Slowly Rotating WNE Stars

• Slowly rotating WNE stars are a subclass of Wolf-Rayet stars with low surface rotational speeds, resulting from efficient angular momentum loss during the post-RSG phase.
• The models employ internal gravity waves and a revised Tayler–Spruit dynamo to reconcile observed slow rotation with theoretical predictions through calibrated parameter regimes.
• Simulation outcomes underscore the critical role of precise mass-loss prescriptions and AM transport parameter tuning in achieving WNE surface velocities below 70 km/s.

Slowly rotating early-type nitrogen-sequence Wolf-Rayet (WNE) stars represent a theoretically predicted subclass of Wolf-Rayet stars that exhibit surface rotational velocities markedly lower than those expected from purely hydrodynamic angular momentum (AM) transport in massive star evolution. Their formation is generally attributed to an evolutionary pathway passing through the red supergiant (RSG) phase, during which steep gradients and mass-loss-driven AM removal facilitate substantial spin-down. Contemporary simulations incorporating additional AM transport via internal gravity waves (IGWs) and the revised Tayler–Spruit dynamo (TSF) have provided the first robust formation channels and physically motivated parameter regimes for the existence of these slowly rotating WNE stars (Si et al., 24 Dec 2025).

1. Angular Momentum Transport Frameworks

Two principal extra AM transport mechanisms are invoked to reconcile slow WNE rotation with evolutionary predictions: IGWs and the TSF dynamo. Both are implemented in stellar evolution codes as additional effective viscosities superposed on the standard suite of hydrodynamic transport processes (Eddington–Sweet circulation and shear/turbulent instabilities).

Internal Gravity Waves (IGWs)

The IGW-driven AM diffusion coefficient adopted is:

$$\nu_{\mathrm{IGW}} = A(2D_{\mathrm{thb}})^{-1/3} \left[ D_{\mathrm{th}} N \left( \frac{\rho_b}{\rho} \right) \left( \frac{r_b}{r} \right)^6 \right]^2 V^{8/3} f^{-4/3} \quad \text{(Eq.~3)}$$

where $A$ is a dimensionless efficiency parameter, $D_{\mathrm{th}}$ is the local thermal diffusivity, $N$ the Brunt–Väisälä frequency, and $V$ the characteristic eddy speed. $f$ denotes a wave-damping integral, and the other terms denote structural variables evaluated at convective boundaries and at each radius.

TSF Dynamo

The AM viscosity for the revised Tayler instability (TSF) is:

$$\nu_{\mathrm{TSF}} = \alpha^3 r^2 \Omega \left( \frac{\Omega}{N_{\mathrm{eff}}} \right)^2 \quad \text{(Eq.~12)}$$

with $\alpha$ a dimensionless dynamo saturation parameter and $N_{\mathrm{eff}}$ the effective Brunt–Väisälä frequency. The minimum shear for instability activation follows:

$$q_{\min} \sim \alpha^{-3} \left( \frac{N_{\mathrm{eff}}}{\Omega} \right)^{5/2} \left( \frac{\eta}{r^2 \Omega} \right)^{3/4} \quad \text{(Eq.~13)}$$

Angular momentum evolution is numerically propagated via the standard 1D diffusion equation:

$A$0

where $A$1\,=\, $A$2 or $A$3, depending on the mechanism modeled.

2. Justification and Calibration of Model Parameters

Parameter tuning is mandatory given the uncertainties in wave and dynamo efficiencies in the massive-star regime:

• For IGWs, maintaining WNE surface velocities $A$4—the fiducial "slow" criterion based on nitrogen-rich B star constraints—requires

$A$5

This is two orders of magnitude greater than the $A$6 values used for low-mass RGB stars, supporting a sensitivity of the mechanism to stellar mass.

• For TSF, direct adoption of the low-mass calibration ($A$7–6) leads to catastrophic core spin-down ($A$8 prematurely), while $A$9 fails to activate the instability. Calibrated grid models identify the optimal value as

$D_{\mathrm{th}}$0

for all $D_{\mathrm{th}}$1–$D_{\mathrm{th}}$2 progenitors.

3. Stellar Evolutionary Setups

The simulations utilize MESA version r12115 ("black_hole" suite), spanning:

• Initial masses: $D_{\mathrm{th}}$3
• Metallicity: $D_{\mathrm{th}}$4
• Initial rotation: $D_{\mathrm{th}}$5
• Convection: MLT ($D_{\mathrm{th}}$6, Schwarzschild), with overshoot $D_{\mathrm{th}}$7
• Mass loss: Dutch prescription ($D_{\mathrm{th}}$8 for $D_{\mathrm{th}}$9), RSG regime amplified by $N$0 over standard de Jager (1988), and further rotation-enhanced as

$N$1

The evolutionary sequence follows ZAMS through core H/He burning and envelope-stripping, tracking the blue supergiant (BSG), RSG, WNL, and WNE stages to core C-burning.

4. Simulation Outcomes: Surface and Core Spin Evolution

IGW-Only Models

• Without IGWs, even with substantial late-stage mass loss, 60 $N$2 models display core rotation $N$3 mid-core He burning, persistent envelope spin-up during expansion, and WNE surfaces exceeding $N$4.
• With $N$5, core rotation is efficiently braked to $N$6 by mid-He burning, and the WNE surface falls well below $N$7, even reaching $N$8 by He depletion. The He-core specific AM drops by $N$9 dex.
• All $V$0 models require $V$1 to ensure WNE surface velocities below $V$2. Higher mass models require slightly less efficient IGW transport as mass loss more effectively removes AM.

IGW Models: Convective-Envelope Excitation

Similar qualitative spin-down is seen when IGW excitation originates in the convective envelope, but this reduction is temporary. As envelope convection recedes in late He burning, wave excitation weakens and partial core spin-up ensues, preventing a persistently slow WNE unless $V$3 is extremely large.

TSF-Only Models

• With $V$4–6, AM transport nearly erases all core spin, yielding nonphysical $V$5 throughout the WNE lifetime.
• With $V$6, the threshold for instability exceeds shear, and AM transport reverts to the baseline hydrodynamic case.
• The intermediate case $V$7 yields core rotation $V$8 at mid-He and WNE surface velocities $V$9, aligning with the predicted slow-rotator class. AM profiles indicate a depletion of core AM by two orders of magnitude during envelope stripping.

Mass Dependence and Comparison to Low-Mass Stars

The efficiency of both IGW and TSF AM transport rises steeply with increasing mass. For low-mass stars, IGW $f$0 and TSF $f$1–6 suffice, while only the much larger values noted above match observations for massive WNE progenitors.

5. Synthesis: Formation and Theoretical Status of Slowly Rotating WNE Stars

Both core-excited IGWs ($f$2) and the revised TSF dynamo ($f$3) can independently and self-consistently spin post-RSG helium cores down, yielding WNE stars with $f$4. This theoretical population matches the properties inferred from nitrogen-rich B stars, and the models predict substantial increases in AM-transport efficiency with mass. The dominant AM-loss channel is envelope mass loss during and after the RSG phase, synergizing with internal transport to drive efficient core and surface spin-down (Si et al., 24 Dec 2025).

6. Caveats, Uncertainties, and Future Directions

Significant uncertainties remain in linking simulation output to observed slow-rotator populations:

• Chemical mixing by IGWs was switched off to focus on pure AM transport. Fully self-consistent coupling of chemical and AM transport by IGWs is not yet realized.
• The $f$5 "slow" surface threshold is based on B star observations. Direct constraints for WNE stars are lacking due to spectroscopic challenges.
• Mass-loss prescriptions, especially for RSGs (where de Jager rates are scaled up by $f$6), introduce model-dependent uncertainties in AM removal and resulting rotation.
• Optimal $f$7 and $f$8 values may shift as mass-loss rates, binary interactions, or metallicity are revised.

Future progress requires direct high-precision rotational velocity measurements of WNE stars and refinement of the IGW/TSF implementation in stellar evolution frameworks, potentially integrating chemical-mixing and AM transport in a fully unified prescription (Si et al., 24 Dec 2025).