# Spin-down and reduced mass loss in early-type stars with large-scale
magnetic fields

Z. Keszthelyi1,2 A. de Koter1,3 Y. Götberg4 G. Meynet5 S.A. Brands1 V. Petit6
M. Carrington7 A. David-Uraz8,9 S.T. Geen1 C. Georgy5 R. Hirschi10,11 J.
Puls12 K.J. Ramalatswa13,14 M.E. Shultz6 A. ud-Doula15 1Anton Pannekoek
Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH,
Amsterdam, The Netherlands
2Center for Computational Astrophysics, Division of Science, National
Astronomical Observatory of Japan, 2-21-1, Osawa, Mitaka, Tokyo 181-8588,
Japan
3Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium
4The observatories of the Carnegie institution for science, 813 Santa Barbara
Street, Pasadena, CA 91101, USA
5Geneva Observatory, University of Geneva, Maillettes 51, 1290 Sauverny,
Switzerland
6Dept. of Physics and Astronomy, Bartol Research Institute, University of
Delaware, 217 Sharp Lab, Newark, DE 19716, USA
7Dept. of Physics and Space Science, Royal Military College of Canada, PO Box
1700, Station Forces, Kingston, ON K7K 0C6, Canada
8Dept. of Physics and Astronomy, Howard University, Washington, DC 20059, USA
9Center for Research and Exploration in Space Science and Technology, and
X-ray Astrophysics Laboratory, NASA/GSFC, Greenbelt, MD 20771, USA
10Astrophysics Group, Keele University, Keele, Staffordshire ST5 5BG, UK
11Institute for Physics and Mathematics of the Universe (WPI), University of
Tokyo, 5-1-5 Kashiwanoha, Kashiwa 277-8583, Japan
12LMU München, Universitätssternwarte, Scheinerstr. 1, 81679 München, Germany
13Dept. of Astronomy, University of Cape Town, Private Bag X3, Rondebosch
7701, South Africa
14South African Astronomical Observatory, PO Box 9, Observatory, 7935, South
Africa
15Dept. of Physics, Penn State Scranton, 120 Ridge View Drive, Dunmore, PA
18512, USA

(2022)

###### Abstract

Magnetism can greatly impact the evolution of stars. In some stars with OBA
spectral types there is direct evidence via the Zeeman effect for stable,
large-scale magnetospheres, which lead to the spin-down of the stellar surface
and reduced mass loss. So far, a comprehensive grid of stellar structure and
evolution models accounting for these effects was lacking. For this reason, we
computed and studied models with two magnetic braking and two chemical mixing
schemes in three metallicity environments with the mesa software instrument.
We find notable differences between the subgrids, which affects the model
predictions and thus the detailed characterisation of stars. We are able to
quantify the impact of magnetic fields in terms of preventing quasi-chemically
homogeneous evolution and producing slowly-rotating, nitrogen-enriched (“Group
2”) stars. Our model grid is fully open access and open source.

###### keywords:

stars: evolution — stars: massive — stars: magnetic field — stars: rotation —
stars: abundances

††volume: 370††journal: Winds of Stars and Exoplanets††editors: A. A. Vidotto,
L. Fossati & J. Vink, eds.

##  1 Introduction

Magnetism is a key component in several astrophysical phenomena. For example,
magnetic fields play a crucial role in regulating star formation and
controlling the formation of neutron stars (e.g., Commerçon et al., 2011;
Takiwaki & Kotake, 2011). A fraction of massive stars (initially $M_{\star}>8$
M⊙) and intermediate-mass stars (initially 3 M${}_{\odot}<M_{\star}<8$ M⊙)
show evidence of stable, globally organised, large-scale magnetic fields that
are understood to be of fossil origin. The exact origin of such fossil fields
is unclear; however, they may result from the pre-main sequence evolution of
the star or from stellar merger events (Schneider et al., 2019). Fossil fields
form a magnetosphere around the star, affecting the wind and stellar rotation.
For their stability, they must be anchored in deep stellar layers. While it is
known that magnetic fields can significantly impact the physics and evolution
of stars, there has been no comprehensive grid of stellar evolution models of
massive stars that would take into account the effects of surface fossil
magnetic fields. We use the mesa software instrument (Paxton et al., 2019) to
compute models and map out a large parameter space. The main new additions in
our calculations are magnetic mass-loss quenching, magnetic braking, and
efficient angular momentum transport (see next section).

##  2 Background

###  2.1 Alfvén radius

The Alfvén radius characterises a critical distance at which the magnetic
energy density and the gas kinetic energy density are equal. ud-Doula et al.
(2009) use a numerical fitting to quantify the Alfvén radius as:

$\frac{R_{\rm A}}{R_{\star}}\approx 1+(\eta_{\star}+0.25)^{x}-(0.25)^{x}\,,$
(1)

with $x=1/4$ and $1/6$ for dipolar and quadrupolar field geometries, which we
assume in our INT and SURF models (see next section), respectively.
$R_{\star}$ is the stellar radius. The equatorial magnetic confinement
parameter $\eta_{\star}$ is defined as:

$\eta_{\star}=\frac{B_{\rm eq}^{2}R_{\star}^{2}}{\dot{M}_{B=0}\cdot
v_{\infty}}\,,$ (2)

where $B_{\rm eq}$ is the equatorial magnetic field strength, $\dot{M}_{B=0}$
is the mass-loss rate in absence of a magnetic field, and $v_{\infty}$ is the
terminal velocity (ud-Doula et al., 2009).

###  2.2 Mass-loss quenching

Large-scale magnetic fields lead to channelling and trapping the wind plasma
within the magnetosphere. To account for the global, time-averaged effect of
this process, the adopted mass-loss rates in our models are reduced by a
parameter $f_{\rm B}$, which is defined following the works of ud-Doula et al.
(2008, 2009):

$f_{\rm B}=\,\frac{\dot{M}}{\dot{M}_{B=0}}=\,1-\sqrt{1-\frac{1}{R_{\rm
c}}}\quad\mathrm{if}\quad R_{\rm A}<R_{\rm K}$ (3)

and

$f_{\rm B}=\,\frac{\dot{M}}{\dot{M}_{B=0}}\,=2-\sqrt{1-\frac{1}{R_{\rm
c}}}-\sqrt{1-\frac{0.5}{R_{\rm K}}}\quad\mathrm{if}\quad R_{\rm K}<R_{\rm A}$
(4)

where $R_{\rm A}$, $R_{\rm K}$, and $R_{\rm c}$ are the Alfvén radius, the
Kepler co-rotation radius, and the closure radius in units of the stellar
radius, respectively (see Keszthelyi et al., 2019, 2020, 2021, and references
therein).

###  2.3 Magnetic braking

Stellar rotation exerts a force on magnetic field lines and causes the field
to bend in the azimuthal direction. The associated Maxwell stresses are very
efficient at transferring angular momentum to the surrounding wind plasma,
which results in slowing the spin of the star. We quantify this process
following the work of ud-Doula et al. (2009), where the total wind and
magnetic field-induced loss of angular momentum can be expressed via a Weber-
Davis (Weber & Davies, 1967) scaling relation:

$\frac{\mathrm{d}J_{\rm
B}}{\mathrm{d}t}=\frac{2}{3}\dot{M}_{B=0}\,\Omega_{\star}R_{A}^{2}\,,$ (5)

with $\mathrm{d}J_{\rm B}/\mathrm{d}t$ the rate of angular momentum loss from
the system, $\Omega_{\star}$ the surface angular velocity, and $R_{A}$ the
Alfvén radius (defined in Equation 1).

Based on how magnetic braking is applied, we split the model calculations into
two branches. In one case, we assume internal magnetic braking (INT models).
These models are solid-body rotating and specific angular momentum is
extracted from each layer of the stellar model. In the other case, we only
allow the model to directly remove specific angular momentum from the upper
envelope of the star (SURF models). Thus, radial differential rotation can
develop in deeper layers in the SURF models.

###  2.4 Chemical mixing

Due to stellar rotation, chemical elements are also mixed in radiative regions
of stars. Consequently, main sequence massive stars can replenish their core
with more hydrogen, and also enrich their surface with core-produced materials
(most importantly, nitrogen). We adopt a diffusive scheme, following the work
of Pinsonneault et al. (1989), to account for these effects.

$\frac{\partial X_{i}}{\partial t}=\frac{\partial}{\partial m}\left[(4\pi
r^{2}\rho)^{2}\,D_{\rm chem}\,\frac{\partial X}{\partial
m}\right]+\left(\frac{\mathrm{d}X_{i}}{\mathrm{d}t}\right)_{\rm nuc}\,,$ (6)

where $X_{i}$ is the mass fraction of a given element $i$, $t$ is the time,
$m$ and $\rho$ are the mass coordinate and mean density at a given radius $r$,
$D_{\rm chem}$ is the sum of individual diffusion coefficients contributing to
chemical mixing, and the final term accounts for nuclear burning.

The detailed mixing processes remain highly uncertain in massive star
evolutionary models. For this reason, we also split the model calculations to
construct $D_{\rm chem}$ in two different ways. In one case (Mix1), we assume
chemical mixing equations that are commonly adopted in mesa. We thus adopt
$D_{\rm chem}$ as a sum of individual diffusion coefficients describing
meridional circulation, shear, and GSF instabilities. We then scale $D_{\rm
chem}$ with commonly used factors in this approach. In the other case (Mix2),
we implement the mixing equations of Zahn (1992) into the mesa code. Here we
use the vertical shear mixing and meridional circulation (both defined
differently than in the above approach) to construct $D_{\rm chem}$. In this
case, scaling factors are not used. We do not change the equations for angular
momentum transport for the sake of a consistent model-to-model comparison.

##  3 Summary of computed models

In total, we computed 8,748 main sequence stellar evolution models, with 3
stellar structure models for each of these corresponding to ZAMS, mid-MS, and
TAMS evolutionary stages. We also generated isochrones. All the computed
models are open access and open source and available on Zenodo at
https://doi.org/10.5281/zenodo.7069766. The initial mass and equatorial
magnetic field strength spans from 3 to 60 M⊙ and from 0 to 50 kG in our grid
of models. Four sub-grids are available, accounting for two magnetic braking
and two chemical mixing schemes, introduced above. Hot star mass-loss rates in
our models are adopted from Vink et al. (2001) and are reduced by a factor of
2 to account for the general trend evidenced from recent clumping-corrected
mass-loss rate determinations. Mass-loss rates lower than the predictions of
Vink et al. (2001) are also expected from new numerical simulations (Björklund
et al., 2020; Krtička et al., 2021). The metallicity in our model grid
corresponds to Solar, LMC, and SMC abundances, respectively.

##  4 Results & Discussion

Figure 1: HRDs of the computed models at solar metallicity with an initial
ratio of 0.5 critical angular velocity within the efficient Mix2 chemical
mixing scheme. Panels from left to right show models with initial equatorial
magnetic field strengths of 0, 0.5, 3 kG, respectively. The colour-coding
corresponds to the surface rotational velocity. The initial masses in solar
units are indicated next to the ZAMS. Between 6 and 25 M⊙, the increment is 1
M⊙. From Keszthelyi et al. (2022). Figure 2: Hunter diagram of magnetic
single-star evolutionary models with $B_{\rm eq,ini}=3$ kG, $\Omega_{\rm
ini}/\Omega_{\rm ini,crit}=0.5$ at LMC ($Z_{\rm ini}=0.0064$) metallicity
within two magnetic braking and two chemical mixing schemes. Models within the
SURF/Mix1 scheme are also shown with white lines since they overlap with the
INT/Mix2 models. Additionally, two sets of non-magnetic (NOMAG) models are
shown within the Mix1 (grey) and Mix2 (black) schemes. Models with initial
masses from 15 to 60 M⊙ are shown. The coloured area corresponds to our
definition of Group 2 stars ($v\sin i<50$ km s-1 and at least 0.1 dex nitrogen
enrichment in spectroscopic units). The colour-coding of the models shows the
logarithmic surface gravity. Observations are shown with circles and squares.
A typical reported uncertainty in the observed nitrogen abundances is about
0.1 dex. From Keszthelyi et al. (2022).

###  4.1 Quasi-chemically homogeneous evolution

When internal chemical mixing is efficient, our models at solar metallicity
with an initial rotation rate of 0.5 critical angular velocity show a blueward
evolution on the HRD (left panel of Figure 1). This is because all stellar
layers have a nearly homogeneous chemical composition. Given the blueward
evolution, the increase in effective temperature will lead to Wolf-Rayet type
mass-loss rates in our models, which can help i) spin down the stellar models
and ii) decrease the surface hydrogen abundance. With an initial equatorial
magnetic field strength of 0.5 kG (middle panel), the models experience a
short blueward evolution. However, within a spin-down timescale, the decrease
of rotational velocity results in less efficient chemical mixing. Therefore,
the models turn to a redward evolution on the main sequence. With $B_{\rm
eq,ini}=3$ kG, the initial blueward evolution is shorter, and most of the main
sequence evolution follows a classical path. We can thus conclude that surface
fossil magnetic fields may play an important role in preventing quasi-
chemically homogeneous main sequence evolution, which may be an important
channel for several astrophysical phenomena (e.g., Yoon et al., 2006).

###  4.2 Magnetic stellar evolution models on the Hunter diagram

The Hunter diagram (Hunter et al., 2008), showing surface nitrogen abundance
as a function of projected rotational velocity, is an important diagnostic
tool to gain insights into the chemical and rotational evolution of stars. The
efficiency of chemical mixing and magnetic braking play an important role in
shaping the quantitative evolution of the models on this diagram. For LMC
metallicity, we assumed a baseline abundance of $\log[N/H]=7.15$. The
uncertainties in the magnetic models may lead to an order of magnitude
difference in the predicted nitrogen abundances for a typical 3 kG initial
equatorial magnetic field strength (Figure 2). Nonetheless, we find that these
magnetic models, regardless of the uncertainties related to mixing and braking
schemes, produce slowly-rotating, nitrogen-enriched stars. These so-called
“Group 2” stars could not be explained with typical single star evolution thus
far, though they are commonly found in spectroscopic samples. We note however
that spectropolarimetric observations and comprehensive magnetic
characterisation of Group 2 stars are largely lacking in the Galaxy (see
however, e.g., Aerts et al., 2014; Martins et al., 2012, 2015) and are
practically unavailable in the Magellanic Clouds. Therefore Group 2 stars
might be prime targets to detect magnetic fields.

##  5 Conclusions

We computed and studied an extensive grid of stellar evolution and structure
models, incorporating the effects of surface fossil magnetic fields. We can
quantitatively demonstrate that i) quasi-chemically homogeneous evolution
could be mitigated and prevented for increasing magnetic field strength, and
ii) slowly-spinning, nitrogen-enriched Group 2 stars on the Hunter diagram
could be produced by magnetic massive stars. The library of stellar models is
available to the community via Zenodo at
https://doi.org/10.5281/zenodo.7069766.

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