Why does the outward increase of angular momentum in an accretion disk make the disk Rayleigh stable? Since the density profile in an accretion disk increases inward in the case of constant accretion rate, there should not be an instability in between consecutive rings. Is that the reason why they are Rayleigh stable?
Hello Aria,
Thanks for giving reference to the above paper. In the paper we find mention in the Introduction part that and I quote, "Chandrasekhar has given a detailed account of the instability of the plane interface between two incompressible and viscous fluids of different densities when the lighter fluid is accelerated into the heavier."
So in our accretion disk also, since the density profile increases inwards, the lighter fluid is being accelerated into the heavier. So in this case how does the instability work and what brings about the stability of the disk??
Kindly clarify.
Okay one more reference -- Advective accretion disks and related problems including magnetic fields Review Article
New Astronomy Reviews, Volume 45, Issues 11–12, December 2001, Pages 663-742 ---- They write 'Development of turbulence in an accretion disk cannot be explained simply. A Keplerian disk is stable in the linear approximation to the development of axially symmetric perturbations, conserving the angular momentum. That is, the disks satisfy the
Rayleigh criterion for stability.' Kindly explain in simple terms.
The Rayleigh criterion, relies on fluid elements retaining their angular momenta.
Following Rayleigh, suppose we interchange the fluid in two rings, of equal masses, at radii r1 and r2 . So the fluid initially at r1 is moved to r2 , conserving its angular momentum L1 . Similarly the fluid initially at r2 is moved to r1 , conserving its angular momentum L2 . Then,
change in KE ∼ (L2^2 − L1^2 )(r1^-2 - r2^-2)
Now a system always wants to go to the lowest energy state. If angular momentum is a increasing function of r, then the swapping of fluid parcels is going to cost the system to increase its energy, so the system will not try to do it and the system is stable to this perturbation.
However in case when angular momentum is a decreasing function of r, then the system can release its energy by swapping of fluid parcels and the system will try to do so, so the system is unstable to this perturbation.
So we can say that when the angular momentum is a increasing function of r in an accretion disk, the disk is Rayleigh stable.
- Badges
- Science topic