AN EMPIRICAL RELATION BETWEEN THE LARGE-SCALE MAGNETIC FIELD AND THE DYNAMICAL MASS IN GALAXIES

Magnetic fields are present on all scales in the universe from planets and stars to galaxies and galaxy clusters, and even at high redshifts. They are important for the continuation of life on the Earth, the onset of star formation, the order of the interstellar medium, and the evolution of galaxies (Beck & Wielebinski 2013). Hence, understanding the universe without understanding magnetic fields is impossible.

The most widely accepted theory to explain the magnetic fields on stars and planets is the dynamo theory. This describes the process through which a rotating, convecting, and electrically conducting fluid can maintain a magnetic field over astronomical timescales (Steenbeck & Krause 1969). A similar process can also explain the large-scale coherent magnetic fields in galaxies (see Widrow 2002 and references therein). It is assumed that such fields arise from the combined action of helical turbulence and differential rotation, a process known as the α-Ω dynamo. While a number of fundamental questions concerning the nature of the galactic dynamo remain unanswered, so far, no observational evidence for the effect of galaxy rotation on the large-scale magnetic field has been found. This motivated our currently reported investigation of a possible connection between the tracers of the large-scale magnetic field strength and the rotation of galaxies. Finding such a correlation observationally is not necessarily straightforward due to the possible dilution by galaxy-galaxy interactions and environmental effects which influence the rotation curves and possibly the magnetic fields. Such disturbing effects had to be taken into account when selecting the sample.

The present study is based on a careful measurement of the galaxy rotation speed as well as the model-free tracer of the large-scale magnetic field strength and the integrated polarized flux density (S_PI). We introduce the sample and the data in Section 2 and describe the v_rot measurements in Section 3. We investigate the possible correlations in Section 4 and discuss and summarize the results in Sections 5 and 6, respectively.

Not many galaxies are found in the literature with known polarized intensity measurements. Table 1 shows the 4.8 GHz integrated polarized intensity measurements for a sample of nearby galaxies, including Local Group galaxies and barred galaxies from Beck et al. (2002). To minimize environmental effects, we excluded galaxies in the Virgo and Ursa Major clusters and those known to be in interacting systems. Measurements with poor signal-to-noise ratio (≤1) were also omitted. We included the Local Group dwarf galaxies from Chyży et al. (2011), as they also show large-scale coherent magnetic fields. Galaxies that fit our selection limits, i.e., with known large-scale magnetic fields and minimum environmental disturbances are listed in Table 1. Each galaxy was observed in polarized light at a linear resolution d smaller than the galaxy optical size (in the sample, d < 0.25 × R₂₅, with R₂₅ as the optical radius). Thus, the large-scale magnetic field with a coherent length l (≃half a galaxy size; e.g., Fletcher 2010) is resolved in all galaxies and the beamwidth depolarization is small (∼d²/l² ≤ 6% in the sample).

The adopted rotation speed v_rot of the sample galaxies was derived in one of three ways, namely, by (I) averaging the rotation speed over the flat part of the rotation curves, (II) using the corrected W₂₀ measurements of HIPASS data, or (III) taking values directly from the literature.

In case I, 10 galaxies had rotation curves derived by Sofue et al. (1999)⁶ ; these are indicated with "S" in Table 1. The measurements in the "flat part" of the rotation curve were averaged to calculate v_rot (see the radial range for individual galaxies in Table 1). The errors were estimated assuming 10 km s⁻¹ measurement errors (Sofue et al. 1997) and 3° errors on the inclination. Five galaxies (with reference indicators "A," "B," "BA," "C," and "CS") were taken from other papers (see references in Table 1). For the SMC, v_rot was derived by a weighted average in the flat part of the rotation curve presented in Bekki & Stanimirović (2009), assuming uncertainties of 6 km s⁻¹ in velocity and 5° in inclination. For NGC 3359, in a similar way, we read the 13 outer data points of the rotation curve in Figure 11 of Ball (1983), excluding the outermost point. For the remaining three galaxies, v_rot was calculated in the same way, but with measurements and errors given in their reference papers.

In case II, we derived v_rot from HI W₂₀ measurements for eight galaxies (indicated by "W" in Table 1). The W₂₀ measurements were taken from Koribalski et al. (2004) and have been corrected for instrumental effects, internal turbulent motion, and inclination following Martinsson et al. (2016), assuming a gas velocity dispersion σ_HI = 10 km s⁻¹ and using the HIPASS spectral resolution of 18 km s⁻¹. The measurement errors on W₂₀ were estimated to be three times the error on the systemic velocity as suggested by Koribalski et al. (2004), and the error on inclination was assumed to be three degrees for all galaxies.

For three galaxies, v_rot is taken from the literature (Mateo 1998, with superscript "M" in Table 1), only correcting for the inclination (case III).

The polarized intensity (PI) provides a measure of the large-scale magnetic field in galaxies. It is defined through the integral over the path length L in the emitting medium

$$\begin{eqnarray}&&{\rm{PI}}=K{\displaystyle \int }_{L}{n}_{{\rm{cr}}}{\bar{B}}^{2}{dl},\end{eqnarray} \tag{ 1 }$$

with n_cr the cosmic-ray electron number density, $\bar{B}$ the large-scale transverse magnetic field strength averaged over the line of sight, and K a dimensional constant (Beck et al. 2003). The above definition is similar to that of the total intensity of the radio continuum emission,

$$\begin{eqnarray}&&{\rm{I}}=K{\displaystyle \int }_{L}{n}_{{\rm{cr}}}{B}^{2}{dl},\end{eqnarray} \tag{ 2 }$$

with B the total transverse magnetic field strength that includes both the large-scale uniform field and the local turbulent field, $\langle {B}^{2}\rangle ={\bar{B}}^{2}+\langle {b}^{2}\rangle$, where $\langle b\rangle$ is the turbulent magnetic field strength. Observationally, the total intensity I is just a measure of the Stokes I parameter, while the linearly polarized intensity PI is a measure of the Stokes Q and U parameters,

$$\begin{eqnarray}&&{\rm{PI}}=\sqrt{{Q}^{2}+{U}^{2}}.\end{eqnarray} \tag{ 3 }$$

The integrated flux densities of the total intensity S_I and linearly polarized intensity S_PI (in units of Jansky) are obtained by integrating the I and PI maps (in Jansky/beam) over the area A,

$$\begin{eqnarray}&&{S}_{{\rm{PI}}}=\int {\rm{PI}}\;\;{dA}=K\int {\displaystyle \int }_{L}{n}_{{\rm{cr}}}\;{\bar{B}}^{2}\;\;{dl}\;{dA},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{S}_{{\rm{I}}}=\int {\rm{I}}\;{dA}=K\int {\displaystyle \int }_{L}{n}_{{\rm{cr}}}\;{B}^{2}\;\;{dl}\;{dA}.\end{eqnarray} \tag{ 5 }$$

For the selected sample of 26 galaxies, we collected S_I and S_PI at 4.8 GHz from the literature. Table 1 shows those fluxes, projected to a common distance of 10 Mpc. Plotting S_PI versus v_rot, we find a tight correlation (Figure 1). The Pearson correlation coefficient is r_p = 0.94 ± 0.04, and the Spearman rank coefficient is r_sp = 0.8 ± 0.0. The correlation is even tighter than that between S_I and v_rot with a rank r_sp = 0.7 ± 0.0 (same figure). A linear fit in logarithmic plane leads to a slope of 4.6 ± 0.3, slightly steeper than that found between S_I and v_rot (4.0 ± 0.3). The ratio of the integrated polarized to total flux density, S_PI/S_I, is not constant and increases with v_rot with a least squares fit slope of 0.6 ± 0.2 and a bisector fit slope of 1.2 ± 0.3.

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Taking into account the well-known correlation between the total star formation rate (SFR) and radio luminosity of galaxies (Condon 1992) and the correlation between SFR and stellar mass (e.g., Sparre et al. 2015) that scales with v_rot according to the Tully–Fisher relation (Tully & Fisher 1977), are the observed correlations ${S}_{{\rm{I}}}-{v}_{{\rm{rot}}}$ and ${S}_{{\rm{PI}}}-{v}_{{\rm{rot}}}$ (Figure 1) merely secondary correlations that represent a more direct relation between SFR and v_rot? We estimated SFR by combining the GALEX far-UV (FUV; Lee 2011; Cortese et al. 2012) and the IRAS 25 μm fluxes (Abrahamyan et al. 2015) and following the calibration relation given by Hao et al. (2011) and Kennicutt et al. (2009):

$$\begin{eqnarray*}\displaystyle \frac{{\rm{SFR}}}{{M}_{\odot }\;{\mathrm{yr}}^{-1}}\; & = & \;{10}^{-43.35}\left[\displaystyle \frac{\nu {L}_{\nu }({\rm{FUV}})}{\mathrm{erg}\;{{\rm{s}}}^{-1}}\right.\\ & & +\left.3.89\times \displaystyle \frac{\nu {L}_{\nu }(25\;\mu {\rm{m}})}{\mathrm{erg}\;{{\rm{s}}}^{-1}}\right].\end{eqnarray*}$$

In a few cases with no GALEX data, only the IR data were used to estimate the SFR following Kennicutt & Evans (2012):

$$\begin{eqnarray*}&&\displaystyle \frac{{\rm{SFR}}}{{M}_{\odot }\;{\mathrm{yr}}^{-1}}={10}^{-42.69}\;\displaystyle \frac{\nu {L}_{\nu }(24\;\mu {\rm{m}})}{\mathrm{erg}\;{{\rm{s}}}^{-1}}.\end{eqnarray*}$$

Table 1 lists the SFR values obtained. We address the question raised by comparing the correlations of S_I and S_PI with SFR and v_rot. We found that SFR is indeed correlated with v_rot with r_sp = 0.67, similar to the ${S}_{{\rm{I}}}-{v}_{{\rm{rot}}}$ correlation (r_sp = 0.72), indicating that the SFR could be the main cause of the observed ${S}_{{\rm{I}}}-{v}_{{\rm{rot}}}$ correlation. This is shown better by a tighter correlation of S_I with SFR (r_sp ∼ 0.9) than v_rot (r_sp ∼ 0.7; Figure 2).

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The ${\rm{SFR}}-{v}_{{\rm{rot}}}$ correlation is, however, not as tight as the ${S}_{{\rm{PI}}}-{v}_{{\rm{rot}}}$ correlation (r_sp = 0.80). Hence, the correlation between SFR and S_PI (Figure 2) may not fully explain the tighter ${S}_{{\rm{PI}}}-{v}_{{\rm{rot}}}$ correlation. A correlation is expected between the integrated polarized flux density S_PI and SFR due to cosmic rays as their number N_cr increases with star formation activity (e.g., Murphy et al. 2008). Otherwise, no correlation is expected between the large-scale field $\bar{B}$ (traced by PI) and SFR, while the turbulent (and total) magnetic field positively correlates with SFR (Tabatabaei et al. 2013a, 2013b). This could also explain that the ${S}_{{\rm{I}}}-{\rm{SFR}}$ correlation is tighter than the ${S}_{{\rm{PI}}}-{\rm{SFR}}$ correlation (Figure 2). We note that SFR is not a directly measured quantity. Potential systematic effects in calibration, like a dependence of the calibration factors on the galaxy properties, could increase uncertainties in SFR.

5.1. The Large-scale Magnetic Field

Equation (4) can be written as ${S}_{{\rm{PI}}}\sim \langle {n}_{{\rm{cr}}}\rangle {\bar{B}}^{2}\;V$, with V as the integration volume. The ratio ${S}_{{\rm{PI}}}/\langle {n}_{{\rm{cr}}}\rangle$ is a measure of the energy of the large-scale magnetic field, E($\bar{B}$)=${\bar{B}}^{2}\;V/8\pi$. Studying the corresponding energy density and the large-scale magnetic field strength is then possible via ${S}_{{\rm{PI}}}/{N}_{{\rm{cr}}}\sim {\bar{B}}^{2}$ with ${N}_{{\rm{cr}}}=\langle {n}_{{\rm{cr}}}\rangle V$. The SFR can be taken as a proxy for N_cr (Section 5), and hence ${S}_{{\rm{PI}}}/{\rm{SFR}}\sim {\bar{B}}^{2}$. Figure 3 (top) shows that S_PI/SFR is correlated with ${v}_{\mathrm{rot}}^{1.8\pm 0.3}$ with r_p = 0.8 ± 0.1 and r_sp = 0.7 ± 0.0. Thus, ${\bar{B}}^{2}\sim \;{S}_{{\rm{PI}}}/{\rm{SFR}}\sim {v}_{{\rm{rot}}}^{2}$ or $\bar{B}\sim {v}_{{\rm{rot}}}$, the large-scale ordered field, is linearly proportional to the rotation speed of the galaxies.

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The relation between N_cr and SFR differs if an equipartition between the energy densities of the cosmic-ray electrons and the total magnetic field holds. The global equipartition condition leads to N_cr ∼ B², with B the mean total magnetic field strength in each galaxy. Independent observations show a global correlation between B and SFR in galaxies, B ∼ SFR^0.25–0.3 (Chyży et al. 2011; Heesen et al. 2014). Thus, N_cr changes with SFR as N_cr ∼ SFR^0.5–0.6. It then follows that ${\bar{B}}^{2}\sim \;{S}_{{\rm{PI}}}/{N}_{{\rm{cr}}}\sim {S}_{{\rm{PI}}}/{\rm{SFR}}$^(0.5–0.6), which is correlated to $\sim {v}_{{\rm{rot}}}^{3.0\pm 0.3}$ with r_p = 0.9 ± 0.1 (Figure 3, middle). Hence, a super-linear correlation holds between $\bar{B}$ and v_rot: $\bar{B}\sim {v}_{{\rm{rot}}}^{1.5}$.

We note that the scatter at the high-v_rot end in Figure 3 (top) is due to the galaxies with the highest SFR, NGC 7552 and NGC 1365, as well as the Seyfert galaxy NGC 5643. The large turbulence due to high SF and active galactic nucleus activities in these galaxies could enhance the generation of turbulent fields by the small-scale dynamo. It could also cause depolarization due to internal Faraday dispersion (Sokoloff et al. 1998) reducing the observed PI and S_PI and shifting the S_PI/SFR ratio to relatively low values.

5.2. Tracing the Dynamo Effect

5.3. Magnetic Energy versus Rotational Energy

The total energy of the large-scale magnetic field scales with S_PI/n_cr or S_PI V/N_cr (Section 1). Assuming N_cr ∼ SFR, the magnetic energy is

$$\begin{eqnarray}&&{{\rm{E}}}_{\bar{B}}\sim \;V\;{S}_{{\rm{PI}}}/{\rm{SFR}},\end{eqnarray} \tag{ 6 }$$

while assuming equipartition leads to

$$\begin{eqnarray}&&{{\rm{E}}}_{\bar{B}}\sim \;V\;{S}_{{\rm{PI}}}/{{\rm{SFR}}}^{0.5-0.6}.\end{eqnarray} \tag{ 7 }$$

To estimate ${{\rm{E}}}_{\bar{B}}$ using Equations (6) and (7), the synchrotron radiating volume V is taken to be the volume of the thick disk with the radius of R₂₅ and scale height of 1 kpc, corrected for inclination. The case of equipartition (Equation (7)) even allows calibration of ${{\rm{E}}}_{\bar{B}}=V\;{\bar{B}}^{2}/8\pi$ in erg through

$$\begin{eqnarray*}&&\bar{B}={\bar{B}}_{0}\;\displaystyle \frac{{S}_{{\rm{PI}}}}{{S}_{{{\rm{PI}}}_{0}}}{\left(\displaystyle \frac{{{\rm{SFR}}}_{0}}{{\rm{SFR}}}\right)}^{0.5-0.6},\end{eqnarray*}$$

with ${\bar{B}}_{0}$, ${S}_{{{\rm{PI}}}_{0}}$, and SFR₀ as the corresponding parameters of a reference galaxy. We took a galaxy residing between the dwarfs and normal spirals, i.e., M33 (${\bar{B}}_{0}=2.5\;\mu$G; Tabatabaei et al. 2008), as the reference galaxy in the sample.

The v_rot data allow estimation of the rotational energy of the galaxies, ${{\rm{E}}}_{{\rm{rot}}}=\frac{1}{2}{I}_{m}{{\rm{\Omega }}}^{2}=\frac{1}{2}{I}_{m}\;{({v}_{{\rm{rot}}}/{R}_{25})}^{2}$, with ${I}_{m}\sim {M}_{{\rm{dyn}}}\;{R}_{25}^{2}$ the moment of inertia. For disks, ${I}_{m}=\frac{1}{2}\;{M}_{{\rm{dyn}}}\;{R}_{25}^{2}$ leading to ${{\rm{E}}}_{{\rm{rot}}}=\frac{1}{4}\;{R}_{25}\;{v}_{{\rm{rot}}}^{4}$.

Figure 4 shows the energy of the large-scale magnetic field ${{\rm{E}}}_{\bar{B}}$ against the rotational energy of the galaxies E_rot. The slope of the ${E}_{\bar{B}}-{E}_{{\rm{rot}}}$ correlation in logarithmic scale is 1.03 ± 0.10 (r_p = 0.91 ± 0.08) for N_cr ∼ SFR, and 1.25 ± 0.08 (r_p = 0.96 ± 0.06) assuming equipartition (the magnetic-to-rotation energy changes as $2\times {10}^{-4}\lt {{\rm{E}}}_{\bar{B}}/{{\rm{E}}}_{{\rm{rot}}}\lt 8\times {10}^{-2}$ with a median value of 0.002). Therefore, the two different assumptions on N_cr do not lead to a significant difference in the ${{\rm{E}}}_{\bar{B}}-{{\rm{E}}}_{{\rm{rot}}}$ relation. These correlations show that galaxies with higher rotational energy have a higher energy of the large-scale magnetic field. Considering that the rotational energy is balanced with the gravitational energy and is constant particularly for non-interacting galaxies, these relations show that the large-scale magnetic field is almost stationary at the current epoch in each galaxy (this does not reject a possibly linear dynamo process in the past). This supports the theoretical predictions that the present-day large-scale dynamos cannot be in their linear (growing) phase.

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Considering active star formation as the main source of the turbulence, the ${\rm{SFR}}-{v}_{{\rm{rot}}}$ correlation implies that the turbulent energy does not change much, as long as the galaxy mass is fixed. Several studies show an equipartition between the turbulent and the total magnetic field energy densities (e.g., Beck 2007, 2015; Tabatabaei et al. 2008) indicating that the total magnetic field also does not evolve further.

To study the relations between galactic magnetic fields and rotation, we selected from the literature observations of the large-scale magnetic field of galaxies that are not in clusters and are not interacting. These show a tight correlation between the polarized flux density and the rotation speed v_rot over three orders of magnitude in dynamical mass. Assuming N_cr ∼ SFR, a linear correlation is found between the large-scale magnetic field strength $\bar{B}$ and v_rot. This correlation is super-linear ($\bar{B}\sim {v}_{{\rm{rot}}}^{1.5}$) assuming equipartition between the cosmic-ray electrons and the total magnetic field. On the other hand, no correlation is found between $\bar{B}$ and Ω, expected from the linear α-Ω dynamo theory. The $\bar{B}-{v}_{{\rm{rot}}}$ correlation suggests that there is a coupling between the large-scale magnetic field and the dynamical mass of galaxies: $\bar{B}\sim {M}_{{\rm{dyn}}}^{0.25-0.4}$. Therefore, faster rotating/more massive galaxies have stronger magnetic fields than slower rotating/less massive galaxies. The $\bar{B}-{v}_{{\rm{rot}}}$ correlation shows that the anisotropic turbulent field dominates the large-scale field in faster rotating galaxies, perhaps due to high streaming velocities of the gas with which the magnetic field is coupled.

A comparison between the magnetic and rotational energies shows that the large-scale magnetic field is stationary and does not evolve further in galaxies with fixed dynamical mass.

This is the first study showing a statistically meaningful effect of the galaxy rotation/mass on the large-scale magnetic fields.

We thank the anonymous referee for valuable comments. We also thank Anvar Shukurov and Andrew Fletcher for stimulating discussions and to John Dickel for providing us with the LMC and SMC data. F.S.T., T.P.K.M., and J.H.K. acknowledge financial support from the Spanish Ministry of Economy and Competitiveness (MINECO) under grant number AYA2013-41243-P. J.H.K. and J.E.B. acknowledge financial support to the DAGAL network from the People Programme (Marie Curie Actions) of the European Unions Seventh Framework Programme FP7/2007-2013/under REA grant agreement number PITN-GA-2011-289313.