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Appendix A: Non-inertial linear response theory for gas instabilities in spherical coordinates

In this Appendix, we develop the mathematics of the linear response theory introduced in Sect. 2. The unstable periphery of a gaseous sphere penetrating an external medium is considered in a non-inertial reference frame, and kinematic and dynamical boundary conditions are considered. An instability criterion is obtained in spherical coordinates and plane geometry limit is considered.

Appendix A.1: Kinematic boundary conditions

Appendix A.1.1: Perturbed surface

In the geometrical framework introduced in Sect. 2.2, we consider a potential-flow type description of the surface of the galaxy in its motion throughout an intra-cluster medium. The distribution of the molecular clouds in the ISM of the galaxy is described with a density distribution ρ = ρ(ξ) in S₁ bordered by a surface Σ surface of the frontier of the domain of existence of the (bound) density function ρ: limξ → ∞ρ< ∞. No singularity is allowed in the potential-density couple satisfying the associated Poisson equation ΔΦ = 4πGρ. This distribution is then perturbed to a new state, corresponding to a new perturbed surface density (where the spherical coordinates introduced above in S₁ have been employed). We will limit ourselves to a linear analysis and we assume the defining equation for the surface Σ(ξ,θ,φ;t) = 0 to be given by Eq. (1).

Appendix A.1.2: Internal gas perturbed potential flow

We will refer to a quantity of the orbiting stellar system as “internal”, e.g. its density ρ_in, velocity potential ϕⁱⁿ, etc. To describe the cold ISM, we will use the solution for the Laplace equation for a stationary expanding/contracting potential flow written as (e.g. Landau & Lifshitz 1959) to which we add the perturbation solution of the Laplace equation proportional to ξ^l, i.e. (with B_lm proportionality coefficients of the spherical harmonic basis): (A.1)where we already excluded terms proportional to ξ^{− 1 − l} in the radial solution of the Laplace equation ϕ ∝ A_lmξ^{− l} + B_lmξ^l by setting their corresponding coefficients A_lm = 0. This is done to avoid divergences as long as we move away from Σ inward into the galaxy. To ensure continuity of the surface element fluids at the surface, we proceed in the standard way (e.g. Batchelor 2000) by evaluating the kinematical boundary conditions (i.e., of the Eulerian derivative at the surface) of the fluid elements at the perturbed surface (see Eq. (1)): (A.2)where ∂_x is a compact notation for the derivative . With and by computing the spatial gradient components in S₁ as , as well as , Eq. (A.2)reduces to an equation for the parameters B_lm:

obtained by Eq. (A.2)with the terms computed above and by simple substitution of the perturbed surface of Eq. (1). This equation can be solved for B_lm as: (A.3)obtained by collecting the common terms. We now linearize the previous result to the first order in η. After a McLaurin expansion in η, we find the following compact form for the coefficients B_lm: (A.4)Inserting Eq. (A.4)in Eq. (A.1)helps us to obtain the final form of the potential vector to the first order as: (A.5)which is Eq. (4) of Plesset (1954). Differently from Plesset (1954), we are here interested in describing the motion of the dwarf galaxy along its orbit in the bath of a hotter, lighter intergalactic medium, or vice versa, the motion of this intergalactic medium impacting the dwarf galaxy in its orbital evolution as it appears in the reference frame S₁. This case has similarity with the problem presented in Paper I and was extensively treated in the context of stellar convection by Pasetto et al. (2014). We adapt their formalism and extend their results to this non-axisymmetric context.

Appendix A.1.3: External gas perturbed potential flow

We will refer to a quantity external to the orbiting system as “outside” the system, e.g. the hot intra-cluster medium density ρ_out, its velocity potential ϕ^out, etc. The potential flow for the hot intergalactic medium written in the reference frame S₁ comoving with the stellar system, , is introduced in the previous section, but see recently also Pasetto et al. (2014), as . To these terms, we add now the term computed above for the expansion/contraction of the galaxy ϕ_v1, and the perturbation solution of the Laplace equation proportional to written as to get (A.6)where in contrast to the previous case of Eq. (A.1)we want here to exclude terms proportional to ξ^l by setting their corresponding coefficients B_lm = 0 in the Laplace equation because we do not want to consider divergences as long as we go far outside the dwarf galaxy away from Σ. Again as in Eq. (A.2), we proceed by evaluating the kinematical boundary conditions of the fluid element at the surface (A.7)where the only difference from the previous Eq. (A.2)is that the velocity potential gradients are now derived as . Considering this difference, we proceed exactly as done above for the , to obtain an equation that is linear in A_lm and that can be solved as: (A.8)We now linearize the previous result to the first order in η. We obtain (A.9)Finally, we obtain the potential velocity in the following simplified form: (A.10)where we have inserted Eq. (A.9)in Eq. (A.6)and accepted minor simplifications.

A sanity check shows that this can be reduced in the unperturbed case, η → 0, to the result . This was already suggested in Paper I and extensively considered in a different context in Pasetto et al. (2014). For v = 0 ∧ η ≠ 0, this reduces to the Eq. (5) of Plesset (1954).

Appendix A.2: Dynamical boundary condition

At the surface radius in which we have chosen to represent the galaxy size, apart from the kinematic boundary condition we want to express the condition of continuity of the stress vector (i.e. the dynamic boundary condition). The stress vector s_out and s_in inside and outside the surface Σ must satisfy the condition ⟨ n,s_out ⟩ _{Σ = 0} = ⟨ n,s_in ⟩ _{Σ = 0} so that for inviscid fluids (s = −pI with I identity matrix) we obtain the standard literature dynamical boundary condition p_out = p_in to be treated now thus accounting for the ram pressure that the galaxy is experiencing in its motion.

Appendix A.2.1: Internal gas pressure equation

Now we need the task to impose the dynamical boundary condition of the external gas medium on the internal stellar system gas at each position of its perturbed scale-radius surface r_s + δr_s. Given the framework developed in Sect. 2.2, we can make use of Eq. (7) of Paper I where the non-inertial character of the system S₁ is taken into account. In this notation, we can compute the velocity for the molecular clouds of the dwarf galaxy ∥ v₁ ∥ as is the internal fluid of the galaxy that is inert with respect to the reference frame S₁ comoving with the stellar system. Equation (7) of Paper I in this case reads: (A.11)where ⟨ a_O′,ξ ⟩ is the projection of the acceleration along the position vector ξ and fⁱⁿ(t) is a constant of the space, not depending on B_lm, which we determine by imposing the boundary condition far away from the ideal radius r_s (at infinity). This is because we assume hydrostatic equilibrium far away from the molecular cloud borders and the function fⁱⁿ is therefore determined by the limit of the previous equation for ∥ ξ ∥ → ∞ as shown in Paper I.

Because the lifetime of a molecular cloud (given the star formation efficiency expected to act in the systems under study, see Fig. 1 of Paper I) is much shorter (<300 Myr) than the time-scale over which the orbital parameters change significantly, we assume the velocity of the fluid impacting the dwarf galaxy molecular clouds to be uniform and constant (in S₁). We also neglect non-orthogonal components of the acceleration that remain constant in time along the lifetime of the molecular clouds. In this case, we write simply ⟨ a_O′,ξ ⟩ = a_O′ξcosϑ with ϑ being the angle between a_O′ and ξ. In general, ϑ ≠ θ apart from particular orbits (or part of them).

To progress with Eq. (A.11)we need to evaluate and to first order in the small parameter η. Differentiating Eq. (A.5) yields: (A.12)to be evaluated at the perturbed location . We expand this to the first order to obtain: (A.13)The procedure advances exactly the same for the gradient components and (A.16)We preferred a slightly longer formalism in the first lines of these equations to show the terms proportional to ξ so that in the second lines we simplify their substitution at the perturbed location, and in the third lines (Eqs. (A.14)–(A.16)) the remaining terms emerge more clearly. Other more compact formulas can be worked out if necessary but reduce the readability. Equation (A.11)to the first order on the perturbation is then: (A.17)From this equation we can calculate the pressure. As a “sanity check”, if we require the reference system to be inertial, then the apparent forces disappear a_O′ = 0 and for a zero flow velocity as well as for the case of no perturbation η = 0 we get , which is the standard literature equation of the expanding/contracting bubble for zero surface tension (e.g. Batchelor 2000). Finally, it is evident that when the perturbation is not null but no velocity fluid is included v = 0 we obtain the results in Plesset (1954). Hence in these cases our results reduce to well-known results in the literature.

Appendix A.2.2: External gas pressure equation

In the external gas case, the pressure equation is again obtained from the Bernoulli equation by adding the inertial term as in the previous section. Here, writing our equations in S₁ instead of S₀ results in a slightly more complex formalism; nevertheless, the procedure is the same as outlined above and the approach will result in an easier physical interpretation of our final results. We consider the terms in the following equation (A.18)which we evaluate at the perturbed location . Here the free function has been previously derived in Paper I (their Eq. (8)), and v_rel is the velocity of the fluid impacting the stellar system in S₁, i.e. the velocity of the stellar system itself (apart from the sign). The reason for calling it v_rel, instead of simply ∥ v_O′ ∥ = v, will be clearer later on.

For each term in Eq. (A.18)to the first order we obtain: Equation (A.18)is simplified by collecting Eqs. (A.19)–(A.22)once the scalar product is taken into account. As before, the solution of Eq. (A.18)can be obtained in terms of the pressure p and it can be simplified by retaining only the first order terms. We have (hereafter we define l₊ ≡ l + 1, l₊₊ ≡ l + 2 and l₋ ≡ l − 1, etc. to minimize the notation) (A.23)Again we can check the validity of this equation by assuming no perturbation η → 0 and l = 0 to prove that it effectively reduces to the Theorem of Sect. 3 in Pasetto et al. (2014) as a particular case.

Appendix A.2.3: Surface of equilibrium

Taking the difference between Eqs. (A.17)and (A.23), we express the continuity condition of the pressure impacting the stellar system from the external gas (the ram pressure condition). The equation of motion for the unperturbed equation is: (A.24)Our disposition of the terms indicates immediately that in S₀, without the motion of the fluid or the sphere, we obtain , which indicates the condition of equilibrium where always ρ_in ≠ ρ_out as . This, for example, may describe a case of a galaxy lying at the centre of a galaxy cluster. Therefore, because we are interested in the growth of the perturbation over an equilibrium state (for at least one instability mode), in the resulting equation we only need to study the terms proportional to the perturbation terms (i.e. the terms containing the spherical harmonics) that we analyse in the next section. We move from this equation to investigate the more interesting case of the differential equation for η = η(t) from which, stability condition for the growth of a perturbation can be derived.

Appendix A.3: Condition for the instability

The condition for the instability is derived by considering only the perturbed terms in difference between Eqs. (A.17)and (A.23). Collecting terms in and its derivatives we obtain an equation of the form for some form of the functions and i = 1,2,3, which suggests that we define two special functions as follows: independent from η or its derivatives. In this way, we obtain an equation for the perturbation η of the form which immediately produces an interesting result as follow: the presence of a preferential direction for the motion of the galaxy along its orbit induces a symmetry on the perturbations. The dependence on the considered azimuthal mode remains, i.e. the dependence on m, nevertheless it becomes independent from the azimuthal direction φ. This is an interesting simplification that is a consequence of the geometry assumed.

The study of the stability of the solution of an equation of the form is better performed if we convert it to an eigen-value problem. To proceed in this way, we collect the terms depending on the perturbation factor η and its derivatives. With the aid of Eqs. (A.25)and (A.26)we put the differential equation in standard form. Hence, the more suitable form for starting our stability analysis obtained by taking only the perturbed terms that differ between Eq. (A.17)and (A.23)and accounting for Eqs. (A.25)and (A.26)is (A.27)Despite its complicated form, this equation is formulated in a suitable way to show that it can reduce to Eq. (13) in Plesset (1954).

Considering we have no known terms in the left-hand side (LHS) of Eq. (A.27), i.e. it is a second order ODE of the type for η = η(t), we can attempt a classical quantum mechanics Wentzel-Kramers-Brillouin (WKB) approach to the solution by making use of the transformation (A.28)where on purpose we choose,

to simplify Eq. (A.27)to a standard eigenvalues problem with slowly varying coefficient: (A.29)whose solution is conveniently carried out in WKB approximation. However, we will accomplish a much simpler task here. We are interested in the condition for which at least one mode is unstable, and the instability of the harmonic oscillator equation Eq. (A.29)is well known to depend on the positivity of the growth factor γ²(θ;t) > 0 where (A.30)This represents the desired equation already presented in a more compact fashion in Eq. (7). For the purpose in this Appendix of recovering some limit-cases we will explicitly keep the terms in Eq. (A.30)written in their full extension. γ² relates to the growth of the perturbation for which we were searching (it is indeed called the growth factor). This completes our theoretical framework and equips us with the tools to investigate the growth of the instabilities by compression or instabilities that lead to star formation.

Appendix A.3.1: Special limits

We start by remarking how in the case of a non-inertial reference frame S₁, the instability condition reduces to the study of the positivity of the last row (i.e. the third) of Eq. (A.30)(that we rewrite in a compact way as (with Eq. (6)of Sect. 2.2): (A.31)This equation was already presented by Plesset (1954) (his Eq. (17) with zero surface tension).

Another important limit to recover is the plane case. In the spherical geometry that we have assumed, the plane case can be achieved by taking l → ∞ and R → ∞ and keeping the wave number of the perturbation, k, constant. This is not a trivial task for the presence of the special functions F₁ and F₂ defined in Eqs. (A.25)and (A.26)whose dependence on l involves the determination of the Euler Gamma function for large values of the index l. We refer the interested reader to Appendix B for the computation of their asymptotic behaviour for large l because of its exclusively mathematical nature. Using the results of Appendix B we can show that Eq. (A.30)behaves in the plane case as (A.32)Then, if we define, as usual, the acceleration to be the previous equation reduces to (A.33)which can be easily interpreted remembering the plane case in the literature as discussed for Eq. (4). We indicated with a_⊥ the acceleration orthogonal to the surface that in the plane case represents the vertical direction. Hence, the first term is exactly the instability criterion for the RT effect where the effective acceleration g_eff = a_⊥ − g has been corrected for the presence of the corrective-term a_⊥. In the same way, the second term retains the key dependencies from the relative velocity v_rel = v_in − v_out between the fluid above and below the surface dividing the two sliding fluids that are the basis of the KH instability. These criteria become equivalent to the RT and KH criteria (apart from the numerical factors 9/4ρ_out) at the stagnation point, where cosϑ = 1 and sin²θ = 0 and at the tangent to the sphere cosϑ = 0 and sin²θ = 1, respectively.

Appendix B: Asymptoctic expansion of the functions F₁ and F₂

We elaborate in this Appendix on more mathematical theorems that can be skipped in a first reading.

We are interested in the limits of the special functions F₁ and F₂ defined and in their asymptotic expansion. A plot of the two functions for the instability mode of interest (l = 2) and the angular dependence of interest , is presented in Fig. B.1, where and respectively.

As the index l tends to approach ∞ we can write for while As already discussed in association with Fig.1, there is no observational evidence of strong azimuthal asymmetries in dwarf galaxies of the LG, thus it is safe to assume the perturbation to be well represented by modes with m = 0. With this assumption, in the stagnation point direction θ = 0 is More cumbersome is the same limit for the case . We obtain where in order to prove this theorem the Stirling expansion for the Gamma function has to be considered.

	Fig. B.1 Special function F1 and F2 for the modal perturbation l = 2. The divergence (dashed vertical line) is located at . Only the angular range of interest is accounted.
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By analogy with the previous proofs we get where we used the “big-O” to express that the limit is increasing to infinity as the power written. Once introduced in Eq. (A.30)this behaviour cancels out to the desired limit offering the finite limit we wrote in Eq. (A.32). With the same abuse of notation now clearly we can write This proves the asymptotic behaviour at the leading order of the special functions F₁ and F₂.

© ESO, 2014