliptic orbits. Newtons laws should let us see why that happens.

So one should be able to arrive at the equation of the ellipse presented in the Wiki page, by means of Newtons laws.

 

[fresh_42]

Newtons laws should let us see why that happens.

I haven't checked the English version, the German does exactly this. Try the second alternative, either by reading just the formulas or translate the page. It worked reasonably well here (translated by Google chrome). A few sentences remained untouched, but it worked sufficiently.

 

[anorlunda]

So one should be able to arrive at the equation of the ellipse presented in the Wiki page, by means of Newtons laws.

There is one thing you can't get from Newton's Laws; the initial conditions. I mean the position, and momentum at the time you start applying Newton's Laws.

With Newton's Laws, an orbit could be elliptical or (nearly) circular. Kepler predated Newton,

n 1601, Johannes Kepler acquired the extensive, meticulous observations of the planets made by Tycho Brahe. Kepler would spend the next five years trying to fit the observations of the planet Mars to various curves. In 1609, Kepler published the first two of his three laws of planetary motion. The first law states:

"The orbit of every planet is an ellipse with the sun at a focus."

 

[Nugatory]

Is it difficult to show that planets follow elliptic orbits around the sun, using Newton's theory?

That's standard fare in the first-semester mechanics class. It's in Kleppner and Kolenkow; and I'd expect to find it any comparable textbook. Googling for "derive Kepler's" brings up many promising-looking links.