2 The birth of strange stars: differential rotation and convection?

The time-scale of PNSs (or PSSs) is of the order of a few seconds, which is three orders of magnitude longer than the dynamical one. It is thus worth studying the dynamical bulk evolution in these stars, such as differential rotation and convection. Unfortunately, no numerical model of supernova explosion known hitherto has included the conversion from PNSs to PSSs, although such a conversion may help in modelling the burst process (e.g., 1, helping to solve the present energetic difficulties in getting type II supernova explosions (Benvenuto & Horvath 1989); 2, giving a reasonable explanation for the second peak of neutrino emission in SN 1987A (Benvenuto & Lugones 1999)). The absence of a numerical model is caused by the lack of (1) a full theory determining the conditions at which the quark matter phase transition occurs and of (2) a detailed understanding of the complex burning process of neutron matter into strange matter.

Nevertheless, some efforts have been made in trying to understand the transition and combustion processes in detail. The phase conversion occurs in two steps: first neutron matter deconfinement occurs on a strong interaction time scale $\sim$10^-23 s, then chemical equilibration of the deconfined quark matter takes place on a weak interaction time scale $\sim$10^-8 s. Additional neutrinos and energy are produced in the second step (Dai et al. 1995). Further calculations of such transitions (Anand et al. 1997; Lugones & Benvenuto 1998) have also considered the effect of strong interactions, the effect of finite temperature and strange quark mass, and the effect of trapped neutrinos. Recently, Benvenuto & Lugones (1999) explored the occurrence of deconfinement transition in a PNS modeled by Keil & Janka (1995) and found that the deconfinement appears as long as the bag constant ${\check B}\leq$ 126 MeV fm^-3. Various estimates of the bag constant indicate that the preferred value of ${\check B}$ lies in the range of 60 MeV fm $^{-3}\leq {\check B} \leq 110$ MeV fm^-3 (Drago 1999), which means that deconfinement is very likely to happen. Such 2-flavor quark matter may be transformed immediately into a 3-flavor one if SQM is absolutely stable. From a kinetic point of view, Olinto (1987) has calculated for the first time the conversion of neutron stars into strange stars, suggesting a deflagration mode with a burning velocity range from 10⁴ kms^-1to a few cm/s. However it is found (Horvath & Benvenuto 1988; Benvenuto et al. 1989; Benvenuto & Horvath 1989) that such slow modes are unstable. This instability would be self-accelerated, and the burning should occur finally in detonation modes, although the transition from deflagration to detonation has not been well understood (Lugones et al. 1994). In order for the combustion to be exothermic, there exists a minimum density $\rho_{\rm c}$ for detonation to be possible, $\rho_{\rm c} \sim 2$ times of nuclear density for some cases (Benvenuto & Horvath 1989). Thus the detonation flame can not reach the edge of the compact core, and the outer part of the PNS would be expelled. As a result, a strange star is formed with an almost "bare'' quark surface which is essential to solve completely the "binding energy'' problem in some current pulsar emission models (Xu et al. 1999, 2001). Both the extra neutrino emissivity and the detonation wave can favour a successful core-collapse supernova explosion.

2.1 Differential rotation

It is expected that PNSs (thus PSSs) have a strong differential rotation (e.g., Janka & Mönchmeyer 1989; Goussard et al. 1998). This issue is uncertain, however, due to the lack of a rotating core model in the pre-supernova evolution simulations. The iron core collapses, being triggered by electron capture (for lower entropy core) and/or photodisintegration (for higher entropy core), almost in the same way as for free fall. Many initial core models of highly evolved massive stars without rotation have appeared in the literature (e.g., Arnett 1977; Bruenn 1985), in which the density $\rho_0$ - radius r₀ relation can be fitted by

$$% \rho_0(r_0) = \sum_{i=1}^{n}\rho_{0i} {\rm e}^{-\alpha_{i} r_0},$$ (1)

where $\rho_{0i}, \alpha_{i}$ are constants. For the first order approximation, we let n=1 in our following discussion. In this case, the Lagrangian mass coordinate ${\cal M}(r_0)$ is

$${\cal M}(r_0) = {4\pi \rho_{01}\over \alpha_1^3} [2 - (2+2\alpha_1r_0+\alpha_1^2r_0^2) {\rm e}^{-\alpha_1r_0}].$$ (2)

Setting ${\cal M}(10^8)\sim 1.4~M_\odot$ and $\rho_0(10^8)\sim 10^8$ g cm^-3 for the core with radius R₀=10⁸ cm, one obtains $\rho_{01}=2~10^{10}$ g cm^-3, $\alpha_1=5.4~10^{-8}$ cm^-1, by Eq. (1) and Eq. (2). We assume that the core before collapse with a uniform rotation of period P₀ can also be approximated by Eq. (1) for lack of a rotation core model in simulations, and we investigate below the collapse of the iron core with the above parameters. Here we also assume that pulsar rotation is mainly the result of angular momentum conservation during the core collapse process, rather than of the kick at birth (Spruit & Phinney 1998).

Based on the approximation, the total rotational energy E₀ of the core is

$$E_0 = {16\pi^3 \rho_{01} \over 3P_0^2}\cdot {\Im(\alpha_1, R_0)\over \alpha_1^5},$$ (3)

and the total angular momentum M₀ is

$$M_0 = {16\pi^2 \rho_{01}\over 3P_0}\cdot {\Im(\alpha_1, R_0)\over \alpha_1^5},$$ (4)

where the function $\Im (\alpha, R)$ is defined by

$$\Im (\alpha, R) 24 - (24 +24\alpha R+ 12\alpha^2 R^2$$ =

$$+ 4\alpha^3 R^3+ \alpha^4 R^4){\rm e}^{-\alpha R}.$$ (5)

It is found that $\lim_{\alpha \to 0}\Im(\alpha, R)/\alpha^5 = R^5/5$. Contrary to the case of the core before collapse, newborn strange stars may be well approximated as objects with homogeneous density (Alcock et al. 1986). For a strange star with mass $M=1.4~M_\odot$ and radius R=10⁶ cm, we find the density $\rho\sim 7~10^{14}$ g cm^-3. During the adiabatic collapse process, in which the entropy of each mass element does not change significantly, toroidal forces are negligible (although a poloidal force can cause a shock wave). The angular momentum of each mass element is thus conserved, and PSSs should be in differential rotation. We assume that the material at a shell with radius r₀ of the iron core before collapse contracts to another shell with radius r in a PSS,

$$r^3 = {3\over 4\pi \rho}{\cal M}(r_0).$$ (6)

According to angular momentum conservation of each mass element, one arrives at the velocity field V dependent on r and $\theta$ (polar angle),

$$V = {2\pi\sin\theta \over P_0} \cdot {r_0^2 \over r},$$ (7)

where r₀ is a function of r through Eq. (6), and the mass conservation law $4\pi r_0^2 \rho_0 {\rm d}r_0=4\pi r^2 \rho {\rm d}r$ has been used. Owing to the momentum transport by neutrinos, magnetic fields, and turbulence, a uniform rotation is approached after a certain time and the final rotational energy $E_{\rm f}$ can be estimated as

$$E_{\rm f} = {16\pi^3 \over 15P^2}\rho R^5\sim 2.3~10^{46}P^{-2},$$ (8)

corresponding to the angular momentum $M_{\rm f}$,

$$M_{\rm f} = {16\pi^2 \over 15P}\rho R^5,$$ (9)

where P is the period of newborn uniformly rotating strange star. According to angular momentum conservation in the collapse process, $M_0=M_{\rm f}$, the rotation period P₀ of the initial iron core before collapse is

$$P_0 = {5 \rho_{01} \Im(\alpha_1, R_0)\over \rho R^5 \alpha_1^5} P \sim 4.68~10^3 P,$$ (10)

for typical parameters, i.e., the core rotates with a period of about 50 s for an initial pulsar period $P\sim 10$ ms. From Eq. (7), one obtains the velocity derivative $\vert\nabla V\vert$,

$$\vert\nabla V\vert = \sqrt{({\partial V\over \partial r})^2 +({1\over r} {\partial V\over \partial \theta})^2},$$ (11)

as a function of $\theta$ and r.

Figure 1 shows the velocity and the velocity derivative of this differential rotation scenario for P=10 ms, respectively, based on Eq. (7) and Eq. (11). We find $\vert\nabla V\vert\sim 10^4$ s^-1 in the outer part of PNSs or PSSs. Such differential rotation may play an important role in the creation of magnetic fields of PSS or PNS, as will be discussed in Sect. 3.

Figure 1: The calculated velocity and velocity derivative profiles for different polar angles $\theta$. Dot-dashed, dotted, dashed, and solid lines are for $\theta =\pi /8, \pi /4, 3\pi /8, \pi /2$, respectively. The lower (upper) 4 lines, scaled by the left (right) ordinate, are for velocity (velocity derivative). P=10 ms in the calculations

Since the dynamo actions are in the outer convective layer with thickness $L\sim l_{\rm p}\sim 10^5$ cm (see Eq. (12) and Sect. 3), we give the velocity difference of differential rotation, $\Delta V=V(r=10^6~{\rm cm})-V(r=9~10^5~{\rm cm})$, as a function of P in Fig. 2.

Figure 2: The velocity difference in the convective outer layer with thickness of $L\sim 1$ km as a function of pulsar initial period P for different polar angles $\theta$. See Fig. 1 for the definition of lines

2.2 Convection

Turbulent convection in PNSs has been extensively investigated before (e.g. Burrows & Lattimer 1988; Wilson & Mayle 1988; Miralles et al. 2000). In the Kelvin-Helmholtz cooling phase, both the negative gradient of the entropy and of the lepton fraction can drive convection in newborn neutron stars, given that the mean free paths of leptons are much smaller than the convection length scale. Protostrange stars form after the SQM phase-transition with a time scale of $\sim R/c \sim 10^{-4}$ s, which is much shorter than that of neutrino diffusion and thermal evolution, assuming that the actual combustion mode is detonation (Benvenuto et al. 1989).

We expect that PSSs are convective, since negative gradients of entropy and neutrino fractions would also appear in the outermost layers which can lose entropy and neutrinos faster than the inner part. Following arguments used by Thompson & Duncan (TD93) in the case of PNSs, we compare the radiative and adiabatic temperature profiles to see whether convection occurs in PSSs. We assume neutrinos in the outer part of PSS are nondegenerate since there the neutrinos are lost rapidly. Iwamoto (1982) has shown that for the mean free path l of the scattering of nondegenerate neutrinos by relativistic degenerate quark matter with temperature T, $1/l\propto T^3$ holds. Thus, the neutrino opacity in SQM scales as T³ (rather than T² as for the case of PNSs), and the radiative temperature profile in PSSs is much steeper, $T(r) \propto p(r)$. We can employ the analytical equation of state for SQM in the case of zero strange quark mass ( $m_{\rm s}=0$) and zero coupling constant ( $\alpha_{\rm s}=0$) of strong interaction (Cleymans et al. 1986; Benvenuto et al. 1989) to estimate the adiabatic temperature-pressure relation. The entropy per baryon is $S \sim 3\pi^2T/\mu$ for the case of chemical potential $\mu\sim 300$ MeV and temperature $T\sim 30$MeV. Therefore $T(r)\sim (p(r)+{\check B})^{1/4}$ for SQM moving with fixed S is much less steep, and we can thus expect that a negative entropy gradient appears in the outer layer which is unstable to convection, similar to the case of PNSs. Accordingly we suggest that Schwarzchild convection exists in PSSs, whereas it will depend on detailed simulations whether Ledoux convection in PSSs can be established. If the timescale $\tau_{\rm NM}$ for neutron matter convection is much smaller than the neutrino diffusion time (a few seconds), we may expect that Ledoux convection takes place in a PSS since the negative gradient of the lepton fraction is nearly the same for PSSs as that for PNSs.

In the following, we try to estimate the properties of convection in PSSs. The local pressure scale height $l_{\rm p}$ in PSS is

$$l_{\rm p} = {p\over \rho g} \sim 2~10^5~{\rm cm},$$ (12)

where $g \sim {GM\over R^2}=1.33~10^{14} M_1~R_6^{-2}$ cm s^-2 ( $M_1=M/M_\odot$, R₆ is the radius R in 10⁶ cm) is the gravitational acceleration, $p = (\rho c^2 - 4{\check B})/3\sim 10^{34}$ dyne cm^-2 is the typical pressure in the outer convective layer, and the typical density there being chosen as $\rho = 5~10^{14}$ g cm^-3 (Alcock et al. 1986). We can assume the thickness L of the outer layer where dynamo action exists to be equal to this scale height, $L\sim l_{\rm p}$, and the temperature gradient $\nabla T$ thus is

$$% \nabla T \simeq {T\over L} \sim 5~10^5\,T_{11}~~{\rm K/cm},$$ (13)

where T₁₁ is the SQM temperature in 10¹¹ K. The numerical equation of state of SQM including the effect of non-zero strange quark mass (Eq. (29) of Chamaj & Slominski 1989) is

$$% \begin{array}{lll} \rho & = & 1.07~10^{14} {\check B}_{60} ... ...\\ & & + 1.34~10^{-14}\,T^2 {\check P}^{0.09619}, \end{array}$$ (14)

where ${\check P}/c^2\equiv p/c^2 + 1.07~10^{14} {\check B}_{60}$( ${\check B}_{60}$ is the MIT bag constant in 60 MeV fm^-3, p is the external pressure). According to Eq. (14) the coefficient of thermal expansion $\alpha$ is

$$% \alpha = - {1\over \rho} ({\partial \rho \over \partial T})... ...18}\,T_{11} {\check P}^{0.09619}\rho_{15}^{-1} ~{\rm K}^{-1}.$$ (15)

$\alpha = 1.33~10^{-14}$ K^-1 for T=10¹¹ K and p=0(Usov 1998).

There are two factors which can cause viscous stresses in a PSS: neutrino transport and quark scattering. Neutrino-induced viscosity dominates in a PSS on scales that are large compared to the neutrino mean-free path l. We use the neutrino mean-free path of nondegenerate neutrino scattering in SQM for a dimensional estimate (Iwamoto 1982),

$$l = 1.7~10^2~\rho_{15}^{-2/3} E_{100}^{-3} ~~{\rm cm},$$ (16)

where $\rho_{15}$ is SQM density in 10¹⁵ g cm^-3 and E₁₀₀ is neutrino energy in 100 MeV. According to Eq. (11) of TD93, the neutrino mean free path in nuclear matter, $l^{\rm N}\sim 10^2~\rho_{15}^{-1/3}$ cm for T=30 MeV, which is of the same order as l. Unfortunately, no well-determined neutrino viscosity in PSS has appeared in the literature. We thus just estimate the neutrino-induced viscosity by a simple kinetic argument (Wilson & Mayle 1988)

$$\nu = {1\over 3} lc\xi = 1.7~10^{10}\rho_{15}^{-2/3} E_{100}^{-3} ~~{\rm cm^2~s^{-1}},$$ (17)

where we have assumed the ratio $\xi$ of the neutrino energy density to the quark one to be $\sim$10^-2. However, the kinematic viscosity due to quark scattering in SQM (Heiselberg & Pethick 1993) is much smaller

$$\nu_{\rm q} \sim 0.1~\rho_{15}^{14/9}~(\alpha_{\rm s}/0.1)^{-5/3}~T_{11}^{-5/3}~~ {\rm cm^2~s^{-1}}.$$ (18)

Therefore turbulent convection may have a scale , which finally should be damped on a small scale by $\nu_{\rm q}$.

Two possibilities arise for the scenario of turbulence in PSSs. The first one is that the neutrino viscosity can effectively inhibit a large-scale convection with length scale L, and local convection with scale will exist. In this case the local thermal diffusivity due to quark scattering in SQM (Heiselberg & Pethick 1993) is

$$\kappa_{\rm q}\sim 0.39~\rho_{15}^{2/3}~(\alpha_{\rm s}/0.1)^{-1}~T_{11}^{-1}~~{\rm cm^2~s^{-1}}.$$ (19)

The Prandtl number is

$$P_{\rm rq}={\nu_{\rm q}\over \kappa_{\rm q}}\sim 0.25,$$ (20)

the Rayleigh number is

$$R_{\rm aq}={\alpha g \nabla T l^4\over \kappa_{\rm q}\nu_{\rm q}} \sim 7.2~10^{16},$$ (21)

and the Coriolis number is

$$\tau_{\rm q}={2\Omega l^2\over \nu_{\rm q}}\sim 2~10^8.$$ (22)

Malkus (1954) had estimated the mean-square values (v²) of the fluctuating velocity from the Boussinesq form of the hydrodynamic equations, and found (Eq. (64) in Malkus 1954)

$$v \sim {\kappa\over 3d}(R_{\rm a}-R_{\rm ac})^{1/2},$$ (23)

where $\kappa$ is the thermal diffusivity, d is the length scale of convection, $R_{\rm a}$ is the Rayleigh number, and $R_{\rm ac}$is the critical Rayleigh number. This relation was supported by experiments (Fig. 2 in Malkus 1954) for $R_{\rm a}=10^5-10^9$. Also, Clever & Busse (1981) found the perturbation energy (v²) increases nearly proportional to $R_{\rm a}-R_{\rm ac}$ at values of $R_{\rm a}>10^3$ for low-Prandtl-number convection ( , see Fig. 11 in Clever & Busse 1981). According to Eq. (23), the turbulent convective velocity on this scale is

$$v_{\rm q}\sim 3.5~10^5~{\rm cm}~{\rm s}^{-1},$$ (24)

if we choose $\kappa=\kappa_{\rm q}$, d=l, $R_{\rm a}=R_{\rm aq}$.

The second scenario is that the neutrino viscosity is not high enough to inhibit the large-scale convection, and convection with scale L is possible. We can also obtain some dimensionless numbers for this case. Since both the thermal energy and momentum in PSSs are transported by neutrinos now, the thermal diffusivity $\kappa$ can be estimated to be (Wilson & Mayle 1988)

$$\kappa = {1\over 3} lc = 1.7~10^{12}\rho_{15}^{-2/3} E_{100}^{-3} ~~{\rm cm^2~s^{-1}},$$ (25)

by a simple kinetic argument. Thus the Prandtl number is

$$P_{\rm r} = {\nu \over \kappa} = \xi \sim 0.1 -0.01.$$ (26)

For a rotating spherical fluid shell with thickness L and angular velocity $\Omega$, the Rayleigh number is

$$R_{\rm a} = {\alpha g \nabla T L^4\over \kappa\nu} \sim 6.0~10^{-23}~~T~L^3.$$ (27)

i.e. $R_{\rm a}\sim 6.0~10^3$ for $T \sim 10^{11}$ K, p=0and $L\sim 10^5$ cm. The Taylor number $\tau^2$ can be calculated by

$$\tau = {2\Omega~L^2\over \nu} = 1.2~10^{-10}~\rho_{15}^{2/3} E_{100}^3~\Omega~L^2.$$ (28)

$\tau\sim 1.2~10^3$ for $\Omega\sim 10^3$ s^-1 and $L\sim 10^5$ cm. The critical Rayleigh number $R_{\rm ac}$ can be estimated (Zhang 1995) by

$$R_{\rm ac} \simeq \sqrt{5 \tau}~10^2\sim 10^4.$$ (29)

Turbulent convection is thus possible for $R_{\rm a}>R_{\rm ac}$with a velocity which can be estimated to be (d=L)

$$v\mathrel{\mathchoice {\vcenter{\offinterlineskip\halign{\hfi... ...style ...$$ (30)

according to Eq. (23). Alternatively, if the mixing-length prescription (Böhm-Vitense 1958, TD92) for a fluid of semidegenerate fermions is adapted, the convective velocity in PSS becomes

$$v_{\rm ml} = ({\Gamma-1\over 2\Gamma} {L_{52}\over 4\pi R^2\r... ... \sim 6.8~10^7 L_{52}^{1/3} \rho_{15}^{-1/3}~~{\rm cm~s^{-1}},$$ (31)

where $\Gamma \equiv {\partial {\rm ln}p\over \partial {\rm ln}\rho} ={\rho c^2 \over \rho c^2 - 4{\check B}}\sim 5$, L₅₂ is the convective luminosity in 10⁵² erg s^-1. We note that $v_{\rm ml}$ and v are nearly of the same order, while for PNSs with $\Gamma=5/3$, the convective velocity is $5.4 ~10^7 L_{52}^{1/3} \rho_{15}^{-1/3}$ cm s^-1 $\sim 10^8$ cms^-1 for $\rho\sim 5~10^{14}$ gcm^-3. Thus both, the convection in PSSs and in PNSs have convective velocities of about 10⁸ cm s^-1 if large-scale convection is possible. Also the overturn timescales of PSSs and of PNSs are of the same order, $\tau_{\rm con}=l_{\rm p}/v\sim$ a few ms, and the Rossby number is $R_{\rm o}=P/\tau_{\rm con}\sim 10^3P_1$ in this possibility (P₁ is the value of the initial period in seconds).

Another cause of turbulent motion in PSS (and PNS) is differential rotation. The Reynolds number ${\cal R}$ of this shear flow is

$${\cal R} = {\nabla{\check U}{\check L}^2\over \nu},$$ (32)

where $\nabla{\check U}$ and ${\check L}$ are typical velocity derivative and length scales, respectively. For the local convection with scale , ${\cal R}_{\rm q}\sim 10^8$. For the convection with scale L, ${\cal R}\sim 5.8~10^3$. According to the experiment by van Atta (1966, Fig. 3 there), turbulence requires Reynolds number ${\cal R}>10^4$, thus in our case small scale ( ) turbulence may appear owing to the differential rotation.