Baltic Astronomy, vol. 24, 194 200, 2015 STAR FORMATION AND GALAXY DYNAMO EQUATIONS WITH RANDOM COEFFICIENTS E. A. Mikhailov 1 and I. I. Modyaev 2 1 Faculty of Physics, M. V. Lomonosov Moscow State University, Bld. 1, Str. 2, GSP-1, Leninskie Gory, 119991 Moscow, Russia; ea.mikhajlov@physics.msu.ru 2 Faculty of Mechanics and Mathematics, M. V. Lomonosov Moscow State University, Bld. 1, GSP-2, Leninskie Gory, 119991 Moscow, Russia; ygrekus@gmail.com Received: 2015 March 25; accepted: 2015 April 20 Abstract. We study the influence of star formation on magnetic field in galaxies. Two approaches have been used to describe this mechanism. The first one uses some averaged approximate kinematic characteristics that appear in the galactic dynamo equations. We use the so-called no-z model that takes into account the fact that galactic disks are quite thin and hence we can consider only the field components that are parallel to the plane of the galaxy. We also use the equation that describes the evolution of magnetic helicity, which can be important for galaxies with intensive star formation. The second approach uses the dynamo equations with random coefficients, which are useful for describing magnetic fields in galaxies with rapidly changing kinematic parameters. Both methods yield similar results: if the surface star formation rate is small, the magnetic field evolution does not change very much. If the surface star formation rate is more than five times higher than in the Milky Way, the field decays. Key words: fields 1. INTRODUCTION stars: formation galaxies: magnetic fields ISM: magnetic The possible effect of star formation on magnetic fields of spiral galaxies has been a topic of extensive debate in the literature (Beck et al. 1996). Star formation changes the ratio between the fractions of different components of the interstellar gas and the averaged characteristics of the medium, such as turbulent velocities of gas motions, their scale length, temperature, etc. The generation of magnetic field is closely linked to interstellar turbulence, and hence the field evolution differs from one galaxy to another. Moreover, the mechanism of this influence is still poorly understood. From observational point of view, the problem is due to the lack of magneticfield data for galaxies where the star formation is very intensive. NGC 6946 (Beck & Hoernes 1996) and NGC 253 (Hlavacek-Larrondo et al. 2011) seem to be the only examples of the influence of star formation on magnetic field, but they are not very clear. The magnetic field of NGC 6946 resides between the material arms,
Star formation and galaxy dynamo equations 195 and this may be due both to star formation and to some other factors. The linear size and other parameters of NGC 253 are very different from those of classic galaxies that are usually studied in terms of galaxy magnetism. As for theoretical aspects, magnetic field generation is described by the socalled dynamo theory, which explains the growth of magnetic field by the transformation of the kinetic energy of turbulent motions into magnetic-field energy (Molchanov et al. 1985). The dynamo equations include the kinematic characteristics of the interstellar medium, but they do not include the star-formation rate or the star-formation density. We therefore should adopt some parameterizations to relate the characteristics of star formation, turbulent velocities, half-thickness of the galaxy and the interstellar gas density. While modeling the magnetic fields we can use two different approaches. On the one hand, we can adopt some approximations for averaged properties of the interstellar medium for galaxies where the star formation rate is quite high. To model the magnetic field, we use the so-called no-z model, which describes magnetic fields in thin disks (Moss 1995; Phillips 2001). It is important to take into account vertical flows in such galaxies and we therefore add an equation that describes the magnetic helicity (Shukurov et al. 2006; Mikhailov 2013). On the other hand, star formation is often connected with some small regions, where the properties of the interstellar medium are very different from those in other parts of the galaxy. For some cases it is useful to assume that these properties can be described by some random laws. We think that both of these methods are useful: the first one can be used to analyze magnetic fields in galaxies where the star formation is distributed throughout the entire system, and the second one can be used to describe magnetic fields in galaxies where the star formation is localized in some small regions. In this paper we show that both approaches yield similar results: if the star formation intensity is not very high, the magnetic field does not change very much. However, above some threshold of surface star formation rate (about five times higher than its value in the Milky Way) the magnetic field decays. This result is quite similar to that obtained in our previous studies (e.g. Mikhailov et al. 2012). 2. DYNAMO MECHANISM AND PARAMETERS The evolution of magnetic field is described by the so-called dynamo theory (Beck et al. 1996). The dynamo mechanism is based on two factors. The first one is differential rotation: it transforms the radial component B r of the galactic magnetic field into the azimuthal component B φ. The second one is the alpha-effect, which characterizes turbulent motions of the interstellar medium and transforms the azimuthal component B φ into the radial component B r. Each of these factors causes the magnetic field to decay, but their combined effect makes the field grow exponentially if their magnitude is higher than some critical value (Arshakian et al. 2009). Another important feature of interstellar magnetohydrodynamics is the magnetic helicity conservation law. Magnetic helicity can be introduced as a scalar product of the magnetic field and its vector potential, and for highly conductive media it is an integral of motion. We can introduce it into the dynamo equations as an extra term added to the kinematic alpha-effect, and consider an extra equation for the evolution of magnetic helicity (Shukurov et al. 2006; Sur et al. 2007).
196 E. A. Mikhailov, I. I. Modyaev To model magnetic fields in galaxies with intensive star formation, we should relate surface star formation rate to volume density, mean velocity, and half-thickness of the interstellar gas, and to vertical flow velocity. While making the parameterizations, we follow Mikhailov (2014). According to Abramova & Zasov (2011) and Zasov & Abramova (2012), the relation between the surface star formation rate, Σ SFR, and the volume density of the interstellar gas is ρ 0.94 Σ SFR, (1) and we therefore can adopt ρ Σ 1.06 SFR. (2) The equations of the dynamo theory include the mean velocity v of the turbulent motions. We assume that the velocity of the turbulent motions of neutral hydrogen is v 1 = 10 km s 1 and that of ionized hydrogen is v 2 = 35 km s 1. It can be shown (Mikhailov 2014) that the fraction κ of HII is related to Σ SFR as κ 12 Σ SFR (3) and the relation for the mean velocity of the turbulent motions is v = 10 (1 + 30 Σ SFR ). (4) If the turbulent velocities change, we can assume that the half-thickness of the galaxy should increase. However, Caldu-Primo et al. (2013) claim that for star formation rates not very high the half-thickness of the galaxy does not change very much. We therefore assume that, if the surface star formation rate is less than 10 times that of the Milky Way (Σ SFR < 0.04 M yr 1 kpc 2 ), the half-thickness of the galaxy remains constant. If Σ SFR > 0.04 M yr 1 kpc 2, we assume that the half-thickness of the galaxy doubles. We assume that the velocity U 0 of the vertical flows, which are important for the helicity evolution, is equal to 0.2 km s 1 in the Milky Way, and, according to Shukurov et al. (2006) and Mikhailov (2014), U 0 0.2 km s 1 = Σ SFR 0.004 M yr 1 2. (5) kpc 3. BASIC EQUATIONS The magnetic field H in galaxies consists of two components (Ruzmaikin et al. 1988): H = B + b, (6) where b is the small-scale field with a scale length of about 100 pc, and B is the regular field. Here we study the regular field. Its evolution can be described by the dynamo mechanism and is based on two effects: differential rotation and α-effect which describes the turbulent motions. We write our equations in terms of the so-called no-z model (Moss 1995; Phillips 2001), which takes into account the fact that the disk of the galaxy is quite thin, the z-component of the magnetic field can be taken from the non-divergence equation, and the z-derivatives of inplane components can be replaced by algebraic expressions: 2 B z 2 π2 B 4h 2. Usually
Star formation and galaxy dynamo equations 197 these equations are written in dimensionless variables (e.g. Moss 1995), but if the parameters of the interstellar medium change, so do the units. The only dimensionless parameter is the normalized magnetic helicity α, and the basic system of equations has the following form (Sur et al. 2007; Mikhailov 2013; Mikhailov 2014): B r ( = Ωl2 h 2 (1 + U0 α )B φ h + η ) π2 4 h 2 B r + η { r (1 r r (rb r))}, (7) B φ α = U 0 h α η = r Ω ( r B U0 r h + η π2 4 (1 + α ) B2 B 2 + 3h B rb φ 8 l B 2 ) B φ + η { r (1 r Ω πr r (1 + α ) Ω + η r (rb r))}, (8) + α R m + { 1 α (r r r r ) where Ω is the angular velocity of galactic rotation; l, the turbulent scale; h, the half-thickness of the galaxy; U 0, the typical vertical flow velocity; α, the magnetic helicity; R m, the magnetic Reynolds number, and η, the turbulent viscosity. We can rewrite the latter variabe as (Arshakian et al. 2009): }, (9) η = lv 3, (10) where v is the velocity of the turbulent motions. Figure 1 shows the results for different values of Σ SFR. We modeled the evolution of magnetic field for short starbursts of different intensities. If the surface star formation rate is not very high (Σ SFR = 0.01 M yr 1 kpc 2 ) the magnetic field grows, although the growth rate is less than that for the Milky Way. If the surface star formation rate is higher than five times that of the Milky Way (Σ SFR = 0.02 M yr 1 kpc 2 ), the field decays. If the surface star formation rate is very high (Σ SFR = 0.1 M yr 1 kpc 2 such rates can be observed for early periods of the galaxy evolution), the field decays to a very low level and cannot grow again until the surface star formation rate decreases. 4. MODEL WITH RANDOM COEFFICIENTS The dynamo equation contains some averaged parameters that characterize the turbulent motions, half-thickness of the galaxy and its rotation. Usually for calm galaxies with no rapid processes, such as star formation, supernova explosions, etc., these parameters describe the magnetic field evolution quite well. However, galaxies in which some active processes are taking place contain small HII regions, where such characteristics as temperature and turbulent velocities differ from the ambient values. We can calculate some effective values of these parameters as in Mikhailov et al. (2012) and Mikhailov (2014). However, the more realistic approach is to use some stochastic model. The lifetime of such regions is not very long, and their localization is quite random. We can therefore use a model with
198 E. A. Mikhailov, I. I. Modyaev 10-5 10-6 B, G 10-7 10-8 10-9 0 1 2 3 4 5 t, Gyr Fig. 1. Magnetic field evolution for the deterministic model with different values of surface star formation rate: Σ SFR = 0.004 M yr 1 kpc 2 (the solid line); Σ SFR = 0.01 M yr 1 kpc 2 (the dashed line); Σ SFR = 0.02 M yr 1 kpc 2 (the dotted line), and Σ SFR = 0.1 M yr 1 kpc 2 (dot-dashed line). random parameter values that change at short time intervals. The parameters may acquire one of two possible values with certain probabilities. The first set of parameter values characterizes neutral hydrogen (HI) with some small admixture of ionized component. The second set of parameters describes regions dominated by the ionized component. We assume that these parameters remain constant between the update times when they change. The probability p of the parameter set associated with HII characterizes the intensity of star formation. As a rough approximation we can assume that p κ and use Eq. (3). To describe the magnetic field we use the following set of equations: B r = Ω l2 h 2 B φ η π2 4h 2 B r, (11) B φ It can be shown that the magnetic field grows if = ΩB r η π2 4h 2 B φ. (12) Ω > π2 v 12 h, else it decays. We assume that v = v 1 with the probability p and v = v 2 with the probability (1 p). The results are illustrated in Table 1 and in Fig. 2. For small surface star formation rates the magnetic field grows, and above a certain critical value (Σ SFR 0.018 M yr 1 kpc 2 ) it decays.
Star formation and galaxy dynamo equations 199 10-7 B, G 10-8 10-9 0.0 0.5 1.0 1.5 2.0 t, Gyr Fig. 2. Magnetic field evolution for the model with random coefficients: Σ SFR = 0.004 M yr 1 kpc 2 (the solid line); Σ SFR = 0.01 M yr 1 kpc 2 (the dashed line), and Σ SFR = 0.02 M yr 1 kpc 2 (the dotted line). Table 1. Growth rate of the magnetic field for different values of Σ SFR. Σ SFR, M yr 1 kpc 2 γ, Gyr 1 0.004 1.43 0.010 0.78 0.020 0.18 Of course, we can use a more complicated model for the magnetic field, e.g., a nonlinear modification of equations (11) (12) or (7) (9), however, the most important features show up even in such a simple model. See Mikhalov & Modyaev (2014) for some mathematical aspects of this problem. 5. CONCLUSIONS The two different methods considered yield quite similar results. If the surface star formation rate is not very high, the magnetic field grows. However, if the surface star formation rate is more than five times higher than that of the Milky Way, the magnetic field decays. These results are close to those obtained in our previous studies (Mikhailov et al. 2012; Mikhailov 2014). We point out that here we analyzed the evolution only of the regular component of the magnetic field; that of the small-scale component can be different. ACKNOWLEDGMENTS. We thank D. D. Sokoloff for his valuable advice in preparing this paper. We thank the Organizing Committee of the MSA-2014 Conference for the opportunity to present our work. E. M. acknowledges the support from the Dynasty Foundation for Noncommercial Programs.
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