Analytic insight into the physics of SASI II. Spiral instability of the prograde mode in a rotating stellar core

Technical Summary: Analytic Insight into the Physics of SASI II. Spiral Instability of the Prograde Mode in a Rotating Stellar Core

Problem Statement
During the core-collapse of a rotating massive star, the Standing Accretion Shock Instability (SASI) drives non-axisymmetric motions that imprint specific signatures on neutrino and gravitational wave signals. While previous studies established that rotation favors the growth of prograde spiral modes, the physical mechanism behind this destabilization remained unexplained, particularly in regimes where rotation rates are slower than the SASI frequency. Furthermore, existing perturbative analyses often relied on cylindrical or spherical-equatorial approximations that neglected the poloidal structure of the flow, limiting their accuracy in interpreting numerical simulations of rotating progenitors. Additionally, the impact of rotation on the oscillation frequencies of dominant SASI modes lacked a robust analytical framework.

Methodology
This study extends the perturbative analysis of stalled accretion shocks in spherical geometry to include the effects of rotation and differential rotation. The approach involves:

1. Stationary Flow Modeling: A simplified model of the equatorial plane is constructed, assuming a perfect gas with an adiabatic index $\gamma = 4/3$, dominated by relativistic electron degeneracy pressure. The flow includes a cooling function dependent on density and pressure, and the gravitational potential is treated in the Newtonian approximation.
2. Linear Perturbation Analysis: The linear perturbations are characterized by spherical harmonics $Y^m_\ell$ with poloidal number $\ell$ and azimuthal number $m$. The analysis utilizes a set of coupled differential equations for mass flux, entropy, specific angular momentum, and vorticity.
3. Conservation Properties: To overcome the difficulty of solving the poloidal structure in a rotating progenitor, the study utilizes a conservation property of radial vorticity. This allows the system to be reduced and solved numerically using a shooting method.
4. Adiabatic Approximation: An adiabatic model is developed to isolate the contribution of adiabatic processes to the destabilization. This reduces the differential system order and facilitates an integral equation formulation for eigenfrequencies, treating the post-shock acoustic region as a self-forced oscillator.
5. Analytical Approximation: The study derives explicit analytical expressions for the eigenfrequencies of the dominant SASI modes ($\ell=m=1$ and $\ell=m=2$) by fitting numerical results across a range of shock radii and rotation rates.

Key Contributions and Results

• Analytical Eigenfrequency Formulas: The paper provides analytical approximations for the real part of the eigenfrequencies ($\omega_r$) of the dominant spiral modes ($\ell=m=1$ and $\ell=m=2$) with approximately 10% accuracy when the shock radius $r_{sh}$ exceeds $\sim 1.5$ times the radius of maximum deceleration ($r_\nabla$). The frequency is expressed as a function of the shock radius and the specific angular momentum $L$:
  $$\omega_{fit}^r(\ell, m) \sim \frac{|v_{sh}|}{r_{sh}} F_{\ell,m}\left(\frac{r_{sh}}{r_\nabla}\right) + \frac{mL}{r_{sh}r_\nabla} G_{\ell,m}\left(\frac{r_{sh}}{r_\nabla}\right)$$
  where $F$ and $G$ are dimensionless functions derived from numerical fits.
• Mechanism of Destabilization: The study explains that the destabilization of prograde modes by differential rotation is driven by phase mixing between the advective forcing and the acoustic structure. Differential rotation increases the radial wavelength of vorticity perturbations associated with the prograde mode. This leads to a better phase match with the large radial scale of the acoustic structure, thereby enhancing the coupling efficiency.
• Role of Rotation Rate:
  • In regimes of modest rotation (where no corotation radius exists), the adverse effect on phase mixing is significant at small radii due to the steep inward increase of the rotation frequency ($\propto 1/r^2$).
  • In regimes of stronger rotation involving a corotation radius, the stationary phase approximation identifies the dominant advective-acoustic coupling as occurring between the corotation zone and the shock.
• Mode Dominance: The analysis confirms that the growth rate of the prograde mode increases approximately linearly with specific angular momentum. As rotation increases, the fundamental mode $\ell=m=2$ can become dominant over $\ell=m=1$. The ratio of their frequencies depends on the rotation rate, potentially increasing by a factor of up to 2.3 with rotation compared to the non-rotating case.
• Independence from Cooling Details: The oscillation frequency of the dominant mode is found to weakly depend on the detailed formulation of neutrino emission (cooling function parameters) provided the shock radius is sufficiently large ($r_{sh} \gtrsim 1.5 r_\nabla$). The properties are best captured when measuring the shock radius in units of $r_\nabla$ rather than the proto-neutron star radius.

Significance
The paper claims to provide the first analytical approximations for SASI eigenfrequencies that explicitly include the effect of rotation. By extending the "self-forced oscillator" framework to rotating flows, it offers a physical explanation for the linear enhancement of prograde mode growth rates by differential rotation, a phenomenon previously observed numerically but not analytically understood. The derived formulas allow for the potential recovery of stellar properties (such as the shock radius and specific angular momentum) from observed neutrino and gravitational wave frequencies. The study also highlights the limitations of previous cylindrical or equatorial approximations by demonstrating the importance of the full poloidal structure in accurately interpreting SASI frequencies in rotating progenitors.