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Theory of Magnetic Fields in Non-Convective Stars

Theory of Magnetic Fields in Non-Convective Stars

Virtual Nordita Dynamo Seminar

Abstract: I will discuss possible pathways for the origin and evolution of magnetic fields in early type stars. The strong surface magnetic fields observed in OBA stars are stable and have relatively simple geometries. These fields are likely inherited or created during or shortly after star formation (fossil fields). Alternatively some could be produced during a stellar merger event. Weak surface magnetic fields are now also observed. These fields could be failed fossils or be generated by a contemporary dynamo in the stellar envelope. Such dynamo can tap into the energy provided by stellar differential rotation in radiative zones and/or the kinetic energy of convective motion in subsurface convective layers. Finally, internal magnetic fields could also be very common in the convective cores of OBA stars, which are likely to be rotating rapidly. This notion is supported by asteroseismic observations of evolved low-mass stars, which also point towards efficient angular momentum transport in stellar interiors. Indeed such coupling might be provided by internal magnetic fields. Predicting the rotation rate and the magnetization of compact remnants (WDs, NSs and BHs) requires that we understand the coupled evolution of magnetism and angular momentum transport in early type stars.

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Matteo Cantiello

June 17, 2021
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  1. M tteo C ntiello - June 15th 2021 Theory of

    Magnetic Fields in non-convective stars Their Origin nd Evolution
  2. Solar-like OBA Stars Fully Convective

  3. Observations M gnetic Fields in OBA St rs Kochuchov et

    al. 2015
  4. Ap/Bp Stars Strong, l rge-sc le, st ble B- ields

    • ~5-10% of OBA stars (Wol 1968, Power 2007, Neiner et al. 2011, Wade et al. 2014) • Detected via Zeeman e ect • Associated with chemical peculiarities • B > 200 G and up to ~40 kG • Mostly Dipolar / Quadrupolar geometry. Small scale subdominant Sikora et al. 2018
  5. Ap/Bp Stars Strong, l rge-sc le, st ble B- ields

    Kochuchov et al. 2015 HD 75049 Surface B- eld derived from Stokes I and V pro les of Si, Cr and Fe lines using Magnetic Doppler Imaging. Field Modulus Horizontal Field Radial Field Field Orientation • ~5-10% of OBA stars (Wol 1968, Power 2007, Neiner et al. 2011, Wade et al. 2014) • Detected via Zeeman e ect • Associated with chemical peculiarities • B > 200 G and up to ~40 kG • Mostly Dipolar / Quadrupolar geometry. Small scale subdominant
  6. Ultra-Weak B- ields Veg , Sirius… (B < few G)

    • Close to detection limit. Ubiquitous? • Present in rapidly rotating stars, but also in slow rotators • Possibly complex geometry • Variability unclear. But spots in Vega seem to vary on short timescale ~days (Petit et al. 2017) Lignieres et al. 2009; Petit et al. 2010, 2011, 2017; Bohm et al. 2015 Vega: B- elds Vega: Spot Distribution
  7. Magnetic Desert No B- ields between ~1 nd ~100 G

    • Doesn’t seem to be a sensitivity issue • Will discuss possible explanations later in the talk Aurière et al. 2007; Lignières et al. 2008; Kholtygin et al. 2010
  8. Origin of B-Fields OBA St rs

  9. Fossil Field Contemporary Dynamo Stellar Merger Featherstone et al. 2009

    Schneider et al. 2019 Kochuchov et al. 2015 Star formation / pre-main sequence convection. Reaches stable equilibria Convective cores, subsurface convection, di erentially-rotating radiative zones Di erential rotation activates dynamo. Relaxes in stable equilibrium Stable Equilibria Unstable Equilibria Interlocked Poloidal + Toroidal B- eld. Evolves on long Ohmic timescale B- eld evolves on short Alfvén timescale. In the presence of rapid rotation the evolution is slowed- down substantially
  10. Theory Stell r M gnetic Fields Credit: Z.-H. Yang et

    al. 2020
  11. Magnetic Field Evolution Field Generation Field Dissipation Lorentz acceleration Gravity

    Pressure gradient
  12. Magnetic Field Evolution • In an non-rotating, unmagnetized star, the

    pressure gradient and gravity balance each other. • Adding an arbitrary magnetic eld, the Lorentz force cause it to move at the Alfvén speed vA • The system evolves on an Alfvén timescale. For 1kG eld this is roughly 10 years • A magnetic equilibrium is reached (fossil eld) if all forces are balanced. At this point the eld evolves on the long Ohmic timescale v A = B 4πρ τ A = R vA τ ohm = R2 η ≫ τ A > τ MS
  13. Ohmic-Diffusion • The typical timescale of magnetic di usion in

    a star • Since the conductivity is high in stars, the magnetic di usivity is generally low • The Ohmic di usion timescale is usually longer than the MS lifetime of OBA stars See e.g Braithwaite & Spruit 2021 τ ohm = R2 ⋆ η ∼ 1010 yrs
  14. Lorentz acceleration Gravity Pressure gradient Coriolis acceleration An arbitrary magnetic

    eld evolves towards an equilibrium on an Alfvén timescale τA. However, rotation adds a term in the momentum equation. If rotation is rapid ( ), the Lorentz force is not balanced by inertia but by the Coriolis force. The result is that the magnetic eld evolves on a much longer timescale Ωτ A ≫ 1 τ evol ∼ τ2 A Ω ≫ τ A Braithwaite & Cantiello 2013 Magnetic Field Evolution with Rotation
  15. Stable Magnetic Equilibria • Purely Poloidal Fields unstable (e.g. Flowers

    & Ruderman 1977) • Purely Toroidal Fields unstable (e.g. Tayler 1973; Bonanno & Urpin 2008) • Stability requires interlocked poloidal+toroidal elds (Prendergast 1956) • Numerical and theoretical work con rms the stability (Braithwaite & Spruit 2004; Braithwaite & Nordlund 2006; Braithwaite 2009; Duez & Mathis 2010) • Axially and non-axially symmetric See e.g Braithwaite 2008; Braithwaite & Spruit 2021 Braithwaite 2007
  16. Stable Magnetic Equilibria • Purely Poloidal Fields unstable (e.g. Flowers

    & Ruderman 1977) • Purely Toroidal Fields unstable (e.g. Tayler 1973; Bonanno & Urpin 2008) • Stability requires interlocked poloidal+toroidal elds (Prendergast 1956) • Numerical and theoretical work con rms the stability (Braithwaite & Spruit 2004; Braithwaite & Nordlund 2006; Braithwaite 2009; Duez & Mathis 2010) • Axially and non-axially symmetric See e.g Braithwaite 2008; Braithwaite & Spruit 2021 Braithwaite 2005
  17. Stable Magnetic Equilibria • Purely Poloidal Fields unstable (e.g. Flowers

    & Ruderman 1977) • Purely Toroidal Fields unstable (e.g. Tayler 1973; Bonanno & Urpin 2008) • Stability requires interlocked poloidal+toroidal elds (Prendergast 1956) • Numerical and theoretical work con rms the stability (Braithwaite & Spruit 2004; Braithwaite & Nordlund 2006; Braithwaite 2009; Duez & Mathis 2010) • Axially and non-axially symmetric See e.g Braithwaite 2008; Braithwaite & Spruit 2021
  18. Stable Magnetic Equilibria • Purely Poloidal Fields unstable (e.g. Flowers

    & Ruderman 1977) • Purely Toroidal Fields unstable (e.g. Tayler 1973; Bonanno & Urpin 2008) • Stability requires interlocked poloidal+toroidal elds (Prendergast 1956) • Numerical and theoretical work con rms the stability (Braithwaite & Spruit 2004; Braithwaite & Nordlund 2006; Braithwaite 2009; Duez & Mathis 2010) • Axially and non-axially symmetric See e.g Braithwaite 2008; Braithwaite & Spruit 2021 Braithwaite & Spruit 2004
  19. Kochuchov et al. 2015 Braithwaite & Spruit 2004 Ap/Bp Stars

    are believed to host such stable magnetic eld con gurations
  20. Flux-Freezing • A magnetic eld in a stable con guration

    is ampli ed through phases of stellar contraction • It assumes the eld is not destroyed by other processes • Can in principle explain the elds of some compact remnants (but see also Makarenko et al. 2021) B end = B ini ( R ini Rend ) 2 Ψ = BR2 = k See e.g. Thompson & Duncan 1993; Cantiello et al. 2016
  21. Dynamo Action • A process converting a source of free

    energy it into magnetic energy (convection, di erential rotation…) • It continuously sustains the magnetic elds against decay • The Rossby number (~ Prot /Pcon ) plays an important role in determining the dynamo eld geometry and amplitude (e.g. Augustson et al. 2016) • A dynamo-generated magnetic eld varies on short timescales See e.g Brandenburg & Subramanian 2005 =
  22. Stellar Mergers • Large fraction of massive stars found in

    binaries (e.g. Sana et al. 2012) • Many of these stars interact during their main sequence (e.g. de Mink et al. 2013) • About 10% are expected to merge (e.g. de Mink et al. 2014) • Lack of magnetic stars in close binaries (with some counter-examples) Carrier et al. 2002; Alecian et al. 2015; Folsom et al. 2013 Mink et al. 2014
  23. Stellar Mergers • 3D MHD Simulation using AREPO (9+8 MSun

    ) • MRI appears to be responsible for eld ampli cation. Final ux corresponds to B~9kG in the relaxed merger product • Could explain Tau Sco age problem • More work needed to study the nature of the dynamo, as well as the long-term stability of the eld produced in such simulations Schneider et al. 2019
  24. Stellar Structure OBA St rs

  25. O B A F G Sp. Type HeI H Fe

    H HeII Convective Core Stellar Structure OBA St rs • Convective cores (CNO cycle) • Envelope Convective Regions driven by H,He,Fe recombination • The majority of these stars are rapidly rotating (P~days) • Lifetimes decrease quickly from ~hundreds of Myr for A stars, to a ~few Myr for massive O stars Cantiello & Braithwaite 2019
  26. Dynamo: Convective Cores • Rossby number in convective cores likely

    << 1 • Equipartition Fields up to ~MG expected (e.g. Featherstone et al. 2009; Augustson et al. 2016) • Magnetic buoyancy too slow to bring eld to the surface (MacGregor & Cassinelli 2003; MacDonald & Mullan 2004) • Descendant of core dynamo magnetic elds might have been detected via asteroseismology (Fuller, MC et al. 2015; Stello, MC et al. 2016; Loi & Papaloizou 2018,2020; Bugnet et al. 2021) See Talk from L. Bugnet Augustson et al. 2016
  27. • Rossby number in convective cores likely << 1 •

    Equipartition Fields up to ~MG expected (e.g. Featherstone et al. 2009; Augustson et al. 2016) • Magnetic buoyancy too slow to bring eld to the surface (MacGregor & Cassinelli 2003; MacDonald & Mullan 2004) • Descendant of core dynamo magnetic elds might have been detected via asteroseismology (Fuller, MC et al. 2015; Stello, MC et al. 2016; Loi & Papaloizou 2018,2020; Bugnet et al. 2021) Stello, MC et al. 2016 Fraction of stars with strong internal B-fields From a sample of 3000+ stars At least 50-60% have strong internal B- fields! Dynamo: Convective Cores
  28. Tayler Saturation is provided by turbulent dissipation of the perturbed

    magnetic energy (in Spruit 2002 the saturation was provided by turbulent dissipation of background magnetic energy) Fuller, Piro & Jermyn 2019 Dynamo: Radiative Regions • Source of free energy: di erential rotation • Magneto-Rotational Instability (e.g. Spruit 1999; Wheeler et al. 2015) • Tayler-Spruit Dynamo (Spruit 2002) • Revised Tayler-Spruit Dynamo (Fuller, Piro & Jermyn 2019) • Seems required to explain slow spins of stellar cores (Cantiello et al. 2014; Fuller+ 2019; Fuller & Ma 2019; Ma & Fuller 2019) Slow spins detected via Asteroseismology. See D. Bowman’s Talk
  29. • Source of free energy: di erential rotation • Magneto-Rotational

    Instability (e.g. Spruit 1999; Wheeler et al. 2015) • Tayler-Spruit Dynamo (Spruit 2002) • Revised Tayler-Spruit Dynamo (Fuller, Piro & Jermyn 2019) • Seems required to explain slow spins of stellar cores (Cantiello et al. 2014; Fuller+ 2019; Fuller & Ma 2019; Ma & Fuller 2019) Slow spins detected via Asteroseismology. See D. Bowman’s Talk Fuller, Piro & Jermyn 2019 Dynamo: Radiative Regions
  30. • Dynamo Action in Subsurface convective zones • Fields can

    easily reach the stellar surface via e.g. magnetic buoyancy • Could explain elds of ~1-10 G in A-Stars, and 10-100 G in OB Stars • The presence of subsurface convection zones could be related to line pro le variability and other surface e ects See talks from A. Fullerton, A. Kholtygin, D. Bowman, G. Rauw Cantiello et al. 2009 Dynamo: Subsurface Convection
  31. Cantiello & Braithwaite 2011,2019 Dynamo: Subsurface Convection • Dynamo Action

    in Subsurface convective zones • Fields can easily reach the stellar surface via e.g. magnetic buoyancy • Could explain elds of ~1-10 G in A-Stars, and 10-100 G in OB Stars • The presence of subsurface convection zones could be related to line pro le variability and other surface e ects See talks from A. Fullerton, A. Kholtygin, D. Bowman, G. Rauw
  32. • Dynamo Action in Subsurface convective zones • Fields can

    easily reach the stellar surface via e.g. magnetic buoyancy • Could explain elds of ~1-10 G in A- Stars, and 10-100 G in OB Stars • Ultra-weak elds observed in A stars could be dynamo-generated Cantiello & Braithwaite 2011,2019 Dynamo: Subsurface Convection 3.8 3.9 4.0 4.1 4.2 4.3 log TeÆ 0.5 1.0 1.5 2.0 2.5 3.0 log L/LØ 1.5MØ 2.0MØ 2.5MØ 3.0MØ 4.0MØ 5.0MØ -0.6 -0.3 0.0 0.3 0.6 0.8 1.1 1.4 1.6 1.9 2.2 2.5 2.8 3.0 Vega Sirius Ø UMa µ Leo Alhena -1.1 -0.6 0.0 0.6 1.1 1.6 2.2 2.8 3.3 log Bs (G)
  33. Magnetic Spots in OBA Stars • Surface Magnetic Spots •

    Since the stellar surface is radiative, these spots should be bright • Might have been detected in OB stars (e.g by MOST/BRITE, Ramiaramanantsoa et al. 2014) • The presence of subsurface convection and their associated magnetic spots could explain some of the variability and activity observed in OB and A stars (e.g. Balona et al. 2011, 2013) Cantiello & Braithwaite 2011
  34. Magnetic Desert Aurière et al. 2007; Lignières et al. 2008;

    Kholtygin et al. 2010
  35. Possible Explanations • “There exists a critical eld strength above

    which stable magnetic con gurations exist and below which any large scale eld con guration is destroyed by some instability” (Aurière et al. 2007) • Aurière et al. suggests the instability could be triggered by di erential rotation • A strong enough magnetic eld can suppress convection (MacDonald & Petit 2013) • The existence of subsurface convection naturally explains the dichotomy, since the value of the critical eld required to shuto these convective layers is close to the upper edge of the magnetic desert (Jermyn & Cantiello 2020) • Because Bcrit initially increases with stellar age, we expect the fraction of stars with strong magnetic elds to decline with age Jermyn & Cantiello 2020
  36. Possible Explanations • “There exists a critical eld strength above

    which stable magnetic con gurations exist and below which any large scale eld con guration is destroyed by some instability” (Aurière et al. 2007) • Aurière et al. suggests the instability could be triggered by di erential rotation • A strong enough magnetic eld can suppress convection (MacDonald & Petit 2013) • The existence of subsurface convection naturally explains the dichotomy, since the value of the critical eld required to shuto these convective layers is close to the upper edge of the magnetic desert (Jermyn & Cantiello 2020) • Because Bcrit initially increases with stellar age, we expect the fraction of stars with strong magnetic elds to decline with age Jermyn & Cantiello 2020
  37. Possible Explanations Jermyn & Cantiello 2020 Fossil Field Strength Incidence

    B crit Preserve Erase B dynamo Amplify Current Field Strength Incidence B crit Preserve Erase B dynamo Amplify NGC 1624-2 Vega • “There exists a critical eld strength above which stable magnetic con gurations exist and below which any large scale eld con guration is destroyed by some instability” (Aurière et al. 2007) • Aurière et al. suggests the instability could be triggered by di erential rotation • A strong enough magnetic eld can suppress convection (MacDonald & Petit 2013) • The existence of subsurface convection naturally explains the dichotomy, since the value of the critical eld required to shuto these convective layers is close to the upper edge of the magnetic desert (Jermyn & Cantiello 2020) • Because Bcrit initially increases with stellar age, we expect the fraction of stars with strong magnetic elds to decline with age
  38. Fossil Field Contemporary Dynamo Stellar Merger Featherstone et al. 2009

    Schneider et al. 2019 Kochuchov et al. 2015 Star formation / pre-main sequence convection. Reaches stable equilibria Convective cores, subsurface convection, di erentially-rotating radiative zones Di erential rotation activates dynamo. Relaxes in stable equilibrium Stable Equilibria Unstable Equilibria Interlocked Poloidal + Toroidal B- eld. Evolves on long Ohmic timescale B- eld evolves on short Alfvén timescale. In the presence of rapid rotation the evolution is slowed- down substantially
  39. Spots in OBA Stars Dynamos in convective cores Dichotomy in

    OBA Stars B- ields Augustson et al. 2016 Aurière et al. 2006; Jermyn & Cantiello 2020 Cantiello & Braithwaite 2011,2019 Driven by buoyant magnetic elds from dynamo in subsurface convection Generate elds up to ~MG. Might survive to later stages of stellar evolution Fossil Fields above a critical magnetic eld able to shuto subsurface convection Dynamos in radiative zones Can explain rotation rate of compact remnants
  40. Thanks!