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13.2 Instabilities of rotating perfect fluid stars

The importance of the canonical energy stems from the fact that it can be used to test the stability of the system. In particular:

Consider a real frequency normal-mode solution to the perturbation equations, a solution of form ξ = ˆξei(ωt+mϕ). One can readily show that the associated canonical energy becomes

[ ] i- Ec = ω ω ⟨ξ,A ξ⟩ − 2 ⟨ξ,B ξ⟩ , (250 )
where the expression in the bracket is real. For the canonical angular momentum we get
[ ] i- Jc = − m ω ⟨ξ,A ξ⟩ − 2 ⟨ξ,B ξ⟩ . (251 )
Combining Equation (250View Equation) and Equation (251View Equation) we see that, for real frequency modes, we will have
-ω Ec = − m Jc = σpJc, (252 )
where σp is the pattern speed of the mode.

Now notice that Equation (251View Equation) can be rewritten as

--Jc--- ⟨ξ,iρvj∇j-ξ⟩ ⟨ˆ ˆ⟩ = − m ω + m ⟨ ˆ ˆ⟩ . (253 ) ξ,ρξ ξ,ρξ
Using cylindrical coordinates, and vj = Ω ϕj, one can show that
[ ] ∗ j i ||ˆ||2 ˆ∗ ˆ − iρξiv ∇j ξ = ρΩ m |ξ| + i(ξ × ξ)z . (254 )
But
| | ||2 ||(ξˆ∗ × ˆξ)z|| ≤ ||ˆξ|| (255 )
and we must have (for uniform rotation)
( 1) Jc ∕m2 ( 1 ) σp − Ω 1 + m- ≤ ⟨-ˆ--ˆ⟩ ≤ σp − Ω 1 − m- . (256 ) ξ,ρ ξ

Equation (256View Equation) forms a key part of the proof that rotating perfect fluid stars are generically unstable in the presence of radiation [45Jump To The Next Citation Point]. The argument goes as follows: Consider modes with finite frequency in the Ω → 0 limit. Then Equation (256View Equation) implies that co-rotating modes (with σp > 0) must have Jc > 0, while counter-rotating modes (for which σp < 0) will have Jc < 0. In both cases Ec > 0, which means that both classes of modes are stable. Now consider a small region near a point where σp = 0 (at a finite rotation rate). Typically, this corresponds to a point where the initially counter-rotating mode becomes co-rotating. In this region Jc < 0. However, Ec will change sign at the point where σp (or, equivalently, the frequency ω) vanishes. Since the mode was stable in the non-rotating limit this change of sign indicates the onset of instability at a critical rate of rotation.


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