Issue |
A&A
Volume 519, September 2010
|
|
---|---|---|
Article Number | A16 | |
Number of page(s) | 6 | |
Section | Stellar structure and evolution | |
DOI | https://doi.org/10.1051/0004-6361/201014222 | |
Published online | 07 September 2010 |
The GSF instability and turbulence do not account for the relatively low rotation rate of pulsars
R. Hirschi1,2 - A. Maeder3
1 - Astrophysics Group, EPSAM Institute, University of Keele, Keele, ST5 5BG, UK
2 -
Institute for the Physics and Mathematics of the Universe, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa 277-8583, Japan
3 -
Geneva Observatory, Geneva University, 1290 Sauverny, Switzerland
Received 9 February 2010 / Accepted 28 April 2010
Abstract
Aims. We examine the effects of the horizontal turbulence in
differentially rotating stars on the Goldreich-Schubert-Fricke (GSF)
instability and apply our results to pre-supernova models.
Methods. We derive the expression for the GSF instability with account of the thermal transport and smoothing of the -gradient by the horizontal turbulence. We apply the new expressions in numerical models of a 20
star.
Results. We show that if
the Rayleigh-Taylor instability cannot be killed by the stabilising thermal and
-gradients,
so that the GSF instability is always there and we derive the
corresponding diffusion coefficient. The GSF instability grows toward
the very latest stages of stellar evolution. Close to the deep
convective zones in pre-supernova stages, the transport coefficient of
elements and angular momentum by the GSF instability can, in small
parts of the star, be larger than the shear instability and even as
large as the thermal diffusivity. However, the zones in which the GSF
instability is acting are extremely narrow and there is not enough time
left before the supernova explosion for a significant mixing to occur.
Thus, the GSF instability remains insignificant for the evolution even
when the inhibiting effects of the
-gradient are reduced by the horizontal turbulence.
Conclusions. We conclude that the GSF instability in
pre-supernova stages is not responsible for the relatively low rotation
rate of pulsars compared to the predictions of rotating star models.
Key words: stars: massive - stars: evolution - stars: interiors - stars: rotation - pulsars: general
1 Introduction
The comparison of the observed rotation rate of pulsars and stellar models in the pre-supernova
stages indicates that most stars are losing more angular momentum than currently predicted (Heger et al. 2000; Hirschi et al. 2004).
Normally, the conservation of the central
angular momentum of a pre-supernova model would lead to a neutron star
spinning with a period of 0.1 ms, which is about two orders of
magnitude faster than the estimate for
the most rapid pulsars at birth. The question has arisen whether some
rotational instabilities may play a role in dissipating the angular
momentum. We can think in particular of the Goldreich-Schubert-Fricke
(GSF) instability (Goldreich & Schubert 1967; Fricke 1968),
which has a negligible effect in the main-sequence phase and which may
play some role in the He-burning and more advanced phases (Heger et al. 2000), in particular when there is a very steep -gradient
at the edge of the central dense core. This instability is generally
not accounted for in stellar modelling. The aim of this article is to
examine whether the GSF instability is important in the pre-supernova
stages when the effect of the horizontal turbulence in rotating stars
is taken into account, which reduces the stabilising effects of the
-gradient.
Section 2 recalls the basic properties of the GSF instability, Sect. 3 those of the horizontal turbulence. The effects of turbulence on the GSF instability are examined in Sect. 4. Section 5 show the results of the numerical models. Section 6 gives the conclusions.
2 The GSF instability and the Solberg-Hoiland criterion
2.1 Recollection of the basics
A rotating star with a distribution of the specific angular momentum j decreasing outwards is subject to the Rayleigh-Taylor instability: an upward displaced fluid element will have a higher j
than the ambient medium and thus it will continue to move outward. In
radiative stable media, the density stratification has a stabilising
effect, which may counterbalance the instability resulting from the
outwards decrease of j. In this respect, the -gradient resulting from nuclear evolution has a strong stabilising effect.
The stability condition is usually expressed by the Solberg-Hoiland criterion, given in the first part of Eq. (1).
The GSF instability occurs when the heat diffusion by the fluid elements reduces the
stabilising effect of the entropy stratification in the radiative layers.
The account of a finite viscosity
together with thermal diffusivity K influences the instability criteria (Fricke 1968; Acheson 1978). These authors found instability for each of the two conditions
where



The viscosity


where the various quantities have their usual meaning.
- The first inequality in Eq. (1)
corresponds to the convective instability predicted by the Solberg-Hoiland criterion taking into
account the efficiency factor
, which considers the radiative losses. For
, a displaced fluid element experiences a centrifugal force stronger than in the surrounding and moves further away. The first criterion in Eq. (1) expresses that instability arises if the T gradient, which accounts for thermal and viscous diffusivity, is insufficient to compensate for the growth of the centrifugal force during an arbitrary small displacement.
- The second inequality in Eq. (1) expresses a baroclinic instability related to the
differential rotation in the direction z.
If a fluid element is displaced over a length
in the z direction so that
, the angular velocity of the fluid element is higher than the local angular velocity. The excess of centrifugal force on this element leads to a further displacement and thus to instability. It has often been concluded from this second criterion that only cylindrical rotation laws are stable (solid body rotation is a special case). This is not correct, because viscosity is never zero. In particular the horizontal turbulence produces a strong horizontal viscous coupling with a large ratio
, which does not favour the instability due to the second condition in Eq. (1).
2.2 The
gradient and the GSF instability
In the course of evolution, a
gradient develops around the convective core (there
the
gradients are also large). The
gradient produces stabilising effects. Endal & Sofia (1978) in their developments surprisingly use the same dependence
on the
-gradient as for the meridional circulation (see also Heger et al. 2000). They apply a velocity of the GSF instability in the equatorial plane given by
where U2(r) is the radial component of the velocity of meridional circulation and HT and Hj are respectively the scale heights of the distributions of T and specific angular momentum j.
Let us focus on the first criterion in Eq. (1), it becomes in this case (Knobloch & Spruit 1983; Talon 1997)






3 The coefficient of horizontal turbulence in differentially rotating stars
The importance of the horizontal turbulence in differentially rotating stars was emphasised by Zahn (1992). There are a number of observational effects supporting its existence, in particular the thinness of the solar tachocline (Spiegel & Zahn 1992), the different efficiencies of the transport of chemical elements and of angular momentum as well the observations of the Li abundances in solar type stars (Chaboyer et al. 1995a,b). In massive stars, the horizontal turbulence increases the mixing of CNO elements in a favourable way with respect to observations (Maeder 2003).
A first estimate of the coefficient
of horizontal turbulence was proposed by Zahn
(1992).
A second better estimate was based on laboratory experiments with a
Couette-Taylor cylinder. It gives in a differentially rotating medium (Richard & Zahn 1999; Mathis et al. 2004)
The latitudinal variations of the angular velocity are of the form



with


where U2 and V2 are the vertical and horizontal components of the velocity of meridional circulation and

The above diffusion coefficient (Eq. (8)) derived from laboratory experiments
is essentially the definition of the viscosity or diffusion coefficient, if the characteristic
timescale of the process is equal to
,
i.e.
![]() |
(9) |
with


![$t_{{\rm diff}} ~ \approx~ \left[ {r}/({\Omega_2 ~ V_2} ) \right] ^{1/2}$](/articles/aa/full_html/2010/11/aa14222-10/img41.png)
This expression, despite its difference with respect to Eq. (8), leads to similar numerical values for the horizontal turbulence in stellar models (Mathis et al. 2004), while the original estimate (Zahn 1992) leads to a coefficient

The expression of
requires that we know the vertical and horizontal components U2 and V2 of the velocity of meridional circulation. If not, some approximations are given in the Appendix.
4 The horizontal turbulence and the GSF instability
We examine what happens to the condition (5)
or Solberg-Hoiland criterion in case of thermal diffusivity and
horizontal turbulence. For that let us start from the Brunt-Väisälä
frequency in a rotating star at co-latitude
If it is negative, the medium is unstable.


For a fluid element moving at a velocity v over a distance





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

The GSF instability problem is 2D with two different coupled geometries: the cylindrical one
associated to the rotation with the restoring force along
and the spherical
one where the entropy and chemical stratification restoring force is along
that explains the sin
in Eq. (11), which gives the radial component of the total restoring force. The following formula for
in spherical geometry in the case
of a shellular rotation
can be obtained:
starting with Eq. (2):


The horizontal turbulence also makes some exchanges between a moving fluid element with composition given by




![]() |
(16) |
One can also write



![]() |
(17) |
to be compared to the first part of Eq. (12). If N2 < 0, the medium is unstable, thus the instability condition at the equator becomes
The situation is similar to the effect of horizontal turbulence in the case of the shear instability (Talon & Zahn 1997).
The turbulent eddies with the largest sizes
are those which give the largest contribution to the vertical transport. For these eddies, the equality in (18)
is satisfied, which gives
The diffusion coefficient by the GSF instability is

We notice several interesting properties.
- 1.
- If
, from Eq. (19) we see that the GSF instability is present in a radiative medium whatever the
- and T-gradients are. Thus, these gradients cannot kill the turbulent transport by the GSF instability. However, the size of the effects has to be determined for any given conditions.
- 2.
- If the diffusion coefficient
by the GSF instability is small with respect to K and
, we have
The assumptionsand
are likely, at least at the beginning of the GSF instability when
starts becoming negative. Nevertheless, these assumptions need to be verified for the cases of interest in the advanced stages.
- 3.
- If
, as is the case in regions surrounding stellar cores, we get from Eq. (19)
(22)
No assumption on the size ofis made here. Due to the fast central rotation,
and the
-gradient in regions close to the central core may be large, thus possibly favouring a significant
.


![]() |
Figure 1:
Properties of a 20 |
Open with DEXTER |
5 Rotating stellar models in the pre-supernova stages
In order to quantitatively examine the importance of the GSF instability, we calculated the
evolution all the way from the Main Sequence to the Si burning stage of a 20
star with
an initial rotation velocity of 150 km s-1 with a metallicity Z=0.002 typical of the
SMC composition. We
chose this composition because the internal
-gradients
are steeper at lower Z (Maeder & Meynet 2001), which would favour the GSF instability. Some data for another 20
model with an initial rotation of 300 km s-1 are also given. Equation (20) was used to determine the occurrence of the GSF instability and the value of
.
The above expression (10) for
is used.
The nuclear network in the advanced phases is the same as in previous models (Hirschi et al. 2004).
![]() |
Figure 2:
Same as for Fig. 1 during the phase of central Ne-burning. The actual mass is 19.412 |
Open with DEXTER |
Figure 1 shows in four panels the
main parameters during the first part of the phase of central
He-burning. We first notice in panel d) the building of a -gradient at the edge of the convective core with a difference of
by about a factor of 20. This makes
in most of the region between the edge of the convective core
at 2.9
and the convective H-burning shell at 5.3
as shown in panel b). However
remains negligible with respect to N2T and
.
In order to understand the reason, we need to look back at Eq. (15):
.
The value of
in the star is too small to allow a significant
value of
.
This means in fact that the centrifugal force
in the deep interior is not strong enough to overcome the stabilising
effects of N2T and
as shown in panel c).
The consequence as illustrated in panel a) is that
remains smaller than
everywhere and is thus insignificant.
We also notice that
is always much smaller
than
and K, which here permits the approximation (21) made above.
![]() |
Figure 3: Same as for Fig. 1 during the phase of central O-burning. The actual mass is no longer changing, the central O-content is X(16O) = 0.692, the central Si-content is X(28Si) = 0.153. |
Open with DEXTER |
Figure 2 shows the same plots
during the stage of central neon burning. We notice an impressive
increase of the central angular velocity and a very small in the envelope, with a difference by a factor of 108 between the two, justifying
the examination of the GSF instability. There are two ``
-walls'', the big one
at 7.2
corresponds to the basis of the H-rich envelope, the other one at 4.8
lies at the basis of the He-burning shell. The values of
become much more negative, but over areas of very limited extensions. Again, the value of
are negligible, in particular compared to the big peak of
at 4.8
.
The result is that
is always smaller than
,
even if very locally it can reach about the same value.
is always at least two or three orders of magnitude smaller than
and K, permitting here the simplification (21).
Figure 3 shows the situation
in the central O-burning stage slightly less than a year before the
central core collapse. Two other small steps in
have appeared near the centre, due to the successive ``onion skins'' of the pre-supernova model.
We notice some new facts. In line with what was already seen for neon burning,
the term
becomes negative only in extremely narrow regions where the GSF instability is acting with a diffusion coefficient
larger than in the previous evolutionary stages. Very locally at the upper and/or
lower edges of intermediate convective zones,
may even become larger than
and K reaching values above 108 cm2 s-1 (there approximation (21) is not valid!). With less than a year left before explosion,
the distance over which a significant spread may occur is about 10-3
.
This is not entirely negligible in the dense central regions, but
remains of limited importance, as shown by panels b) and d) where we notice that the
-walls remain unmodified despite the locally large
.
We may wonder whether higher initial rotation velocities lead to different results. Figure 4 shows the various panels for a similar star in the He-burning phase with an initial rotation velocity of 300 km s-1. We see that the central rotation velocity is about the same as for the previous case of lower rotation and, in this stage which determines the further evolution, there is no significant difference in the various properties.
![]() |
Figure 4:
Same as for Fig. 1 for an initial velocity
of 300 km s-1. The star is in the stage of central He-burning with
|
Open with DEXTER |
6 Conclusions
We have examined the effects of the horizontal turbulence on the GSF instability. This instability is present as soon as
is smaller than zero,
whatever the effects of the stabilising
-gradients.
On the whole, the numerical models of rotating stars show that
the diffusion coefficient by the GSF instability grows toward the very
latest stages of stellar evolution, but the zones over which it is
acting are extremely narrow and there is not enough time left before
the supernova explosion for a significant mixing to occur. Thus, even
when the inhibiting effect of the -gradient is reduced by horizontal turbulence, the GSF instability is unable to smooth the steep
-gradients and to significantly transport matter.
We conclude that the amplitude and spatial extension of the GSF instability makes it impossible to reduce the angular momentum of the stellar cores in the pre-supernova stages by two orders of magnitude. Therefore, other mechanisms such as magnetic fields (Spruit 2002; Maeder & Meynet 2004; Mathis & Zahn 2005; Zahn et al. 2007) and gravity waves (Talon & Charbonnel 2005; Mathis et al. 2008) must be investigated further.
Appendix A: some approximations for meridional circulation
The coefficient
requires the knowledge of the components
U2 and V2 of the meridional
circulation because of the horizontal turbulence.
If the solutions of the fourth order system of equations governing
meridional circulation are not available, some approximations may be
considered. We note that the same problem would occur for Eq. (4) by Endal & Sofia (1978). As shown by stellar models, the orders of magnitude of U2 and V2 are the same. The numerical models give in general
and
.
Using these orders of magnitude in Eq. (8), we get
For U2, various expressions can be used taking into account the amount of differential rotation (Maeder 2009). We can also get an order of magnitude using the approximation for a mixture of perfect gas and radiation with a local angular velocity

![]() |
||
![]() |
(A.2) |
where the various quantities have their usual meaning. Acknowledgements
We thank the referee, Dr Stéphane Mathis, for his careful reading of the manuscript and his valuable comments. R. Hirschi acknowledges support from the Marie Curie International Incoming Fellowship nb. 221145 within the 7th European Community Framework Programme and from the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.
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All Figures
![]() |
Figure 1:
Properties of a 20 |
Open with DEXTER | |
In the text |
![]() |
Figure 2:
Same as for Fig. 1 during the phase of central Ne-burning. The actual mass is 19.412 |
Open with DEXTER | |
In the text |
![]() |
Figure 3: Same as for Fig. 1 during the phase of central O-burning. The actual mass is no longer changing, the central O-content is X(16O) = 0.692, the central Si-content is X(28Si) = 0.153. |
Open with DEXTER | |
In the text |
![]() |
Figure 4:
Same as for Fig. 1 for an initial velocity
of 300 km s-1. The star is in the stage of central He-burning with
|
Open with DEXTER | |
In the text |
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