Magnetic Fields in Elliptical Galaxies: An Observational Probe of the Fluctuation Dynamo Action

The source of hot gas in ellipticals can be internal or external to the galaxy. Internally, the hot gas can be due to outflows from evolved stars and Type Ia supernova explosions. Externally, some of the circumgalactic gas (expelled from the galaxy by winds driven by frequent Type II supernova explosions at earlier stages of galactic evolution) can fall back into the galaxy, as well as gas accreted from the galaxy group or cluster. The hot gas in elliptical galaxies is observed via its X-ray emission. The relationship between the X-ray luminosity L_X and optical luminosity (in the B band) L_B of bright elliptical galaxies is observationally found to be ${L}_{{\rm{X}}}\propto {L}_{B}^{2.2}$ (O'Sullivan et al. 2001). This confirms the nonstellar origin of the X-ray emission, otherwise the relationship would have been linear. Gas mass can be inferred from X-ray observations. The total mass of the hot gas is about $1 \% \mbox{--}2 \%$ of the total stellar mass (Mathews & Brighenti 2003).

The turbulence in spiral galaxies that merge to form an elliptical galaxy decays. Numerical simulations of decaying three-dimensional isotropic turbulence in an isothermal gas suggest that the turbulent velocity $\,{u}_{\mathrm{rms}}$ decays as a power law, ${(t/{t}_{0})}^{-{\eta }_{u}}$, with the exponent ${\eta }_{u}$ in the range $0.4\mbox{--}0.6$ and t₀ being the eddy turnover time (Mac Low et al. 1998; Stone et al. 1998; Subramanian et al. 2006; Sur 2019). Considering ${\eta }_{u}$ as 0.6, the rms turbulent velocity in the parent spiral galaxy, ${u}_{\mathrm{rms}}\simeq 10\,\mathrm{km}\,{{\rm{s}}}^{-1}$ decreases to $1\,\mathrm{km}\,{{\rm{s}}}^{-1}$ in $5\times {10}^{8}\,\mathrm{yr}$ (where the eddy turnover time is calculated as follows: ${t}_{0}\approx {\ell }_{0}/{u}_{{\rm{rms}}}\approx 100\,{\rm{pc}}/10\,{\rm{km}}\,{{\rm{s}}}^{-1}\simeq {10}^{7}\,{\rm{yr}}$). However, we aim to look for magnetic fields in quiescent elliptical galaxies that probably had a major merger a few gigayears earlier, so the turbulence due to parent spirals must have significantly decayed by then. Also, there is a lower level of turbulence (than in the parent spiral galaxies) maintained in quiescent elliptical galaxies due to continuous energy injection by Type Ia supernova explosions. This is also seen in observations. Each explosion of a Type Ia supernova enriches the medium with roughly $0.7\,{\text{}}{M}_{\odot }$ of iron, and iron abundance in the hot gas of ellipticals increases from the outskirts to the center of the galaxy, where the density is highest, so the supernova activity is highest (Mathews & Brighenti 2003). Further, the observed iron abundance can be used to estimate the supernova explosion rate in ellipticals (${10}^{-3}\,{\mathrm{yr}}^{-1}$). The rates are a factor of ∼30 lower than in spiral galaxies ($3\times {10}^{-2}\,{\mathrm{yr}}^{-1}$). Thus, the turbulence decays immediately after the merger of spiral galaxies but is then maintained at a certain level by continuous energy injection by Type Ia supernova explosions in quiescent elliptical galaxies. Type Ia supernova explosions will drive irrotational turbulence (negligible vorticity) in the ISM of elliptical galaxies. But this irrotational turbulence interacts with the multiphase ISM (see Li et al. 2020b, and references therein, for reasons justifying the presence of inhomogeneous, multiphase gas in ellipticals) to generates vortical motions required for the fluctuation dynamo action.

To estimate the properties of turbulence in elliptical galaxies, we closely follow Moss & Shukurov (1996) ⁷ with two major changes. First, they consider two drivers of turbulence: supernova explosions and random star motions, whereas we only consider Type Ia supernova explosions since the length scales associated with random star motions would be much smaller than those due to supernova explosions. Second, they consider acoustic turbulence but we consider hydrodynamic turbulence, because Mathews & Brighenti (1997) showed that the acoustic modes will quickly decay and cannot maintain the pervasive turbulence.

X-ray observations suggest a temperature T of the order of ${10}^{7}\,{\rm{K}}$ for the diffuse interstellar gas in elliptical galaxies (Mathews & Brighenti 2003). This implies a sound speed ${c}_{s}\approx \sqrt{{k}_{{\rm{B}}}T/{m}_{{\rm{p}}}}\approx 300\,{\rm{km}}\,{{\rm{s}}}^{-1}$, where ${k}_{{\rm{B}}}=1.38\times {10}^{-16}\,{\rm{erg}}\,{{\rm{K}}}^{-1}$ is the Boltzmann constant and ${m}_{{\rm{p}}}=1.67\times {10}^{-24}\,{\rm{g}}$ is the proton mass (assuming that most of the mass of the hot gas is dominated by protons). We now use the sound speed and Sedov–Taylor blast wave self-similarity solution (Ostriker & McKee 1988) to calculate the driving scale of the turbulence. The radius of a supernova remnant R as a function of time t is then

$$\begin{eqnarray}&&R={\left(\kappa \displaystyle \frac{{E}_{\mathrm{SN}}}{{\rho }_{0}}\right)}^{1/5}{t}^{2/5},\end{eqnarray} \tag{ 1 }$$

where ${E}_{\mathrm{SN}}$ is the energy of a supernova explosion, ${\rho }_{0}$ is the average gas density of the ambient medium, and $\kappa \sim 2$ for a monatomic ideal gas. Now, the radius R is roughly equal to the driving scale of the turbulence ${{\ell }}_{0}$, when the velocity of the shock front is equal to c_s . Thus, $t\approx R/{c}_{s}$. This gives

$$\begin{eqnarray}&&{{\ell }}_{0}\simeq {\left(\kappa \displaystyle \frac{{E}_{\mathrm{SN}}}{{\rho }_{0}}\right)}^{1/3}\displaystyle \frac{1}{{c}_{s}^{2/3}}.\end{eqnarray} \tag{ 2 }$$

Using ${E}_{\mathrm{SN}}={10}^{51}\,\mathrm{erg}$, ${\rho }_{0}={m}_{{\rm{p}}}\,{n}_{0}=1.67\times {10}^{-27}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ (assuming an ambient number density, ${n}_{0}={10}^{-3}\,{\mathrm{cm}}^{-3}$, Forman et al. 1985), and ${c}_{s}=300\,\mathrm{km}\,{{\rm{s}}}^{-1}$, we obtain ${{\ell }}_{0}\simeq 300\,\mathrm{pc}$. This scale is slightly larger than that in spiral galaxies ($\simeq 100\,\mathrm{pc}$) because the ambient medium is less dense.

The supernovae survive longer in ellipticals than in spirals, but their frequency is much lower and so they inject less turbulent kinetic energy into the medium. To calculate the turbulent velocity, we assume that about $1 \%$ of the total supernova energy is converted to the kinetic energy of the turbulence (Dyson & Williams 1997). The rate at which supernovae inject energy into the medium can be balanced with the rate of gain of kinetic energy of the medium as

$$\begin{eqnarray}&&\displaystyle \frac{{v}_{0}^{2}}{{{\ell }}_{0}/{v}_{0}}\approx f{\nu }_{\mathrm{SN}}{E}_{\mathrm{SN}}{M}_{\mathrm{gas}}^{-1},\end{eqnarray} \tag{ 3 }$$

where v₀ is the turbulent velocity, f is the fraction of energy that is injected into the medium, ${\nu }_{\mathrm{SN}}$ is the frequency of supernova explosions, and ${M}_{\mathrm{gas}}$ is the mass of the hot gas. Using ${{\ell }}_{0}=300\,\mathrm{pc}$, $f=0.01$, ${\nu }_{\mathrm{SN}}={10}^{-3}\,{\mathrm{yr}}^{-1}$(Cappellaro et al. 1999), ${E}_{\mathrm{SN}}={10}^{51}\,\mathrm{erg}$, and ${M}_{\mathrm{gas}}={10}^{10}\,{\text{}}{M}_{\odot }$, we obtain ${v}_{0}\simeq 2.5\,\mathrm{km}\,{{\rm{s}}}^{-1}$. This is lower than that in spiral galaxies ($\simeq 10\,\mathrm{km}\,{{\rm{s}}}^{-1}$).

The estimated driving scale of turbulence ($300\,\mathrm{pc}$) is roughly similar to that obtained in Moss & Shukurov (1996) ($350\,\mathrm{pc}$), but the turbulent velocity we get ($2.5\,\mathrm{km}\,{{\rm{s}}}^{-1}$) is smaller than their estimate ($60\,\mathrm{km}\,{{\rm{s}}}^{-1}$). This is because we consider that Type Ia supernova explosions drive hydrodynamic turbulence rather than the acoustic turbulence, which they assume. The acoustic turbulence decays very quickly due to relatively high viscosity in the ISM of elliptical galaxies. For a fully ionized gas, with electron number density ${n}_{{\rm{e}}}$ and temperature T, the viscosity ν based on the kinetic theory is estimated as (Brandenburg & Subramanian 2005)

$$\begin{eqnarray}&&\nu \simeq 6.5\times {10}^{22}{\left(\displaystyle \frac{T}{{10}^{6}{\rm{K}}}\right)}^{5/2}\left(\displaystyle \frac{{n}_{{\rm{e}}}}{1\,{\mathrm{cm}}^{-3}}\right)\,{\mathrm{cm}}^{2}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 4 }$$

where we assumed a Coulomb logarithmic factor of order unity. For the core for an elliptical galaxy, ${n}_{{\rm{e}}}=0.1\,{\mathrm{cm}}^{-3}$ and $T={10}^{7}\,{\rm{K}}$, we obtain $\nu \simeq 2\times {10}^{26}\,{\mathrm{cm}}^{2}\,{{\rm{s}}}^{-1}$. The diffusive length scale ${l}_{\nu }={(\nu /t)}^{1/2}$ in an eddy turnover time ${t}_{0}=300\,\mathrm{pc}/2.5\,\mathrm{km}\,{{\rm{s}}}^{-1}\simeq {10}^{7}\,\mathrm{yr}$ is $3\times {10}^{5}\,\mathrm{cm}$, which is much smaller than the driving scale ($\sim 300\,\mathrm{kpc}$). Thus, high viscosity dissipates acoustic waves within a very short distance. Plasma effects due to the presence of magnetic fields can reduce the viscosity calculated via Equation (4), but not sufficiently to account for such a large difference. Furthermore, the dissipation of acoustic waves might provide an additional heating mechanism for the medium (details of the mechanism are given by Fabian et al. 2005; Zweibel 2020).

The driving scale of turbulence ${{\ell }}_{0}\simeq 300\,\mathrm{pc}$ and the rms velocity ${v}_{0}\simeq 2.5\,\mathrm{km}\,{{\rm{s}}}^{-1}$ are global values calculated from average quantities close to the core of a typical elliptical galaxy. ⁸ The eddy turnover time can be calculated as ${t}_{0}\approx {{\ell }}_{0}/{v}_{0}\simeq 1.2\times {10}^{8}\,\mathrm{yr}$. The typical value of magnetic Reynolds numbers can also be calculated. Spitzer resistivity η for a plasma at a temperature T can be estimated from (Brandenburg & Subramanian 2005)

$$\begin{eqnarray}&&\eta \simeq {10}^{4}{\left(\displaystyle \frac{T}{{10}^{6}{\rm{K}}}\right)}^{-3/2}\,{\mathrm{cm}}^{2}\,{{\rm{s}}}^{-1},\end{eqnarray} \tag{ 5 }$$

where we assumed the Coulomb logarithmic factor of order unity. Using $T={10}^{7}\,{\rm{K}}$, we obtain $\eta \simeq 3\times {10}^{2}\,{\mathrm{cm}}^{2}\,{{\rm{s}}}^{-1}$. For ${{\ell }}_{0}=300\,\mathrm{pc}$ and ${v}_{0}=2.5\,\mathrm{km}\,{{\rm{s}}}^{-1}$, we obtain $R{e}_{{\rm{M}}}={\ell }_{0}{v}_{0}/\eta \simeq {10}^{24}$. The estimated $R{e}_{{\rm{M}}}$ is considerably greater than $R{e}_{{\rm{M}}}^{({\rm{crit}})}$ ($\approx {10}^{2}$), which is required for the fluctuation dynamo action. Even though the flow is very viscous, such a high magnetic Reynolds number ensures fluctuation dynamo action in random flows. Thus, it is reasonable to expect fluctuation dynamo-generated magnetic fields in elliptical galaxies.

Since the gas and stellar densities in elliptical galaxies vary with radius (usually the variation is described by the King profile with a central core radius, see Statler 2012) the properties of the ISM turbulence would also change. However, we are interested mostly in the core region of elliptical galaxies where the density of the hot gas and supernova activity are higher and thus the expected magnetic field strength and the probability for an unambiguous detection are also higher. The magnetic field strength may be further enhanced close to the core due to the compression by cooling flows.

The rms value of fluctuation dynamo-generated magnetic field $\,{b}_{\mathrm{rms}}$ is in near-equipartition with the turbulent kinetic energy (Haugen et al. 2004; Federrath et al. 2011; Tzeferacos et al. 2018; Seta et al. 2020),

$$\begin{eqnarray}&&\,{b}_{\mathrm{rms}}\approx 0.5{\left(4\pi \rho \right)}^{1/2}{v}_{0},\end{eqnarray} \tag{ 6 }$$

where ρ is the density of the medium and v₀ is the turbulent velocity. This is probably representative of magnetic field strengths in the core of elliptical galaxies. In the core of a typical elliptical, with $\rho =0.1{m}_{{\rm{p}}}=1.67\times {10}^{-26}\,{\rm{g}}\,{\mathrm{cm}}^{-3}$ (Forman et al. 1985) and ${v}_{0}=2.5\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (as obtained in Section 2.1), we obtain $\,{b}_{\mathrm{rms}}\simeq 0.2\,\mu {\rm{G}}$.

The gas in the centers of elliptical galaxies quickly loses its energy via X-ray emission. Since the central core cools quickly, the weight of the outer region will cause the surrounding material to flow inward. This is referred to as the cooling flow (Fabian 1994). Cooling flows can also be inferred from observations of O vi line emission from elliptical galaxies (Bregman et al. 2001, 2005). The magnetic field would be further enhanced by such cooling flows due to compression. By flux conservation under spherically symmetric compression, the field strength b grows as ${\rho }^{2/3}$, where ρ is the gas density. In cooling flows, the density can be enhanced by a factor of 10 or more (Mathews & Brighenti 1997). Thus, the magnetic field with magnitude approximately equal to $0.2\,\mu {\rm{G}}$ can be further amplified to be around $1\,\mu {\rm{G}}$. Mathews & Brighenti (1997), using a detailed model of spherical cooling flow, suggest a radial dependence for the magnetic field strength of the form $b(r)\propto (1\mbox{--}10)\,{r}^{-1.2}\,\mu {\rm{G}}$. The radial dependence of thermal electron number density is ${n}_{{\rm{e}}}(r)\propto 0.1{r}^{-1.5}\,{\mathrm{cm}}^{-3}$ (Mathews & Brighenti 2003). Thus, we do expect a significant Faraday rotation of the polarized light from background radio sources passing through the foreground elliptical galaxy (further discussed in Section 4 and Section 5.1).

The polarization angle of linearly polarized synchrotron radiation rotates when the radiation passes through a magnetized medium containing thermal electrons. The amount of rotation is a function of the wavelength of the radiation and the Faraday rotation measure, RM, which in terms of convenient units is written as

$$\begin{eqnarray}&&\displaystyle \frac{\mathrm{RM}}{1\,\mathrm{rad}\,{{\rm{m}}}^{-2}}=K{\int }_{0}^{L/1\,\mathrm{pc}}\displaystyle \frac{{n}_{{\rm{e}}}}{1\,{\mathrm{cm}}^{-3}}\displaystyle \frac{{b}_{\parallel }}{1\,\mu {\rm{G}}}{\text{}}{dl},\end{eqnarray} \tag{ 9 }$$

where ${n}_{{\rm{e}}}$ is the thermal electron number density, b_∥ is the magnetic field parallel to the line of sight, L is the path length, and $K=0.81$ when the units are scaled as indicated. For a cosmologically distant source at a redshift z, the RM in the observer frame is further reduced by a factor of ${(1+z)}^{2}$ from that given by Equation (9) (which is the RM in the rest frame).

The standard deviation of fluctuations in rotation measure ${\sigma }_{\mathrm{RM}}$ can be estimated from the two-point correlation function of RM, which in turn can be estimated from the two-point correlation of the magnetic field. For the simplest case, this can be done by assuming that the number density of thermal electrons is uniform. Assuming a constant ${n}_{{\rm{e}}}$ in Equation (9), Shukurov & Sokoloff (2007) demonstrated that

$$\begin{eqnarray}&&\displaystyle \frac{{\sigma }_{\mathrm{RM}}}{1+{z}^{2}}=\displaystyle \frac{{\left(2\pi \right)}^{1/4}}{{3}^{1/2}}{{Kn}}_{e}\,{b}_{\mathrm{rms}}{\left(L{{\ell }}_{b}\right)}^{1/2},\end{eqnarray} \tag{ 10 }$$

where $\,{b}_{\mathrm{rms}}$ is the rms magnetic field strength in the region, L is the path length, and ${{\ell }}_{b}$ is the correlation length of the random magnetic field. In the central region, assuming that the fluctuation dynamo is active (see Section 2.3), ${n}_{{\rm{e}}}=0.1\,{\mathrm{cm}}^{-3}$ (Mathews & Brighenti 2003), $\,{b}_{\mathrm{rms}}=1\,\mu {\rm{G}}$, ${{\ell }}_{b}=100\,\mathrm{pc}$ (${{\ell }}_{0}/{{\ell }}_{b}\simeq 3\mbox{--}4$ and ${{\ell }}_{0}\simeq 300\,\mathrm{pc}$), and $L=10\,\mathrm{kpc}$ (since the emission is considered close to the central region). For $z\approx 0$, these values yield ${\sigma }_{\mathrm{RM}}\simeq 75\,\mathrm{rad}\,{{\rm{m}}}^{-2}$. We expect that this can be observed in the case of good signal-to-noise ratio. However, the number density decreases with the radius and this might further decrease ${\sigma }_{\mathrm{RM}}$ at larger radii.

The analysis can be extended to include the contribution from a varying thermal electron number density (see Felten 1996 and Appendix A in Bhat & Subramanian 2013). Assuming the King profile for electrons in an elliptical galaxy, ${n}_{{\rm{e}}}={n}_{{\rm{e}}}(0){\left(1+{\left(r/a\right)}^{2}\right)}^{-3/4}$, where ${n}_{{\rm{e}}}(0)$ is the electron number density at $r=0$ and a is the core radius, and also assuming that $\,{b}_{\mathrm{rms}}\propto {n}_{{\rm{e}}}^{\gamma }$, ${\sigma }_{\mathrm{RM}}$ at $r=0$ is

$$\begin{eqnarray}&&\displaystyle \frac{{\sigma }_{\mathrm{RM}}(0)}{1+{z}^{2}}=\displaystyle \frac{{\left(2\pi \right)}^{1/4}}{{3}^{1/2}}{{Kn}}_{{\rm{e}}}(0)\,{b}_{\mathrm{rms}}{\left(a{{\ell }}_{b}\right)}^{1/2}{\left(\displaystyle \frac{{\rm{\Gamma }}\left(\tfrac{3}{2}(\gamma +1)-0.5\right)}{{\rm{\Gamma }}\left(\tfrac{3}{2}(\gamma +1)\right)}\right)}^{1/2}.\end{eqnarray} \tag{ 11 }$$

Using ${n}_{{\rm{e}}}(0)=0.1\,{\mathrm{cm}}^{-3},\,{b}_{\mathrm{rms}}=1\,\mu {\rm{G}},a=3\,\mathrm{kpc},{{\ell }}_{b}=100\,\mathrm{pc},$ and $\gamma =2/3$ (motivated by the flux-freezing or cooling-flow constraint, i.e., $b\propto {\rho }^{2/3}$), we obtain ${\sigma }_{\mathrm{RM}}(0)\simeq 35\,\mathrm{rad}\,{{\rm{m}}}^{-2}$ at $z\approx 0$. ${\sigma }_{\mathrm{RM}}$ as a function of the radial distance in the plane of the sky r_⊥ is

$$\begin{eqnarray}&&{\sigma }_{{\rm{RM}}}({r}_{\perp })=\displaystyle \frac{{\sigma }_{{\rm{RM}}}(0)}{1+{z}^{2}}{\left[1+{\left({r}_{\perp }/a\right)}^{2}\right]}^{-\tfrac{1}{4}{\rm{\Gamma }}\left(\tfrac{3}{2}(\gamma +1)-1\right)}.\end{eqnarray} \tag{ 12 }$$

For ${\sigma }_{\mathrm{RM}}(0)/(1+{z}^{2})=35\,\mathrm{rad}\,{{\rm{m}}}^{-2}$ and $\gamma =2/3$, we obtain ${\sigma }_{\mathrm{RM}}({r}_{\perp })=35\,\mathrm{rad}\,{{\rm{m}}}^{-2}{\left(1+{\left({r}_{\perp }/a\right)}^{2}\right)}^{-0.22}$.

Therefore, we expect significant fluctuations in Faraday rotation from at least the core of an elliptical galaxy (usually defined as the radius at which the slope of the surface brightness profile changes dramatically from low values).

We confirm our expectations for ${\sigma }_{\mathrm{RM}}$ using numerical simulations of the nonlinear fluctuation dynamo (numerically solving the Navier–Stokes equation with a prescribed random forcing and induction equation for an isothermal gas in a periodic domain as described in Haugen et al. 2004). The complete details of the simulations are described in Section II of Seta et al. (2020). The forcing is solenoidal, nearly incompressible, and the Mach number of the flow is less than 0.1. We note that in numerical simulations of (Type Ia) supernova-driven turbulence, the overall Mach number is typically $\lt 0.1$ (Li et al. 2020a). The initial conditions of our simulations are as follows: a uniform density, zero velocity, and a very weak random magnetic field with no net flux. As the turbulence is driven numerically within the domain, the density remains roughly the same (since the gas is isothermal, the turbulent driving is near-incompressible, and the Mach number is less than unity), and the velocity reaches a statistically steady state after about two turbulent eddy turnover times (e.g., Federrath et al. 2009; Price et al. 2011). The magnetic field first grows exponentially (kinematic stage) and then saturates (saturated stage) due to the back reaction of the flow via the Lorentz force induced by the magnetic field (Figure 1 in Seta et al. 2020). We use the magnetic field from the saturated stage of the simulation with the forcing wavenumber ${k}_{{\rm{F}}}=5$, Reynolds number $Re=283$, and magnetic Reynolds number ${{Re}}_{{\rm{M}}}=2261$ in a periodic box of nondimensional size $L=2\pi$ with 512³ points (Seta et al. 2020). This means that there are five velocity correlation cells along each direction in the domain and a total of 125 cells in the whole cube. The simulations are for an isothermal gas with incompressible forcing. Thus, the electron number density is roughly uniform within the domain. To consider the effect of the thermal electron number density distribution on the rotation measure distribution, we consider the following physically motivated distributions:

• 1.  
  a uniform ${n}_{{\rm{e}}}$ as suggested by isothermal simulations,
• 2.  
  ${n}_{{\rm{e}}}$ following the King profile similar to the gas density, i.e., ${n}_{{\rm{e}}}{(r)={n}_{{\rm{e}}}{(0)(1+(r/a)}^{2})}^{-3/4}$, where a is the core radius,
• 3.  
  ${n}_{{\rm{e}}}$ proportional to ${b}^{3/2}$ as suggested by the magnetic flux-freezing or cooling-flow condition (extreme situation, ignores the turbulent nature of the medium).

We assume that the background polarized source density is uniform and the only effect the radiation has while passing through the simulation box is the rotation of its polarization angle, which is quantified by RM. With the fluctuation dynamo-generated magnetic field b (normalized to $\,{b}_{\mathrm{rms}}=1$ over the numerical domain) for each ${n}_{{\rm{e}}}$ (normalized to $\langle {n}_{{\rm{e}}}\rangle =1$ over the numerical domain) given above, we calculate the RM at each point within a face of the domain as

$$\begin{eqnarray}&&\mathrm{RM}({x}_{i},{y}_{i})=K\displaystyle \sum _{i=1}^{512}{n}_{{\rm{e}}}({x}_{i},{y}_{i},{z}_{i})\,b({x}_{i},{y}_{i},{z}_{i})\,{\text{}}{dz},\end{eqnarray} \tag{ 13 }$$

where $({x}_{i},{y}_{i},{z}_{i})$ is a coordinate on the grid and ${\text{}}{dz}=2\pi /512$ is the grid spacing. This involves looking along the sightlines through the core of an elliptical galaxy and gathering statistics from the periodic box. We show the generated $\mathrm{RM}$ maps in Figure 2 for all three cases: ${n}_{{\rm{e}}}$ constant (a), ${n}_{{\rm{e}}}$ King profile (b), and ${n}_{{\rm{e}}}\propto {b}^{3/2}$ (c). The shape and size of structures for a constant ${n}_{{\rm{e}}}$ (Figure 2(a)) and for ${n}_{{\rm{e}}}$ following the King profile (Figure 2(b)) are very similar, and for the case of the King profile, the intensity of $\mathrm{RM}$ decreases as a function of the distance from the center. The structures in the third case, ${n}_{{\rm{e}}}\propto {b}^{3/2}$ (Figure 2(c)), are larger than in the other two, with very high $\mathrm{RM}$ values. On larger scales, the structures look similar for all three cases because of the same magnetic field, but the amplitude and small-scale $\mathrm{RM}$ structures vary depending on the ${n}_{{\rm{e}}}$ distribution.

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Magnetic fields generated by nonlinear fluctuation dynamo simulations tend to follow a power-law spectrum with slope 3/2 in the kinematic stage and −5/3 in the saturated stage (Figure 2 in Seta et al. 2020). In Figure 3(a), we show the one-dimensional (angle-integrated) power spectrum M_k of $\mathrm{RM}$ using the magnetic field in the saturated stage for the three different electron density distributions discussed above. The slope of the power spectra is similar for constant ${n}_{{\rm{e}}}$ and for the case of ${n}_{{\rm{e}}}$ following the King profile (red and blue lines in Figure 3(a)), ${k}^{-0.8}$, and the power spectrum is much flatter for the case with ${n}_{{\rm{e}}}\propto {b}^{3/2}$, ${M}_{k}\propto {k}^{-0.5}$. Even though all three cases have the same magnetic field, the $\mathrm{RM}$ power spectra can be different. Then we calculate the second-order structure function ${\mathrm{SF}}_{\mathrm{RM}}(r)$ as (Monin & Yaglom 1971)

$$\begin{eqnarray}&&{\mathrm{SF}}_{\mathrm{RM}}(r)=\langle | \mathrm{RM}({\boldsymbol{x}}+{\boldsymbol{r}})-\mathrm{RM}({\boldsymbol{x}}){| }^{2}\rangle ,\end{eqnarray} \tag{ 14 }$$

where ${\boldsymbol{x}}$ is the position in the two-dimensional $\mathrm{RM}$ plane and $r=| {\boldsymbol{r}}|$ is the length of the displacement vector. Since we have periodic boundary conditions, we consider only half of the periodic domain, i.e., r varies from 0 to $L/2$. The scale at which the slope of the structure function of $\mathrm{RM}$ changes, referred to as the break scale, is usually related to the driving (or outer) scale of the turbulence (Haverkorn et al. 2008; Anderson et al. 2015). The structure function of $\mathrm{RM}$ can be used to calculate the $\mathrm{RM}$ correlation function ${C}_{\mathrm{RM}}(r)$ as (Hollins et al. 2017)

$$\begin{eqnarray}&&{C}_{\mathrm{RM}}(r)=1-\displaystyle \frac{{\mathrm{SF}}_{\mathrm{RM}}(r)}{2{\sigma }^{2}},\end{eqnarray} \tag{ 15 }$$

where $2{\sigma }^{2}$ is the value of ${\mathrm{SF}}_{\mathrm{RM}}$ at which $\mathrm{RM}$ is no longer correlated, which would also be twice the standard deviation of $\mathrm{RM}$, if there are a sufficient number of correlation cells in the numerical domain. The break scale can also be determined from the correlation function of $\mathrm{RM}$. It is the length scale at which the correlation function crosses zero for the first time starting from $r=0$. Figure 3(b) shows the correlation function of $\mathrm{RM}$ for different electron density distributions. Even though the turbulence is driven on the same scale ($0.2L$) for all three cases, the break scale is different. The correlation length of $\mathrm{RM}$ can be calculated by integrating the correlation function as

$$\begin{eqnarray}&&{{\ell }}_{b}=\int C(r)\,{\text{}}{dr}.\end{eqnarray} \tag{ 16 }$$

Using Equation (16), we calculate the correlation length of $\mathrm{RM}$ for all three cases and find ${{\ell }}_{b}/L=0.125\pm 0.07,0.080\pm 0.006$, and 0.052 ± 0.07. For this analysis, we see that the correlation length not only depends on the thermal electron density distribution but is also different from the break scale.

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Figure 4 shows the probability density function (PDF) of $\mathrm{RM}$ for the three electron number density distributions. In all cases, the mean $\mathrm{RM}\approx 0$ and the distribution is symmetric, because there is no large-scale field in the domain. Figure 4(a) shows the PDFs for two distributions of ${n}_{{\rm{e}}}$: uniform and following a King profile with $a=L/4=1.25{{\ell }}_{0}$. Both distributions are close to a Gaussian distribution (${ \mathcal N }$) with different standard deviations ${\sigma }_{\mathrm{RM}}$. The one with the King profile has a smaller standard deviation. The distributions agree well with the analytical expressions, Equations (10) and (11), respectively. Figure 4(b) shows the $\mathrm{RM}$ distribution when ${n}_{{\rm{e}}}\propto {b}^{3/2}$. The distribution is clearly non-Gaussian and can be approximated by a product of a Cauchy–Lorentz distribution and a Gaussian function (${ \mathcal L }{ \mathcal N }$). The Gaussian or non-Gaussian nature of the $\mathrm{RM}$ distribution in these cases follows from Equation (9) or Equation (13). In an isotropic random magnetic field, the polarization angle performs a random walk as it rotates randomly along the path length. When ${n}_{{\rm{e}}}$ does not depend on b directly, $\mathrm{RM}$ depends on the first power of the magnetic field and the random walk is a Brownian motion, which gives rise to a Gaussian distribution. For the case where ${n}_{{\rm{e}}}\propto {b}^{3/2}$, $\mathrm{RM}$ depends on a higher power of the magnetic field, and since the underlying magnetic field is intermittent, the resulting $\mathrm{RM}$ distribution is non-Gaussian. In the case of a non-Gaussian distribution, it is difficult to associate a single number (for example ${\sigma }_{\mathrm{RM}}$) to the distribution, but the calculated standard deviation for a non-Gaussian distribution would be far higher than that for a Gaussian distribution (compare the x-axes in Figures 4(a) and (b)). The calculated standard deviation of the $\mathrm{RM}$ distribution shown in Figure 4(b) is approximately equal to $170\,\mathrm{rad}\,{{\rm{m}}}^{-2}$, whereas the deviation for both the Gaussian distributions in Figure 4(a) is less than $100\,\mathrm{rad}\,{{\rm{m}}}^{-2}$. We show that the standard deviation of fluctuations in $\mathrm{RM}$ is significantly different for different thermal electron number density distributions for a given fixed magnetic field.

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Furthermore, to account for the effects of beam smearing, we perform a Gaussian smoothing of the $\mathrm{RM}$ maps with a beam size (or smoothing scale) ${l}_{\mathrm{beam}}$ ranging from 0.0 (no smoothing) to $L/10$ (beam size one-tenth of the numerical domain). The dependence of the standard deviation of $\mathrm{RM}$ on ${l}_{\mathrm{beam}}$ is shown in Figure 5. For all three cases of the thermal electron density distribution, ${\sigma }_{\mathrm{RM}}$ decreases (exponentially for high ${l}_{\mathrm{beam}}$), but the difference between the three cases still persists even for strong smoothing of the $\mathrm{RM}$ maps.

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Thus, the fluctuations in rotation measure should not be directly associated with the fluctuations in the magnetic fields without considering the fluctuations in thermal electron density. This sets an expectation for future observations.

The most direct approach would be to measure RM variance using a grid of polarized background radio sources observed through the ISM of a foreground elliptical galaxy. This requires a sufficient number of background sources to enable separation of RM fluctuations arising within and external to the target galaxy. This approach is unlikely to be feasible for any individual galaxy, even if using future highly sensitive radio telescopes, because the sky density of background polarized sources (e.g., Hales et al. 2014 and Figure 3 from Hales 2013) will be too low to yield a sufficient statistical sample behind even an arcminute-sized galaxy (see Section 5.4 in Seta 2019, for an estimate to observe $\mathrm{RMs}$ from background extragalactic sources seen through the nearest elliptical galaxy Maffei 1). A more promising approach could be to identify elliptical galaxies located in the foreground of extended polarized radio sources (e.g., lobes), through which spatially resolved RM fluctuations may even be discerned. Depolarization in this scenario would also yield constraints on ${\sigma }_{\mathrm{RM}}$.

Similarly, polarized fast radio bursts (FRBs) could be used to gather RM statistics. The event rate of FRBs (e.g., Petroff et al. 2019; Cordes & Chatterjee 2019) is probably insufficient to sample many independent sightlines through any individual foreground galaxy within a realistic timeframe. Furthermore, it may be unrealistic to consider gathering statistics of host-galaxy RM contributions to study ellipticals; while the emission mechanism(s) of FRBs and their identification with any particular astronomical object or phenomenon remain unknown, their compelling association with young neutron stars (Metzger et al. 2019) and with environments that scatter radio waves (CHIME/FRB Collaboration et al. 2019) appears to argue against an origin within old stellar populations. Instead, a plausible approach might be to compare a control sample of RMs from FRBs with those located behind elliptical galaxies.

The approaches above could also be attempted without prior knowledge of the foreground elliptical galaxies by searching for correlations with various line tracers. For example, a similar correlation has been identified between polarization properties of background sources and Mg ii absorption line systems, indicating a likely association with magnetic outflows from intervening star-forming galaxies (Kim et al. 2016).

Laing (1988) and Garrington et al. (1988) studied radio polarization from Fanaroff–Riley Class II (FRII) sources and observed that the level of fractional polarization was significantly higher on the jet side closer to us than on the counter-jet side pointing away from us. This is referred to as the Laing–Garrington effect. Assuming that both jets have similar intrinsic properties, the difference in fractional polarization can be attributed to depolarization by a foreground screen arising from the X-ray-emitting halo of the elliptical galaxy, where the radiation from the more distant side travels more through the screen and is thus more depolarized.

If the elliptical galaxy acts as a Faraday screen, i.e., a medium with magnetic fields and thermal electrons but lacking relativistic electrons (this is a justified assumption since an elliptical galaxy has a scarcity of cosmic-ray electrons), and if it is assumed that the jet does not significantly disturb the ISM, then we can use historical observations of the Laing–Garrington effect to probe magnetic fields in elliptical galaxies. We investigate this observational approach below using available historical data.

For a Faraday screen, assuming a Gaussian distribution of Faraday depths, the degree of polarization p at wavelength λ is given by (Burn 1966; Sokoloff et al. 1998) ⁹

$$\begin{eqnarray}&&p={p}_{0}\exp (-2{\sigma }_{\mathrm{RM}}^{2}{\lambda }^{4}),\end{eqnarray} \tag{ 17 }$$

where p₀ is the maximum degree of polarization and ${\sigma }_{\mathrm{RM}}$ is the standard deviation of the fluctuations in rotation measure due to the screen. The depolarization $\mathrm{DP}$ (see Figure 7 in Basu et al. 2019, for $\mathrm{DP}$ calculated from MHD simulations) between the wavelengths ${\lambda }_{1}$ and ${\lambda }_{2}$ with ${\lambda }_{1}\gt {\lambda }_{2}$ is given by

$$\begin{eqnarray}&&\mathrm{DP}=\exp (-2{\sigma }_{\mathrm{RM}}^{2}({\lambda }_{1}^{4}-{\lambda }_{2}^{4})).\end{eqnarray} \tag{ 18 }$$

Using Equation (18), we obtain the standard deviation in the rotation measure distribution for the jet ${\sigma }_{{\mathrm{RM}}_{{\rm{j}}}}$ and counter-jet ${\sigma }_{{\mathrm{RM}}_{\mathrm{cj}}}$. Then, assuming that the difference is due to the halo of an elliptical galaxy, we derive the standard deviation in the rotation measure distribution for the galaxy ${\sigma }_{{\mathrm{RM}}_{\mathrm{ellip}}}$ from the standard deviation of rotation measure for the jet and counter-jet as follows:

$$\begin{eqnarray}&&{\sigma }_{{\mathrm{RM}}_{\mathrm{ellip}}}=\sqrt{{\sigma }_{{\mathrm{RM}}_{\mathrm{cj}}}^{2}-{\sigma }_{{\mathrm{RM}}_{{\rm{j}}}}^{2}},\end{eqnarray} \tag{ 19 }$$

Finally, ${\sigma }_{{\mathrm{RM}}_{\mathrm{ellip}}}$ is used to estimate the magnetic field strength $\,{b}_{\mathrm{rms}}$ using Equation (10) for uniform thermal electron number density and Equation (11) for the King profile.

Garrington & Conway (1991) made a similar estimate for a typical single source at low redshift and another source at high redshift assuming uniform thermal electron density. In Table 1, we do this calculation for all the 31 sources studied by Garrington et al. (1991) with varying path lengths ($L$ ∼ projected linear size, $\mathrm{LS}$ in Table 1) and two different models of thermal electron density (uniform ${n}_{{\rm{e}}}$ and ${n}_{{\rm{e}}}$ following the King profile). For the case of uniform ${n}_{{\rm{e}}}$, we assume ${n}_{{\rm{e}}}$ to be ${10}^{-2}\,{\mathrm{cm}}^{-3}$, and for the case of the King profile, we assume ${n}_{{\rm{e}}}$ at radius $r=0$ to be ${10}^{-2}\,{\mathrm{cm}}^{-3}$ and the core radius $a=3\,\mathrm{kpc}$ (see Table 4 in Garrington & Conway 1991). For both cases, we assume the magnetic field correlation length ${{\ell }}_{b}=100\,\mathrm{pc}$ (as expected from the fluctuation dynamo action). The estimated magnetic field for the case of uniform n_e, ${b}_{{\rm{rms}}}({\rm{uni}})$, and that at around $r=0$ for the case of the King profile, ${b}_{{\rm{rms}}}({\rm{KP}},r=0)$, are given in the last two columns of Table 1. The rms magnetic field ranges from $0.14\,\mu {\rm{G}}$ to $1.33\,\mu {\rm{G}}$ for the case of uniform n_e, and from $1.36\,\mu {\rm{G}}$ to $6.21\,\mu {\rm{G}}$ for the case of the King profile. The magnetic field strength in the sample roughly agrees with the value estimated in Section 2.3. We propagate the uncertainties reported in the observed values of the depolarization fraction (Garrington et al. 1991) to magnetic field strengths. The uncertainties in field strengths range from $5 \%$ to $42 \%$. The uncertainty reported here is due to the measurement error; there can be additional systematics due to variations in ${{\ell }}_{b}$. Figures 6(a) and 7(a) shows histograms of magnetic fields strengths in host ellipticals obtained using ${b}_{{\rm{rms}}}({\rm{uni}})$ and ${b}_{{\rm{rms}}}({\rm{KP}},r=0)$ (last two columns of Table 1). Figures 6(b) and 7(b) show two-dimensional histograms of the calculated magnetic field versus redshift. From the two-dimensional histograms, it appears that the magnetic field is somewhat stronger in elliptical galaxies at lower redshifts. However, we caution that the sample size is small and has not been controlled for any selection biases. The typical strengths are broadly consistent with the simulated values shown in Figure 1 (also in agreement with small-scale simulations in Section 4), which is an encouraging sign. However, a direct comparison of the redshift trends is not possible, because the observed sample is small and its volume completeness is difficult to judge.

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Table 1. List of Sources, Their Redshift z, Projected Linear Size $\mathrm{LS}$, and Depolarization Fractions for Jets ${\mathrm{DP}}_{{\rm{j}}}$ and Counter-jets ${\mathrm{DP}}_{\mathrm{cj}}$ with Their Respective Uncertainties (Garrington et al. 1991)

Name	z	$\mathrm{LS}$	${\mathrm{DP}}_{{\rm{j}}}$	${\mathrm{DP}}_{\mathrm{cj}}$	${\sigma }_{{\mathrm{RM}}_{{\rm{j}}}}$	${\sigma }_{{\mathrm{RM}}_{\mathrm{cj}}}$	${\sigma }_{{\mathrm{RM}}_{\mathrm{ellip}}}$	$\,{b}_{\mathrm{rms}}(\mathrm{uni})$	$\,{b}_{\mathrm{rms}}(\mathrm{KP},r=0)$
	#	(kpc)	#	#	$({\rm{rad}}\,{{\rm{m}}}^{-2})$	$({\rm{rad}}\,{{\rm{m}}}^{-2})$	$({\rm{rad}}\,{{\rm{m}}}^{-2})$	$(\mu {\rm{G}})$	$(\mu {\rm{G}})$
0017+15	2.012	88	0.61 ± 0.02	0.21 ± 0.07	12.48 ± 0.41	22.17 ± 2.37	18.33 ± 2.88	0.29 ± 0.05	1.82 ± 0.29
0123+32	0.794	192	0.96 ± 0.04	0.89 ± 0.03	3.59 ± 1.83	6.06 ± 0.88	4.88 ± 1.73	0.15 ± 0.05	1.36 ± 0.48
0225-01	2.037	131	0.85 ± 0.03	0.08 ± 0.04	7.16 ± 0.78	28.21 ± 2.79	27.29 ± 2.89	0.35 ± 0.04	2.66 ± 0.28
0232-04	1.436	110	0.81 ± 0.03	0.16 ± 0.05	8.15 ± 0.72	24.03 ± 2.05	22.60 ± 2.19	0.49 ± 0.05	3.42 ± 0.33
0838+13	0.684	89	0.73 ± 0.04	0.25 ± 0.02	9.96 ± 0.87	20.90 ± 0.60	18.37 ± 0.83	0.93 ± 0.04	5.83 ± 0.26
0850+58	1.322	129	0.57 ± 0.04	0.15 ± 0.03	13.31 ± 0.83	24.45 ± 1.29	20.51 ± 1.63	0.45 ± 0.04	3.42 ± 0.27
1023+06	1.699	112	0.45 ± 0.03	0.25 ± 0.06	15.86 ± 0.66	20.90 ± 1.81	13.61 ± 2.88	0.24 ± 0.05	1.68 ± 0.36
1115+53	1.235	77	0.86 ± 0.03	0.18 ± 0.04	6.89 ± 0.80	23.24 ± 1.51	22.20 ± 1.60	0.68 ± 0.05	4.00 ± 0.29
1218+33	1.519	79	0.56 ± 0.02	0.07 ± 0.04	13.52 ± 0.42	28.94 ± 3.11	25.60 ± 3.52	0.61 ± 0.08	3.63 ± 0.50
1226+10	2.296	40	0.69 ± 0.04	0.04 ± 0.02	10.81 ± 0.84	31.85 ± 2.47	29.95 ± 2.65	0.59 ± 0.05	2.48 ± 0.22
1241+16	0.557	116	0.90 ± 0.06	0.41 ± 0.04	5.76 ± 1.82	16.76 ± 0.92	15.74 ± 1.18	0.81 ± 0.06	5.84 ± 0.44
1258+40	1.659	172	0.84 ± 0.04	0.11 ± 0.05	7.41 ± 1.01	26.37 ± 2.72	25.31 ± 2.84	0.37 ± 0.04	3.22 ± 0.36
1318+11	2.171	37	0.97 ± 0.04	0.06 ± 0.06	3.10 ± 2.10	29.77 ± 5.29	29.61 ± 5.32	0.65 ± 0.12	2.65 ± 0.48
1323+65	1.618	71	0.73 ± 0.03	0.18 ± 0.03	9.96 ± 0.65	23.24 ± 1.13	21.00 ± 1.29	0.49 ± 0.03	2.75 ± 0.17
1634+17	1.897	55	0.77 ± 0.08	0.27 ± 0.07	9.07 ± 1.80	20.31 ± 2.01	18.17 ± 2.42	0.39 ± 0.05	1.95 ± 0.26
1634+58	0.985	73	0.84 ± 0.10	0.25 ± 0.04	7.41 ± 2.53	20.90 ± 1.21	19.54 ± 1.61	0.78 ± 0.06	4.46 ± 0.37
1656+57	1.281	33	0.90 ± 0.04	0.10 ± 0.10	5.76 ± 1.22	26.93 ± 5.85	26.31 ± 5.99	1.19 ± 0.27	4.55 ± 1.04
1709+46	0.806	103	0.56 ± 0.02	0.15 ± 0.03	13.52 ± 0.42	24.45 ± 1.29	20.37 ± 1.57	0.83 ± 0.06	5.62 ± 0.43
1732+16	1.270	132	0.68 ± 0.05	0.08 ± 0.04	11.02 ± 1.05	28.21 ± 2.79	25.97 ± 3.07	0.59 ± 0.07	4.53 ± 0.53
1816+47	2.225	49	0.82 ± 0.06	0.06 ± 0.03	7.91 ± 1.46	29.77 ± 2.65	28.70 ± 2.77	0.53 ± 0.05	2.48 ± 0.24
0107+31	0.689	364	0.65 ± 0.03	0.39 ± 0.02	11.65 ± 0.62	17.22 ± 0.47	12.69 ± 0.86	0.31 ± 0.02	4.00 ± 0.27
0712+53	0.064	49	0.34 ± 0.03	0.28 ± 0.03	18.44 ± 0.75	20.03 ± 0.84	7.82 ± 2.80	1.33 ± 0.48	6.21 ± 2.22
0824+29	0.458	129	0.86 ± 0.05	0.44 ± 0.02	6.89 ± 1.33	16.08 ± 0.45	14.53 ± 0.80	0.81 ± 0.04	6.15 ± 0.34
0903+16	0.411	280	0.65 ± 0.02	0.38 ± 0.03	11.65 ± 0.42	17.46 ± 0.71	13.00 ± 1.03	0.53 ± 0.04	5.87 ± 0.46
1001+22	0.974	576	0.50 ± 0.04	0.36 ± 0.02	14.78 ± 0.85	17.94 ± 0.49	10.17 ± 1.51	0.15 ± 0.02	2.35 ± 0.35
1055+20	1.111	399	0.88 ± 0.07	0.52 ± 0.11	6.35 ± 1.97	14.35 ± 2.32	12.87 ± 2.77	0.20 ± 0.04	2.60 ± 0.56
1354+19	0.720	376	0.59 ± 0.05	0.23 ± 0.03	12.89 ± 1.04	21.52 ± 0.95	17.23 ± 1.42	0.41 ± 0.03	5.24 ± 0.43
1548+11	1.901	379	0.89 ± 0.13	0.33 ± 0.04	6.06 ± 3.80	18.69 ± 1.02	17.68 ± 1.69	0.15 ± 0.01	1.89 ± 0.18
1618+17	0.555	430	0.84 ± 0.05	0.62 ± 0.03	7.41 ± 1.27	12.27 ± 0.62	9.78 ± 1.24	0.26 ± 0.03	3.64 ± 0.46
1830+28	0.594	217	0.81 ± 0.03	0.75 ± 0.04	8.15 ± 0.72	9.52 ± 0.88	4.92 ± 2.08	0.18 ± 0.07	1.74 ± 0.74
2325+29	1.015	425	0.83 ± 0.05	0.60 ± 0.04	7.66 ± 1.24	12.69 ± 0.83	10.11 ± 1.40	0.16 ± 0.02	2.24 ± 0.31

Note. The calculated standard deviations of the rotation measure in jets ${\sigma }_{{\mathrm{RM}}_{{\rm{j}}}}$, counter-jets ${\sigma }_{{\mathrm{RM}}_{\mathrm{cj}}}$, elliptical hosts ${\sigma }_{{\mathrm{RM}}_{\mathrm{ellip}}}$, and the rms magnetic field in ellipticals with their respective uncertainties for the uniform ${n}_{{\rm{e}}}$, $\,{b}_{\mathrm{rms}}(\mathrm{uni})$, and for the ${n}_{{\rm{e}}}$ following the King profile, $\,{b}_{\mathrm{rms}}(\mathrm{KP},r=0)$, are also given.

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The Laing–Garrington effect can only be employed for elliptical galaxies with an active galactic nucleus (AGN). The jets might contaminate the radio signal as well as the turbulence and magnetic fields in such ellipticals. Thus, it may not be trivial to disentangle the influence of the jet from the magnetic field properties that would arise purely under the influence of supernova-driven turbulence in the hot ISM of elliptical galaxies. In general for ellipticals with an AGN at the center, the galaxy+jet system can give rise to large-scale $\mathrm{RM}$ gradients and asymmetric $\mathrm{RM}$ distributions with nonzero mean $\mathrm{RM}$ (as shown in Figure 8 of Laing et al. 2008, in comparison to our Figure 4). Although the magnetic field strength obtained from the fluctuation dynamo analysis in quiescent ellipticals (Section 2) and that using the Laing–Garrington effect for ellipticals with jets are similar, the correlation scale of the magnetic field can be widely different for the two cases. Moreover, the depolarization asymmetry in jets can also be due to an extended disk of magneto-ionic medium surrounding elliptical galaxies (Gopal-Krishna 1997), the absence of a spherically symmetric halo (Laing et al. 2008) or a difference in Faraday rotation due to shocked and unshocked medium (Guidetti et al. 2012). To robustly study fluctuation dynamos in elliptical galaxies using the Laing–Garrington effect, a statistically large sample of sources may be required.

The detection of radio synchrotron radiation could be attempted. We are interested in estimating the synchrotron flux in quiescent elliptical galaxies for which the contribution due to star formation is negligible and that due to the central AGN is absent. Assuming that there is a significant population of relativistic electrons, the flux density from the central region of an elliptical galaxy can be estimated by considering the relationship between minimum-energy magnetic field strength and radio luminosity for a spherical radio source. Following Pacholczyk (1970), assuming an unresolved source with spectral index ${S}_{\nu }\propto {\nu }^{-0.8}$ over frequencies between $10\,\mathrm{MHz}$ and $100\,\mathrm{GHz}$, and assuming an ion to electron energy density ratio of 100, the predicted spectral flux density is approximately

$$\begin{eqnarray}&&12\,\mathrm{nJy}{\left(\displaystyle \frac{{b}_{\mathrm{rms}}}{1\mu {\rm{G}}}\right)}^{3.5}{\left(\displaystyle \frac{R}{1\mathrm{kpc}}\right)}^{3}{\left(\displaystyle \frac{\nu }{1\mathrm{GHz}}\right)}^{-0.8}{\left(\displaystyle \frac{{D}_{L}}{100\mathrm{Mpc}}\right)}^{-2},\end{eqnarray} \tag{ 20 }$$

where R is the radius of the volume, ν is the observing frequency, and D_L is the luminosity distance. Assuming $\nu =100\,\mathrm{MHz}$, $R=1\,\mathrm{kpc}$ from earlier, $\,{b}_{\mathrm{rms}}\approx 0.5\,\mu {\rm{G}}$ from Section 2.3, and a distance of $50\,\mathrm{Mpc}$ ($z\approx 0.01$), this equates to approximately 25 nJy, which is unlikely to be feasible with even the next generation of radio telescopes. However, it is plausible that this could be detected by stacking the radio emission over a large sample of suitably selected elliptical galaxies.

In apparent contrast with the estimate of flux density above, Nyland et al. (2017) found that cores of early-type galaxies are bright in synchrotron radiation because stars are still being formed in the central regions of the galaxies. They also found a deficiency in radio emission compared to the infrared emission. One possible explanation for this deficiency is the presence of significant magnetic field strengths in the medium. ¹⁰ Nyland et al. (2017) estimated magnetic fields in the range $4\mbox{--}85\,\mu {\rm{G}}$, with a median of $15\,\mu {\rm{G}}$ (an order of magnitude higher than that in the nearby spiral galaxies), but to estimate the magnetic field strength they assumed that those early-type galaxies are very similar to present-day spiral galaxies. Their assumptions included energy equipartition between cosmic rays and magnetic fields (see Seta & Beck 2019 for pros and cons of the equipartition assumption), ratio of the number density of cosmic-ray protons to that of cosmic-ray electrons of 100, and disk-like or cylindrical geometry of the synchrotron-emitting region with gas scale heights similar to those of a spiral galaxy. Additionally, the emission due to accretion by the central black hole might also contaminate the total synchrotron intensity. Based on our simple estimates from Section 2.3 and the indirect observational result from Section 5.2, we argue that magnetic field strengths in elliptical galaxies are probably an order of magnitude lower than those in spirals and certainly not higher.

Strong gravitational lens systems could be used to measure differences in RM or depolarization between multiple (and/or possibly spatially resolved) images of a background polarized radio source that have been lensed by a foreground elliptical galaxy (Greenfield et al. 1985; Patnaik et al. 1999; Narasimha & Chitre 2004). ¹¹ A mass model for the lensing system is required. A complication with this approach is that the Local Group or cluster environment will also contribute toward ${\sigma }_{\mathrm{RM}}$.

Greenfield et al. (1985) studied the radio polarization signal from two gravitationally lensed images of the quasar 0957+561 ($z\approx 1.41$) at multiple wavelengths and found that the rotation measures of the two images differ by $100\,\mathrm{rad}\,{{\rm{m}}}^{-2}$. This difference can be attributed to the magnetic field in the ISM of the lensing (cD) galaxy at $z\approx 0.36$. So, to estimate the magnetic field in the galaxy using Equation (9) for the quasar at a redshift z, we can write $\mathrm{RM}$ in terms of mean quantities as

$$\begin{eqnarray}&&\mathrm{RM}\approx \left(0.81\displaystyle \frac{\langle {n}_{{\rm{e}}}\rangle }{\,{\mathrm{cm}}^{-3}}\displaystyle \frac{\langle b\rangle }{\,\mu {\rm{G}}}\displaystyle \frac{L}{\,\mathrm{pc}}\right)/{\left(1+z\right)}^{2},\end{eqnarray} \tag{ 21 }$$

where $\langle \cdots \rangle$ denotes the average over the path length. Assuming $\langle {n}_{{\rm{e}}}\rangle \approx 0.01\,{\mathrm{cm}}^{-3}$ and $L\approx 30\,\mathrm{kpc}$, for $\mathrm{RM}=100\,\mathrm{rad}\,{{\rm{m}}}^{-2}$ and $z=1.41$, we obtain $\langle b\rangle \simeq 2.5\,\mu {\rm{G}}$. This is comparable in strength to magnetic fields observed in the Milky Way and nearby spiral galaxies (Fletcher 2010; Haverkorn 2015; Beck 2016). Also, since $\langle b\rangle$ is not close to zero, it refers to the presence of a large-scale or ordered field. Furthermore, such strong ordered fields are not expected in cD galaxies, and thus the $\mathrm{RM}$ difference cannot be completely associated with the galaxy. The nonzero strong ordered field could be due to the contribution from the cluster's magnetic field, which even if random (on scales of $10\,\mathrm{kpc}$, Subramanian et al. 2006), can act as a coherent field over the scale of the galaxy. This argument is supported by the mass modeling of the system, which suggests that the lens must also have some contribution from the cluster to explain the lensing observations (Greenfield et al. 1985). Thus, it then becomes difficult to differentiate between the contribution to the $\mathrm{RM}$ difference due to the galaxy's magnetic field and that due to the cluster's.

Alternatively, rather than examining RMs directly, RM variance could be probed by observing depolarization similar to Section 5.2 but over the spatial extent of the foreground lensing elliptical galaxy within an extended background radio source (Strom & Jaegers 1988; Laing 1988; Garrington et al. 1988). More recent attempts to statistically measure $\mathrm{RM}$ excess due to intervening galaxies (Farnes et al. 2014; Malik et al. 2020; Lan & Prochaska 2020) can also be extended to study magnetic fields in elliptical galaxies by considering the morphology of the intervening galaxies in the statistical sample.

Observations must carefully consider selection effects so as to minimize potential bias from coherent magnetic fields and from unrelated RM fluctuations. The observations should ideally target giant (core) ellipticals, which as a class exhibit negligible rotation (Kormendy et al. 2009). These galaxies are typically located within the central regions of clusters, so it will be important to discriminate between RM fluctuations arising in the interstellar and intracluster gas (Baldry et al. 2006; Kraljic et al. 2018). Selection should also ideally target ellipticals in hydrostatic equilibrium (Statler 2012; Penoyre et al. 2017). Note that a simple selection on red galaxies could lead to misleading results (e.g., Evans et al. 2018). Evolution with redshift in the source sample (van Dokkum et al. 2010; Penoyre et al. 2017) will need to be taken into account when compiling statistics on saturation RM variance.

Some of the approaches described in the sections above could be tested now using a combination of data from surveys such as the Sloan Digital Sky Survey (York et al. 2000) and NRAO VLA Sky Survey (Condon et al. 1998), or in the near future using, for example, forthcoming data from the Very Large Array Sky Survey (Lacy et al. 2020) and eventually from the Square Kilometre Array (Taylor et al. 2015, Heald et al. 2020).