Issue 
A&A
Volume 564, April 2014



Article Number  A7  
Number of page(s)  9  
Section  Extragalactic astronomy  
DOI  https://doi.org/10.1051/00046361/201323325  
Published online  26 March 2014 
Online material
Appendix A: Expressions of the enthalpy
We assume a polytropic equation of state , where C is a positive constant and γ the adiabatic index. The speed of sound is defined by , so that when γ ≠ 1, the enthalpy can be written as: (A.1)In the case of an isothermal equation of state (γ = 1), the speed of sound is constant and entirely fixed by temperature. The isothermal equation of state can be written and the enthalpy yields: (A.2)In both cases, the first order perturbation of the enthalpy can be written as: (A.3)Indeed, the perturbed fluid remains polytropic, so up to the first order in ρ_{1}/ρ_{0}, when γ ≠ 1,
Appendix B: Discriminant of the dispersion relation
Appendix B.1: The discriminant is always positive
The dimensionless dispersion relation introduced in Sect. 3 (Eq. (22)) involves two dimensionless quantities, α and β. Considering that (B.1)the discriminant can be written as a second order polynomial expression in β: (B.2)In turn, the discriminant Δ′ of this latter polynomial expression is always negative: (B.3)Consequently, as β is a real quantity, the discriminant Δ is always positive and always real.
Appendix B.2: Two roots are real
There are four solutions for x, which can be either real or with a nonzero imaginary part, depending on the sign of . But out of these four solutions, two are always real. Indeed, is always positive:

If , .

If α > 0, β has to be negative because of Eq. (B.1), so Δ = α^{2} − 4β > α^{2} and .
Appendix B.3: A condition for stability
All roots are real and the system is stable when and only when is also positive, i.e., when and . These two conditions can be expressed as conditions on the total wavenumber k: The second condition encompasses the first one, and is analogous to the Jeans criterion. All roots are thus real and the system is stable for , with .
Appendix C: About the fastest growing mode
Appendix C.1: A parametric expression for the fastest growing mode
At a given radius R and for a given k_{R}, the quantity introduced in Sect. 3.1 is minimal when (C.1)When k_{z} ≠ 0, the latter condition yields (C.2)Using the expressions of α and β and developing the intermediate expressions, the second equation leads to the following parametric equation describing the fastest growing mode: (C.3)
Appendix C.2: Intersection with the line k_{z} = k_{R}
In order to characterize the shape of the perturbations, it would be of interest to determine the intersection of the curve describing the fastest growing mode with the line k_{z} = k_{R}, which corresponds to spherical perturbations. At the intersection,
and k_{R} satisfies (C.4)It reduces to the following polynomial expression, which can be solved numerically: (C.5)where k_{0} = ω_{0}/c_{0} is a characteristic wavenumber depending on the position R through c_{0}.
© ESO, 2014
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