# Are pulsars spun up or down by SASI spiral modes?

Rémi Kazeroni, Jérôme Guilet, Thierry Foglizzo

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Rémi Kazeroni, Jérôme Guilet, Thierry Foglizzo. Are pulsars spun up or down by SASI spiral modes?. Monthly Notices of the Royal Astronomical Society, 2017, 471 (1), pp.914-925. 10.1093/mn-ras/stx1566. hal-01645468

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# Are pulsars spun up or down by SASI spiral modes?

Rémi Kazeroni,<sup>1,2*</sup> Jérôme Guilet<sup>2</sup> and Thierry Foglizzo<sup>1</sup>

Accepted 2017 June 20. Received 2017 June 20; in original form 2017 January 23

## ABSTRACT

Pulsars may either be spun up or down by hydrodynamic instabilities during the supernova explosion of massive stars. Besides rapidly rotating cases related to bipolar explosions, stellar rotation may affect the explosion of massive stars in the more common situations where the centrifugal force is minor. Using 2D simulations of a simplified set-up in cylindrical geometry, we examine the impact of rotation on the standing accretion shock instability (SASI) and the corotation instability, also known as low-T/|W|. The influence of rotation on the saturation amplitude of these instabilities depends on the specific angular momentum in the accretion flow and the ratio of the shock to the neutron star radii. The spiral mode of SASI becomes more vigorous with faster rotation only if this ratio is large enough. A corotation instability develops at large rotation rates and impacts the dynamics more dramatically, leading to a strong one-armed spiral wave. Non-axisymmetric instabilities are able to redistribute angular momentum radially and affect the pulsar spin at birth. A systematic study of the relationship between the core rotation period of the progenitor and the initial pulsar spin is performed. Stellar rotation rates for which pulsars are spun up or down by SASI are estimated. Rapidly spinning progenitors are modestly spun down by spiral modes, less than ~30 per cent, when a corotation instability develops. Given the observational constraints on pulsar spin periods at birth, this suggests that rapid rotation might not play a significant hydrodynamic role in most core-collapse supernovae.

**Key words:** hydrodynamics – instabilities – shock waves – stars: neutron – stars: rotation – supernovae: general.

## 1 INTRODUCTION

The study of the birth properties of pulsars is of considerable importance for the understanding of core-collapse supernovae. These properties provide meaningful information on the progenitors as well as the physics of the explosion. In particular, the initial pulsar spin may be related to the core rotation period of the progenitor which can affect the dynamics of the core-collapse and the explosion. A majority of young neutron stars are considered to be slow rotators, with periods of a few tens to a few hundreds of milliseconds. Such a broad distribution can be inferred from observations of known pulsars whose age can be estimated (Vranesevic et al. 2004; Popov & Turolla 2012) as well as population synthesis studies (Faucher-Giguère & Kaspi 2006). This distribution sets a strong constraint on the explosion mechanism, suggesting that either explosions result from progenitors with slowly rotating cores at the point of collapse or there exists an efficient spin-down mechanism.

The inner angular momentum profile of massive stars is still poorly constrained. Recent asteroseismic observations of low-mass red giants (Beck et al. 2012; Mosser et al. 2012) have revealed that the core of these stars seem to rotate significantly slower than predicted by stellar evolution calculations (Cantiello et al. 2014; Deheuvels et al. 2014). These computations including angular momentum transport due to magnetic torques predict pulsar spin periods of 10–15 ms, rather fast compared to the observations (Heger, Woosley & Spruit 2005). The inclusion of internal gravity waves in these computations of stellar evolution could affect the distribution of angular momentum (Fuller et al. 2015). Binary interaction during the progenitor lifetime may also influence the rotation profile (Sana et al. 2012). Even in the last stage of stellar evolution, during core collapse, hydrodynamic instabilities can induce transverse mass motions that can impact the initial rotation period of the neutron star (Wongwathanarat, Janka & Müller 2010, 2013).

The most promising framework to explain core-collapse supernovae is the delayed neutrino-driven mechanism (Bethe & Wilson 1985; Janka, Melson & Summa 2016; Müller 2016). In this scenario, a fraction of the copious number of neutrinos emitted from the forming neutron star and the accretion layer interacts

<footnote n="1"><sup>1</sup>Laboratoire AIM, CEA/DRF-CNRS-Université Paris Diderot, IRFU/Département d'Astrophysique, CEA-Saclay F-91191, France</footnote>

<footnote n="2"><sup>2</sup>Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany</footnote>

<footnote>* E-mail: kazeroni@MPA-Garching.MPG.DE</footnote>
with the matter in the gain layer, where neutrino absorption dominates neutrino emission, to revive the stalled accretion shock. Except at the low-mass end of supernova progenitors, massive stars do not explode in spherical symmetry (Liebendörfer et al. 2001; Kitaura & Hillebrandt 2006). Multidimensional simulations of core-collapse supernovae have emphasized the supportive role of hydrodynamical instabilities for the shock revival. Two types of instabilities can break the spherical symmetry of the collapse and lead to an asymmetric explosion: the neutrino-driven convection (Herant et al. 1994; Janka & Mueller 1996) and the standing accretion shock instability (SASI) (Blondin, Mezzacappa & DeMarino 2003).

Simulating robust explosions from ab initio models remains a theoretical and numerical challenge. Even though state-of-the-art 3D models seem to be close to the explosion threshold (Hanke et al. 2013; Lentz et al. 2015; Melson et al. 2015; Janka et al. 2016), the explosion energies obtained only cover the lower side of the range $10^{50}-10^{51}$ erg inferred from observations (Müller 2015). Several ingredients have been considered for their ability to ease the explosion, such as rotation (Iwakami, Nagakura & Yamada 2014; Nakamura et al. 2014), magnetic fields (Guilet, Foglizzo & Fromang 2011; Obergaulinger, Janka & Aloy 2014), pre-collapse inhomogeneities (Couch & Ott 2013; Müller & Janka 2015) and changes in neutrino-nucleon interaction rates (Melson et al. 2015). Our work focuses on the impact of stellar rotation and covers a wide range of rotation rates from slow to fast spinning cases, the former having received little attention in 3D models so far. Rotation may facilitate standard core-collapse supernovae in several ways. The critical neutrino luminosity threshold for explosion decreases with the specific angular momentum injected (Iwakami et al. 2014; Nakamura et al. 2014). Rotation favours the development of prograde SASI spiral modes (Blondin & Mezzacappa 2007; Yamasaki & Foglizzo 2008), even if the centrifugal force is weak. Above a certain rotation rate, the development of a strong one-armed spiral mode related to a corotation instability may help the shock revival in situations where the neutrino heating is not sufficient (Takiwaki, Kotake & Suwa 2016). Such rotations may be related to magnetorotational bipolar explosions with possibly higher explosion energies than standard core-collapse supernovae (Moiseenko, Bisnovatyi-Kogan & Ardeljan 2006; Burrows et al. 2007; Takiwaki, Kotake & Sato 2009) and could be linked to the formation of magnetars with possible consequences on long gamma-ray bursts (Metzger et al. 2011).

Besides its influence on the shock dynamics, rotation may also impact the pulsar spin at birth. Blondin & Mezzacappa (2007), using a 3D adiabatic set-up, showed that SASI spiral modes have the potential to spin up a neutron star born from a non-rotating progenitor. This surprising consequence of SASI was confirmed numerically by Fernández (2010), analytically by Guilet & Fernández (2014) and experimentally by Foglizzo et al. (2012, 2015) who reproduced the redistribution of angular momentum by a spiral mode in a shallow-water analogue of SASI. A SASI spiral mode could spin up a neutron star born from a non-rotating progenitor to periods in the range from 50 ms to 1 s (Guilet & Fernández 2014). However, SASI can also develop sloshing modes that do not allow for an angular momentum redistribution. Kazeroni, Guilet & Foglizzo (2016, hereafter Paper I) investigated the conditions for a spiral mode to prevail and showed that its emergence requires a minimum ratio between the initial shock radius and the proto-neutron star (PNS) radius. Regarding rotating progenitors, Blondin & Mezzacappa (2007) pointed out that SASI may reduce the neutron star spin compared to a simple estimate based on angular momentum conservation during the collapse. This could even give birth to

a counter-rotating neutron star if SASI redistributes enough angular momentum. Nevertheless, it is still unclear how efficiently a neutron star could be spun up or down by a SASI spiral mode throughout the range of stellar rotation rates.

The purpose of this paper is twofold. Using a parametric study of an idealized model, we aim at better describing the influence of rotation on the shock dynamics, focusing on both SASI and the corotation instability. Then, we propose to study the effect of spiral modes on the natal pulsar spin in order to assess to what extent these instabilities could spin the pulsar up or down. The paper is organized as follows. The physical and numerical models are described in Section 2. Section 3 is dedicated to the interplay between the instabilities and rotation. The relationship between the iron core rotation rate and the initial pulsar spin is examined in Section 4. The results are analysed in Section 5 and conclusions are drawn in Section 6.

## 2 METHODS

The idealized set-up considered in this work is dedicated to the study of SASI in its simplest form. It is very similar to the one described in Paper I, but includes the progenitor rotation. In short, the model is restricted to the equatorial plane of a massive star described in cylindrical geometry. This allows non-axisymmetric modes to develop despite the 2D geometry. The accretion flow is modelled by a perfect gas with an adiabatic index $\gamma = 4/3$. The flow is decelerated by a shock wave of initial radius $r_{sh0}$ with an incident Mach number $M_1 = 5$. The matter is then accreted subsonically on to the surface of a PNS. Its radius, defined as $r_*$, delimits the inner boundary of the domain. Heating by neutrino absorption is ignored in order to suppress the neutrino-driven convection for the sake of simplicity. A cooling function is used to mimic the energy losses due to neutrino emission, with the approximation $\mathcal{L} \propto \rho P^{3/2}$ (Blondin & Mezzacappa 2006) where $\rho$ and $P$, respectively, stand for the density and the pressure.

Once the non-rotating stationary flow has relaxed on the grid for a few hundreds of numerical timesteps (panel a of Fig. 1), a constant amount of specific angular momentum is injected through the outer boundary of the domain, similarly to the approach considered by Iwakami et al. (2014). The specific angular momentum $j$, defined as $j = v_\phi(r)r = \Omega(r)r^2$ where $v_\phi$ and $\Omega$ are the azimuthal velocity and the rotational velocity, respectively, is conserved if the evolution is axisymmetric. In this study, we consider a range of rotation rates covering three orders of magnitude: $10^{13} \text{ cm}^2 \text{ s}^{-1} \le j \le 10^{16} \text{ cm}^2 \text{ s}^{-1}$. Note that $j = 10^{16} \text{ cm}^2 \text{ s}^{-1}$ would correspond to a spin period of 0.6 ms for a pulsar of 10 km radius if the angular momentum were conserved down to its surface. According to stellar evolution calculations, the specific angular momentum contained in-between the enclosed masses $1.3 M_\odot$ and $2 M_\odot$ is such that $j \sim 10^{14}-10^{15} \text{ cm}^2 \text{ s}^{-1}$ if the magnetic field is taken into account and $j \sim 10^{16}-10^{17} \text{ cm}^2 \text{ s}^{-1}$ otherwise (Heger et al. 2005). Among the few 3D simulations including rotation, some considered very fast rotation rates such as: $j = 1.2-6.3 \times 10^{16} \text{ cm}^2 \text{ s}^{-1}$ (Nakamura et al. 2014), $j = 3.1 \times 10^{16} \text{ cm}^2 \text{ s}^{-1}$ (Kuroda, Takiwaki & Kotake 2014) and $j = 3.9 \times 10^{16} \text{ cm}^2 \text{ s}^{-1}$ (Takiwaki et al. 2016). However, slower rotations have also been considered: $j = 10^{15} \text{ cm}^2 \text{ s}^{-1}$ (Blondin & Mezzacappa 2007), $j = 1-4 \times 10^{15} \text{ cm}^2 \text{ s}^{-1}$ (Endeve et al. 2012) and $j \le 10^{16} \text{ cm}^2 \text{ s}^{-1}$ (Iwakami et al. 2014).

We let the 1D rotating flow relax for a few more hundreds of numerical timesteps (panel a of Fig. 1) in order to reach an equilibrium before density perturbations are introduced to trigger SASI. Rota-
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Time evolution of the stationary flow and early non-linear phase of the instability in the case $R = 3$ and $j = 10^{16} \text{ cm}^2 \text{ s}^{-1}$. The quantities are normalized by the initial shock radius $r_{sh0}$. Panel (a): after the non-rotating stationary flow has relaxed on the grid for a few tens of milliseconds, a constant specific angular momentum is injected through the outer boundary (dot-dashed line). The mean shock radius (cyan curve) increases by less than 10 per cent. Once the 1D rotating flow has relaxed, density perturbations are injected (dashed line). Panel (b): the amplitude of the Fourier coefficient $c_1$ of the shock deformation begins to grow about an advection time after the perturbations have been injected (red crosses). A measurement of the oscillation frequency and the growth rate of the mode $m = 1$ of SASI is performed during the well-defined exponential growth of the instability in the linear regime (fit represented by the black line). Panels (c) and (d): the evolution of the real part (c) and the imaginary part (d) of the Fourier coefficient (red crosses) are fitted by the function (2) during the exponential growth of the coefficient (black lines).</figcaption></figure>

tion induces an increase of the shock radius of less than 10 per cent in the axisymmetric phase of the simulations. Similar to the approach considered in Paper I, two density perturbations at pressure equilibrium are launched through the outer boundary to initiate the two counter-rotating spiral modes $m = \pm 1$, where $m$ is the azimuthal wavenumber of the unstable mode of SASI. The amplitude of the perturbation is $10^{-4}$ and the asymmetry between the two counter-rotating spiral modes is $\epsilon = 0.25$ (as defined by equations 2 and 3 in Paper I).

When converting to physical units, we use the same normalization values as in Paper I: a PNS with radius $r_* = 50$ km, a neutron star of mass $M_* = 1.3 M_\odot$ and a constant mass accretion rate $\dot{M} = 0.3 M_\odot \text{ s}^{-1}$ which are typical values for the stalled shock phase of a core-collapse supernova during the first second after bounce. As in Paper I, the parameter $R$ defines the radii ratio: $R \equiv r_{sh0}/r_*$. For non-rotating cases, the threshold $R \gtrsim 2$ is a necessary condition for a spiral mode to prevail in the non-linear regime (Paper I). Below this ratio, a sloshing mode dominates the dynamics even if a spiral mode was artificially initiated in the linear regime of SASI. In this study, we consider the following radii ratios: $R = \{1.67, 2, 2.5, 3, 4, 5\}$. In total, 140 simulations are performed to cover the parameter space.

As in Paper I, the numerical simulations are performed with a version of the RAMSES code (Teyssier 2002; Fromang, Hennebelle & Teyssier 2006) adapted to cylindrical coordinates. RAMSES is a second-order finite volume code and employs the MUSCL-Hancock scheme. We use the HLLD Riemann solver (Miyoshi & Kusano 2005) and the monotonized central slope limiter. Except for the introduction of rotation, the numerical set-up is the same as in Paper I. Periodic boundary conditions are used in the azimuthal direction while a reflecting condition is employed at the inner boundary, $r = r_*$, and a constant inflow set by the 1D stationary flow at the outer boundary, $r = r_{out}$. The only modification to the previous set-up concerns the numerical resolution employed. A uniformly spaced grid is used with 1600 cells in the azimuthal direction. The number of radial cells is 150 below the shock for $R = 3$ and the grid cell size is maintained constant when $R$ varies. The outer boundary is located at $r_{out} = 6-10 r_{sh0}$. This represents a total of 1600–2730 cells in the radial direction. The value of $r_{out}$ is set so that the shock wave is always contained within the simulation grid even for the high rotation cases where it expands significantly (see Section 3.2).

# 3 INFLUENCE OF ROTATION ON THE SHOCK WAVE DYNAMICS

## 3.1 Overlapping instabilities

The accretion flow described in the previous section is unstable to SASI when rotation is neglected (Paper I). The centrifugal force $\Omega(r)^2 r$ is not dominant in the dynamics for the range of rotation rates considered in this study. The advection time from the shock radius to the PNS surface is only increased by 30 per cent over the range of rotation rates considered in our study, while the shock radius expands by less than 10 per cent. The centrifugal force is quadratic with respect to the rotation rate and remains much smaller than gravity $|GM_{NS}/r^2|$:

$$ \frac{j^2/r^3}{GM_{NS}/r^2} = \frac{2T}{W} = 1.2 \times 10^{-1} \left( \frac{j}{10^{16} \text{ cm}^2 \text{ s}^{-1}} \right)^2 \times \left( \frac{50 \text{ km}}{r} \right) \left( \frac{1.3 M_\odot}{M_{NS}} \right). \quad (1) $$

where $T$ and $W$, respectively, stand for the rotational kinetic energy and the gravitational potential energy.

In order to estimate the growth rates $\omega_i$ and oscillation frequencies $\omega_r$ of SASI from our simulations, an azimuthal Fourier decomposition of the shock deformation is computed. The time evolution of the Fourier coefficients $c_m$ is then fitted during the linear phase of SASI using a function of the form

$$ f(t) = A \cos(\omega_r t + \Phi) \exp(\omega_i t), \quad (2) $$

where $A$ and $\Phi$ represent the amplitude and the phase (see Fig. 1 for an example of this method).

The perturbative study of Yamasaki & Foglizzo (2008) revealed that the linear effect of the rotation on the growth rate of SASI is due to a Doppler shift of the frequencies. This shift induces a linear increase of the growth rate of prograde spiral modes with respect to the specific angular momentum (Fig. 2). Our set of simulations confirms the linear increase of the growth rate as shown in the case $R = 3$ (Fig. 3, upper panel), and also demonstrates a linear increase of the oscillation frequency (Fig. 3, middle panel). Recently, Blondin et al. (2017) obtained similar results in their simulations and showed that the linear trends hold for the 2D cylindrical geometry as well
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Growth rates of the most unstable modes as a function of the specific angular momentum for different values of R. The growth rate is normalized by the post-shock velocity divided by the shock radius. Solid lines are for the spiral modes *m* = 1 and dashed lines for the spiral modes *m* = 2.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Impact of rotation on the linear and the non-linear regimes of the most unstable mode of SASI (*m* = 1) for *R* = 3. Upper panel: the growth rate of the spiral mode depends linearly on *j* as shown both by a perturbative analysis (red curve) and simulations of the linear regime (blue curve). The values are normalized by the post-shock velocity divided by the shock radius. Middle panel: the oscillation frequency increases linearly with *j* as shown by the perturbative analysis (red curve). This trend is similarly obtained from numerical simulations of the linear regime (blue curve). In the non-linear regime, the oscillation frequency decreases with *j* due to the expansion of the shock wave (green curve). Lower panel: the corotation radius increases with *j* as shown both by the perturbative analysis (red curve) and simulations (blue curve). It is slightly higher in the non-linear phase (green curve) than in the linear regime. The black line represents the PNS radius below which no corotation exists. The values are normalized by the initial shock radius.</figcaption></figure>

as the 2D and 3D spherical geometries. The comparison with the perturbative analysis shows a 8 per cent discrepancy of growth rates at a low rotation rate and up to 15 per cent at a high rotation rate. The discrepancy for oscillation frequencies is significantly smaller, less than 1 per cent at low rotation and less than 10 per cent at high rotation. The increase of the discrepancy with rotation is not explained. It is reminiscent of the discrepancy observed by Foglizzo et al. (2007) in the simulations of Blondin & Mezzacappa (2006), which was also largest for the largest shock distance. Note that Blondin et al. (2017) also observed a departure from the linear trend at high rotation rates.

A corotation radius $r_{co}$ is defined by

$$r_{\text{co}} = \sqrt{\frac{mj}{\omega_r}}. \qquad (3)$$

This delimits a region inside which the flow rotates faster than the spiral pattern. Such a region can only exist if the rotation rate is high enough (Fig. 3, lower panel). Note that a corotation radius can emerge in the non-linear regime although it does not exist in the linear regime. This is a consequence of a wider post-shock region in the non-linear regime which increases the advection time and diminishes the oscillation frequency (Fig. 3, middle panel). The oscillation frequency in the non-linear regime is estimated by computing the rotation frequency of the triple point. For $R = 3$, a corotation radius can appear with a specific angular momentum 30 per cent lower than predicted by the linear analysis. Its emergence above the inner boundary, located at $r = r_*$, does not seem to affect the linear increase of the growth rates and the oscillation frequencies (Fig. 3).

Takiwaki et al. (2016) identified an explosion aided by a corotation instability in a simulation of a fast rotating case ($j = 3.9 \times 10^{16} \text{ cm}^2 \text{ s}^{-1}$) where a strong open one-armed spiral wave generates turbulent kinetic energy that complements the energy deposition by neutrinos and successfully revives the stalled shock. Such a situation seems to occur only if the dynamics is SASI-dominated as in the $27 M_{\odot}$ progenitor of their study. The mechanism responsible for SASI has been well explained as an advective-acoustic cycle (Foglizzo et al. 2007; Guilet & Foglizzo 2012). By contrast, the origin of the corotation instability remains uncertain. In a different context of an isolated neutron star rotating differentially, it has been proposed that this instability is generated by an energy transfer occurring across a band around a corotation radius where acoustic waves could be amplified (Watts, Andersson & Jones 2005; Passamonti & Andersson 2015). Numerically, it has been observed that the corotation instability can develop when the ratio $T/W$ exceeds a value of the order of 1 per cent both in differentially rotating neutron stars (Shibata, Karino & Eriguchi 2002, 2003) and in rotating collapsing stellar cores (Ott et al. 2005; Kuroda et al. 2014). Despite quite different mechanisms from a theoretical point of view, Kuroda et al. (2014) could not clearly distinguish SASI from the corotation instability in their simulations of a $15 M_{\odot}$ progenitor.

Although a spiral mode $m = 1$ always dominates the non-linear shock dynamics in our simulations (Fig. 4, upper panels), a tightly wound spiral flow is observed in the post-shock region at a large enough rotation rate (Fig. 4, middle right panel) and resembles the open one-armed spiral wave identified by Takiwaki et al. (2016). Such a flow pattern is never witnessed in cases where a corotation radius cannot be well defined (Fig. 4, lower panels), that is to say where SASI seems to be the only possible instability. This open one-armed spiral pattern is thus suggestive of a distinct instability.
<figure><img src="image_6.png" /><figcaption>**Figure 4.** A SASI-dominated case with $R = 5$ and $j = 10^{15} \text{ cm}^2 \text{ s}^{-1}$ (left-hand panels) is compared to a case dominated by a corotation instability with $R = 5$ and $j = 6 \times 10^{15} \text{ cm}^2 \text{ s}^{-1}$ (right-hand panels). Upper panels: entropy snapshots of the non-linear dynamics. Both cases are dominated by a spiral mode $m = 1$ and a greater shock expansion is reached in the faster rotating case. The black dashed lines delimit the region zoomed in the snapshots below. Middle panels: pressure variation with respect to the azimuthal averaged value computed as $(P - \langle P \rangle_\phi)/P$. A tightly wound spiral wave is observed only if the rotation is fast enough. The black dashed lines delimit the region zoomed in the snapshots below. Lower panels: rotation frequency defined as $v_\phi/r$. The black lines indicate a corotation radius that delimits the regions in which the flow has a higher rotation frequency than the spiral mode. A well-defined corotation develops only for high enough rotation rates and could trigger a second instability overlapping with SASI. The corotation instability seems to generate a strong open one-armed spiral wave.</figcaption></figure>

It is however not clear yet whether such a feature is a systematic signature of the corotation instability.

## 3.2 Shock evolution

In this section, we focus on the influence of rotation on the shock wave dynamics. For this purpose, the mean radius $\langle r_{\text{sh}} \rangle$ of the shock

and its degree of asymmetry $\Delta r$, defined as the dominant coefficient of the Fourier transform of the shock deformation, are time-averaged over the non-linear regime of the simulations. The degree of asymmetry corresponds to the saturation amplitude of the dominant mode of SASI and may set the magnitude of the neutron star kick (Scheck et al. 2004, 2006). Fig. 5 shows the influence of rotation on the dynamics compared to non-rotating cases in situations
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Influence of rotation on the non-linear regime of SASI for different values of R. Upper panel: saturation amplitude of the mode $m = 1$ as a function of $j$. Lower panel: mean shock radius as a function of $j$. The dashed curve shows the effect of the centrifugal force in 1D simulations. The black circled points denote that a corotation emerges in the linear regime (open circles) or in the non-linear regime (filled circles). In both panels, the values are averaged in the non-linear regime and normalized by the ones measured in the non-rotating cases. To reduce the variability of the results, the non-rotating simulations are repeated five times with slightly different initial perturbation amplitudes. The simulations where $j = 10^{15} \text{ cm}^2 \text{ s}^{-1}$ are also repeated five times to assess the stochasticity of our results. The vertical bars indicate the minimum and maximum values obtained for each value of R.</figcaption></figure>

where SASI is the only instability involved. For each value of R, five simulations of the non-rotating case are performed in order to estimate the variability of the results, as discussed later in this section. These simulations differ only by a minor change of the initial perturbation amplitude. Both the shock expansion and the degree of asymmetry are non-monotonic functions of the rotation rate. In fact, these quantities increase with the angular momentum only if the condition $R > 2$ is fulfilled. In the absence of a corotation radius and if $R > 2$, rotation increases the shock expansion and the degree of asymmetry of the SASI spiral mode by 15 per cent and 30 per cent, respectively. The larger the radii ratio, the lower the angular momentum threshold for rotational effects on SASI.

Below the ratio $R = 2$, it was observed in Paper I that sloshing modes dominate the non-linear regime of SASI when rotation is neglected. For $R = 2$, a minimal specific angular momentum of $j = 10^{15} \text{ cm}^2 \text{ s}^{-1}$ is necessary to destabilize a sloshing mode into a spiral mode. Surprisingly, the shock expansion and the degree of asymmetry, respectively, diminish by 10 per cent and 20 per cent between $j = 10^{15} \text{ cm}^2 \text{ s}^{-1}$ and $j = 3 \times 10^{15} \text{ cm}^2 \text{ s}^{-1}$. This decrease is related to the non-linear saturation amplitude since the growth rates of the most unstable modes are still increasing functions of the rotation rate (Fig. 2).

Interestingly, the critical radii ratio required for the rotation to increase the degree of asymmetry is similar to the ratio $R = 2$

<figure><img src="image_8.png" /><figcaption>**Figure 6.** Same as in Fig. 5 for the corotation instability. In this case, the simulations where $j = 10^{16} \text{ cm}^2 \text{ s}^{-1}$ are repeated five times.</figcaption></figure>

identified in Paper I. We suggest that this may not be a coincidence and we rather propose a mechanism for the symmetry breaking between counter-rotating spiral modes, which can explain the equality of these critical ratios. For this purpose, we will assume that the rotation induced by a spiral mode has a similar effect on its saturation amplitude as the rotation of the progenitor. For $R > 2$, if a spiral mode has a slightly larger amplitude than the counter-rotating spiral, the rotation it induces will cause it to grow to even larger amplitudes until it dominates the dynamics. On the other hand for $R < 2$, the induced rotation would cause the amplitude of the spiral to diminish and therefore tend towards a state where the two counter-rotating spirals have the same amplitude, i.e. a sloshing mode.

These results suggest that rotation might not always ease the shock revival in ab initio 3D simulations of core-collapse novae. On the one hand, if $R > 2$ rotation enhances the spiral mode of SASI which may facilitate the explosion (Fernández 2015). On the other hand, for low radii ratios, rotation reduces the amplitude of the spiral mode, disfavouring the shock revival. To assess the stochasticity of our results, we performed five different realizations of each simulation where $j = 10^{15} \text{ cm}^2 \text{ s}^{-1}$, varying the initial perturbation amplitude by a few per cent. The vertical bars in the Fig. 5 indicate that the shock expansion and the degree of asymmetry can vary, respectively, by 10 per cent and 20 per cent. These fluctuations are as large as the effect of rotation on the shock dynamics. Thus, a single ab initio 3D simulation including rotation may not be sufficient to assess the impact of rotation on SASI and the explosion due to the stochastic nature of the non-linear regime of SASI. The consequences of the stochasticity on the explodability were pointed out by Cardall & Budiardja (2015) who showed that a critical neutrino luminosity cannot be well defined for SASI-dominated cases. The same conclusion may be valid for rotation, whose impact on the dynamics may be altered by stochasticity.

Finally, the presence of a corotation radius greatly impacts the dynamics as shown in Fig. 6. The shock expansion and the degree
<table><caption>**Table 1.** This table summarizes the properties of the dynamics depending on the regime considered. The shock radius (second column) and the degree of asymmetry (third column) are normalized by the initial shock radius $r_{sh0}$. From top to bottom, the different regimes are: centrifugal force studied in 1D simulations, SASI in non-rotating progenitors (see Paper I), SASI without a corotation instability and SASI with an overlapping corotation instability.</caption><thead><tr><th></th><th>⟨r<sub>sh</sub>⟩/r<sub>sh0</sub></th><th>Δr/r<sub>sh0</sub></th></tr></thead><tbody><tr><td>Centrifugal force (1D)</td><td>1.05</td><td>—</td></tr><tr><td>SASI without rotation</td><td>1.3–2.3</td><td>30–60 per cent</td></tr><tr><td>SASI without a corotation</td><td>1.4–2.8</td><td>40–80 per cent</td></tr><tr><td>SASI with a corotation</td><td>2–4</td><td>60–200 per cent</td></tr></tbody></table>

of asymmetry increase more steeply when SASI overlaps with a corotation instability. This second instability increases the mean shock radius and the saturation amplitude by a factor up to 3 and 4, respectively, for $R = 5$, compared to non-rotating progenitors. For cases where $R \le 2$, the corotation instability modifies the influence of rotation on the dynamics. While the degree of asymmetry decreases with rotation when the corotation is absent, it increases when the corotation radius exceeds the PNS radius. The impact of corotation is more important with greater values of $R$ because the frequency of the fundamental mode is decreased and the minimal rotation rate for a corotation to emerge is thus reduced. The amplification of the spiral mode by the corotation instability is larger than the stochastic fluctuations as illustrated by the vertical bars in Fig. 6. The spiral mode becomes more vigorous in the presence of a corotation instability and this may have consequences on the explosion mechanism (Takiwaki et al. 2016). The diversity of the effects of rotation on the shock dynamics is summarized in Table 1.

# 4 PULSAR SPIN

## 4.1 Pulsar spin estimate

In Paper I, we showed that the spin periods measured in our simulations of a cylindrical set-up agree relatively well with the analytical estimate of Guilet & Fernández (2014) adapted to this geometry. These results are based on the idealized scenario proposed by Blondin & Mezzacappa (2007) in which the mass cut radius coincides with the surface where the angular momentum changes sign. In that study, the derivation of the pulsar spin neglected the angular momentum of the iron core collapsed into a PNS and the parameter space was restricted to a single rotation rate. In order to clarify the issue of a pulsar spin-up or spin-down, we propose to disentangle the respective contributions of the angular momentum contained initially in the core and the quantity redistributed by a spiral mode.

Considering a spiral mode in a rotating iron core, the total angular momentum of a neutron star at birth, noted $L_{NS}$, can be estimated by

$$L_{NS} \equiv L_{\text{CORE}} - \Delta L. \qquad (4)$$

where $\Delta L$ corresponds to the angular momentum redistributed by a spiral wave and $L_{CORE}$ to the quantity contained in the core prior to collapse. The angular momentum redistribution can be evaluated by

$$\Delta L \equiv L_{\text{SPIRAL}} - L_0, \qquad (5)$$

where $L_0$ represents the angular momentum contained in the stationary flow and $L_{SPIRAL}$ the quantity stored below the shock wave.

As in Paper I, we will assume that the quantity opposite to $L_{SPIRAL}$ is accreted on to the PNS. To evaluate this quantity, we first compute the mean radial profile of the angular momentum density:

$$l_z(r, t) \equiv H r^2 \int_0^{2\pi} \rho v_\phi d\phi, \qquad (6)$$

where $H$ is an arbitrary parameter that corresponds to the height of the cylinder. The results are independent of its value, which can be set so that $\dot{M} = 0.3 M_{\odot} \text{ s}^{-1}$. $L_{SPIRAL}$ is computed using the following formula:

$$L_{\text{SPIRAL}} \equiv \frac{1}{T} \int_{t_0}^{t_0+T} \int_{r_{\text{cut}}}^{r_{\text{sh}}} l_z(r, t) \, dr \, dt, \qquad (7)$$

where $t_0$ corresponds to the initial time of the non-linear regime, $T$ to the time interval considered and $r_{cut}$ to the mass cut radius. This radius is defined by computing the radius above which the mean radial profile of the angular momentum density in the non-linear regime is greater than the initial one. This method gives an upper bound on the amount of angular momentum redistributed by a spiral mode.

The inner boundary of our model does not include the neutron star. An additional assumption on the PNS interior is needed to evaluate the quantity $L_{CORE}$. The PNS is assumed to result from the contraction of the inner core (1.3 $M_{\odot}$), which roughly corresponds to a mass coordinate of 1500 km (see e.g. Marek & Janka 2009), into a 50 km radius object. We will consider idealized density and angular velocity profiles of the inner 1500 km region. The density profile is defined by

$$\rho_{\text{CORE}}(r) = \begin{cases} \rho_0 & \text{if } r < r_{\text{int}}, \\ \rho_0 \left(\frac{r_{\text{int}}}{r}\right)^3 & \text{if } r \ge r_{\text{int}}, \end{cases} \qquad (8)$$

with $r_{int} = 300$ km. For $\rho_0 = 4 \times 10^9 \text{ g.cm}^{-3}$, the mass contained in the sphere of 1500 km radius is about $1.3 M_{\odot}$. This profile is chosen to mimic more realistic conditions while being consistent with the use of a constant mass accretion rate at the outer boundary. The angular velocity profile is defined by

$$\Omega_{\text{CORE}}(r) = \begin{cases} \Omega_0 & \text{if } r < r_0, \\ \Omega_0 \left(\frac{r_0}{r}\right)^2 & \text{if } r \ge r_0, \end{cases} \qquad (9)$$

where $r_0 = 1000$ km and $\Omega_0 = j/r_0^2$. Such a profile is intended to imitate the results of stellar evolution calculations (see e.g. Ott et al. 2006) while being consistent with the assumption of a constant specific angular momentum injected at the outer boundary. The angular momentum of the core, which would be conserved during an axisymmetric collapse, can be computed as follows:

$$L_{\text{CORE}} \equiv 4\pi \int_{0}^{1500 \text{ km}} \rho_{\text{CORE}}(r) r^2 \Omega_{\text{CORE}}(r) r^2 dr. \qquad (10)$$

## 4.2 Angular momentum redistribution in the post-shock region

The angular momentum redistribution due to a spiral mode in the post-shock flow is displayed in Fig. 7. This redistribution significantly exceeds the initial angular momentum only for slow rotation rates, i.e. $j \lesssim 10^{14} \text{ cm}^2 \text{ s}^{-1}$, and if a SASI spiral mode emerges, i.e. if $R > 2$. In this regime, the quantity $L_{SPIRAL}$ does not depend on the angular momentum injected and equals the non-rotating case.

As the rotation rate increases, the ratio $\Delta L/L_0$ converges towards a constant value. The higher the radii ratio, the higher this asymptotic value. The break of slopes is induced by the corotation that sets
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Angular momentum redistribution in the post-shock flow, defined by the ratio $\Delta L/L_0$ for different values of R. The black circled points denote that a corotation emerges in the linear regime (open circles) or in the non-linear regime (filled circles).</figcaption></figure>

a minimal limit to the ratio between the angular momentum redistributed and the initial quantity. As in Paper I, we observe that high radii ratios are more efficient at redistributing angular momentum. This is a consequence of the higher saturation amplitudes associated with high radii ratios.

## 4.3 Influence of spiral modes on the pulsar spin

The importance of the spiral modes in the angular momentum budget of the collapse of a massive star depends both on the rotation rate and the radii ratio (Fig. 8). In the cases where the angular momentum redistributed, $\Delta L$, exceeds the one contained in the core, $L_{CORE}$, SASI can give birth to counter-rotating neutron stars. This is observed for high radii ratios, $R \ge 3$, and low rotation rates: $j \le 1 - 5 \times 10^{14} \text{ cm}^2 \text{ s}^{-1}$. In this region of the parameter space, non-axisymmetric modes are the dominant contribution to the angular momentum budget.

On the contrary, we identify two distinct regions of the parameter space where non-axisymmetric modes do not play a dominant role on the neutron star spin. The first one corresponds to low radii ratios: $R \le 2$. In such cases, rotation hardly affects SASI (Fig. 5) and the saturation amplitude of the spiral is too low to drive significant angular momentum redistribution (Guilet & Fernández 2014). The second one is related to the corotation instability. The amount of angular momentum redistributed can only account for less than 30 per cent of the core one (Fig. 9). This indicates that the corotation instability may only modestly spin down the neutron star and does not represent an efficient braking mechanism.

Note that in the case considered by Blondin & Mezzacappa (2007): $R = 5$ and $j = 10^{15} \text{ cm}^2 \text{ s}^{-1}$, the angular momentum in the core overcomes the quantity redistributed by SASI (Fig. 8) and this does not lead to a counter-rotating neutron star if the core angular momentum is taken into account in the derivation of the neutron star spin period. Counter-rotating neutron stars may only concern progenitors with a rotation rate at least 2–5 times slower.

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Ratio between the angular momentum redistributed by a spiral mode and the quantity contained in the core prior to collapse. Above the horizontal black line, the spiral mode mostly sets the spin period and these situations give birth to a counter-rotating neutron star. The black circled points denote that a corotation emerges in the linear regime (open circles) or in the non-linear regime (filled circles).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Figure 9.** Average ratio between the angular momentum redistributed by the spiral mode and the quantity contained in the iron core, computed over cases where a corotation radius emerges. This ratio reflects the efficiency of the corotation instability to spin down the pulsar for different values of R.</figcaption></figure>

The angular momentum of the neutron star $L_{NS}$ can be converted into a rotation frequency $f_{NS}$ using the relation:

$$f_{NS} = \frac{L_{NS}}{2\pi I_{NS}}, \qquad (11)$$

where $I_{NS} \sim 10^{45} \text{ g cm}^2$ represents the moment of inertia of the neutron star. The rotation frequency of the core is defined in a similar way by considering the frequency a neutron star would have assuming a spherically symmetric collapse of the iron core: $f_{CORE} = L_{CORE}/(2\pi I_{NS})$. Fig. 10 shows how a spiral mode influences the relationship between the core rotation frequency and the neutron star one. On the one hand, the results confirm that the neutron star spin period is almost not affected in cases with a low radii ratio or a high rotation rate (around the black dashed line). On
<figure><img src="image_12.png" /><figcaption>**Figure 10.** Relationship between the rotation frequency inferred from a spherically symmetric collapse ($f_{\text{CORE}}$) and the one that accounts for the development of spiral modes ($f_{\text{NS}}$). The black dashed line illustrates the absence of angular momentum redistribution ($f_{\text{NS}} = f_{\text{CORE}}$). Neutron stars above and below this line are, respectively, spun up and down by SASI. The upper grey dot-dashed line indicates the maximal rotation frequency that seems compatible with the observations ($f \lesssim 100$ Hz). Counter-rotating neutron stars correspond to the cases below the lower grey dot-dashed line. The horizontal dashed lines refer to the rotational frequencies obtained in simulations without rotation. The angular momentum redistribution at low rotation rates (isolated points) is very similar to the non-rotating cases.</figcaption></figure>

the other hand, when the radii ratio is sufficiently high and the core rotation frequency is below 100 Hz, the spiral mode is able to significantly impact the neutron star periods. The frequencies obtained are consistent with pulsar frequencies at birth deduced from observations: $f_{\text{NS}} \lesssim 100$ Hz. Both neutron star spinning-down (below the black dashed line) and neutron star spinning-up (above this line) are possible.

Considering a high enough radii ratio, e.g. $R \ge 3$, the influence of a spiral mode on the pulsar spin can be described as follows. For a non-rotating progenitor, SASI is able to redistribute angular momentum and spin up a neutron star to rotation frequencies of several tens Hz (horizontal dashed lines in Fig. 10). The direction of rotation of the SASI spiral mode is stochastic, so is the neutron star one (see Paper I). As long as the core rotation rate is negligible, the angular momentum redistribution proceeds very similar to the non-rotating case and the initial pulsar period is close to the one obtained without progenitor rotation (isolated points in Fig. 10). In these cases, the direction of rotation of the pulsar is still stochastic since the rotation rate is not high enough to significantly affect the growth rates of the spiral modes (Yamasaki & Foglizzo 2008) and to determine the direction of rotation of the spiral. In this range of rotation rates, the neutron star is thus not necessarily counter-rotating. For higher rotation rates, as intuitively expected, we observe that the core angular momentum required to balance the quantity redistributed by SASI is similar to the amount of angular momentum stored in a SASI spiral mode of a non-rotating case (see fig. 8 of Paper I

and Guilet & Fernández 2014 for an estimate). In such a case, the neutron star could be spun down to no rotation. Finally, as the rotation rate increases from this specific case, the spin down driven by the spiral mode is less and less significant and reaches an asymptotic efficiency in cases dominated by a corotation instability (Fig. 9).

## 5 DISCUSSION

The systematic exploration of the parameter space enables us to quantify together the effects of stellar rotation on the shock wave dynamics and the natal neutron star spin (Fig. 11). The results confirm that previous studies dedicated to the neutron star spin-up born from non-rotating progenitors (Fernández 2010; Guilet & Fernández 2014; Kazeroni et al. 2016) can naturally be extended to slowly rotating progenitors, i.e. $j \lesssim 10^{14} \text{ cm}^2 \text{ s}^{-1}$. Below the grey dashed line in Fig. 11, the direction of rotation of the spiral mode is not determined by the progenitor rotation. This is compatible with the characteristics of the mechanism forming a SASI spiral mode in non-linear regime and investigated in Paper I.

If the radii ratio is such that $R < 3$, the angular momentum redistribution by a spiral mode is negligible and the pulsar spin results from the conservation of the angular momentum of the iron core. The shock radius only increases compared to the non-rotating case if a corotation instability develops. Such situations correspond

<figure><img src="image_13.png" /><figcaption>**Figure 11.** Different regimes in the parameter space $(R, j)$ regarding the influence of rotation on the shock dynamics and the neutron star spin. The spiral mode of SASI is affected by rotation in cases above the orange dot-dashed line. The amplitude of the spiral mode is either increased or decreased depending on the value of $R$. In cases above the purple dot-dashed line, a corotation radius exceeds the PNS one and a corotation instability overlaps with SASI. Below the grey dashed line, the direction of rotation of the spiral mode is not determined by rotation. In this region, neutron stars may either have the same direction of rotation as the progenitor or the opposite one. The three blue dashed lines indicate a neutron star spin-down of 30 per cent, 50 per cent and 100 per cent compared to a spherically symmetric collapse. The latter value corresponds to a non-rotating neutron star.</figcaption></figure>
to very fast rotation rates, i.e. $j \ge 10^{16} \text{ cm}^2 \text{ s}^{-1}$. For the rest of this section, we restrict ourselves to large radii ratios $R \ge 3$. In these cases, rotational effects may already become significant for a lower specific angular momentum.

As the radii ratio increases, this diminishes the minimum thresholds for rotation to boost the spiral of SASI (above the orange dot-dashed line) and to trigger a corotation instability (above the purple dot-dashed line). These trends can be explained by an increase of the saturation amplitude and a lowering of the oscillation frequency with $R$. Besides, the maximum rotation rate to obtain a given spin-down efficiency increases with $R$ (dashed blue curves). The intersection between the region of the parameter space in which SASI is impacted by rotation and the one in which a significant spin-down operates, e.g. 50 per cent, is relatively narrow (Fig. 11). Indeed, in the range of rotation rates where SASI is the only instability the dynamics is modestly impacted by stellar rotation. For example, a case with $R \sim 4-5$ and $j \sim 5 \times 10^{14} \text{ cm}^2 \text{ s}^{-1}$ leads to a 10–15 per cent increase of the shock radius and a maximum spin-down of 50 per cent.

Fig. 10 indicates that no progenitor with a specific angular momentum higher than $j \sim 10^{15} \text{ cm}^2 \text{ s}^{-1}$ could give birth to a neutron star with a rotation frequency lower than 100 Hz. This suggests that if fast rotating progenitors may ease the shock revival due to a corotation instability, the resulting natal neutron star spin period may be too short to match the distribution inferred from observations (Fig. 11). A corotation instability does not seem to be compatible with a minimal spin period of 10 ms, at least when the magnetic field is neglected.

# 6 CONCLUSION

The interplay of SASI and stellar rotation has been investigated by conducting a set of parametric numerical simulations of an idealized model in cylindrical geometry. The formalism considered in this study enabled us to isolate and study the development of hydrodynamical instabilities in rotating collapsing stellar cores. Our approach is an essential complement to state-of-the-art simulations of core-collapse supernovae where a given physical effect cannot easily be disentangled from the complexity of the modelling. A comparison between our simulations of the linear phase of SASI and the perturbative analysis by Yamasaki & Foglizzo (2008) showed a good agreement, with discrepancies smaller than 15 per cent. In the non-linear regime, the influence of rotation on the shock wave dynamics as well as the angular momentum redistribution by one-armed instabilities have been addressed in a quantitative manner. Our main results can be summarized as follows.

(1) In the SASI-dominated regime, the saturation amplitude and the shock expansion do not depend monotonously on the rotation rate. These quantities either increase or decrease with increasing specific angular momentum, depending on the shock to the neutron star radii ratio (Fig. 5). The critical radii ratio above which rotation boosts the spiral mode is similar to the threshold above which a spiral mode emerges non-linearly in non-rotating progenitors (Paper I). This equality may be related to the symmetry breaking mechanism between counter-rotating spiral modes investigated in our previous study.

(2) For fast enough rotation rates, $j \gtrsim 10^{15}-10^{16} \text{ cm}^2 \text{ s}^{-1}$, a corotation radius emerges above the PNS surface (Fig. 11). As a consequence, a corotation instability overlaps with SASI (Fig. 4) and the spiral mode is greatly amplified (Fig. 6) compared to cases where SASI is the only instability. In the latter regime, the shock radius

increases by 10–15 per cent compared to SASI without initial rotation while it can be amplified by 50–200 per cent in the presence of a corotation instability (Table 1).

(3) The angular momentum redistribution induced by spiral modes in rotating collapsing iron cores has been examined. We performed a wide coverage of the parameter space to evaluate how much the natal pulsar spin period deviates from a simple estimation based on the conservation of angular momentum. For a high enough radii ratio, i.e. $R \gtrsim 3$, SASI has the potential to significantly affect the pulsar spin if the rotation rate corresponds to a period of at least 10 ms inferred from a spherically-symmetric collapse. Below such a rotation rate, the rotation frequency is mostly set by SASI (Fig. 10) and seems to be in agreement with observational data (Faucher-Giguère & Kaspi 2006; Popov & Turolla 2012). In that region of the parameter space, SASI may either spin up the neutron star or spin it down, leading to a counter-rotating body. For the slowest rotation rates considered, i.e. $j \lesssim 10^{14} \text{ cm}^2 \text{ s}^{-1}$, the direction of rotation of the spiral mode and thus the one of the neutron star is independent of the stellar rotation.

(4) The spin-down process is much less efficient in the range of rotation rates related to the corotation instability (Fig. 11). The magnitude of the spin-down increases with the radii ratio but does not exceed ~30 per cent for $R = 5$ (Fig. 9). Our results based on purely hydrodynamical simulations suggest that spiral modes of SASI and the corotation instability may not be able to reconcile fast rotating iron cores with the distribution of pulsar spins at birth estimated from observations.

Our study is based on several simplifications made to deepen our physical understanding of hydrodynamical instabilities. In order to transpose our findings to more comprehensive models, we shall first discuss the importance of the neglected ingredients. The numerical domain was restricted to the equatorial plane of a massive star in cylindrical geometry to allow spiral modes to emerge in 2D and to perform a wider coverage of the parameter space. Some artefacts of this simplified set-up cannot be excluded and would deserve a numerical confirmation in 3D. It has been demonstrated with various degrees of sophistication that SASI-dominated models do exist, both in 2D axisymmetric cases (Müller, Janka & Heger 2012) and in 3D spherical simulations (Hanke et al. 2013; Melson et al. 2015). A strong SASI spiral mode activity can precede the shock revival as observed in a simulation of a $20 M_\odot$ progenitor (Melson et al. 2015). We expect our findings to depend weakly on the geometry considered. Without rotation, the minimal radii ratio $R$ for a spiral mode to prevail over sloshing modes is expected to be slightly lower in 3D spherical geometry than in 2D cylindrical one (Paper I). This is a consequence of a shorter advection time and a higher growth in spherical geometry. For a given rotation rate, the growth rate of the most unstable mode is slightly higher in 3D spherical geometry (Blondin et al. 2017). Spiral modes may thus be more easily destabilized in spherical geometry. Among the other simplifications of our model, the absence of neutrino heating is of particular importance. As the post-shock region expands, neutrino-driven convection is favoured against SASI by the long advection time that decreases the SASI growth rates and increases the $\chi$ parameter (Foglizzo, Scheck & Janka 2006). This suggests that there exists a maximal radii ratio $R$ above which the dynamical evolution might be dominated by neutrino-driven convection. That would set an upper limit on the efficiency of the spin-down process driven by a spiral mode at large rotation rates (Fig. 9). Iwakami et al. (2014) showed that for fast enough rotation rates a SASI-dominated model may transition to a convection-dominated model due to the
expansion of the gain region induced by rotation. These points may help to delimit the parameter space (R,j) in which our approach remains valid when neutrino heating is included.

Our results are intended to serve as guidelines for ab initio sim-
ulations of core-collapse supernovae that include stellar rotation.
Covering a wide parameter space, this study can be helpful to antic-
ipate the impact of rotation on complex simulations both regarding
the dynamics of SASI and the induced angular momentum redistri-
bution (Fig. 11). To this end, it may be fruitful to better constrain
the radii ratio which is time and progenitor dependent. A single ab
initio simulation may not be enough to fully capture the impact of
rotation on core-collapse supernovae. Depending on the radii ratio,
rotation may either increase or decrease the amplitude of the SASI
spiral mode. Moreover, several realizations of a given simulation
may be needed to account for the stochastic nature of the non-linear
regime of SASI (Fig. 5).

Spiral modes do not seem to constitute a relevant spin-down
mechanism for the fastest rotation rates considered in this study.
These cases, which could in principle give birth to millisecond
pulsars, undergo a maximum spin-down of 30 per cent. That effect
is insufficient to match the observed pulsar periods of several tens
to hundreds of milliseconds. Ott et al. (2006) already pointed out
the lack of known spin-down mechanisms robust enough to spin
down a millisecond pulsar to a period of at least ten milliseconds.
The corotation instability does not seem to stand as a promising
candidate and rapid rotation might not constitute a key player in the
mechanism of the majority of core-collapse supernovae when the
magnetic field is neglected.

Despite the simplicity of our set-up, the simulations bring to
light the diversity of hydrodynamical processes at play. The over-
lap between SASI and the corotation instability, already noticed by
Kuroda et al. (2014), deserves further investigations to clearly dis-
tinguish between these instabilities at least in their linear regimes.
Moreover, the picture may evolve when a magnetic field with
fast rotation is considered. In addition to the potential develop-
ment of the magnetorotational instability (Akiyama et al. 2003;
Obergaulinger et al. 2009), the dynamics of the corotation instabil-
ity may be altered by the magnetic field (Fujisawa 2015). This work
will be extended to study whether the magnetic field could affect
the angular momentum redistribution process and the natal pulsar
spin.

ACKNOWLEDGEMENTS

We thank Marc Joos, Sébastien Fromang and Matthias González
for their help with the code. We acknowledge insightful discussions
with Bernhard Müller, Thomas Janka and Luc Dessart. This work
was granted access to the HPC resources of TGCC/CINES under the
allocations x2015047094 and t2016047094 made by GENCI (Grand
Équipement National de Calcul Intensif). This work is part of ANR
funded project SN2NS ANR-10-BLAN-0503. JG acknowledges
support from the Max-Planck-Princeton Center for Plasma Physics.

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