The elliptical orbits and inverse-square law forces.

[d-richard]

Greetings,
I read a chapter in the book College Physics by Vuille and Serway explaining the laws of planetary motion in terms of the universal law of gravitation. The first law stated that planets move in an elliptical orbit and the book's explanation was that all objects bound by an inverse-square law move in ellipses if orbiting, not in circles. Could anyone please explain why inverse-square law forces cause this effect? Thanks.

 

[A.T.]

in ellipses if orbiting, not in circles.

Circles are ellipses.

 

[UltrafastPED]

Kepler's Laws of Planetary Motion: http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html

and: http://csep10.phys.utk.edu/astr161/lect/history/kepler.html

Newton and Kepler: http://csep10.phys.utk.edu/astr161/lect/history/Newtonkepler.html

And here is a derivation of Kepler from Newton:
https://www.uu.edu/dept/math/SeniorPapers/09-10/DavisEmily.pdf

 

[d-richard]

thanks for the links, and yeah I forgot that circles are ellipses. Thanks for reminding me

 

[pmacias]

Hello there,

I can provide an explanation for why inverse-square law forces result in elliptical orbits. The inverse-square law is a fundamental law in physics that states that the force between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. In the case of planetary motion, this means that the gravitational force between a planet and the sun decreases as the distance between them increases.

Now, when a planet is in motion around the sun, it experiences a continuous change in direction due to the gravitational force acting upon it. This force is always directed towards the sun, but as the planet moves further away from the sun, the force decreases due to the inverse-square law. As a result, the planet's velocity also decreases, causing it to move in a curved path, ultimately resulting in an elliptical orbit.

In simpler terms, the inverse-square law causes the strength of the gravitational force to decrease as the distance between the two objects increases, resulting in a curved path of motion rather than a straight line. This is why planets, and other objects bound by the inverse-square law, move in elliptical orbits.

I hope this explanation helps to clarify the relationship between inverse-square law forces and elliptical orbits. If you have any further questions, please don't hesitate to ask. Happy exploring!