[Paper Review] Stellar evolution with rotation and magnetic fields II: General equations for the transport by Tayler--Spruit dynamo

A. Maeder, G. Meynet|ArXiv.org|Apr 21, 2004

Stellar, planetary, and galactic studiesPhysics and Astronomy19 references79 citations

TL;DR

This paper develops a general formulation for the Tayler–Spruit dynamo that consistently accounts for both chemical (μ) and thermal (T) stratification gradients and non-adiabatic effects in differentially rotating stars, avoiding the simplifying asymptotic approximations of prior work. The general solution shows that magnetic fields enforce near-solid-body rotation, reducing internal mixing and surface nitrogen enrichment compared to models with rotation alone, suggesting magnetic fields may not explain observed N/C excesses in OB stars.

ABSTRACT

We further develop the Tayler--Spruit dynamo theory, based on the most efficient instability for generating magnetic fields in radiative layers of differentially rotating stars. We avoid the simplifying assumptions that either the $μ$-- or the $T$--gradient dominates, but we treat the general case and we also account for the nonadiabatic effects, which favour the growth of the magnetic field. Stars with a magnetic field rotate almost as a solid body. Several of their properties (size of the core, MS lifetimes, tracks, abundances) are closer to those of models without rotation than with rotation only. In particular, the observed N/C or N/H excesses in OB stars are better explained by our previous models with rotation only than by the present models with magnetic fields that predict no nitrogen excesses. We show that there is a complex feedback loop between the magnetic instability and the thermal instability driving meridional circulation. This opens the possibility for further magnetic models, but at this stage we do not know the relative importance of the magnetic fields due to the Tayler instability in stellar interiors.

Motivation & Objective

• To develop a consistent, general formulation of the Tayler–Spruit dynamo that accounts for both μ-gradient and T-gradient effects in stellar interiors, avoiding prior simplifying assumptions.
• To incorporate non-adiabatic effects—particularly radiative losses—that influence magnetic field growth and stability.
• To assess whether the asymptotic solutions of Spruit (2002) are sufficient or if the general solution is required for accurate modeling of stellar evolution with magnetic fields.
• To investigate the feedback between the Tayler instability and meridional circulation, and the resulting equilibrium in differential rotation.
• To evaluate the observational implications of magnetic field transport, especially regarding surface abundances and main-sequence lifetimes in massive stars.

Proposed method

• Derives a general expression for magnetic diffusivity η that includes both μ-gradient and T-gradient contributions, using the Alfvén frequency and Brunt–Väisälä frequencies Nμ and NT.
• Introduces a non-adiabatic correction to the magnetic instability growth rate by accounting for radiative losses through the thermal diffusivity K and magnetic diffusivity η.
• Establishes general equations for angular momentum and chemical element transport via the magnetic field, derived from the balance between magnetic diffusion and meridional circulation.
• Applies the general solution to a 15 M⊙ model, comparing results with asymptotic solutions and models without magnetic fields.
• Compares the characteristic velocities of magnetic instability (Umagn) and meridional circulation (Ucirc) to assess their relative importance and potential equilibrium.
• Uses numerical modeling to evaluate the impact of the magnetic field on rotation profiles, core size, and surface abundances, particularly nitrogen and helium.

Experimental results

Research questions

• RQ1 How does the inclusion of both μ-gradient and T-gradient effects alter the transport efficiency of the Tayler–Spruit dynamo compared to prior asymptotic approximations?
• RQ2 What is the role of non-adiabatic effects in stabilizing or enhancing the magnetic field growth in radiative zones?
• RQ3 Can a stable equilibrium between the Tayler instability and meridional circulation be achieved, and what does it imply for the differential rotation profile?
• RQ4 How do magnetic fields affect the internal mixing and surface abundance anomalies (e.g., N/C excess) observed in OB stars?
• RQ5 To what extent do magnetic fields in the general solution reduce surface enrichment compared to models with rotation alone?

Key findings

• The general solution consistently recovers Spruit’s (2002) asymptotic results in limiting cases (Nμ ≫ NT or Nμ ≪ NT), but shows quantitative differences in intermediate regimes where both gradients are significant.
• Numerical models of a 15 M⊙ star show that magnetic fields enforce near-solid-body rotation, reducing differential rotation compared to models with rotation only.
• The magnetic field suppresses internal mixing and surface nitrogen enrichment, resulting in N/C ratios that are lower than in models with rotation alone—contrary to observations of OB stars.
• The characteristic velocity of the magnetic instability (Umagn) and meridional circulation (Ucirc) are comparable in magnitude at certain radii, indicating a potential feedback equilibrium with reduced differential rotation.
• In the deep envelope, Ucirc > Umagn, while closer to the surface, Umagn > Ucirc, suggesting a transition zone where the two processes compete.
• The equilibrium rotation profile is likely intermediate between solid-body rotation (magnetic field models) and strong differential rotation (rotation-only models), implying more mixing than current magnetic models predict.