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Epicyclic frequency

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Characteristic of accretion discs
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In astrophysics, particularly the study of accretion disks, the epicyclic frequency is the frequency at which a radially displaced fluid parcel will oscillate. It can be referred to as a "Rayleigh discriminant". When considering an astrophysical disc with differential rotation Ω {\displaystyle \Omega } , the epicyclic frequency κ {\displaystyle \kappa } is given by

κ 2 ≡ 2 Ω R d d R ( R 2 Ω ) {\displaystyle \kappa ^{2}\equiv {\frac {2\Omega }{R}}{\frac {d}{dR}}(R^{2}\Omega )} , where R is the radial co-ordinate.

This quantity can be used to examine the 'boundaries' of an accretion disc: when κ 2 {\displaystyle \kappa ^{2}} becomes negative, then small perturbations to the (assumed circular) orbit of a fluid parcel will become unstable, and the disc will develop an 'edge' at that point. For example, around a Schwarzschild black hole, the innermost stable circular orbit (ISCO) occurs at three times the event horizon, at 6 G M / c 2 {\displaystyle 6GM/c^{2}} .

For a Keplerian disk, κ = Ω {\displaystyle \kappa =\Omega } .

Derivation

An astrophysical disk can be modeled as a fluid with negligible mass compared to the central object (e.g. a star) and with negligible pressure. We can suppose an axial symmetry such that Φ ( r , z ) = Φ ( r , − z ) {\displaystyle \Phi (r,z)=\Phi (r,-z)} . Starting from the equations of movement in cylindrical coordinates : r ¨ − r θ ˙ 2 = − ∂ r Φ r θ ¨ + 2 r ˙ θ ˙ = 0 z ¨ = − ∂ z Φ {\displaystyle {\begin{aligned}{\ddot {r}}-r{\dot {\theta }}^{2}&=-\partial _{r}\Phi \\r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}&=0\\{\ddot {z}}&=-\partial _{z}\Phi \end{aligned}}}

The second line implies that the specific angular momentum is conserved. We can then define an effective potential Φ e f f = Φ − 1 2 r 2 θ ˙ 2 = Φ − h 2 2 r 2 {\displaystyle \Phi _{eff}=\Phi -{\frac {1}{2}}r^{2}{\dot {\theta }}^{2}=\Phi -{\frac {h^{2}}{2r^{2}}}} and so : r ¨ = − ∂ r Φ e f f z ¨ = − ∂ z Φ e f f {\displaystyle {\begin{aligned}{\ddot {r}}&=-\partial _{r}\Phi _{eff}\\{\ddot {z}}&=-\partial _{z}\Phi _{eff}\end{aligned}}}

We can apply a small perturbation δ r → = δ r e → r + δ z e → z {\displaystyle \delta {\vec {r}}=\delta r{\vec {e}}_{r}+\delta z{\vec {e}}_{z}} to the circular orbit : r → = r 0 e → r + δ r → {\displaystyle {\vec {r}}=r_{0}{\vec {e}}_{r}+\delta {\vec {r}}} So, r → ¨ + δ r → ¨ = − ∇ → Φ e f f ( r → + δ r → ) ≈ − ∇ → Φ e f f ( r → ) − ∂ r 2 Φ e f f ( r → ) δ r − ∂ z 2 Φ e f f ( r → ) δ z {\displaystyle {\ddot {\vec {r}}}+\delta {\ddot {\vec {r}}}=-{\vec {\nabla }}\Phi _{eff}({\vec {r}}+\delta {\vec {r}})\approx -{\vec {\nabla }}\Phi _{eff}({\vec {r}})-\partial _{r}^{2}\Phi _{eff}({\vec {r}})\delta r-\partial _{z}^{2}\Phi _{eff}({\vec {r}})\delta z}

And thus : δ r ¨ = − ∂ r 2 Φ e f f δ r = − Ω r 2 δ r δ z ¨ = − ∂ r 2 Φ e f f δ z = − Ω z 2 δ z {\displaystyle {\begin{aligned}\delta {\ddot {r}}&=-\partial _{r}^{2}\Phi _{eff}\delta r=-\Omega _{r}^{2}\delta r\\\delta {\ddot {z}}&=-\partial _{r}^{2}\Phi _{eff}\delta z=-\Omega _{z}^{2}\delta z\end{aligned}}} We then note κ 2 = Ω r 2 = ∂ r 2 Φ e f f = ∂ r 2 Φ + 3 h 2 r 4 {\displaystyle \kappa ^{2}=\Omega _{r}^{2}=\partial _{r}^{2}\Phi _{eff}=\partial _{r}^{2}\Phi +{\frac {3h^{2}}{r^{4}}}} In a circular orbit h c 2 = r 3 ∂ r Φ {\displaystyle h_{c}^{2}=r^{3}\partial _{r}\Phi } . Thus : κ 2 = ∂ r 2 Φ + 3 r ∂ r Φ {\displaystyle \kappa ^{2}=\partial _{r}^{2}\Phi +{\frac {3}{r}}\partial _{r}\Phi } The frequency of a circular orbit is Ω c 2 = 1 r ∂ r Φ {\displaystyle \Omega _{c}^{2}={\frac {1}{r}}\partial _{r}\Phi } which finally yields : κ 2 = 4 Ω c 2 + 2 r Ω c d Ω c d r {\displaystyle \kappa ^{2}=4\Omega _{c}^{2}+2r\Omega _{c}{\frac {d\Omega _{c}}{dr}}}

References

  1. p161, Astrophysical Flows, Pringle and King 2007


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