Including Magnetic Effects in 1-D Stellar Models
Greg Feiden & Brian Chaboyer (Dartmouth)
Three color EUV image of the Sun. White lines show a model of the Sun’s magnetic field
(Schrijver & Title 2011)
Measurements of average magnetic fields in M dwarfs
Reiners 2012
Observed large scale magnetic geometries (Donati 2011)
Symbol size: mean large-scale field strength
Symbol Shape: degree of axisymmetry (decagons=purely axisymmetric)
Color: field configuration (blue=toroidal; red=poloidal)
Browning (2008): 3D MHD simulations of a 0.3M⦿ star
Browning (2008): 3D MHD simulations of a 0.3M star
R=0.88R★
R=0.24 R★
Magnetic fields posses both small scale structure and large scale ordering, with more energy
associated with the large scale fields. Strong axisymmetric toroidal fields (with ~ 20% of the
total magnetic energy) are found at all depths, with typical <BΦ> = 10kG
Observations of double lined eclipsing binaries suggest that stars
are inflated compared to stellar models
Single Age and Metallicity
5 – 10% deviation
1 Gyr, Solar Comp.
Number of authors have suggested that magnetic fields are inflating stars
(e.g. Ribas 2006; Lopez-Morales 2007; Morales et al. 2008; Chabrier et al. 2007;
Mullan & MacDonald 2001)
KOI-126 – A Triple System
Video courtesy of Josh Carter
Carter et al. (2011, Science, 331, 562)
Match made in the heavens
KOI-126 A
KOI-126 B & C
Age = 4.1± 0.6 Gyr
Feiden, Chaboyer, & Dotter (2011, ApJ, 740, L25)
Match not made in the heavens
KOI-126 A
CM Dra (Lacy 1977; Morales et al. 2009)
[Fe/H] = -0.30 (Rojas-Ayala et al; Terrien et al. 2012)
Age = 4.1± 0.6 Gyr
Feiden, Chaboyer, & Dotter (2011, ApJ, 740, L25)
Multiple Metallicities and Ages:
mean absolute error in the models is 2.3%, most
stellar radii fit models to within 4%
Feiden & Chaboyer (2012, ApJ, 757, 42)
Magnetism?
Effects of Magnetic Fields on Stellar Structure
 Magnetic fields suppress
thermal convection
(Thompson 1951;
Chandrasekhar 1961; Gough
& Tayler 1966; Mullan &
MacDonald 2001, 2010)
 Surface spots reduce flux
across a given surface area
(Hale 1908; Spruit 1982;
Chabrier et al. 2007)
 Surface faculae increase flux
across a given surface area
(Spruit 1977, Foukal et al.
2006)
Image taken by B. De Pontieu with the
Swedish 1-m solar telescope.
Self-Consistent
Magnetic Stellar Evolution
Models
… in 1-Dimension
Basic Equations
Basic Equations
new thermodynamic
state variable
f can range from 0 (fluid parcel carries its original
magnetic energy as it moves) to 1 (magnetic energy
of a fluid parcel is always equal to its surroundings;
and is valid for a perfectly conducting plasma)
Lydon & Sofia (1995, ApJS, 101, 357)
Magnetic Field Radial Profile
Turning on the Magnetic Field
Numerical Tests
Test the models by comparing to
detached eclipsing binaries with well
determined masses, radii,
ages and compositions
EF Aquarii: a 1.24 & 0.95 M⦿ detached eclipsing binary
Feiden & Chaboyer 2013,
ApJ, 765, 86,
Photospheric field strengths of
1.6 kG (γ = 2) and 2.6kG (γ = 4/3) for EF Aqr A and
3.2 kG (γ = 2) and 5.5kG (γ = 4/3) for EF Aqr B
X-ray emission and Ca II K line core emission suggest
actual magnetic fields of about 1 kG (EF Aqr A) and 3 kG (EF Aqr B)
YY Geminorum M = 0.599+/- 0.004
associated with the Castor AB quadruple system
[Fe/H] = +0.1 0.2 dex and an age of 360 Myr
Peak field
13 kG
Peak field
500 kG
Dipole Profile
Magnetic models also match the observed effective temperature, and Li abundance
Log N(Li) = 0.11 (Barrado y Navascues et al. 1997), while standard models predict that
surface Li should be completely depleted after about 15 Myr.
Predicted Surface Magnetic Field Strengths
compared to Observations
Peak interior magnetic field strengths in the models are ~ 104 to 105 gauss,
which are similar to those found in 3D MHD models of stellar dynamos
Can the predicted surface magnetic field strengths be
reduced?

We assumed ideal MHD (perfectly conducting fluid); finite electrical
conductivity affects the magnetic inhibition of convection (MacDonald &
Mullen 2009, 2010, 2013; it also makes it more difficult for the dynamo
mechanism to operate).

In our formulation, the most significant implication for finite conductivity is
that f (which determines the flux of magnetic energy between a convecting
bubble and the surroundings) is no longer 1

In the extreme (non-physical) case of f = 0, predicted model radii inflate by
3% (for a 0.4 M⦿ model) to 9% (for a 0.9 M⦿ model) compared to f = 1
models

Future work could look at relating the free electron fraction to conductivity
and use this to determine f
Turbulent Dynamo

Brandenburg & Subramanian (2005)
and Brown et al. (2010) have
suggested that the the physical
source of the solar and stellar
dyanamo is turbulent convection,
and not the shear induced by
rotation (Parker 1955)

Generation of magnetic fields will
suppress the turbulent velocities in
the convection zone

Reformulated our mixing length
equations to incorporate this idea
into our magnetic stellar models
Turbulent Dynamo

Brandenburg & Subramanian (2005)
and Brown et al. (2010) have
suggested that the the physical
source of the solar and stellar
dyanamo is turbulent convection,
and not the shear induced by
rotation (Parker 1955)

Generation of magnetic fields will
suppress the turbulent velocities in
the convection zone

Reformulated our mixing length
equations to incorporate this idea
into our magnetic stellar models
YY Gem (0.6M⦿; rotational
dynamo required a 4kG field
to match the observations)
Fully Convective Stars: CM Dra, M = 0.21 & 0.23 M⦿
[Fe/H] = -0.30±0.15 dex; Age derived from
common proper motion white dwarf
turbulent dynamo
B = Λ Bequipartition
Surface magnetic field ~ 3kG
X-ray luminosity suggest that CM Dra has an average
surface magnetic field strength between 1 - 4 kG
Interior Structure: M=0.231M⦿
Gaussian profile
All magnetic models (gaussian, dipole & turbulent dynamo)
have a very similar temperature gradient near the surface.
Radii of fully convective stars compared to standard models
Observational data are from Morales et al 2009 (CM Dra); Carter et al. 2011 (KOI-126);
Doyle et al. 2011 (Kepler-16B) and Orosz et al 2012 (Kepler-38B)
Summary & the Future

1-D Dartmouth stellar evolution code has been modified to include the
effects of a prescribed magnetic field

For stars with radiative cores 1-D models which include the effects of
magnetic fields due to a turbulent dynamo can fit the observed properties
(mass, radius, Teff, surface magnetic field strength, and Li abundance) of
eclipsing binaries (strength of magnetic field near the surface key
parameter which controls the change in radii)

Radiative core models which assumed the dynamo is sourced by rotation
predicted surface magnetic field strengths which are higher than observed

Magnetic models with convective cores only match observed radii with
very large magnetic fields, which are inconsistent with the predictions from
turbulent dynamo simulations

Future: Closer interaction with 3-D magnetic-hydro simulations to use
realistic magnetic field topologies/strengths in stellar models (3-D stellar
models?)