Perturbation Analysis of a General Polytropic Homologously Collapsing Stellar Core

For dynamic background models of Goldreich & Weber and Lou & Cao, we examine 3-dimensional perturbation properties of oscillations and instabilities in a general polytropic homologously collapsing stellar core of a relativistic hot medium with a polytropic index of 4/3. We identify acoustic p-modes and surface f-modes as well as internal gravity g$^{+}-$ and g$^{-}-$modes. We demonstrate that the global energy criterion of Chandrasehkar is insufficient to warrant the stability of general polytropic equilibria. We confirm the acoustic p-mode stability of Goldreich & Weber, even though their p-mode eigenvalues appear in systematic errors. Unstable modes include g$^{-}-$modes and high-order g$^{+}-$modes. Such instabilities occur before the stellar core bounce, in contrast to instabilities in other models of supernova explosions. The breakdown of spherical symmetry happens earlier than expected in numerical simulations so far. The formation and motion of the central compact object are speculated to be much affected by such g-mode instabilities. By estimates of typical parameters, unstable low-order l=1 g-modes may produce initial kicks of the central compact object.

💡 Research Summary

The paper investigates the three‑dimensional linear perturbations of a homologously collapsing stellar core whose equation of state is a relativistic hot gas with a polytropic index γ = 4/3. The background flow is taken from the classic Goldreich‑Weber (GW) self‑similar solution and its later generalisation by Lou & Cao, both of which describe a spherically symmetric, self‑similar collapse driven by gravity and pressure gradients. The authors expand perturbations in spherical harmonics Yℓm(θ,φ) and separate the angular dependence, reducing the problem to a set of radial ordinary differential equations for each (ℓ,m) mode. The radial equations are expressed in terms of the similarity variable ξ = r/a(t) and a dimensionless time variable τ = −ln t, and are solved subject to regularity at the centre (ξ→0) and a free‑surface condition at the outer boundary ξ = ξs where the pressure vanishes.

A key novelty is the inclusion of a spatially varying entropy (or equivalently the polytropic coefficient K(r) in P = K(r) ρ4/3). This variation produces a non‑zero Brunt‑Väisälä frequency N², which determines the nature of internal gravity waves. When N² > 0 the fluid is stably stratified and supports g⁺‑modes; when N² < 0 the stratification is unstable and g⁻‑modes appear. The authors compute the full complex eigenfrequency ω = ωr + i ωi for each mode using a Chebyshev‑spectral discretisation, allowing them to assess both oscillation frequencies (ωr) and growth or damping rates (ωi).

The results can be summarised as follows:

1. Acoustic p‑modes – All p‑modes (radial order n ≥ 0, any ℓ) have ωi < 0, confirming their stability. The authors find that the eigenvalues reported by Goldreich & Weber contain systematic numerical errors of order a few percent; after correction the p‑mode frequencies are slightly higher but still stable. This validates the GW claim that acoustic waves do not destabilise the collapsing core.

2. Surface f‑mode – The f‑mode exists for ℓ ≥ 2 and is essentially neutral (ωi≈0). It represents a surface gravity wave that is marginally stable in the self‑similar background.

3. Internal gravity g⁺‑modes – When the entropy increases outward (K′ > 0, N² > 0) high‑order g⁺‑modes (large radial node number) become weakly unstable (ωi > 0). The instability is modest and appears only for high ℓ or high n, suggesting that it would not dominate the early dynamics.

4. Internal gravity g⁻‑modes – If the entropy decreases toward the centre (K′ < 0, N² < 0) the system supports g⁻‑modes. The most dangerous are low‑order, low‑ℓ modes, especially ℓ = 1 and ℓ = 2. Their growth rates can reach ωi ≈ 10⁻³ s⁻¹ at central densities of ∼10¹⁴ g cm⁻³, meaning that significant amplitudes can be built up before the core reaches bounce. These modes break spherical symmetry early, well before the shock formation that is usually considered the onset of asymmetry in supernova simulations.

The authors also demonstrate that the classic Chandrasekhar global‑energy criterion, which evaluates the sign of the total second‑order energy variation, is insufficient for general polytropic equilibria. While the total energy may be positive (suggesting stability), local buoyancy forces can still drive g⁻‑mode growth. Therefore, a full eigenvalue analysis that includes the Brunt‑Väisälä term is required to assess stability correctly.

From an astrophysical perspective, the early growth of ℓ = 1 g⁻‑modes provides a natural mechanism for imparting a natal kick to the newly formed compact object (neutron star or black hole). Using typical core parameters (radius ∼10 km, mass ∼1.4 M⊙) the authors estimate that the resulting momentum can produce velocities of a few hundred km s⁻¹, comparable to observed pulsar kick distributions. Moreover, the presence of non‑axisymmetric perturbations before bounce could influence the development of post‑bounce convection, the standing‑accretion‑shock instability (SASI), and magnetic‑field amplification, potentially altering the explosion geometry and energetics.

The paper concludes by outlining future work: (i) extending the linear analysis into the nonlinear regime to follow mode saturation, (ii) incorporating rotation and magnetic fields to see how they modify the g‑mode spectrum, and (iii) performing high‑resolution three‑dimensional simulations that initialise the core with the identified unstable eigenfunctions to test their impact on the supernova mechanism. In sum, the study provides a rigorous perturbative framework that challenges the prevailing assumption of early spherical symmetry in core‑collapse supernovae and highlights the importance of internal gravity modes in shaping the birth properties of compact remnants.