Keplers Laws of Planetary Motion: A Step-by-Step Guide

[Gwilim]

[problem]

State Keplers laws of planetary motion.

The motion of a planet about the Sun, assumed to be fixed at the origin, may be approximated by

r''= -ur^(-3) r

for u=y ms, where y is the universal gravitational constant and ms is the mass of the Sun. Derive the energy equation for this system, and by considering hXr''. where h is the angular momentum vector, obtain the Lenz-Runge vector. Now show that the path of the planet has polar equation

l/r = 1 - ecos(theta)

for suitable l and e.

[/problem]

There's so much I don't understand there it isn't even funny (obviously Keplers laws themselves will be no problem to learn, applying them however is something else). If someone can show me how to solve this type of problem once, I might be able to learn from there.

 

[tiny-tim]

Hi Gwilim! How's it going?

The motion of a planet about the Sun, assumed to be fixed at the origin, may be approximated by

r''= -ur^(-3) r

Hint: the trick with equations involving r'' and r is to introduce r' …

in other words, how do you integrate r'.r or r''.r' ?

(and r'' x r ?)

 

[Gwilim]

Umm I think I'll go learn a bit about vectors and then come back to this