GALACTIC DYNAMO AND SPIRAL ARMS - 3D MHD SIMULATIONS D. ELSTNER Astrophysikalisches Institut, Potsdam, Germany1
H. LESCH Sternwarte der Universitat Munchen, Germany2
SUSANE VON LINDEN Landessternwarte, Heidelberg, Germany3 KATARZYNA OTMIANOWSKA-MAZUR AND M. URBANIK Astronomical Observatory Jagiellonian University, Krakow, Poland 4 S u m m a r y : In the present project we investigate the evolution of a three-dimensional (3D), large-scale galactic magnetic field under the influence of gas flows in spiral arms and in the presence of dynamo action. Our principal goal is to check how the dynamical evolution of gaseous spiral arms affects the global magnetic field structure and to what extent our models could explain the observed spiral patterns of polarization B-vectors in nearby galaxies. A two-step scheme is used: the N-body simulations of a two-component, self-gravitating disk provide the time-dependent velocity fields which are then used as the input to solve the mean-field dynamo equations. We found that the magnetic field is directly influenced by large-scale non-axisymmetric density wave flows yielding the magnetic field locally well-aligned with gaseous spiral arms in a manner similar to that discussed already by Otmianowska-Mazur et al. 1997. However, an additional field amplification, introduced by a non-zero a-term in the dynamo equations, is required to cause a systematic increase of magnetic energy density against the diffusive losses. Our simulated magnetic fields are also used to construct the models of a high-frequency (Faraday rotation-free) polarized radio emission accounting for effects of projection and limited resolution, thus suitable for direct comparisons with observations. Key
w o r d s : galactic magneticfields,numerical simulations, MHD galactic dynamo
1. INTRODUCTION The origin of large-scale galactic spiral-like magnetic fields possessing a substantial radial component despite the differential rotation shear (Neininger, 1992; Neininger et al, 1991) can be explained by either small scale phenomena related to turbulent motions (called the turbulent dynamo Elstner et al., 1990) or large-scale radial shear flows in bars and spiral arms (Lesch and Chiba, 1997). Although the question which of them is dominant is still vigorously debated (Zweibel, 1996), the existence of 1 2 3 4
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the field component concentrated in density wave compression regions, following the direction of dust lanes to local details, is well established by high-resolution observations in M51 (Neininger and Horellou, 1996). In many other galaxies such correlation between dust and magnetic field is missing (Beck et al. 1996). The magnetic field evolution under a combined action of turbulent amplification and large-scale flows is not yet well studied, though some results have already begun to appear (Panesar and Nelson, 1992). In our previous paper (Otmianowska-Mazur et al, 1997henceforth: [OM1997]) we made a first, simplified approach to the problem, studying the influence of density-wave flows neglecting the turbulent dynamo process. In this work, we make a more physically realistic approach, accounting for small-scale turbulent gas dynamics as well. 2. MODEL DESCRIPTION Our experiment consists of two basic steps: the three-dimensional N-body calculations yielding a realistic gas velocity field and of a hydrodynamical model of the magnetic field evolution using the gas flow picture from the first step (see OM1997 for more details). The N-body code uses the particle-mesh scheme. The galaxy consisted of two essential components: a collisionless self-gravitating disk composed of stars as well as highly inelastically colliding molecular clouds moving in the gravitational potential of the stellar population with addition of an analytically introduced contribution due to the bulge and dark matter halo. In our model the molecular clouds were assumed to represent the large scale velocity field of the gas, controlling the magnetic field evolution via the flux freezing effect. The energy density of galactic magnetic field is two orders of magnitude smaller than the rotational energy density of a galactic disk, which allows us to use kinematical approach of magnetic lines passively following the gas motions (OM1997). We modeled the magnetic field evolution by analyzing numerically the time dependent solutions of the dynamo equation:
where B is the magnetic induction, v is the large-scale velocity of the gas, a is the dynamo coefficient describing the mean helicity of turbulent motions (the a-effect) and r is the turbulent diffusion coefficient. Calculations for the magnetic field are done with a three-dimensional extension of the explicit finite difference code on a staggered grid described in Elstner et al., 1990. The code preserves V • B = 0. The computations are performed in cylindrical coordinates. To model the observed radio and polarization properties the disk was split into a large number of small elements in cylindrical coordinates. Knowing the local values of magnetic field components in each volume element we computed its contribution to Stokes parameters /, Q and U (see Urbanik et al., 1997for details). The beam-smoothed contributions of all elements yielded the distributions of Stokes parameters /, Q and U over the visible disk, then converted to the form of
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maps of total power, polarized intensity and polarization angles as seen by the radio telescope observing with an assumed resolution. As we analyzed only the brightness distributions in arbitrary units, we avoided assumptions concerning the magnetic field strengths and cosmic ray electron densities in physical units by analyzing the brightness maps in arbitrary units in which the maximum total power brightness was always 10000. The distributions of the simulated gas density used for a comparison with the polarization maps were also inclined to the sky plane by the same value as the magnetic disk, integrated along the line of sight and convolved with the same telescope beam. 3. MODEL INPUT PARAMETERS The input parameters adopted for our N-body simulations are summarized in Table 1. To model galaxy with spiral structure a massive bulge is used (OM1997 and von Linden et al, 1998). This ensured the development of a pure spiral, unbarred pattern. The simulations were performed with about 38000 stellar particles and 19000 molecular clouds. Table 1 presents the masses and scale sizes of components of the model galaxy. Table 1. Input parameters for the spiral galaxy model
dark-halo disk gas bulge
Mass in 1010 Mo :
Scale length in kpc:
9.6 6.0 0.8 6.0
15.0 6.0 12.0 1.6
Table 2. The magnetic field evolution model parameters Model
A B C
arr
azz
a^ kms-1
n
kms - 1
cm2/s
0 5 5
0 -10 -10
1.5 x 1026 1.5 x 1026 7.5 x 1026
Table 2 presents the input parameters for the second step of the model, describing the magnetic field evolution. The number of grid points in all cases is 101 in the radial and azimuthal direction and 21 points in the vertical direction. The distance between the grid points in the radial direction is 150 pc yielding 30 kpc as the diameter of the galactic disk. Along the Z axis the step is 50 pc, what gives 0.5 kpc for the galactic disk half-thickness. The initial magnetic field was an axisymmetric quadrupole. It Studia geoph. et geod. 42 (1998)
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consisted of a toroidal field with the same direction as galactic rotation and of a poloidal component symmetric with respect to the galactic plane. Three experiments with different values of the turbulence coefficients a and 77 were performed (see Table 2). The first of them (A) was made with n = 1.5 x 1026 cm2 s-1 constant in the whole disc and no a effect. The next two experiments (B and C) were done with diagonal components of the antisymmetric (with respect to the galactic plane) a-tensor: arr = a^ = 5 km s-1 and azz = -10 km s-1 constant above the galactic plane. Notice that the rotation was counterclockwise. The diffusion coefficient was 5 times higher in case C than in B. The adopted time step was 105yr for the magnetic field calculation by updating every 100 time steps a new velocity field. To model the synchrotron polarization we assumed the cosmic-ray electrons having an uniform distribution in the disk with a truncation radius of 15 kpc, and a Gaussian vertical distribution with a scale height of 1 kpc. The electron energy spectrum corresponded to the nonthermal spectral index anth=0.8. The telescope beam had a spatial resolution of 1 kpc and that is typically attained in VLA observations of nearby galaxies. The model galaxy was inclined by 30°, a typical value for nearby face-on spirals. 4. RESULTS Our numerical simulations yielded the 3D galactic magnetic field structures at every time step of 107 yr, polarization maps from chosen magnetic field distributions, and the magnetic energy changes in time. Fig. 1 presents the total magnetic energy E normalized to its initial value E0 in the logarithmic scale. In the case A the total magnetic field energy decreases after 0.2 x 109 yr. The simulations with a non-zero a give a constant growth of E even when a higher diffusion coefficient is assumed. The growth of energy is fastest in the experiment B with E exceeding 20 times E0 after 600 million years. Fig. 2a, b compares, respectively, the magnetic field structure in the galactic plane and the gas density distribution for models A (diffusion but no a-effect) and B (the same diffusion coefficient and a no-zero a), at the evolutionary time of t — 2.1 x 108 yr. In both cases a good alignment of magnetic vectors with gaseous spiral arms is attained, especially left of the centre, where the magnetic field follows precisely a sharp bend of the outermost arm. The magnetic field structure of model C looks very similar to that of model B. The opposite direction of the magnetic field in model A and B is due to the initial condition with the same positive sign for BT and Bv. The field of model A results from the initial Br with a negative B^,. The field of model B induces with the a-effect from the initially positive Bv a negative Br. Independently of the induction process amplifing the field, the structure of the field is in all models mainly influenced by the gas velocity. In both cases the beam-smoothed polarized intensity distributions (Fig. 3a, b) form clearly visible ridges along spiral arms following their sudden bends. The Bvectors are parallel to the gaseous and polarization ridges, thus the concentration 376
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Fig. 1. The total magnetic energy E (in a logarithmic scale) normalized to the initial energy Eo for cases A, B, and C described in Table 2.
Fig. 2. The magnetic field vectors in the disk plane of the galaxy, superimposed onto the gas density distribution (greyscale) at time t = 2.1 x 108 yr for model A without a-effect (left panel) and model B with a-effect (right panel)
of magnetic fields in gas compression regions and its alignment in different regions of the galactic plane (see Fig. 2) remain visible also when the integration through a whole disk thickness is applied. This means that our models do not show strong Studia geoph. et geod. 42 (1998)
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Fig. 3, The contours of the polarized intensity and polarization B-vectors superimposed onto the gas density distribution all integrated along the line of sight and smoothed to the telescope beam of 1 kpc for the same cases as in Fig. 1. The galaxy is inclined by 30° to the line of sight with the major axis (x-axis) running horizontally
variations of the field structure with the height above the disk plane which might completely obscure observed signatures of magnetic spiral arms. The interrelations between the pitch angles tjj of polarization B-vectors in the presence of a-effect and those of gaseous spiral arms are illustrated in Fig. 4a, b at a more advanced evolutionary stage after t = 3.8 x 108 yr. In cylindrical coordinates used the logarithmic spiral with V=const corresponds to straight lines inclined by i/>. Though the evolution started with an initially azimuthal field, most of the disk surface is occupied by negative pitch angles of the same sign as for gaseous arms. However, the absolute values of ^ are rather loosely associated with the gas density maxima. The mean pitch angle of gaseous spiral pattern is about —25°. This value is attained by polarization B-vectors in the dense parts of spiral arms, however large negative tjj are present also over large areas of interarm regions, especially in the outer disk. In the inner disk at ln(r) of about 1,5 highly negative magnetic pitch angles at azimuthal angles 15°-20° also extend far into the interarm region. The discussed magnetic pitch angle distributions do not however result solely from the dynamo action, as they look very similar with and without an a-term. Some differences are visible only statistically in the histogram shown in Fig. 4a. Though both pixel-to-pixel distributions of V> have a similar shape, the dynamo action causes a slight increase in number of points having highly negative values (< —10°) with a smaller number of those showing $ in the range from -10 to +10°. The effect is 378
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Fig. 4. a) The values of magnetic pitch angles with symbol size proportional to the pitch angle absolute value overlaid onto an outline of the gas density maxima for the model with a-effect (B) at the evolutionary time t = 3.8 x 10a yr. Circles and stars denote positive and negative pitch angles respectively. The plot is made in the frame of azimuthal angle counted anticlockwise from the x-axis and natural logarithm of the galactocentric radius. The median filter was applied to the density distribution to enhance the spiral arms, b) The directions of polarization B-vectors in the same coordinates overlaid of the outline of gaseous spiral arms
not strong and not obviously associated with spiral arms or interarm regions. The inclusion of the dynamo action also slightly increases the number of map points with low polarized brightness at the expense of the high-brightness ones, implying relatively more flux coming from diffuse, extended regions than in case of a pure density wave action (Fig.5b). 5. DISCUSSION AND CONCLUSIONS Our present work demonstrates that also in the presence of density wave flows the a-effect is necessary for the magnetic field growth in time. However, in the present version of the model the dynamo action influences only slightly the pitch angle distribution. In both cases, with and without the dynamo action, the magnetic field forms polarized ridges along the gas compression regions (traced in real galaxies by heavy dust lanes), with polarization B-vectors following precisely the spiral arms. Both effects were observed in M51 by Neininger and Horellou (1996). Studia geoph. et geod. 42 (1998)
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Fig. 5. The histograms comparing the values of magnetic pitch angles (a) and values of polarized intensity (b) for models with (B) and without (A) the a-effect (see legends inside the Figure). Pitch angles and polarized signal are counted in bins with widths 10° and 400 map units, respectively. The maps of polarized brightness were scaled to the same integrated flux, the bins with signal weaker than 400 map units were ignored
The dynamo action is likely to generate three-dimensional magnetic field structures with plane-parallel geometry of magnetic lines changing with the distance from the plane. Thus, in reality the observed pattern of polarization B-vectors may be different from the field structure in the galactic plane. Therefore we applied the polarization modeling techniques yielding the polarized signal integrated along the line of sight. All our conclusions are based on polarization pictures which would be seen by an observer measuring the Stokes parameters integrated through a whole disk thickness. In contrast to classical density wave models (e.g. Otmianowska-Mazur and Chiba, 1993) the magnetic field obtained in this work does not exhibit dramatic changes of B-vector pitch angles between arms and interarm regions. In this respect our work more closely reproduces observations showing no strong arm/interarm variations of magnetic pitch angles in real galaxies (Krause et al., 1989; Beck and Hoernes, 1996). This success is due to a more realistic picture of the gas flow in the galactic disk and constitutes a step toward the reproduction of real galactic magnetic fields showing many signatures of the dynamo action (e.g. Urbanik et al., 1991) but remaining parallel to the spiral arms also away from them. Manuscript received: 31st January, 1998;
Revisions accepted: 11th May, 1998
References Beck, R. and Hoeraes, P., 1996: Magnetic spiral arms in the galaxy NGC 6946", Nature 379, 47 Beck, R., Brandenburg, A., Moss, D., Shukurov, A. and Sokoloff, D.D., 1996: Galactic Magnetism: Recent Developments and Perspectives, Ann. Rev. Astron. Astrophys. 34, 155 380
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Galactic Dynamo and Spiral Arms ... Elstner, D., Meinel, R. and Riidiger, G., 1990: Galactic Dynamos with non-sharp boundaries, Geophys. Astrophys. Fluid Dynamics 50, 85 Krause, M., Beck, R. and Hummel, E., 1989: The magnetic field structures in two nearby spiral galaxies - part two - the bisymmetric spiral magnetic field in M81, Astron. and Astrophys. 217, 17 Lesch, H. and Chiba, M., 1997: On the origin and evolution of Galactic magnetic fields, Fund. Cosm. Phys. 18, 273 Neininger, N., Klein, U., Beck, R. and Wielebinski, R., 1991: Correlation of magnetic and optical structure in the barred spiral galaxy M83, Nature 352, 781 Neininger, N., 1992: The magnetic field structure of M51, Astron. and Astroph. 263, 30 Neininger, N. and Horellou, C., 1996: High resolution radio continuum polatization of M51, in Polarimetry of the Interstellar Medium (ed.: W.G. Roberge and D.C.B. Whittet), San Francisco, Astr. Soc. Pacific 97, 592 Otmianowska-Mazur, K. and Chiba, M., 1995: Numerical simulation of large-scale magnetic field evolution in spiral galaxies Astron. and Astrophys. , 301, 41 Otmianowska-Mazur, K., von Linden, S., Lesch, H. and Skupniewicz, G., 1997: Global three-dimensional simulations of magnetic field evolution in a galactic disk. I. Barred galaxies, Astron. and Astrophys. 323, 56 Panesar J.S and Nelson, A.H., 1992: Numerical models of 3-D galactic dynamos, Astron. and Astrophys. 264, 77 Urbanik, M., Elstner, D. and Beck, R., 1997: Observational signatures of helical galactic magnetic fields, Astron. and Astrophys. 326, 465 von Linden, S., Otmianowska-Mazur, K., Lesch, H. and Skupniewicz, G., 1998: Global three-dimensional simulations of magnetic field evolution in a galactic disk. II Gas rich galaxies", Astron. and Astrophys., in press. Zweibel, E., 1996: Slow and steady spirals, Nature 383, 758
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