During the core collapse of a massive star just before its supernova explosion the amplification of asymmetric motions by the standing accretion shock instability SASI imprints on the neutrino flux and the gravitational waves a frequency signature carrying direct information on the explosion process. The physical interpretation of this multimessenger signature requires a detailed understanding of the instability mechanism. A perturbative analysis is used to characterize the properties of SASI and assess the effect of the region of neutronization above the surface of the protoneutron star. The eigenfrequencies of the most unstable modes are compared to those obtained in an adiabatic approximation where neutrino interactions are neglected above the neutrinosphere. The differential system is solved analytically using a Wronskian method and approximated asymptotically for a large shock radius. The oscillation period of SASI is well fitted with a simple analytic function of the shock radius the radius of maximum deceleration and the mass of the protoneutron star. The oscillation period is weakly dependent on the parametrized cooling function which however affects the SASI growth rate. The general properties of SASI eigenmodes are described using an adiabatic model. In this approximation the eigenvalue problem is formulated as a selfforced oscillator. The forcing agent is the radial advection of baroclinic vorticity perturbations and entropy perturbations produced by the shock oscillation. The differential system defining the eigenfrequencies is reduced to a single integral equation. Its analytical approximation sheds light on the radially extended character of the region of advectiveacoustic coupling. The simplicity of this adiabatic formalism opens new perspectives to investigate the effect of stellar rotation and nonadiabatic processes on SASI.
['Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Solar and Stellar Astrophysics']
Analytic insight into the physics of SASI I. Shock instability in a nonrotating stellar core
{'I. Shock instability in a non-rotating stellar core': 'T. Foglizzo  \nUniversité Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France e-mail: foglizzo@cea.fr  \nReceived 18 September 2024; accepted 23 October 2024', 'ABSTRACT': 'Context. During the core collapse of a massive star just before its supernova explosion, the amplification of asymmetric motions by the standing accretion shock instability (SASI) imprints on the neutrino flux and the gravitational waves a frequency signature carrying direct information on the explosion process.  \nAims. The physical interpretation of this multi-messenger signature requires a detailed understanding of the instability mechanism. Methods. A perturbative analysis is used to characterize the properties of SASI, and assess the e ff ect of the region of neutronization above the surface of the proto-neutron star. The eigenfrequencies of the most unstable modes are compared to those obtained in an adiabatic approximation where neutrino interactions are neglected above the neutrinosphere. The di ff erential system is solved analytically using a Wronskian method and approximated asymptotically for a large shock radius.  \nResults. The oscillation period of SASI is well fitted with a simple analytic function of the shock radius, the radius of maximum deceleration and the mass of the proto-neutron star. The oscillation period is weakly dependent on the parametrized cooling function which however a ff ects the SASI growth rate. The general properties of SASI eigenmodes are described using an adiabatic model. In this approximation the eigenvalue problem is formulated as a self-forced oscillator. The forcing agent is the radial advection of baroclinic vorticity perturbations and entropy perturbations produced by the shock oscillation. The di ff erential system defining the eigenfrequencies is reduced to a single integral equation. Its analytical approximation sheds light on the radially extended character of the region of advective-acoustic coupling. The simplicity of this adiabatic formalism opens new perspectives to investigate the e ff ect of stellar rotation and non-adiabatic processes on SASI.  \nKey words. hydrodynamics - instabilities - shock waves - stars: neutron - supernovae: general - neutrinos', '1. Introduction': 'The explosive death of massive stars is sensitive to the development of hydrodynamical instabilities which break the spherical symmetry of the stellar core, a ff ect the e ffi ciency of neutrino energy absorption and generate turbulence (Müller 2020; Janka et al. 2016; Burrows & Vartanyan 2021). A spherically symmetric scenario based on radial motions seems possible only for the lightest progenitors (Kitaura et al. 2006; Stockinger et al. 2020). Asymmetric motions contribute to the kick and spin of the neutron star (Müller et al. 2019; Janka 2017), the emission of gravitational waves (Kotake & Kuroda 2017) and a modulation of neutrino emission (Müller 2019b; Tamborra & Murase 2019). The correlation of instability signatures in the gravitational waves and neutrino signals has been considered by Drago et al. (2023) to improve the detection e ffi ciency of gravitational waves from nearby supernovae. Ultimately, the information encoded in the gravitational waves and neutrino signals can be used to recover the properties of the dying star and its explosion mechanism (Powell & Müller 2022). The computational cost of 3D numerical simulations precludes a systematic coverage of the large parameter space describing the initial conditions of the stellar core-collapse. Understanding the underlying mechanism of instabilities is essential to extrapolate the results of sparse numerical simulation, evaluate the impact of additional physical ingredients, and design e ff ective prescriptions for parametric studies (Müller 2019a).  \nAmong the hydrodynamical instabilities at work during the phase of stalled accretion shock, the Standing Accretion Shock Instability (Blondin et al. 2003) is able to introduce coherent transverse motions with a large angular scale, growing over a timescale related to the advection time from the shock to the neutron star surface. The mechanism of SASI has been described as an advective-acoustic cycle between the shock and the vicinity of the proto-neutron star (Foglizzo et al. 2007; Fernández & Thompson 2009b; Scheck et al. 2008). The analytical understanding of SASI eigenfrequencies in the WKB approximation is however restricted to the limit of a large shock radius for high frequency overtones (Foglizzo et al. 2007) while the lowest frequency fundamental mode is often the most unstable. A fully analytic solution including the fundamental mode was obtained only in a very idealized model where the stationary flow is plane parallel and uniform except in a compact region of deceleration mimicking the vicinity of the neutron star (Foglizzo 2009) (hereafter F09). This toy model neglected the flow gradients that are actually extended all the way from the shock to the neutron star and the non-adiabatic character of the neutrino processes taking place in the vicinity of the neutron star. Despite these limitations, this model illustrated the interplay of the advective-acoutic and the purely acoustic cycles, and the expected phase mixing taking place at high frequency. It was used by Guilet & Foglizzo (2012) to interpret the frequency spacing of the eigenspectrum in the framework of the advective-acoustic mechanism rather than  \na purely acoustic process. The physical understanding of SASI led Müller & Janka (2014) to define an empirical formula for the SASI oscillation period, in order to interpret the modulation of the neutrino signal. Directly proportional to an approximation of the advection time from the shock to the neutron star surface, it was tested on a 2D numerical simulation of the collapse of a 25 M ⊙ progenitor. However its generality for other progenitors, proposed by Müller (2019b), has not been established yet.  \nThe present study aims at improving our understanding of the fundamental mode of SASI in spherical geometry in order to better identify the physical parameters governing its oscillation frequency. Simple cooling functions mimicking neutrino emission will be used to assess the role of the cooling region. An adiabatic approximation will also be used to assess the role of the advective-acoustic coupling in the radially extended region between the shock and the proto-neutron star. The adiabatic approximation is motivated by the adiabatic simulations of Blondin & Mezzacappa (2007) and by the shallow water experiment (Foglizzo et al. 2012, 2015), which both suggest that several properties of SASI may be understood using adiabatic equations. The adiabatic approximation has also been used to analyze the impact of general relativistic corrections on SASI eigenfrequencies (Dunham et al. 2024).  \nThe set of di ff erential equations defining the eigenfrequencies of spherical accretion is recalled in Sect. 2 with particular attention to the radial extension of the non-adiabatic cooling layer and its impact on the oscillation period of SASI. The eigenfrequencies calculated in the adiabatic approximation are compared to those including neutrino losses in Sect. 3. The adiabatic model is formulated as a self-forced oscillator in Sect. 4, with eigenfrequencies defined by an integral equation. An analytic approximation of this equation is obtained and analyzed in Sect. 5. Conclusions and perspectives are formulated in Sect. 6.', '2.1. Stationary flow': 'We use the same general framework as Blondin et al. (2003); Foglizzo et al. (2007); Blondin et al. (2017) to describe the phase where the accretion shock stalls at the radius r sh. Neutrino absorption is neglected and neutrino emission near the protoneutron star radius r ns is idealized with a cooling function of the density ρ and pressure p :  \nL = -A cool ρ β -α p α . (1)  \nThe collapsing stellar core immediately after bounce is modelled as a perfect gas with an adiabatic index γ = 4 / 3, dominated by the degeneracy pressure of relativistic electrons. The gravitational potential Φ ≡ -GM ns / r is assumed to be dominated by the mass M ns of the proto-neutron star in the Newtonian approximation for simplicity. The self-gravity of the accreting gas and the increase of M ns with time are neglected. The dimensionless measure of the entropy S is defined by:  \nS ≡ 1 γ -1 log p ρ γ . (2)  \nArticle number, page 2 of 16  \nFig. 1. Ratios r peak / r ns (thick solid lines) and r ∇ / r ns (thin dotted lines) depending on the coe ffi cients ( α, β ) which define the cooling function in Eq. (1). The contour lines are calculated for γ = 4 / 3 in spherical geometry, with r sh / r ns = 10, without dissociation ( ε = 0). The values of r peak / r ns and r ∇ / r ns are indicated with the same colour code. The cooling parameters ( α, β ) = (3 / 2 , 5 / 2), (6 , 1) and (5 , 6) used in Fig. 3 are indicated with crosses for reference, connected by grey lines. The black dotted line marks the threshold β = α above which the advection time from r peak to r ns is finite and r ∇ = r ns.  \n<!-- image -->  \nα  \nThe stationary flow is described by the mass conservation, the entropy equation and the Euler equation:  \nρ v = GLYPH<18> r sh r GLYPH<19> g ρ sh v sh , (3)  \n∂ S ∂ r = L pv , (4)  \n∂ ∂ r v 2 2 + c 2 γ -1 + Φ ! = L ρ v , (5)  \nwhere c ≡ ( γ p /ρ ) 1 / 2 is the sound speed and v < 0 the radial velocity. The geometrical parameter g = 2 accounting for the spherical geometry is noted as a parameter in this section and Sect. 3 to keep track of its impact on the equations in the spirit of Walk et al. (2023). The subscript "sh" in Eq. (3) refers to quantities immediately below the shock, and the subscript "1" refers to pre-shock quantities. The jump conditions at the shock follow from the conservation of mass flux and momentum flux, taking into account the energy lost across the shock through nuclear dissociation. This latter is modeled as in Fernández & Thompson (2009b); Fernández et al. (2014) by the parameter ε measuring the energy loss ∆ e disso per unit of mass in units of the pre-shock kinetic energy density v 2 1 / 2 as in Huete et al. (2018):  \nε ≡ ∆ e disso v 2 1 / 2 . (6)  \nThe e ff ect of ε on the post-shock Mach number M sh and the Rankine-Hugoniot conditions is recalled in Appendix A, following Eqs. (A4-A6) of Foglizzo et al. (2006). The pre-shock deceleration e ff ect of pressure is neglected, with v 2 1 ∼ 2 GM ns / r sh. Defining the Mach number M≡ | v | / c as positive, we assume a strong adiabatic shock M 1 ≫ 1 in the numerical calculations of this section.  \nThe adiabatic compression of the post-shock flow by the gravitational potential Φ produces an inward increase of the enthalpy c 2 / ( γ -1) and thus an increase of the gas temperature T ∝ c 2 according to the Bernoulli equation (5). In our model of stationary  \naccretion the temperature profile displays a maximum at a radius noted r peak where the energy losses by neutrino emission balance the adiabatic heating due to the gravitational compression. The width of the cooling layer depends a priori on the parameters ( α, β ) of the cooling function L , the mass accretion rate and the geometry. With α = 3 / 2 and β = 5 / 2, Walk et al. (2023) noted that both the radius r peak ∼ 1 . 2 r ns of maximum temperature and the typical width ∆ r peak ∼ 1 . 5 r ns of the temperature profile measured at half maximum seemed surprisingly independent of both the mass accretion rate and the shock radius, and also seemed independent of the geometry. A closer inspection confirms that the location of r peak varies by less than 1% between the cylindrical and spherical geometries, and varies by less than 0 . 5% with r sh for r sh / r ns > 3.  \nFigure 1 compares r peak to the radius of maximum deceleration noted r ∇ , with r ∇ < r peak over a large range of parameters ( α, β ) except for β approaching unity. Noting that r ∇ = r ns for α < β (Foglizzo et al. 2007), Fig. 1 indicates that r peak / r ∇ ∼ 1 . 2 for ( α, β ) = (3 / 2 , 5 / 2), and r peak / r ∇ ∼ 1 . 0 for ( α, β ) = (6 , 1). In Sect. 2.2 the parameters ( α, β ) are varied continuously from (3 / 2 , 5 / 2) to (6 , 1) and from (3 / 2 , 5 / 2) to (5 , 6) in order to evaluate their impact on SASI properties.', '2.2. Perturbed flow': 'The perturbations of the stationary flow are characterized by the wavenumbers ℓ, m of spherical harmonics and a complex eigenfrequency ω . The eigenfrequency is independent of the azimuthal wavenumber | m | ≤ ℓ as established in Foglizzo et al. (2007). We use the same physical variables as in Foglizzo et al. (2006) guided by analytical simplicity: the perturbation of the Bernoulli constant δ f , the perturbed mass flux δ h , the perturbed dimensionless entropy δ S . The perturbation δ K is a combination of perturbed entropy δ S and the quantity δ w ⊥ defined from the radial component of the curl of the vorticity perturbation δ w ≡ ∇ × δ v :  \nδ f ≡ v δ vr + δ c 2 γ -1 , (7)  \nδ h ≡ δ vr v + δρ ρ , (8)  \nδ S ≡ 1 γ -1 δ c 2 c 2 -δρ ρ , (9)  \nδ w ⊥ ≡ r ( ∇ × δ w ) r , (10)  \n= 1 sin θ " ∂ ∂θ ( δ w φ sin θ ) -im δ w θ # , (11)  \nδ K ≡ rv δ w ⊥ + ℓ ( ℓ + 1) c 2 γ δ S , (12)  \nWe introduce the quantity δ A defined with the divergence of the ortho-radial velocity δ v ⊥ ≡ (0 , δ v θ, δ v φ )  \nδ A ≡ r 2 ∇ · δ v ⊥ , (13) = r sin θ " ∂ ∂θ ( δ v θ sin θ ) + im δ v φ # . (14)  \nWe establish in Appendix B that δ A and δ w ⊥ are related to the perturbation δ vr of the radial velocity as follows:  \nδ vr = r δ w ⊥ ℓ ( ℓ + 1) -1 ℓ ( ℓ + 1) ∂δ A ∂ r . (15)  \nThis equation invites us to interpret δ A /ℓ ( ℓ + 1) as the potential defining the compressible part of the perturbed velocity, and  \nr δ w ⊥ /ℓ ( ℓ + 1) as the rotational contribution to the radial velocity perturbation. Using in Appendix B the transverse components of the Euler equation (Eqs. B.10-B.11) with Eq. (12), we note that δ A is related to δ K and δ f by  \nδ K = i ωδ A + ℓ ( ℓ + 1) δ f . (16)  \nThe di ff erential system satisfied by ( δ A , δ h , δ S , δ K ) is as follows:  \n1 - M 2 v ∂ ∂ r + i ω c 2 ! δ A ℓ ( ℓ + 1) = -δ h  \n-γ -1 + 1 M 2 ! δ S γ + δ K v 2 ℓ ( ℓ + 1) , (17)  \ni  \nω  \nc  \n2  \n!  \n2  \nω  \n-  \n2  \nω  \n2  \nc  \nv  \nℓ  \n(  \nδ  \nℓ  \nA  \n+  \n1)  \n-i ω v 2 δ S + i ω v 2 c 2 δ K ℓ ( ℓ + 1) , (18)  \n∂ ∂ r -i ω v ! δ S = δ L pv ! , (19)  \n∂ ∂ r -i ω v ! δ K ℓ ( ℓ + 1) = δ L ρ v ! . (20)  \nIt includes explicit non-adiabatic terms only in the Eqs. (19-20) governing δ S and δ K . In Eq. (18) the Lamb frequency ω Lamb associated to the spherical harmonic of order ℓ defines the turning point of non-radial acoustic waves:  \nω 2 Lamb ≡ ℓ ( ℓ + 1) c 2 r 2 (1 - M 2 ) . (21)  \nThe boundary conditions at the shock are reformulated from Eqs. (28,29,E7,E8) in Foglizzo et al. (2006) using Eq. (16):  \nδ A sh ℓ ( ℓ + 1) = -1 -v sh v 1 ! v 1 ∆ ζ , (22)  \nδ h sh = 1 -v sh v 1 ! ∆ v v sh , (23)  \nδ S sh = γ v 1 c 2 sh ( i ω + ω Φ ) ∆ ζ 1 -v sh v 1 ! 2 , (24)  \nδ K sh = -ℓ ( ℓ + 1) ∆ ζ c 2 sh γ [ ∇ S ] sh 1 . (25)  \n∆ ζ and ∆ v ≡ -i ω ∆ ζ are the shock displacement and velocity. The reference frequency ω Φ in Eq. (24) is defined by:  \nω Φ r sh | v sh | ≡ 1 2 v 1 v sh 2 r sh v 2 1 d Φ d r -4 v sh v 1 1 -v sh v 1 + v sh v 1 γ M 2 sh r sh [ ∇ S ] sh 1 GLYPH<16> 1 -v sh v 1 GLYPH<17> 2 . (26)  \nω Φ defines the threshold frequency separating the regime ( ω ≪ ω Φ ) where the phase of δ S sh is opposed to ∆ ζ , from the regime ( ω ≫ ω Φ ) where it is set by ∆ v .  \nThe first two di ff erential equations (17-18) are transformed into the following second order di ff erential equation:  \n       1 - M 2 v ∂ ∂ r + i ω c 2 ! 2 + ω 2 -ω 2 Lamb v 2 c 2        δ A = ∂ ∂ r r δ w ⊥ v -ℓ ( ℓ + 1) γ -1 γ δ L pv ! , (27)  \nWe recognize on the left-hand-side of Eq. (27) an acoustic oscillator modified by advection, with a forcing on the right-handside driven by vorticity perturbations and by the perturbation of  \nLamb  \n2  \n1  \n- M  \nv  \n2  \n∂  \nr  \n∂  \n+  \nh  \n=  \nδ  \nFig. 2. Growth rate (solid lines) and oscillation frequency (dashed lines) of the fundamental SASI mode, normalized by | v sh | / r sh, calculated with ε = 0 for the cooling parameters (3 / 2 , 5 / 2) (purple) and (6 , 1) (orange) and displayed as functions of r sh / r ∇ . The analytic fitting formula (31) for ω r is displayed with a green dashed line.  \n<!-- image -->  \nneutrino cooling. Interestingly, part of this forcing can be studied analytically in the adiabatic approximation defined in Sect. 3. The variables δ vr , δρ , δ c , δ p are deduced from δ A , δ h , δ S , δ K using Eqs. (B.4-B.7) also recalled in Appendix B. In particular using Eq. (B.4) to express the lower boundary condition δ vr ( r ns) = 0 leads to  \nGLYPH<16> c 2 δ h + c 2 δ S -δ f GLYPH<17> ns = 0 , (28)  \nWith a vanishing sound speed c ns = 0 resulting from the hypothesis of stationary flow, the lower boundary condition δ vr ( r ns) = 0 is equivalent to δ f ns = 0.  \nThe boundary value problem is solved numerically using a shooting method from the shock to the inner boundary, iterating over the value of the complex eigenfrequency ω . The mathematical singularity at r ns is overcome numerically by using log M as integration variable like in Foglizzo et al. (2007). The inner boundary is then defined as the point where the Mach number has reached a su ffi ciently small value ( M∼ 10 -9 ).  \nThe analysis of Foglizzo et al. (2007) noted the similar properties of the SASI harmonics as functions of r sh / r ∇ with ( α, β ) = (3 / 2 , 5 / 2) and (6 , 1). We extend this analysis for the fundamental mode and remark in Fig. 2 that the normalized growth rate and oscillation frequency are asymptotically independent of ( α, β ) for r sh / r ∇ ≫ 1, i.e. when the cooling layer is thin compared to the shock radius. This surprising property is checked by varying continuously ( α, β ) from (3 / 2 , 5 / 2) to (6 , 1) and from (3 / 2 , 5 / 2) to (5 , 6) in Fig. 3. It is remarkable that the normalized oscillation frequency seems independent of the cooling process down to a very low ratio r sh / r ∇ : the following analytic formula ω fit r provides an approximation for ω r within 3% for 1 . 8 < r sh / r ∇ < 10:  \nτ adiab adv ≡ r sh | v sh | log r sh r ∇ , (29)  \nR 1 ≡ log r sh r ∇ -1 ! , (30)  \nω fit r ≡ 2 π τ adiab adv GLYPH<16> 0 . 56773 + 0 . 28628 R 1 -0 . 031763 R 2 1 GLYPH<17> . (31)  \nThe adiabatic advection time τ adiab adv corresponds to a velocity profile increasing linearly with radius as expected asymptotically in spherical geometry for γ = 4 / 3 (Eq. 19 in Walk et al. (2023)).  \nFigures 2 and 3 also show that the detailed cooling process does a ff ect the growth rate of SASI (solid lines) more than its oscillation frequency (dashed lines) when r sh < 4 r ∇ .  \n<!-- image -->  \nα  \nFig. 3. Growth rate (solid lines) and oscillation frequency (dashed lines) of the fundamental SASI mode, normalized by | v sh | / r sh, calculated with r sh / r ∇ = 2 (purple lines) and r sh / r ∇ = 4 (orange lines) for a range of cooling parameters ( α, β ) varying continuously from (3 / 2 , 5 / 2) to (6 , 1) (thin lines) and from (3 / 2 , 5 / 2) to (5 , 6) (thick lines), as indicated in Fig. 1.Fig. 4. Variation of the ratio T SASI /τ adiab adv associated to the fundamental SASI mode ℓ = 1, calculated for the cooling parameters (3 / 2 , 5 / 2) (purple) and (6 , 1) (orange) and displayed as functions of r sh / r ∇ . The vertical lines highlight the range of r sh / r ns in the analysis of model s25 in Müller & Janka (2014). The analytic fitting formula (31) is displayed with a green dashed line.  \n<!-- image -->  \nThe important role of r ∇ rather than r ns was also noted in Scheck et al. (2008) where the strength of SASI in numerical simulations seemed correlated with the abruptness of the deceleration close to the proto-neutron star and confirmed in the toy model used by F09 and Guilet & Foglizzo (2012).  \nWe remark on Fig. 4 that the oscillation period T SASI ≡ 2 π/ω r of the fundamental ℓ = 1 SASI mode can be significantly longer than the approximate advection time τ adiab adv for a small ratio r sh / r ∇ . The relationship between these two timescales is further discussed in a simplified model in Sect. 5.3.  \nThe modest impact of the specificities of the parametrized cooling process on the fundamental SASI eigenfrequency en-  \nFig. 5. Estimate of the relative error between the oscillation period of the neutrino signal in model s25 in Müller & Janka (2014) and in analytical estimates. The empirical formula (32) is shown with a purple line. The result of the perturbative calculation is shown for r ∇ = r ns without dissociation (orange line, Eq. 33), and for r ∇ = 1 . 3 r ns with dissociation prescribed by Eq. (35) (green line, Eq. 36). The sensitivity to the estimate of r ∇ is shown with dashed lines for r ∇ = 1 . 25 r ns (khaki) and r ∇ = 1 . 35 r ns (blue)  \n<!-- image -->  \nFig. 6. E ff ect of dissociation, measured by ε , on the oscillation frequency of SASI for r sh / r ∇ = 1 . 8 (dashed lines), 1 (solid lines) and 2 . 3 (dotted lines) for ( α, β ) = (3 / 2 , 5 / 2) (purple lines) and ( α, β ) = (6 , 1) (orange lines). The analytical formula (34) is indicated with crosses.  \n<!-- image -->  \nε  \ncourages us to apply our simple model to a more realistic physical model of core-collapse in Sect. 2.3. This also invites us to look for a deeper analytic understanding of the SASI mechanism using an adiabatic approximation, in Sect. 3.', '2.3. Comparison of the perturbative model with the empirical formula proposed by Müller & Janka (2014)': 'The adiabatic approximation of the advection time was also used in the empirical formula (33) describing the oscillation period T SASI in the core-collapse simulation of a 25 M ⊙ progenitor (Müller & Janka 2014), inspired by the physics of the advectiveacoustic cycle. r ∇ was approximated with r ns and the time varia-  \nof the mass of the proto-neutron star was neglected:  \nT MJ SASI ≡ 19ms GLYPH<18> r sh 100km GLYPH<19> 3 2 log r sh r ns ! . (32)  \nThe SASI modulation of the neutrino signal was identified in their model s25 between t = 0 . 12s and t = 0 . 45s post-bounce. The shock radius r sh decreases from 125km to 55km and the ratio r sh / r ns varies between 1 . 8 and 2 . 3 according to their Figs. 3 and 6. The formula (32) captures well the r 3 / 2 sh -dependence which is the main source of variability of T SASI during the collapse. The overall accuracy of this formula is ∼ 26% for model s25 as shown in Fig. 5.  \nThe expected variation of T SASI estimated from Eq. (31) is as follows:  \nT SASI = T MJ SASI × 1 . 82 GLYPH<16> 1 . 7 M ⊙ M ns GLYPH<17> 1 2 0 . 56773 + 0 . 28628 R 1 -0 . 031763 R 2 1 . (33)  \nAccording to Fig. 5 the overall accuracy of formula (33) applied to the model s25 is improved to ∼ 10%, which is remarkable given the simplicity of the perturbative model which does not involve any adjusted parameter and neglects dissociation at the shock. The mass increase M ns / M ⊙ ∼ 1 . 7 -2 estimated from Fig. 2 in Müller & Janka (2014) contributes to a 8% decrease of T SASI. A more physical estimate should also take into account dissociation, which can significantly increase T SASI when r sh is large, and the distinction between r ns and r ∇ , which can significantly decrease T SASI when r sh / r ns is smallest.  \nFollowing Fig. 6, the impact of dissociation on the oscillation frequency of SASI is tentatively approximated as  \nω r ( ε ) = ω r (0) -      2 GM ns r 3 sh       1 2 ε (0 . 19817 + 0 . 74804 ε ) , (34)  \nWe note that this analytical formula is tested only in the narrow range 1 . 8 < r sh / r ∇ < 2 . 3 and meant as an order of magnitude estimate. The dissociation parameter is approximated according to Fig. 12 in Huete et al. (2018):  \nε ∼ 0 . 5 GLYPH<18> r sh 150km GLYPH<19> 1 . 3 M ⊙ M ns ! , (35)  \nwith a saturation at ε ∼ 0 . 5 due to partial dissociation of α -nuclei (Fernández & Thompson 2009a). Thus Eq. (34) is rewritten as  \n10ms T SASI = 10ms T 0 SASI -1 . 069 ε (0 . 19817 + 0 . 74804 ε ) GLYPH<16> r sh 100km GLYPH<17> 3 2 GLYPH<16> 1 . 7 M ⊙ M ns GLYPH<17> 1 2 , (36)  \nwith  \nT 0 SASI 10 . 42ms ≡ GLYPH<16> r sh 100km GLYPH<17> 3 2 GLYPH<16> 1 . 7 M ⊙ M ns GLYPH<17> 1 2 log GLYPH<16> r sh r ∇ GLYPH<17> 0 . 56773 + 0 . 28628 R 1 -0 . 031763 R 2 1 . (37)  \nAccording to Fig. 5, taking into account dissociation does improve the accuracy of the empirical formula only for a limited range of prescribed ratio 1 . 25 < r ∇ / r ns < 1 . 35. Improving the accuracy of the present perturbative model beyond the ± 10% level seems elusive given its many approximations regarding the microphysics, neutrino interactions and gravity. Its main improvement compared to the empirical formula (32) is the fact that it contains physically consistent estimates of the impact of the mass, the dissociation at the shock and the SASI phase inferred  \nfrom the perturbative analysis. Each of them can potentially affect the oscillation period by several 10% according to Eq. (33) and Figs. 4, 6. By neglecting these e ff ects, the empirical formula Eq. (32) is not expected to maintain its ∼ 30% accuracy for other progenitors than s25. The accuracy of Eq. (36) should be tested on other numerical simulations of core collapse, remembering that our prescription for dissociation is very crude.', '3. Adiabatic model': 'The adiabatic character of the flow refers here to the region between the shock and the inner boundary, while non adiabatic processes associated to nuclear dissociation and neutrino emission are incorporated in the boundary conditions. The production of entropy perturbations by the shock displacement ∆ ζ is taken into account. Nuclear dissociation across the shock is formally taken into account using the parameter ε (Eq. 6), but it is set to zero in the illustrative Figures of this Section. The goal of this Section is to gain some analytical understanding on the e ff ect of the physical parameters involved in the SASI mechanism.', '3.1. Explicit expressions in the adiabatic approximation': 'With L = 0 the di ff erential equations (19-20) describe the advection of perturbations produced by the shock. Using the boundary conditions (24-25) with ∇ S = 0:  \nδ S = δ S she i ω R sh d r v , (38) δ K = 0 . (39)  \nδ S sh is defined by Eq. (24) with a simpler expression for ω Φ :  \nω Φ r sh | v sh | ≡ 1 2 v 1 v sh -g 1 -v sh v 1 . (40)  \nUsing Eq. (12) with δ K = 0 and Eq. (38) gives the explicit expression for δ w ⊥  \nδ w ⊥ = -ℓ ( ℓ + 1) c 2 γ rv δ S . (41)  \nThe vorticity associated to the entropy perturbation is also explicitly calculated in Appendix C:  \nδ wr = 0 , (42)  \nδ w θ = -imc 2 rv sin θ δ S γ , (43)  \nδ w φ = c 2 rv ∂ ∂θ δ S γ . (44)  \nTogether with Eq. (12), Eqs. (43-44) demonstrate that δ K = 0 can be interpreted as a consequence of the baroclinic production of vorticity.  \nEq. (16) implies that the wave action δ f /ω is directly related to δ A :  \nδ f i ω = -δ A ℓ ( ℓ + 1) , (45)  \nThe transverse velocity components ( δ v θ, δ v φ ) are deduced from δ A using the transverse Euler equations Eqs. (B.10,B.11) with  \nArticle number, page 6 of 16  \nEqs. (43,44). δ vr is deduced from Eqs. (15) and (41):  \nδ vr = -c 2 γ v δ S -1 ℓ ( ℓ + 1) ∂δ A ∂ r , (46)  \nr δ v θ = -1 ℓ ( ℓ + 1) ∂δ A ∂θ , (47)  \nr δ v φ = -im ℓ ( ℓ + 1) δ A sin θ . (48)  \nThe adiabatic solution can be viewed as the asymptotic limit of the model with a cooling function when r peak ∼ r ns, corresponding to α ≪ 1 and β ≫ 1 according to Fig. 1. For the sake of simplicity we explore the adiabatic solution with an inner boundary defined by δ vr = 0, formulated by Eq. (28) with c ns defined as the adiabatic sound speed at the inner boundary. This prescription could be improved to better account for non adiabatic processes in the cooling layer. The adiabatic simulations in Blondin &Mezzacappa (2007) also used δ vr = 0 at the inner boundary.', '3.2. Quantitative comparison of the eigenfrequencies with and without the region of non-adiabatic cooling': 'The eigenfrequencies corresponding to the adiabatic model are solved numerically and compared in Fig. 7 to the eigenfrequencies of the non-adiabatic formulation for di ff erent values of the shock radius. The overall trends suggest that the main properties of SASI can be qualitatively understood by focusing on adiabatic processes. The fundamental mode is the most unstable one for a small shock radius, and becomes dominated by higher overtones for a larger shock radius. Among the most visible di ff erences with the non-adiabatic model, the adiabatic simplification underestimates the frequency by a factor ≤ 1 . 3 for a large shock radius r sh ∼ 10 r ns. The growth rate of the most unstable mode is overestimated by a factor ≤ 1 . 4 for a small shock radius and underestimated by a factor ≤ 1 . 3 for a very large shock radius.', '4.1. Derivation of the second order differential equation': 'For the sake of mathematical simplicity we use the radial coordinate X defined as in Foglizzo (2001),  \ndX ≡ v 1 - M 2 d r . (49)  \nThe second order di ff erential equation (27) is simplified into:  \n       ∂ ∂ X + i ω c 2 ! 2 + ω 2 -ω 2 Lamb v 2 c 2        δ A = ∂ ∂ X r δ w ⊥ v . (50)  \nUsing Eq. (41) the forcing term of Eq. (50) is thus proportional to the entropy perturbation δ S sh produced by the shock:  \n       ∂ ∂ X + i ω c 2 ! 2 + ω 2 -ω 2 Lamb v 2 c 2        δ A ℓ ( ℓ + 1) = -F S δ S sh , (51)  \nF S ≡ ∂ ∂ X e i ω R sh d r v γ M 2 . (52)  \nWe show in Appendix C that the forced oscillator equation can be rewritten as follows, using boldface for vector quantities:  \n       ∂ ∂ X + i ω c 2 ! 2 + ω 2 -ω 2 Lamb v 2 c 2        δ L = ∂ ∂ X r δ w v , (53)  \n= ∂ ∂ X rv δ w c 2 M 2 , (54)  \nFig. 7. Oscillation frequency (upper plot) and growth rate (lower plot) of the modes ℓ = 1 calculated in units of the post-shock frequency v sh / r sh, for γ = 4 / 3, as a function of the shock distance in the model with cooling using ( α, β ) = (3 / 2 , 5 / 2) (dashed lines) and in the adiabatic approximation (solid lines). The fundamental mode (in red) and the first three overtones (green, orange, blue) are displayed. The grey horizontal line in the upper plot indicates the Lamb frequency at the shock. The fundamental mode becomes dominated by higher overtones as its frequency becomes too low for acoustic propagation ( ω r < ω sh Lamb ).  \n<!-- image -->  \nwhere δ L ≡ r × δ v is the perturbed specific angular momentum vector and δ w ≡ ∇× δ v is the perturbed vorticity vector. We note from Eqs. (43-44) that rv | δ w | / c 2 is conserved when advected. The relation between δ S sh and δ A sh is deduced from Eqs. (22) and (24):  \nℓ ( ℓ + 1) c 2 sh δ S sh γδ A sh = -1 -v sh v 1 ! ( i ω + ω Φ ) . (55)  \nThe components of transverse vorticity at the shock are related to the components of perturbed specific angular momentum according to Eqs. (43-44) and Eqs. (47-48):  \nrv δ w θ δ L θ ! sh = rv δ w φ δ L φ ! sh = -ℓ ( ℓ + 1) c 2 sh δ S sh γδ A sh . (56)  \nThe boundary conditions at the shock and at the inner boundary are written in Appendix C as follows:  \n∂δ A ∂ X sh = i ω v 2 sh δ A sh " 1 -2 v sh v 1 + ( γ -1) M 2 sh 1 -v sh v 1 !# -h 1 + ( γ -1) M 2 sh i δ A sh r sh v sh v 1 2 v sh -2 ! ,  \n(57)  \n∂δ A ∂ X ns = i ω c 2 ns δ A ns -ℓ ( ℓ + 1) 1 - M 2 ns M 2 ns δ S sh γ e i ω R ns sh d X v 2 . (58)  \nThey can also be written with the variables r δ v θ, r δ v φ and δ w θ, δ w φ , using Eqs. (43-44) and (47-48). Equations (51) and (53) are thus equivalent for ℓ ≥ 1. The same advective-acoustic cycle can either be considered as an entropic-acoustic cycle or as a vortical-acoustic cycle.  \nWe define a new perturbative variable δ Y as follows:  \nδ Y ≡ δ f i ω e i ω R sh d X c 2 = -δ A ℓ ( ℓ + 1) e i ω R sh d X c 2 . (59)  \nY 0 is defined as the solution of the homogeneous equation associated to Eq. (51) with δ S = 0 and δ K = 0, and satisfying the inner boundary condition (58):  \n       ∂ 2 ∂ X 2 + ω 2 -ω 2 Lamb v 2 c 2        Y 0 = 0 , (60)  \n∂ Y 0 ∂ X ns = i ω c 2 ns Y 0( r ns) . (61)  \nWe refer to Y 0 as the acoustic structure of the post-shock cavity modified by the radial velocity, in the absence of interaction with advected perturbations. The structure of the acoustic equation and its modification by the advection velocity v is more easily recognized when rewriting Eq. (60) with the variable r :  \n       ∂ 2 ∂ r 2 + ∂ log ∂ r 1 - M 2 v ! ∂ ∂ r + ω 2 -ω 2 Lamb c 2 (1 - M 2 ) 2        Y 0 = 0 . (62)  \nIdentifying the physics of SASI with a forced oscillator enables us to evaluate the e ffi ciency of the coupling depending on two e ff ects:  \n(i) the amplitude of the forcing term ( ∝ 1 / M 2 in Eq. 52),  \n(ii) the phase match between the forcing term (advected wavelength) and the oscillator (acoustic wavelength).  \nThe forcing amplitude is strongest where M is smallest, in the close vicinity of the proto-neutron star, but a strong phase mixing is expected there due to decrease of the radial wavelength ( ∝ 2 π | v | /ω r ) of advected perturbations. A trade-o ff between the e ff ects (i) and (ii) favours advected perturbations with a low enough frequency, coupled to the acoustic structure in a region far enough from the proto-neutron star.  \nThis description as a forced oscillator had been proposed in the context of radial Bondi accretion accelerated into a black hole (Foglizzo 2001, 2002) and in a planar adiabatic toy model of SASI with a localized region of feedback (F09). The present work is the first formulation where the radially extended character of the advective-acoustic coupling is taken into account in a shocked decelerated accretion flow in spherical geometry.  \nThe quantitative evaluation of the coupling e ffi ciency is analyzed in the next Sect. 4.2 using a classical resolution of the forced oscillator with the Wronskian method.', '4.2. Integral equation defining the eigenfrequencies': "We derive in Appendix D the integral equation defining the eigenfrequencies, which is equivalent to the full di ff erential system and boundary conditions. It is formulated here with the variable r :  \na ' 1 Y sh 0 + a ' 2 r sh ∂ Y 0 ∂ r ! sh = -M 2 sh e R ns sh i ω v d r 1 -M 2 Y ns 0 -Z sh ns ∂ ∂ r GLYPH<18> Y 0e R sh i ω M 2 1 -M 2 d r v GLYPH<19> M 2 sh M 2 e R sh i ω v d r d r , (63)  \nwith a ' 1 , a ' 2 defined by:  \na ' 1 ≡ ( γ -1) M 2 sh + i ω i ω + ω Φ v sh v 1 1 -v sh v 1 , (64)  \na ' 2 ≡ -1 - M 2 sh GLYPH<16> 1 -v sh v 1 GLYPH<17> ( i ω + ω Φ ) r sh v sh . (65)  \nThe first two terms associated to Y 0 on the left-hand-side of Eq. (63) are acoustic, marginally modified by advection. The terms on the right-hand-side involve the phase oscillations of the advected perturbations. Despite the integration by part we recognize in the integral the forcing term of Eq. (52) multiplied by the derivative of the homogeneous solution Y 0. As expected in Sect. 4.1 for the classical problem of a forced harmonic oscillator, this integral characterizes the e ffi ciency of the forcing which depends on both the amplitude profile of the forcing δ F and the matching of its phase compared to the phase of the oscillator Y 0. An analytic approximation of this integral equation is obtained in Sect. 5 in the asymptotic regime where the acoustic radial structure is non-oscillatory.  \nWe checked numerically that the eigenfrequencies satisfying the single equation (63) are strictly the same as obtained in Fig. 7 from the solution of the fourth order adiabatic di ff erential system (17-20) with L = 0, with the boundary conditions defined by Eq. (28) and Eqs. (22-25) with ∇ S = 0.", '5.1.1. Stationary flow': 'We focus on the case of a strong shock M 1 ≫ 1 and large shock radius r sh ≫ r ns, with the post-shock energy density described by the Bernoulli equation (5) is dominated by the enthalpy and the gravitational contributions:  \nc 2 γ -1 ∼ GM ns r . (66)  \nThe kinetic energy density associated to the radial velocity is a minor contribution which is even more negligible when the photodissociation across the shock is taken into account. The parameter ε impacts the post-shock Mach number and velocity according to Eqs. (A.2-A.3):  \nM 2 sh ∼ γ -1 2 γ (1 -ε ) , (67)  \nv sh v 1 ∼ ( γ -1)(1 -ε ) 2 + ( γ -1)(1 -ε ) ≤ γ -1 γ + 1 ,  \n(68) a ≡ α v + 1 2 , (75)  \nc 2 sh r sh GM ns ∼ 4 γ ( γ -1)(1 -ε ) GLYPH<2> 2 + ( γ -1)(1 -ε ) GLYPH<3> 2 .  \n(69) b ≡        α v + 1 2 ! 2 + ℓ ( ℓ + 1)        1 2 . (76)  \nArticle number, page 8 of 16  \nFig. 8. Solution of the homogeneous equation Y 0 associated to the fundamental mode ℓ = 1 (solid orange line) for r sh / r ns = 5. It is very well approximated analytically by Eq. (73) (dashed orange line with crosses). The radial profile of the amplitude of the forcing term in Eq. (63) (purple solid line) is compared to its analytical approximation (dashed purple line with crosses) using Eqs. (72) and (74). The Mach number profile in the adiabatic model (green solid line) is compared to its analytical approximation (Eqs. 72, green dashed line with crosses). The Mach profile of the flow including cooling is shown for reference (dark green line).  \n<!-- image -->  \nsh  \nEq. (66) allows for a power law approximation of the radial velocity profile with an exponent α v deduced from Eq. (A.11). It depends on both the adiabatic index and the geometry g = 2:  \nα v ≡ 1 γ -1 -g , (70)  \nv = v sh r r sh ! α v , (71)  \nM sh M ∼ GLYPH<18> r sh r GLYPH<19> α v + 1 2 . (72)  \nWe note that Eq. (66) is valid only for r ns ≤ r ≪ r sh, and becomes inaccurate in the vicinity of the shock where the sound speed is particularly sensitive to photodissociation as seen in Eq. (69).', '5.1.2. Acoustic structure of the oscillator': 'Taking advantage of the low frequency of the fundamental mode for ℓ ≥ 1 we approximate in Appendix E the homogeneous solution Y 0 as a linear combination of power laws independent of the frequency for ω ≪ ω Lamb  \nY 0 ∼ r r ns ! a        GLYPH<18> r ns r GLYPH<19> b + b -a b + a r r ns ! b        , (73)  \n∂ Y 0 ∂ r ∼ b -a r ns r r ns ! a -1        r r ns ! b -GLYPH<18> r ns r GLYPH<19> b        . (74)  \nwhere the exponents a , b are defined by:  \nThe exponents ± b correspond to the two components of the acoustic structure Y 0, with the component decreasing inward ( + b ) associated to the evanescent profile of pressure perturbations at the shock, and the component decreasing outward ( -b ) associated to the evanescent profile of pressure perturbations at the inner boundary.  \nThis approximate solution of Y 0 satisfies the lower boundary condition only in the asymptotic limit r ns ≪ r sh.', '5.1.3. Forcing by advected perturbations': 'Weuse the following approximation of the advection time τ adv( r ) profile:  \nτ adv( r ) ≡ Z r sh d r v , (77)  \n∼ 1 α v -1 r sh | v sh | " GLYPH<18> r sh r GLYPH<19> α v -1 -1 # if α v , 1 , (78)  \n∼ r sh | v sh | log r sh r if α v = 1 , (79)  \nIn particular Eq. (79) is applicable for γ = 4 / 3. In this regime the advection time τ adv is asymptotically dominated by the inner region. A numerical calculation indicates that this analytical formula slightly underestimate the adiabatic advection time by less than 6% in the range 1 < r sh / r ns < 100.  \nWe check the accuracy of our approximation of M and Y 0 in Fig. 8.', '5.2. Analytic expression of the fundamental eigenfrequency': "We approximate Eq. (63) using Eq. (72) and Eq. (73) with M 2 ≪ 1 except near the shock. We use the approximations (73) and (74) of Y 0 and ∂ Y 0 /∂ r and note that ( ∂ log Y 0 /∂ log r )sh ∼ 1 / ( b + a ) with a , b defined by Eqs. (75) and (76). We define N ≡ a ' 2 + a ' 1 / ( b + a ) and introduce the normalized eigenfrequency Z ≡ i ω r sh / | v sh | :  \nN ( Z ) ≡ γ -1 b + a M 2 sh + 1 - M 2 sh -Z b + a v sh v 1 GLYPH<16> 1 -v sh v 1 GLYPH<17> GLYPH<16> Z + ω Φ r sh | v sh | GLYPH<17> . (80)  \nUsing the variable x ≡ r / r ns, for ℓ ≥ 1 the eigenfrequency equation (63) is approximated by  \n-x 3 a -b -1 sh Z x sh 1 GLYPH<16> x b -x -b GLYPH<17> e i ω R sh d r v d x x 3 a = N i ω r sh | v sh | ! + 2 b ℓ ( ℓ + 1) M 2 sh x a + b sh e i ωτ ns adv . (81)  \nThe approximation of the eigenfrequency with γ = 4 / 3 allows for an explicit calculation of the integral involved in Eq. (81). The following equation defining the eigenfrequencies for ℓ ≥ 1 is expressed using the complex amplification factor Q ( ω ) per advective-acoustic cycle:  \nQ ( Z ) ≡ 2 b GLYPH<16> r sh r ns GLYPH<17> 2 -b GLYPH<26> 1 + h ( Z + 2) 2 -b 2 i M 2 sh ℓ ( ℓ + 1) x 3 sh GLYPH<27> [1 -( Z + 2 -b ) N ] ( Z + 2 + b ) -Z + 2 -b x 2 b sh , (82)  \nQ i ω r sh | v sh | ! e i ωτ ns adv = 1 , (83)  \nFig. 9. Eigenfrequency of the adiabatic solution (green lines) based on the integral formulation (63) compared to its integrated approximation (83) (orange lines) based on the approximate description of the flow profile v , M and acoustic structure Y 0. The estimate of the oscillation frequency 2 π/τ ns adv neglecting the phase of Q is shown as a dashed blue line. For reference the eigenfrequency of the flow with cooling using ( α, β ) = (3 / 2 , 5 / 2) is also shown (purple lines with crosses).  \n<!-- image -->  \nFig. 10. Amplitude |Q| (solid lines) and normalized phase φ Q / 2 π (dashed lines) of the complex amplification factor associated through Eqs. (86-87) to the fundamental eigenfrequency ω calculated with cooling (purple lines with crosses) and in the adiabatic approximation (green lines), using the same adiabatic estimate of the advection time τ ns adv from Eq. (79). The orange line shows |Q| and φ Q / 2 π deduced from the analytic estimate (orange lines) using Eq. (82) with ω satisfying Eq. (83).  \n<!-- image -->  \nsh  \nns  \nns  \nwhere b 2 = 1 + ℓ ( ℓ + 1) and the advection time τ ns adv from the shock to the inner boundary is approximated by Eq. (79). The numerical solution of this equation is compared to the exact solution of Eq. (63) in Fig. 9.  \nThe good agreement obtained with the analytic expression (83) demonstrates that the approximation of the flow profile v , M and the approximation of the homogeneous solution Y 0 is su ffi cient to capture the essence of the instability in the adiabatic model and identify the leading contributions in the integral expression (63). The frequency is overestimated by up to 20% and the growth rate is overestimated by up to 26%. The region of radial propagation of acoustic waves was neglected for the sake of simplicity in the analytical approximation (73) of Y 0, without much consequences on the approximation of the fundamen-  \ntal frequency ω r because the radial extension of this region is modest compared to the region a ff ecting the phase of advected perturbations.  \nThe larger inaccuracy ∼ 76% of the simple formula ω r ∼ 2 π/τ ns adv , also shown in Fig. 9, is mainly due to neglecting the frequency dependence of the phase of Q noted φ Q . The complex Eq. (83) is equivalent to the following set of two real equations:  \nω i = log |Q| ns ,  \nω r = 2 π -φ Q τ ns .  \nτ adv (84) adv (85)", '5.3. Oscillation period of the advective-acoustic cycle in the adiabatic approximation': 'Comparing ω r and 2 π/τ ns adv in Fig. 9, it is particularly striking that the oscillation period T SASI ≡ 2 π/ω r is significantly longer than the adiabatic advection time τ ns adv . This contrasts with the simulations in Scheck et al. (2008) where T SASI ∼ τ ns adv , in line with the analytic toy model of F09. Not only does T SASI di ff er from τ ns adv by more than 76%, but T SASI is actually 76% longer than the longest advection timescale τ ns adv , excluding the possibility that the oscillation period could coincide with any advection time. As explained in Scheck et al. (2008), the relation between T SASI and τ ns adv does depend on the phase φ Q of the complex e ffi ciency Q .  \nThe adiabatic approximation teaches us that even in a simple adiabatic model the phase shift φ Q should not be neglected a priori. This was also true in the calculation of Sect. 2.3 with nonadiabatic cooling. The value of |Q| and φ Q can be deduced from ω and τ ns adv from Eqs. (84-85):  \n|Q| = e ω i τ ns adv , (86) φ Q = 2 π -ω r τ ns adv . (87)  \nThis characterisation of the fundamental eigenfrequency is calculated in Fig. 10 in the adiabatic approximation and compared to the analytic estimate deduced from Eqs. (82-83). For reference it is also displayed in the flow with cooling, using the same adiabatic advection time, which should not be confused with the actual advection time taking into account the cooling layer. In the adiabatic flow the winding angle of the spiral pattern of advected perturbations from the shock to the neutron star is Φ adv ≡ 2 π -φ Q . We note in Fig. 10 that φ Q < π , implying that the radial structure of advected perturbations of entropy or vorticity contains at least a change of sign as seen in Fig. 2 in Blondin & Shaw (2007), Fig. 1 in Fernández (2010), and Fig. 10 in Buellet et al. (2023).', '5.4. Comparison to the analytic model in Foglizzo (2009)': 'The adiabatic formulation incorporates major improvements compared to the adiabatic toy model used by F09; Sato et al. (2009); Guilet & Foglizzo (2012): the plane parallel model allowed for explicit analytic solutions and a deeper understanding of the instability mechanism, but some simplifications were difficult to justify: (i) the uniform character of the flow between the shock and the region of deceleration implied that the vertical size of the acoustic and advected cavities were identical, thus overestimating the impact of radial acoustic propagation on the phase of the solution, (ii) the maximum strength Q ∼ 2 of the advective-acoustic cycle was arbitrarily set by the specific value  \n(0 . 75) chosen for the ratio c 2 sh / c 2 out . This limited the relative effect of the purely acoustic cycle on the most unstable modes, (iii) the optimum value of the vertical wavenumber compared to the horizontal one was set by the specific value chosen for L / H = 4 in F09; Sato et al. (2009) and L / H = 6 in Guilet & Foglizzo (2012), (iv) the width of the coupling region described by H ∇ / H was a also a free parameter of the model.  \nThe present adiabatic model o ff ers a new framework overcoming these shortcomings, still simple enough to allow for analytical results:  \n-the spherical geometry is taken into account (rather than a plane parallel approximation in F09),  \n-the adiabatic heating and deceleration are produced in a self consistent manner by the radial convergence in 3D and by the proto-neutron star gravity (rather than localized at the lower boundary by some external potential with adjustable depth c 2 sh / c 2 out and width H ∇ / H in F09),  \n-the gravity produced by the neutron star at the shock is not neglected, allowing for both displacement and velocity e ff ects for the production of entropy at the shock (rather than ignoring displacement e ff ects with ω Φ = 0 in F09).  \nThe adiabatic model sheds light on some objections by Blondin & Mezzacappa (2006) regarding the advective-acoustic mechanism in a non-rotating flow:  \n(i) the fact that the advection time may be longer than the oscillation period does not contradict the advective-acoustic mechanism.  \n(ii) the fact that the pressure field shows no evidence for a radially localized coupling radius is explained by the radially extended character of the coupling process, as seen in the integral in Eq. (63). The feedback should refer to the radially extended pressure structure rather than the radial propagation of an acoustic wave.', '6. Conclusions and perspectives': '- 1. The oscillation period of the fundamental SASI mode can be estimated from the shock radius r sh, the radius of maximum deceleration r ∇ and the central mass M ns.\n- 2. A possible improvement to the analytic formula used by Müller & Janka (2014) is proposed, based on the perturbative analysis without any adjustment on a specific numerical simulation.\n- 3. An adiabatic model of the shock dynamics incorporating non adiabatic processes in the boundary conditions is able to capture the general properties of SASI eigenmodes.\n- 4. In the adiabatic approximation the SASI mechanism can be described as a self-forced oscillator. The forcing by advected perturbations is distributed from the shock surface to the cooling layer rather than inside the cooling layer.\n- 5. An analytic description of the properties of the fundamental mode of SASI in the adiabatic approximation is obtained in the asymptotic limit of a large ratio r sh / r ∇ .  \nImproving the accuracy of the estimation of the SASI oscillation period from a perturbative analysis requires (i) a more accurate prescription for partial dissociation at the shock and in the flow, as described in Appendix A in Fernández & Thompson (2009a), and (ii) a better characterization of the parametrized cooling function leading to a realistic value of the ratio r ∇ / r ns.  \nThe adiabatic framework opens a new path to investigate analytically the physical e ff ect of stellar rotation in the equatorial  \nplane (paper II), and address the intriguing results of Walk et al. (2023) obtained by comparing SASI properties in cylindrical and spherical geometries.  \nWe note that the adiabatic approximation does account for the growth rate and oscillation period within ∼ 30%, leaving room for additional non-adiabatic e ff ects. By decreasing the velocity and Mach number near the proto-neutron star they increase the amplitude of the forcing term and also increase the phase mixing near the proto-neutron star. Beside a ff ecting the trade-o ff between these two e ff ects, non-adiabatic e ff ects modify the amplitude of δ S and δ K during their advection, add a source of feedback in the cooling layer and modify the acoustic equation. A quantitative assessment of some of these e ff ects can be obtained within the formalism of the self-forced oscillator, by incorporating them in the lower boundary condition.  \nAcknowledgements. The anonymous referee is thanked for constructive suggestions. 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T., Kresse, D., et al. 2020, MNRAS, 496, 2039  \nTamborra, I. & Murase, K. 2019, in Supernovae. Series: Space Sciences Series  \n- of ISSI, ed. A. Bykov, C. Roger, J. Raymond, F.-K. Thielemann, M. Falanga,\n- &R. von Steiger, Vol. 68, 87-107  \nWalk, L., Foglizzo, T., & Tamborra, I. 2023, Phys. Rev. D, 107, 063014', 'Appendix A: Stationary accretion': 'The loss of energy through nuclear dissociation is measured by the parameter ε defined by Eq. (6). It decreases the value of the post-shock Mach number M sh below the reference adiabatic value M ad, and a ff ects the Rankine-Hugoniot conditions according to Eqs. (A4-A6) of Foglizzo et al. (2006):  \nM 2 ad ≡ 2 + ( γ -1) M 2 1 2 γ M 2 1 -γ + 1 , (A.1)  \nε =       1 + 2 γ -1 1 M 2 1             1 -M 2 sh M 2 ad             1 -M 2 sh M 2 1       , (A.2)  \nv sh v 1 = M 2 sh M 2 1 1 + γ M 2 1 1 + γ M 2 sh , (A.3)  \nc sh c 1 = M sh M 1 1 + γ M 2 1 1 + γ M 2 sh . (A.4)  \nThe nuclear binding energy of the iron nuclei is 8 . 8MeV per nucleon, or 1 . 77 × 10 52 erg / M ⊙ . If nuclear dissociation was complete across the shock this would translates into a dissociation parameter scaling linearly with r sh:  \nε = 0 . 7 GLYPH<18> r sh 150km GLYPH<19> 1 . 3 M ⊙ M ns ! . (A.5)  \nWe note that our definition of ε in Eq. (6) follows Huete et al. (2018), which di ff ers from Eq. (4) in Fernández & Thompson (2009b) by a factor 2. A fraction of the dissociation of iron nuclei is radially distributed between the shock and the neutron star surface, as calculated in Fernández & Thompson (2009a). Taking into account incomplete dissociation, the value of ϵ formulated in Eq. (35) is a rough estimate deduced from Eq. (51) and Fig. 12 in Huete et al. (2018) for r sh < 175km. It is smaller than Eq. (A.5) by a factor ∼ 1 . 4 and saturates at ϵ ∼ 0 . 5 in exploding models.  \nThe value of M sh is expressed as a function of M 1 and ε ( r sh) using Eq. (A.2):  \nM 2 sh M 2 ad = 1 2 + 1 -2 ε 1 + 2 γ -1 1 M 2 1 -M 2 ad 2 M 2 1 1 +          GLYPH<18> 1 -M 2 ad M 2 1 GLYPH<19> 2 + 4 ε M 2 ad M 2 1 1 + 2 γ -1 1 M 2 1          1 2 , (A.6) ∼ 1 -ε if M 1 ≫ 1 (A.7)  \nThe jump condition (A.3) is thus a function of ε ( r sh) using Eq. (A.6), with the simple following expression for a strong shock:  \nv 1 v sh ∼ 1 + 2 (1 -ε )( γ -1) , (A.8)  \n∼ 1 + 6 1 -ε . (A.9)  \nThe definition of the cooling function (1), the dimensionless entropy (2) and the mass conservation (3) are used to eliminate the variables ρ, p , M , v from the stationary equations (4-5):  \nρ ρ sh = c c sh ! 2 γ -1 e -( S -S sh) , (A.10)  \nr g c γ + 1 γ -1 M = r g sh c γ + 1 γ -1 sh e S -S sh . (A.11)  \nArticle number, page 12 of 16', 'Appendix B: Differential system and boundary conditions of the perturbed accretion': 'We rewrite the di ff erential system satisfied by δ f , δ h (Eqs. (E2E3) in Foglizzo et al. (2006)), using the radial coordinate X defined as in Foglizzo (2001):  \ndX ≡ v 1 - M 2 d r . (B.1)  \nThus  \n∂ ∂ X + i ω c 2 ! δ f i ω = δ h + γ -1 + 1 M 2 ! δ S γ  \n+  \n1 - M 2 i ω v δ L ρ v ! , (B.2)  \n∂ ∂ X + i ω c 2 ! δ h = i ω v 2       ω 2 -ω 2 Lamb ω 2 c 2 δ f -δ S       + 1 - M 2 v 2 i δ K ω r 2 (B.3)  \nThe set of equations (A1-A4) in Foglizzo et al. (2007) are repeated here for completeness:  \nδ vr v = 1 1 - M 2 δ h + δ S -δ f c 2 ! , (B.4)  \nδρ ρ = 1 1 - M 2 -M 2 δ h -δ S + δ f c 2 ! , (B.5)  \nδ c 2 c 2 = γ -1 1 - M 2 δ f c 2 - M 2 δ h - M 2 δ S ! , (B.6)  \nδ p γ p = 1 γ -1 δ c 2 c 2 -δ S γ , (B.7)  \nδ L ρ v ! = ∇ S c 2 γ " ( β -1) δρ ρ + α δ c 2 c 2 -δ vr v # , (B.8)  \nδ L pv ! = ∇ S " ( β -1) δρ ρ + ( α -1) δ c 2 c 2 -δ vr v # . (B.9)  \nThe perturbed mass conservation and transverse components of the perturbed Euler equation are:  \n-i ωδ v θ + v δ w φ + 1 r ∂ ∂θ δ f = c 2 γ 1 r ∂δ S ∂θ , (B.10)  \n-i ωδ v φ -v δ w θ + im r sin θ δ f = c 2 γ im δ S r sin θ , (B.11)  \nWe eliminate δ v θ and δ v φ in the system of Eqs. (12), (14), (B.10) and (B.11) and obtain Eq. (16).  \nThe derivative of δ A is calculated using Eqs. (16), (20) and (B.2), leading to the di ff erential system (17-20) satisfied by δ A , δ h , δ S , δ K . The second derivative of δ A is calculated using Eqs. (12), (20) and (B.3), resulting in Eq. (27).  \nUsing Eq. (16), Eqs. (B.4-B.7) are rewritten as follows:  \nδ vr v = δ K ℓ ( ℓ + 1) v 2 -1 ℓ ( ℓ + 1) v ∂δ A ∂ r -δ S γ M 2 , (B.12)  \nδρ ρ = 1 c 2 v ∂ ∂ r -i ω ! δ A ℓ ( ℓ + 1) -γ -1 γ δ S , (B.13)  \n1 γ -1 δ c 2 c 2 = 1 c 2 v ∂ ∂ r -i ω ! δ A ℓ ( ℓ + 1) + δ S γ , (B.14)  \nδ p γ p = 1 c 2 v ∂ ∂ r -i ω ! δ A ℓ ( ℓ + 1) , (B.15)  \nUsing Eq. (B.12) with Eq. (12), we can express δ vr with δ A and δ w ⊥ and obtain Eq. (15). We note that this relation is equivalent to the radial component of the vector calculus relation ∇ 2 v = ∇ ( ∇ · v ) - ∇ × ( ∇ × v ).  \nThe baroclinic production of vorticity by the advection of entropy perturbations can be calculated in the same way as Eqs. (E17-E19) in Foglizzo et al. (2005), using the conservation of the tangential component of the velocity across the shock:  \nδ v θ sh = v 1 -v sh r sh ∂ ∆ ζ ∂θ , (B.16)  \nδ v φ sh = v 1 -v sh r sh im ∆ ζ sin θ . (B.17)  \nThe vorticity produced at the shock is defined by Eqs. (B.16B.17), the transverse components (B.10-B.11) of the Euler equation at the shock, together with the angular derivative of Eqs. (22):  \nδ wr sh = 0 , (B.18)  \nδ w θ sh = -c 2 sh γ im r sh v sh sin θ GLYPH<16> δ S sh + [ ∇ S ] sh 1 ∆ ζ GLYPH<17> , (B.19)  \nδ w φ sh = c 2 sh γ 1 r sh v sh ∂ ∂θ GLYPH<16> δ S sh + [ ∇ S ] sh 1 ∆ ζ GLYPH<17> . (B.20)', 'Appendix C: Adiabatic model': 'Following Eqs. (B5-B7) in Foglizzo (2001), the di ff erential equation describing the specific vorticity in a spherical adiabatic flow can be integrated as  \nδ wr = GLYPH<18> r sh r GLYPH<19> 2 δ wr she i ω R sh d r v , (C.1)  \nδ w θ = 1 rv       ( rv δ w θ )sh -c 2 -c 2 sh sin θ im δ S sh γ       e i ω R sh d r v , (C.2)  \nδ w φ = 1 rv       ( rv δ w φ )sh + c 2 -c 2 sh sin θ ∂ ∂θ δ S sh γ       e i ω R sh d r v . (C.3)  \nUsing Eqs. (B.18-B.20) with ∇ S = 0, together with Eqs. (C.1C.3) gives the expression (42-44) of the vorticity perturbation throughout the flow.  \nWith δ Y defined by Eq. (59), the expressions of δ A sh, δ h sh are deduced from Eqs. (22-24):  \n1 -v sh v 1 ! ∆ ζ = δ Y sh v 1 , (C.4)  \nδ h sh = -i ω v 1 v sh δ Y sh , (C.5)  \nδ S sh γ = δ Y sh c 2 sh ( i ω + ω Φ ) 1 -v sh v 1 ! . (C.6)  \nWe rewrite Eq. (17) using the definitions of X and δ Y with δ K = 0:  \nδ Y sh = -δ A sh ,  \nℓ ( ℓ + 1) (C.7) ∂δ A ∂ r ! sh = ℓ ( ℓ + 1) v sh 1 - M 2 sh       δ Y sh i ω v 2 sh v sh v 1 + M 2 sh ! -δ S sh γ       1 M 2 sh + γ -1             , (C.8) ∂δ Y ∂ X ! sh = -1 - M 2 sh ℓ ( ℓ + 1) v sh ∂δ A ∂ r ! sh + i ω c 2 sh δ Y sh , (C.9)  \nwhich results in Eq. (57).  \nAt the inner boundary in the adiabatic approximation, we use the condition δ vr = 0:  \nδ h ns + δ S ns = δ f c 2 ns . (C.10)  \nThe entropy is simply advected from the shock (38) and we express δ h with δ Y and ∂δ Y /∂ X using Eq. (17):  \nδ S ns = δ S she R ns sh i ω d r v , (C.11)  \nY  \n∂δ  \n∂  \nX  \n+ δ S γ e R sh i ω c 2 d X 1 M 2 + γ -1 ! . (C.12)  \nThe inner boundary condition (C.10) is thus reduced to Eq. (58). Multiplying Eq. (51) by m / ( ω sin θ ) and using Eqs. (43) and (48):  \n       ∂ ∂ X + i ω c 2 ! 2 + ω 2 -ω 2 Lamb v 2 c 2        r δ v φ = -∂ ∂ X r δ w θ v . (C.13)  \nEquivalently, using Eq. (44) and (47) and the derivative of Eq. (51) with respect to θ leads to:  \n       ∂ ∂ X + i ω c 2 ! 2 + ω 2 -ω 2 Lamb v 2 c 2        r δ v θ = ∂ ∂ X r δ w φ v , (C.14)  \nwhich is equivalent to Eq. (C.13) given the spherical symmetry of the stationary flow.', 'Appendix D: Equation defining the eigenfrequencies in the adiabatic model': 'The di ff erential equation satisfied by δ Y is deduced from Eqs. (51) and (59):  \n      ∂ 2 ∂ X 2 + ω 2 -ω 2 Lamb v 2 c 2       δ Y = δ F , (D.1)  \nδ F ≡ e R sh i ω c 2 d X F S δ S sh . (D.2)  \nWe define Y 0 as the solution of the homogeneous equation satisfying the inner boundary condition of pure acoustic waves (i.e. without entropy and vorticity perturbations), and Y -another homogeneous solution such that their Wronskien is W :  \n∂ Y 0 ∂ X ! ns = i ω c 2 ns Y ns 0 , (D.3)  \nW ≡ Y 0 ∂ Y -∂ X -Y -∂ Y 0 ∂ X , (D.4)  \nδ Y = Y -d -+ Z ns Y 0 W δ F d X ! -Y 0 d 0 + Z sh Y -W δ F d X ! , (D.5)  \nwhere δ F ≡ F S δ S sh is the forcing term on the right hand side of Eq. (51) for the variable δ Y defined by . Using Eq. (D.5) at the upper boundary:  \nδ Y sh = Y sh -d -+ Z sh ns Y 0 W δ F d X ! -d 0 Y sh 0 , (D.6)  \n∂δ Y ∂ X ! sh = ∂ Y -∂ X ! sh d -+ Z sh ns Y 0 W δ F d X ! -d 0 ∂ Y 0 ∂ X ! sh , (D.7)  \nArticle number, page 13 of 16  \nR  \nsh  \ni  \nω  \n2  \nc  \nd  \nX  \n=  \nh  \ne  \nδ  \nUsing Eq. (D.5) at the lower boundary:  \nwith a 1 , a 2 , a 3 defined by:  \nδ Y ns = d -Y ns --Y ns 0 d 0 -Z sh Y -W δ F d X ! ,  \nns (D.8) ∂δ Y ∂ X ! ns = d -∂ Y -∂ X ! ns -∂ Y 0 ∂ X ! ns d 0 -Z sh ns Y -W δ F d X ! . (D.9)  \nUsing Eq. (D.3), the lower boundary condition (58) translates into  \nd -" ∂ Y -∂ X ! ns -i ω c 2 ns Y ns -# = δ F sh(1 - M 2 ns ) M 2 sh M 2 ns e R ns sh i ω v 2 d X . (D.10)  \nWe note using Eq. (D.4) and (D.3) that  \n∂ Y -∂ X ! ns -i ω c 2 ns Y ns -= W Y ns 0 . (D.11)  \nThus  \nWd -= Y ns 0 δ F sh(1 - M 2 ns ) M 2 sh M 2 ns e R ns sh i ω v 2 d X . (D.12)  \nEliminating d 0 between Eqs. (D.6) and (D.7) and using Eq. (D.4):  \nY sh 0 ∂δ Y ∂ X ! sh -∂ Y 0 ∂ X ! sh δ Y sh = " Y sh 0 ∂ Y -∂ X ! sh -∂ Y 0 ∂ X ! sh Y sh -# d -+ Z sh ns Y 0 W δ F d X ! , (D.13) = Wd -+ Z sh ns Y 0 δ F d X . (D.14)  \nThe eigenfrequencies are thus defined by  \nY sh 0 ∂δ Y ∂ X ! sh -∂ Y 0 ∂ X ! sh δ Y sh = Y ns 0 δ F sh(1 - M 2 ns ) M 2 sh M 2 e R ns sh i ω v 2 d X + Z sh Y 0 δ F d X . (D.15)  \nns ns  \nReplacing the forcing term by its expression (D.2),  \nY sh 0 ∂δ Y ∂ X ! sh -∂ Y 0 ∂ X ! sh δ Y sh = δ F sh        Y ns 0 (1 - M 2 ns ) M 2 sh M 2 ns e R ns sh i ω v 2 d X + Z sh ns Y 0e R sh i ω c 2 d X ∂ ∂ r       M 2 sh M 2 e R sh i ω v d r       d r        . (D.16)  \nUsing Eqs. (24) and (57) this equation takes the form  \na 1 Y sh 0 + a 2 r sh ∂ Y 0 ∂ r ! sh + a 3 Y ns 0 = Z sh ns Y 0e R sh i ω c 2 d X ∂ ∂ r       M 2 sh M 2 e R sh i ω v d r       d r ,  \nArticle number, page 14 of 16  \na 1 ≡ GLYPH<16> ∂δ Y ∂ X GLYPH<17> sh δ F sh ,  \n(D.18)  \n= 1 + ( γ -1) M 2 sh -i ω i ω + ω Φ v sh v 1 1 -v sh v 1 , (D.19)  \na 2 ≡ -1 - M 2 sh r sh v sh δ Y sh δ F sh , (D.20)  \n= -1 - M 2 sh GLYPH<16> 1 -v sh v 1 GLYPH<17> ( i ω + ω Φ ) r sh v sh , (D.21)  \na 3 ≡ -(1 - M 2 ns ) M 2 sh M 2 ns e R ns sh i ω v 2 d X . (D.22)  \nAfter one integration by parts:  \nY sh 0 ∂δ Y ∂ X ! sh -∂ Y 0 ∂ X ! sh δ Y sh = δ F sh GLYPH<26> Y sh 0 -Y ns 0 M 2 sh e R ns sh i ω v 2 d X -Z sh ns ∂ ∂ r GLYPH<18> Y 0e R sh i ω c 2 d X GLYPH<19> M 2 sh M 2 e R sh i ω v d r d r        . (D.23)  \nThe equation defining the eigenfrequencies becomes Eq. (63) with a \' 1 ≡ a 1 -1 and a \' 2 ≡ a 2 defined by Eqs. (64-65).', 'Appendix E: Approximation of the adiabatic stationary flow and the homogeneous perturbative solution': 'The dissociation measured by the parameter ε a ff ects the relation between | v sh | and the local free fall velocity, as deduced from Eq. (68) for a strong shock:  \n| v sh | r sh 2 GM ns ! 2 ∼ ( γ -1)(1 -ε ) 2 + ( γ -1)(1 -ε ) ,  \n1 (E.1) c 2 sh r sh GM ns = 1 M 2 sh v 2 sh r sh GM ns . (E.2)  \nEquation (E.2) is transformed into Eq. (69) using Eqs. (E.1) and (67). We use a power law approximation of the enthalpy profile deduced from the Bernoulli equation for r ≪ r sh and M 2 ≪ 1:  \nc 2 " 1 + ( γ -1) M 2 2 # = ( γ -1) GM ns r + c 2 sh       1 + ( γ -1) M 2 sh 2       -( γ -1) GM ns r sh , (E.3) c 2 ∼ ( γ -1) GM ns . (E.4)  \nr  \nThe mass conservation and the adiabatic hypothesis in Eq. (A.11) imply the power law approximations (71) and (72) for the velocity and Mach number profiles.  \nFor γ = 4 / 3 we approximate  \nM 2 ∼ M 2 sh r r sh ! 3 , (E.5)  \nv ∼ v sh r r sh , (E.6)  \ni ω Z sh M 2 1 - M 2 d r v ∼ -M 2 sh i ω r sh | v sh | Z sh x 2 d x 1 - M 2 sh x 3 , (E.7)  \n(D.17) ∼ i ω r sh 3 | v sh | log       1 - M 2 1 - M 2       .  \nsh (E.8)  \nThe oscillatory phase associated to δ f in Eq. (63) is made of a product of ( i ω ) with two explicit contributions and a third contribution from the definition of δ Y in Eq. (59). The sum of these three contributions is:  \nZ sh M 2 1 - M 2 d r v + Z sh M 2 1 - M 2 d r v + Z sh d r v = Z sh 1 + M 2 1 - M 2 d r v . (E.9)  \nWith α v = 1 and ˜ r ≡ r / r sh,  \n1 + M 2 1 - M 2 v sh v ∼ 2 M 2 sh ˜ r 2 1 - M 2 sh ˜ r 3 + 1 ˜ r . (E.10)  \nThe integral can be estimated as follows:  \nZ sh 1 + M 2 1 - M 2 d r v ∼ r sh | v sh |       log r sh r ns -2 3 log 1 - M 2 sh 1 - M 2 ns       , (E.11)  \n∼ r sh | v sh |       log r sh r ns + 2 M 2 sh 3       (E.12)  \nWith a strong adiabatic shock M 2 sh ∼ 1 / 8, the correction to the advection timescale τ ns adv is thus of order 12% for r sh / r ns = 2 and 5% for r sh / r ns = 5. This correction is smaller if dissociation diminishes the value of M 2 sh .  \nThe contribution of the integral inside the radial derivative in Eq. (63) is limited to a contribution of order M 2 sh :  \nZ r ω d X c 2 = ω Z r M 2 1 - M 2 d r v ,  \nsh sh (E.13) ∼ ω r sh | v sh | M 2 sh α v + 2        1 -r r sh ! α v + 2        . (E.14)  \nWith ω r ∼ 2 π | v sh | / r sh for a small shock radius this phase shift is not negligible since it reaches 0 . 74 ∼ π/ 4 at the inner boundary and it is linear in ω .  \nWe integrate Eq. (49) using Eq. (71) and neglecting M 2 ≪ 1:  \nX sh ≡ r sh v sh α v + 1 , (E.15)  \nX X sh ∼ r r sh ! α v + 1 . (E.16)  \nThe two solutions Y ± 0 of the homogeneous equation (60) are approximated as power laws with exponents α ± .  \nY ± 0 ( r ) ≡ r r ns ! α ± . (E.17)  \nInjecting Y ± 0 ( r ) into Eq. (60),  \nα ± ( α ± -α v -1) r r sh ! -2 α v -2 = -      ω 2 r 2 sh c 2 -ℓ ( ℓ + 1) r 2 sh r 2       GLYPH<18> v sh v GLYPH<19> 2 . (E.18)  \nFor ℓ ≥ 1, the restoring force in Eq. (60) is independent of the frequency in the region where ω ≪ ω Lamb. Thus, using Eq. (E.4) for ℓ ≥ 1,  \nα ± ( α ± -α v -1) = ℓ ( ℓ + 1) - M 2 sh ω r sh v sh ! 2 r r sh ! 3 , (E.19) ∼ ℓ ( ℓ + 1) if ω ≪ ω Lamb . (E.20)  \nIn this region the approximate solution is α ± ≡ a ± b with a , b defined by Eqs. (75) and (76). The homogeneous solution Y 0 for ℓ ≥ 1 is a linear combination of power laws Y ± 0 satisfying the lower boundary condition (61):  \nY 0( r ) = r r ns ! a -b + R ns r r ns ! a + b , (E.21)  \n1 - M 2 ns v ns ∂ Y 0 ∂ r ! ns = i ω c 2 ns Y ns 0 . (E.22)  \nWith M ns ≪ 1 and using Eq. (72),  \na -b + ( a + b ) R ns = M 2 sh r ns r sh ! 2 + α v i ω r sh v sh (1 + R ns) , (E.23)  \nR ns = a -b - M 4 γ + 1 sh GLYPH<16> r ns r sh GLYPH<17> γ + 3 + α v (3 -γ ) γ + 1 i ω r sh v sh a + b + M 4 γ + 1 sh GLYPH<16> r ns r sh GLYPH<17> γ + 3 + α v (3 -γ ) γ + 1 i ω r sh v sh . (E.24)  \nFrom Eq. (E.24) we conclude that the coe ffi cient R ns is asymptotically independent of ω when r ns ≪ r sh.  \nR ns ∼ -a -b a + b . (E.25)  \nThe acoustic function is independent of the frequency in the asymptotic limit r ns ≪ r sh for low frequency perturbations ℓ ≥ 1 driven by advection such that ω r sh / | v sh | ≤ 1, as described by Eqs. (73) and (74).  \nUsing the index 0 to denote the acoustic perturbation associated to the homogeneous solution, we note from Eqs. (46) with δ S = 0 and Eq. (59) that the relation between ∂ Y 0 /∂ r and δ v 0 r is  \nδ v 0 r = -1 ℓ ( ℓ + 1) ∂δ A 0 ∂ r , (E.26)  \n= ∂ Y 0 ∂ r -i ω M 2 1 - M 2 Y 0 v ! e -R sh i ω M 2 1 -M 2 d r v . (E.27)  \nThe fact that Eq. (74) imposes ∂ Y 0 /∂ r = 0 is approximately compatible with δ v 0 r ∼ 0 to the extent that M ns ≪ 1.', 'Appendix F: Approximation of the eigenfrequency equation for γ = 4 / 3': 'The advection time for α v = 1 is approximated using Eq. (79):  \ne R r sh i ω vr d r = x x sh ! -i ω r sh | v sh | . (F.1)  \nDefining δ z ≡ i ω r sh / | v sh | + 2 -b and noting that a = 1, the leading terms in Eq. (81) for ℓ ≥ 1 are :  \nN ( δ z -2 + b ) + 2 b M 2 sh ℓ ( ℓ + 1) x δ z -3 sh = Z x sh 1 GLYPH<18> x sh x GLYPH<19> δ z 1 x 2 b -1 ! d x x , (F.2)  \nThe integral on the right hand side can be calculated explicitly. If δ z , 0,  \nZ x sh 1 GLYPH<18> x sh x GLYPH<19> δ z 1 x 2 b -1 ! d x x = 1 δ z        1 -δ zx -2 b sh 2 b + δ z -2 bx δ z sh 2 b + δ z        , (F.3)  \nArticle number, page 15 of 16  \nUsing Eq. (F.3) in Eq. (F.2) we obtain the following equation defining δ z :  \n[1 -δ zN ( b -2 + δ z )] GLYPH<18> 1 + δ z 2 b GLYPH<19> -δ z 2 bx 2 b sh = x δ z sh       1 + δ z 2 b M 2 sh ℓ ( ℓ + 1) x 3 sh GLYPH<18> 1 + δ z 2 b GLYPH<19>       . (F.4)  \nReintroducing the normalized eigenfrequency Z = i ω r sh / | v sh | ,  \nx b -2 sh 2 b        [1 -( Z + 2 -b ) N ( Z )] ( Z + 2 + b ) -Z + 2 -b x 2 b sh        = x Z sh        1 + h ( Z + 2) 2 -b 2 i M 2 sh ℓ ( ℓ + 1) x 3 sh        . (F.5)  \nThe approximate advection time τ sh adv from the shock to the inner boundary, defined by Eq. (79), is introduced to obtain Eqs. (8283).'}