1. Stability of magnetic fields in stars Vienna 11 th September 2007 Jonathan Braithwaite CITA, Toronto

2. Magnetic fields in non-convective stars ● Which stars can we consider non-convective? – (envelope of) main-sequence A stars – white dwarfs – neutron stars – solar core, etc. ● Same principles apply to all, but observation of A stars is easiest! ● Observations show a tendency for: – steady, large-scale magnetic fields – lack of differential rotation ● Given the lack suitable self-regenerative processes, we need a field in stable equilibrium

3. Equilibrium and stability ● Various equilibrium fields can be constructed ● Certain equilibrium field configurations have been shown to be unstable, including: – All purely poloidal fields 1, – All purely toroidal fields 2. 1. Wright 1973, Markey & Tayler 1973. 2. Tayler 1973.

4. Equilibrium and stability ● Various equilibrium fields can be constructed ● Certain equilibrium field configurations have been shown to be unstable, including: – All purely poloidal fields 1, – All purely toroidal fields 2. ● A roughly axisymmetric twisted-torus poloidal-toroidal field is probably the simplest, most fundamental stable equilibrium ● It evolves on an Alfven timescale out of an arbitrary turbulent magnetic field 3 1. Wright 1973, Markey & Tayler 1973. 2. Tayler 1973. 3. Braithwaite & Spruit 2004, Braithwaite & Nordlund 2006 shading represents toroidal component

5. Shape of stable torus field Braithwaite & Nordlund 2006 Forms in a star, out of an arbitrary initial magnetic field

6. Questions ● Are there other, more complex stable equilibria? (cf. observations of non-dipolar Ap stars) ● If so, which initial turbulent fields will evolve into which equilibria? And what strength field is produced? Possible factors include: – central concentration, i.e. radial profile of field strength – coherence length of turbulent field, ● Does rotation have any effect on the stable equilibria available?

7. Simulations to investigate magnetic field stability in a star ● Use numerical magnetohydrodynamics (MHD) to follow evolution of an initially random “turbulent” field into a stable equilibrium ● Model star as ball of self-gravitating ideal gas (polytropic index n=3, similar to an A star) in a box ● Use “stagger-code” (see e.g. Nordlund & Galsgaard 1995), a high-order finite-difference MHD code

8. Coherence length of initial field ● Initial random field contains wavenumbers up to k max ● Simulations run with R * k max /2  = 1.5, 3, 6 and 12. ● Helicity defined as H  ∫A.B dV, where B = curl A. It is conserved in the limit of infinite conductivity ● Higher k max means lower helicity, because different regions cancel each other out ● Lower initial helicity results in a lower- energy field, since the equilibrium is the lowest energy state at that value of helicity Braithwaite & Helfield, in prep. Above right: magnetic energy against time Below right: magnetic helicity against time

9. Radial profile of initial field ● Run simulations where initial field is tapered as B ~   ● If star forms from a uniform magnetised cloud and flux loss fraction is independent of radius, we expect p=2/3 ● Try values p = 1/4, 1/3, 1/2, 2/3.

10. Simulations with different values of p ● B ~   ● If p ≥ 0.5, dipolar torus field does form 1 ● Otherwise, some more complicated equilibrium is found ● At p ≥ 0.5, the dipolar field diffuses outwards ● It goes eventually into a non-axisymmetric equilibrium ● Non axi-symmetric equilibria seem to consist of a twisted flux tube(s) close to the surface of the star 1. Braithwaite & Helfield, in prep.

11. The effect of rotation ● Does rotation affect the stability of magnetic field configurations in a star? ● Rotation has a tendency for stabilisation, because the Coriolis force acts perpendicular to the velocity ● Poloidal fields are unstable in non-rotating stars; does rotation stabilise them?

12. Instability of a poloidal magnetic field A purely poloidal field is unstable, as one half of the star can rotate with respect to the other, and the magnetic energy outside the star goes down. But what about a rotating star? A poloidal field inside the star with a potential (zero current) field in the atmosphere

13. Why should rotation not stabilise a poloidal field? ● Result: rotation only slows decay of poloidal field. ● In a stably stratified star, Coriolis force has no effect on motion at the equator (like on Earth!), which is precisely where this instability is strongest.

14. Conclusions ● Stable mixed-poloidal-toroidal equilibrium produced from random initial field (Braithwaite & Spruit 2004) ● Strength of resulting field depends on initial helicity ● Whether simple or more complex equilibrium is reached depends on the radial profile of initial field ● Near the threshold the particular form of the initial field may be important. May have something to do with why only some A stars are magnetic. ● Poloidal fields unstable even in rotating stars; since purely toroidal fields are also known to be unstable (Tayler 1973, Braithwaite 2006), stable fields should be mixed poloidal-toroidal in both non- rotating and rotating stars

15. Ongoing and future projects, open questions ● Stability and longevity of non-axisymmetric equilibria ● Axisymmetric equilibria: possible toroidal/poloidal ratios ● Effect of magnetic field on star's moment of inertia, could be important for: – whether magnetic and rotation axes should tend towards being aligned or at 90 degrees – emission of gravitational waves by young magnetically- deformed neutron stars – free precession of white dwarfs and neutron stars ● Why are only some A stars magnetic? ● Magnetism in O/B stars – differential rotation?