B 'A Brief History of Time' question on gravity

[Nitram]

I'm reading through Stephen Hawking's 'A Brief History of Time' and came across this sentence in the second chapter:

" If the law were that the gravitational attraction of a star went down faster or increased more rapidly with distance, the orbits of the planets would not be elliptical, they would either spiral into the sun or escape from the sun ."

I think the choice of wording is poor but I can see that if gravity increased with distance and was proportional to say, $r^2$ or $r^3$ then the distant stars would cause the Earth to escape from its current orbit around the Sun. However, if gravity was proportional to $r^{-4}$ or $r^{-5}$ why would the Earth spiral into the Sun? The Earth would experience a smaller gravitational force from the Sun. Would it be because there are effectively no forces from the distant stars and these are the forces that give the Earth its orbital velocity around the Sun? So the Earth's orbital velocity would gradually decrease until it 'fell' into the Sun.

 

[PeroK]

You could take a look at Betrand's theorem:

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

 

[Nugatory]

Would it be because there are effectively no forces from the distant stars and these are the forces that give the Earth its orbital velocity around the Sun? So the Earth's orbital velocity would gradually decrease until it 'fell' into the Sun.

No, what’s going on has nothing to do with the distant stars. Even if the force were very weak, we could still drop an object straight into the sun if it weren’t also moving sideways. One way of thinking about it: a stable orbit requires centrifugal force to exactly balance the gravitational force. Too little centrifugal force and the object falls into the sun; too much and it escapes. When you work through the math (that’s the Bertrand’s Theorem that @PeroK linked) it turns out that only a $1/r^2$ force allows that balance.

I have appealed to “centrifugal force” here, but be aware that it’s a somewhat dubious notion. It’s OK for this handwaving answer, but it’s not a substitute for doing the math properly in an inertial frame)

 

[jbriggs444]

No, what’s going on has nothing to do with the distant stars. Even if the force were very weak, we could still drop an object straight into the sun if it weren’t also moving sideways. One way of thinking about it: a stable orbit requires centrifugal force to exactly balance the gravitational force. Too little centrifugal force and the object falls into the sun; too much and it escapes. When you work through the math (that’s the Bertrand’s Theorem that @PeroK linked) it turns out that only a $1/r^2$ force allows that balance.

I have appealed to “centrifugal force” here, but be aware that it’s a somewhat dubious notion. It’s OK for this handwaving answer, but it’s not a substitute for doing the math properly in an inertial frame)

"Spiral into the sun seems" seems extreme. Given conservation of energy and conservation of angular momentum, there is a range of orbital radii which are permissible. For a wide range of force laws and initial conditions, you can't spiral in and you can't escape.

For most of these force laws and most initial conditions you will not have simple closed orbits that arrive back at their starting point. Whether to call these orbits "stable" is a different question. I'd call them stable but not closed.

 

[Nugatory]

For a wide range of force laws and initial conditions, you can't spiral in and you can't escape.

gah - yes, I did a brain slide from no stable (against perturbation) and closed orbits into no orbits.