GENERATION OF A SEED MAGNETIC FIELD AROUND FIRST STARS: THE BIERMANN BATTERY EFFECT

According to ADS10, the equation of magnetic field generation is given as

$$\begin{eqnarray} &&\frac{\partial \bm {B}}{\partial t}=\bm {\nabla }\times \left(\bm {v}\times \bm {B}\right) -\frac{c}{e n_{\rm e}^2}\bm {\nabla }n_{\rm e}\times \bm {\nabla }p_{\rm e} -\frac{c}{e}\bm {\nabla }\times \bm {f}_{{\rm rad}}, \end{eqnarray} \tag{ 1 }$$

where $\bm {B}$, $\bm {v}$, and e denote the magnetic flux density, the fluid velocity, and the elementary charge, respectively. p_e and n_e represent the pressure and the number density of electrons. $\bm {f}_{{\rm rad}}$ is the radiation force acting on an electron. The first term on the right-hand side is the advection term of magnetic flux, whereas the second term describes the Biermann battery term (Biermann 1950), which was not included in our previous study (ADS10). The third term is the radiation term which represents the momentum transfer from photons to gas particles. Note that the dissipation term due to the resistivity of the gas is omitted since it is negligible in comparison with the other terms (ADS10).

The radiation force on an electron, $\bm {f}_{{\rm rad}}$, involves two processes. The first one is the contribution from Thomson scattering, $\bm {f}_{\rm rad,T}$. $\bm {f}_{\rm rad,T}$ is given by

$$\begin{eqnarray} &&\bm {f}_{\rm rad,T} = \frac{\sigma _{\rm T}}{c}\int _0^{\nu _{L}}\bm {F}_{0\nu }d\nu + \frac{\sigma _{\rm T}}{c}\int _{\nu _{L}}^{\infty }\bm {F}_{0\nu }\exp[- \tau _{\nu _{L}}a(\nu)] d\nu, \nonumber \\ \end{eqnarray} \tag{ 2 }$$

where σ_T denotes the Thomson scattering cross section, $\bm {F}_{0\nu }$ is the unabsorbed energy flux density, ν_L is the Lyman-limit frequency, $\tau _{\nu _{L}}$ denotes the optical depth at the Lyman limit regarding the photoionization, and a(ν) is the frequency dependence of the photoionization cross section, which is normalized at the Lyman-limit frequency.

Another source of the radiation force is the momentum transfer from photons to electrons through the photoionization process. The force, expressed as $\bm {f}_{\rm rad,I}$, is

$$\begin{eqnarray} &&\bm {f}_{{{\rm rad}},I} = \frac{1}{2}\frac{n_{{\rm H}\,{\mathsc i}}}{c n_{\rm e}} \int _{\nu _{L}}^{\infty } \sigma _{\nu _{L} } a(\nu) \bm {F}_{0\nu }\exp [- \tau _{\nu _{L}} a(\nu)] d\nu, \end{eqnarray} \tag{ 3 }$$

where $\sigma _{\nu _{L}}$ is the photoionization cross section at the Lyman limit and $n_{\rm H\,\mathsc {i}}$ represents the number density of neutral hydrogen atoms.

In order to assess the electron number density, we also solve the following photoionization rate equation for electrons:

$$\begin{eqnarray} \frac{\partial y_{\rm e}}{\partial t} + (\bm {v} \cdot \bm {\nabla }) y_{\rm e} &=& \frac{\Gamma }{n_{\rm H}} - \alpha _{\rm B} y_{\rm e} y_{\rm p} n_{\rm H} + k_{\rm coll}y_{\rm e}y_{\rm H\,\mathsc{i}} n_{\rm H}, \end{eqnarray} \tag{ 4 }$$

where α_B and k_coll denote the case B recombination rate and collisional ionization rate per unit volume, respectively. y_e, y_p, and $y_{\rm H\,\mathsc{i}}$ is the number fraction of electrons, protons, and neutral hydrogen atoms, respectively. n_H is the number density of hydrogen nuclei and $\bm {v}$ denotes the velocity of the fluid. The photoionization rate per unit volume, Γ, is also obtained by the formal solution of the radiation transfer equation:

$$\begin{eqnarray} &&\Gamma = n_{\rm H\,\mathsc{i}}\int _{\nu _{L}}^{\infty }\sigma _{\nu _{L}} \frac{{F}_{0\nu }}{h\nu }\exp [- \tau _{\nu _{L}}a(\nu)] d\nu. \end{eqnarray} \tag{ 5 }$$

We solve the ordinary set of hydrodynamics equations:

$$\begin{eqnarray} \frac{D\rho }{Dt}&=&-\rho \bm {\nabla }\cdot \bm {v}, \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} \frac{D\bm {v}}{Dt}&=&-\frac{1}{\rho }\bm {\nabla }p-\bm {\nabla }\phi _{\rm DM}, \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} \rho \frac{D\epsilon }{Dt}&=&G-L-p\bm {\nabla }\cdot \bm {v}, \end{eqnarray} \tag{ 8 }$$

and the equation of state

$$\begin{eqnarray} &&p=(\gamma -1)\rho \epsilon, \end{eqnarray} \tag{ 9 }$$

where ρ, p, and are the density, pressure, and the specific energy of the fluid, respectively. γ denotes the specific heat ratio. G and L are the radiative heating rate and cooling rate per unit volume, respectively. ϕ_DM is the gravitational potential of the dark matter halo (see Section 2.4). The feedback from the magnetic fields to the fluid is neglected in Equation (7), since we consider the generation of very weak magnetic fields. We also omit the self-gravitational force of gas, which is unimportant as long as we consider gas with n_H ≲ 10³ cm⁻³.

In order to perform hydrodynamics simulations, we use the Cubic Interpolated Profile (CIP) method (Yabe & Aoki 1991). The CIP scheme basically tries to solve not only the advection of physical quantities but also the derivatives of the quantities. Using this scheme, we can capture spatially sharp profiles of fluids that are always expected in problems including the propagation of ionization fronts. In this paper, the CIP scheme is applied to the advection terms of Equations (4) and (6)–(8).

Using the formal solution of the radiation transfer equations, the radiative heating rate G is assessed as

$$\begin{eqnarray} &&G=n_{\rm H\,\mathsc{i}}\int _{\nu _{L}}^{\infty }\frac{F_{0\nu }}{h\nu }h\left(\nu -\nu _{L}\right) \sigma _{\nu _{L}}a(\nu)\exp [- \tau _{\nu _{L}}a(\nu)]d\nu.\nonumber \\ \end{eqnarray} \tag{ 10 }$$

The cooling rate, L, is given as

$$\begin{eqnarray} &&L=L_{\rm coll}n_{\rm e}n_{\rm H\,\mathsc{i}}+L_{\rm rec}n_{\rm e}n_{\rm p} +L_{\rm exc}n_{\rm e}n_{\rm H\,\mathsc{i}}+L_{\rm ff}n_{\rm e}n_{\rm p},\nonumber \\ \end{eqnarray} \tag{ 11 }$$

where L_coll, L_rec, L_exc, and L_ff are the cooling coefficients associated with collisional ionization, recombination, collisional excitation, and free–free emission. These cooling rates are taken from the compilations in Fukugita & Kawasaki (1994).

We consider a mini-halo in the IGM at redshift z ≃ 20 exposed to an intense radiation flux from a nearby first star. We assume that the dark matter density profile of the mini-halo is described by the Navarro–Frenk–White profile (Navarro et al. 1997),

$$\begin{eqnarray} &&\rho _{\rm DM}\left(r \right) = \frac{\rho _{\rm s}}{\left(r/r_{\rm s}\right) \left(1+r/r_{\rm s}\right)^2}, \end{eqnarray} \tag{ 12 }$$

where r is the radial distance from the center of the halo. ρ_s and r_s are a characteristic density and radius, which are determined by a halo mass M_halo collapsing at redshift z (Prada et al. 2011). We assume the gas density profile is a core-halo structure with the ρ∝r⁻² envelope. The core radius r_c is determined by the core density and the total gas mass (M_gas = (Ω_B/Ω_M)M_halo). We study various cases of core densities (n₀ cm⁻³), the halo mass M_halo, and the distances between the source star and the halo center (D kpc). We assume stationarity of the mini-halo with respect to the source star. This assumption is based upon the fact that the change in distance between the star and the halo due to the relative motion is smaller than D if we consider cosmic expansion. We remark that in case where the neighboring overdense region and the source star are contained in the same halo of >10⁷ M_☉, a change of the distance due to the velocity dispersion of the halo could have a significant effect, especially for low-mass source stars.

Initially, the ambient gas is assumed to be neutral when the source star is turned on. The initial number density of the ambient gas in the intergalactic space is n_IGM = 10⁻² cm⁻³. We assume that the initial temperature is 500 K. We perform two-dimensional simulations in cylindrical coordinates assuming axial symmetry. As shown in Figure 1, the computational domain is 100 pc × 200 pc in the R–z plane. We consider source stars of various masses, M_* = 500, 300, 120, 60, 25, and 9 M_☉. The luminosities, effective temperatures, and ages of these stars are taken from the table of Schaerer (2002). The incident radiation from the source star is assumed to be perpendicular to the left edge of the computational domain.

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We employ four models (A–D) of different M_halo,n₀ and D listed in Table 1; these are plausible values of the mini-halos in the standard ΛCDM cosmology (see Section 3.2). The number of grids we use in these simulations is basically 250 × 500, which is confirmed to be enough to obtain physical results by the convergence study (see Section 3.4).

First, we consider a first star of 500 M_☉ (t_age = 2 × 10⁶ yr). Figure 2 shows the results for model A. The two columns correspond to the snapshots at 0.1 Myr and 2 Myr. The top row shows the color contour of the mass density of the gas. The gas density distribution is isotropic at 0.1 Myr, while it is highly disturbed by the radiation flux from the left at 2 Myr. We can also find a shock front at 2 Myr inside the core of the overdense region.

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The second and third rows show the maps of the temperature and the electron density. Clearly, an ionization front is generated by the flux from the left, and it propagates into the core of the cloud. It is also worth noting that the patterns of temperature and electron density distribution are similar but not identical to each other. A slight difference between the two contour maps leads directly to the nontrivial Biermann battery term.

The bottom row shows the distribution of the generated magnetic field strength. As expected, we obtain strong magnetic fields in the neighborhood of the ionization front. In addition, strong magnetic fields are produced at the center of the core where the density is the highest. The peak field strength is as strong as 6 × 10⁻¹⁸ G in this case.

In Figure 3, we show the time evolution of the peak magnetic field strength in the simulated region (red curve). We also plot the peak magnetic field generated by the Biermann battery term (blue) as well as that generated by the radiation processes (green). The field strength grows almost linearly in this model, and the final strength is as large as ∼6 × 10⁻¹⁸ G. We also find that the radiation process is less important than the Biermann battery effect. However, it is still noteworthy that the difference between the two contributions is only a factor of ∼10, although the natures of these processes are very different from each other.

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We also investigate the other models of n₀ and D listed in Table 1. The snapshots at 2 Myr for these models (B, C, and D from top to bottom) are shown in Figure 4. The left column shows the magnetic field strength, whereas the right column illustrates the number density of electrons. In model B, the generated magnetic field is of the order of ∼10⁻¹⁹ G (top left panel), which is smaller than that in model A. Since the ionization front is not trapped by the dense core in model B (top right), the source terms of Equation (1) become very small as soon as the ionization front passes through the core. As a result, the magnetic field does not have enough time to grow. In models C and D, the core is 10 times more distant from the source star than models A and B. Therefore, the ionization front is trapped at lower density regions and becomes less sharp than that in model A. Consequently, the generated magnetic field is smaller than that in model A (middle and bottom rows).

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We also plot the time evolution of the peak field strengths of models A–D in Figure 5. Models A, C, and D mostly increase monotonically, whereas model B has a clear plateau/decline. Such different behavior also comes from the fact that the ionization front immediately passes through the core in model B. In such a case, the time for the magnetic field to grow is not enough as stated above. In addition, the fluids that host the generated magnetic fields expand due to the thermal pressure of the photoionized gas. Such expansion results in the slight decline of the magnetic field strength.

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The dependence on the source stellar mass M_* is also studied. We employ six models of M_* = 500, 300, 120, 60, 25, and 9 M_☉, while the other parameters are the same as in model A. Figure 6 shows the peak magnetic field strength as a function of M_*. The peak field strength of each run is evaluated when it gets to the time for the death of the source star. The peak field strength basically increases as the stellar mass becomes more massive. However, the difference between the field strength of M_* = 9 M_☉ and 500 M_☉ is a factor of nine, which is a small difference for such a mass difference. The reason for this behavior comes from the fact that the more massive the stars are, the more ionizing photons they emit but also the shorter lifetime they have. These two competing effects cancel each other.

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In any case, the generated magnetic field strengths stay around 10⁻¹⁸–10⁻¹⁷ G if we consider stellar masses of 9 M_☉–500 M_☉, a range wide enough for first stars.

The parameters n₀ and D we employ in this paper are reasonable values. The baryonic density of the virialized halo at z = 20 is ∼1 cm⁻³, and the sizes of the first halos are ≲ 100 pc. In addition, the radii of cosmological H ii regions in the numerical simulations are a few kpc (e.g., Yoshida et al. 2007). Thus, our models A–D are reasonable for the standard cosmological model.

To summarize the results of this section, the magnetic field strength generated around a first star is 10⁻¹⁹ G < B < 10⁻¹⁷ G for M_* = 500 M_☉, and a factor of a few smaller for less massive stars.

In addition, we also make mention of the coherence length of the magnetic field. Due to limited computational resources, we employ a rather small box size (∼200 pc). The resultant coherence length of the magnetic field clearly exceeds the box size in most of the cases. In addition, if the ionization structure is more or less similar to that of our previous results in ADS10, the coherence length will be as large as ∼1 kpc, which is the size of the shadow.

Since the equation for magnetic field generation, Equation (1), includes spatial derivatives in its source terms, we have to pay attention to the effects of the cell size on our numerical results. We check the numerical convergence of the magnetic field strengths for model A. We perform runs with N_R × N_z = 63 × 125, 125 × 250, 250 × 500 (canonical), 400 × 800, 500 × 1000, where N_R and N_Z are the number of grids in the R-axis and the z-axis, respectively. In Figure 7, the magnetic field probability distribution functions (PDFs) of these runs are plotted. The abscissa is the absolute value of the magnetic field strength, while the vertical axis represents the fraction of grid cells that fall in the range of [B, B + ΔB], where ΔB = 5 × 10⁻¹⁹ G. The peak field strength declines as the number of grids increases. However, the peak field strength converges at ∼6 × 10⁻¹⁸ G and the PDFs show similar distributions for N_R × N_Z ⩾ 250 × 500. Therefore, the numerical results of the simulations converge very well at N_R × N_Z = 250 × 500, which we use in other runs.

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