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Critical Condition of Core-Collapse Supernovae I: One Dimensional Models

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A fitted scaling relation now predicts the critical neutrino luminosity for supernova explosions from four inputs.

desk verdict A careful 1D parameter study that usefully extends the critical luminosity framework to finite pre-shock Mach number, with a real but openly acknowledged numerical caveat at low M that should temper the headline 44% shift. read the letter →

arxiv 2411.16857 v1 pith:GRTSJDN6 submitted 2024-11-25 astro-ph.HE

David Pochik , Todd Thompson This is my paper · ORCID
classification astro-ph.HE
keywords core-collapsesupernovaecriticalneutrinoluminosityantesonicconditionaccretionshockheatingproto-neutronstarpre-shockMachnumberSi/Ointerface
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks what sets the exact boundary between a stalled accretion shock and a successful supernova explosion in the neutrino-heating mechanism. Using time-dependent one-dimensional accretion models with a general equation of state, neutrino heating and cooling, and a controlled Mach number for the pre-shock flow, it derives a scaling relation for the critical neutrino luminosity $L_\nu^{\mathrm{crit}}$ as a function of accretion rate, proto-neutron star radius and mass, and average neutrino energy. It finds that pressurized pre-shock inflow lowers the critical luminosity by up to roughly 44 percent relative to free-fall models, connecting this to entropy jumps across compositional interfaces like the Si/O layer. Among three proposed explosion diagnostics, the antesonic ratio varies least along the critical curve, staying near $0.215 \pm 0.01$. If correct, the relation gives a fast quantitative way to rank progenitors by explodability and prepares the same measurements for multi-dimensional models.

What carries the argument

The central object is the critical curve in the $(L_\nu, \dot{M})$ plane, whose normalization is mapped by raising $L_\nu$ in small steps until the stalled shock advects outward. The controlling device is the outer-boundary Mach number $\mathcal{M}$, which sets the pre-shock pressure through $P_{\mathrm{go}}\simeq \rho v^2/(\Gamma_{\mathrm{go}}\mathcal{M}^2)$ and thereby the thermal content of the accreted matter, while the neutrino optical depth is held fixed at $\tau=2/3$ by adjusting ghost-zone densities. The main diagnostic is the antesonic ratio $\max(c_s^2/v_{\mathrm{esc}}^2)$, the squared ratio of post-shock sound speed to escape velocity, whose near-constant value along the critical curve anchors the paper's comparison of explosion conditions.

What would settle it

Run the $\mathcal{M}=1.3$ series again with pre-shock neutrino heating forcibly set to zero; if $L_\nu^{\mathrm{crit}}$ rises back toward the $\mathcal{M}=2.0$ values, the low-Mach lowering is a numerical artifact rather than the pressurized-inflow effect.

Watch

Extended reading notes

Core claim

Across a grid of spherically symmetric, time-dependent accretion models with a general equation of state and optically thin neutrino heating and cooling, the paper finds that the explosion threshold is a critical neutrino luminosity $L_\nu^{\mathrm{crit}}$ and fits it as $$L_\$nu^{{\mathrm{crit}}$} = 32.8 \left(\frac{\dot{M}}{0.5\,M_\odot\,\mathrm{s}^{-1}}\right)^{1.29} \left(\frac{R_\star}{30\,\mathrm{km}}\right)^{n_{R_\star}} \left(\frac{M_\star}{1.4\,M_\odot}\right)^{n_{M_\star}} \left(\frac{\langle\epsilon_{\nu_e}\rangle}{12.6\,\mathrm{MeV}}\right)^{n_\epsilon} $10^{{51}}$\,\mathrm{erg\,s}^{-1},$$ with $n_{R_\star}=-2.58(\dot{M}/0.5)^{-0.16}$, $n_{M_\star}=1.97(\dot{M}/0.5)^{-0.15}$, and $n_\epsilon=-1.69(\dot{M}/0.5)^{0.06}$ at $\mathcal{M}=2.0$. The key physical claim is that the thermal content of the pre-shock flow, parameterized by the outer-boundary Mach number $\mathcal{M}$, changes the normalization: lowering $\mathcal{M}$ from 2.0 to 1.3 reduces $L_\nu^{\mathrm{crit}}$ by about 44 percent, while raising it from 2.0 to 3.0 changes it by about 12 percent, with convergence to pressureless free-fall at high $\mathcal{M}$. The paper further argues that accretion of compositional interfaces such as the Si/O layer therefore promotes explosion two ways at once: a drop in $\dot{M}$ and a rise in entropy that lowers the critical curve. When the antesonic, advection-to-heating timescale, and force-explosion conditions are compared, the antesonic ratio shows the least variation along the critical curve, staying near $\max(c_s^2/v_{\mathrm{esc}}^2)\simeq 0.215\pm0.01$.

Load-bearing premise

The load-bearing assumption is that one outer-boundary Mach number captures the thermal state of the infalling material while the free-nucleon suppression factor completely stops pre-shock neutrino heating; for the lowest-Mach models that suppression fails, so spurious heating may contaminate the pre-shock entropy and partly produce the lowered critical luminosity the paper attributes to pressurized inflow.

Editorial extensions

If this is right

  • At fixed accretion rate, pressurized pre-shock inflow lowers the neutrino luminosity required to explode, so models that assume pressureless free-fall overestimate how hard it is to explode realistic progenitors.
  • Accretion of an Si/O compositional interface pushes the shock toward explosion through two simultaneous effects: $\dot{M}$ decreases and higher-entropy material lowers $L_\nu^{\mathrm{crit}}$ at fixed $\dot{M}$.
  • Along the critical curve, the antesonic ratio stays near $0.215\pm0.01$, varying only about 3-5 percent depending on $\mathcal{M}$, making it the most stable of the tested explosion diagnostics.
  • Shock oscillations can temporarily push any diagnostic past its nominal critical value without producing an explosion, so time-averaged rather than instantaneous thresholds are needed near criticality.
  • The paper anticipates that the fitted power-law exponents may also depend on $R_\star$, $M_\star$, and $\langle\epsilon_\nu\rangle$ themselves, so the relation is a reference point rather than a universal scaling.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extension: if the roughly 44 percent normalization shift survives in 2D and 3D, then entropy jumps at compositional interfaces matter as much as $\dot{M}$ drops for predicting which progenitors explode, and progenitor-to-explosion mapping should carry both.
  • Extension: a single pre-shock Mach number may be too coarse; a two-parameter family that varies entropy and infall speed separately would test whether the effect is truly pressurized inflow or a more general thermal-content dependence.
  • Extension: the near-constant antesonic ratio suggests a practical explosion probe for simulations and eventually observations: record $\max(c_s^2/v_{\mathrm{esc}}^2)$ behind the shock and flag values above roughly 0.215, with oscillations time-averaged.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This paper presents spherically symmetric, time-dependent Athena++ accretion models of the stalled core-collapse supernova shock with a general equation of state and neutrino heating/cooling. Varying the accretion rate, neutrino luminosity, average neutrino energy, PNS radius and mass, and the pre-shock Mach number at fixed neutrino optical depth tau = 2/3, the authors compute the critical neutrino luminosity L_nu^crit separating accretion from explosion. They fit the numerical critical curves to a power-law scaling relation (Eq. 22) with Mdot-dependent exponents (Eqs. 23-25), and they test three proposed explosion diagnostics: the antesonic condition, the advection/heating timescale criterion, and the force explosion condition. The main qualitative claims are that low pre-shock Mach number lowers the normalization of L_nu^crit and that, across the explored model space, the antesonic ratio shows the smallest relative variation, with a near-critical value of about 0.215 +/- 0.01. The low-M behavior is interpreted as evidence that accretion of higher-entropy material across compositional interfaces, such as the Si/O interface, can promote explosion in addition to the well-known decrease in Mdot.

Significance. If the results hold, this is a useful quantitative map of the 1D critical condition and a natural baseline for the planned 2D and 3D extension. The paper has real strengths: the simulations enforce tau = 2/3 through a feedback loop, use stabilized checkpoint models to step in L_nu, check resolution convergence of L_nu^crit to about 1%, and test several published explosion criteria on the same time-dependent data set. The comparison of the antesonic, timescale, and force conditions is informative even where it shows that none of the criteria is exactly constant. However, the central scaling relation is a least-squares fit with no reported uncertainties, and the low-M branch of the study is acknowledged by the authors themselves to be affected by imperfect suppression of pre-shock neutrino heating. The significance is therefore conditional: the framework and diagnostics are valuable, but the quantitative low-M shift and the fit parameters need additional support before the results can be used as calibrated predictions.

major comments (3)
  1. [Sec. 3.2.1, Eqs. (22)-(25)] The central product of the paper is the power-law scaling relation for L_nu^crit, but the text reports no fit uncertainties, no goodness-of-fit statistic, and no number of fitted model points. Equations (23)-(25) claim weak Mdot dependence of the exponents based on curves like Figure 8, yet no error bars are shown on those fits, and Table 1 calibrates the normalization for each Mach number using a single point. In addition, Eq. (22) contains no M term at all, even though M is a headline input parameter; the M dependence is handled by separately rescaling the prefactor at three discrete values. I request that the authors report the fit covariances and residuals, state the fitting range and number of points, and either extend Eq. (22) to include M explicitly or clearly restrict its validity to M = 2.0.
  2. [Sec. 3.1 and Fig. 17 caption] The manuscript explicitly acknowledges that for M = 1.3 the free-nucleon suppression factor chi_N in Eq. (17) does not drive the pre-shock Qdot to zero; Figure 1 shows Qdot slowly approaching zero in the pre-shock region, and the Figure 17 caption states that the M = 1.3 profile has a non-zero entropy gradient because Qdot is inadequately suppressed. This contamination is load-bearing for the paper's main quantitative claim, because the 44% decrease in L_nu^crit between M = 2.0 and M = 1.3 (Table 1, Figure 7) is the basis for the compositional-interface interpretation in Section 3.6. Spurious pre-shock heating can raise the entropy and lower the ram pressure of material entering the shock, thereby lowering L_nu^crit artificially. I request a control experiment in which the heating and cooling rates are artificially set to zero for r > R_shock for all M, together with a quantitative estimate of the integrated spurious pre-shock heating relative to the accretion enthalpy flux, so that the physical part of the low-M shift can be separated from the numerical artifact.
  3. [Sec. 3.3, Figure 9] The abstract's statement that the antesonic ratio shows the least variation across the model space is based on Figure 9, but the comparison mixes time-averaged maxima for oscillatory models with final-time maxima for non-oscillatory models, and the bars show that instantaneous values can exceed the critical value without leading to explosion. The quoted range 0.215 +/- 0.01 also includes the M = 1.3 models affected by the pre-shock heating problem discussed above. I ask for a version of Figure 9 and the associated statistics restricted to M >= 2.0, and for a statement of how the time-averaging window affects the 3-5% variation quoted in the conclusions. Without this, the 'least variation' claim is not cleanly separated from the grid of models and diagnostics used to define it.
minor comments (6)
  1. [General] There are several typos, including 'relativistc' in Section 2.2 and 'timscles' in Section 3.4; a careful proofreading pass is needed.
  2. [Eq. (34)] The notation \dot{M}^{1.0} is ambiguous; it should be written as \dot{M}/(1 M_sun/s) or defined explicitly before use, as is done in the text but not in the equation.
  3. [Figure 5 caption] The caption states that R_shock stabilizes at ~150-200 km and ~600 km for models that are not the fiducial Rstar = 30 km models; please clarify in the caption which parameter sets produce these radii, since the range quoted in the text is much smaller.
  4. [Section 2.4 and Section 3.2] The distinction between the first exploding model L_nu^crit and the near-critical stable model L_nu^{crit,n} is clear, but the reader would benefit from an explicit statement of why different sections use one or the other; currently the transition between Sections 3.2 and 3.3 is abrupt.
  5. [Data availability] The data availability statement says the implementation and data are available upon request; depositing the problem generator and model outputs in a persistent repository would strengthen reproducibility, especially since readers cannot otherwise check the fit values in Eq. (22).
  6. [Figure 12 caption] The caption attributes the black profile to Eqs. (28)-(29) and the blue profile to Eq. (30), but the latter is defined together with Eq. (31); please correct the cross-reference.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Lcrit is measured from explosion runs, Eq. 22 is an explicit least-squares fit, and the three explosion-condition tests do not impose the conditions they claim to test.

full rationale

All load-bearing numbers are measured, not imposed. Lcrit is defined operationally in Section 3.2 as the smallest Lnu at which the stalled shock advects through the outer boundary (vr,out > 0), and every critical-curve point comes from time-dependent Athena++ runs. Equation 22 and the exponents in Eqs. 23-25 are explicitly a least-squares fit to those measured values ('we determine the power law scalings ... by using a least squares method to fit Lcrit'), so there is no fitted input renamed as a prediction: the fit is presented as a fit, and Table 1's 'eqn.' rows are the same fit re-normalized at each M, not independent evidence. The tests of the antesonic condition, timescale ratio, and FEC evaluate Eqs. 26, 28-31, and 33 using simulation fields; none of these conditions is used as the explosion criterion or imposed as a constraint, so the finding that the antesonic ratio varies least is a comparison of diagnostics, not a tautology. The paper cites Pejcha & Thompson (2012) and Raives et al. (2018, 2021) for the theoretical antesonic benchmark, and one author (Thompson) is shared, but these citations supply an external parameter-free threshold (3/16 for isothermal free-fall) that is tested against the data rather than assumed; Eq. 22 does not depend on it. The acknowledged failure of chi_N (Eq. 17) to fully suppress pre-shock heating for M=1.3 (Section 3.1, Figure 1 caption, Figure 17 caption) is a stated numerical-fidelity caveat and could affect the validity of the low-M normalization, but it is not a circular reduction of a result to its own input. No step reduces, by construction, to inputs; score 0.

Assumptions & free parameters 9 free parameters · 7 assumptions · 0 invented entities

The central claim is a numerical scaling relation, so the fitted power-law coefficients and the tau = 2/3 choice carry much of the load. The paper introduces no new physical entities, but it relies on standard neutrino microphysics, an optically thin transport assumption, and an ad hoc pre-shock thermal-content parameterization whose suppression function fails in part of the parameter space it is meant to describe.

free parameters (9)
  • power law normalization C = 32.8
    Least-squares normalization in equation 22 for L_crit at the fiducial point Mdot = 0.5 Msun/s, R* = 30 km, M* = 1.4 Msun, <epsilon> = 12.6 MeV, M = 2.0.
  • Mdot exponent = 1.29
    Fitted exponent in equation 22 giving L_crit proportional to Mdot^1.29.
  • R* exponent normalization = -2.58
    Fitted exponent in equation 23, n_R* = -2.58 (Mdot/0.5)^0.16.
  • R* exponent Mdot scaling = 0.16
    Fitted power of Mdot in the R* exponent, equation 23.
  • M* exponent normalization = 1.97
    Fitted exponent in equation 24, n_M* = 1.97 (Mdot/0.5)^-0.15.
  • M* exponent Mdot scaling = -0.15
    Fitted power of Mdot in the M* exponent, equation 24.
  • epsilon exponent normalization = -1.69
    Fitted exponent in equation 25, n_epsilon = -1.69 (Mdot/0.5)^0.06.
  • epsilon exponent Mdot scaling = 0.06
    Fitted power of Mdot in the epsilon exponent, equation 25.
  • per-M normalization = not reported; see Figure 7
    Equation 22 is rescaled to match L_crit at Mdot = 0.5 for each M (M = 1.3, 2.0, 3.0), adding a fitted normalization for each Mach number.
assumptions (7)
  • domain assumption Neutrino heating and cooling rates of Qian and Woosley (1996), modified for eta_e != 0, are accurate for the accretion flow.
    Used in Section 2.2 for Qdot and Ye evolution; these rates set the energy deposition that defines L_crit.
  • domain assumption Optically thin neutrino transport with fixed tau = 2/3 captures the critical condition.
    Section 2.3; they tune ghost-zone density to maintain tau = 2/3, whereas in reality tau varies with the flow and PNS properties.
  • ad hoc to paper The pre-shock thermal content can be parameterized by a single Mach number M at the outer boundary.
    Section 2.3, equation 21; this is the paper's new parameterization and is not derived from progenitor models.
  • ad hoc to paper The free-nucleon fraction suppression chi_N adequately models the drop in heating across the shock.
    Section 2.2, equation 17; known to fail for low M where pre-shock Qdot is not fully suppressed.
  • ad hoc to paper The density suppression f_sup = exp(-rho/rho0) prevents spurious high-density heating and cooling.
    Section 2.2, following Fernandez (2012); an ad hoc numerical fix.
  • domain assumption L_nu_e = L_bar_nu_e = L_nu and eta_nu_e = eta_bar_nu_e = 0.
    Section 2.2; simplifies the neutrino spectra and is not derived from a transport solution.
  • standard math The general EoS from Timmes and Swesty (2000) via Coleman (2020) is valid in the modeled regime with rho below about 1e12 g/cm3.
    Section 2.1 and 2.2; background tool for the hydrodynamic closure.

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Cite this review

Pith. "Pith review of Critical Condition of Core-Collapse Supernovae I: One Dimensional Models." pith.science (2026). https://pith.science/paper/GRTSJDN6

@misc{pith2026241116857,
  author       = {Pith},
  title        = {Pith review of: Critical Condition of Core-Collapse Supernovae I: One Dimensional Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GRTSJDN6}},
  note         = {Machine review of arXiv:2411.16857}
}
abstract

When the core of a massive star collapses, neutrino heating can energize the stalled accretion shock, leading to a successful supernova. The critical condition that characterizes the transition from accretion to explosion is a central topic of study and is often characterized by a critical proto-neutron star (PNS) neutrino luminosity $L_\nu^{\rm crit}$, which depends on the post-collapse mass accretion rate $\dot{M}$ from the progenitor. We examine the critical condition by solving the spherically symmetric time-dependent Euler equations with a general equation of state and realistic microphysics for a range of $\dot{M}$, average neutrino energy $\left< \epsilon_{\nu}\right>$, luminosity $L_{\nu}$, PNS radius $R_{\star}$, mass $M_{\star}$, and pre-shock Mach number $\mathcal{M}$ for a fixed neutrino optical depth from the PNS surface of $2/3$. We derive $L_{\nu}^{\mathrm{crit}}$ as a function of the input parameters. We show that pressurized pre-shock flow, as parameterized by low $\mathcal{M}$, changes the normalization of the critical condition because accretion of higher entropy shells at later times after collapse leads to lower $L_{\nu}^{\mathrm{crit}}$. We connect this finding to the onset of explosion due to compositional interface accretion. Across our parameter space, we test critical conditions that have been proposed in the literature, including the ``antesonic" condition, the ``force explosion condition," and the heuristic heating-advection timescale condition. We discuss how shock oscillations impact these critical conditions. Compared to other explosion conditions, we find that the antesonic ratio shows the least variation across the model space we explore. This work is preparatory for similar experiments in 2D axisymmetry and 3D.

Figures

Figures reproduced from arXiv: 2411.16857 by the authors.

Figure 1
Figure 1. Steady state profiles with fiducial inputs at Lν = 30 × 1052 and M˙ = 0.7 M⊙ s −1 for ρ, T, P, vr, Ye, and Q˙ using M = 1.3 (solid), 2.0 (dashed), and 3.0 (dotted). The red profiles in the bottom left panel are cs, and the cyan profiles in the top right panel are Pram. Rshock is characterized by the strong discontinuity in the profiles, which is where the accretion flow transitions from super- to sub-sonic. Due to t… view at source ↗
Figure 2
Figure 2. Time steady solutions for ρ, vr, T, Ye, P, Q˙ , the antesonic ratio c 2 s/v2 esc, ϵ, and the Bernoulli integral B from a series of simulations using M˙ = 1.0 M⊙ s −1 and M = 2.0 with fiducial inputs. The black profiles show model evolution as Lν is increased over the range [50, 70] × 1051 ergs s−1 in increments of 1051 ergs s−1 . The blue profile is evaulated at Lν = 77 × 1051 ergs s−1 , which is just beneath L crit… view at source ↗
Figure 3
Figure 3. vr profiles for the exploding model, L crit ν = 77.2× 1051 ergs s−1 , from the series shown in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Critical curve for M = 2.0 with fiducial in￾puts. Red dots represent models with strong shock oscil￾lations that lead into explosions. Red stars are explosions that occur without oscillations. The blue shaded area shows regions with stable oscillatory models, while els…
Figure 5
Figure 5. Figure 5: Examples of stable and oscillatory accretion so￾lutions. (Left) Shock velocities Vshock for models of varying time-dependent characteristics. All models here use M⋆ = 1.4 M⊙ and ⟨ϵνe ⟩ ≃ 12.6 MeV. The stable, non-oscillatory profile (solid black) uses M = 1.3, M˙ = 0.4…
Figure 6
Figure 6. Figure 6: L crit ν -M˙ analysis for various input parameters at M = 2.0. (Left) L crit ν at R⋆ = 30 km and M⋆ = 1.4 M⊙ with ⟨ϵνe ⟩ ≃ 10.5 MeV (orange), ⟨ϵνe ⟩ ≃ 12.6 MeV (black), and ⟨ϵνe ⟩ ≃ 15.1 MeV (grey). (Middle) L crit ν at R⋆ = 30 km and ⟨ϵνe ⟩ ≃ 12.6 MeV with M⋆ = 1.2 M⊙…
Figure 7
Figure 7. Figure 7: Critical curves with fiducial inputs for M = 1.3 (solid), M = 2.0 (dashed), and M = 3.0 (dotted). The grey profiles of corresponding linestyle are the linear fittings provided by equation 22. Each linear fitting is scaled with L crit ν data at M˙ = 0.5 M⊙ s −1 from eac…
Figure 8
Figure 8. Figure 8: Absolute value of L crit ν power law scaling terms n for R⋆ (solid black, equation 23), M⋆ (solid red, equation 24), and ⟨ϵνe ⟩ (solid green, equation 25) as functions of M˙ . The dashed lines show the linear fits to the data. M. However, the absolute change in vesc wi…
Figure 9
Figure 9. Figure 9: Maximum antesonic ratio for stable models with fiducial inputs at L crit,n ν using M = 1.3 (solid line), M = 2.0 (dashed line) and M = 3.0 (dotted line). Blue stars represent oscillatory stable models, which use the time-averaged value of the maximum antesonic ratio. R…
Figure 10
Figure 10. Figure 10: Antesonic ratio quantities (equation 27), i.e., P, ρ, Γ, and RA for near-critical fiducial models with M = 1.3 (solid), M = 2.0 (dashed), and M = 3.0 (dotted) plotted over M˙ . Similar to [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]
Figure 11
Figure 11. Figure 11: (Top) Time averaged cs and vesc as functions of Lν and M˙ evaluated at RA for M = 2.0. The dots show the evolution of cs and vesc with Lν at fixed M˙ , where cs and vesc decrease in value and reach a minimum near the critical luminosity. As M˙ is increased, this minim…
Figure 12
Figure 12. Figure 12: (Top) Maximum advection and heating timescale ratios for M = 1.3 (solid), M = 2.0 (dashed), and M = 3.0 (dotted) for fiducial near-critical stable models. The black profile is calculated with equations 28 and 29, and the blue profile is calculated with equations 30. T…
Figure 13
Figure 13. Figure 13: Maximum mass enclosed in the gain region Mgain for fiducial near-critical models. The behavior in Mgain profiles directly corresponds to the maximum timescale ra￾tios with τadv,2 (see [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]
Figure 15
Figure 15. Figure 15: Ψ as a function of time. (Top) ˜ Ψ for a series ˜ of models at M˙ = 0.3 M⊙ s −1 and M = 2.0 with fiducial inputs. Ψ oscillates around 3 ˜ .0 × 10−3 for stable models. At L crit ν = 16.2 × 1051 ergs s−1 (blue), Ψ is driven up in value. ˜ The red dashed line at Ψ = 0 sh…
Figure 16
Figure 16. Figure 16: Ψ for near-critical models with fiducial inputs as ˜ a function of M˙ at M = 1.3 (solid, magenta cap), M = 2.0 (dashed, cyan cap), and M = 3.0 (dotted, green cap). Blue stars indicate oscillatory stable models, which use the time￾averaged value of Ψ. Red dots indicate…
Figure 17
Figure 17. Figure 17: (Top) S profiles (in units of kb baryon−1 ) as functions of mass (in units of M⊙) for pre-collapse Solar metallicity progenitors of MT = 11 M⊙ (black), MT = 15 M⊙ (red), and MT = 20 M⊙ (blue) from the Woosley & Heger (2007) progenitor data sets. The first steep gradie…

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Pith tools

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