Issue |
A&A
Volume 509, January 2010
|
|
---|---|---|
Article Number | A35 | |
Number of page(s) | 6 | |
Section | Stellar structure and evolution | |
DOI | https://doi.org/10.1051/0004-6361/200912887 | |
Published online | 14 January 2010 |
Magnetorotational instability in proto-neutron stars
V. Urpin
Departament de Física Aplicada, Universitat d'Alacant,
Ap. Correus 99, 03080 Alacant, Spain
A.F. Ioffe Institute of Physics and Technology, 194021 St. Petersburg, Russia
Isaac Newton Institute of Chile, Branch in St. Petersburg,
194021 St. Petersburg, Russia
Received 14 July 2009 / Accepted 1 October 2009
Abstract
Context. Magneto-rotational instability (MRI) has been
suggested to lead to a rapid growth of the magnetic field in core
collapse supernovae and produce departures from spherical symmetry that
are important in determining the explosion mechanism.
Aims. We address the problem of stability in differentially
rotating magnetized proto-neutron stars at the beginning of their
evolution.
Methods. To do this, we consider a linear stability taking into account non-linear effects of the magnetic field and strong gravity.
Results. Criteria for MRI are derived without simplifying
assumptions about a weak magnetic field. In proto-neutron stars, these
criteria differ qualitatively from the standard condition
where
is the angular velocity and s
the cylindrical radius. If the magnetic field is strong, the MRI can
occur only in the neighbourhood of the regions where the spherical
radial component of the magnetic field vanishes. The growth rate of the
MRI is relatively low except for perturbations with very small scales
which usually are not detected in numerical simulations. We find that
MRI in proto-neutron stars grows more slowly than the double diffusive
instability analogous the Goldreich-Schubert-Fricke instability in
ordinary stars.
Key words: stars: neutron - stars: rotation - stars: magnetic field - supernovae: general - instabilities
1 Introduction
There is a growing amount of evidence that core-collapse supernovae are
asymmetric and that the core-collapse mechanism itself is responsible for
the asymmetry (see Buras et al. 2003; Akiyama et al. 2003, for more details).
Several possibilities are explored to account for this observed asymmetry.
One is associated with the influence of rotation on convection, which seems
to be inevitable during the early evolution of proto-neutron stars (PNS).
Convective motions in PNSs are very fast (
108-109 cm/s) and,
therefore, the convective turnover time is short,
1-10 ms
(see, e.g., Burrows & Lattimer 1986). Nevertheless, if angular momentum
is conserved, the collapsing core can spin up to very short periods
5-10 ms and generate strong differential rotation. Such fast rotation
modifies convection and makes convective motions anisotropic and constrained
to the polar regions (Fryer & Heger 2000; Miralles et al. 2004). This
mechanism is a natural way to create anisotropic energy and momentum
transport by convective motions, that only requires that the angular
velocity be of the order of the Brunt-Väisälä frequency.
The other possibility to create asymmetry is the effect of jets (see, e.g., Khokhlov et al. 1999; Wheeler et al. 2002). Even though the mechanism of jet formation is still unclear, it seems that MHD jets are common in systems where a central body accretes matter with angular momentum and magnetic field (see, e.g., Meier et al. 2001), and a core-collapse supernova is such a system. Calculations have established that nonrelativistic axial jets originating within the collapsed core can initiate a bipolar asymmetric supernova explosion that is consistent with observations (Hwang et al. 2000).
Another way to generate asymmetry is associated with the magnetic field that can be an important ingredient of the explosion mechanism (Bisnovatyi-Kogan 1971; Kundt 1976). The toroidal magnetic field can be amplified by differential rotation to such high values that it becomes dynamically important (Ardelyan et al. 2005). The effect of the magnetic field on asymmetry of supernovae was considered by Wheeler et al. (2000, 2002) who found that it is possible to produce both a strong toroidal field and an axial jet. Two-dimensional MHD simulations of core collapse indicate that the shape of shock waves and the neutrinosphere can be modified by the effect of the magnetic field (Kotake et al. 2004; Takiwaki et al. 2004).
The possible presence of a magnetic field and differential rotation in a
core-collapse supernova favours magnetorotational instability (MRI),
which can enhance turbulent transport and amplify the magnetic field.
This instability was considered by Akiyama et al. (2003) in the context of
core collapse. The authors argued that instability must occur
in core collapse and that it has the capacity to produce fields that are
sufficiently strong to affect, if not cause, the explosion. Thompson et al.
(2005) constructed one-dimensional models, including rotation and magnetic
fields, to study the mechanism of energy deposition. They explored several
mechanism for viscosity and argue that turbulent viscosity caused by the MRI
can be most effective. Numerical simulations provide contradictory
conclusions regarding the importance of MRI in core collapse. Moiseenko
et al. (2006) claim that MRI has been found in their 2D simulations, and
that it is responsible for a strong amplification of the poloidal magnetic
flux. However, what these authors call MRI is different to
standard MRI considered by Velikhov (1959; see also Balbus & Hawley 1991). For instance, the instability found by Moiseenko et al. (2006) starts to develop only when the ratio of the toroidal and poloidal fields reaches a
value of a few tens. On the other hand, the onset of standard MRI
in 2D does not depend on the toroidal field at all. The dependence on the
ratio of the toroidal and poloidal field is more typical for Tayler
instability (Tayler 1973), which is more relevant to the topology of the
magnetic field than differential rotation. Therefore, it is possible that
Moiseenko et al. (2006) incorrectly identify instability, and MRI
does not occur in their simulations. Apart from that, Moiseenko et al. (2006)
attributed a rapid growth of the toroidal and poloidal fields to a dynamo
driven by the magnetorotational instability. This also rise some doubts
because of Cowlings's anti-dynamo theorem which states that an
axisymmetric dynamo cannot exist (see, e.g., Shercliff 1965). Two-dimensional
simulations of core collapse with a strong magnetic field have been
performed by Sawai et al. (2005, 2008). They found that the magnetic
field can play an important role in the dynamics of the core only if the
poloidal field of the progenitor is strong enough (
1012-1013 G),
but MRI was not seen in the considered models. The magnetic field is
amplified mainly by field compression and field wrapping in these simulations.
On the contrary, Shibata et al. (2006) claim that they found MRI in
their simulations of magnetorotational core collapse in general relativity.
These authors paid attention to resolution in order to resolve unstable
MRI modes and they claim that amplification of the magnetic field in the
considered models is caused by MRI. Note, however, that the poloidal
field obtained in their models is of the order of that estimated from
conservation of magnetic flux in core collapse, and a more refined
analysis is required to determine the mechanism of amplification. Fryer &
Warren (2004) argued that it is difficult to produce magnetic fields in excess
of 1014
G even if MRI occurs in core collapse because rotation is not
sufficiently fast. A detailed study of the magnetorotational core
collapse has been performed by Obergaulinger et al. (2006a,b, 2009). The initial magnetic field was purely poloidal in their models with a strength ranging from 1010 to 1013
G. Such fields are much higher than those estimated to exist in
realistic stellar cores, but the authors wanted to investigate the
principal effects of a magnetic field. The initial magnetic field is
amplified by differential rotation in these simulations, giving rise to
astrong toroidal component. The poloidal component grows mainly by
compression
during collapse and does not change significantly after core bounce.
The authors also found that extended regions exist where the criterion
of MRI is satisfied at various epochs. However, the growth rate of this
instability is typically too small, except for a few models
with a strong initial magnetic field B=1012 G.
In this paper, we study the effect of MRI on core-collapse supernova. Since the magnetic field can be sufficiently strong, the criterion of instability is derived taking into account terms depending on the Alfven frequency. We derive the criterion that applies to any rotation profile but a special consideration is made for the case of shellular rotation that often is used to mimic rotation of proto-neutron stars. We address only the axisymmetric instability because numerical simulations of a magnetic core-collapse are usually done in 2D. The main goal of this study is to show that the effect of MRI on proto-neutron stars often is overestimated.
2 The growth rate of convective and magnetorotational instabilities
We assume that the initial PNS is restricted by the radius of neutrino sphere. The PNS has a high-entropy mantle, so that the outer part of the star is initially at a relatively large radius. It takes a few tenths of a second for the neutrinos in the high-entropy mantle to leak from the star and for the mantle to collapse to the canonical radius. In this paper, we study MRI in relatively deep layers of the PNS where the density is comparable to (or higher than) the nuclear density.
Consider a PNS rotating with angular velocity
;
(s,
,
z) are cylindrical coordinates. We explore the Boussinesq
approximation and assume that the magnetic energy is small compared to the
thermal one. In the unperturbed state, the star is in hydrostatic equilibrium,
![]() |
(1) |
where






We consider MRI assuming that the basic state is quasi-stationary. This is justified if the growth rate of the MRI (





We consider axisymmetric short-wavelength perturbations with spatial and temporal dependence
where
is the wave-vector. Small perturbations will be indicated by a
subscript 1. Then, the linearized MHD equations read in a
short-wavelength approximation
![]() |
|
![]() |
(2) |
![]() |
(3) |
![]() |
(4) |
![]() |
(5) |
where







The matter is assumed to be in chemical equilibrium, thus the density
is a function of the pressure p, temperature T and lepton fraction Y.
In the Boussinesq approximation, perturbations of the pressure are small and,
therefore,
can be expressed in terms of T1 and Y1,
![]() |
(6) |
where




where

The dispersion equation corresponding to Eqs. (2)-(8) is
![]() |
(9) |
where
![]() |
Here

![\begin{displaymath}q^{2} = (k_{z}^{2} \Omega_{\rm e}^{2} - s k_{s} k_{z}\Omega^{...
...} \cdot [ \vec{G} - \vec{k}
(\vec{k} \cdot \vec{G})/k^{2} ] ,
\end{displaymath}](/articles/aa/full_html/2010/01/aa12887-09/img51.png)
where




Equation (9) has an unstable solution if either b2 or b0 is negative, or
![]() |
(10) |
In the limit of a weak field, these conditions yield
![]() |
(11) |
The first inequality is the criterion of convection modified by rotation, and the second condition represents the criterion of MRI.
The solution of Eq. (9) is
![]() |
(12) |
If

![]() |
(13) |
The solutions



For the purpose of illustration, we plot in Fig. 1 the dependence of
on
for
the unstable magnetorotational mode. Its growth rate is given by
Eq. (12) with the upper sign. Even if stratification is
negligible, the growth rate of this mode is typically low and
in a weak magnetic field (
). Stratification can substantially decrease the growth rate and this is seen very well from
the figure. For example, the growth rate decreases approximately by a factor of two if
.
At
,
stable stratification completely suppresses the magnetorotational instability of the considered perturbations.
![]() |
Figure 1:
The dependence of |
Open with DEXTER |
In the convective zone where
,
the quantity
is also
negative and, hence,
.
Therefore, the magnetorotational
instability does not occur in convectively unstable regions, and these two
instabilities are spatially separated if the magnetic field is weak (see also
Obergaullinger et al. 2009).
The value of
is of the order of 1-10 m s-1
in collapsing cores (see Thompson et al. 2005). Therefore, rotation has an
important impact on convection only if
is of
the same order of magnitude,
rad/s (Miralles et al.
2004). This value can be reached in PNS if
rotation of the progenitor was very fast (Villain et al. 2004). On the
contrary, MRI can have an important influence even
if rotation is slower.
3 The criteria of magnetorotational instability
The condition of MRI (second inequality (10))
depends on the direction of
and can be written as follows
![]() |
(14) |
where
![]() |
|||
![]() |
In these expressions, we denote
![]() |
(15) |


Since the dependence of F on the direction of
is
simple, we can obtain that F reaches its minimum at
The value of F corresponding to these kz2/k2yields the following condition of instability
![]() |
(17) |
By taking the curl of Eq. (1), it can be readily obtained that the condition of hydrostatic equilibrium leads to
where

We consider only the magnetic field satisfying the condition

![]() |
|||
![]() |
|||
![]() |
(19) |
The two conditions of instability follow straightforwardly from the above expression:
Conditions (20) and (21) look like the Solberg-Høiland conditions
(Tassoul 2000), but with additional terms due to the chemical
composition gradients and the magnetic field. If the magnetic field is
weak and
then Eqs. (20) and (21) yield
![]() |
(22) |
The first criterion is similar to the Schwarzschild criterion for convection modified by rotation. However, the Schwarzschild criterion at B=0 involves the angular momentum gradient, whereas the criterion of magnetorotational instability depends on the angular velocity gradient, as was noted by Balbus (1995). If rotation is cylindrical with


Taking into account that
,
Eq. (21) can be transformed into
![]() |
|||
![]() |
(23) |
where
is the angle between vectors
and
.
If the Alfven frequency is small, then the criterion of MRI reads
![]() |
(24) |
and is different from the usually used condition







![]() |
(25) |
This condition depends on the direction of



4 Instability in core-collapse supernovae
The occurence of MRI is sensitive to the rotation profile. It follows
from both theoretical modelling and analytic consideration that core
collapse of a rotating progenitor results in differential rotation of a
protoneutron star (Zwerger & Müller 1997; Dimmelmeier et al. 2002;
Müller et al. 2004; Obergaulinger et al. 2006a,b). Many studies of MRI model rotation of the collapsing core by a shellular profile with
where r is the spherical radius (see, e.g., Akiyama et al. 2003; Thompson et al. 2005; Sawai et al. 2005). Such rotation can be justified if the progenitor rotates with an angular velocity that depends on r alone (Mönchmeyer & Müller 1989).
Then, if the angular momentum is conserved, it can be shown that
rotation of the collapsing core is shellular at the beginning of
evolution, at least (see, e.g., Akiyama et al. 2003). However, there arestudies that assume a cylindrical rotation for the initial profile (see, e.g., Obergaulinger et al. 2006a,b).
We assume that the angular velocity is low compared to the Keplerian
one and little departure occurs from a spherical geometry. Then,
and
are approximately radial, and we have
![]() |
(26) |
where

![]() |
Figure 2:
The dependence of the critical value of
|
Open with DEXTER |
Substituting expressions (26) into Eqs. (20), (21), we obtain
![]() |
(27) | ||
![]() |
|||
![]() |
(28) |
Condition (28) can be satisfied only if the angular velocity is higher than the Brunt-Väisälä frequency which is rather high in core-collapse supernovae. For example, according to calculations by Thompson et al. (2005), the value of



If rotation is approximately shellular as is often assumed and
,
then we have
and
where
.
Then, Eq. (28) can be transformed into
![]() |
(29) |
It turns out that stratification strongly suppresses MRI for such a rotation profile because


In Fig. 2, we plot the critical value of
that discriminates
between stable and unstable regions as a function of
for several
values of the angle
.
The region above the corresponding curve is
magnetorotationally unstable.
For any given
and negative
,
there exists a range of
where the instability can arise. If
is greater than
(or comparable to)
then the instability occurs over a rather wide
range of
.
If
(which is more typical
for PNSs) then the instability arises only in a very narrow range of
.
This dependence can be easily understood from Eq. (29). Even if
,
MRI occurs only in those regions where the first term
on the l.h.s. of Eq. (29) is small. This occurs in the neighbourhood of the
line
![]() |
(30) |
Condition (30) implies that the field line is perpendicular to the radius and, hence, at any magnetic topology, the instability occurs only near the region where Br = 0. How extended this region is depends on the relation between


There is no generally accepted point of view regarding topology of the
magnetic field in core-collapse supernovae. For illustration,
we consider the simplest configuration with
![]() |
(31) |
where f and F are functions of the spherical radius that satisfy the divergence-free condition

![]() |
||
![]() |
(32) |
Substituting these expressions into Eq. (29), we obtain
![]() |
(33) |
where






![]() |
(34) |
5 Conclusion
Interest in MRI in core collapse
supernova is due to the fact that it can generate a strong
magnetic field and produce significant departures from spherical
symmetry. These departures are crucial for the explosion mechanism. In
the present paper, we have derived criteria of MRI in proto-neutron
stars taking into account the effect of a non-axial magnetic field.
Criteria have been obtained in a form analogous to the Solberg-Høiland
criteria but including terms containing the magnetic field. It turns
out that the criterion of MRI in proto-neutron stars can differ
from the standard condition
even in a weak magnetic field due to strong gravity and the gradient of the
lepton fraction. If the Alfven frequency
is
small compared to the angular velocity then the instability occurs in
the region where the component of
perpendicular to
has a negative
projection on
whereas the component
parallel
to
is unimportant for instability. In the case of slow rotation,
when departures from sphericity are small (
), the criterion
reduces to a simple inequality
(see, e.g., Urpin 1996). For instance, shellular rotation with
which is often used in modelling proto-neutron stars does not
satisfy this condition. Therefore, MRI does not occur if the
proto-neutron star rotates shellularly and the wavelength of perturbations
is such that the Alfven frequency is smaller than
.
Only detailed
calculations of rotational core collapse can give the answer to whether the
condition of MRI is fulfilled in proto-neutron stars and, generally,
this answer should depend on rotation of the progenitor. Note that the
velocity of unstable perturbations is approximatelly perpendicular to the
radius because gravity strongly suppresses motion in the radial direction
and, likely, the radial turbulent transport should be suppressed when the
instability saturates.
However, MRI can arise in proto-neutron stars even if the neccesary
condition that determines the onset of instability in a weak field is not
fullfied. This occurs if the magnetic field is very strong or the
wavelength of perturbations is small such that
.
The latter inquality is equivalent to
![]() |
(35) |
where










It appears that the importance of MRI in core collapse can be
overappreciated since its growth rate is relatively low and reaches the value
only for perturbations with a wavelength
(see Eq. (35)). Apart from this, gravity strongly suppresses development of
any perturbations with non-radial wavevectors. Only perturbations with
can be unstable but hydrodynamic motion for such
perturbations has a small radial component and turbulent transport should
be inefficient radially. Also, it is possible that MRI occurs only in
not very extended regions of the proto-neutron star that may diminish
substantially its effect on core collapse. Note that since the wavelength
is small, one needs a very high resolution in numerical simulations
to see the most rapidly growing modes (Obergaulinger et al. 2009).
Perturbations with longer
grow substantially slower.
Our conclusion is in contrast to the results obtained by Masada et al.
(2007) who considered axisymmetric and nonaxisymmetric magnetorotational
instability of PNSs taking into account dissipative processes. These authors
also used a local approximation in the stability analysis. In the local
analysis, however, the shape of perturbations is assumed to be unchanged and,
therefore, this approach applies only if the growth rate of instability is
greater than the rate with which perturbations change their shape. In the
case of MRI, the growth rate is smaller than (or, at maximum, comparable
to)
whereas perturbations change their shape because of
differential rotation with a rate
(see, e.g.,
Balbus & Hawley 1992, for details). Therefore, the results obtained for a
nonaxisymmetric MRI in the local approximation raise some doubts. As far as
axisymmetric instability is concerned, Masada et al. (2007) were confused
when identifying different modes in the dispersion equation. In the
dissipative case, they obtained a dispersion equation of the seventh order
which describes seven different modes. Certainly, different modes can be
unstable
in different conditions, but the authors considered only one criterion
(Eq. (32) of their paper). Unfortunately, this criterion does not correspond
to MRI but describes a secular instability that is a magnetic analogy of
the well known Goldreich-Schubert-Fricke (GSF) instability. The magnetic
analogy of this instability was first considered by Urpin (2006) for the case
of ordinary stars, and Masada et al. (2007) obtained the criterion of the
same instability modified for the conditions of PNSs. In contrast to MRI,
this instability is dissipative and disappears if diffusive coefficients go
to zero. Stratification has a weak impact on this dissipative instability,
and it can arise in PNSs. Note, however, that diffisive processes can reduce
the stabilizing influence of stratification on the MRI as well.
As was
noted by Masada et al. (2007), diffusion of heat and leptons can influence
buoyancy if the characteristic diffusion timescale is shorter than the
buoyant frequency .
Since the diffusion timescale is determined
mainly by lepton diffusion which is slower than thermal diffusion, this
condition is equivalent to
where
is the
coefficient of lepton diffusivity. In deep layers with a density
1014 g/cm3, the value of
is
cm2 s-1 (see Eq. (45) by Masada et al. 2007).
Hence, diffusion begins to suppress the stabilizing effect of
stratification when the wavelength of perturbations is shorter than
cm (we suppose
s-1).
The effects considered in this paper are important for longer
wavelengths for the standard pulsar magnetic field.
As was mentioned above, MRI is not the only instability caused by
differential rotation in proto-neutron stars. The analogy of the GSF
dissipative instability can occur in both magnetic and non-magnetic PNSs.
In non-magnetic stars, this instability arises if the angular velocity depends
on the vertical coordinate z (Urpin 2007), and the shellular rotation is a
particular case of such rotation. Since neutrino transport in PNSs is much
more efficient than radiative transport in ordinary stars, the dissipative
instability can be rather fast. If
,
the condition of the instability reads
![]() |
(36) |
where











References
- Akiyama, S., Wheeler, J. C., Meier, D., & Lichtenstadt, I. 2003, ApJ, 584, 954 [NASA ADS] [CrossRef] [Google Scholar]
- Ardelyan, N., Bisnovatyi-Kogan, G., & Moiseenko, S. 2005, MNRAS, 359, 333 [NASA ADS] [CrossRef] [Google Scholar]
- Balbus, S. 1995, ApJ, 453, 380 [NASA ADS] [CrossRef] [Google Scholar]
- Balbus S., & Hawley J. 1991, ApJ, 376, 214 [NASA ADS] [CrossRef] [Google Scholar]
- Balbus S., & Hawley J. 1992, ApJ, 400, 610 [CrossRef] [Google Scholar]
- Bisnovatyi-Kogan, G. 1971, Sov. Astron., 14, 652 [NASA ADS] [Google Scholar]
- Buras, R., Rampp, M., Janka, H.-Th., & Kifonidis, K. 2003, PRL, 90, 1101 [Google Scholar]
- Burrows, A., & Lattimer, J. M. 1986, ApJ, 307, 178 [NASA ADS] [CrossRef] [Google Scholar]
- Dimmelmeier, H., Font, J. A., & Müller, E. 2002, A&A, 393, 523 [Google Scholar]
- Fryer, C. L., & Heger, A. 2000, ApJ, 541, 1033 [NASA ADS] [CrossRef] [Google Scholar]
- Fryer, C. L., & Warren M. 2004, ApJ, 601, 391 [NASA ADS] [CrossRef] [Google Scholar]
- Hwang, U., Holt, S., & Petre, R. 2000, ApJ, 537, L119 [NASA ADS] [CrossRef] [Google Scholar]
- Khokhlov, A., Höflich, P., Oran, E. S., et al. 1999, ApJ, 524, L107 [NASA ADS] [CrossRef] [Google Scholar]
- Kotake, K., Sawai, H., Yamada, S., & Sato, K. 2004, ApJ, 608, 391 [NASA ADS] [CrossRef] [Google Scholar]
- Kundt, W. 1976, Nature, 261, 673 [NASA ADS] [CrossRef] [Google Scholar]
- Masada, Y., Sano, T., & Shibata, K. 2007, ApJ, 655, 447 [NASA ADS] [CrossRef] [Google Scholar]
- Meier, D., Koide, S., & Uchida, Y. 2001, Science, 291, 84 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
- Miralles, J. A., Pons, J. A., & Urpin, V. 2000, ApJ, 543, 1001 [NASA ADS] [CrossRef] [Google Scholar]
- Miralles, J. A., Pons, J. A., & Urpin, V. 2002, ApJ, 574, 356 [NASA ADS] [CrossRef] [Google Scholar]
- Miralles, J. A., Pons, J. A., & Urpin, V. 2004, A&A, 420, 245 [Google Scholar]
- Moiseenko, S., Bisnovatyi-Kogan, G., & Ardelyan, N. 2006, MNRAS, 370, 501 [NASA ADS] [Google Scholar]
- Mönchmeyer, R., & Müller, E. 1989, in Timing Neutron Stars, ed. H. Ögelman, & E. P. J. van den Heuvel, NATO ASI Ser. C, 262 (Dordrecht: Kluwer), 549 [Google Scholar]
- Müller, E., Rampp, M., Buras, R., Janka, H.-T., & Shoemaker, D. 2004, ApJ, 603, 221 [NASA ADS] [CrossRef] [Google Scholar]
- Obergaulinger, M., Aloy, M. A., & Müller, E. 2006a, 450, 1107 [Google Scholar]
- Obergaulinger, M., Aloy, M. A., Dimmelmeier, H., & Müller, E. 2006b, 457, 209 [Google Scholar]
- Obergaulinger, M., Cerda-Duran, P., Müller, E., & Aloy, M. A. 2009, A&A, 498, 241 [Google Scholar]
- Sawai, H., Kotake, K., & Yamada, S. 2005, ApJ, 631, 446 [NASA ADS] [CrossRef] [Google Scholar]
- Sawai, H., Kotake, K., & Yamada, S. 2008, ApJ, 672, 465 [NASA ADS] [CrossRef] [Google Scholar]
- Shercliff, J. A. 1965, A textbook on magnetohydrodynamics (Oxford: Pergamon) [Google Scholar]
- Shibata, M., Liu, Y. T., Shapiro, S., & Stephen, B. 2006, PRD, 74, 4026 [Google Scholar]
- Takiwaki, T., Kotake, K., Nagataki, S., & Sato, K. 2004, ApJ, 616, 1086 [NASA ADS] [CrossRef] [Google Scholar]
- Tassoul, J.-L., 2000, In Stellar Rotation (Cambridge: Cambridge Univ. Press) [Google Scholar]
- Tayler, R. 1973, MNRAS, 161, 365 [NASA ADS] [CrossRef] [Google Scholar]
- Thompson, T., Quataert, E., & Burrows, A. 2005, ApJ, 620, 861 [NASA ADS] [CrossRef] [Google Scholar]
- Urpin, V. 1996, MNRAS, 280, 149 [NASA ADS] [CrossRef] [Google Scholar]
- Urpin, V. 2006, A&A, 447, 285 [Google Scholar]
- Urpin, V. 2007, A&A, 469, 639 [Google Scholar]
- Urpin, V., Chanmugam, G., & Sang, Y. 1994, ApJ, 433, 780 [NASA ADS] [CrossRef] [Google Scholar]
- Velikhov E. 1959, Sov. Phys. JETP, 9, 995 [Google Scholar]
- Villain, L., Pons, J., Cerdá-Durán, P., & Gourgoulhon, E. 2004, A&A, 418, 283 [Google Scholar]
- Wheeler, J. C., Meier, D. L., & Wilson, J. R. 2002, ApJ, 568, 807 [NASA ADS] [CrossRef] [Google Scholar]
- Wheeler, J. C., Yi, I., Höflich, P., & Wang, L. 2000, ApJ, 537, 810 [NASA ADS] [CrossRef] [Google Scholar]
- Zwerger, T., & Müller, E. 1997, A&A, 320, 209 [Google Scholar]
All Figures
![]() |
Figure 1:
The dependence of |
Open with DEXTER | |
In the text |
![]() |
Figure 2:
The dependence of the critical value of
|
Open with DEXTER | |
In the text |
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