3 Dynamo actions in proto-strange stars

Magnetic fields in a static plasma of finite electric conductivity $\sigma$ are subject to diffusion and dissipation, the timescale of which is $\tau_{\rm ohm}={{\check L}^2/\kappa_{\rm m}} ={4\pi\sigma {\check L}^2/c^2}$ ( $\kappa_{\rm m}=c^2/(4\pi \sigma)$ is the magnetic diffusivity). However, in a flowing plasma, magnetic fields may be created with certain velocity fields. This is the so-called "dynamo'' process, which converts kinetic energy into magnetic energy (e.g., Moffatt 1978). The equation describing the time-variation of the magnetic field ${\vec B}$ governs the dynamo action

$${\partial {\vec B} \over \partial t} = \nabla \times ({\vec v}\times {\vec B})+{\cal R}_{\rm m}^{-1}\nabla^2 {\vec B},$$ (33)

where ${\cal R}_{\rm m}={\tau_{\rm ohm}/\tau_{\rm v}} ={\check U} {\check L}/\kappa_{\rm m}$ is the magnetic Reynolds number ( $\tau_{\rm v}={\check L}/{\check U}$ is the advection timescale). The magnetic diffusivity can be estimated

$$\kappa_{\rm m} = {c^2\over 4\pi \sigma} = 5.8~10^{-3} \rho_{... ... (\alpha_{\rm s}/0.1)^{5/3} T_{11}^{5/3}~~ {\rm cm^2~s^{-1}}$$ (34)

for SQM (here $\sigma$ is calculated from Eq. (39) and Eq. (28) of Heiselberg & Pethick 1993). Thus ${\cal R}_{\rm m}\sim 10^{16}$ for PSSs. But for PNSs, the magnetic diffusivity is (TD93)

$$\kappa_{\rm m}^{\rm N} = 1.5~10^{-5} \rho_{15}^{-1/3}~~ {\rm cm^2~s^{-1}},$$ (35)

for neutron matter with an electron fraction of 0.2, which is about two orders smaller than $\kappa_{\rm m}$. From Eq. (17) and Eq. (34), one obtains the magnetic Prandtl number $P_{\rm m}$

$$P_{\rm m} = {\nu \over \kappa_{\rm m}} = 3.0~10^{12}~\rho_{15}^{2/9}E_{100}^{-3}~ (\alpha_{\rm s}/0.1)^{-5/3}~T_{11}^{-5/3}.$$ (36)

3.1 An estimate of field strength

Thompson & Duncan (TD93) concluded that the dominant kinetic energy to be converted into magnetic energy in the dynamo action of PNSs is the convective energy since the initial pulsar rotation period is probably much larger than the overturn time ($\sim$1 ms) of a convective cell, although large-scale $\alpha-\Omega$ dynamo action is essential for neutron stars with (i.e., initial period 1 ms) to produce very high fields ("magnetars'', Duncan & Thompson 1992). However, here we suggest that rotation can not be neglected even for pulsars with typical initial periods, because the differential rotation energy density ${E_{\rm d}-E_{\rm f}\over R^3} \sim 8~10^{31}$ erg cm^-3is even larger than the turbulent energy density $\rho v^2/2 \sim 5~10^{30}$ erg cm^-3 for P=10 ms. If the large-scale convection scenario discussed in Sect. 2.2 is possible, most of the differential rotation energy may be converted to magnetic energy by dynamo action. Let's first estimate the total differential rotation energy $E_{\rm d}$ in the case when the angular momentum of each mass element is conserved in the collapse,

$$E_{\rm d} = {16\pi^3\rho\over 3P_0^2}\int_0^R r_0^4 {\rm d}r \sim 3.1~10^{46}P_1^{-2}~~{\rm erg}.$$ (37)

For a typical initial period P=10 ms, one gets $E_{\rm d}\sim 10^{50}$ erg. The dominant energy in the core-collapse supernovae is the gravitational energy $E_{\rm g}\sim 0.6~GM^2/R \sim 3 ~10^{53}$ erg for $M=1.4~M_\odot$ and R=10⁶ cm, if a strange star is residual. We thus find that a significant part ($\sim$0.03% for P=10 ms) of the gravitational energy has to be converted to the rotation energy if angular momentum is conserved. If most of this differential rotation energy would be converted to magnetic energy by $\alpha-\Omega$ dynamo action, ${B^2\over 8\pi}\sim {E_{\rm d}-E_{\rm f}\over R^3}$, one obtains the saturation magnetic field in the interior of the stars,

$$B_{\rm sat}\sim 4.3~10^{14} P_1^{-1}~~{\rm G},$$ (38)

which is near the value estimated by Thompson & Duncan (TD93) who assumed that most of the convective energy is converted to the magnetic one. $B_{\rm sat}\sim 10^{16}$ G for a typical period P=10 ms. Certainly, the actual "dipole'' magnetic field $B_{\rm p}$ of pulsars should be only a fraction of $B_{\rm sat}$. For example, in the upper convection zone of the sun, the rms value of the magnetic field is only $10^{-2} B_{\rm sat}$. If the same is assumed for the pulsars, the rms value of fields in PSSs or PNSs should be $\sim$ 10¹² P₁^-1 G. However, the poloidal magnetic field should be another fraction of this rms value, thus

$$B_{\rm p} \mathrel{\mathchoice {\vcenter{\offinterlineskip\ha... ...riptstyle ...$$ (39)

Observations for ordinary pulsars show that the fields are distributed from $\sim$10¹¹ G to 5.5 10¹³ G (PSR J1814-1744). Therefore, it is likely the initial period P of ordinary pulsars could be in the range of a few hundreds to decades of milliseconds. If P<10 ms, much stronger dipole fields (e.g., $\sim$ 10¹⁴-10¹⁵G for magnetars) can be generated.

In the other scenario, where diffusivities based on neutrino scattering inhibit large-scale convection, the differential rotation is damped due to the momentum transport by neutrinos. In this case the dynamo may be driven dominantly by turbulent convection, and the field strength can be estimated as $B=\sqrt{4\pi\rho v_{\rm q}^2}\sim 10^{13}$ G, assuming an equilibrium state between kinetic and magnetic energy. The efficiency of converting differential rotation energy into magnetic energy by dynamo action in this scenario may be much smaller than in the case where convection with the large scale Lexists since a significant fraction of differential rotation energy is likely to be converted into thermal energy due to the high neutrino viscosity.

Thus, dynamo action may amplify significantly the field before PSSs cools down to temperatures T smaller than the critical one, $T_{\rm c}$. When $T<T_{\rm c}$, CSC appears, and the field would exist as a fossil one for a very long time since $\tau_{\rm ohm}\rightarrow \infty$ if $\sigma\rightarrow \infty$. In fact, Alford et al. (2000) investigated recently the effect of CSC on the magnetic fields, and found that, unlike the conventional superconductors where weak magnetic fields are expelled by the Meissner effect, color superconductors can be penetrated by external magnetic fields and such fields can exist stably on a timescale longer than the cosmic age.

Equation (39) may have an observational consequence for pulsar statistics, which in turn could test Eq. (39). In the magnetic dipole model of pulsars, $(3.2~10^{19})^2 P{{\rm d}P\over {\rm d}t}=B_{\rm p}^2$ (note: fields are assumed not to decay in SQM), and the rotation period P is a function of time t due to energy loss. P(0) denotes the initial period of pulsars. Considering Eq. (39), one has $P(t){{\rm d}P(t)\over {\rm d}t} = \zeta P(0)^{-2}$, the solution of which is

$$P(t)^2=P(0)^2+2\zeta~P(0)^{-2}t,$$ (40)

where $\zeta \sim 4.3~10^{-18}~{\rm s^3}$. Observational tests of this equation are highly desirable.

The condition $E_{\rm d}<E_{\rm g}$ gives a limit for the initial pulsar period P. Based on Eq. (37), one gets P>0.2 ms. Actually, $P\gg 0.2$ ms since the efficiency of converting gravitation energy into differential rotation energy during collapse may be very small. It is thus doubtful that supernovae can produce pulsars with submilliseconds periods. For recycled millisecond pulsars the above estimate is not relevant.

3.2 Fast dynamos

The magnetic field amplification processes in newborn pulsars are essentially fast dynamos because of the high magnetic Reynolds numbers ( ${\cal R}_{\rm m}\sim 10^{16}{-}10^{17}$ for large-scale convection, ${\cal R}_{\rm m}\sim 10^{10}$ for local turbulence) of both PSSs and PNSs. Unfortunately, the question of whether or not fast dynamo exist has not been answered theoretically although many numerical and analytical calculations strongly support the existence of kinematic fast dynamos for given sufficiently complicated flows (Soward 1994; Childress & Gilbert 1995). It is worth studying fluids without magnetic diffusion since the diffusion timescale is much longer than the advection timescale, $\tau_{\rm ohm}\gg \tau_{\rm v}$, for fast dynamos. The complex flows, such as stretch-twist-fold, may effectively amplify the field in this case. The amplified strong magnetic fields are concentrated in filaments with radii $l_{\rm f}$, which can be estimated to be $l_{\rm f} \sim {\cal R}_{\rm m}^{-1/2}{\check L} \sim 0.01/\sqrt{{\cal R}_{\rm m16}}$ mm ( ${\cal R}_{\rm m}={\cal R}_{\rm m16}~10^{16}$) by equating the diffusion timescale of filament field, $l_{\rm f}^2/\kappa_{\rm m}$, to the advection timescale $\tau_{\rm v}$. Fast dynamos in PNSs have been considered by Thompson & Duncan (TD93) who suggested PNS dynamos as the origins of pulsar magnetism. As the fluid parameters (e.g., ${\cal R}_{\rm m}$, $\nu$, $\kappa$) of both PSSs and PNSs are similar, fast dynamos may also work for newborn strange stars.

There are three timescales in the fast dynamo of PSSs: the diffuse timescale $\tau_{\rm ohm}={\check L}^2/\kappa_{\rm m}$, the advection timescale $\tau_{\rm v}={\check L}/{\check U}$, and the buoyancy timescale $\tau_{\rm b}$,

$$\tau_{\rm b} \approx 4c\sqrt{\pi\over 3GM}RL^{1/2}\rho^{1/2}B^{-1}$$

$$\sim 10~{\rm ms}~R_6L_5^{1/2}\rho_{15}^{1/2}B_{16}^{-1},$$ (41)

( $R=R_6\times 10^6$ cm, $L=L_5\times10^5$ cm, B=B₁₆ 10¹⁶ G). $B_{16}\sim 1.12~\rho_{15}^{1/2}v_8$ ( $v=v_8\times 10^8$ cm s^-1) if we assume that kinetic and magnetic energy densities are in equilibrium. Filaments can be generated inside the convective layer since turbulent convection is violent on small scales ($\sim$100 cm). , which means that filaments rise to the stellar surface by magnetic buoyancy as soon as strong fields are created by fast dynamos. Magnetic buoyancy flow or convection could create and amplify the poloidal (transverse) field from the azimuthal (aligned) field, at the same time, the aligned field could also be created and amplified from the transverse field by differential rotation as discussed in Sect. 2.1. Thus these processes may complete a "dynamo cycle'' in newborn pulsars.

Let's give some estimates for fast dynamos. There are mainly two types of flows in PSSs: eddy (convection) and shear (differential rotation). Actually, these flows are coupled. However, we may deal with them separately in order to have an overview of the field generation processes. For pure straining motion with velocity field ${\cal V}$,

$${\cal V} = \tau_{\rm v}^{-1} (-x,y,0),$$ (42)

which represents eddy flow to some extent, the magnetic field can be amplified considerably to $B_{\rm max}^{\rm e}=B_0{\cal R}_{\rm m}^{1/2}$ at a timescale $t_{\rm max}^{\rm e}={1\over 2}\tau_{\rm v}{\rm ln}{\cal R}_{\rm m}^{1/2}$ (Soward 1994). For an initial field strength $B_0\sim 10^{10}$ G, $B_{\rm max}^{\rm e}\sim 10^{18}{\cal R}_{\rm m16}^{1/2}$ G and $t_{\rm max}^{\rm e}\sim 18 \tau_{\rm v3}{\rm ln}{\cal R}_{\rm m16}^{1/2}$ ms ( $\tau_{\rm v}=\tau_{\rm v3}\times 10^{-3}$ s). On another hand, the linear shear flow (Soward 1994),

$${\cal V} = T_{\rm v}^{-1} (y,0,0),$$ (43)

can amplify the field to $B_{\rm max}^{\rm s}\sim B_0{\cal R}_{\rm m}^{1/3} \sim 2~10^{15}{\cal R}_{\rm m16}^{1/3}$ G at a growth time $t_{\rm max}^{\rm s}\sim T_{\rm v}{\cal R}_{\rm m}^{1/3} \sim 21T_{\rm v4}{\cal R}_{\rm m16}^{1/3}$ s for B₀=10¹⁰ G ( $T_{\rm v}=10^{-4}T_{\rm v4}\sim {L/(V(R)-V(R-L))}$, see Fig. 2). The timescale for field generation is thus from 10^-5 s (for ${\cal R}_{\rm m}\sim 10^{10}$) to 10 s (for ${\cal R}_{\rm m}\sim 10^{16}$), while the amplified fields could be 10¹² (for ${\cal R}_{\rm m}\sim 10^{10}$) to 10¹⁸(for ${\cal R}_{\rm m}\sim 10^{16}$) G in filaments. The magnetic fields emerging from under the stellar surface are likely to be much smaller than $B_{\rm max}$ in filaments because the filaments will expand to an approximately homogeneous field near and above the surface owing to magnetic pressure. Incoherent or coherent superposition of this buoyant field flux might result in the observed dipole moment of pulsars (TD93).

Differential rotation may play an essential role for the generation of large-scale magnetic fields through $\alpha-\Omega$dynamo process, but even in the absence of differential rotation, large-scale magnetic fields may be created. Owing to the alignment of small-scale convection rolls parallel to the axis of rotation, global magnetic fields can be generated as has been shown in various dynamo models (see, for instance, Busse 1975).