Statistical tests of galactic dynamo theory

Spiral galaxies contain magnetic fields that are coherent on scales larger than the outer scale of turbulence, the so-called large-scale, or mean magnetic fields (Beck & Wielebinski, 2013; Beck, 2015). Mean-field turbulent dynamo theory is the leading theory to explain the prevalence and properties of large-scale magnetic fields in galaxies (Ruzmaikin et al., 1988; Beck et al., 1996; Brandenburg & Subramanian, 2005; Klein & Fletcher, 2015). The galactic dynamo theory has proved to be successful in explaining the overall large-scale magnetic properties of a generic spiral galaxy as well as those of a selection of nearby galaxies. Dynamo models contain a number of parameters that are poorly constrained by theory and observations as they require detailed knowledge on the size and shape of the galactic ionized layers, the magnitude and spatial distribution of the turbulent speeds and scales, as well as their variations with galactic azimuth and radius (e.g., Brandenburg et al., 1993; Shukurov & Sokoloff, 1998; Moss, 1998; Rohde et al., 1999; Moss et al., 2007). Statistical studies of the large-scale magnetic fields in samples of galaxies are limited by the fact that they require observations of synchrotron emission and Faraday rotation at sensitivity and resolution currently achievable only for a modest number of nearby galaxies (Fletcher 2010, Van Eck et al. 2015, (hereafter VBSF), Tabatabaei et al. 2016).

The radio telescopes of a new generation would significantly expand the observational database (e.g., Gaensler et al., 2004) to make such comparisons feasible. However, approaches to such comparisons of theory with observations need to be developed now, but VBSF demonstrated that correlations between individual parameters of spiral galaxies (such as the rotational shear rate) and their magnetic fields are not easy to detect because even the simpler properties of the global magnetic structures are sensitive to a relatively large number of diverse galactic parameters. As a result, the scatter between the remaining parameters hides the expected correlation. In addition, the observational data need to be reduced in a coherent and systematic manner to admit comparison with theory. Therefore, a statistical comparison of dynamo theory with observations requires, apart from a suitable reduction of the observational data (as suggested by VBSF, ), a careful approach.

Another obstacle in observational verifications of galactic dynamo theory is that interstellar random magnetic fields usually exceed the mean field in strength (Ruzmaikin et al., 1988; Beck et al., 1996), so that dedicated techniques are required to deduce the parameters of the global magnetic structures observed. Moreover, galactic magnetic fields, despite being dominated by an axisymmetric structure (in agreement with predictions of the dynamo theory) are further modified by the spiral arms and various asymmetries of the host galaxies. Therefore, careful identification of the underlying axially symmetric magnetic field in the observations is required before the theory can be meaningfully compared with observations. Among approaches suggested to quantify the large-scale structures in the observed distributions of synchrotron emission and Faraday rotation are the expansion of the azimuthal patterns in Faraday rotation into Fourier series in azimuthal angle (Sokoloff et al., 1992a; Berkhuijsen et al., 1997; Fletcher et al., 2004, 2011) and wavelet analysis (Frick et al., 2000; Patrikeev et al., 2006).

In this paper we compare predictions of galactic dynamo theory with the pitch angles of the large-scale magnetic fields in those galaxies where observational data have been interpreted appropriately. The pitch angle of the mean magnetic field is defined as $p=\arctan({B}_{r}/{B}_{\phi})$, with $-90^{\circ}<p\leq 90^{\circ}$, where $(r,\phi,z)$ are cylindrical polar coordinates with the $z$-axis parallel to the galactic angular velocity $\bm{\Omega}$, and $\bm{B}$ is the regular magnetic field. This is the acute angle between the mean magnetic field and the tangent to the circumference in the galactic plane, and its negative values signify a trailing spiral. There are several advantages in using the magnetic pitch angle to test the galactic dynamo theory (Baryshnikova et al., 1987; Krasheninnikova et al., 1989). Firstly, $p$ is readily predicted by the standard non-linear mean-field dynamo theory, with the need to fix the values of only a small number of parameters (Sur et al. 2007,Chamandy et al. 2014 (hereafter CSSS), Chamandy & Taylor 2015 (hereafter CT)). Secondly, the closely related position angle of the polarized synchrotron emission is directly observable, though it must be corrected for Faraday rotation. In contrast, the strength of the mean magnetic field depends on additional parameters involving less certain physics. Its observational determination relies on the questionable assumption of an energy or pressure equipartition between cosmic rays and magnetic fields (Beck & Krause, 2005; Stepanov et al., 2014) when deduced from the synchrotron intensity, or the implicit assumption of the statistical independence of the magnetic, thermal-electron and cosmic-ray density fluctuations when obtained from Faraday rotation (Beck et al., 2003). As a result, the regular magnetic field magnitude ${B}$ determined from observational data is arguably less reliable than its inferred pitch angle. Moreover, ${B}$ is sensitive to details of the dynamo non-linearity and the extent to which the dynamo is supercritical, to which $p$ is less sensitive.

Fletcher (2010) (see also Klein & Fletcher 2015), briefly assessed the ability of dynamo theory to explain some features of galactic magnetic fields, but they restricted the comparison to the kinematic, or linear regime of dynamo action where magnetic field still grows exponentially. Nearby galaxies definitely have the dynamo action saturated since the energy density of the large-scale magnetic field is comparable to the turbulent energy density, so nonlinear, steady-state solutions of the dynamo equations are more relevant in such comparisons. As pointed out by Elstner (2005) and studied in detail by CT, magnetic pitch angles are predicted to be smaller in magnitude in the saturated state than in the kinematic regime. VBSF expanded the database compiled by Fletcher (2010) and performed a statistical analysis of the data, including an assessment of the level of agreement between the data and solutions of a simple analytical non-linear dynamo model. The magnetic pitch angles predicted by the dynamo model used by VBSF had magnitudes much too small to explain observations, confirming concerns voiced previously (Elstner, 2005). CT explored the parameter space of the dynamo models more extensively to find magnetic pitch angles comparable to those observed can be obtained in standard dynamo models but not with the canonical, solar-neighborhood parameter values used by VBSF. Our goal here is to develop further such a comparison using essentially the same dynamo model as CT (with a few refinements). Given the deliberate simplicity of the galactic dynamo models used, the incompleteness of the galactic dynamo theory in general, and imperfections in the observational data, it is unrealistic to expect an agreement between the theory and observations at the level of rigorous statistical tests, and such comparisons would necessarily remain qualitative at least until more extensive, homogeneous and reliable observational data become available.

The paper is organized as follows. In Section 2, we briefly discuss the data set. We then present dynamo solutions in Section 3 illustrating both numerical and analytical results in various regions of the parameter space. Sections 2 and 3 present only a brief discussion, and the reader is referred to VBSF and CT for more detail. A discussion of the galactic model used as input to the model can be found in Section 4. Section 5 presents the main results, comparing the magnetic pitch angles obtained in the dynamo theory and observations, while Section 6 explores effects of varying model parameters. We discuss the implications of our results and the significance of the remaining discrepancies between the theory and the data in Section 7. We formulate our conclusions in Section 8.

Our sample contains ten nearby galaxies selected because observations of their synchrotron emission and, especially, Faraday rotation, have been interpreted with sufficient detail and reliability in terms of the pitch angle of the large-scale magnetic field as to admit comparison with theory. In particular, the azimuthal variation of the synchrotron polarization angle in most of them has been interpreted in terms of the magnitudes of the Fourier components of the product $n_{\mathrm{e}}{{\mbox{\boldmath$B$}}}h$, where $n_{\mathrm{e}}$ is the thermal electron density and $h$ is the scale height of the ionized layer. This allows a reliable estimate of the pitch angle of the axisymmetric part of the large-scale magnetic field (the azimuthal wave number $m=0$) whose parameters can be predicted by the dynamo theory. The mode $m=1$ corresponds to a bisymmetric structure, most often associated with an overall asymmetry of the pattern rather than dynamo action, whereas the mode $m=2$ is usually attributed to a perturbation induced by a two-armed spiral pattern. In all cases where magnetic field structure is known for both the disk and halo, we use the disk data.

Table 1 presents parameters of the magnetic fields in the sample galaxies, both as observed and as obtained from various versions of the dynamo models discussed in Section 3. Table 2 contains the galactic parameters used as an input to the dynamo models. We mostly use the data compiled by VBSF but with a few notable exceptions. A typographical error in the magnetic pitch angle of IC 342 has been corrected: in the outermost annulus, $p=-18^{\circ}\pm 2^{\circ}$ (c.f. Gräve & Beck, 1988). We exclude the galaxies M94 and NGC 4414 from the sample since the radial range over which the data is averaged is not stated in the original work (VBSF). NGC 253 is included, however, with the radial range $1.4$–$6.7\,{\rm kpc}$ (Heesen et al., 2009). For M33, we have selected the observational model from Tabatabaei et al. (2008) that contains $m=0$ and $m=2$ azimuthal modes but no vertical field component ${B}_{z}$, as this type of magnetic field is more likely to be maintained in a thin gas layer (this choice is different from that of VBSF). To determine the H I and star formation rate surface densities in the outer annulus of NGC 6946, exponentials were fitted to the data from the two innermost annuli.

We divide the sample into three groups based on the galaxy morphology and magnetic field fitting technique. Group I consists of M31, M33, and M51, galaxies without a prominent bar, for which azimuthal Fourier components ($m=0$, $m=1$, and $m=2$) have been fitted to the multi-frequency data on the synchrotron polarization angle in the galactic disk. Following VBSF, we compare our dynamo model with the axisymmetric ($m=0$) component. Group II consists of M81, NGC 253, NGC 1566, NGC 6946, and IC 342. M81 has no bar whereas the others are SAB galaxies but with a small bar. In each case, the bar does not extend to the midpoint of the inner radial range considered. The bar radii are about $2.2\,{\rm kpc}$ for NGC 253 (Sorai et al., 2000), $1.7\,{\rm kpc}$ for NGC 1566 (Combes et al., 2014), $1.8\,{\rm kpc}$ for NGC 6946 (Fathi et al., 2007, and references therein), and $1.5\,{\rm kpc}$ for IC 342 (Crosthwaite et al., 2000). For M51 and NGC 253, disk and halo magnetic field components were fitted separately (Fletcher et al., 2011; Heesen et al., 2009), while for M31, such a separation was considered but found to be unnecessary (Fletcher et al., 2004). Finally, Group III consists of two barred galaxies, NGC 1097 and NGC 1365, with bars as large as about $10\,{\rm kpc}$ and $11\,{\rm kpc}$, respectively, thus extending into most (NGC 1365) or the whole (NGC 1097) of the range of galactocentric distances where the pitch angle of the mean magnetic field is known (Beck et al., 2005). The structure of the mean magnetic field in and near the bar is affected by the associated strongly nonaxisymmetric gas flow (Moss et al., 2001, 2007), so we do not expect our dynamo model, designed for axially symmetric galaxies, to be very successful at explaining such fields.

The dynamo model is essentially that of CT and CSSS, with a few refinements, and we refer the reader to those papers for more information, but here we introduce it only briefly. We solve three coupled dynamo equations in the slab approximation, where radial derivatives are neglected. This approximation has been shown to produce accurate results when compared to higher dimensional solutions (Chamandy, 2016). The radial and azimuthal components of the mean-field induction equation and the dynamical quenching equation, which stems from magnetic helicity balance (Shukurov et al., 2006), are then given by

with $l$ the turbulent correlation length, $\alpha$ equal to the sum of kinetic and magnetic contributions,

and the field strength corresponding to energy equipartition with turbulent kinetic energy given by

with $\rho$ the density of ionized gas and $u$ is the rms turbulent velocity. The turbulent diffusivities $\eta$ and $\kappa$ are assumed to be independent of $z$. We adopt the standard estimate

with both $u$ and the correlation time of interstellar turbulence $\tau$ assumed constant. The mean vertical velocity and kinetic $\alpha$ effect are given by

and

respectively, with $U_{\mathrm{0}}$ and $\alpha_{\mathrm{0}}$ both parameters with dimensions of velocity. Note that we have neglected ${B}_{z}\partial{U}_{\phi}/\partial z$ in equation (2), and assumed ${B}_{z}^{2}\ll{B}_{r}^{2}+{B}_{\phi}^{2}$ in equation (3), which are appropriate for a thin disk. We have also assumed ${U}_{r}=0$. Vacuum boundary conditions, ${B}_{r}={B}_{\phi}=0$ at $z=\pm h$, are used, along with $\partial^{2}\alpha_{\mathrm{m}}/\partial z^{2}=0$ at $z=\pm h$. The boundary condition on $\alpha_{\mathrm{m}}$ leaves unconstrained the helicity flux through the disk surface.

In this section we derive the dynamo parameters (4) for the sample galaxies using the observational data of Table 2. The galactic rotation curves and, hence, the rotation $\Omega(r)$ and shear rates $q(r)$ are reasonably well constrained; these parameters are the same as those in VBSF.

A suitable estimate of the magnitude of the mean vertical velocity at the disk surface, $U_{\mathrm{0}}$, can be found in Appendix B1 of VBSF. With a typographic error in that paper corrected, we have

where $h$ is the scale height of the ionized gas layer normalized by $0.5\,{\rm kpc}$, $\Sigma_{\mathrm{I}}$ is the HI surface density in units of $M_{\odot}\,{\rm pc}^{-2}$, $\Sigma_{\mathrm{*}}$ is the surface density of star formation rate in units of $M_{\odot}\,{\rm pc}^{-2}\,{\rm Gyr}^{-1}$, and $n_{\mathrm{h}}$ is the number density of the hot phase normalized by $10^{-3}\,{\rm cm^{-3}}$.

This form of $U_{\mathrm{0}}$ leads to $V=U_{\mathrm{0}}/u$ proportional to $h/u$. Then, since $H\propto h/u$, virtually all features of the dynamo solutions depend only on the ratio $h/u$, rather than $h$ and $u$ separately. The only variable that depends on $h$ and $u$ differently is $B_{\mathrm{eq}}=u\sqrt{2\pi\Sigma_{\mathrm{I}}/h}$. The value of $B_{\mathrm{eq}}$ enters the steady-state magnetic field strength but not the pitch angle (7) and (12c) and the kinematic growth rate (10c) of the mean magnetic field.

In our fiducial model we assume that the ionized disk is flared, with an exponential increase in the disk scale height with galactocentric radius. For the Milky Way, we adopt the exponential scale length of $10\,{\rm kpc}$, as in the H I disk model of Kalberla & Dedes (2008). We also set $h|_{r=8\,{\rm kpc}}=0.4\,{\rm kpc}$ for the Milky Way, consistent with observed values of the warm neutral medium in the solar neighborhood (Lockman, 1984) (see also Sect. VI.2 of Ruzmaikin et al., 1988). This exponential relation is then rescaled using the $r_{25}$ radius. Adopting $r_{25}=16\,{\rm kpc}$ for the Milky Way (Bigiel & Blitz, 2012), we have

with $h_{\mathrm{0}}=(0.4\,{\rm kpc})\exp(-8\,{\rm kpc}/10\,{\rm kpc})\approx 0.18\,{\rm kpc}$ and $r_{\mathrm{0}}=10\,{\rm kpc}\,(r_{25}/16\,{\rm kpc})$.

We adopt a turbulent speed that does not vary within or between galaxies, $u=10\,{\rm km\,s^{-1}}=\mathrm{const}$, corresponding to transonic turbulence in the warm interstellar gas of a temperature about $10^{4}\,{\rm K}$. There is some evidence of variations of the turbulent speed with galactocentric radius, discussed in Section 6.3, but it is perhaps not conclusive enough.

Figure 3 shows the key parameters of the galaxy models obtained from the data of Table 2. Columns correspond, from left to right, to galaxy Groups I–III, as defined in Section 2. The top row presents the radial variation of $R_{U}=3HV$, a dimensionless quantity that determines the importance of the galactic outflow via equations (7), (9) and (11)). Note that $3HV\ll 1$ in all cases for which the required data is available, so outflows hardly affect the dynamo action in the model adopted. For those galaxies where $\Sigma_{\mathrm{*}}$ is not listed in Table 2, we adopt $U_{\mathrm{0}}=0$. Given the uncertainties involved in our estimate of the outflow speed, equation (14), we return to address the role of outflows in Section 7. The bottom row of Figure 3 shows the radial profile of $h(r)/u$ used in our fiducial model.

For the turbulent correlation time $\tau$ we adopt $14\,{\rm Myr}$, which is slightly larger than the value of $\tau=l/u\approx 10\,{\rm Myr}$ with $l\approx 100\,{\rm pc}$ and $u\approx 10\,{\rm km\,s^{-1}}$ often adopted in galactic dynamo models, but in the middle of the physically plausible range of 3–$30\,{\rm Myr}$. This choice leads to a good overall agreement with the observational data on the magnetic pitch angle while producing positive local kinematic growth rates for all data points. Restricting $\tau$ to be the same for all galaxies is probably too stringent, as $\tau$ is likely to depend on other factors such as the star formation rate and the local sound speed. However, to keep the model, analysis, and presentation simple, we prefer to adopt a constant $\tau$ in our fiducial model, but discuss the implications of relaxing this constraint in Section 6.2.

Only dynamo solutions with a positive local growth rate $\gamma$ of equation (10c) are physically significant as only then the model produces a non-zero mean magnetic field. In order to satisfy this constraint for all the data points, we are forced to adopt $C_{\alpha}=2$, rather than the standard value of unity. With $C_{\alpha}=1$, eight of the $29$ data points have decaying magnetic fields in the kinematic regime, $\gamma<0$. According to equation (7), the magnetic pitch angle $p$ in the saturated regime is insensitive to the value of $C_{\alpha}$, and numerical solutions confirm this. A change from $C_{\alpha}=2$ to $C_{\alpha}=4$ in the numerical solution does not produce more than a few per cent difference in $p$. Thus, in Regions (a) and (b) of parameter space, shown in Figure 1, $\alpha_{\mathrm{k}}$ is enhanced by a factor of $2$, but in Region (c) the upper limit $\alpha_{\mathrm{k}}=u$ is achieved, corresponding to $K=1$. In each case where $\mathrm{Co}>1$, the amplitude of $\alpha_{\mathrm{k}}$ reaches the upper limit $Ku$. Therefore, for fiducial model parameters, the relevant regions of parameter space are (a) and (c) only. The amplitude of $\alpha_{\mathrm{k}}$ reaches the upper limit of $10\,{\rm km\,s^{-1}}$ for only five of the $29$ data points, all at galactocentric distances $r\lesssim 4$–$5\,{\rm kpc}$: in the innermost annulus of M51, at all radii in NGC 1097, and in the innermost annulus of NGC 1365.