- #1
pop_ianosd
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- TL;DR Summary
- I'm trying to understand why a stability criterion derived in a course for circular orbits in a general force field is valid.
I'm also trying to obtain a simple argument for why circular orbits in an inverse-cubed force field are unstable.
In this chapter, the stability of an object orbiting in a circular orbit of radius [itex]r_c[/itex] in an arbitrary force field [itex]f[/itex] is considered.
The author arrives at the equation of a harmonic oscillator, for small deviations [itex]x[/itex] from the circular orbit:
[tex] \ddot{x} + \left[-3\frac{f(r_c)}{r_c} - f'(r_c)\right]x= 0 [/tex]
From here follows that if the sign of the coefficient of [itex]x[/itex] is positive the orbit is stable.
My most specific question is about the case where [itex]f = -\frac{c}{r^3}[/itex]. The author claims that in order to prove that it is unstable, we must keep the second order term when approximating the equation (308). That equation, with such a specific force would look like this:
[tex] \ddot{x} = \frac{h^2 - c}{(r_c + x)^3} [/tex]
Since we are considering circular orbits, we have [itex]-\frac{h^2}{r_c^3} = -\frac{c}{r_c^3}[/itex] (eq. 306 in the link), so the right term in the equation above is always zero, and I therefore don't see how keeping the second order term would help.
My question is: what am I doing wrong, or do you have a better guess for what the author meant.
The way I would prove instability in this case is to derive the solutions to the unapproximated equation, and observe that the initial conditions that produce bounded solutions sit on a line, and not in a volume, in the initial conditions space. But I'm not too satisfied with this argument, as it deals with the concept of boundness rather than stability, and it just doesn't seem very elegant.
Returning to the criterion above, I'm also not convinced of it. I have the intuition that it would be a necessary condition for stability, but I wonder whether it is sufficient.
The author arrives at the equation of a harmonic oscillator, for small deviations [itex]x[/itex] from the circular orbit:
[tex] \ddot{x} + \left[-3\frac{f(r_c)}{r_c} - f'(r_c)\right]x= 0 [/tex]
From here follows that if the sign of the coefficient of [itex]x[/itex] is positive the orbit is stable.
My most specific question is about the case where [itex]f = -\frac{c}{r^3}[/itex]. The author claims that in order to prove that it is unstable, we must keep the second order term when approximating the equation (308). That equation, with such a specific force would look like this:
[tex] \ddot{x} = \frac{h^2 - c}{(r_c + x)^3} [/tex]
Since we are considering circular orbits, we have [itex]-\frac{h^2}{r_c^3} = -\frac{c}{r_c^3}[/itex] (eq. 306 in the link), so the right term in the equation above is always zero, and I therefore don't see how keeping the second order term would help.
My question is: what am I doing wrong, or do you have a better guess for what the author meant.
The way I would prove instability in this case is to derive the solutions to the unapproximated equation, and observe that the initial conditions that produce bounded solutions sit on a line, and not in a volume, in the initial conditions space. But I'm not too satisfied with this argument, as it deals with the concept of boundness rather than stability, and it just doesn't seem very elegant.
Returning to the criterion above, I'm also not convinced of it. I have the intuition that it would be a necessary condition for stability, but I wonder whether it is sufficient.