cortar la estrella por la mitad a lo largo de un plano paralelo al eje de simetrÃa, y rotar cada mitad en 90⦠en direcciones opuestas, lo cual reducirÃa la energÃa.
Asociación Argentina de Astronomía BAAA, Vol. 54, 2011 J.J. Clariá, P. Benaglia, R. Barbá, A.E. Piatti & F.A. Bareilles, eds.
PRESENTACIÓN MURAL
Formal proof of the Flowers-Ruderman instability mechanism in magnetic stars P. Marchant1 , A. Reisenegger1 & T. Akgün1,2 (1) Departamento de Astronomía y Astrofísica, Pontificia Universidad Católica de Chile, Casilla 306, Santiago, Chile (2) Barcelona Supercomputing Center - Centro Nacional de Supercomputación, C/ Gran Capità 2-4, Barcelona, 08034, Spain Abstract. It has been know for decades that stable magnetic fields exist on some stars. However, it has not been possible to construct analytically a field configuration that can be proved to be stable. It is then constructive to study general processes that can drive an instability, since this can provide clues on how to construct an equilibrium. An example of this was given in 1977, when Flowers and Ruderman described a perturbation that destabilized a purely dipolar magnetic field. They considered the effect of cutting the star in half along a plane parallel to the symmetry axis and rotating each half 90◦ in opposite directions, which would cause the energy of the magnetic field in the exterior of the star to be greatly reduced. We formally solve for the energy of the external magnetic field and check that it decreases monotonously along the entire rotation. Finally, we consider the stabilizing effect of adding a toroidal field by studying the internal energy perturbation when the rotation is not done along a sharp cut, but with a continuous displacement field that switches the direction of rotation across a region of small but finite width. Resumen. Se ha sabido por décadas que campos magnéticos estables existen en algunas estrellas. Sin embargo, no ha sido posible construir analíticamente un campo cuya estabilidad pueda ser demostrada. Es entonces constructivo estudiar procesos generales que pueden conducir a una inestabilidad, ya que éstos pueden entregar claves sobre como construir un equilibrio. Un ejemplo de esto fue dado en 1977, cuando Flowers y Ruderman describieron una perturbación que desestabiliza un campo puramente dipolar. Ellos consideraron el efecto de cortar la estrella por la mitad a lo largo de un plano paralelo al eje de simetría, y rotar cada mitad en 90◦ en direcciones opuestas, lo cual reduciría la energía del campo magnético afuera de la estrella. Resolvemos formalmente la energía del campo magnético externo y verificamos que ésta decrece de forma monótona durante toda la rotación. Finalmente, consideramos el efecto estabilizante de añadir un campo toroidal estudiando la perturbación de la energía dentro de la estrella cuando la rotación no se realiza a través de un corte fino, sino con un desplazamiento continuo que cambia su dirección de rotación a través de una región de grosor pequeño, pero finito.
81
P. Marchant et al.
82 1.
Introduction
Large-scale magnetic fields are present in many stellar objects, and they appear to be long-lived since they do not evolve in a timescale accessible to observations. This is contrary to our common experience with the magnetic field of the sun, where small scale structure and a dynamic nature are the norm. These objects are mostly stably stratified so dynamo effects are expected to be irrelevant in keeping the strength of the magnetic field, which should be in a state of stable equilibrium. It is then important to study possible equilibrium configurations that could be present in these objects, as magnetic fields can have an important impact on angular momentum transport and diffusion processes in stars. Even though these long-lived fields have been known to exist for more than half a century, it has not been possible to find an analytic model for a field that has been shown to be in stable equilibrium. However, stable configurations have been found to exist via numerical calculations, where an initially random field usually evolves into an approximately axisymmetric configuration that is a combination of toroidal and poloidal components of similar energies (Braithwaite & Spruit 2004). In these simulations, once a stable configuration has been achieved, the decay of the field is driven by Ohmic dissipation, and it can be seen to evolve in a timescale comparable to the lifetime of the star. The study of general instabilities that can affect magnetic fields can give us clues of the properties required by an equilibrium configuration. With regard to the stability of purely poloidal fields, Flowers & Ruderman (1977) argued that any poloidal field that has the shape of a dipole outside of the star should be unstable. In an analogous way to a couple of aligned magnets, one half of the star could turn with respect to the other producing a quadrupole (as shown in figures 1 and 2). In this contribution, we summarize our formal proof of the Flowers-Ruderman instability for a pure dipole and our study of the stabilizing effect of a toroidal field (Marchant et al. 2011).
N
N
S S
N
S
N
S
Figure 1. A couple of aligned magnets is in an unstable state of equilibrium. Figure adapted from Braithwaite & Spruit (2006)
2.
Figure 2. In a similar way to the aligned magnets, each half of a star with a dipole field could rotate in opposite directions, until it reaches a state of stable equilibrium.
Formal proof of the instability
To prove the Flowers-Ruderman instability, the following simplifying assumptions are made:
Formal proof of Flowers-Ruderman instability
83
• The star is perfectly spherical, and all hydrodynamic quantities are functions only of r. • Magnetic flux is completely frozen into the fluid. • Outside the star there is a perfect vacuum. Under these conditions, only the energy of the magnetic field outside the star changes, and this was calculated as Z h i 1 E= (1) B2 dV = E0 1 + (A − 1) sin2 Ω , 8π r>R
where E0 is the initial energy of the dipole outside the star, Ω is the angle of rotation of each half of the star, and A is a numerical constant that results from an infinite series that could not be summed analytically. Also, a quantity Υ defined as Υ=
R3 8π
Z
4π
Br2
r=R
dΩ,
(2)
was considered. Since the field is only displaced on the surface, Υ is a constant through the entire rotation. It was shown that due to the conservation of Υ, the final energy is smaller than the initial one, which implies that the constant A in (1) is smaller than 1, and the energy decreases monotonically with the angle of rotation. Moreover, the conservation of Υ allowed us to obtain lower and upper bounds for this number, 0.5463 < A < 0.5466, but in principle this could be solved with arbitrary precision. Using perturbation theory, it was seen that all the net work done was due to surface currents induced by the perturbation. 3.
Stabilizing effect of a toroidal field
With a toroidal field, a sharp cut is not possible. However, one can take both halves of the star to rotate in opposite directions, with a thin transition region in which the direction of rotation switches continuously (see figure 3). In this case, the “bending” of toroidal field lines opposes the displacement, so for a sufficiently strong toroidal field we would expect the field to be stable to the FlowersRuderman instability mechanism. We considered particular models for the poloidal and toroidal components with variable strength, which resulted in a dipole field outside the star. The poloidal field was given by BP = ∇α × ∇φ,
#
"
35Bp 2 6 r 4 3 r6 α(r) = + sin2 θ, r − 16 5 R2 7 R4
(3)
and the toroidal field was modeled as contained in a torus, as shown in figure 4, with the strength of the field given by BT = BT cos2
ρπ ˆ φ 2µR
(4)
where the meaning of ρ and µ is illustrated in figure 4. Under these conditions, we estimated that the system is stable when the ratio of poloidal to total magnetic energy satisfies EP /ET . 0.96. This means that a very weak toroidal field is sufficient to stabilize the star.
P. Marchant et al.
84
µR
ρ ϕ
(1 − µ)R
Figure 3. Performing a smooth cut to the star due to the presence of a toroidal field.
4.
Figure 4. The toroidal field was modeled as contained in a torus of radius µR.
Conclusions
We showed that a pure dipole field is unstable to the Flowers-Ruderman instability mechanism by direcly evaluating the external energy of the magnetic field for an arbitrary angle of rotation for each half of the star, and proving that it decreases monotonously along the entire rotation. To study how a toroidal field could stabilize the star against the perturbation described by the Flowers-Ruderman instability, we considered particular models for the toroidal and poloidal components of the field. In this case, the cut had to be performed smoothly, as shown in figure 3. Under these conditions, the field was stable to the Flowers-Ruderman instability when the ratio of poloidal to total magnetic energy satisfies EP /ET . 0.96, so a very weak toroidal field is sufficient to stabilize the star. Agradecimientos. This project was supported by FONDECYT Regular Project 1060644, FONDECYT Regular Project 1110213, FONDAP Center for Astrophysics (15010003), Proyecto Basal PFB-06/2007 and Proyecto Límite VRI 2010-15. We also thank SOCHIAS for providing funding to assist to the SOCHIAS-AAA meeting at San Juan, Argentina. References Braithwaite, J. and Spruit, H.C., 2004, Nat, 431, 891 Braithwaite, J. and Spruit, H.C., 2006, A&A, 450, 1097 Flowers, E. and Ruderman, M.A., 1977, ApJ, 215, 302 Marchant, P., Reisenegger, A. and Akgün, T., 2011 MNRAS, 412, 2426
