Letter | Published:

Detection of microgauss coherent magnetic fields in a galaxy five billion years ago

Nature Astronomyvolume 1pages621626 (2017) | Download Citation


Magnetic fields play a pivotal role in the physics of interstellar medium in galaxies1, but there are few observational constraints on how they evolve across cosmic time2,3,4,5,6,7. Spatially resolved synchrotron polarization maps at radio wavelengths reveal well-ordered large-scale magnetic fields in nearby galaxies1,8,9 that are believed to grow from a seed field via a dynamo effect10,11. To directly test and characterize this theory requires magnetic field strength and geometry measurements in cosmologically distant galaxies, which are challenging to obtain due to the limited sensitivity and angular resolution of current radio telescopes. Here, we report the cleanest measurements yet of magnetic fields in a galaxy beyond the local volume, free of the systematics traditional techniques would encounter. By exploiting the scenario where the polarized radio emission from a background source is gravitationally lensed by a foreground galaxy at z = 0.439 using broadband radio polarization data, we detected coherent μG magnetic fields in the lensing disk galaxy as seen 4.6 Gyr ago, with similar strength and geometry to local volume galaxies. This is the highest redshift galaxy whose observed coherent magnetic field property is compatible with a mean-field dynamo origin.

We measured the magnetic field in a z = 0.439 galaxy using Karl G. Jansky Very Large Array (VLA) broadband (1–8 GHz) polarization observations of the strong lensing system CLASS B1152+199. The polarized background source at z = 1.019 was lensed by the late-type galaxy into two images (A and B) separated by 1.56” at impact parameters 6.5 and 2.6 kpc (refs 12,13), as shown in Fig. 1. The lensing galaxy has been classified as a late-type star-forming disk galaxy with a substantial interstellar medium (ISM), based on strong O[ii] emission, and large hydrogen, Mgii and dust column densities seen towards the lensed quasar images12,13,14,15. Characterization of the lensing galaxy enables one to establish links between its galaxy properties and magnetic fields. This is not achievable by standard Faraday rotation measure grid studies (inference of magnetic fields in distant galaxies using Faraday rotation—the integral of the line-of-sight magnetic field weighted by thermal electron density—of polarized radio sources behind them) based on samples of intervening galaxies of vastly different masses, types and viewing geometries. These studies have yielded only statistical detections of magnetic fields associated with cosmologically distant galaxies3,4,5,6,7 without reliable field strength and geometry estimations.

Fig. 1: The 5 GHz total intensity radio contour of the gravitational lensing system CLASS B1152+199 overlaid on the HST F814W image.
Fig. 1

The lensed images are labelled A and B, whereas the lensing galaxy is labelled G. The blue radio contours are at 0.2, 2.0 and 20.0 mJy beam−1, respectively, with the lowest contour level corresponding to a 10σ detection. The resolution of the radio image is 0.4” × 0.29”, as indicated by the black ellipse in the lower left corner. Images A and B lie 1.1” and 0.45” away from the centre of the lensing galaxy G, which correspond to projected distances of 6.5 and 2.6 kpc for a cosmology with H 0 = 69.6 km s−1 Mpc−1, Ω M = 0.286 and Ω vac = 0.714.

Due to the achromatic and non-polarizing nature of lensing, the Faraday rotation and wavelength-dependent beam depolarization characterized by Faraday dispersion (σRM describes the fluctuations in the magnetic field and electron density on scales smaller than the telescope beam in a foreground medium, which lead to cancellation of the polarization vectors of neighbouring sightlines within a beam, reducing the net measured polarization) of the two lensed images should remain unchanged unless there are propagation effects along ray bundles through different parts of the lensing galaxy16,17. Since Faraday effects intrinsic to the background source and that produced in the Milky Way foreground can be assumed to have negligible differences for all lensed images (see Methods), differential Faraday rotation and depolarization between lensed images A and B carry information on large- and small-scale magnetic fields, respectively, in the lensing galaxy. Unlike the standard Faraday rotation measure grid technique, the lensing approach can deliver Faraday rotation and Faraday dispersion produced by a distant intervening galaxy free of contamination from the background source and the Milky Way. Previous efforts to measure the differential Faraday rotation of images in strongly lensed systems were not successful since they relied on narrowband radio polarization data that were difficult to interpret17,18. With broadband polarization data of CLASS B1152+199 and the rotation measure synthesis technique19, we obtained Faraday depth spectra of the lensed images, as shown in Fig. 2. Differences in the gas density and the line-of-sight projection of the large-scale magnetic field in the lensing galaxy along the two light paths produce a shift in the observer’s frame of 510 rad m−2 in the location of the main peak in the Faraday depth spectra. In addition, the presence of random magnetic fields and high gas column density along sightline B (ref. 14) can lead to large Faraday rotation fluctuations on scales smaller than the telescope beam, which further depolarize and disperse incident background radiation. This can cause the broader peak and lower fractional polarization of the dominant Faraday component of image B compared with image A.

Fig. 2: Faraday depth spectra of images A (red) and B (blue) of the gravitational lensing system CLASS B1152+199 computed using the rotation measure synthesis technique, followed by deconvolution using the RM-Clean algorithm50.
Fig. 2

The rms-noise levels in the spectra are 2 × 10−4 and 7 × 10−4 for images A and B, respectively. The dominant Faraday component of image A is located at +9.7 ± 0.5 rad m−2, and the dominant Faraday component of image B is located at +517 ± 3 rad m−2. The corresponding intrinsic polarization position angles (de-rotated to λ = 0 m) of image A (+24.2 ± 0.4°) and image B (+26 ± 2°) are consistent within their 1σ uncertainties, which strongly suggests that the dominant polarized components of the lensed images originate from the same part of the background source. The main Faraday component of image B is broader (full width at half maximum (FWHM) ≈ 191 rad m−2) than that of image A (FWHM ≈ 129 rad m−2) and it is about 50% less polarized than image A. This amount of additional broadening exceeds that inferred for circumgalactic medium of intervening galaxies7, as our sightlines probe through the more turbulent galactic disk at small impact parameters. Finally, we note that a weaker Faraday component is visible in the positive tail of the main peak in both spectra.

To quantify the appearance of the Faraday depth spectra and to extract physical properties of the magnetized gas in the lensing galaxy, we performed a direct fit to the observed fractional Stokes parameters Q/I and U/I as a function of frequency of the two images independently across 1–8 GHz. The complex fractional polarization spectrum P of each lensed image can be best modelled by two distinct polarized components that have Faraday rotation and depolarization effects occurring external to synchrotron emitting regions20 (see Methods)

$$P=\sum _{k=1}^{k=2}{p}_{0,k}{e}^{2i({\theta }_{0,k}+{\rm{R}}{{\rm{M}}}_{k}{c}^{2}/{\nu }^{2})}{e}^{-2{\sigma }_{{\rm{R}}{\rm{M}},k}^{2}{c}^{4}/{\nu }^{4}}$$

where ν is the observing frequency, and p 0,k , θ 0,k , RM k and σRM,k are the intrinsic fractional polarization, intrinsic polarization angle, Faraday rotation and Faraday dispersion of the k-th components, respectively. The resulting best-fit parameters to the polarization spectra of images A and B are listed in Table 1 and the best-fit curves are displayed in Supplementary Fig. 1. The two polarized components obtained by independent fits to the lensed images have intrinsic fractional polarization and polarization angles that differ by less than 1.4σ (see Methods). This strongly indicates that in polarization, we detected the same components of the background source in both lensed images. Therefore, differential magnification and time delay effects (this lensing system did not exhibit any variability through a six month radio monitoring campaign21), which alter the intrinsic polarization properties of the images, are unimportant in this lensing system and are not further considered in our analysis. The differential Faraday rotation (ΔRM) and differential Faraday dispersion ((ΔσRM 2)1/2—the additional Faraday dispersion experienced by image B compared with image A) between the lensed images are clean in situ measures of the lensing galaxy’s large-scale and small-scale magnetic field properties, respectively (see Methods). The lensing galaxy acts as a Faraday screen that produces rest-frame |ΔRM| = |ΔRMobs|(1+z)2 of 1,040 ± 60 rad m−2 and (ΔσRM 2)1/2 of 100 ± 30 rad m−2 between the same polarized components of the two lensed images (computed using the more strongly polarized component), broadly similar in magnitude to the |ΔRM| and (ΔσRM 2)1/2 values reported in the Milky Way and in the nearby Large Magellanic Cloud22,23,24. The fact that the rest-frame |ΔRM| exceeds (ΔσRM 2)1/2 by a factor of ten is strong evidence for the presence of coherent magnetic fields: a purely random field characterized by an RM fluctuation of 100 rad m−2 would only produce a small residual Faraday rotation after passing through a large number of turbulent cells along the line of sight (the expected RM after passing through N turbulent cells in a region without coherent fields is ~ σRM/√N rad m−2).

Table 1 Best-fit parameters and reduced χ2 of maximum likelihood fits to broadband polarization spectra of the lensed images in CLASS B1152+199

To estimate the large-scale magnetic field strength and geometry in the lensing galaxy, we constructed a model for the lensing galaxy’s magnetized ISM. We assumed that the galaxy hosts a disk magnetic field that has either axisymmetric or bisymmetric geometry (anti-symmetric with respect to the rotational axis by 180°; ref. 25), and the strength falls off exponentially with a radial scale length in the range of 5 kpc < r MAG < 20 kpc (ref. 1). We assumed that the ionized gas column density towards image A through the lensing galaxy G \(\left({{\rm{N}}}_{{\rm{e}},{\rm{A}}}=\mathop{\int }\limits_{{\scriptstyle \begin{array}{c}{\rm{S}}{\rm{i}}{\rm{g}}{\rm{h}}{\rm{t}}{\rm{l}}{\rm{i}}{\rm{n}}{\rm{e}}\,{\rm{A}}\,{\rm{t}}{\rm{h}}{\rm{r}}{\rm{o}}{\rm{u}}{\rm{g}}{\rm{h}}\,{\rm{G}}\\ \end{array}}}{n}_{{e}}{\rm{d}}{l}\right)\) ranges between 5 and 300 pc cm−3 (this corresponds to the range of electron column density through a Milky Way-type galaxy with an inclination of 33°; ref. 26). Furthermore, we adopted a differential electron column density (Ne,B−Ne,A) of 156 pc cm−3 by assuming that 10% of the X-ray absorbing gas column is ionized (see Methods). The positions of the lensed images on the sky are converted into locations within the disk of the lensing galaxy using an inclination angle of 33° and a position angle of the line-of-node of −63°, derived from isophote fitting to the Hubble Space Telescope image13 and an assumed intrinsic axial ratio for spiral galaxies of 0.15 (ref. 27). We computed the range of plausible magnetic field strength in the lensing galaxy at the location of image B for values of r MAG and Ne,A in the parameter space considered. Figure 3 shows the resulting magnetic field strength at the galacto-centric radius of image B (2.6 kpc) in the lensing galaxy for the axisymmetric and bisymmetric field cases. The mean coherent magnetic field strength is 8 μG for the axisymmetric case and 11 μG for the bisymmetric case. We note that true field strengths could differ from the reported values by at most 40% if we consider a range of possible ionization fractions within its 90% confidence range (see Methods). To account for the measured rest-frame differential RM for a specific r MAG and Ne,A, a bisymmetric field requires a higher field strength—up to a factor of 1.7 more—than an axisymmetric field.

Fig. 3: Coherent magnetic field strength (|B c(r B)|) at the galacto-centric radius of image B (r B = 2.6 kpc) in the lensing galaxy.
Fig. 3

a,b, |B c(r B)| is shown as a function of the electron column density along sightline A (Ne,A) and the radial magnetic field scale length r MAG for the case of an axisymmetric field (a) and a bisymmetric field (b) for 5 kpc < r MAG < 20 kpc and 5 pc cm−3 < Ne,A < 300 pc cm−3. The magnetic field strength ranges between 4 and 16 μG for an axisymmetric field and between 7 and 17 μG for a bisymmetric field in the parameter space probed. The 1σ uncertainties associated with |B c(r B)| range from 0.3 to 1 μG.

While Faraday rotation is a measure of coherent magnetic fields, Faraday dispersion σRM encodes random magnetic field information in the lensing galaxy on scales smaller than ~20 pc—the projected lensed image size at the redshift of the lensing galaxy at milli-arcsec resolution13. Considering the case where only isotropic and homogenous fluctuations in magnetic fields contribute to σRM, the random magnetic field strength B r in the lensing galaxy on scales smaller than 20 pc is found to decrease with increasing Ne,A from 6 μG to 3 μG, with a mean of 4 μG (see Methods).

According to arguments given in ref. 28, conservation of magnetic helicity imposes a lower limit of 1.6 on the random-to-coherent magnetic field ratio (B r/B coherent) in galaxies for a dominant helicity scale of 100 parsecs. If magnetic fluctuations in the lensing galaxy follow the Kolmogrov spectrum, the random field strength scales as B rl 1/3, and we can extrapolate B r from smaller than 20 pc scales and obtain B r on 100 pc scales to be between 11 and 23 μG in the parameter space probed. For the axisymmetric field case, our observations imply a B r/B coherent ratio that satisfies this criterion in over 40% of the parameter space. However, for the bisymmetric field case, our observations yield a ratio that falls below this limit across the entire parameter space considered. More generally speaking, since the estimated strength for a bisymmetric field exceeds that of an axisymmetric field, for any given random field strength, the helicity conservation criterion on B r/B coherent can always be satisfied more easily by an axisymmetric field. Hence, an axisymmetric large-scale field in the lensing galaxy is favoured over a bisymmetric one.

We examined whether the mean-field dynamo is responsible for the observed coherent large-scale magnetic field in the lensing galaxy. Both the strength and the preferred field geometry in the lensing galaxy are compatible with a mean-field-dynamo-generated magnetic field. The mean-field-dynamo model predicts that the axisymmetric mode is the strongest mode excited in a galactic disk and that the random field is expected to dominate over the coherent field29, as observed here. Ongoing star formation in this z = 0.439 galaxy, indicated by strong O[ii] lines in its spectrum12,30, and subsequent supernova explosions can drive turbulence and therefore the alpha-effect, which is a necessary ingredient for the mean-field dynamo11,29. Assuming Milky Way-value dynamo parameters, the dynamo can produce fields of similar strength and coherency in a galaxy by z = 0.439 (ref. 31).

We have presented direct measurements of μG magnetic fields with a significant coherent component in a galaxy when the universe was two-thirds of its current age. Magnetic field properties in this cosmologically distant galaxy have comparable strength and geometry to galaxies in the present day universe1,8. Since the mean-field dynamo has less time to operate in a young galaxy, the existence of coherent μG magnetic fields in this z = 0.439 galaxy has implications on the mean-field dynamo e-folding time τ dynamo: if the disk of the lensing galaxy settled into equilibrium at z ~ 2 (ref. 32), and magnetic field saturation occurred before z = 0.439, τ dynamo has an upper limit of \(\frac{5.8}{\mathrm{ln}({B}_{z=0.439}/{B}_{{\rm{s}}{\rm{e}}{\rm{e}}{\rm{d}}})}{\rm{G}}{\rm{y}}{\rm{r}}\), where B seed is the seed magnetic field of the mean-field dynamo. For B seed ~ 3 × 10−16 G (ref. 33), τ dynamo < 2.4 × 108 years, which is consistent with the typical rotational period of a spiral galaxy. Future such measurements of galactic magnetic fields at higher redshifts will provide a discriminant between the mean-field dynamo and other magnetic field generation mechanisms, such as the cosmic-ray driven dynamo34.


Observations and data reduction

The lensing system CLASS B1152+199 was observed on 13 November 2012 in the A array configuration with the VLA. Data were taken in L (0.994–2.006 GHz), S (1.988–4.012 GHz) and C bands (3.988–6.012 GHz, 5.988–8.012 GHz), providing a continuous frequency coverage from 1 to 8 GHz. Data were recorded in 16 spectral windows in each band, with each spectral window (SPW) further divided into 64 1 MHz channels in L band and 64 2 MHz channels in S and C bands. The total on-source time across all bands was 30 min. Data calibration and reduction were carried out using the Common Astronomy Software Applications (CASA)35. Visibilities affected by radio frequency interference were excised using the automated flagging algorithm 'rflag' in CASA with additional manual flagging. The standard VLA primary calibrator 3C286 was used for bandpass and absolute flux density calibrations. The absolute polarization angles were calibrated to 3C286 as well, whose angle is assumed to be constant (+33°) across 1–8 GHz (ref. 36). Polarization leakage terms were determined from the unpolarized source J0713+4349. Using the secondary calibrator J1158+2450, we determined the time-dependent antennae gains of each SPW separately. Three rounds of phase-only self-calibration were performed on data in each SPW to reach the theoretical noise levels ~70 μJy beam−1, ~20 μJy beam−1 and ~15 μJy beam−1 across L, S and C bands, respectively.

Imaging and deconvolution were performed using the CASA task ‘clean’. Supplementary Fig. 2 shows the total intensity and polarized intensity images of the system centred at 5 GHz across a 2 GHz bandwidth. For the purpose of rotation measure synthesis and polarization analysis, we imaged Stokes I, Q and U of the lensing system averaged in 8 MHz channels between 1 and 8 GHz using robust weighting. The native resolutions of these channel maps range from 2.36" × 1.75" at 1.006 GHz to 0.23" × 0.22" at 7.991 GHz. Since both lensed images remain unresolved at the highest observing frequency, the two lensed images were subsequently fitted as point sources in each 8 MHz channel map in all Stokes using the MIRIAD37 task ‘imfit’. We note that since images A and B are separated by 1.56", they are blended at the lowest frequencies. If one followed the standard practice of smoothing all channel maps down to the angular resolution of the lowest frequency channel map before polarization analysis, it would lead to the loss of spatial information of the corresponding Faraday components. We chose to fit for the flux densities at the native resolution of the channel maps to retain spatial information of the Faraday components.

Faraday rotation synthesis

The frequency coverage of 1–8 GHz with 8-MHz-wide channels yields a FWHM of the rotation measure spread function of 110 rad m−2. The channel width limits our sensitivity to Faraday depths with a magnitude of less than 6 × 105 rad m−2. The highest observing frequency sets our sensitivity to extended structures in Faraday depth space: the sensitivity drops by 50% for structures with extents greater than 2,230 rad m−2 (ref. 19).

Stokes QU fitting

Since extracting properties of the underlying Faraday structures based on RM synthesis alone has considerable ambiguities38,39, we directly fitted the observed fractional Stokes Q and U as a function of frequency to various models of the line-of-sight synchrotron-emitting and Faraday-rotating magnetized gas to reliably extract properties of the magneto-ionic medium being probed40.

The lensed polarized source itself is modelled as being either Faraday thin (polarized emission emitted at a single Faraday depth) or Faraday thick (polarized emission emitted at multiple Faraday depths from either a combination of multiple Faraday thin components or polarized emissions that have been Faraday dispersed and depolarized). Specifically, we considered the following models for the background source: (1) a single Faraday thin component; (2) a single polarized component propagating through an inhomogeneous external Faraday screen; (3) two spatially unresolved Faraday thin components; (4) two spatially unresolved polarized components propagating through two different inhomogeneous external Faraday screens; (5) a Burn slab, which consists of regular fields and well-mixed thermal and cosmic-ray electrons producing Faraday depolarization; (6) a slab with both regular and random magnetic fields in a volume of well-mixed thermal and cosmic-ray electrons; and (7) three spatially unresolved Faraday thin components.

The complex polarized emission from the lensed source subsequently propagates through the lensing galaxy’s ISM, acting as an external Faraday screen that could produce additional Faraday rotation and depolarization because of the presence of coherent and small-scale magnetic fields. Finally, this polarized emission penetrates the magneto-ionic medium of the Milky Way, producing further Faraday rotation.

We assumed that the intergalactic medium contributes a negligible amount to the observed Faraday rotations41 and that the variation of the Milky Way foreground Faraday rotation on the arcsec scale is negligible42. In addition, we assumed that random magnetic fields in the inhomogeneous Faraday screens fluctuate on scales much smaller than the telescope beam, such that depolarization effects can be described analytically. We note that isolating RM contributions from Faraday screens associated with different parts of a sightline is not possible even with Stokes QU fitting because one effectively measures a single net Faraday rotation and external Faraday dispersion from all the screens. This is the precise reason why we require closely spaced sightlines provided by strong gravitational lensing systems to separate out the Faraday rotation and Faraday dispersion produced in the lensing galaxy, as Faraday effects produced by the background source and the Milky Way have negligible differences for all lensed images.

We used Faraday components identified in the deconvolved Faraday depth spectrum as initial guesses for the fitted parameters. We decided on the best-fit model by comparing the reduced χ2 of the models using the f-test. We selected the least complex model that minimizes the reduced χ2: a more complex model is only favoured when the f-test suggests that it improves the fit by at least 2.3σ (~98% confidence level) over the simpler model. A lensed polarized source described by model (4) including effects of the lensing galaxy and the Milky Way with complex polarization in the form of equation (1) was found to provide the best fit to the observed broadband polarization data. The Stokes QU data and the best-fit curves to the broadband polarized spectra of image A and B are presented in Supplementary Fig. 1.

We note that unlike narrowband polarization data, which can suffer from nπ ambiguity (where multiple rotation measure values can provide equal good fits to the data), our broadband polarization data with fine frequency sampling can eliminate this problem, thus providing unique fits to the observations.

Finally, we repeated Stokes QU fitting using only data between 2 and 8 GHz, as blending of the lensed images below 2 GHz may affect the results. We found the best-fit model and the corresponding best-fit parameters to be consistent within uncertainties with results obtained using 1–8 GHz data. Hence, blending of the lensed images at low frequencies does not affect our Stokes QU fitting results.

Polarization properties of the lensed background source

Independent Stokes QU fits to the lensed images A and B yield consistent polarization properties for the lensed background source: the best-fit values for p 0, component 1 are different at the 1.4σ level, whereas θ 0, component 1, p 0, component 2 and θ 0, component 2 are all consistent within their uncertainties. In addition, the derived Faraday rotation difference of the two polarized components (RM component 1−RM component 2) for image A is in excellent agreement with that obtained for image B, which further strengthens the argument that we have detected the same components of the background source in both lensed images in polarization.

Probe of the large-scale magnetic fields

As both polarized components inferred from Stokes QU fitting experience the same amount of differential Faraday rotation and dispersion (which is not expected if the light bundle passes through clumpy turbulent local features such as an H ii region), the observed differential Faraday rotation and dispersion is most likely produced by coherent magneto-ionic structures on large scales. Even in the unlikely case that the light path through image B does penetrate an H ii region, the magnetic field strength and direction inferred from Faraday rotation would still be representative of the galactic-scale fields in the lensing galaxy, as has been shown for Galactic H ii regions43.

Estimating the differential electron column density

To convert the differential hydrogen column density14 |∆NH| = (0.48 ± 0.04) × 1022 cm−2 into a differential electron column density, we adopted an ionization fraction of 10%, the Galactic empirical value determined by comparing X-ray absorption column densities and dispersion measures towards radio pulsars44. This resulted in a differential electron column density ∆Ne = Ne,B−Ne,A of 156 ± 13 pc cm−3. We note that this value implicitly takes into account the free electrons provided by helium ionization. The ionization fraction used for the conversion is reasonable for our lensing system, which qualifies as a damped Lyman-α system15 (a quasar absorption line system with a HI column density exceeding 2 × 1020 cm−2). Since the ionization fraction in sub-damped Lyman-α systems at z ~ 0.5 is found to be less than 15% (ref. 45), damped Lyman-α systems in the same redshift range are expected to be even less ionized due to self-shielding46. Based on Cosmic Origin Spectrograph observations, the system has recently been identified as a ghostly damped Lyman-α system, where the expected Lyman-α absorption trough is filled by Lyman-α emission from the lensing galaxy47. The uncertainty in the adopted ionization fraction \(1{0}_{-3}^{+4} \% \) (ref. 44; 90% confidence level) has the following impact on the estimated magnetic field strengths: if the ionization fraction is 7%, the reported field strengths in this paper will be a factor of 1.4 higher, whereas if the ionization fraction is 14%, the reported field strengths in this paper will be a factor of 0.7 lower.

Determining the large-scale coherent magnetic field strength

We constructed a model for the magneto-ionic medium in the lensing galaxy to translate the rest-frame differential Faraday rotation into a magnetic field strength estimate. We assumed that the lensing galaxy hosts large-scale magnetic fields of either axisymmetric or bisymmetric geometry, with its strength B c(r) following an exponential with radial scale length r MAG: \({B}_{{\rm{c}}}(r)={B}_{0}\,{\rm{e}}{\rm{x}}{\rm{p}}(-r/{r}_{{\rm{M}}{\rm{A}}{\rm{G}}})\). Given a value of the electron column density along sightline A Ne,A between 5 and 300 pc cm−3, the electron column density along sightline B was determined by Ne,B = ∆Ne+Ne,A. In the thin disk approximation, and under the assumption that magnetic fields and electron densities are uncorrelated, the differential Faraday rotation between images A and B can be expressed as \({\rm{\Delta }}{\rm{R}}{\rm{M}}\approx k({{\rm{N}}}_{{\rm{e}},{\rm{B}}}{B}_{||,{\rm{B}}}-{{\rm{N}}}_{{\rm{e}},{\rm{A}}}{B}_{||,{\rm{A}}})\), where k = 0.812 rad m−2 pc−1 cm3 μG−1 and B ||,A and B ||,B are line-of-sight projections of the lensing galaxy’s magnetic field in μG along image A and B, respectively. The coherent disk magnetic field strength B 0 can be estimated by

$${B}_{0}\approx \frac{{\rm{\Delta }}{\rm{R}}{\rm{M}}}{k\,\sin \,i\,|{C}_{1}{{\rm{N}}}_{{\rm{e}},{\rm{A}}}{e}^{-\frac{{r}_{{\rm{A}}}}{{r}_{{\rm{M}}{\rm{A}}{\rm{G}}}}}-{C}_{2}{{\rm{N}}}_{{\rm{e}},{\rm{B}}}{e}^{-\frac{{r}_{{\rm{B}}}}{{r}_{{\rm{M}}{\rm{A}}{\rm{G}}}}}|}$$

Here, i is the inclination of the disk of the lensing galaxy, while C 1 and C 2 are constants whose values depend on the field geometry.

For an axisymmetric field, \({C}_{1}=\cos \,{p}_{0}\cos \,{\theta }_{{\rm{A}}}+\sin \,{p}_{0}\sin \,{\theta }_{{\rm{A}}}\) and \({C}_{2}=\cos \,{p}_{0}\cos \,{\theta }_{{\rm{B}}}+\sin \,{p}_{0}\sin \,{\theta }_{{\rm{B}}}\), where p 0 is the pitch angle adopted to be −20°, typical for galaxies observed in the local volume1,48,49 and θ A and θ B are the azimuthal angles of the sightlines A and B in the frame of the lensing galaxy. For a bisymmetric field, \({C}_{1}=\cos \,{p}_{1}\cos \,{\theta }_{{\rm{A}}}\cos ({\theta }_{{\rm{A}}}-{\beta }_{1})+\sin \,{p}_{1}\sin \,{\theta }_{{\rm{A}}}\cos ({\theta }_{{\rm{A}}}-{\beta }_{1})\) and \({C}_{2}=\cos \,{p}_{1}\cos \,{\theta }_{{\rm{B}}}\,\cos ({\theta }_{{\rm{B}}}-{\beta }_{1})+\sin \,{p}_{1}\sin \,{\theta }_{{\rm{B}}}\,\cos ({\theta }_{{\rm{B}}}-{\beta }_{1})\), where the p 1 is adopted to be −20°, and β 1, which determines the azimuth where the bisymmetric mode is maximum, is assumed to be 0° so that it gives the lower limit of the bisymmetric field strength B 0.

Determining the small-scale random magnetic field strength

For isotropic and homogenous magnetic field fluctuations, the random magnetic field strength B r in the lensing galaxy on scales smaller than 20 pc can be estimated using1

$${B}_{{\rm{r}}}=\frac{\sqrt{3fN{\rm{\Delta }}{\sigma }_{{\rm{R}}{\rm{M}}}^{2}}}{k\sqrt{{{\rm{N}}}_{{\rm{e}},{\rm{B}}}^{2}-{{{\rm{N}}}^{2}}_{{\rm{e}},{\rm{A}}}}}=\frac{1.02\times {10}^{3}}{\sqrt{{{\rm{N}}}_{{\rm{e}},{\rm{B}}}^{2}-{{{\rm{N}}}^{2}}_{{\rm{e}},{\rm{A}}}}}\sqrt{\frac{(f\,/\,0.05)(L\,/\,1\,{\rm{k}}{\rm{p}}{\rm{c}})}{(l\,/\,2\,{\rm{p}}{\rm{c}})}}(\frac{\sqrt{{\rm{\Delta }}{\sigma }_{{\rm{R}}{\rm{M}}}^{2}}}{100\,{\rm{r}}{\rm{a}}{\rm{d}}\,{{\rm{m}}}^{-2}})\,\mu {\rm{G}}$$

We assumed a filling factor f ≈ 0.05 and a path length L of 1 kpc through the lensing galaxy such that for a turbulent cell size l ~ 2 pc a sightline contains N ~ 5 × 102 cells. This yields a random magnetic field strength ranging between 3 and 6 μG for 5 pc cm−3 < Ne,A < 300 pc cm−3. The dependence of B r on the assumed path length L is not strong: a factor of four difference in the assumed L will only result in a factor of two difference in B r.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Additional information

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Author information


  1. Max Planck Institute for Radio Astronomy, Auf dem Hügel 69, Bonn, D-53121, Germany

    • S. A. Mao
    • , O. Wucknitz
    • , A. Basu
    •  & R. Beck
  2. National Radio Astronomy Observatory, PO Box O, Socorro, NM, 87801, USA

    • C. Carilli
  3. Cavendish Astrophysics Group, Cambridge, CB3 0HE, UK

    • C. Carilli
  4. Dunlap Institute for Astronomy & Astrophysics, University of Toronto, Toronto, ON, M5S 3H4, Canada

    • B. M. Gaensler
  5. Department of Physics and Astronomy, Rutgers University, Piscataway, NJ, 08854, USA

    • C. Keeton
  6. Department of Physics, University of Toronto, Toronto, ON, M5S 1A7, Canada

    • P. P. Kronberg
  7. Department of Astronomy, The University of Wisconsin, Madison, WI, 53706, USA

    • E. Zweibel
  8. Department of Physics, The University of Wisconsin, Madison, WI, 53706, USA

    • E. Zweibel


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S.A.M. led the VLA proposal and observations, performed the data reduction, analysis and interpretation, and wrote the paper. C.C. and B.M.G. contributed to the VLA proposal and interpretation of the data. O.W. and C.K. contributed to the interpretation of the data from the lensing perspective. P.P.K. and E.Z. contributed to the VLA proposal. A.B. and R.B. contributed to the interpretation of the data. All authors discussed the results, interpretations and presentation of the paper.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to S. A. Mao.

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