Centrifugal Magnetospheres in Magnetic OB Stars

• Centrifugal magnetospheres are dense, wind-fed plasma regions confined by strong magnetic fields and sustained by centrifugal forces in rotating OB stars.
• They are characterized by closed magnetic loops between the Alfvén and Kepler radii, supporting rigid corotation and distinctive Balmer and radio emission signatures.
• MHD simulations and analytic models reveal that plasma accumulation, steady leakage, and centrifugal breakout regulate mass-loss and angular momentum evolution in massive stars.

Centrifugal magnetospheres (CMs) are dense regions of magnetically confined, wind-fed plasma supported by centrifugal force in the circumstellar environments of rapidly rotating, strongly magnetic stars. The defining property of a CM is that closed magnetic field loops, originating from the stellar surface, extend beyond the Keplerian co-rotation radius ($R_{\rm K}$), such that wind plasma trapped on these loops is forced into rigid rotation and is supported against gravity primarily by centrifugal force. CMs represent a special subset of massive-star magnetospheres and are critical to understanding the mass-loss, angular momentum evolution, and multiwavelength emission—particularly Balmer line and radio signatures—of magnetic OB stars.

1. Fundamental Parameters and Existence Criteria

The structure and existence of CMs are governed by the interplay between magnetic confinement, radiative wind driving, and stellar rotation. The principal parameters and radii are:

• Magnetic confinement parameter ($\eta_*$):

$$\eta_* = \frac{B_{\rm eq}^2 R_*^2}{\dot{M} v_\infty}$$

where $B_{\rm eq}$ is equatorial field strength (typically $B_{\rm p}/2$ for a dipole), $R_*$ stellar radius, $\dot{M}$ wind mass-loss rate, $v_\infty$ wind terminal speed.

• Alfvén radius ($R_{\rm A}$):

The equatorial extent of closed magnetic loops,

$$R_{\rm A} \simeq 0.3 R_* + ( \eta_* + 0.25 )^{1/4} R_*$$

or, in the $\eta_* \gg 1$ limit, $R_{\rm A} \simeq \eta_*^{1/4} R_*$ (Shultz et al., 2022).

• Kepler co-rotation radius ($R_{\rm K}$):

Where gravitational and centrifugal accelerations balance for co-rotating material,

$$R_{\rm K} = \left( \frac{G M_*}{\Omega^2} \right)^{1/3} = W^{-2/3} R_*$$

with $W \equiv v_{\rm rot} / v_{\rm orb}$ and $v_{\rm orb} = \sqrt{G M_*/R_*}$ (Shultz et al., 2014).

Criterion for CM formation:

$R_{\rm A} > R_{\rm K}$

Stars with sufficiently strong fields (large $\eta_*$) and fast rotation (large $W$) produce CMs (Petit et al., 2012, Shultz et al., 2022).

2. Structure and Dynamics of Centrifugal Magnetospheres

In the aligned dipole case, closed loops between $R_{\rm K}$ and $R_{\rm A}$ accumulate wind-fed plasma in a disk- or torus-like structure about the magnetic/rotational equator. For oblique (tilted) dipoles, accumulation surfaces become warped, and plasma localizes in two "clouds" at the intersection of the rotational and magnetic equators (ud-Doula et al., 2023).

Density structure: MHD simulations and analytic models show that plasma accumulates most efficiently just above $R_{\rm K}$, with the radial density profile falling steeply ($\rho \propto r^{-5}$ to $r^{-6}$), as expected from magnetic tension versus centrifugal support (ud-Doula et al., 2023, Berry et al., 2022). The CM is highly stratified, with densities exceeding the ambient wind by 2–4 orders of magnitude.

Stability and leakage: Theoretical maximum densities are set by the balance between magnetic tension and centrifugal force, but observations indicate that actual densities are typically well below the catastrophic "breakout" limit, implying slower, quasi-stationary leakage instead of episodic ejections (Shultz et al., 2014, Shultz et al., 2020).

3. Plasma Feeding and Loss Mechanisms

CMs are fed by radiatively driven stellar winds channeled along magnetic field lines toward the magnetic equator. Mass can escape the CM via several mechanisms:

• Centrifugal Breakout (CBO): Plasma accumulates until centrifugal stress overcomes magnetic tension, leading to magnetic reconnection and ejection of material along newly opened field lines. The critical breakout density is

$$n_{\rm break} \simeq \frac{B^2}{4\pi m_{\rm p} v_{\rm rot}^2}$$

(Shultz et al., 2014). For typical OB star parameters, CBO predicts $n_e \sim 10^{14-15}$ cm$^{-3}$.

• Continuous Leakage: Observed CM densities ($n_e \sim 10^{12.2-12.8}$ cm$^{-3}$) are 1–2 orders of magnitude below the CBO threshold (Shultz et al., 2014). Diffusive or small-scale reconnection processes and leakage through cusp regions are favored to explain steady state, with leakage rates $$\dot{M}_{\rm leak} \sim 10^{-12} - 10^{-11} \, M_\odot \,{\rm yr}^{-1}$$, much less than wind feeding rates (Shultz et al., 2014, Owocki et al., 2017).
• Diffusion-plus-drift: Semi-analytic models capture equilibrium between wind input and loss via cross-field diffusion and radial centrifugal drift, predicting power-law radial density profiles and matching observed emission measures (Owocki et al., 2017).

4. Observational Diagnostics: Balmer Line and Radio Emission

Balmer Lines (e.g., H$\alpha$)

• Criteria for emission: All stars with $B_{\rm K} \gtrsim 100$ G (field at $R_{\rm K}$) and $R_{\rm A}/R_{\rm K}\gtrsim6$ display H$\alpha$ emission; none with lower values do, regardless of $\dot{M}$ or $L$ (Shultz et al., 2020).
• Line profiles: CMs exhibit broad, double-peaked H$\alpha$ emission with peak separations corresponding to rigid corotation at $R_{\rm K}$. Emission wings typically extend beyond $\pm v\sin i$, modulated synchronously with rotation, and profiles are nearly scale-invariant when normalized in velocity and flux (Shultz et al., 2014, Shultz et al., 2020).
• Empirical scalings: Maximum H$\alpha$ equivalent width correlates tightly with $R_{\rm A}/R_{\rm K}$, $B_{\rm K}$, and the normalized area of the accumulation region, but not with $T_{\rm eff}$, $L$, or $\dot{M}$ (Shultz et al., 2020).

Radio Emission

• Centrifugal breakout and electron acceleration: Radio emission (gyrosynchrotron) arises from electrons accelerated to relativistic energies by magnetic reconnection during CBO events (Owocki et al., 2022, Leto et al., 7 Nov 2025). The radio luminosity scales as

$$L_{\rm rad} \propto B_{\rm p}^{2}\;R_{*}^{4}\;\Omega^{2} \propto \frac{B_{\rm p}^{2}\,R_{*}^{4}}{P_{\rm rot}^{2}}$$

(Leto et al., 7 Nov 2025, Shultz et al., 2022).

• Efficiency: Acceleration efficiency—fraction of thermal electrons converted to relativistic—is $10^{-6}$ to $10^{-2}$, higher in denser magnetospheres (Leto et al., 7 Nov 2025).

Electron Scattering and Photometric Modulation

• Light curves: CM clouds can eclipse the stellar disk, producing periodic dips; for optically thick ($\tau_{\rm K} \gtrsim 1$) regions, electron scattering off-limb produces characteristic emission bumps in the light curve, in direct analogy with $\sigma$ Ori E and related stars (Berry et al., 2022, Berry et al., 2023).

5. Theoretical Models and Simulations

Rigidly Rotating Magnetosphere (RRM) Model

• The RRM model (Townsend & Owocki) predicts the equilibrium surfaces where plasma can accumulate in minima of the combined gravitational plus centrifugal potential, conforming to the observed locations of dense CM clouds. This framework provides analytic surface density distributions ($\sigma(r) \propto (R_{\rm K}/r)^p$, $p \sim 5-6$ via CBO scaling), matching 3D MHD results and empirical Balmer-line emission (ud-Doula et al., 2023, Berry et al., 2022).

Magnetohydrodynamic (MHD) Simulations

• Full 3D MHD simulations confirm and extend the RRM, capturing field line warping, cloud "wing" formation at intersections of the magnetic and rotational equators, and the onset of CBO. Simulations reveal a strong azimuthal density asymmetry and field distortion as mass accumulates, especially in oblique rotators (ud-Doula et al., 2023).

Population Studies and Magnetic Confinement–Rotation Diagram

• The distribution of magnetic OB stars in the $R_{\rm A}$ vs. $R_{\rm K}$ plane (the confinement–rotation diagram) visually separates CM-hosting stars (upper right, $R_{\rm A} > R_{\rm K}$) from dynamical-magnetosphere stars and correlates with observed H$\alpha$ or radio emission (Petit et al., 2012, Shultz et al., 2022).

6. Observational Constraints and Empirical Trends

Large-scale surveys and monitoring campaigns confirm:

• CM incidence: $\sim$10% of B-type stars have kG-strength fields; of these, $\sim$25% (i.e., $\sim$2.5% overall) host CMs (Shultz et al., 2014).
• Thresholds for emission: H$\alpha$ only appears for $R_{\rm A} > 20 R_*$, $R_{\rm K} < 3 R_*$, $B_{\rm K}\gtrsim100$ G (Shultz et al., 2014, Shultz et al., 2020).
• Scale invariance: Normalized H$\alpha$ profiles across very different CMs collapse to a universal shape, indicating shared underlying density and temperature structure (Shultz et al., 2020).
• No large amplitude variability: CMs appear stable over long timescales, inconsistent with sporadic, catastrophic CBO (Shultz et al., 2014, Shultz et al., 2020).

7. Outstanding Problems and Implications

Despite successful unification of CM phenomenology under the RRM+CBO paradigm, outstanding issues remain:

• Density discrepancy: Observed plasma densities are systematically lower than the analytic CBO "breakout" limit, requiring efficient, continuous plasma leakage—likely via small-scale reconnection, turbulence, or cusp outflows (Shultz et al., 2014, Owocki et al., 2017).
• Calibration of opacity and mass-loading: Matching photometric light curves and Balmer emission requires marginally optically thick CMs ($\tau_{\rm K}\sim1$), with higher mass-loading efficiency than some theoretical norms (Berry et al., 2023).
• Role of field obliquity and multipolar topologies: Deviations from perfect dipolar symmetry affect cloud location and density, complicating the mapping of observed emission variations (ud-Doula et al., 2023, Berry et al., 2022).
• Connection to non-thermal emission: Centrifugal breakout provides a direct link between stellar rotation, field strength, and non-thermal radio (and potentially X-ray) emission, with scaling relations matching observations over five decades in luminosity (Leto et al., 7 Nov 2025, Shultz et al., 2022, Owocki et al., 2022).

---

Centrifugal magnetospheres thus emerge as rotationally powered, magnetically regulated, and wind-fed environments whose properties can be quantitatively predicted from first principles and have been empirically validated via population studies, spectroscopy, and photometry. The precise balance of plasma accumulation, leakage, and breakout shapes both the multiwavelength emission and the long-term angular momentum evolution of magnetic, massive stars.