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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)

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Measurements of average magnetic fields in M dwarfs Reiners 2012

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Symbol size: mean large-scale field strength Symbol Shape: degree of axisymmetry (decagons=purely axisymmetric) Color: field configuration (blue=toroidal; red=poloidal) Observed large scale magnetic geometries (Donati 2011)

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Browning (2008): 3D MHD simulations of a 0.3M star

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R=0.24 R ★ R=0.88R ★ 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 = 10kG

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5 – 10% deviation 1 Gyr, Solar Comp. Observations of double lined eclipsing binaries suggest that stars are inflated compared to stellar models Single Age and Metallicity 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)

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KOI-126 – A Triple System Video courtesy of Josh Carter Carter et al. (2011, Science, 331, 562)

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Match made in the heavens Age = 4.1± 0.6 Gyr KOI-126 A KOI-126 B & C Feiden, Chaboyer, & Dotter (2011, ApJ, 740, L25)

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Match not made in the heavens Age = 4.1± 0.6 Gyr KOI-126 A CM Dra (Lacy 1977; Morales et al. 2009) [Fe/H] = -0.30 (Rojas-Ayala et al; Terrien et al. 2012) Feiden, Chaboyer, & Dotter (2011, ApJ, 740, L25)

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Feiden & Chaboyer (2012, ApJ, 757, 42) Multiple Metallicities and Ages: mean absolute error in the models is 2.3%, most stellar radii fit models to within 4%

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Magnetism?

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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.

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Self-Consistent Magnetic Stellar Evolution Models … in 1-Dimension

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Basic Equations

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Lydon & Sofia (1995, ApJS, 101, 357) 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)

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Magnetic Field Radial Profile

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Turning on the Magnetic Field

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Numerical Tests

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Test the models by comparing to detached eclipsing binaries with well determined masses, radii, ages and compositions

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EF Aquarii: a 1.24 & 0.95 M detached eclipsing binary 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) Feiden & Chaboyer 2013, ApJ, 765, 86,

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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 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. Dipole Profile Peak field 13 kG Peak field 500 kG

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Predicted Surface Magnetic Field Strengths compared to Observations Peak interior magnetic field strengths in the models are ~ 10 4 to 10 5 gauss, which are similar to those found in 3D MHD models of stellar dynamos

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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

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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

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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)

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Fully Convective Stars: CM Dra, M = 0.21 & 0.23 M X-ray luminosity suggest that CM Dra has an average surface magnetic field strength between 1 - 4 kG turbulent dynamo B = Λ B equipartition Surface magnetic field ~ 3kG [Fe/H] = -0.30±0.15 dex; Age derived from common proper motion white dwarf

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Interior Structure: M=0.231M Gaussian profile All magnetic models (gaussian, dipole & turbulent dynamo) have a very similar temperature gradient near the surface.

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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)

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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?)

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Magnetic mapping of solar-type stars Pascal Petit figure: © M. Jardine.

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