# An integral condition for core-collapse supernova explosions

## Abstract

Here, we derive an integral condition for core-collapse supernova (CCSN) explosions and use it to construct a new diagnostic of explodability. The fundamental challenge in CCSN theory is to explain how a stalled accretion shock revives to explode a star. In this manuscript, we assume that the shock revival is initiated by the delayed-neutrino mechanism and derive an integral condition for spherically symmetric shock expansion, v _{s} > 0. One of the most useful one-dimensional explosion conditions is the neutrino luminosity and mass-accretion rate ($${L}_{\nu }\mbox{--}\dot{{ \mathcal M }}$$) critical curve. Below this curve, steady-state stalled solutions exist, but above this curve, there are no stalled solutions. Burrows & Goshy suggested that the solutions above this curve are dynamic and explosive. In this manuscript, we take one step closer to proving this supposition; we show that all steady solutions above this curve have v _{s} > 0. Assuming that these steady v _{s} > 0 solutions correspond to explosion, we present a new dimensionless integral condition for explosion, Ψ > 0. Ψ roughly describes the balance between pressure and gravity, and we show that this parameter is equivalent to the τ condition used to infer the $${L}_{\nu }\mbox{--}\dot{{ \mathcal M }}$$ critical curve. The illuminating difference is that there is a direct relationship between Ψ and v _{s}. Below the critical curve, Ψ may be negative, positive, and zero, which corresponds to receding, expanding, and stalled-shock solutions. At the critical curve, the minimum Ψ solution is zero; above the critical curve, Ψ _{min} > 0, and all steady solutions have v _{s} > 0. Using one-dimensional simulations, we confirm our primary assumptions and verify that Ψ _{min} > 0 is a reliable and accurate explosion diagnostic.

- Authors:

- Florida State Univ., Tallahassee, FL (United States)
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- National Science Foundation (NSF); USDOE

- OSTI Identifier:
- 1352363

- Report Number(s):
- LA-UR-15-25862

Journal ID: ISSN 1538-4357; TRN: US1701010

- Grant/Contract Number:
- AC52-06NA25396

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- The Astrophysical Journal (Online)

- Additional Journal Information:
- Journal Name: The Astrophysical Journal (Online); Journal Volume: 834; Journal Issue: 2; Journal ID: ISSN 1538-4357

- Publisher:
- Institute of Physics (IOP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTRONOMY AND ASTROPHYSICS; astronomy and astrophysics; hydrodynamics; methods: analytical; methods: numerical; shock waves; supernovae: general

### Citation Formats

```
Murphy, Jeremiah W., and Dolence, Joshua C.
```*An integral condition for core-collapse supernova explosions*. United States: N. p., 2017.
Web. doi:10.3847/1538-4357/834/2/183.

```
Murphy, Jeremiah W., & Dolence, Joshua C.
```*An integral condition for core-collapse supernova explosions*. United States. doi:10.3847/1538-4357/834/2/183.

```
Murphy, Jeremiah W., and Dolence, Joshua C. Tue .
"An integral condition for core-collapse supernova explosions". United States.
doi:10.3847/1538-4357/834/2/183. https://www.osti.gov/servlets/purl/1352363.
```

```
@article{osti_1352363,
```

title = {An integral condition for core-collapse supernova explosions},

author = {Murphy, Jeremiah W. and Dolence, Joshua C.},

abstractNote = {Here, we derive an integral condition for core-collapse supernova (CCSN) explosions and use it to construct a new diagnostic of explodability. The fundamental challenge in CCSN theory is to explain how a stalled accretion shock revives to explode a star. In this manuscript, we assume that the shock revival is initiated by the delayed-neutrino mechanism and derive an integral condition for spherically symmetric shock expansion, vs > 0. One of the most useful one-dimensional explosion conditions is the neutrino luminosity and mass-accretion rate (${L}_{\nu }\mbox{--}\dot{{ \mathcal M }}$) critical curve. Below this curve, steady-state stalled solutions exist, but above this curve, there are no stalled solutions. Burrows & Goshy suggested that the solutions above this curve are dynamic and explosive. In this manuscript, we take one step closer to proving this supposition; we show that all steady solutions above this curve have vs > 0. Assuming that these steady vs > 0 solutions correspond to explosion, we present a new dimensionless integral condition for explosion, Ψ > 0. Ψ roughly describes the balance between pressure and gravity, and we show that this parameter is equivalent to the τ condition used to infer the ${L}_{\nu }\mbox{--}\dot{{ \mathcal M }}$ critical curve. The illuminating difference is that there is a direct relationship between Ψ and vs. Below the critical curve, Ψ may be negative, positive, and zero, which corresponds to receding, expanding, and stalled-shock solutions. At the critical curve, the minimum Ψ solution is zero; above the critical curve, Ψmin > 0, and all steady solutions have vs > 0. Using one-dimensional simulations, we confirm our primary assumptions and verify that Ψmin > 0 is a reliable and accurate explosion diagnostic.},

doi = {10.3847/1538-4357/834/2/183},

journal = {The Astrophysical Journal (Online)},

number = 2,

volume = 834,

place = {United States},

year = {Tue Jan 10 00:00:00 EST 2017},

month = {Tue Jan 10 00:00:00 EST 2017}

}

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