Generation of Seed Magnetic Field around First Stars: the Biermann Battery Effect
aa r X i v : . [ a s t r o - ph . C O ] A ug Generation of Seed Magnetic Field around First Stars:the Biermann Battery Effect
Kentaro Doi and Hajime Susa Department of Physics, Konan University, Okamoto, Kobe, Japan
ABSTRACT
We investigate generation processes of magnetic fields around first stars. Sincethe first stars are expected to form anisotropic ionization fronts in the surround-ing clumpy media, magnetic fields are generated by effects of radiation force aswell as the Biermann battery effect. We have calculated the amplitude of mag-netic field generated by the effects of radiation force around the first stars inthe preceding paper, in which the Biermann battery effects are not taken intoaccount. In this paper, we calculate the generation of magnetic fields by theBiermann battery effect as well as the effects of radiation force, utilizing theradiation hydrodynamics simulations. As a result, we find that the generatedmagnetic field strengths are ∼ − G − − G at ∼ Subject headings: early universe—HII regions —radiative transfer — magneticfields
1. Introduction
According to the theoretical studies in the last decade, first stars are expected to bevery massive ( & M ⊙ ) (e.g., Bromm, Coppi & Larson 2002; Nakamura & Umemura 2001;Abel, Bryan, Norman 2002; Yoshida 2006). Recent studies which properly address the ac-cretion phase of first star formation also revealed that the primary star formed in the center [email protected] [email protected] & M ⊙ , although significant fractionof first stars are less massive ( . M ⊙ )(Stacy et al. 2010; Clark et al. 2011a,b; Greif et al.2011). In any case, star formation episodes in the very early universe are different from thatin local galaxies. One of the reasons of this difference is that the primordial gas clouds thathost the first stars lack heavy elements, though they are most efficient coolants in interstellarclouds at T . ∼ n H < cm − , which is muchhotter than the local interstellar molecular gas clouds. Consequently, the Jeans mass of thecollapsing primordial gas is much larger than that of the interstellar gas, which leads to theformation of very massive stars(e.g., Omukai 2000).Another important difference between the star formation sites in the early universe andlocal molecular clouds is the strengths of magnetic fields. Typical field strength B ∼ µ G inthe local molecular gas results in the formation of jets from protostars and regulate the gravi-tational collapse of cloud cores. The effects of magnetic field on the star formation in the earlyuniverse have been studied from theoretical aspects. First of all, the coupling of the magneticfield with the primordial gas was studied by a detailed chemical reaction model(Maki & Susa2004, 2007). They found that magnetic field is basically frozen-in the primordial gas duringits collapse, differently from the local interstellar gas(e.g., Nakano & Umebayashi 1986a,b).Under the assumption of the flux freezing condition, the dynamical importance of the mag-netic field is also investigated by several authors. In case the magnetic field is stronger than ∼ − G at n H = 10 cm − , field strength is amplified to ∼ G at n H = 10 cm − which isenough to launch the bipolar outflows, and to suppress the fragmentation of the accretiondisk(Machida et al. 2006, 2008). Tan & Blackman (2004) estimated the condition for themagnetorotational instability (MRI) to be activated in the accretion disk around the proto-star, by comparing the Ohmic dissipation time scale with the growth time scale of MRI. Theyfound that the condition is B & − G at n H = 10 cm − which corresponds to B & − Gat n H = 10 cm − . We also remark that the turbulent motion powered by the accreting gascan amplify the initial field strength faster than the simple flux freezing, although the effectsare still under debate. Thus, the magnetic field could be of importance at the final phase ofthe star formation process in primordial gas clouds, if B & − − − G at n H = 10 cm − ,i.e. at the initial phase of the collapse of primordial gas in the mini-halos.In addition, recent theoretical studies suggested that the heating by the ambipolar dif-fusion process in star-forming gas clouds could change the thermal evolution of the prestellarcore in case the field strength is as strong as 10 − G at IGM comoving densities (Schleicher et al.2009; Sethi et al. 2008). This process also might leads to the formation of massive blackholes(Sethi et al. 2010) since such heating can shut down the H cooling and open the pathof the atomic cooling(e.g., Omukai & Yoshii 2003). 3 –In any case, it is important to determine the magnetic field strengths in star forminggas clouds in the early universe, in order to quantify the effects of magnetic fields on theprimordial star formation. In spite of such potential importance, initial seed magnetic fieldstrengths are still unknown observationally. Only the observations on the distortion of thecosmic microwave background spectrum(e.g., Barrow et al. 1997; Seshadri & Subramanian2009), and the measurements of the Faraday rotation in the polarized radio emission fromdistant quasars (e.g., Blasi et al. 1999; Vall´ee 2004) imply rather mild upper limits on thefield strengths in IGM, at the level of B ∼ − G. On the other hand, various theoreticalstudies predicted that it is as small as . − G at IGM densities. For instance, there aremodels generating the magnetic field by the Biermann battery(Biermann 1950) during thestructure formation (Kulsrud et al. 1997; Xu et al. 2008). Recent numerical simulation bySur et al. (2010) suggests that strong magnetic field emerges during collapse of turbulentprestellar cores of primordial gas due to the Biermann battery and the turbulent dynamoaction. There is also a number of models that the fields are generated just after the bigbang(e.g., Turner& Widrow 1988; Ichiki et al. 2006).It is also suggested that reionization of the universe inevitably generates magnetic fields.Gnedin et al. (2000) have shown that the considerable Biermann battery term arises at theionization fronts in their cosmological simulations. They predict ∼ − G at δρ/ρ ≃ .It also is suggested that in the neighborhood of luminous sources like QSOs(Langer et al.2003) or first stars (Ando et al. (2010); here after ADS10) magnetic field could be generatedthrough the momentum transfer process from ionizing photons to electrons, since ∇ × E =0 is satisfied at the borders between the shadowed regions and ionized regions. ADS10predicted B ∼ − G at IGM densities at z = 20, however, they did not take into accountthe Biermann battery effect, since they assume that the gas is isothermal and static.In this paper, we extend our previous study (ADS10) to investigate the generation pro-cess of magnetic fields due to the ionizing radiation from first stars with more precision. Wetake into consideration not only the effects of radiation force but also the Biermann batterymechanism, utilizing the two-dimensional radiation hydrodynamics simulations. Then wediscuss whether the magnetic field strength obtained in our study could be important forsubsequent star formation process. In section 2, we describe the basic equations and thesetup of our model. We show the results of our calculations in section 3. Sections 4 and 5are devoted to the discussions and summary. 4 –
2. Basic equations & Model2.1. Equation of magnetic field generation
According to ADS10, the equation of magnetic field generation is given as ∂ B ∂t = ∇ × ( v × B ) − cen ∇ n e × ∇ p e − ce ∇ × f rad (1)where B , v and e denote the magnetic flux density, fluid velocity, and the elementary charge,respectively. p e and n e represent the pressure and the number density of electrons. f rad isthe radiation force acting on an electron. The first term on the right-hand side is theadvection term of magnetic flux, whereas the second term describes the Biermann batteryterm (Biermann 1950), which was not included in our previous study (ADS10). The thirdterm is the radiation term which represents the momentum transfer from photons to gasparticles. Remark that the dissipation term due to the resistivity of the gas is omitted, sinceit is negligible in comparison with the other terms (ADS10). The radiation force on an electron, f rad involves two processes. The first one is thecontribution by the Thomson scattering, f rad , T . f rad , T is given by f rad , T = σ T c Z ν L F ν dν + σ T c Z ∞ ν L F ν exp [ − τ ν L a ( ν )] dν, (2)where σ T denotes the cross section of the Thomson scattering, F ν is the unabsorbed energyflux density, ν L is the Lyman-limit frequency, τ ν L denotes the optical depth at the Lymanlimit regarding the photoionization, and a ( ν ) is the frequency dependence of photoionizationcross section which is normalized at the Lyman limit frequency.Another source of the radiation force is the momentum transfer from photons to electronsthrough the photoionization process. The force expressed as f rad , I , is f rad , I = 12 n HI cn e Z ∞ ν L σ ν L a ( ν ) F ν exp [ − τ ν L a ( ν )] dν, (3)where σ ν L is the photoionization cross section at the Lyman limit and n HI represents thenumber density of neutral hydrogen atoms. 5 –In order to assess the electron number density, we also solve the following photoionizationrate equation for electrons: ∂y e ∂t + ( v · ∇ ) y e = Γ n H − α B y e y p n H + k coll y e y HI n H , (4)where α B and k coll denote the case B recombination rate and collisional ionization rate perunit volume, respectively. y e , y p and y HI is the number fraction of electrons, protons andneutral hydrogen atoms, respectively. n H is the number density of hydrogen nuclei and v denotes the velocity of the fluid. The photoionization rate per unit volume, Γ, is alsoobtained by the formal solution of the radiation transfer equation:Γ = n HI Z ∞ ν L σ ν L F ν hν exp [ − τ ν L a ( ν )] dν. (5) We solve the ordinary set of hydrodynamics equations:
DρDt = − ρ ∇ · v , (6) D v Dt = − ρ ∇ p − ∇ φ DM , (7) ρ DǫDt = G − L − p ∇ · v , (8)and the equation of state p = ( γ − ρǫ, (9)where ρ , p , and ǫ 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 rateper unit volume, respectively. φ DM is the gravitational potential of the dark matter halo(see2.4). The feedback from magnetic fields to the fluid is neglected in equation(7), since weconsider the generation of very weak magnetic fields. We also omit the self gravitationalforce of gas, which is unimportant as long as we consider the gas with n H . cm − .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 theadvection 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 the problems 6 –including the propagation of ionization fronts. In this paper, the CIP scheme is applied tothe advection terms of equations (4) and (6)-(8).Using the formal solution of radiation transfer equations, the radiative heating rate G is assessed as G = n HI Z ∞ ν L F ν hν h ( ν − ν L ) σ ν L a ( ν ) exp [ − τ ν L a ( ν )] dν. (10)The cooling rate, L , is given as L = L coll n e n HI + L rec n e n p + L exc n e n HI + L ff n e n p , (11)where L coll , L rec , L exc and L ff are the cooling coefficients regarding the collisional-ionization,the recombination, the collisional excitation and the free-free emission. These cooling ratesare taken from the compilations in Fukugita & Kawasaki (1994). We consider a mini-halo in the IGM at redshift z ≃
20 exposed to an intense radiationflux from a nearby first star. We assume that the dark matter density profile of the mini-halois described by the NFW profile (Navarro et al. 1997) ρ DM ( r ) = ρ s ( r/r s ) (1 + r/r s ) , (12)where r is the radial destance from the center of halo. ρ s and r s are a characteristicdensity 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 ρ ∝ r − 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 ( n cm − ), the halo mass M halo and distances between the source star and halo center ( D kpc). We assume stationarityof the mini-halo with respect to the source star. This assumption is based upon the factthat the the change of distance between the star and the halo due to the relative motion issmaller than D , if we consider the cosmic expansion. We remark that in case the neighboringoverdense region and the source star is contained in a same halo of > M ⊙ , change of thedistance due to the velocity dispertion of the halo could have significant effect especially forlow mass source stars. 7 –Initially, the ambient gas is assumed to be neutral when the source star is turned on. Theinitial number density of the ambient gas in the intergalactic space is n IGM = 10 − cm − . Weassume that the initial temperature is 500K. We perform two-dimensional simulations in thecylindrical coordinates assuming axial symmetry. As shown in Figure 1, the computationaldomain is 100 pc ×
200 pc in R − z plane. We consider source stars of various masses, M ∗ = 500,300,120,60, 25 and 9 M ⊙ . The luminosities, effective temperatures and the agesof these stars are taken from the table of Schaerer (2002). The incident radiation from thesource star is assumed to be perpendicular to the left edge of the computational domain.We employ four models (A-D) of different M halo ,n and D listed in Table 1, these areplausible values of the minihalos in standard ΛCDM cosmology(see section 3.2). The numberof grids we use in these simulations is basically 250 ×
500 which is confirmed to be enoughto obtain physical results by the convergence study (see section 3.4).
3. Results3.1. Typical results
First, we consider a first star of 500 M ⊙ ( t age = 2 × yr). Figure 2 shows the resultsfor the model A. Two columns correspond to the snapshots at 0 . . × − G in this case.In Figure 3, we show the time evolution of the peak magnetic field strength in thesimulated region (red curve). We also plot the peak magnetic field generated by the Biermannbattery term (blue) as well as the one by the radiation processes (green). The field strengthgrows almost linearly in this model, and the final strength is as large as ∼ × − G. 8 –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 afactor of ∼
10, although the nature of these processes are very different from each other. n , D , M ∗ We also investigate the other models of n and D listed in Table 1. The snapshots at2Myr of these models (B,C and D from top to bottom) are shown in Figure 4. The leftcolumn shows the magnetic field strength, whereas the right column illustrates the numberdensity of electrons. In the model B, the generated magnetic field is of the order of ∼ − G(top left panel), which is smaller than that in the model A. Since the ionization front is nottrapped by the dense core in the 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, themagnetic field does not have enough time to grow. In the models C and D, the core is 10times more distant from the source star than the models A and B. Therefore, the ionizationfront 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 the model A (middle andbottom row).We also plot the time evolution of the peak field strengths of the models A-D in Figure5. The models A, C and D mostly increase monotonically, whereas the model B has a clearplateau/decline. Such different behavior also comes from the fact that the ionization frontimmediately passes through the core in the model B. In such case, the time for the magneticfield to grow is not enough as stated above. In addition, the fluids that host the generatedmagnetic fields expand due to the thermal pressure of photoionized gas. Such expansionresults in the slight decline of the magnetic field strength.The dependence on the source stellar mass M ∗ is also studied. We employ six modelsof M ∗ = 500,300,120,60,25 and 9 M ⊙ , while the other parameters are same as the model A.Figure 6 shows the peak magnetic field strength as a function of M ∗ . The peak field strengthof each run is evaluated when it gets to the time for the death of the source star. The peakfield strength basically increases as the stellar mass becomes more massive. However, thedifference between the field strength of M ∗ = 9 M ⊙ and 500 M ⊙ is a factor of 9, which isa small difference for such a mass difference. The reason of this behavior comes from thefact that the more massive the stars are, the more ionizing photons they emit, but also theshorter lifetime they have. These two competing effects cancel with each other.In any case, the generated magnetic field strengths stay around 10 − − − G, if we 9 –consider the stellar mass of 9 M ⊙ - 500 M ⊙ , which range is wide enough for first stars.The parameters n and D we employ in this paper are reasonable values. The baryonicdensity of the virialized halo at z = 20 is ∼ − , and the sizes of the first halos are . − G < B < − G for M ∗ = 500 M ⊙ , and a factor of a few smallerfor less massive stars. In addition, we also remark that the coherence length of the magnetic field. Due to thelimited computational resource, we employ rather small box size ( ∼ ∼ Since the equation of the magnetic field generation (1) includes spatial derivatives in itssource 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 the model A. Weperform runs with N R × N z = 63 × , × , × , × , × N R , N Z is the number of grids in R -axis and z -axis, respectively. In Figure 7, themagnetic field probability distribution functions (PDFs) of these runs are plotted. The axis ofabscissas is the absolute value of the magnetic field strength, while the vertical axis representsthe fraction of grid cells that fall in the range of [ B, B + ∆ B ], where ∆ B = 5 × − G. Thepeak field strength decline as the number of grids increases. However, the peak field strengthconverges ∼ × − G and PDFs show similar distributions for N R × N Z ≥ × N R × N Z = 250 ×
4. Discussions
In this paper, we investigate the magnetic field generated by first stars. As a result, themaximal magnetic field strength is . − G, mainly generated by the Biermann batterymechanism.In fact, the order of magnitude of the magnetic field generated by the Biermann batterycould be assessed as B ∼ cn e e (cid:16) n e ∆ r (cid:17) (cid:16) p e ∆ r (cid:17) sin θ t age ∼ . × − G (cid:18) t age (cid:19) (cid:18) sin θ . (cid:19) (cid:18) ∆ r (cid:19) − (cid:18) T K (cid:19) , (13)where ∆ r denotes the typical length of n e and p e change significantly, θ is the angle between ∇ n e and ∇ T . The resultant value of above equation is roughly consistent with the resultsof our numerical simulations.We find the relative importance of the Biermann battery effect versus the radiativeprocesses in this paper. However, this is only relevant for present setup, because the twoeffects depend differently on various parameters. In particular, the Biermann battery effectdo not depend on the flux of the source star directly (see eq.13), whereas the radiation forceis proportional to the flux. This means if consider the magnetic field generation process inthe very neighbor of the source objects, such as the accretion disks of the protostars/blackholes, radiative processes could play central roles in the generation of magnetic field. Wewill study this issue in the near future.The field strength obtained in this paper is similar to the results of Xu et al. (2008),in which they investigated magnetic fields in collapsing mini-halos/prestellar cores. If weassume that the magnetic field generated around first stars are brought into another prestellarcore, and evolve during the collapse of the primordial gas in a similar way to Xu et al.(2008), the magnetic field will be amplified up to ∼ − G at 10 cm − . However, thismagnetic field will not affect subsequent star formation, since the magnetic field strengthrequired for jet formation(Machida et al. 2006) and MRI activation(Tan & Blackman 2004)is 10 − − − G at 10 cm − , if we assume simple flux freezing condition. On the otherhand, recent studies suggest that weak seed magnetic field is amplified by the turbulenceduring the first star formation(Schleicher et al. 2010; Sur et al. 2010). In Sur et al. (2010),weak seed magnetic fields are exponentially amplified by small-scale dynamo action if theyemploy sufficient numerical resolutions. In this case, the magnetic fields generated by thefirst stars in this paper might be amplified and affect subsequent star formation. However,these studies assume a priori given turbulence and initial magnetic field. To try to settle 11 –this issue, we need cosmological MHD simulations with very high resolution.
5. Summary
In summary, we have investigated the magnetic field generation process by the radiativefeedback of first stars, including the effects of radiation force and the Biermann battery. Asa result, we found 10 − G . B . − G on the boundary of the shadowed region, if we takereasonable parameters expected from standard theory of cosmological structure formation.The resultant field strength with a simple assumption of flux freezing suggests that the suchmagnetic field is unimportant for the star formation process. However, it could be importantif the magnetic field is amplified by turbulent motions of the star forming gas could.We appreciate the anonymous referee for helpful comments. We also thank N.Tominagaand M.Ando for fruitful discussions. This work was supported by Ministry of Education,Science, Sports and Culture, Grant-in-Aid for Scientific Research (C), 22540295.
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This preprint was prepared with the AAS L A TEX macros v5.2.
Model A B C D M halo [ h − M ⊙ ] 5 × × × × n [cm − ] 10 1 10 1 D [kpc] 0.2 0.2 2 2Magnetic field [G] 5 . × − . × − . × − . × − Table 1: The peak magnetic field strengths of four models of M ∗ = 500 M ⊙ . 14 –Fig. 1.— Schematic view of the computational domain. The shadowed region is formedbehind the over dense region , while the other region is exposed to radiation field. 15 –Fig. 2.— Two snapshots (left:0.1Myr, right:2Myr) for the model A are shown. Four rowscorrespond to the mass density of gas [gcm − ](top), the gas temperature [K], the numberdensity of electron [cm − ], and the magnetic field strength [ G ], respectively. Black solid linesrepresent the position of the ionization fronts. 16 – -21 -20 -19 -18 -17 B [ G ] time[yr]Frad+BatteryFradBattery Fig. 3.— Time evolution of the peak magnetic field for the model A (red). Other two curvescorrespond to the field generated by Biermann battery (blue) and radiation pressure (green). 17 –Fig. 4.— Snapshots at 2Myr for the models B,C and D are shown. Left column : magneticfiled strength, B [G] . Right column: electron density, n e [cm − ]. 18 – -21 -20 -19 -18 -17 B [ G ] time[yr] Model AModel BModel CModel D Fig. 5.— Time evolution of the peak magnetic field for the models A-D. -18 -18 -18 -18 -17
0 100 200 300 400 500 B [ G ] Stellar Mass