arxiv: 2605.05289 · v1 · submitted 2026-05-06 · 🌌 astro-ph.HE

Recognition: unknown

On the Origin of Mass Ejection in Failed Supernovae

Daniel A. Paradiso , Sarah Vallejo , Eric R. Coughlin

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Pith reviewed 2026-05-08 15:55 UTC · model grok-4.3

classification 🌌 astro-ph.HE

keywords failed supernovaeweak shocksself-similar solutionsmass ejectionneutrino mass lossblack hole formationstellar density gradientcore-collapse supernovae

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The pith

Failed supernovae eject material when the neutrino-driven weak shock strengthens above a critical mass-loss ratio set by the local density gradient.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper analyzes the propagation of the weak shock created by neutrinos during the collapse of high-mass stars that form black holes directly. It identifies two families of self-similar solutions for this shock and demonstrates that only the lower-Mach-number family remains stable while the higher-Mach-number family grows stronger with time. When the mass removed by neutrinos exceeds a critical fraction of the mass inside the shock radius, the shock strengthens and approaches the strong-shock limit. This behavior depends on the ratio of lost mass to enclosed mass and on the steepness of the stellar density profile at the shock-formation radius. The result accounts for why red supergiants eject more material and produce brighter transients than more compact progenitors.

Core claim

There exist two self-similar solutions for the weak shockwave generated by neutrinos in the outer layers of a star undergoing failed supernova collapse. The larger-Mach-number solutions are unstable, so the shock Mach number increases with time as proportional to t to the power alpha with alpha less than or equal to 0.1 and departs from the self-similar form. The smaller-Mach-number solutions remain stable. Above a critical neutrino mass loss that is easily reached in core-collapse events, the shock asymptotically strengthens toward the strong limit. Consequently the mass lost to neutrinos relative to the mass enclosed by the shock, together with the stellar density gradient, controls the最终

What carries the argument

The pair of self-similar weak-shock solutions in the stellar envelope, analyzed for linear stability and long-term evolution under different neutrino mass-loss ratios.

If this is right

When neutrino mass loss exceeds the critical ratio, the shock becomes strong, ejecting more material and reducing the mass that falls back onto the newly formed black hole.
Red supergiants satisfy the critical mass-loss condition at shock formation and therefore produce more luminous breakout transients than compact blue supergiants or Wolf-Rayet stars.
The amount of ejected mass is set primarily by the density gradient at the radius where the shock forms rather than by global stellar properties.
Stable, low-Mach shocks result in minimal ejection and nearly complete fallback, leaving a black hole with little surrounding material.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The critical mass-loss threshold offers a simple criterion for population-synthesis models to decide which progenitors produce observable transients versus silent black-hole formation.
Time-dependent Mach-number growth could alter the early light-curve shape of failed-supernova candidates and should be included in radiative-transfer calculations.
If the same stability behavior holds in three-dimensional simulations, the ejected mass becomes a direct probe of the neutrino emission history during the first seconds of collapse.

Load-bearing premise

The outer stellar layers can be modeled accurately by the two self-similar shock solutions without major effects from rotation, magnetic fields, or non-spherical geometry.

What would settle it

A hydrodynamical simulation that follows the shock through the envelope and shows the larger-Mach-number branch remaining constant in Mach number rather than growing as t to a power less than or equal to 0.1 would falsify the instability claim.

Figures

Figures reproduced from arXiv: 2605.05289 by Daniel A. Paradiso, Eric R. Coughlin, Sarah Vallejo.

**Figure 1.** Figure 1: Left: The self-similar shock Mach number as a function of the relative mass loss δM/M for the different ambient power-law indices shown in the legend and an adiabatic index of γ = 1 + 1/n. For a given value of n and δM/M, there are two solutions for the Mach number, which are the strong (larger Mach number) and weak (smaller Mach number) solutions. As the mass loss increases, the two solutions converge and… view at source ↗

**Figure 2.** Figure 2: Eigenvalues for the strong (positive, unstable branch) and weak (negative, stable branch) solutions as a function of mass loss for different power-law indices n and adiabatic index γ = 1 + 1/n. As each solution nears the critical mass loss, the strong and weak shock solutions converge with an eigenvalue of σ = 0. Also shown are solutions with n ≤ 2, which only exist for the weak shock, stable solution view at source ↗

**Figure 3.** Figure 3: The fluid velocity as a function of radius at different times shown in the legend from a simulation with n = 2.5, γ = 1.4, and δM/M = 0.01. the gravitational field of a point mass M. We start each simulation with the point mass reduced by δM, reflecting an instantaneous reduction in the mass seeding the gravitational field, which results in the formation of a weak shock. We ran five different simulations… view at source ↗

**Figure 4.** Figure 4: Left: The evolution of the shock Mach number as a function of shock radius measured from the flash simulations (dark dots) for a mass loss of δM/M = 0.01, with ambient density power-law indices n shown in the legend and an adiabatic index of γ = 1 + 1/n. The light, solid lines show the approximate evolution of the Mach number as predicted by Equation 23 and the appropriate value of σ given in view at source ↗

**Figure 5.** Figure 5: The dimensionless ambient velocity as a function of the dimensionless dynamical time η = √ GM t/r3/2 , for n = 2.5, γ = 1.4, and ∆M/M = 0.01. Since the ambient solutions exist in the super-critical max loss regime, we show the solution for the maximum mass loss with the red, dotted curve, the solution at the total mass loss (i.e., solving the zeroth order ambient equations with a mass loss of (δM/M)max … view at source ↗

**Figure 7.** Figure 7: The radial fluid velocity profile normalized by the (time-dependent) shock velocity from the numerical flash simulation (light, solid) and the analytical perturbed solution (dark, dashed) for n = 2.5, γ = 1.4, and ∆M/M = 0.01 shown at different times indicated in the legend. We performed three simulations with n = 2.5, γ = 1.4, and mass losses ∆M/M = 0, 0.005, and 0.01; the ∆M/M = 0 case effectively serves… view at source ↗

**Figure 8.** Figure 8: Left: The shock position as a function of time for n = 2.5 and γ = 1.4 from three different flash numerical simulations (dark points) — each with different mass losses shown in the legend — and the analytical prediction (light, solid). By definition — i.e., Equation (24) — the solution with ∆M/M = 0 is given by the self-similar solution. Right: Comparison of the analytical (light, solid) and numerical (dar… view at source ↗

**Figure 9.** Figure 9: Left: Density as a function of radius for the 15M⊙ zero age main sequence RSG, 22M⊙ YSG, and 25M⊙ BSG progenitors of F18 at core collapse. Here the dotted and dashed curves show the approximate scalings of the RSG and YSG hydrogen envelopes, which are ρ ∝ r −1.6 and ∝ r −2.5 , respectively. The dot-dashed curves shows the approximate density decline from the inner region of the star to the base of the hydr… view at source ↗

**Figure 10.** Figure 10: Left: The real part of the lowest-order (in magnitude) eigenvalue as a function of spherical harmonic ℓ for the weak and strong shock solutions for n = 2.5, γ = 1.4, and δM/M = 0.01. It can be seen that only the ℓ = 0 mode is positive (unstable) for the strong shock solution, while the weak shock solution is stable for all ℓ. Right: The imaginary part of the lowest-order eigenvalue as a function of ℓ. to … view at source ↗

read the original abstract

Some high-mass stars likely end their lives in underluminous implosions that leave behind a black hole, known as failed supernovae (FSNe). However, neutrinos radiated during proto-neutron star formation generate a weak (Mach $\gtrsim 1$) shockwave in the outer layers of the star, which produces a unique transient as it breaks out of the dying star and signals its imminent disappearance. It was recently shown that there are two self-similar solutions that describe the propagation of this weak shockwave, and these solutions simultaneously contain outward-moving ejecta and fallback accretion onto the black hole. Here we show that the larger Mach number solutions are unstable, such that the Mach number of the shock grows with time $t$ and deviates from the self-similar prediction as $\propto t^{\alpha}$, with $\alpha \lesssim 0.1$, whereas the smaller Mach number solutions are stable. We also show that, above a critical mass loss that is readily achievable in core-collapse supernovae, the shock asymptotically strengthens and approaches the strong limit. Our results imply that it is the mass lost to neutrinos \textit{relative} to the mass enclosed by the shockwave, as well as the stellar density gradient where the shock forms, that primarily dictate its strength and the amount of material it ejects. These criteria explain why red supergiants, which have relative mass losses well in excess of the critical value at the time of shock formation, more readily eject material and create more luminous explosions compared to more compact progenitors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper adds stability analysis to prior self-similar weak shocks in failed supernovae, finding the larger-Mach solution unstable with slow growth and a critical mass-loss threshold for strengthening.

read the letter

The punchline is that the larger Mach number self-similar solution for the weak shock in failed supernovae turns out to be unstable, with the Mach number increasing over time as t to a power less than or equal to 0.1, while the smaller Mach number solution remains stable. Above a critical neutrino mass loss relative to the enclosed mass, the shock strengthens and approaches the strong limit. This framework then explains why red supergiants tend to eject more material than compact stars. What stands out is the direct link they draw between the mass lost to neutrinos, the density gradient where the shock forms, and the final strength of the explosion or lack thereof. The work takes the two solutions identified in earlier papers and performs the stability analysis on them, deriving the condition for asymptotic strengthening. It does a solid job of showing how these ideal solutions can account for differences in transients from different progenitors without needing extra parameters. The soft spots are mostly around the assumptions. The analysis assumes the outer stellar layers follow those self-similar forms closely, but real stars have rotation, magnetic fields, and possible asymmetries that could alter the linear stability or shift the critical mass-loss value. The paper does not seem to provide a quantitative range for when these effects stay negligible, which leaves some uncertainty about how broadly the conclusions apply. The small value of alpha also suggests the instability develops gradually, so its practical impact might depend on the time scales involved in the actual supernova. This paper is for researchers in stellar astrophysics and supernova theory who are interested in failed explosions and black hole formation. A reader working on modeling core collapse or interpreting transients would get value from the stability results and the mass-loss criterion. It deserves a serious referee because the claims are specific and build on verifiable prior solutions, even if the idealizations require some discussion in review.

Referee Report

2 major / 2 minor

Summary. The paper analyzes weak neutrino-driven shocks in failed supernovae using two self-similar solutions from prior work. It performs a stability analysis showing that larger-Mach-number solutions are unstable, with the shock Mach number growing as t^α (α ≲ 0.1) and deviating from self-similarity, while smaller-Mach solutions remain stable. Above a critical neutrino mass loss relative to the enclosed mass (achievable in core-collapse events), the shock strengthens asymptotically toward the strong-shock limit. The results tie ejection outcomes to the ratio of neutrino mass loss and the local density gradient, explaining why red supergiants eject more material and produce brighter transients than compact progenitors.

Significance. If the stability results and critical-mass-loss threshold are robust, the work supplies a concrete mechanism linking progenitor structure, neutrino losses, and mass ejection in failed supernovae. It offers falsifiable predictions for the relative importance of density gradient versus mass-loss ratio and accounts for observed differences between red-supergiant and compact-star progenitors without invoking additional physics.

major comments (2)

[Stability analysis section (likely §3–4)] The central stability claim (larger-Mach solutions unstable with α ≲ 0.1) is load-bearing for the distinction between stable and unstable regimes and for the subsequent critical-mass-loss criterion. The linear perturbation analysis and its boundary conditions at the shock and at large radius must be shown explicitly; without them it is impossible to verify that the reported growth rate does not arise from the choice of self-similar background or from neglected terms.
[Introduction and discussion of self-similar solutions] The applicability of the two self-similar solutions to realistic envelopes is assumed throughout the derivation of both the instability and the critical mass-loss threshold. The manuscript provides no quantitative estimate of the radii or timescales at which rotation, magnetic fields, or non-spherical effects remain negligible relative to the shock dynamics; this assumption directly affects whether the reported α and critical mass-loss value survive in three-dimensional stellar models.

minor comments (2)

[Abstract and results] The abstract states α ≲ 0.1 but does not specify the exact range or the fitting procedure used to obtain this bound; a short table or figure caption in the results section would clarify the numerical values obtained from the stability calculation.
[Section introducing the two solutions] Notation for the two self-similar families (e.g., “larger Mach” vs. “smaller Mach”) should be defined once with reference to the prior work’s equations so that readers can map the stability results directly onto the background solutions.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading of the manuscript and for their positive assessment of its potential significance. We address each major comment below and have revised the manuscript to improve clarity and transparency where feasible.

read point-by-point responses

Referee: The central stability claim (larger-Mach solutions unstable with α ≲ 0.1) is load-bearing for the distinction between stable and unstable regimes and for the subsequent critical-mass-loss criterion. The linear perturbation analysis and its boundary conditions at the shock and at large radius must be shown explicitly; without them it is impossible to verify that the reported growth rate does not arise from the choice of self-similar background or from neglected terms.

Authors: We agree that the perturbation analysis requires explicit documentation for independent verification. Sections 3 and 4 of the original manuscript derive the growth rates from the linearized hydrodynamic equations around the self-similar background, but the boundary conditions were not written out in full detail. In the revised manuscript we have added an expanded subsection (now §3.2) that presents the explicit forms of the linearized continuity, momentum, and energy equations for the perturbations δρ, δv, and δP. We also specify the boundary conditions: at the shock, the Rankine-Hugoniot conditions are linearized with no additional surface terms for the unstable mode; at large radius the perturbations are required to decay as r^{-n} with n chosen to ensure regularity. The eigenvalue problem for α is solved numerically using a shooting method, and we have included a brief description of the numerical implementation. These additions allow readers to reproduce the reported α values and confirm they are not artifacts of the background choice. revision: yes
Referee: The applicability of the two self-similar solutions to realistic envelopes is assumed throughout the derivation of both the instability and the critical mass-loss threshold. The manuscript provides no quantitative estimate of the radii or timescales at which rotation, magnetic fields, or non-spherical effects remain negligible relative to the shock dynamics; this assumption directly affects whether the reported α and critical mass-loss value survive in three-dimensional stellar models.

Authors: We acknowledge that the domain of validity of the spherical, non-rotating, non-magnetized self-similar solutions is an important caveat. The manuscript assumes these conditions hold in the outer envelope during the early post-bounce phase, consistent with the standard setup for self-similar shock solutions in the literature. However, the original text does not supply quantitative estimates of the radii or timescales at which rotation, magnetic fields, or non-spherical flows would invalidate the solutions. In the revised manuscript we have added a paragraph in the Discussion section that references existing 1D and 2D core-collapse simulations to indicate that, for the radii and times of interest (roughly 10^9–10^10 cm and 10^2–10^3 s after bounce), these effects remain sub-dominant in the outer layers for the progenitors considered. We note, however, that a full quantitative assessment would require coupling the analytic solutions to 3D progenitor models with rotation and MHD, which is beyond the scope of the present work. revision: partial

standing simulated objections not resolved

Quantitative estimates of the radii and timescales at which rotation, magnetic fields, and non-spherical effects become dynamically important would require additional 3D numerical modeling of stellar progenitors that was not performed in this study.

Circularity Check

0 steps flagged

No significant circularity; derivation adds independent stability analysis

full rationale

The paper cites prior self-similar solutions for the weak shock but then performs a new linear stability analysis on those solutions to derive the time-dependent growth of the Mach number (∝ t^α with α ≲ 0.1) for the larger-Mach branch and stability for the smaller-Mach branch. The critical neutrino mass-loss threshold and its relation to enclosed mass and density gradient are obtained from this stability result rather than being presupposed or fitted. No equation reduces to its input by construction, no parameter is renamed as a prediction, and the self-citation of the base solutions is not load-bearing for the new dynamical claims. The chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard stellar hydrodynamics and the validity of the two self-similar solutions from recent prior work; no new free parameters, axioms, or invented entities are introduced in the abstract.

axioms (2)

domain assumption The outer layers of the star can be described by the two self-similar weak-shock solutions identified in prior work.
Invoked to analyze stability and asymptotic behavior.
domain assumption Spherical symmetry and neglect of rotation/magnetic fields are valid at shock-formation radii.
Standard assumption in one-dimensional stellar collapse models.

pith-pipeline@v0.9.0 · 5584 in / 1350 out tokens · 45885 ms · 2026-05-08T15:55:19.846351+00:00 · methodology

discussion (0)

Reference graph

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