A&A 408, 331-335 (2003)
DOI: 10.1051/0004-6361:20030917
V. Urpin
A. F. Ioffe Institute for Physics and Technology,
SU 194021, St. Petersburg, Russia
Isaak Newton Institute of Chili, Branch in St. Petersburg,
Russia
Received 3 February 2003 / Accepted 22 May 2003
Abstract
Young neutron stars formed in core collapse or originating
in the merger of a binary neutron star system may possess
significant differential rotation. We consider the stability
properties of such hot (non-superfluid) differentially rotating
neutron stars. Compared to ordinary stars, the criteria of
hydrodynamic instability of neutron stars can be different because
generally the thermal diffusivity and viscosity are of the same
order of magnitude and are relatively large. For example, the
well-known Goldreich-Schubert instability can manifest itself only
in rapidly rotating stars and is strongly suppressed if rotation is
slow. The rotational instabilities in neutron stars can grow on the
thermal or dynamical timescale.
Key words: MHD - stars: neutron - stars: rotation - stars: pulsars: general
It is generally believed that young neutron stars formed in the collapse of a massive star should be rapidly rotating due to the conservation of the angular momentum of the collapsing core. Most likely, this rotation is non-uniform because even if the progenitor rotates uniformly, collapse will generate a significant amount of differential rotation. Numerical simulations of rotating core collapse indicate that the angular velocity in a newly formed neutron star should be appreciably non-uniform (see Mönchmeyer & Müller 1989; Janka & Mönchmeyer 1989; Zwerger & Müller 1997; Rampp et al. 1998). During a convective stage that lasts 30-40 s (Miralles et al. 2000, 2002) and seems to be inevitable in protoneutron stars, the original distribution of the angular velocity may evolve to even a more complex one. During this stage, the angular momentum is basically transported by convection and neutron finger instability and can be very rapid. Perhaps, turbulent Reynolds stresses result in the angular velocity dependent on both the radial and polar coordinates like that of the Sun where the departure from a uniform rotation reaches about 30%. When convection is exhausted and the neutron star becomes transparent to neutrinos it is still rather hot and, therefore, the matter is likely to be in a normal (non-superfluid) state in the core. The non-superfluid neutrino transparent stage can last from hours to thousands of years depending on the model of cooling and superfluidity, and our consideration will concern this stage. Note that accretion of supernova remnant material may also drive differential rotation during this stage (Watts & Andersson 2002). Differential rotation can also be maintained by stellar oscillations. For instance, recent papers on r-modes argued that the modes can drive differential rotation via non-linear effects (Rezzolla et al. 2000; Levin & Ushomirsky 2001).
There are only a few possibilities that can be contemplated as mechanisms of the angular momentum transport during the neutrino-transparent non-superfluid stage: viscosity, hydrodynamic and hydromagnetic instabilities, and large scale motions caused by the magnetic field if the latter has been generated in the core by convection. Viscosity seems to be not very efficient at large scales because it can redistribute the angular momentum only on a relatively long timescale. The characteristic timescale of viscous dissipation, , can be estimated as where L is the lengthscale of the angular velocity and is the kinematic viscosity. If the star is transparent to neutrinos, viscosity is mainly determined by the scattering of neutrons and has been calculated in the classical paper by Flowers & Itoh (1976). A simple fitting formula for their numerical results has been obtained by Cutler et al. (1990). Using this fit and assuming that L is comparable to the neutron star radius, we can estimate as 10^{2}- 10^{3} yrs that probably is comparable to (or longer than) the duration of a non-superfluid stage.
If there is a sufficiently strong magnetic field in the neutron star then large-scale motions induced by the field can redistribute the angular momentum by magnetic braking (Shapiro 2000). Differential rotation twists lines of a frozen-in poloidal magnetic field and amplifies the toroidal field. This process generates Alfvén waves, which can transport the angular momentum within the star and even carry out some angular momentum from the star to the surrounding plasma. Note, however, that the efficiency of this mechanism is very sensitive to assumptions regarding the density of external plasma, . For example, a simple estimate of the magnetic braking timescale proposed by Shapiro (2000) shows that this process is more efficient than viscous damping only at even if the magnetic field is as strong as 10^{12} G, where is the density of the neutron star. For smaller , Alfvén waves inside the neutron star dissipate on a viscous timescale.
The angular momentum can also be transported by turbulent motions if differential rotation is unstable. Since a spatial dependence of the angular velocity is likely to be complex, the neutron star can be subject to hydrodynamic instabilities. The stability properties of differentially rotating neutron stars and radiative zones of ordinary stars can be essentially different because the viscous and thermal timescales are generally relatively short for small-scale disturbances in neutron stars and are generally comparable. This can change qualitatively, for example, the criterion of the well-known Goldreich-Schubert instability (Goldreich & Schubert 1967) that arises on the thermal timescale.
Hydrodynamic instabilities can be of importance for neutron stars originating in the merger of a binary neutron star system as well. In their numerical simulations, Rasio & Shapiro (1999) and Shibata & Uryu (2000) found that coalescence will form a differentially rotating remnant with the core rotating faster than the envelope. Note that differentially rotating neutron stars can support significantly more rest mass than their uniformly rotating counterparts (Baumgarte et al. 2000). Hydrodynamic instabilities in the remnant will destroy differential rotation and may lead to delayed collapse and a delayed gravitational wave burst.
In this paper, we derive the stability criteria for differentially rotating neutron stars. Our consideration assumes the neutron stars that are relatively cold to be transparent to neutrinos but are still sufficiently hot to be in a normal (non-superfluid) state. This evolutionary stage begins at the age 40 s and can last from hours to 10^{3} years depending on the model. The paper is organized as follows. In Sect. 2, we derive the dispersion equation governing the rotational modes of a differentially rotating neutron star. In Sect. 3, the conditions of instability are obtained. The growth rate of instability is calculated in Sect. 4. A discussion of the results is given in Sect. 5.
Consider the core of a neutron star rotating with the angular
velocity
,
where (s, ,
z) are
cylindrical coordinates. We consider axisymmetric short wavelength
perturbations with the space-time dependence
where
is the
wavevector,
.
Small perturbations will
be indicated by subscript 1, whilst unperturbed quantities will
have no subscript, except for indicating vector components when
necessary. In the unperturbed state, the star is in hydrostatic
equilibrium,
(1) |
We use the Boussinesq approximation since the e-folding time
of short wavelength instabilities associated with shear is typically
much longer than the period of a sound wave with the same wavelength.
Then, the linearized equations governing the behaviour of small
perturbations in the lowest order in
read
(2) |
(3) |
(4) |
The neutrino emissivity depends on details of the internal structure
but can generally be expressed in terms of
and T,
(see, e.g., Maxwell 1979). Then,
(5) |
(6) |
Q^{2} | = | ||
= | |||
= |
Equation (6) describes three low-frequency modes that can
exist in rotating non-magnetic neutron star. The sound waves are
excluded from consideration in the Boussinesq approximation.
The condition that at least one of the roots of Eq. (6) has
a positive real part (unstable mode) is equivalent to one of the
following inequalities
(7) |
Substituting the values of b_{0}, b_{1} and b_{2}, we can
rewrite the second condition (7) as
(8) |
(9) |
(10) |
Q^{2} < 0, | (11) |
(12) |
(13) |
(14) |
Consider the criteria (8) and (10) in more detail. In neutron
stars transparent to neutrinos, we have typically
.
Since
and
increase
rapidly when the wavelength decreases, neither of the conditions (8) or (10) can be satisfied for very short wavelengths. Therefore,
we consider Eqs. (8) and (10) at
(15) |
(16) |
(17) |
(18) |
(19) |
The conditions (17) and (18) depend on the direction of .
To obtain the true necessary conditions of instability, we have to
minimize these expressions as functions of .
For both the
inequalities (17) and (18), a dependence on
has the same
form and can be represented as
F(k_{s}, k_{z}) = A k_{s}^{2} - C k_{z} k_{s} + D k_{z}^{2} < 0, | (20) |
(21) |
C^{2} > 4 A D. | (22) |
(23) |
Solving the inequality (23) for
,
we
obtain that Eq. (23) is fulfilled if
(24) |
(25) |
In the case of a rapidly rotating star,
,
we have
and
,
and Eqs. (24) and (25) yield
(26) |
(27) |
(28) |
Consider the growth rate of rotational instabilities.
Since the coefficients of Eq. (6) are real there exist
three real roots or one real and two complex conjugate roots.
The number of roots with a positive real part is determined
by Routh criterium (DiStefano III et al. 1994),
which states that the number of unstable modes of a cubic
Eq. (6) is given by the number of changes of sign in the
sequence
(29) |
The roots
(i=1,2,3) of the cubic Eq. (6) can
be represented as
.
The expressions
for x_{i} are
(30) |
(31) |
q | = | ||
p | = |
Under the condition (15) (or (16)), we have from the expressions (30), (31) with the accuracy in terms
,
(32) |
(33) |
(34) |
Contrary to the classical Goldreich-Schubert instability that
arises on the thermal time scale, the rotational instabilities
in neutron stars can grow either on the thermal or dynamical
timescale. If
then the growth
rate is
(35) |
(36) |
(37) |
We considered hydrodynamic stability of differentially rotating neutron stars during the evolutionary stage when they are non-superfluid and transparent to neutrinos. Differentially rotating neutron stars can be formed either in core collapse or in the merger of binary neutron stars. If the Rayleigh stability criterion is fulfilled then the rotational instabilities can arise only if the angular velocity depends on z. However, the instabilities caused by this dependence in neutron stars turns out to be qualitatively different from those in ordinary stars. The classical Goldreich-Schubert instability that likely operates in ordinary stars can arise at any . In neutron stars, the rotational instabilities can appear only if exceeds some threshold value. This value depends on the angular velocity of a neutron star and can be small for rapidly rotating stars with or large for slowly rotating stars with . The difference to ordinary stars is caused by a particular character of kinetic processes in neutron stars where viscosity is large and generally comparable to the thermal diffusivity. Unfortunately, calculations of the Brunt-Vaisala frequency, , in young neutron stars have some problems because it depends on a poorly known equation of state. However, most likely this frequency is of the order of 10^{2}-10^{3} s^{-1}. Therefore, the critical value of the period, discriminating between the rapid unstable and slow stable rotation in neutron stars, is approximately 0.1-0.01 s.
Contrary to the classical Goldreich-Schubert instability, the rotational instabilities in neutron stars can arise either on the thermal or dynamical timescale. If the condition (37) is fulfilled then the growth time of instability is comparable to the rotation period. Otherwise, the growth rate is determined by the thermal effects, and the corresponding growth time is . For perturbations with km, the growth time is 100 days, whereas perturbations with m grow on a timescale 10 s.
In the present paper, we have addressed the behaviour only of axisymmetric perturbations. It is clear, however, that the obtained results can apply to nonaxisymmetric perturbations with the azimuthal wavelength much longer than the vertical or radial ones, . Note that, generally,non-axisymmetric perturbations can also undergo other instabilities (e.g., the r-mode instability, see Andersson et al. 1999). Interaction of the neutron star crust with turbulent motions caused by rotational instabilities may result in small irregularities of the measured periods of young pulsars.
Acknowledgements
This work has been supported by the Spanish Ministery of Science and Technology (grant AYA2001-3490-C02-02).