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 ρ₁/ρ₀, when γ ≠ 1,

and 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 non-zero 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 = α² − 4β > α² 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 k₀ = ω₀/c₀ is a characteristic wavenumber depending on the position R through c₀.

© ESO, 2014