A Why must β be a consistent rational number across all circular orbit radii?

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For orbits to be closed, the parameter β must be a rational number, and it must remain consistent across all circular orbit radii. This consistency is necessary because rational numbers are disconnected from one another, meaning that between any two rational numbers, there exist non-rational numbers. Therefore, if β were to change with varying radius, it would imply a continuous transition through non-rational values, which is not possible. The discussion emphasizes that maintaining the same rational value for β ensures the integrity of the circular orbit across different radii. Understanding this concept is crucial for grasping the implications of Bertrand's theorem in orbital mechanics.
Kashmir
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Wikipedia on Bertrands theorem, when discussing the deviations from a circular orbit says:
>..."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
 
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Kashmir said:
"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 ?
As the quote states, rational numbers are totally disconnected from one another, meaning that between two rational numbers you have non-rational numbers. So the rational number β cannot change continuously, while you continuously vary the radius.
 
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