The Core-Collapse Supernova Critical Condition and the Birth of Millisecond Proto-Magnetar Winds

Matthias J. Raives

Caltech Tea Talk | Jan 25 2021
Todd A. Thompson, Sean M. Couch, Matthew S. B. Coleman, Johnny P. Greco, Ondřej Pejcha

Why do supernovae explode?

What role does turbulence play in aiding explosion?

How do proto-magnetars spin down after birth?

Can proto-magnetar winds produce $r$-process elements?

Supernovae

Credit: NASA

Timeline of Explosion

Stars with $M>8M_\odot$ fuse progressively heavier elements in their cores until they reach iron peak elements.

The core becomes too massive to support itself againts its own gravity, and the star begins to collapse.

Credit: Janka et al (2007)

The Supernova Problem

If this steady state continues indefinitely, the PNS will collapse into a black hole.

What revives the shock — Neutrino heating

How much heating is required to revive the shock?

Why do some models explode when others fail?

Physically, what distinguishes failed explosions from successful ones?

Credit: Sukhbold et al (2016)

The Supernova Critical Condition

During the steady state accretion phase, there is some $\dot{M}$ and some $L_{\nu,\rm core}$, set by progenitor properties and fundamental physics.

Burrows & Goshy (1993) showed that steady accretion is only possible below a certian “critical” $L_{\nu,\rm core}^{\rm crit}(\dot{M})$.

Credit: Burrows & Goshy (1993)

An “Antesonic” Condition

Simplest version of the problem: pressureless free-fall onto an isothermal gas
Combine the Euler equations for isothermal accretion flow $$v_r\frac{{\rm d}v_r}{{\rm d}r} = -\frac{1}{\rho}\frac{{\rm d}P}{{\rm d}r} - \frac{GM}{r^2}$$ with the shock jump conditions for pressureless free-fall onto a standing shock $$\rho_1v_1 = \rho_2 v_2$$ $$\rho_1v_1^2 + P_1 = \rho_2 v_2^2$$ and the resulting equations are valid only for $$\frac{c_T^2}{v_{\rm esc}^2}\leq \frac{3}{16}$$

Time-Dependent Simulations

In real supernovae, we start with a stable solution, but cross the critical curve as $\dot{M}$ and $L_{\nu,\rm core}$ decrease with time.

Our analysis of the critical condition assumes a steady-state.

We need time dependent simulations to resolve this issue.

$\dot{M}$ is instantaneously decreased at the outer boundary; the new accretion rate propogates inwards.

When the new $\dot{M}$ hits the shock, it moves outwards to adjust to the new steady-state solution.

If the new $\dot{M}$ is small enough, no solution can be found, and the shock moves out to “infinity”, and the solution transitions into a wind

Turbulence

Supernova simulations in 2D and 3D explode more readily than in 1D.

Why? 2D/3D motions like turbulence and convection play an important role.

We want a critical condition that can naturally take these effects into account

We introduce turbulence as a pressure term $P_{\rm turb}=\nabla\cdot\widetilde{\mathbf{R}}$, where $\widetilde{\mathbf{R}}$ is the Reynolds stress tensor:

$\widetilde{R}_{rr} = \bar{\rho}\widetilde{v_\mathrm{turb}^2}$

where $v_{\rm turb}$ is the radial turbulent velocity, and the tilde denotes the density-weighted average. The Euler equation becomes

$$\widetilde{v}_r\frac{{\rm d}\widetilde{v}_r}{{\rm d}r} = -\frac{1}{\overline\rho}\frac{{\rm d}(P_{\rm gas} + \widetilde{R}_{rr})}{{\rm d}r} - \frac{GM}{r^2} - \frac{1}{\overline\rho}\frac{2\widetilde{R}_{rr} - \widetilde{R}_{\theta\theta}-\widetilde{R}_{\phi\phi}}{r}$$

When turbulence is present, the critical solution has $\frac{c_T^2}{v_{\rm esc}^2} < \frac{3}{16}$ — it explodes more easily.

Analytically, we can show that the critical condition is

$$\frac{c_T^2}{v_{\rm esc}^2} + \frac{v_{\rm turb}^2}{v_{\rm esc}^2} = \frac{1}{32}\left( 3 + 16\frac{v_{\rm turb}^2}{v_{\rm esc}^2} + \sqrt{9 - 32\frac{v_{\rm turb}^2}{v_{\rm esc}^2}} \right)$$

This isn’t really illuminating in the same way the “pure thermal” antesonic condition is.

In the limit of small $v_{\rm turb}$…

When turbulence is present, the critical solution has $\frac{c_T^2}{v_{\rm esc}^2} < \frac{3}{16}$ — it explodes more easily.

Analytically, we can show that the critical condition is (in the limit of small $v_{\rm turb}$)

$$\frac{c_T^2}{v_{\rm esc}^2} + \frac{1}{3}\frac{v_{\rm turb}^2}{v_{\rm esc}^2} \approx \frac{3}{16}$$

Turbulence “makes up for” some of the missing thermal pressure

Turbulence Lowers the Critical Curve

Turbulent velocities $\frac{v_{\rm turb}}{c_T}\sim 0.5-1$ expected from simulations.

Proto-Magnetar Winds

Credit: S. Habbal, M. Druckmüller and P. Aniol

Motivation

Neutrino mechanism cannot power high luminosity events like GRBs and SLSNe.

Magnetocentrifugal forces dominate proto-neutron star winds

Rapid magnetic braking can taps huge reservoir of rotational energy — B field forces co-rotation out to Alfvén radius $R_A\gg R_0$.

But… existing models assume 2D axisymmetry, dipole spindown, and cannot simulate long timescales

Goal: derive $\dot{E}(t)$, $\dot{J}(t)$, $\dot{M}(t)$

$$E_{\rm rot}\sim MR^2\Omega^2 \sim 10^{53}{\:\rm erg}\;\left(\frac{P}{1\:{\rm ms}}\right)^{-2}$$ $$\dot{E}_{\rm SD}\sim \dot{M}R_A^2\Omega^2 \gg \dot{M}R_0^2\Omega^2$$

My Models

We simulate multidimensional models of a magnetized, rotating wind with an isothermal EOS in athena++.

In 2D, the initial dipole is aligned with the axis of rotation, but in 3D, we can freely tilt the magnetic axis.

Nucleosynthesis in PNS Winds

Core fusion only produces elements up to ${}^{56}{\rm Fe}$. Heavier elements are formed by neutron capture processes

$\beta$ decay can be slow or rapid compared to neutron capture rate

$r$-process efficacy can be described by a single parameter (Hoffmann 1997)

$\zeta \sim \frac{S^3}{t_{\rm dyn} Y_e^3}$ $\qquad\zeta>10^9$ required for 3rd peak

This condition is not met for normal PNS winds — entropy is too low, dynamical time is too long, electron fraction is too high, or some combination.

What about magnetars? Closed field loops will trap matter temporarily. Neutrino heating increases $S$.

Summary

Why do supernovae explode? Steady-state shocked accretion flow becomes impossible. The flow must transition into a wind.

What role does turbulence play in aiding explosion? Turbulence lowers the critical sound speed required for explosion.

How do proto-magnetars spin down after birth? My models account for tilt between the rotation and magnetic axes. They can simulate to late times and do not assume dipole spindown. I provide $\dot{E}(t)$, $\dot{J}(t)$ across a wide parameter space.

DVD Extras

Let us briefly consider the effects of rotation. In our simplified model, all rotation does is add a centrifugal term to the momentum equation,

$$a_{\rm cen} = \frac{v_\phi^2}{r}$$

The crtical condition, meanwhile, becomes

$$\frac{c_T^2}{v_{\rm esc}^2} + \frac{1}{3}\frac{v_{\phi}^2}{v_{\rm esc}^2} \approx \frac{3}{16}$$

This is basically the same as the critical condition with turbulence!

Turbulence and rotation modify the momentum equation in very different manners — turbulence, as an effective pressure, and rotation as an effective potential.

We suppose that other contributors to explosion will affect the critical condition in the same manner

$$\frac{c_T^2}{v_{\rm esc}^2} + \frac{1}{3}\frac{v_{\rm turb}^2}{v_{\rm esc}^2} \approx \frac{3}{16}$$

We suppose that other contributors to explosion will affect the critical condition in the same manner

$$\frac{c_T^2}{v_{\rm esc}^2} + \frac{1}{3}\frac{v_{\phi}^2}{v_{\rm esc}^2} \approx \frac{3}{16}$$

We suppose that other contributors to explosion will affect the critical condition in the same manner

$$\frac{c_T^2}{v_{\rm esc}^2} + \frac{1}{3}\frac{v_{\rm aco}^2}{v_{\rm esc}^2} \approx \frac{3}{16}$$

We suppose that other contributors to explosion will affect the critical condition in the same manner

$$\frac{c_T^2}{v_{\rm esc}^2} + \frac{1}{2}\frac{v_{A}^2}{v_{\rm esc}^2} \approx \frac{3}{16}$$

We suppose that other contributors to explosion will affect the critical condition in the same manner

$$\frac{c_T^2}{v_{\rm esc}^2} + \frac{1}{2}\frac{v_{A}^2}{v_{\rm esc}^2} \approx \frac{3}{16}$$

This is not a precise prediction, but a framework for understanding the fundamental physics of the critical condition.