Linear analysis of the standing accretion shock instability with non-spherical accretion flows

Kazuya Takahashiwith Yu Yamamoto, Shoichi Yamada

(Waseda Univ.)

Outline

Introduction

My research

Results

Standing Accretion Shock Instability (SASI)

New pointsHow?

Non-spherical structures in progenitors

Summary

Introduction

Standing Accretion Shock Instability (SASI):* One of the hopeful mechanisms for successful explosion of core-collapse supernovae* Multi-dimensional fluid instability, which enhances the neutrino heating because the fluid particles dwell the gain region much longer due to the complex flow patterns.

(Blonding et al. 2003)(Iwakami et al. 2014)

Introduction

The neutrino heating mechanism is thought to be the most promising one for core-collapse supernovae. However, successful explosions have not been obtained yet.Recently, the non-spherical structure in progenitors arises as a new key, which was caused by nuclear burnings in the shells and following violent convections (e.g. Arnett & Meakin 2011, Chatzopoulos et al. 2014).

(Arnett & Meakin 2011)

Fe core

Si burning

O burning

Introduction

Actually, we recently showed by linear analysis that such a non-spherical fluctuations in convectiving shells can be amplified during the infall onto the stalled shock, which suggests that fluctuations in the progenitor may affect the shock dynamics (KT & Yamada 2014).

(KT & Yamada 2014)

Adding perturbations by hand, Couch & Ott (2013, 2014) and Muller & Janka (2014) showed that perturbations in convecting shells can facilitate the shock revival even for a progenitor that fails to explode without the fluctuation.

Introduction

In most cases, however, numerical simulations are carried out with the steady upstream flows, partly because the star evolutions are calculated in 1D and hence the progenitors have spherically symmetric structures.

Muller & Janka (2014) reported that large scale perturbations (l = 1, 2) especially contribute the shock revival, which may have enhanced the Standing Accretion Shock Instability (SASI).

Introduction

Previous analytical studies on SASI (e.g. a series of papers by Yamasaki & Yamada, or the group of Foglizzo) have assumed the spherically symmetric steady flows in front of the standing shock.

We need the analytical study on SASI for non-steady and non-spherical upstream flows

How to analyze SASI

(cf. e.g. Yamasaki & Yamada 2007)Global linear analysis on the post-shock flow

We solved an eigenvalue problem. i.e., We seek the eigenfunctions of the flow between the shock surface and the proto-neutron star surface.

Outer Boundary →

Inner Boundary →

Connected by the Rankine-Hugoniot relations at the standing shock

How to analyze SASI

Linearized equations:

Initially, no perturbation:

: vector that denotes perturbations

B.C. at the shock surface; connected by the Rankine-Hugoniot relations

: Matrices whose elements are the background quantities

B.C. at the PNS surface; imposed

: ONLY ONE condition is imposed

How to analyze SASI

Laplace-transformed linearized equations:

B.C. at the shock surface; connected by the Rankine-Hugoniot relations

B.C. at the PNS surface; imposed

: ONLY ONE condition is imposed

(Laplace transformed)

Laplace-transformation with respect to time:

How to analyze SASI

Laplace-transformed linearized equations:

B.C. at the shock surface; connected by the Rankine-Hugoniot relations

B.C. at the PNS surface; imposed

0. We know the upstream flows (given).1. They are connected downstream with the linearized Rankine-Hugoniot relations provided a value of s and a one parameter is given: the perturbation of shock radius.

2. We solve the linearized cons. laws from the shock surface to PNS.

3. Seek the parameter that gives the consistent value at PNS surface

How to analyze SASI

Laplace-transformed linearized equations:

B.C. at the shock surface; connected by the Rankine-Hugoniot relations

B.C. at the PNS surface; imposed

0. We know the upstream flows (given).1. They are connected downstream with the linearized Rankine-Hugoniot relations provided a value of s and a one parameter is given: the perturbation of shock radius.

2. We solve the linearized cons. laws from the shock surface to PNS.3. Seek the parameter that gives the consistent value at PNS surface

For each value of s, we obtain the eigenvalue: Laplace-transformed value of the shock radius.Repeating this procedure for a series of s, we obtain the evolution of the shock radius by performing the inverse Laplace transform.

Basic equations that describe downstream flow:

Formulation

Continuity eq.

Momentum eq.

Energy eq.

EoS

Eq. for electron number

nucleon, nuclei (NSE, ideal Boltzmann gas), photons (ideal Bose gas), electrons and positrons (ideal Fermi gas)

Neutrino transport: Light bulb + geometric factorcf. Onhishi et al. (2006), Scheck et al. (2006)

Linearize the eqs. under the assumptions:* Background flow is spherically symmetric.* Decompose with the spherical harmonics Y_{lm} as our previous work

Formulation

And then Laplace-transform the linearized eqs.

: whose elements are written by the background flow

FormulationLinearization:* Background flow is spherically symmetric* Unperturbed shock surface is spherical:

* Non-spherical components are decomposed with

Continuity eq.

Momentum eq.

Momentum eq.

Momentum eq.

Energy eq.

Electron number

FormulationLaplace-transformed linearized eqs:

One parameter family of

c: vector, R: matrix, whose components are written by the background flow

: perturbations in front of the shock (given)

Downstream flow across the shock is given by

Formulation

Outer boundary(determined by one parameter)

Inner boundary condition

Basic equation

We solve the eingenvalue problem that is given by

Mode Analysis

×

×

××

×

×

Re

Im

(Growth rate)

(oscillate frequency)

SASI modes correspond to the poles of the Laplace transformed perturbed shock radius (= eigenvalue) in the complex plane

Mode Analysis

SASI modes correspond to the poles of the Laplace transformed perturbed shock radius (= eigenvalue) in the complex plane

×

×

××

×

×

Re

Im

(Growth rate)

(oscillate frequency)

searchpath

Along the path, we solve the eigenvalue problem for each s.

⇒ we can easily find the poles by seeing the rapid changes of the eigenvalue around them.

Mode Analysis

SASI modes correspond to the poles of the Laplace transformed perturbed shock radius (= eigenvalue) in the complex plane

×

×

××

×

×

Re

Im

(Growth rate)

(oscillate frequency)

searchpath

Along the path, we solve the eigenvalue problem for each s.

⇒ we can easily find the poles by seeing the rapid changes of the eigenvalue around them.

ReIm

Model setup

Background steady flow:

(for every quantities: rho, v, etc.)

Perturbations (2 patterns):

(The distance between the shock front and PNS ～ 50 km: rather small)

Results

××

××

×

×

Re

Im

×

(oscillate frequency)

(Growth rate)

External forces do not affect the intrinsic SASI modes, which is obtained by the impulse force

It only adds the simply oscillate mode whose frequency is the same as the input one.

newly added mode by the external force

Intrinsic modes

Results

Intrinsic eigenmode

Mode of the external force

G: Green function that describes the response to the impulsive forcef : External force

××

××

×

×

Re

Im

×

Why?

Results

The most rapidly-growing mode appears to be driven by the purely-acoustic cycle.

One of the 4 advective-acoustic modes grows exponentially whereas the other modes are decaying ones.The growth rate of the advective-acoustic mode is, however, much smaller than the purely-acoustic-cycle modes.

About the intrinsic modes:We found 17 growing modes for the range of ω > 1 [ms]. (#modes for larger frequencies seem to be countably infinite)And we identified the most rapidly growing mode.

Summary

Oscillations do not affect the SASI activity, which only add a simple oscillate mode, as long as there are only growing or decaying modes.

* How about other backgrounds, inner boundary conditions, or higher ells?

For a background, where the distance between the shock front and PNS is rather small, the purely acoustic cycle appears to be more efficient than the advective-acoustic cycle.

* The exponentially growing or decaying modes in SASI do NOT resonate with the simply oscillating force.* How about the evolutionary paths of each mode?

Appendix

Aim & Scope

Investigating the impact of fluctuating upstream flows on shock dynamics (SASI) in core-collapse supernovae analytically in detail (linear analysis).

Upstream (pre-shock flow)Consistently, we began the study of the growth and time-variability of fluctuation in super-sonic flows.

Downstream (post-shock flow)Then, we investigate the impact on shock dynamics, SASI.

Published

In preparation

(KT & Yamada 2014)

Aim & Scope

Investigating the impact of fluctuating upstream flows on shock dynamics (SASI) in core-collapse supernovae analytically in detail (linear analysis).

Upstream (pre-shock flow)Consistently, we began the study of the growth and time-variability of fluctuation in super-sonic flows.

Downstream (post-shock flow)Then, we investigate the impact on shock dynamics, SASI.

Published

In preparation

(KT & Yamada 2014, ApJ)

Rankine-Hugoniot relations:

Formulation

U: conservative variable, F: corresponding flux,q: velocity of the shock, : jump across the shockn: normal vector of the shock surface

Shock surface:

Yamamoto et al. (2013)