[Paper Review] Stellar evolution with rotation and magnetic fields:I. The relative importance of rotational and magnetic effects

A. Maeder, G. Meynet|ArXiv.org|Sep 24, 2003

Stellar, planetary, and galactic studiesPhysics and Astronomy8 references115 citations

TL;DR

This paper investigates the relative impact of rotational and magnetic effects in stellar evolution, focusing on the Tayler–Spruit dynamo mechanism. It demonstrates that magnetic field instabilities driven by differential rotation dominate over meridional circulation in transporting angular momentum and chemical elements, with magnetic diffusion coefficients orders of magnitude larger than those from rotation or turbulence, especially in radiative zones. The study establishes a physical criterion for magnetic field existence based on energy availability from differential rotation, showing that strong internal fields can form even in regions with strong compositional gradients, while surface fields remain weak due to insufficient differential rotation near the surface.

ABSTRACT

We compare the current effects of rotation in stellar evolution to those of the magnetic field created by the Tayler instability. In stellar regions, where magnetic field can be generated by the dynamo due to differential rotation (Spruit 2002), we find that the growth rate of the magnetic instability is much faster than for the thermal instability. Thus, meridional circulation is negligible with respect to the magnetic fields, both for the transport of angular momentum and of chemical elements. Also, the horizontal coupling by the magnetic field, which reaches values of a few $10^5$ G, is much more important than the effects of the horizontal turbulence. The field, however, is not sufficient to distort the shape of the equipotentials. We impose the condition that the energy of the magnetic field created by the Tayler--Spruit dynamo cannot be larger than the energy excess present in the differential rotation. This leads to a criterion for the existence of the magnetic field in stellar interiors. Numerical tests are made in a rotating star model of 15 M$_{\odot}$ rotating with an initial velocity of 300 km$\cdot$s$^{-1}$.

Motivation & Objective

• To assess the relative importance of rotational instabilities versus magnetic field effects in stellar evolution.
• To determine whether magnetic fields generated by the Tayler–Spruit dynamo can significantly influence angular momentum and chemical element transport in radiative zones.
• To establish a physical criterion for the existence of magnetic fields in rotating stars based on energy constraints from differential rotation.
• To evaluate the role of magnetic fields in shaping stellar structure and evolution, particularly in massive 15 M⊙ stars.
• To explore the implications of magnetic field dominance for future stellar evolution models.

Proposed method

• The study uses analytical and numerical models of a 15 M⊙ star rotating at 300 km·s⁻¹ to compare characteristic timescales of meridional circulation and magnetic instability.
• It applies the Tayler–Spruit dynamo model, where differential rotation winds up weak toroidal magnetic fields, leading to instability and amplification of horizontal magnetic fields.
• The growth rate of the magnetic instability is calculated using σ = ω_A² / Ω, with ω_A being the Alfvén frequency, and the Coriolis force is accounted for via the reduction factor ω_A / Ω.
• The model distinguishes between case 0 (dominant μ-gradient) and case 1 (dominant thermal gradient) for instability conditions, using N_T² and N_μ² as stability criteria.
• A physical criterion for magnetic field existence is derived from energy balance: the magnetic field energy cannot exceed the available energy from differential rotation, leading to the condition |q|/3 > (Ω/N_T)^{1/4} (K/r²N_T)^{1/4}.
• Numerical tests are performed across radial shells to assess whether the instability condition is met at each point in the star’s interior, particularly near the core and in outer layers.

Experimental results

Research questions

• RQ1 How do the timescales of meridional circulation and magnetic field instability compare in rotating stars?
• RQ2 To what extent do magnetic fields dominate over rotational instabilities in transporting angular momentum and chemical elements?
• RQ3 What physical conditions are required for the Tayler–Spruit dynamo to generate a magnetic field in stellar interiors?
• RQ4 Why are strong surface magnetic fields not observed in massive stars despite strong internal fields?
• RQ5 How does the energy budget of differential rotation constrain the maximum possible magnetic field strength?

Key findings

• The growth rate of the magnetic instability (σ = ω_A² / Ω) is much faster than that of thermal or meridional instabilities, making magnetic effects dominant in angular momentum and element transport.
• Magnetic diffusion coefficients for angular momentum and chemical elements are several orders of magnitude larger than those from meridional circulation or shear mixing, especially in radiative zones.
• Magnetic instability is present throughout the radiative envelope except in the very outer layers, where differential rotation is too weak to sustain the Tayler–Spruit dynamo.
• The criterion for magnetic field existence, derived from energy balance, is satisfied in the core and inner radiative zone, but fails near the surface, explaining the lack of strong surface magnetic fields in massive stars.
• In regions with strong μ-gradients (case 0), the magnetic field still develops due to high differential rotation, demonstrating that compositional stratification alone does not suppress the instability.
• The model suggests that a state of marginal equilibrium is reached during main sequence evolution, where differential rotation is just sufficient to sustain the magnetic field, preventing further growth.