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Astron. Astrophys. 340, 447-456 (1998)

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2. Pulsar mode

If, following Usov (1992) and Duncan & Thompson (1992), we assume that some pulsars are born with period, [FORMULA]ms, the initial Rotational Kinetic Energy (RKE) of such pulsars would be

[EQUATION]

where [FORMULA] is the moment of inertia of the NS in units of [FORMULA] g cm2 and [FORMULA] is the cyclic frequency in units of [FORMULA]. It follows that, the magnetic dipole luminosity could be as large as

[EQUATION]

where [FORMULA] is the surface magnetic field in units of [FORMULA]G. The time scale for electromagnetic mode is thus

[EQUATION]

It is well known that rapidly spinning pulsars may radiate more of gravitational radiation than electromagnetic radiation (Ostriker & Gunn 1969) because of equatorial ellipticity ([FORMULA]). For a pulsar with a solid crust, the major source of [FORMULA] could be the dynamical anisotropies associated with the magnetic field. On the other hand, note that, a freshly born pulsar is believed to be very hot with core temperature well above 10 MeV. This means that the crust will be liquid (Meszaros & Rees 1992) and the solid crust must be formed when they become sufficiently cool by intense [FORMULA] radiation to temperature, probably, below, 1 MeV. Since the whole NS behaves like a self-gravitating liquid at this stage, a large RKE is likely to introduce an equatorial ellipticity far exceeding what is obtained by considering only the magnetic field as the source of such ellipticity.

First note that, the mean density of a canonical NS with mass [FORMULA] and radius 10 Km is [FORMULA] g cm-3. Then the critical cyclic frequency at which mass sdedding occurs is [FORMULA] Hz, which corresponds to a critical period of [FORMULA] ms. In our attempt to first understand the problem within the purely classical paradigm, we ignore the complexities and uncertainties associated with the actual equation of state (EOS) of hot and liquid NS (for a cold NS with solid crust, the cold EOS permits value of [FORMULA] ms). A value of [FORMULA], thus may not at all be allowed for the liquid like hot NS. If the actual value of [FORMULA]Hz, the rotating configuration will be described by Maclaurian Spheroids with the two equatorial principal axes [FORMULA], where c is (not to be confuse with speed of light) length of the principal axis along the rotation vector (Chandrasekhar 1969). Equatorial eccentricity being zero, there will not be any gravitational radiation in such a case.

However, since in the present problem, we are working in a region [FORMULA] (actually by considering [FORMULA], one, unphysically overshoots [FORMULA] barrier), we are not necessarily dealing with axisymmetric configurations because for [FORMULA], the Maclaurian spheriods are dynamically unstable and tends to degenerate into Jacobi Ellipsoids with [FORMULA] (Chandrasekhar 1969 and ref. therein) with equatorial ellipticity

[EQUATION]

Along the Jacobi sequence, [FORMULA] increases monotonically in keeping with the increasing angular momentum. In fact, Poincare (see Chandrasekhar 1969) showed that, the Jacobian sequence eventually bifurcates into a new sequence of pear shaped configuration where

[EQUATION]

In this above limit the value of [FORMULA]. Consequently, even when, we exclude the regime of extreme ellipticity by hand, we find that such hot and fluid-like ultrafast pulsars may emit superstrong gravitational radiation with a luminosity (Ostriker & Gunn 1969):

[EQUATION]

where [FORMULA]. It is trivial to see that the corresponding instantaneous time scale for emission of gravitational radiation would be

[EQUATION]

Since [FORMULA], the NS would primarily emit its RKE by gravitational radiation within a time [FORMULA]s rather than by any significant relativistic wind, and acquire a value of [FORMULA] ms for which the gravitational radiation mode will be quenched. The consequent value of [FORMULA] would also be insignificant for the requirement of GRB problem once P degrades to this range.

It may be relevant to point out that in a recent work extending the Usov-type mechanism (Blackman & Yi 1998), the putative hot ms pulsars have been classified into [1] Supercritical strong field rotator (SPS) for [FORMULA], ([FORMULA] being the critical luminosity over which pulsar spin down is dominated by emission of gravitational radiation) and [2] Subcritical strong field rotators (SBS) with [FORMULA]. For the SPS, the initial spindown is dominated by strong gravitational radiation with e-folding [FORMULA]s. This is somewhat analogous to our conclusion reached above. It is only with reference to the latter class, the SBS, with a relatively lower value of [FORMULA], one may endeavour to correlate observation of long GRBs. But, it is seen that, the peak luminosity in such cases would be well below [FORMULA] erg/s, which is grossly insufficient for GRB970508 and 971214 ([FORMULA]).

Also, recently, in several important theoretical works it has been pointed out that (Andersson, Kokkotas, & Schutz 1998, Lindblom, Owen & Morsink 1998) even when the value of [FORMULA], or even if the above crude discussion on probable emission of gravitational radiation from the rotating Jacobi Ellipsoids were not precise, the NS may spin down by emitting gravitational radiation. When the NS is sufficiently hotter than [FORMULA] K, and one would consider the overall configuration of the fluid to be broadly spheriodal, some degree of non-axisymmetry will set in because of velocity and density perturbations (r-mode). Lindblom et al (1998) estimate that because of r-mode instability, the NS would spin sown to a period [FORMULA] by the joint action of neutrino viscosity and gravitational radiation within a neutrino emission dominated cooling time, [FORMULA], to a temperature [FORMULA] K. However, this paper used a somewhat old formula for NS cooling which gives a very large value of [FORMULA] yr. On the other hand, the more recent work of Andersson et al. (1998), which claims to be more accurate than the previous one, points out that, if one uses more recent work on NS cooling by direct URCA process (Lattimer et al. 1994), the value of [FORMULA] could be as small as [FORMULA] s. It is clear that, in this case, because of rapid decrease in the value of [FORMULA], the value of [FORMULA] would again be insignificant.

Even in the former case of a supposed very high value of [FORMULA], the value of [FORMULA] may be much less than what is implied by Eq. (6). This is because of the fact that, in the theories of formation of superstrong NS magnetic field, the field is not generated spontaneously and automatically at the moment of birth of the nascent proto-NS. Instead, it is believed to be generated either by differential rotation (Kluzniak & Ruderman 1997) or by dynamo amplification (Duncan & Thompson 1992). The latter process results in a faster generation of magnetic field and is presumably caused by vigourous convection driven by strong neutrino flux [FORMULA] erg/s/cm2. One may have such a strong neutrino flux only if [FORMULA]s. If the neutrino cooling is too slow, [FORMULA] yr, the value of [FORMULA] will accordingly be too low, and no strong magnetic field may at all be generated . In this case, the Usov type models will be irrelevant.

Consequently, assuming a NS is born with a [FORMULA]ms-period, it spins down to 10-20 ms range by emitting an energy [FORMULA] erg in the form of gravitational wave energy and probably much more energy in the form of neutrinos. Thus, because of the r-mode instability a supposed ultrfast pulsar will hardly have an opportunity to acquire a large [FORMULA] independent of the actual value of [FORMULA].

Note that, in contrast, an old and recycled pulsar with spin [FORMULA]ms will be very cold with a thick and solid crust whose value of [FORMULA] may indeed be derermined by B and could be very low. Such cold stars with [FORMULA] will have superfluid interior, where the r-mode gets completely supressed (Andersson et al. 1998, Lindblom et al. 1998).

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© European Southern Observatory (ESO) 1998

Online publication: November 9, 1998
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