How Massive Are the Superfluid Cores in the Crab and Vela Pulsars and Why Their Glitch-Events Are Accompanied with under and Overshootings?

The Crab and Vela are well-studied glitching pulsars and the data obtained so far should enable us to test the reliability of models of their internal struc-tures. Very recently it was proposed that glitching pulsars are embedded in bimetric spacetime: their incompressible superfluid cores (SuSu-cores) are embedded in flat spacetime, whereas the ambient compressible and dissipative media are enclosed in Schwarzschild spacetime. In this letter we apply this model to the Crab and Vela pulsars and show that a newly born pulsar initially of 1.25 M  and an embryonic SuSu-core of 0.029 M  could evolve into a Crab-like pulsar after 1000 years and into a Vela-like pulsar 10,000 years later to finally fade away as an invisible dark energy object after roughly 10 Myr. Based thereon we infer that the Crab and the Vela pulsars should have SuSu-cores of 0.15 M  and 0.55 M  , respectively. Furthermore, the under- and overshootings phenomena observed to accompany the glitch events of the Vela pulsar are rather a common phenomenon of glitching pulsars that can be well-explained within the framework of bimetric spacetime.

[1]- [7] and the references therein). In Table 1 we summarize their basic observational data relevant for the present discussion. The SuSu-Scenario relies on solving the TOV equation in combination with the equations of torque balance between the incompressible superfluid cores, whose dynamics obey the Onsager-Feymann equation, and an overlying shell of compressible and dissipative matter (see Sec. 2 and Eq. 10 in [8]).
In [9] these equations were solved at the background of a bimetric space-time (see Figure 1). Unlike the original model [8], in which the spin-down of the Su-Su-core is set to follow a priori given sequence of values { } The strategy of obtaining the optimal values here relies on using a global iterative solution procedure that takes the following constraints into account (see also Table 1): 1) The elements of the sequence n g ∆Ω     Ω   must fulfill the three conditions: , which means that the media in both the core and in the surrounding shell must have identical rotational frequency initially.
as otherwise the magnetic field would fail to spin-down the crust and therefore to surpass cr ∆Ω required for triggering a prompt spin-down of the core into the next lower energy state.
Indeed, one possible sequence which fulfills the above-mentioned constraints, though it might not be unique, is shown in Figure 2.
2) The initial conditions used here are   ∆Ω Ω starts increasing to reach 4 × 10 −9 after approximately 1000 yr, (which corresponds to the Crab phase/blue-star) and 8.15 × 10 −7 after 11,000 yr (which corresponds to Vela phase/red-star). 3) The elements of the sequence { } n am Ω are obtained through the energy balance equation: where am I is the inertia of the ambient compressible dissipative medium, which, due to the increase of the SuSu-core, must decrease on the cosmic time and  The radius of the core is set to increases as dictated by the Onsager-feynmann equation: denote the velocity vector, the vector of line-element, the reduced Planck constant, mass of the superfluid particle pair and the number of vortices, respectively (see [11] for further details). Imposing zero-torque condition on the incompressible SuSu-core, i.e., We then obtain the following recursive relation: and n c R correspond to the cross-sectional area of the Su-Su-core and to the corresponding radius, respectively. The increase in the dimension of the core implies that the matter in the geometrically thin boundary layer between the SuSu-core and the ambient medium should undergo a crossover phase transition into an incompressible superfluid, whose total energy density saturates around the critical value 0 6 cr ρ ρ ≈ (see [12] and the references therin). The growth of the core proceeds on the cosmic time scale and ends when the pulsar has metamorphosed entirely into a maximally compact invisible dark energy object and therefore becomes observationally indistinguishable from a stellar black hole.

Solution Procedure & Results
The set of equations consists of the TOV equation for modeling the compressible dissipative matter in the shell overlaying the incompressible gluon-quark superfluid core, whereas the latter is set to obey the zero-torque condition and to dynamically evolve according to the Onsager-Feymann equation (for further details see Sec. 2 and Eq. 10 in [8]).
The global iteration loop is designed here to find the optimal values of the parameters: 0 1 , α α , the elements of the sequence  Figure 5 we show the time-development of the rotational frequencies of the core, the ambient medium and of the magnetic field during the first 10 to 100 years after the birth of the pulsar. The long-term evolution of the magnetic field and the growing mass of the core and of the entire object are shown in Figure 6 and Figure 7. Here the mass of the pulsar's core grows with time to reach 0.15M  after 1000 years and reaches 0.55M  after 11,000 years; hence reproducing the exact total masses of 1.4M  for the Crab and 1.8M  for the Vela pulsars as revealed from observations. The relative ratio of inertia of both cores reads 3 20 Crab Vela Due to the incompressible, superfluid and supreconducting character of the core, the evolution of the magnetic field is solely connected to the dynamics of the ambient compressible and dissipative matter in the shell as well as to its dimensions (see [13] for further details on the physical aspects of compressibility of fluid flows). As the mass and dimension of the core grow with time, the surrounding shell must shrink. In this case, conservation of the magnetic flux should strengthen the magnetic field intensity. This interplay between the loss of magnetic energy due to loss of rotational energy and enhancement by conservation of magnetic flux in combination with dynamo action and other mechanisms,     where R  denotes the pulsar's radius. Assuming M E to roughly decay as the rotational energy E Ω , then we obtain: where α Ω is constant coefficient.
On the other hand, dynamo action in combination with magnetic flux conservation and other enhancement mechanisms would contribute positively to the magnetic field, that can, for simplicity absorbed in the term: α is set to ensure that the magnetic field remains in the very sub-equipartition regime. Hence the interplay between magnetic loss and enhancement would yield an effective magnetic field that evolves according to: Consequently, our model predicts that the decreasing volume of the shell enclosing the ambient medium in combination with dynamo action in the boundary layer could potentially be the mechanism that keeps the decay of magnetic fields in pulsars extremely weak.
In fact our model predicts the glitch activity of a newly born pulsar, which evolves into a Crab phase, followed by a Vela phase and finally by an invisible phase, to be approximately two orders of magnitude larger than it was estimated by other models (see Figure 8 to be compared to [1] [5]). According to our Figure 8. The glitch activity of a newly born pulsar versus cosmic time. In the very early times, the pulsar underwent millions of glitches, though the total ejected rotational energy was relatively very low. These activities start to be significant as the pulsar ages and become maximally effective between 100 and 60,000 years, followed by a decreasing phase, during which time-passages between successive glitches become increasingly longer. model pulsar may undergo millions or up to billions of glitches during their luminous life time with passages of time between two successive glitch events that range from nanoseconds in the very early time up to hundreds or even thousands of years toward the end of their luminous life times (see Figure 4). The vast difference in the evolution of glitch activity between the two approaches here may be attributed to the strong non-uniformity of time-duration between glitch events. Moreover, the model also predicts the occurrence of under-and overshootings that have been observed to accompany the glitch events in the Vela pulsar (see [14] and the references therein). In the case of the Vela, when the core expels certain number of vortices and moves to the next lower energy state, the enhanced rotational energy of the matter in the BL amplifies the magnetic field strength. Due to the non-locality of magnetic fields 1 , this enhancement is communicated to the crust via Alfven waves, A V , whereas the excess of rotational energy is communicated via shear viscosity with an effective propagational velocity vis V . As these two speeds are generally different with A vis V V > in most cases, the time-delay in the arrival of communication enforces the crust to react differently. Specifically, the arrival of magnetic enhancement prior to the rotational one leaves the crust subject to an enhanced magnetic braking and therefore to a stronger reduction of its rotational frequency (see the top panel of Figure 9).
Indeed, in the case of Vela, the propagational speed of Alfven waves may be esitmated to be of order 8   In fact, the under-and overshooting here may indicate that MFs are insensitive to the momentary rotational frequency of the crust, but rather to the activity and dynamics of the matter in the BL.
Extending this analysis to both the Crab and Vela pulsars, the relative time-delays is expected to be: Finally, although the physics is entirely different, the situation here is strikingly similar to action of the solar dynamo, which is considered to be located in the so-called tachcline between the rigid-body rotating core and the overlying convection zone [16].