- #1

Kashmir

- 468

- 74

>..."The next step is to consider the equation for ##u## under small perturbations ##{\displaystyle \eta \equiv u-u_{0}}## from perfectly circular orbits"

(Here ##u## is related to the radial distance as ##u=1/r## and ##u_0## corresponds to the radius of a circular orbit ) ...>"The solutions are

##{\displaystyle \eta (\theta )=h_{1}\cos(\beta \theta )}##">"For the orbits to be closed, ##β## must be a rational number. What's more, **it must be the same rational number for all radii**, since β cannot change continuously; the rational numbers are totally disconnected from one another"Why does ##\beta## have to be the **same** rational number for all radii at which a circular orbit is possible ?

I understand why it should be rational, but why the same number for all radii?

Link: https://en.m.wikipedia.org/wiki/Bertrand's_theorem