Issue 
A&A
Volume 496, Number 2, March III 2009



Page(s)  317  332  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/20079095  
Published online  30 January 2009 
Luminosity of a quark star undergoing torsional oscillations and the problem of ray bursts
J. Heyvaerts^{3}  S. Bonazzola^{1}  M. Bejger^{1,2}  P. Haensel^{2}
1  LUTh, UMR 8102 du CNRS, Pl. Jules Janssen, 92195
Meudon, France
2 
N. Copernicus Astronomical Center, Polish
Academy of Sciences, Bartycka 18, 00716 Warszawa,
Poland
3 
Observatoire Astronomique de Strasbourg,
11 rue de l'Université, 67000 Strasbourg, France
Received 18 November 2007 / Accepted 2 December 2008
Abstract
Aims. We discuss whether the windingup of the magnetic field by differential rotation in a newborn quark star can produce a sufficientlyhigh, energy, emission rate of sufficiently long duration to explain long gammaray bursts.
Methods. In the context of magnetohydrodynamics, we study the torsional oscillations and energy extraction from a newborn, hot, differentiallyrotating quark star.
Results. The newborn compact star is a rapid rotator that produces a relativistic, leptonic wind. The star's torsional oscillation modulates this wind emission considerably when it is odd and of sufficient amplitude, which is relatively easy to reach. Odd oscillations may occur just after the formation of a quark star. Other asymmetries can cause similar effects. The buoyancy of woundup magnetic fields is inhibited, or its effects are limited, by a variety of different mechanisms. Direct electromagnetic emission by the torsional oscillation in either an outside vacuum or the leptonic wind surrounding the compact object is found to be insignificant. In contrast, the twist given to the outer magnetic field by an odd torsional oscillation is generally sufficient to open the star's magnetosphere. The Poynting emission of the star in its leptonic environment is then radiated from all of its surface and is enhanced considerably during these open episodes, tapping at the bulk rotational energy of the star. This results in intense energy shedding in the first tens of minutes after the collapse of magnetized quark stars with an initial poloidal field of order of 10^{14} Gauss, sufficient to explain long gammaray bursts.
Key words: gamma rays: bursts  elementary particles  stars: general  magnetic fields  magnetohydrodynamics (MHD)  pulsars: general
1 Introduction and motivation
Understanding the physical nature of ray bursts (GRBs) in a way that is consistent with observations of their entire evolution remains a challenging mystery. A vast literature on the subject exists and a large number of models have been proposed to explain this phenomenon (for a review see e.g., Zhang & Mészáros 2004; and Mészáros 2006). In this paper, we limit ourselves to the case of long GRBs, of duration of between about 10 s and 1000 s. There are several basic facts to be explained. First, the release of about 10 ^{49}10^{51} erg in rays, of mean power higher than 10^{48} erg s^{1}. The violent energy outflow, which eventually transforms into a GRB, must originate in a compact volume of linear size , because of the observed millisecond variability of the GRBs. To achieve the observed bulk Lorentz factor , an energy outflow of 10^{51} erg should have a restmass load of only 10 . In other words, the baryon wind associated with long GRBs is only 10 s^{1}. Therefore, within the inner engine of GRBs the separation of light from the matter is realized, producing the most luminous electromagnetic explosions in the Universe.
Quark stars are hypothetical stars that consist of deconfined quarks (the structure of these stars was studied in detail by Haensel et al. 1986; and Alcock et al. 1986a). They are presumably born in special supernovae, such as SNIc, from the collapse of very massive Wolf Rayet stars (Paczynski & Haensel 2005). The quark star itself forms in a second collapse a few minutes after the newborn protoneutron star simultaneously deleptonizes and spins up. Alternatively, a quark star could result from the collapse of accreting neutron stars in Xray binaries (Cheng & Dai 1996). The collapse of stars of an initial mass less than 30 is expected to result in the formation of a compact object rather than a black hole (Fryer & Kalogera 2001). Some authors claim that this limit is in fact 50 (Gaensler et al. 2005), which implies that a large fraction of the progenitors of SNIc's should eventually evolve into compact stars. Among those, a possibly nonnegligible fraction could be quark stars (Paczynski & Haensel 2005).
Bare quark stars differ from the normal, nucleonic, neutron stars in that the surface of a bare quark star (a strange star) plays the role of a membrane from which only leptons and photons can escape. This property was noted already in the early papers about quark stars (Alcock et al. 1986a; Haensel et al. 1991). The quark star surface then effectively separates baryonic from both leptonic matter and radiation. The exterior of a quark star is free of baryons, since the latter would be accreted onto the star and converted into deconfined quarks, resulting in a release of energy. The close environment of a newlyborn quark star is therefore expected to be baryonfree.
If quark stars do indeed exist, they would be prime candidates for emitting relativistic winds with small baryonic pollution. The importance of small baryonic pollution in the context of GRB fireball models was already emphasized by Haensel et al. (1991), when discussing the collision of quark stars as an inner engine of short gammaray bursts. Models of the emission of gamma radiation in a GRB (Dar & De Rújula 2004; Zhang & Mészáros 2004; De Rújula 1987) require that the central engine emits a bulk relativistic outflow with a Lorentz factor ranging from 100 to 1000 (Dar 2006; Mészáros 2006). Such a high Lorentz factor can only be reached if the baryon content of the outflow is very low. For example Paczynski (1990) has shown that a radiationdriven wind can reach a Lorentz factor of 100 only if the luminosity injected in the wind exceeds by a factor 10^{2} the rest mass energy blown away per second. Bucciantini et al. (2006) indicated (using the results of 1D calculation of relativistic winds by Michel 1969) that low baryonic pollution is necessary to obtain high Lorentz factors in a centrifugallydriven magnetized wind. Dessart et al. (2007) reached the same conclusion. For this reason, the quark star model is preferable because the low baryonic pollution of a quark star's environment ensures that energy can be deposited cleanly outside the star in the form of accelerated electronpositron pairs and ray radiation without unnecessarily accelerating baryons. This is why we strongly favour a quark star model and consider that the central engine of a long GRB may be such a star. The fact that quark stars could be the source of GRBs was first suggested by Alcock et al. (1986b). Our calculations are however not specific to a quark star, except in Sects. 3.1 and 3.2. They would also apply to a strongly magnetized neutron star.
The way in which the relativistic wind energy from the quark star is eventually released as gamma ray radiation is modeldependent and could be due to a range of environments at a distance far larger than the lightcylinder radius from the star. Low baryonic pollution is needed only within a few lightcylinder radii from the central engine, a few thousand km, where the magnetized relativistic wind is expected to be accelerated to the required high Lorentz factors. Since the collapse from protoneutron star to quark star is slightly delayed, no thick envelope is expected to be present in the rather small region where the leptonic wind is accelerated.
Newly formed quark stars could be at the origin of GRBs because of the sudden transformation of hadronic matter into deconfined quarkmatter when a neutron star or a protoneutron star collapses to a quark star. Different ways in which such a collapse could be triggered have been described by a number of authors (Wang et al. 2000; Berezhiani et al. 2002; Ouyed & Sannino 2002; Berezhiani et al. 2003; Lugones et al. 2002; Cheng & Dai 1996; Dai & Lu 1998a; Haensel & Zdunik 2007; Drago et al. 2007,2004b,2006; Ouyed et al. 2002; Paczynski & Haensel 2005; Drago et al. 2004a; Bombaci & Datta 2000; Cheng & Dai 1998a,b).
Another promising subclass of models for a GRB central engine is based on the existence of a rapidly (millisecond period) and differentially rotating compact star endowed with a strong (10^{15} to 10^{17} Gauss) surface magnetic field, or spontaneously developing such a field from differential rotation. This millisecond magnetar model of the GRB central engine was first proposed by Usov (1992), who suggested that the GRB energy release derived its origin from the pulsar activity of a millisecondperiod compact object with a dipole field of 10^{15} Gauss. The idea of a millisecond magnetar has been revisited and discussed by many authors (Zhang & Mészáros 2002; Thompson 1994; Zhang & Mészáros 2001). Following Kluzniak & Ruderman (1998), a number of authors proposed differential rotation as a mechanism for strengthening the toroidal magnetic field in the interior of a newlyborn neutron star (Ruderman et al. 2000) or in an accreting neutron star that develops differential rotation as a result of an rmode instability (Spruit 1999). Amplified magnetic fields, of the order of 10^{17} Gauss, would then be brought to the star's surface by buoyancy forces where the energy would be emitted in the form of bursts and Poynting flux. Ruderman et al. (2000) suggested that the emergence of strong fields would generate episodic, pulsar activity from the open magnetosphere of the object. Dai & Lu (1998b) considered this model for quark stars. However, it would be interesting to identify mechanisms that could generate the GRB phenomenon in objects with fields weaker than 10^{17} Gauss. In this paper, we propose such a mechanism.
A newlyborn, compact star is entirely fluid. It supports differential rotation and internal oscillations of large amplitude that act on its internal magnetic field. In particular, differential rotation in the star would create a toroidal field from the poloidal component. This process is often referred to as an process, or windingup of the magnetic field. The woundup field and the differential rotation velocity constitute an energy reservoir into which electromagnetic emission could tap. More generally, the star's internal motions may have some effect on the Poynting power radiated. The main problem of the quarkmagnetar model is to explain how the existence of the woundup magnetic field could affect the star's emission, either by direct extraction of the associated energy or by any other means.
Our aim in this paper is to discuss the efficiency of the various ways in which a differentiallyrotating, magnetized, compact star could shed energy in its environment.
We restrict our attention to the case of aligned rotators in which the magnetic dipole axis is parallel to the rotation axis. The magneticfield distribution in the star, although it may be locally structured, is regarded as smoothly distributed on a larger scale.
We first show that field windingup by differential rotation is not a steady, but oscillating process (Sect. 2). When much of the available kinetic energy of the differential rotation has been transformed into toroidal magnetic energy, the magnetic tension forces react back and reverse the motion. By means of this mechanism, any initial differential rotation develops into a torsional, standing, Alfvén wave in the star. In the absence of losses, the star would oscillate, in the rest frame accompanying the average rotation, like a torsion pendulum (see for instance, Bonazzola et al. 2007). In Sect. 2, we determine the amplitude of the wave and in Sect. 3.1 we discuss its damping in a quark star as a result of second viscosity. This damping is found to be small.
We then discuss various mechanisms by which the energy of the torsional oscillation could emerge from the star. Buoyancy is a possibility. It may however be quenched if, while the buoyant matter moves, some of the weak reactions between quarks remain frozen, which occurs if the matter is in a coloursuperconducting state with a large enough gap, and the strange quark is sufficiently massive (Sect. 3). Alternatively, buoyancy may be inhibited by magnetic stratification or, if it develops, it could only redistribute flux in the star if the latter is magnetized in bulk.
Since the internal magnetic field is timevariable, it could conceivably act as an antenna and radiate a largeamplitude, electromagnetic wave in an external vacuum. We calculate this radiation in (Sect. 4.1) and find that the emitted power is insufficient to produce a GRB. Radiation in a leptonic medium surrounding the star is shown to be equally inefficient (Sect. 4.2).
We next consider the modulation by differential rotation of the DC Poynting emission of the fast, aligned magnetic rotator (Sect. 5). We find that an even oscillation, in which the southern hemisphere oscillates in phase with respect to the northern hemisphere and the magnetic structure has a dipolartype of symmetry causes a negligible modulation of the energy output. However, an odd torsional oscillation, in which the southern hemisphere oscillates in phase opposition with respect to the northern hemisphere, easily causes the star's magnetosphere to be blown open in a timedependent way. An even oscillation acting on a magnetospheric structure that would not be strictly symmetrical with respect to the equator has a similar effect. These openings of the magnetosphere cause a modulation of the power emitted in the relativistic wind blown by the fast rotator that is large enough to meet the requirements of energy and time scale necessary to explain the GRB phenomenon, even for moderately magnetized stars, with a field of order of a few 10^{14} Gauss (Sects. 5.3, 5.4). A collapse that is strictly symmetrical with respect to the equator would not excite odd oscillations. However, the existence of a kick received by neutron stars at their birth indicates that a supernova collapse is in reality not strictly symmetrical. We show that even a weak, odd oscillation is sufficient to open the star's magnetosphere during several tens of minutes after the collapse.
We use a Gaussian CGS system of units throughout the paper and spherical polar coordinates based on the rotation axis, r, , and , where is the colatitude. The corresponding unit vectors are , , and .
2 Torsional oscillation in the star
Differential rotation necessarily induces, in an highly magnetized and conducting star, a torsional oscillation. Such Alfvénic oscillations in fluid, magnetized, compact stars were studied by, e. g., Bastrukov & Podgainy (1996), Rincon & Rieutord (2003), and Bonazzola et al. (2007). We assume that inside the star the poloidal part of the magnetic field is timeindependent. This is reasonable because the strange matter is but weakly compressible. We also assume that perfect MHD is valid.
2.1 Magnetic diffusion time scale
Perfect MHD is a good approximation when the magnetic
diffusion time
is longer than the timescale of the considered phenomenon.
The time
depends on the electrical conductivity
and
the gradient lengthscale of the field which we assume to equal the star's radius R,
such that
.
The electric conductivity is the sum of the
electronic and quark conductivity,
and
.
The electron fraction in quark matter depends considerably on the physical conditions.
The quark conductivity was calculated for normal quarks by Heiselberg & Pethick (1993).
Dynamical screening of transverse interactions by the Landau damping of
the exchanged gluons is important in determining quark mobility.
The result can be expressed, for normal, massless quarks and a strong coupling
constant supposedly equal to 0.1, as:
where is the baryonic number density, n_{0} the nuclear density (0.17 fm^{3}), and T_{10} = T/10^{10} K. If quarks are coloursuperconducting in a colourflavour locked (CFL) state, charge neutrality of the quark component alone is enforced because pairing is maximized when n_{d}=n_{u}=n_{s}. This happens when the mass of the strange quark is insufficiently large compared to the gap . In the absence of electrons, the CFL coloursuperconductor is an electric insulator (Alford et al. 2007). Electron suppression results in a difference between the chemical potentials of s and d quarks, which must remain limited. Electrons are suppressed (Rajagopal & Wilczek 2001) only if:
where is a mean quark chemical potential. When the mass of the strange quark is so large that the inequality (2) is not satisfied, electrons remain present in the coloursuperconducting quark matter and this matter is then a conductor with a conductivity which may be far larger than Eq. (1). If quarks are in a twoflavour coloursuperconductor (2SC) state, only d and u quarks of two colours are paired, and electrons are present in quark matter. Then differs only by numerical factors from Eq. (1) and does not vanish. The magnetic diffusion timescale associated with the scale length R and the conductivity (1) is about s for and T_{10}=1. This is a lower bound to the true magnetic diffusion timescale, which is longer than this value when the electronic conductivity does not vanish. This time is long enough for perfect MHD to be valid on the timescale of a burst. The exception to this general conclusion is when quarks are in a colour CFLsuperconductor state in most of the star, and the s quark is so light that inequality (2) holds true, in which case quark matter is an insulator. The star's interior is then electrically inactive and none of the phenomena described in Sects. 2.2 and 5.3 occurs. In this paper, we assume that the star's interior is a good electrical conductor, which implies that for the electrical conductivities implied by Eq. (1) perfect MHD holds true.
2.2 Period and amplitude of the torsional oscillation
In perfect MHD, the evolution equations for the velocity
and for the magnetic field
inside the
star can be written, in a Galilean rest frame, as:
where , and U are the baryonic mass density, the enthalpy (the fluid is supposed to be barotropic), and the gravitational potential, respectively. The equations for the toroidal components of and can be written in the form:
We note that for rigid rotation, i.e. for with constant , the r.h.s. of Eq. (6) vanishes, so that no windingup of the field in this case occurs. Windingup results from differential rotation with depth or latitude, or both. The ProudmanTaylor theorem does not apply when nonpotential forces, such as the magnetictension force, are exerted onto the fluid and the motions are nonstationary. It is however possible that when nonpotential forces are small compared to pressure forces and the timescale of the internal motions is long compared to the star rotation period, these motions organize themselves such that becomes a function of the axial distance only. In that case, the field winding is only possible when the poloidal magnetic field has a component perpendicular to the rotation axis. The buildup of the toroidal field causes a reaction magnetictension force (the terms on the right hand side of Eq. (5)) to grow. A torsional oscillation then develops. The system of Eqs. (5) and (6) for the azimuthal velocity and field inside the star has been solved numerically with appropriate boundary conditions (Bonazzola et al. 2007).
We define R to be the star's radius and
R_{10} = R/(10 km).
The period
of the torsional oscillation is esssentially
that of an Alfvén wave with a node at both poles, that is, with a wavelength in a poloidal field
Gauss.
We adopt
Gauss as a reference value since
fields of order of between
and 10^{14} Gauss
are commonplace in isolated neutron stars (Haberl 2007) and fields
of several 10^{14} Gauss are reported to be
typical of anomalous Xray pulsars and soft gammaray bursters (Ziolkowski 2002).
We assume that the field
is initially rooted deep inside the star.
This view is supported by simulations of collapse (Obergaulinger et al. 2006a), which indicate that the
magnetic field after collapse is concentrated in the inner core.
The density of the medium inside the star is of the
order of the mean star density
,
M being the mass of the star and
its volume.
The period
of the torsional oscillation is
that of an Alfvén wave of wavelength
in a medium of
density :
This result may also be obtained from a linearization of Eqs. (5), (6). The magnetic field, instead of pervading all of the star, could conceivably be present only in some superficial layer where the density is , although the aforementioned simulations do not support this. In this case, would be smaller than given by the expression in Eq. (7) by a factor . The surface density of a quark star is , where is the bag constant. For MeV fm^{3}, this surface density is g cm^{3}, which is about 0.6 times the average density . For comparable poloidal fields, the period of a superficial torsional wave is only slightly less than the period given by Eq. (7) of a global torsional wave.
The wave amplitude is set by the amount of energy initially stored in the collapse as kinetic energy of the differential rotation. We assume to be a fraction of the order of a few percent of the star's total rotational energy, W_{*}. Simulations by Obergaulinger et al. (2006b,a) indicated that does not exceed 10% after the collapse to a neutron star. Burrows et al. (2007) found that the value of is less constrained. We adopt as a representative upper limit. We define I_{*} to be the moment of inertia of the star and I_{45} = I_{*}/(10^{45} g cm^{2}), P_{*} to be the star's average rotation period, and P_{*}(ms) its value in milliseconds. Obergaulinger et al. (2006b,a) found that when the collapsed core reaches quasiequilibrium, P_{*} most often ranges in value between 5 and 40 millisec. Burrows et al. (2007) found that 2 millisec is a lower bound. Since the quark star forms from the hot neutron star after a second collapse, its rotation accelerates by a factor 1.5, according to the moments of inertia calculated by Bejger & Haensel (2002). Thus, P_{*} = 3 ms would be a representative value of its spin period. The rotational energy of the star being , the kinetic energy available from the differential rotation is . This is the energy of the torsional oscillation, if it is global. For , P_{*}= 3 ms, and I_{45} = 1, erg.
The oscillating toroidal field in the star
is at its maximum amplitude
when the energy of the torsional oscillation
is entirely in magnetic form, which implies that, for a global oscillation
of a star of volume V,
.
Thus:
For P_{*}= 3 ms, , and the other parameters equal to unity, Gauss. Thus, for a poloidal magnetic field of order of a few 10^{14} Gauss, the torsional Alfvénic oscillation is nonlinear. Equation (8) indicates the typical amplitude of the toroidal field inside the star. The field distribution has however a certain profile with depth and latitude, so that Eq. (8) is not a precise estimate of the subsurface toroidal field.
The matter's angular velocity in the star
is
,
where
varies
with position and time and has
a null time average value. Indicating time averaging by brackets, we have
(9) 
From , it is found that the rms value of the timevarying angular velocity equals and the velocity amplitude of the torsional wave close to the surface is about a factor of times the rotational velocity:
When magnetic flux is present only in a superficial layer of mass m, only that part of the kinetic energy of differential rotation that develops in this layer feeds the energy of the torsional oscillation. If this energy is distributed proportionally to mass, the estimate given by Eq. (10) of the oscillation's amplitude remains valid. The radial component of the current in the star is then:
(11) 
If this component of the current does not vanish, a DC current could flow from the star to the magnetospheric lepton plasma. In Sect. 5.3, we consider the consequences of these currents.
3 Energyextraction mechanisms
The energy of the torsional oscillation could leak out of the star by a number of different mechanisms. We consider these mechanisms in turn and discuss whether they could represent the origin of long GRBs.
3.1 Viscous damping of the torsional oscillation
The oscillation could be damped in the star by viscous friction or Ohmic dissipation and then escape by means of heat conduction and thermal radiation from the surface. The surface of a quark star at temperatures K is an efficient source of photons and e^{+}e^{} pairs (see e.g. Aksenov et al. (2003) and references therein). However, Ohmic dissipation is negligible under the conditions assumed in Sect. 2.1. The shear viscosity of normal quark matter has been calculated by Heiselberg & Pethick (1993). The viscosity of quark matter with unpaired components is of a comparable order of magnitude. The Reynolds number for a scale R and a velocity given by Eq. (10) is found to be of order 10^{14}. Thus, shearviscous dissipation is negligible.
Even superfluid quarkmatter
suffers bulk viscosity (Madsen 1992).
An Alfvén wave, being noncompressive,
is not damped by bulk viscosity at the linear approximation, but
the Alfvénic torsional oscillation is nonlinear.
By nonresonant coupling, its magneticpressure gradients generate
a compressive oscillation.
Bulk viscosity acting on this
compressive part of the nonlinear oscillation causes a damping which can
be calculated by solving the MHD equations perturbatively to second order.
We simplified the calculation of this damping by considering
an homogeneous medium of mass density contained in a Cartesian box with unperturbed density
and magnetic
field
and an extension
in the zdirection.
The solution to first order is the noncompressive standing Alfvén wave.
We define
and
to be the Alfvén and the sound
speed in the unperturbed medium respectively,
the toroidal magnetic amplitude of the wave and
the coefficient of bulk viscosity. The
secondorder solution brings in the following damping time:
The bulk viscosity in quarkmatter depends on the finite time required by quarks to return to the weakinteraction equilibrium after the flavour equilibrium is disturbed by the leptonless strangenesschanging reaction
Any compression of the medium causes such a disturbance, because the s quark is more massive than the u and d quarks. For coloursuperconducting quark matter, the existence of a gap drastically reduces the reaction rate when . The gap energy , which is uncertain, is between 1 and 50 MeV (Madsen 2000). According to Madsen (1992), the bulk viscosity coefficient of quark matter experiencing an harmonic density perturbation of pulsation is:
where T is the temperature of the medium. Equation (14) is valid only when the Fermi energies of the s and dquarks differ by less than . The coefficients and are:
In Eqs. (15), (16), h is the Planck's constant, c the speed of light, the Boltzmann's constant, the mass of the strange quark, and the Fermi energy of the d quarks, which in a 1 quark star of radius 10 km is 196 MeV. The mass is expressed in units of 100 MeV by
The rate of the reactions represented by Eq. (13) depends on the weakcoupling parameter (Madsen 1992):
(18) 
The time needed to reestablish the equality of the chemical potentials of the s and d quarks after a perturbation (the strangeness equilibration time) is . Writing T = T_{9} 10^{9} K, the strangeness equilibration time is:
This is far shorter than the period of the torsional wave, which means that s and d quarks always remain close to the equilibrium of the reaction given by Eq. (13) when the quarks are in a normal state. For these representative numbers, remains small. This legitimates Eq. (14), in which can also be neglected so that:
(20) 
If the quarks are in a coloursuperconducting state with a gap , is reduced by (Madsen 2000). For MeV and K, the relaxation time of reactions (13) remains less than the period of the torsional wave. For lower temperatures, it rapidly becomes much longer. For example, at 10^{9} K with a gap of 1 MeV, this time is s, which is so long compared to the period of the wave that bulk viscosity is quenched. The damping time in a 10^{14} Gauss poloidal field with a toroidal magnetic field amplitude of 10^{16} Gauss is given by Eq. (12). For normal quarks this time is:
(21) 
This is much longer than the torsional wave period, owing to the fact that the compression in this wave is small, so that bulkviscous damping is negligible.
3.2 Flux emergence by magnetic buoyancy
The internal magnetic field can emerge through the surface of the star and expand into the quasivacuum outside as an electromagnetic signal. This point of view is adopted in the models of a number of authors such as Kluzniak & Ruderman (1998), Ruderman et al. (2000), Spruit (1999), and Dai & Lu (1998b): amplified magnetic fields, supposedly of order 10^{17} Gauss, would be brought to the surface of a neutron star by buoyancy forces and the energy will be emitted in the form of bursts and Poynting flux.
We estimate the flux emergence time for a given magnetic field, assuming that nothing
opposes buoyancy.
Since the field in the woundup flux tubes is
essentially toroidal, these tubes may be regarded as thin circular annuli centred
on the axis. The rapid rotation of the star inhibits their expansion or contraction
perpendicular to the axis, so that the flux tubes move parallel to it towards the closest pole. Their
length l remains constant.
Consider a
flux tube of a small cross section S and length l, threaded by a field B.
Under perfect MHD conditions,
it conserves its magnetic flux and its baryonic content.
For subsonic motions, it also remains
in pressure equilibrium with its environment.
We define
to be the material pressure,
to be the magnetic field in this environment
at an altitude z along the polar axis, and
to be the material pressure
and
the magnetic field in the tube when it reaches the altitude z.
Total pressure equilibrium implies that:
The pressure P_{c} in the inner regions of a quark star is about 10^{35} erg cm^{3}. Since the magnetic pressure of a field of 10^{16} Gauss is far lower, the difference between the matter densities and in the tube and in its environment can be calculated perturbatively:
(23) 
where we have used dP/d . The difference in magneticenergy density should be added to derive the difference in total energy density. Denoting by (g_{z}) the component of gravity parallel to the rotation axis, the vertical motion of the flux tube is described by the equation
The existence of the buoyancy instability depends on the distribution of the magnetic field in the star. If the field intensity increases with altitude such that is always less than at a higher level, the distribution of flux is stable. By contrast, if the flux tube moves in an unmagnetized environment, buoyancy can only be inhibited by density differences between the magnetized and nonmagnetized medium. These differences may result from a situation of nonequilibrium of weak interactions in the moving fluid, as described below. For , if we assume that remains almost constant during the motion and that and , we find from Eq. (24) that the buoyancy time needed to raise the tube by about a stellar radius is:
This is the time required to bring an isolated flux tube to the surface when the toroidal field has reached the value . After the sudden formation of the quark star, the differential rotation causes this toroidal field to increase in strength as , where is the maximum toroidal field developed in the torsional oscillation represented by Eq. (8). Buoyancy starts to be effective only when the growing toroidal field has reached a value such that its buoyancy time in Eq. (25) calculated for B(t), has become shorter than the age t of the newborn quark star. For standard stellar parameters and , P_{*}= 3 ms and , this occurs at the buoyancy starting time, s.
The ratio of the buoyancy time
(Eq. (25)) to the
period of the torsional oscillation
(Eq. (7)) is:
where is the poloidal field and is the total field. When the woundup magnetic field strength is close to its maximum value (Eq. (8)), . This implies that when nothing opposes buoyancy, woundup fields in isolated flux tubes float to the star's surface in a time shorter than the period of the oscillation.
However, buoyancy motions are reduced or quenched when the ascending
magnetized matter becomes denser, at the same total pressure,
than matter in its neighbourhood. This may happen
when the magnetic field pervades the entire volume of the star
and the field intensity increases with the altitude z.
Another effect opposing buoyancy is when
reactions such as Eq. (13) or the decay reactions
cannot reach equilibrium in the buoyancy time . The relaxation time of the strangenesschanging reaction (13) in normal quark matter is about T_{10}^{2} s (Sect. 3.1). For coloursuperconducting quarkmatter with a gap energy , the time to achieve equilibrium of the same reactions is lengthened by a factor (Madsen 2000). It may become longer than both and if the gap is sufficiently large. Similarly, the relaxation time of the quark decay reactions in Eq. (27) is, for normal quarks, T_{10}^{4} s (Iwamoto 1983). For coloursuperconducting quarkmatter, this time is lengthened by a factor .
The temperature is the parameter controlling whether chemical equilibrium of the reactions in Eqs. (13) and (27) can be achieved on a given timescale. When a protoneutron star collapses into a quark star, an energy of about 10^{53} erg is released, which is reflected in the initial temperature of the newborn object of K. The star is then opaque to neutrinos (Steiner et al. 2001). It cools by emitting thermal and 's from a neutrinosphere, thermal photons of frequency higher than the plasma frequency and lepton pairs. According to Usov (2001), thermal, photon emission dominates over lepton emission at K. At 10^{11} K, the photon emissivity is barely smaller than that of the blackbody. Adding neutrino and antineutrino thermal emission, the net effective emissivity at this temperature is a little less than a factor of two higher than the blackbody emissivity. The star then cools to about K in 0.5 s.
Does the nonequilibrium of the strangenesschanging reactions or the reactions
suppress the ascent of magnetized matter to the surface?
If quarks are in a coloursuperconducting state with a gap ,
the matter in the buoyant tube
retains its original strangeness during its ascent if
,
where
is the
relaxation time of the reactions in Eq. (13)
in normal, quark matter (Eq. (19)). This condition is satisfied when:
The time is definitely longer than at K, the temperature when buoyancy starts, meaning that in the absence of a gap, the reactions in Eq. (13) remain in equilibrium. At this temperature and for B_{16} = 1, the inequality of Eq. (28) holds true when is larger than about 14 MeV, which is plausible because the gap could be as high as 50 MeV (Madsen 2000). For a gap larger than 14 MeV, the reactions in Eq. (13) will remain out of equilibrium during the buoyancy motion. Similarly, the decay reactions in Eq. (27) remain frozen during buoyancy motions if . This inequality is satisfied without the need for a gap when T_{10} < B_{16}^{ 0.25}, which is not quite the case for K and B_{16} =1; this implies that, at this temperature and in the absence of a gap, decay reactions remain more or less in equilibrium during buoyancy. In the presence of a gap, the inequality is satisfied if the gap is such that:
At K and for B_{16} = 1, the inequality (29) holds true when is larger than 12 MeV. For this or a larger gap, the reactions in Eq. (27) keep out of equilibrium during the buoyancy motion. It is surprizing that the gap values which freeze the reactions in Eqs. (13) and (27) on the timescale are so close. This results from the fact that the relaxation time of reactions in Eq. (13) lengthens more rapidly with gap energy than that for the decay reactions in Eq. (27).
We therefore have two situations. The gap is either less than 10 MeV and both reactions in Eqs. (13) and (27) reach equilibrium on a timescale shorter than the buoyancy timescale when buoyancy starts. In this case, chemical nonequilibrium has no role in limiting buoyancy. Otherwise, the gap exceeds 14 MeV and both reactions remain frozen on the buoyancy timescale. We disregard any intermediate situation.
Depressurized, nonequilibrated, quark matter weighs
more than equilibrated matter at the same pressure
because its energy density is not minimal.
For a gap larger than 14 MeV, buoyancy is quenched
when the total energy density
in the rising flux tube
(including its magnetic energy density) exceeds
the total energy density
in the ambient medium.
To illustrate this, we consider a flux tube reaching a region in the star, at an altitude of z,
where its material pressure is less than at the altitude z_{1} where it started its ascent.
We assume that during its motion the weak reactions remain frozen.
The difference between the mass density
of equilibrated matter at this pressure
and the mass density
of the frozen matter at the same pressure
is expressed by Haensel & Zdunik (2007) as:
(30) 
They calculated for cold quarkmatter in which only the decay reactions (27) are frozen. However we are interested in a situation where the reactions in Eqs. (13) and (27) are both frozen. We find that in this case:
where is defined by Eq. (17). The field of a flux tube that is still buoyant at an altitude zwhere the ambient magnetic field is must be such that
where is the material pressure in the rising flux tube at this altitude. Since , we can assume to be almost equal to the total external pressure. For isolated flux tubes moving through an unmagnetized medium, Eq. (32) becomes:
(33) 
where is the mass density in the equilibrated unmagnetized environment. At the star's surface , being the bag constant. We define (50 MeV fm^{3}). We assume that MeV. If quark matter is to be more stable than nucleonic matter, should not exceed 1.6 for free quarks confined in the bag, and 1.4 for the QCD coupling constant 0.2 (see, e.g., Fig. 8.2 in Haensel et al. (2007)). With given by Eq. (31), the minimum value of a field that would be buoyant near the surface, at a level z, is:
Deeper inside the star, at a level z_{1}, the field in this same flux tube had a value due to the conservation of its magnetic flux and quark number content. From Eq. (22), . The pressure deep inside the star is taken to be . The pressure close to the star's surface is the bag constant . The minimum buoyant field deep inside the star, , is then:
Taking the moment of inertia of the star to be I_{*} = 0.4 M R^{2}, the ratio of the toroidal field generated by the torsional oscillation (Eq. (8)) to is:
Buoyancy is quenched when . For P_{*}= 3 ms, , , and , this occurs (provided that the coloursuperconductivity gap exceeds 14 MeV) when
(37) 
An important question is whether the buoyant flux is entirely expelled out of the star with the leptonic wind or whether, although the magnetic field partly emerges, it remains rooted in the subsurface layers.
This depends on how rapidly the magnetic field can diffuse through quark matter, which itself depends on its electrical conductivity and the gradient lengthscale l_{M} of the field. Since the magnetic field decreases in the flux tube during its ascent, its cross section at the surface cannot be smaller than when it started. Since rapid rotation prevents radial motions perpendicular to the rotation axis, the field scale length l_{M} of buoyant flux tubes cannot diminish. The fact that the magnetic diffusion timescale is about 10^{11} s for conducting quark matter implies that during the first few minutes after the formation of the strange star, the magnetic flux emerging through the star's surface as a result of buoyancy remains rooted in quark matter at starspots. In this case, magnetic activity from the torsional oscillations, as described below, continues after the first burst of magnetic buoyancy has brought the inner magnetic field closer to the surface.
If matter is magnetized in bulk, buoyancy assumes the form of a convective instability. When it develops, the more magnetized material is brought to the star's surface, while the less magnetized material sinks deeper into the star. This results in a redistribution of magnetic field in the star, not in a net loss of flux. The end result of the field redistribution should be close to a state of marginal buoyancy instability. A fraction of the surface magneticflux tubes should emerge from the star, baryonic matter draining down along the field as it emerges. However, since this matter cannot diffuse out of the field, the emerging magnetic loops remain connected to the subsurface flux. The magnetized volume experiences little change in this process, so that a substantial part of the star's volume remains magnetized, if it was initially, and magnetic activity from the torsional oscillation persists after the flux redistribution.
In the following, we consider cases when the initial field stratification in the star is stable against buoyancy or when the coloursuperconductivity gap is larger than 14 MeV and the mass of the strange quark sufficiently high to inhibit the buoyancy of woundup magnetic fields. Our results also apply to when the star was initially magnetized throughout a substantial fraction of its volume and remained so after a short, first episode of buoyancy.
3.3 Direct magnetic dipole radiation
The timedependent, internal, stellar magnetic field may be a source of electromagnetic emission
from the star's environment, whether a vacuum or a leptonic plasma.
For example, if the newborn star is an oblique rotator (Usov 1992), it will emit
electromagnetic radiation due to the rotation
of its magnetic dipole.
We define
to be the angle between the magnetic and rotation axes,
the polar field,
and
the star's rotation rate. The power emitted in vacuo by
the magnetic dipole rotation is (Landau & Lifshitz 1975):
An orthogonal rotator with a polar field of 10^{15} Gauss and a rotation period of 3 ms would radiate a flux of , similar to that required to explain the highenergy photon emission of the gammaray burst. For a polar field of only 10^{14} Gauss the emitted power should decline to , which is insufficient to account for a GRB. We note that this emission taps the rotational energy of the compact star, which is about erg for a rotation period of 3 ms. Even at this high rate, the radiation of a 10^{15} Gauss millisec magnetar should last for about 10^{3} s. In the next section, we discuss whether the internal star's torsional oscillation could somehow act as a substitute for a rotating, magnetic dipole.
4 Radiation by torsional oscillation
The collapse leads to a state of differential rotation in the star, the angular velocity varying either with depth or with latitude or both. In an aligned rotator, somewhat analogously to the rotating oblique dipole, the oscillating internal toroidal magnetic field may act as an antenna generating a largescale electromagnetic wave in the star's environment at the period of the torsional star's oscillation. We calculate in Sect. 4.1 the power emitted in a vacuum environment. The radiation in a leptonic wind is considered in Sect. 4.2.
4.1 Radiation in a vacuum
To study the electromagnetic emission from the compact star driven by the
torsional oscillation, we begin by calculating the electromagnetic field
in an external vacuum.
Maxwell's equations are solved outside the star under the
boundary conditions
that
and the tangential components of the electric field
are continuous at the star's surface.
The matching of the conditions at the star's surface requires neither
nor to be continuous, since a
surface current could support a sharp discontinuity in these components.
We denote by a superscript < (or >)
quantities relevant to the inside (or the outside) of the star.
The electric field just below the star's surface is given by the law of perfect conductivity:
(39) 
Since the velocity of the fluid in the star is assumed to be azimuthal only, the condition that the tangential components of the electric field are continuous reduces to and:
In the presence of a torsional oscillation in the rotating star, Eq. (40) has both a timevarying and a constant component. The latter determines the timeindependent, external electric field, while the former determines the outside radiation caused by the torsional oscillation. The boundary condition in Eq. (40) determines completely the solution in the vacuum outside the star. We calculate the electromagnetic field radiated out of the star by the internal torsional oscillation of pulsation , assuming axisymmetry. The toroidal field is then the only timedependent component of the outer magnetic field. Omitting for simplicity the superscripts > which refer to the outside region, the equations for the electromagnetic fields in this region are Ampere's and Faraday's equations in a vacuum, given by:
The only nonvanishing and timedependent component of the magnetic field is the azimuthal one. It is useful to introduce an angular potential such that
The system (41), (42) reduces to an equation for alone, which, for harmonic timedependance in the form of , translates into the Helmholtz equation for :
The operator is the ordinary scalar Laplacian. We expand in spherical axisymmetric harmonics. The solution for each harmonic component of degree of is, to an arbitrary multiplicative factor:
where is given by the dispersion law of freespace electromagnetic waves, that is . In Eq. (45), is a semiinteger Hankel function and is the Legendre polynomial of order . The value of is the lowest value of pertinent to our problem. In fact Eq. (6) illustrates that inside the star is generated by terms. The dipole components of the poloidal magnetic field correspond to the lowest value of , since and , where is the polar field. Similarly, the lowest value of for the timedependent rotation velocity at the star's surface is , when , being the velocity of the timedependent part of the rotation at the equator. This angular variation in corresponds to a constantamplitude modulation of the rigidbody rotationrate at the star's surface: . Such an oscillation would be induced by variations in the fluid's angular velocity with depth. Differential rotation in latitude corresponds to . Since the vector product of and the poloidal magnetic field is expanded in spherical harmonics with , we now restrict our attention to emission in the mode. The semiinteger Hankel functions can be expressed as a sum of a finite number of simple terms. For , the solution for which behaves as an outgoing wave at infinity is:
where m_{0} is a complex factor. The complete solution can be derived from Eqs. (41)(43) and is written in the following form, where B_{0} is a complex amplitude:
The relations (47)(49) solve the system of Eqs. (41), (42). The complex amplitude B_{0} is determined from the boundary conditions, such that the component of the electric field is continuous at the star's surface (Eq. (40)). The timedependent field component of Eq. (49) must match the corresponding timedependent part of the field component just below the star's surface, which requires that:
where is the polar field and is the timedependent part of the component of the azimutal velocity at the equator. By matching the two members of Eq. (50) we derive the complex wave amplitude B_{0}. Since the star's radius is far smaller than the wavelength of the emitted wave, Eq. (50) should be evaluated to the dominant order in the small parameter , providing:
The velocity amplitude of the torsional oscillation is given by Eq. (10) and . The modulus of B_{0} is then:
The magnetic amplitude of the wave is far smaller than the subsurface magnetic field because of the significant impedance mismatch between the star's interior and the outside vacuum. Denoting by the Alfvén speed inside the star, the ratio of these impedances is . The peculiarities of the spherical wave solution for are also responsible for the smallness of this amplitude, which is not set by assuming continuity of . The correct boundary condition is Eq. (50) and its fulfilment implies that is discontinuous at the star's surface.
It is interesting to evaluate the Poynting power radiated off the star's surface.
If the lowfrequency wave emission can
be represented by radiation in vacuo, the solution of Eqs. (47)(49)
provides an upper bound to
the power that may be dissipated in the star's environment and radiated away as X and photons. The radial component
of the Poynting vector associated with the lowfrequency radiation is
where the superscript * designates the complex conjugate and the real part of a complex number. The power radiated as Poynting flux by the torsional oscillation in its supposedly vacuum environment can be calculated from Eq. (53) by integrating over the star's surface, taking Eqs. (47)(49) into account. A number of simplifications occur in this calculation, which finally infers that:
The magnetic amplitude B_{0} of the wave is given by Eq. (51) and is given by Eq. (10). We then have:
Numerically, the power amounts to:
A glance at Eqs. (38) and (55) indicates that much less energy is radiated away in an outside vacuum by the torsional oscillation than by an oblique, rotating, magnetic dipole. The Poynting power radiated by the torsional wave is smaller than the emission of a rotating dipole for several reasons. First, the radiation is quadrupolar instead of dipolar. Then, the field B_{0} is only of the order 10^{8} Gauss for the values of parameters adopted as representative (Eq. (52)), much less than the polar field of an ordinary pulsar or magnetar. Finally, the period is of the order of a few seconds, much longer than the rotation period of the newborn compact star. As a result, the power in Eq. (56) falls short by many orders of magnitude of the observed power of radiation in a long GRB!
4.2 Radiation in a leptonic wind
Could the presence of a circumstellar, leptonic plasma drastically change the power radiated by the torsional oscillation? This plasma originates in charges, electrons, and positrons that have passed the bag of the quark star. We note that at a distance from the star of larger than the lightcylinder radius, the plasma cannot be in rigid corotation, but must flow outward (Goldreich & Julian 1969). Since the wavelength associated with the frequency of the torsional oscillation is much larger than the lightcylinder radius of the rapidly spinning star, the wave emitted by this oscillation propagates into the wind driven by the rapid global rotation. We have to determine the amount of energy of the torsional oscillation radiated per second in these conditions. We assume that the wind has already reached its terminal velocity at the surface of a sphere of radius comparable to that of the lightcylinder, . Since the ratio of the torsional oscillation period to the spin rotation period is large, the torsional oscillation appears, on both of the scales of the spin period and the lightcylinder radius, as a quasistatic perturbation. Its effect is not only to emit a signal that assumes the character of a wave at distances larger than , but it also modulates the wind in which it propagates as a result of the variations imposed on the conditions of its lauching. These modulational effects are distinct from emission of lowfrequency radiation in a given wind. Much of the action causing wind modulation occurs below or close to the lightcylinder and will be discussed in Sect. 5.
We now calculate the emission by the torsional oscillation
in an expanding, possibly resistive, leptonic wind.
The conductivity
of the medium is assumed to be real.
This is because the wave would be highly nonlinear if
the gyrofrequency
of leptons in the wave's magnetic field
was much larger than the wave frequency.
As a result, the effect of the plasma current
on the real part of the index of refraction would become negligible and the wave
would force its way nonlinearly through the leptonic environment
(Asseo et al. 1975; Salvati 1978). We may then restrict our consideration
to resistive effects.
The modulus of the wind speed
is assumed to be constant, both in time and space, and
oriented radially outwards:
.
We assume the wind to be ultrarelativistic
and to have a velocity equal to the speed of light.
When taking the wind to be radial, we assume that corotation is lost at distances of the order
or larger than
.
The background, magnetic field in the
wind is severely wound up by the star's rotation. At distances much larger
than the lightcylinder radius, its azimuthal component dominates over the poloidal
component and declines proportionally to 1/r.
We thus neglect the poloidal field component
and assume the unperturbed magnetic field
to be azimuthal, so that:
Specifically, we consider . The electric current associated with Eq. (57) is radial. For the adopted angular profile it reduces to zero almost everywhere, except at both the polar axis and the equatorial plane. Heyvaerts & Norman (2003) demonstrated that the magnetic field in a perfect MHD wind asymptotically becomes potential almost everywhere, the current being confined to boundary layers about the polar axis and at surfaces where the poloidal polarity reverses. Our choice of complies with this. The singularity at the polar axis represents the current carried by a jet, while the change in the direction of the field at the crossing of the equator is caused by the change in polarity of the poloidal field at . Ohm's law infers that:
By considering the divergence of Eq. (58) and solving the resulting differential equation for the charge density , it is found that the latter vanishes in the unperturbed wind. The unperturbed electric field then vanishes too. We define , , , , and to be the perturbations of the electric and magnetic field, current density, charge density, and leptonic fluid velocity, respectively. It is sufficient to describe the lepton's dynamics at the inertialess (also called forcefree) approximation. The magneticfield perturbation is toroidal in the considered geometry and the electric force is a negligible secondorder term. The perturbed Maxwell equations, Ohm's law, and the dynamical equation in the zeroinertia limit can be written (using the compact notation for time derivatives) as:
The toroidal components of Eqs. (59) and (63) imply that the electric field is only poloidal. The other two components of Eq. (63) infer the fluid velocity once the solution for the other unknowns has been found. The current is eliminated by taking the vector product of Eq. (60) and . As a result the conductivity is eliminated from the equations describing the perturbation. With Eq. (59), this infers an equation for the magnetic perturbation, which is expressed most accurately in terms of the angular potential of this perturbation (Eq. (43)). For harmonic timedependence , it is found that satisfies the Helmholtz equation (Eq. (44)). Regardless of the conductivity, the perturbations propagate in this geometry as electromagnetic waves, provided the inertialess limit is considered. The complete solution is identical to the vacuum result (Eqs. (47)(49)) as is of course the Poynting flux at the star's surface and the emitted Poynting power (Eq. (55)).
5 Modulation by the torsional oscillation of the energy emitted in the rotator's wind
A rapidlyspinning aligned rotator emits
a wind carrying power in electromagnetic, potential, thermal and kinetic energy form.
The contributions of these different forms of energy depend on the distance
to the star. Some forms of energy may dissipate en route or at terminal shocks, producing
observable X and
radiation. Close to the compact star, much of this flux
is in Poynting form because the kinetic energy remains low while
the wind has not yet been effectively accelerated.
The thermal and gravitational energy fluxes often constitute
but a little part of the total energy flux.
The energy output of the star in its wind environment then enters
the latter as DC Poynting flux,
the radial component of which is given in terms of the field components just above
the star's surface by:
(65) 
The boundary condition at the star's surface implies that . We define to be the subsurface fluid velocity and the radial field component, which is continuous across the star's surface. We then find that:
The value of at the base of a relativistic wind is given approximately by Eq. (67) below. This can be sketchily explained as follows: due to the rotation of the star and the effect of flux freezing, a toroidal field is generated on open field lines from the poloidal field. The knowledge that the foot point of a field line is rotating is propagated along this line at a finite velocity , by means of convective transport and propagation as an Alfvénic signal. The field line thus curves away from the sense of rotation at an angle of to the radial direction, such that . Since the wind is relativistic and the Alfvén speed in the tenuous external plasma is close to the speed of light, and the toroidal field just above the star's surface is:
A more precise justification of the approximate relation in Eq. (67) is omitted for conciseness. Then, from Eq. (66):
This flux is only emitted from the polar caps, the regions on the stellar surface connected to open, field lines. The magnetosphere is closed, where the apex of the local field line is at a distance Dsmaller than the lightcylinder radius . When field lines are dipolar, the polar caps extend to a colatitude , which we refer to as classical, and described by:
Under certain conditions however, the magnetospheric field may depart considerably from dipolarity (Sect. 5.3). When the flux is distributed on the star as a dipolar field, the radial, field component varies with as , being the field at the pole. By considering to be the solid body rotation velocity at the angular speed , we obtain, by integrating over the colatitudes corresponding to the two polar caps, the DC Poynting power emitted by the star under these conditions:
The power represented by Eq. (70) is comparable to the power emitted by an oblique, rotating dipole (Eq. (38)). This is a classical result (see for example Michel 1991). With a rotation period of 3 ms, the power erg s^{1}. This is insufficient to match the high luminosity of a GRB unless fields in excess of 10^{15} Gauss are involved (Usov 1992). Could differential rotation drastically change this result?
5.1 Quasistatic modulation of the wind
We assume that the star experiences a torsional oscillation. The velocity then differs from the solidbody rotation velocity and varies with time. Because the period of the oscillation is much longer than the mean spin period, this causes a quasistatic change in both the structure of the magnetosphere and the polar cap angle. It even causes a change, which we neglect, in the shape of the light cylinder. The structure of the magnetosphere and the energy output of the wind adjust to equilibrium values corresponding to the instantaneous velocity profile on the star's surface.
This profile may be even with respect to the equator, or odd, or a mixture of both. An even oscillation is one in which the azimuthal velocity perturbation is symmetric with respect to the equator, i.e. where the timevarying azimuthal velocity is in phase at two points positioned symmetrically with respect to the equator. An odd oscillation is one in which it is antisymmetrical, i.e. where the velocity is in phaseopposition at two points symmetric with respect to the equator. It may appear that nature should provide only even profiles, by a principle, or rather a postulate, of northsouth symmetry. However, this symmetry is broken in the case of the collapse of supernovae. It is indeed well known that newborn neutron stars receive a kick, that is, a net thrust from the collapse. This would be forbidden by the principle of northsouth symmetry. Therefore, it cannot be excluded that, similarly, odd modes of torsional oscillation are present in the initial excitation of a newborn compact star. The amplitude of these modes is expected to be small, but we show below that it need not be large to produce important effects. The kick received by a newborn neutron star produces a velocity of the order of 200500 km s^{1}. This corresponds to an asymmetry in the momentum emission of the order of km s^{1}. The supernova explosion emits a momentum per steradian of about km s^{1}/4 . The asymmetry in the momentum emission appears to be a fraction of between a few 10^{4} and a few 10^{3} of the total. A similar fraction of the total star's rotational energy may appear after the collapse in the form of odd differential rotation.
5.2 Modulation of the wind by an even oscillation
An even torsional oscillation has but little effect on the structure of the magnetosphere and wind
because the two footpoints of a closed field line follow the same motion exactly
if, as assumed in this subsection,
the poloidal field lines are
strictly symmetric with respect to the equator. Otherwise,
these field lines would undergo a twist in the presence of
an even torsional oscillation, because their footpoints would not be at exactly
opposite latitudes, and would be carried in the azimuthal direction at different
angular velocities.
It is difficult to realistically anticipate the degree
af asymmetry in poloidal field lines. It could vary from very little
to a complete absence of symmetry.
The degree of asymmetry necessary for the magnetosphere to open
will be estimated in Sect. 5.3.
Strictly symmetric field lines whose footpoints are moved by an even torsional oscillation
are, however, not twisted and, as a result, no poloidal electric current is driven
in the magnetosphere. Nevertheless, because the rotation rate on the star varies with colatitude,
the GoldreichJulian charge distribution in the magnetosphere differs slightly
from the case of solid body rotation, as well as the
DC Poynting flux (Eq. (68)). Using as a model
the following even differential rotation:
we calculate from Eq. (68) and integrate over the classical polar caps to derive the emitted power :
(72) 
where
(73) 
The wind power is modulated slightly at a level of , which is about 10 % for a period of 3 ms and . The change in the timeaveraged power is at the 0.1% level for the same figures. These small changes cannot account for the existence of a gammaray burst.
5.3 Magnetosphere opening by an odd oscillation
An odd oscillation differs from an even one in that the two footpoints of a closed field line experience differential motion in longitude, introducing a twist in this field line. This causes a poloidal current to flow in the closed magnetosphere and drastically changes its structure. When the twist exceeds a threshold of order , the magnetosphere opens. Field expansion by the shearing of the footpoints of field lines was first discussed in the context of solar flares (Low 1990; Heyvaerts et al. 1982; Aly 1985). It was established that it occurs in Cartesian geometry with a direction of invariance by theorems constraining the properties of linetied forcefree fields (Aly 1990,1985) and by numerical simulations (Biskamp & Welter 1989). The same process has been considered also in the case of axisymmetric structures extending above a spherical surface on which the field lines are tied. In general, it was demonstrated that rapid expansion occurs when a finite shear is reached (Aly 1995). This is also supported by numerical simulations (Mikic & Linker 1994). Full opening occurs for a finite twist (of order ) in some specific examples (LyndenBell & Boily 1994; Wolfson 1995). In the present context, the lightcylinder radius imposes a limit on the distance to the apex of closed field lines, such that field opening is even easier when magnetospheric inflation proceeds. When the field opening becomes almost complete, it causes a growth in the polar caps and the emitted wind power. In this case, the role of the torsional oscillation is not to modulate the energy output by the addition of its own electromagnetic emission but to open the door for a more significant wind emission from the central object. This enhanced wind emission would acquire its energy directly from the rotational kinetic energy of the star, not only from the energy of the differential rotation, and lasts for as long as the torsional oscillation survives with sufficient amplitude.
The magnetosphere opens if the difference in longitude between the two conjugate footpoints of a field line exceeds
typically half a turn (
).
To justify this statement,
one should try to solve for the structure of the magnetosphere as a function of the
difference in longitude between the footpoints of field lines.
By assuming axisymmetry, the poloidal
field is represented by a flux function
,
so that the total magnetic field can be written as:
Any field line follows a surface of constant a (a magnetic surface) because the magnetic flux being transmitted through a circle perpendicular to and centred on the polar axis passing at , is . The perturbations of the closed magnetosphere of the star are of quasistatic nature (Sect. 5.1). Neglecting the particle's inertia, the force equation for the instantaneous equilibrium is the forcefree equation:
The electromagnetic state of the magnetosphere is not described by the magnetic field alone but also by the electric potential . Equation (75) is supplemented by the timeindependent Maxwell's equations:
The components of Eq. (75) can be expressed in terms of the functions , , and , the latter being defined by:
The current through the circle perpendicular to and centred on the polar axis passing at is J = I/2. We refer to I as the poloidal current. In an axisymmetric state, the electric field is poloidal. The toroidal component of Eq. (75) then shows that the gradients of I and aare everywhere parallel, which implies that I is a function of a: . The poloidal part of Eq. (75) can then be written as:
where is the scalar Laplacian and
(81) 
Equation (80) indicates that the gradients of U and aare everywhere parallel, which implies that U is a function of a, . The rotation rate of the matter, , is given by the electric drift velocity of particles and is found to be:
(82) 
The projection of Eq. (80) onto provides the socalled pulsar equation (Michel 1991):
This equation has a singularity at the lightcylinder, which causes any field line reaching this limit to diverge (Contopoulos et al. 1999). To determine approximately which field lines become open, it suffices to solve Eq. (83) for and confirm which field lines reach a distance larger than . In this limit, Eq. (83) reduces to:
It is shown in Appendix A that, for a selfsimilar model of the magnetospheric field, there is no solution to Eq. (84) with closed field lines when the twist exceeds .
We represent odd differential rotation by the following simple
model for surface differential rotation:
The magnetic flux distribution on the surface of the star is a function of the colatitude , so that is a known function of a, . The twist at time t associated with Eq. (85) is:
It is implied here that the footpoint P_{2} is at the colatitude and that P_{1} and P_{2} are at the same longitude at t=0. In the model presented in Appendix A, we regard at time t as being half the maximum twist implied by Eq. (86). The limit value of for the twist is reached for a rather small amplitude of the torsional oscillation because the period of the latter is long compared to the spin period. Similarly to Eq. (10), we parametrize as in terms of the fraction of the star's rotational energy available in this odd oscillation mode. The magnetosphere is in an open magnetic configuration when:
As explained in Sect. 5.1, a fraction 10^{4} of the star's rotational energy may be stored in odd torsional oscillation modes. At this level of excitation of odd modes, the magnetosphere would be open during a large fraction of the oscillation period. There would be no opening only when is very small, i.e.:
We assume that odd oscillation modes are initially excited to a level higher than the limit indicated by Eq. (88).
A similar result is obtained when an even oscillation is considered
(with an amplitude given by the larger value indicated in Eq. (10)) but the dipolarlike magneticfield lines are not strictly
symmetric with respect to the equator.
This would happen if, for example, the field is a noncentred dipole.
We define
to be the difference of the
absolute values of the latitudes of two conjugate footpoints.
The twist experienced by these footpoints
will be larger than ,
and thus the magnetosphere will open,
when
,
which translates into the condition:
For Eq. (89) to be satisfied for typical values of P_{*}, , and , it suffices that the dipole field be decentred by a fraction of a few 10^{3} of the stellar radius. Since the odd torsional oscillation has an amplitude larger than indicated by Eq. (88) or the magnetic geometry is northsouth asymmetric to a degree larger than indicated by Eq. (89), the magnetosphere will alternate during the oscillation cycle between a classical state, which we refer to as closed, and an open state.
For an odd torsional oscillation, the configuration is closed when the condition of Eq. (87) is not satisfied. It becomes an open state, where all field lines are open and carry winds, when the twist is sufficiently large for Eq. (87) to be satisfied. During closed episodes, the polar caps opening is limited to , and during open episodes . There is a transitory state which we neglect because it lasts much less than a wave period. In an open state, the power fed by the compact star into its relativistic wind is much larger than the classical value given by Eq. (70). This may be the reason why the emitted power is enhanced considerably in the first moments after the collapse, an enhancement that should decline as the star's rotation decelerates and last at most until the odd mode amplitude has decreased below the limit fixed by Eq. (88). We calculate the lifetime of odd oscillations and their associated emission in Sect. 5.4. The idea that a GRB would be the result of pulsartype emission from a compact star with an entirely open magnetosphere was considered by Ruderman et al. (2000), who however regard the expansion of the magnetosphere as being caused by magnetic buoyancy rather than by twisting, as we suggest in this paper.
The power emitted at time t is calculated by integrating the Poynting flux
given in Eq. (68)
over the windemitting star surface, where
is given by (see Eq. (85)):
When the magnetoshere is in a closed state, the emitted power is, neglecting terms of order :
Similarly, when the magnetosphere is open, it is given by:
Numerically, with 10^{14} Gauss and R = 10 km:
(93) 
The power emitted during the open episodes is larger than that emitted during the closed episodes by a factor of . For P_{*}= 3 ms, this factor is about 110. For the same rotation period and B_{p14} = 3, reaches erg s^{1}, which is enough to explain the GRB emission, allowing for a conversion factor from kinetic wind energy to radiation that is smaller than unity.
5.4 Damping of rotation and odd oscillation
Neither the amplitude of the odd torsional oscillation
nor the rapid stellar rotation will last long in the presence of such large losses.
A rapidly spinning star with an odd oscillation is characterized by two parameters,
the average starrotation rate
and the amplitude of the
differential rotation
.
Due to wind losses, both decrease in time.
To calculate their evolution, a model of the internal magnetic field of the star is required.
Although this is not an accurate model
for the windingup of the field when the rotation depends only on the distance
to the axis, we shall assume for simplicity
that the unperturbed, magnetic field is uniform in the star and parallel to the
rotation axis, i.e. that
.
The normal component of this field
on the star's surface equals that of a dipolar field and the inner field
equals the outer field at the z>0 pole.
It is convenient for us to use the cylindrical coordinates
D, ,
and z, the parameter r representing the spherical distance to the star's centre,
R the radius of the star, and
the colatitude.
Since the fluid
is almost incompressible with a uniform mass density ,
we assume as in Sect. 2 that its velocity
is azimuthal and can be written as:
The velocity is supposedly odd in z. When writing the even part of the rotation velocity as , we neglect the even torsional modes, which play no role in the magnetosphere opening in the case of a northsouth, symmetrical, magnetic structure. The magnetic field in the presence of the perturbation develops an azimuthal component equal to:
(95) 
For the assumed, uniform, unperturbed field, Eqs. (5), (6) can be written as:
If were, at any given time, structured according to the ProudmanTaylor theorem, it would be cylindrical, that is, it would depend only on the distance to the rotation axis. Then, for a completely axial field as assumed above, the term on the righthand side of Eq. (97) would vanish. The assumption of a uniform axial field, however, is only applied to simplify the following calculations. Nothing constrains the structure of the field inside the star, especially when it is not dynamically significant. For nonaxial , Eqs. (96), (97) still provide a sketchy representation of the torsional oscillation: the operator must be assumed to represent ( ) (see Eqs. (5), (6)) and the variations in along a field line should be ignored, although they exist. Using Eqs. (96), (97) is equivalent here to replacing a curved poloidal magnetic field in a cylindrical velocity field with a uniform axial field in a zdependent velocity field. An odd torsional oscillation in a cylindrical velocity field would correspond to northsouth asymmetric poloidal field lines. The simple approach implied by Eqs. (96), (97) should be sufficient for our purposes of estimating the damping time of the torsional oscillation. These equations hold for a linear as well as for a nonlinear axisymmetric perturbation of an incompressible medium. The velocity perturbation satisfies the Alfvén propagation equation:
A standing wave solution of Eq. (98) is
where k and are related by the dispersion relation and . The wavenumber k could depend on the distance D to the axis, each cylindrical magnetic surface then having its own oscillation period. To keep things simple, we assume that k is a constant equal to , , and is an odd function of z. The proper choice of in Eq. (99) ensures that the velocity field of the perturbation at the star's surface coincides with Eq. (90), that is:
Equations (97) and (99) infer the magnetic perturbation to be:
To obtain the angularmomentum balance equation for both the z>0 and z<0 hemispheres, the torques acting on each must be calculated. Each hemisphere experiences volume torques exerted by magnetic tension and surface torques caused by the drag produced by the emission of Poynting energy at the star's surface. The angular momentum J_{+} of the z>0 hemisphere and the angular momentum J_{} of the z< 0 hemisphere can be calculated from Eqs. (94), (99), and (100). They are:
(102)  
(103) 
The moments of inertia I_{1} and I_{2} are:
The numerical value of the integral on the righthand side of Eq. (105) is 0.187. The moment of inertia I_{*} of the entire star with respect to the rotation axis is I_{*} = 2 I_{1}. The torque T_{B+} exerted by magnetic tension on the z>0 hemisphere can be calculated from the Lorentz force density (Eq. (96)). If is given by Eq. (101), this implies that:
(106) 
The magnetic tension torque T_{B} exerted on the z < 0 hemisphere is T_{B} =  T_{B+}. The Poynting torque d exerted on the strip of the star's surface between colatitudes and in the z>0 hemisphere is related to the Poynting power d emanating from that strip by d , where is the angular velocity of the fluid at that colatitude and at that time (see Eqs. (90), (68) and (99), (100)). The torque d on the strip between and in the z < 0 hemisphere is similarly calculated:
The total Poynting torques and on the z>0 and z<0 hemispheres are derived by integrating Eqs. (107) and (108), respectively, over colatitudes from zero to the polar cap angle . When the magnetosphere is closed, . When it is entirely open, . The angularmomentum balance equation for the z>0 and z<0 hemisphere are given respectively by:
By adding the expressions in Eqs. (109) and (110), we derive an equation for the time evolution of the global rotation :
By substracting Eqs. (110) from (109) we derive after some algebra:
A characteristic damping time appears to be defined by:
When the magnetosphere is completely open, Eqs. (111), (112) reduce to:
From Eqs. (104), (105), we find that the damping times for and in the open regime are, respectively:
When the fastspinning aligned rotator experiences episodes of magnetospheric opening, its evolution consists of a succession of open (or high) states and closed, classical (low) states. During open periods, Eqs. (114), (115) apply and the energy output, of the order indicated by Eq. (92), is considerable. During these periods, the open field should occasionally reconnect, attempting to return to a closed structure, but, once reformed, the latter is again blown open after the very short time needed to build a twist again of approximately half a turn. Large irregular variability is then expected during these open periods, down to the millisecond timescale, which is the time to cross through a lightcylinder size, expected to be representative of the equatorial current sheet, at the speed of light. We note that the closed episodes are initially short in duration when is of the order of 10^{4} as supected. When a total time of order of has been spent in the open state, the oscillation has damped to an amplititude insufficient to open the magnetosphere, and the average rotation has been substantially reduced. Neglecting the weak damping experienced during the closed episodes, the spin rate and the largest amplitude of the differential rotation vary as
where t is the cumulated time spent in the open state. The emitted power scales as , and declines in a time of approximately . The GRB would disappear from view in about this time, which, for a polar field of Gauss, is about 20 min. The openings completely cease when the maximum twist falls below half a turn, that is, from Eq.(86), when . This happens after a time such that
The mean rotation rate at time is given by Eq. (118). We define , which, from Eqs. (116), (117), is nearly unity. Relating to as in Eq. (87), we obtain, for , . Numerically:
(121) 
For and s (i.e. half the period (7) for Gauss) ms. After spending a time in the open state, the high episodes cease and the object becomes a pulsar with a period of the order of 5 millisec. As indicated above, the time of high activity is about 3.5 (Eq. (113)), which is comparable to the observed timescale of long duration GRBs when the field of the compact star is somewhat higher than 10^{14} Gauss. For high fields, of order 10^{15} Gauss, this timescale is about 100 s.
6 Conclusion
It is natural to consider that a newborn quark star experiences differential rotation, causing its internal woundup toroidal field to increase in strength to about 10^{16} Gauss. This motion then develops into a magnetic torsional oscillation, which could be the origin of longduration ray bursts. We have indeed shown that an odd oscillation of small amplitude, which should be easily reached, is sufficient to open the star's magnetosphere. A similar effect would also result from other causes of northsouth asymmetries. The rapid rotation then drives a relativistic wind from the entire stellar surface. When the star is a quark star, this wind is entirely leptonic. We have calculated the Poynting power released and the timescale of this phenomenon, which meet the observational constraints if the polar field of the quark star is of the order of a few 10^{14} Gauss and its initial rotational angular velocity is of the order of 300 Hz. Large amplitude variations in the light curve on timescales ranging from minutes to milliseconds is a natural outcome of this process.
Acknowledgements
We are very grateful to L. J. Zdunik for his help in clarifying the conditions under which magnetic buoyancy is inhibited by the finite relaxation time of weak reactions among quarks. M.B. was partially supported by the LEA AstroPF collaboration and the Marie Curie IntraEuropean Fellowships MEIFCT2005023644 and ERG2007224793 within the 6th and 7th European Community Framework Programmes. This work was supported in part by the MNiSW grant N20300632/0450.
Appendix A: Magnetosphere opening: an example
We describe the asymptotic properties
of the solutions to Eq. (84)
in the context of a selfsimilar model and show
that there is a limit twisting for closed solutions to exist.
We define A to be the equatorial value of the flux function a (Eq. (74)).
When a field line on the magnetic surface a is twisted, there is a relation between
its twist ,
which is
the difference in longitude between its footpoints,
and the poloidal current I(a).
The differential equation of a field line is indeed:
Using Eqs. (74) and (79), we evaluate the change in longitude accumulated following a field line on the magnetic surface a from one of its footpoints P_{1} to the other P_{2}:
where d is the line element along the poloidal field line a. It is the twist , not the poloidal current I, which is known. The relation Eq. (A.2) does not infer I(a) directly when is known, because the line of constant aon which the integration in Eq. (A.2) is to be carried is unknown before the problem expressed in Eq. (84) has been solved. To determine under which conditions a dipolar field line closing at a few stellar radii would be inflated sufficiently by the magnetospheric current to reach the lightcylinder, we attempt to identify separable solutions to Eq. (84) (LyndenBell & Boily 1994; Wolfson 1995; Bardou & Heyvaerts 1996) of the form:
Wolfson (1995) numerically studied the solutions of Eq. (84) under the ansatz (A.3). We show here that the solutions must open when the twist reaches a finite value. The largest value of a being the total star flux A, , and . Similarly g(0) = 0, since there is no flux through a circle of zero radius centred on the polar axis. The apex of field line a is at a distance D(a), such that D(a) = R (A/a)^{1/p}: the more inflated the magnetosphere, the smaller the parameter p. We will then deal with the small p limit. A solution of the form of Eq. (A.3) cannot match any given flux distribution on the star, nor any given twist . The constraint of Eq. (A.2) can only be satisfied in an average sense. Using Eq. (A.3) in Eq. (84) the following equation is obtained:
The lefthand side is a function of , while the righthand side is a function of a. Both then equal a common constant, K. This means that for the ansatz of Eq. (A.3) to be satisfied when the similarity exponent is p, the current function I(a) must be:
The dipolar angular function is recovered for p=1 and I = K = 0. For nonvanishing K, the constraint of Eq. (A.2) becomes, for solutions of the form Eq. (A.3):
The limits on the integral at the denominator reflects the fact that all field lines span the interval in (Eq. (A.3)). Equation (A.6) is consistent with Eq. (A.5) only when is independent of a, which corresponds to the peculiar twist profile in which one hemisphere rotates like a solid body and the other in an opposite sense. We define to be this twist. We may think of as being some average on one hemisphere of a more realistic twist profile. K is related to by:
Having chosen a value of this relation infers K for a given p. Given this relation, the value of p itself results from the need for the solution of the angular function to satisfy the three requirements , g(0) = 0, and , the last one resulting from the symmetry of magnetic surfaces with respect to the equator. A secondorder differential equation accepting only two boundary conditions, the extra condition eventually determines the value of p. To express this condition explicitly, Eq. (A.4) for the angular function g must be solved in the limit of small p. In terms of the variable , Eq. (A.4) can be written:
For small p the second term of Eq. (A.8) is negligible. The exponent (1 + 2/p) being very large, the third term of (A.8) essentially vanishes wherever g < 1. It remains nonnegligible only in the vicinity of the equator (x =0), where g reaches unity. The solution g(x) is then almost a linear function for all x, except in a small region about x = 0. This allows us to simplify Eq. (A.8) in the small p limit as:
This equation has a first integral. The condition that g'(0) = 0 at x = 0, where g must equal unity, can be satisfied by an appropriate choice of the integration constant, giving:
Wherever g is sufficiently less than unity, . However, we know that in these regions the modulus of the slope should be unity, because g has already been recognized to be a linear function varying from g = 0 at x =1 to very nearly g=1 at x =0. The relation between K and p is then, in the small p limit:
We should now establish the relation between and p resulting from Eq. (A.7). To calculate the angular integral at the denominator, we must solve Eq. (A.10) when . Taking Eq. (A.11) into account, changing the unknown function gfor h such that g = (1 ph) and making use of the fact that the limit for p approaching 0 of (1+ py)^{1/p} is , it can be shown that the solution of Eq. (A.10) is in this limit:
Since p is small, . The integral that appears in Eq. (A.7) can then be calculated, resulting in:
Thus, the exponent p approaches zero as the twist approaches and the magnetosphere swells boundlessly in this limit. For our purpose, it is sufficient that a field line extends farther than the lightcylinder for it to open. We can then safely adopt the limit of a twist of a half a turn in causing an almost complete opening.
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