Stable magnetic fields in stars

Jon Braithwaite

Magnetism on the main sequence

Ap stars – observational properties

� Peculiar metallic abundances.

� Variability:

� Regular. Periods from 12 hrs to several decades.

� All quantities have same period --> oblique rotator.

� Magnetic field:

� Static. No changes seen over fifty years.

� Strong. From 300 to 30,000 gauss.

� Large-scale, roughly dipolar.

� Age: Hubrig et al. (2000) show that Ap stars are all at least 30% of way through MS lifetime.

Summary

� Observations: around 5% of main-sequence A stars have a strong, steady, large-scale magnetic field.

� Possible explanations: Core-dynamo or fossil-field?

� Problem: fossil-field model requires stable field, but there is no known stable field.

� Solution: use numerical magnetohydrodynamics to follow evolution of arbitrary initial field into a stable configuration.

How magnetic fields are measured

� Circular polarisation in wings of spectral lines (via Zeeman effect) yields average line-of-sight component of magnetic field, Bs.

� If field > 2 kG, and line splitting exceeds thermal and rotational broadening, modulus of field can be obtained, Bmod.

� These (and two or three related quantities) are measured as the star rotates.

� A model is made of the magnetic field, (e.g. dipole + quadrupole) and the free parameters are found by fitting to the observations.

Example: observations of 53 Cam

Longitudinal field and field modulus, Bagnulo et al. 2001.

Shape of magnetic field

observed

The field on 53 Cam (Bagnulo et al. 2001) is roughly dipolar but has a significant quadrupolar component.

Origin of Ap-star magnetic fields

Two possibilities:

Core-dynamo modelThe field is being continually regenerated

Fossil-field modelThe field is left over from formation of star

The core-dynamo model

� Field produced in convective core, then rises to surface1.

� Issues:

� Lack of correlation between field strength and rotation (in contrast to late-type stars).

� Field observed has a dipole axis not parallel to rotation axis.

� Mechanism is unclear by which field rises to surface quickly enough.

� Why do only 5% of main-sequence A stars have a magnetic field?

Figure taken from MacGregor & Cassinelli 2003

1. See e.g. Meyer 1994, MacGregor & Cassinelli 2003

The fossil-field model

� Field is left over from formation.

� Field is in equilibrium, and stable against arbitrary perturbation.

� Problems:

� The existence of a stable field configuration has not yet been proven.

� Why are not all A-stars magnetic?

Equilibrium and stability

� An arbitrary magnetic field is not in equilibrium, as the Lorentz and pressure forces will not generally be balanced. It will change on an Alfvén time-scale.

� Various equilibrium fields have been found, but none have been shown to be stable.

� Certain field configurations have been shown to be unstable, including:

� All purely poloidal fields1,

� All purely toroidal fields2.

1. Wright 1973, Markey & Tayler 1973.2. Tayler 1973.

Instability of toroidal field

A magnetic field exerts a positive pressure perpendicular to its direction.

In a toroidal field, this creates a situation like that in a compressed spinal column.

Consequently: any purely toroidal field is subject to instability on or near the magnetic axis.

Instability of a poloidal magnetic

fieldA 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.

A poloidal field of uniform direction inside the star with a potential (zero current) field in the atmosphere

Possible stable field

By adding a toroidal component, rotation of magnets is prevented.

Magnetic helicity

�

Defined as B.A dV, integrated inside a magnetic surface.

� Conserved in situations where reconnection can be ignored, i.e. where diffusion has little effect.

� Has been useful in contexts such as fusion reactors.

� Arbitrary magnetic field with non-zero magnetic helicity cannot decay beyond lowest energy state with that value of magnetic helicity.

� Stable field configuration therefore expected to exist in an environment with low diffusivity.

Philosophy of this research

To find an equilibrium field and show that it is stable.

Model a star containing an arbitrary initial magnetic field, and follow the evolution of this field in time, using numerical magnetohydrodynamics (MHD).The field will either decay to zero, or find a stable state.

Aim:

Method:

The model

� Star modelled as self-gravitating ball of plasma, (a polytrope of index n=3,) embedded in a hot, non-conductive atmosphere.

� Fitted into cubic computational box of side 4.5R*.

Initial magnetic field

� Calculated from random vector potential, with length-scales down to a few grid spacings.

� Weak: ratio of thermal to magnetic energy densities = 100. But stronger than in real star, to speed things up.

Numerical code

� Adapted from a code by Nordlund & Galsgaard (1995).

� Grid-based, finite-difference MHD code.

� Cartesian coordinates:

� No singularities or special points,

� Simpler, and therefore faster,

� Some wastage at corners of computational box.

Results: Initial evolution of field

� Main result: field decays quickly at first, but stops decaying after a small number of Alfvén-crossing-times.

� Field formed is a mixed poloidal-toroidal torus shape, roughly axisymmetric.

� The shape of the field produced is independent of the initial field, except that it can be either a right- or left-handed torus.

Creation of stable field

Shape of stable fieldField lines plotted in red go through the centre of the star and emerge in the atmosphere.

Blue field lines are confined to the stellar interior.

Core of torus plotted in grey as a visual aid.

Stereographic version can be found on MPA website

Shape of stable field

Toroidal component is confined to the loops of poloidal field which are closed within the star.

Poloidal field lines (in red) which go through the stellar surface, and torus field lines (blue) confined to the interior. Core of torus shown in grey.

Summary of main result

� Stable field which evolves is a mixed poloidal-toroidal torus shape, roughly axisymmetric. It may be offset from the centre of the star.

� The field on the stellar surface looks like a dipole, possibly somewhat offset from the centre, agreeing with observations. The field below the surface is however not a simple dipole.

� The stable field could also exist inside a white dwarf or neutron star.

Slow outwards diffusion of stable field

� Owing to finite conductivity, field decays on an Ohmic diffusion time-scale (much longer than the Alfvén time-scale).

� As it decays, it moves outwards.

� Field strength on surface increases.

Field diffusing outwards

Figure: sequence of frames at equal time intervals.Azimuthal average of poloidal (in red) and toroidal (in blue) components.

Toroidal field threads through loops of poloidal field.

As field diffuses outwards, poloidal component extends into atmosphere.

Increase of surface field strength

Field strength on surface increases as field diffuses outwards.

Time-scale: 2x109 yrs.

Decay of the field at the end of its long period of diffusion

� At some point after the field has been slowly diffusing outwards for a long time, it becomes unstable.

� At first, circular shape of torus is lost.

� Field quickly decays away to nothing.

Final decay

of field

Why does field suddenly decay?

� As field diffuses outwards, toroidal field dies away.

� Purely poloidal field is unstable.

Reason for field's instability

� When torus is first formed, energy of toroidal component is greatest.

� Later, energy is mainly in poloidal component. Fraction of energy in poloidal component,

with non-conducting (solid line) and conducting (dotted line) atmosphere.

Point where field becomes unstable

Conclusions

� Stable field will form in a star from any initial field, provided that its magnetic helicity is not zero.

� Field on surface is chiefly dipolar.

� Field strength on surface increases.

� Suddenly decays after a (long) time.

Implications for theory of magnetic stars

� Fossil-field versus core-dynamo models:

� Lack of known stable field was the main problem for the fossil-field model. This problem has now been solved.

� Fossil-field model can also be used to explain magnetic fields in:

� white dwarfs, which have no convective core and therefore no dynamo-model alternative,

� highly magnetised neutron stars (known as `magnetars').

Unanswered questions

� Why are only 5% of MS stars in this mass range observably magnetic?

� There is evidence that magnetic A stars are the progenitors of magnetic white dwarfs. Are the higher-mass magnetic O and B stars the progenitors of magnetars?

Three studies of 53 Cam

a – Displaced-dipole model (Landstreet 1988)

b – Dipole-quadrupole model (Bagnulo et al. 2001)

c – Zeeman-Doppler imaging (Kochukhov et al. 2004)