Magnetic field structure and halo in NGC 4631
aa r X i v : . [ a s t r o - ph . GA ] D ec Astronomy&Astrophysicsmanuscript no. NGC4631 c (cid:13)
ESO 2018October 3, 2018
Magnetic field structure and halo in NGC 4631 ‹ Silvia Carolina Mora ‹‹ and Marita Krause ‹‹‹ Max-Planck-Institut f¨ur Radioastronomie, Auf dem H¨ugel 69, 53121 Bonn, GermanyReceived 2013 / Accepted 2013
ABSTRACT
Context.
All of the edge-on spiral galaxies observed so far present a similar magnetic field configuration, which consists of a plane-parallel field in the disk and an X-shaped field at larger z-distances from the plane of the galaxy. Only NGC 4631 seems to have adi ff erent field orientation in its disk. Along the eastern and western halves of the disk of NGC 4631 the magnetic field orientation isparallel to the galactic plane, but in the central region of the disk a vertical field seems to dominate. Aims.
In order to clarify whether NGC 4631 has a unique magnetic field configuration in the central region along its disk, we presenthigh-resolution Faraday-corrected polarization data.
Methods.
Radio continuum observations of NGC 4631 at 4.85 GHz were performed with the VLA. In addition, observations weremade with the E ff elsberg telescope at 4.85 GHz and at 8.35 GHz. These were analyzed together with archival VLA-data at 8.35 GHz.The single-dish and interferometer data were combined to recover the missing zero-spacings. Results.
We determined an integrated total spectral index of α tot “ ´ . ˘ .
04 and a nonthermal integrated spectral index of α nth “ ´ . ˘ .
03. The vertical scale heights in NGC 4631 vary significantly in di ff erent regions within the galaxy and their meanvalues at 4.85 GHz are with 2.3 kpc (370 pc) for the thick (thin) disk higher than the mean values found so far in six other edge-onspiral galaxies. This may originate in the tidal interaction of NGC 4631 with its neighbouring galaxies. The rotation measures arecharacterized by a smooth large-scale distribution. Along the galactic plane the degree of Faraday depolarization is significantly high.We estimated a total magnetic field strength in the disk of NGC 4631 of B t « ˘ µ G and an ordered field strength of B ord « ˘ µ G.The total field strengths in the halo are of the order of the total magnetic field strength in the disk, whereas the ordered field strengthsin the halo seem to be higher than the value in the disk.
Conclusions.
The derived distribution of rotation measures implies that NGC 4631 has a large-scale regular magnetic field configu-ration. Despite the strong Faraday depolarization along the galactic plane and the strong beam depolarization in the transition zonebetween the disk and halo, our research strongly indicates that the magnetic field orientation along the central 5-7 kpc of the disk isalso plane-parallel. Therefore, we claim that NGC 4631 also has a magnetic field structure plane-parallel along its entire disk, similarto all other edge-on galaxies observed up to now.
Key words.
Galaxies: individual: NGC 4631 - Galaxies: halos - Galaxies: magnetic fields - Galaxies: interactions - Galaxies: spiral -Radio continuum: galaxies
1. Introduction
In external galaxies the best tracer of the magnetic field is syn-chrotron radiation produced by relativistic electrons moving in amagnetic field. These particles emit electromagnetic waves in acharacteristic nonthermal frequency spectrum and with a polar-ization direction perpendicular to the magnetic field. The planeof polarization of the electromagnetic waves rotates as the lin-early polarized emission passes through a magnetized plasmaalong the line of sight (LOS). This phenomenon is known asthe Faraday e ff ect and the angle by which it rotates ( ∆ ψ ) ina Faraday thin medium is proportional to the rotation measure(RM): ∆ ψ “ λ ¨ RM . Furthermore, the RM is proportional tothe line-of-sight integral over the density of thermal electronsmultiplied by the strength of the regular field component paral-lel to the LOS: RM ş n e B q dl (Gardner & Whiteoak 1966).With the RM distribution one can correct the polarization an-gles for Faraday rotation and also obtain information about the ‹ Based on observations with the 100 m telescope of the MPIfR(Max-Planck-Institut f¨ur Radioastronomie) at E ff elsberg and the VLAoperated by the NRAO. The NRAO is a facility of the National ScienceFoundation operated under agreement by Associated Universities, Inc. ‹‹ e-mail: [email protected] ‹‹‹ e-mail: [email protected] strength (if we have knowledge of n e ) and the direction of themagnetic field component along the LOS: the sign of the rota-tion measures indicates the direction of the B q -field. By defini-tion, positive values indicate that the B q -field is directed towardsthe observer and negative values indicate that the field is directedaway from the observer. In addition, rotation measure values addup for parallel fields along the LOS, but may cancel for antipar-allel fields of equal strength.The best way to study magnetic fields in galaxies is withradio observations of the continuum emission in the cm-wavelength regime. The total intensity of the synchrotron emis-sion measures the strength of the total magnetic field, while thelinearly polarized intensity reveals the strength and the structureof the ordered field perpendicular to the LOS.Observations of face-on spiral galaxies show that the mag-netic field lines in the disk follow a spiral structure similar tothe optical spiral morphology (Beck et al. 1996). Where spiralarms are visible, the fields tend to be more ordered in the inter-arm regions. In all of the edge-on spiral galaxies studied up tonow, the large-scale ordered fields in the disk lie plane-parallelalong the disk of the galaxy and in the halo they have an X-shaped morphology (Krause 2009), sometimes with nearly verti-cal field components above and below the central region (e.g., inNGC 5775, Soida et al. 2011). The disk-parallel magnetic field is the expected edge-on projection of the spiral magnetic diskfield seen in face-on galaxies. This large-scale magnetic field inthe disk is believed to be amplified by the action of a large-scale α Ω dynamo (Ruzmaikin et al. 1988).The galaxy NGC 4631 is known for the strong vertical mag-netic fields in its radio halo (Hummel et al. 1991). It has beenargued for quite some time that NGC 4631 seems to be theonly edge-on galaxy that does not exhibit a plane-parallel fieldin its disk. However, the large-scale vertical field in the disk inNGC 4631 seems to be restricted to the central region (Krause2003). Outside of a radius of about 2.5 kpc the field is plane-parallel in the western and eastern halves of the disk. In its outerhalo it does present the typical X-shaped configuration seenin the other edge-on galaxies (Golla & Hummel 1994; Krause2009). Furthermore, above and below the galactic center the halofield is orientated perpendicular to the plane of the galaxy.In addition, NGC 4631 may be undergoing considerablestar formation activity; this is indicated by the high star forma-tion e ffi ciency (Krause 2012), the blue color (B-V), the strongH α emission (Golla 1999), and the small ratio S µ m { S µ m (Hummel et al. 1991) of this galaxy. Hence, it was argued thatthe field in the central region of the disk might be wind-drivenand that it may be related to the high star-forming activity inthis galaxy (Golla & Hummel 1994). However, there are otheredge-on galaxies with higher star formation activity that stillpresent plane-parallel magnetic fields in their disks, for exam-ple NGC 253 (Krause 2012; Heesen et al. 2009).The rotation curve of NGC 4631 rises nearly rigidly in theinner 2.5 kpc (Golla & Wielebinski 1994). Therefore, Krause(2009) has suggested that the di ff erential rotation in this regionmight be too low to trigger a large-scale α Ω dynamo and thusthat there may be no amplification of the plane-parallel mag-netic field in the disk. Nevertheless, it still remains unclear whyNGC 4631 may have a unique magnetic field orientation in thecentral region of its disk. However, Irwin et al. (2012) alreadynoticed in their 6 GHz EVLA observations that the magneticfield (uncorrected for Faraday rotation) along the plane of thegalaxy appears to be parallel to the disk.The galaxy NGC 4631 exhibits an extraordinary prominentradio halo (Ekers & Sancisi 1977; Wielebinski & von Kap-Herr1977; Hummel & Dettmar 1990). In addition, the halo has beenobserved in other interstellar medium components, for exam-ple in X-ray emitting gas (Wang et al. 2001; Yamasaki et al.2009), dust (Martin & Kern 2001; Neininger & Dumke 1999),and molecular gas (Rand 2000; Irwin et al. 2011). The centralregion has an interesting structure consisting of three collinearemission peaks (Ekers & Sancisi 1977; Duric et al. 1982; Golla1999). The easternmost radio peak of the triple source coin-cides with the huge HII region CM 67 (Crillon & Monnet 1969;Roy et al. 1991). According to Krause et al. (1994), the nonther-mal radio continuum, CO, and HII line emission in this areaare physically connected and form a huge star forming regionin NGC 4631.The interaction of NGC 4631 with its neighbours is arguedto be responsible for the large extent and asymmetries of itshalo. The galaxy NGC 4631 is located in a group environmentwhere many other galaxies reside (Giuricin et al. 2000). Amongothers, there is the dwarf elliptical galaxy NGC 4627 (2 . to thesoutheast). The interaction between NGC 4631 and NGC 4656was discovered by Roberts (1968), who mapped the emissionfrom these galaxies in the 21 cm hydrogen radio line. Lateron, Weliachew et al. (1978) and Rand (1994) observed thesegalaxies with the Westerbork Synthesis Radio Telescope and they found an HI bridge emerging from NGC 4631 towardsNGC 4656, an HI concentration stretching out to the south ofNGC 4631, and two elongated spurs on the northern extensionof NGC 4631 (one oriented north-south and the other extend-ing to the north-east), labeled 1 to 4 in the above mentionedpapers. Combes (1978) presented a model for the tidal inter-action of the three dominant galaxies NGC 4631, NGC 4656,and NGC 4627. Furthermore, gravitational interaction is alsoknown to modify galactic magnetic fields (Drzazga et al. 2011;Chy˙zy & Beck 2004).In this paper we present observations of NGC 4631 withthe E ff elsberg 100 m telescope at 4.75 GHz and 8.35 GHz,and with the VLA at 4.75 GHz. The VLA observations werecombined with previously published VLA observations byGolla & Hummel (1994) and then merged with the E ff elsbergsingle-dish maps (see Sect. 2). The results for total intensity, haloproperties, polarized intensity (including Faraday rotation anddepolarization), and the magnetic field strengths are presented inSect. 3, followed by the discussion in Sect. 4. The conclusionsare summarized in Sect. 5.The parameters of NGC 4631 assumed throughout this studyare presented in Table 1. The dynamical center refers to the lo-cation of the central concentration of mass in NGC 4631, whichaccording to Irwin et al. (2011) is best represented by the IR cen-ter at 2 µ m. Table 1.
Parameters of NGC 4631, as obtained from the literature.
Parameter Value
Morphological type SBcdDynamical Center a α “ h m s δ “ ˝ . ˝ Position angle of major axis 85 ˝ Assumed distance b . fl
370 pc)
2. Observations and data reduction
Observations with the 100 m E ff elsberg telescope of NGC 4631at 4.85 GHz ( λ ˆ scannedarea, centered on the dynamical center of the galaxy (see Table1). The data were reduced with the NOD2 software package(Haslam 1974) in which scanning e ff ects due to weather con-ditions, receiver instabilities, and the radiation frequency inter-ference (RFIs) were removed. Flux-calibration was done by us-ing the radio source 3C286 on the scales of Baars et al. (1977).The U- and Q-maps were used to determine the linearly po-larized intensity I pol “ a Q ` U and the polarization angle ψ “ { p U { Q q . The linear resolution of the maps is 147 half-power beam width (HPBW) and the rms noise values are400 µ Jy { beam in total intensity and 73 µ Jy { beam in polarizedintensity.Single dish observations with the E ff elsberg 100 m telescopeat 8.35 GHz ( λ taken with a map size of 30 ˆ each, scanned in two orthog-onal directions to reduce scanning e ff ects. Flux-calibration wasagain done by using the radio source 3C286 on the scales ofBaars et al. (1977). High-frequency noise was removed from themaps together with a slight smoothing to 85 HPBW in orderto increase the signal-to-noise ratio. Again, the polarized inten-sity and polarization angles were determined from the U- andQ-maps. The rms noise values in the maps are 350 µ Jy { beam intotal intensity and 70 µ Jy { beam in polarized intensity.The E ff elsberg Stokes I maps together with the linear po-larization at λ ˝ , and give the apparent magnetic field orientation since theyhave not been corrected for Faraday rotation (see Sect. 3.5). Inall of the maps presented in this article the dynamical center ofNGC 4631 is indicated by an “x” and the beam-area is shown inthe left-hand corner of each image. D E C L I NA T I O N ( J ) RIGHT ASCENSION (J2000)12 42 30 15 00 41 45 3032 363432302826 0 20 40 60 80
Fig. 1. E ff elsberg Stokes I Map at λ p fl µ Jy { beam q . The colorscalegives the flux density in mJy / beam. The angular resolution is 85 .The rms noise ( σ ) is of 350 µ Jy { beam. Contour levels correspond to σ ¨ p´ , , , , , , , , q . The “x” pinpoints the dynamicalcenter. ) The galaxy NGC 4631 was observed for 9 h at 4.85 GHz inApril 1999 with the VLA telescope in its D-array configura-tion (Project AD 896, here called data set VLA ). These ob-servations were pointed about 70 west of the dynamical cen-ter of NGC 4631. The data were calibrated and reduced inthe AIPS data processing package. The nearby point source1225 +
368 was used for phase and instrumental polarization cal-ibration. The polarization position angle and flux-calibrationwere done by using 3C286 according to the flux scale pub-lished by Baars et al. (1977). The UV data were self-calibratedfor phase and amplitude. The maps of the three Stokes parame-ters I, U, and Q have a linear resolution of 12 . ˆ .
08 HPBW.The polarized intensity and polarization angles were determinedfrom the U- and Q-maps. The map of the total intensity togetherwith the linear polarization is shown in Fig. 3. The rms noise D E C L I NA T I O N ( J ) RIGHT ASCENSION (J2000)12 42 45 30 15 00 41 45 3032 38363432302826 0 50 100 150 200
Fig. 2. E ff elsberg Stokes I map at λ p fl µ Jy { beam q . The colorscale corre-sponds to the flux density in mJy / beam. The half-power beam width is147 and the rms noise ( σ ) is 400 µ Jy { beam. Contour levels are givenby σ ¨ p´ , , , , , , , , , , q . values are 26 µ Jy { beam in total intensity and 16 µ Jy { beam inpolarized intensity. In Table 2 we list all VLA data available atthese two frequency ranges, together with their distinct phasecenters and beam sizes. D ec li n a t i o n ( J ) Right ascension (J2000)12 42 25 20 15 10 05 00 41 55 50 45 4032 35 0034 300033 300032 300031 3000 0 100 200
Fig. 3.
VLA map at λ µ Jy { beam. Thelength of the vectors is proportional to the polarized intensity p fl . µ Jy { beam q . The half-power beam width is 12 . ˆ .
08 andthe rms noise ( σ ) is 26 µ Jy { beam. Contour levels are given by σ ¨p´ , , , , , , , , q . Primary beam correction was ap-plied. An additional VLA total intensity map of NGC 4631 at λ . These observations were conducted in D-array configuration with 7.2 h of net observing time and werepointed northeast of the dynamical center of NGC 4631. SinceGolla & Hummel (1994) used a zero-spacing flux density for theimaging of the total intensity map, the same was done for the Table 2.
Data used according to λ and telescope, with its correspond-ing HPBW and pointing position (P.P.). C = near dynamical center ofNGC 4631; NE = northeast of dynamical center; W = west of dynami-cal center; S.P. = Stokes Parameter. λ S.P. E ff elsberg VLA VLA HPBW
P.P.
HPBW
P.P.
HPBW
P.P. C - - 12 NEQ 85 C - - 12 NEU 85 C - - 12 NE6.2 cm I 147 C . ˆ . W 12 NEQ 147 C . ˆ . W - -U 147 C . ˆ . W - - imaging of our Stokes I VLA map so that these two maps couldbe combined into a mosaic. With prior regridding and smooth-ing, the two VLA total intensity maps were mosaiced using theAIPS task LTESS. This task already applies the primary beamcorrection (PBC), which corrects for the primary beam attenua-tion with a cut-o ff parameter which we set to 25% of the beamsensitivity at the phase center.The E ff elsberg and the VLA data at λ ff elsbergmap was recovered with this procedure. The u,v-overlap adopted(UVRANGE) between the two data sets lies within 0.725 k λ and 0.775 k λ . This range was determined based on two crite-ria. Firstly, the normalizing scale factor derived by AIPS for thecombination should approximate the resolution di ff erence be-tween the two data sets. Secondly, the integrated flux densityof the Stokes I merged map should be similar to that of theE ff elsberg map.The object at position α “ h m s ; δ “ ˝ is identified as a background source according toHummel & Dettmar (1990). With our E ff elsberg maps we de-termined that it has a flux density of 6.5 mJy (8 mJy) at 3.6 cm(6.2 cm). We tried to extract this bright source in both high- andlow-resolution maps before merging them. However, in the low-resolution maps it is di ffi cult to separate the flux of this sourcefrom the extended flux of NGC 4631 because of the size of thebeams. Thus, this extraction heavily distorted our merged mapscausing negative bowls to appear below the galaxy.Data from the VLA at λ ) were available fromGolla & Hummel (1994). The phase center was chosen to be lo-cated northeast of the dynamical center of NGC 4631, as Gollaand Hummel sought to study the radio spur in the northeasternhalo. Prior to merging with the single-dish data, the interferom-eter maps were corrected for the primary beam attenuation andcut at 25% of the sensitivity level at the beam center. For themerging of the data at this frequency the UVRANGE was be-tween 1.1 and 1 . λ . In this case we recovered about 20% ofthe flux density of NGC 4631 found in the E ff elsberg map withinthe area of the VLA primary beam.Consequently, we produced maps in polarized intensity(I pol ), polarization angle ( ψ ), and polarization degree (P) at bothfrequencies. The polarization angles were clipped at the 2 σ levelof the I pol map. The polarization degrees were clipped at the 2 σ level of both the polarized intensity and the total intensity map.In addition, we smoothed our merged maps in all Stokes param-eters to a resolution of 25 and produced maps of the I pol , ψ ,and P at this angular resolution. The noise values of the maps are shown in Table 3. Values for each map were estimated bycalculating the average noise around the galaxy in small areascarefully selected to be free of emission. Table 3.
Noise values according to λ for maps at an angular resolutionof 12 . ˆ .
08 HPBW, with primary beam correction.
Map λ λ I merged [ µ Jy { beam] 20 23Q merged [ µ Jy { beam] 15 13U merged [ µ Jy { beam] 14 14I pol [ µ Jy { beam] 15 14
3. Results
The maps of the VLA or merged total intensity emission ofNGC 4631 at 4.85 GHz and 8.35 GHz are presented in con-tours in Figs. 3, 4, 5, 8, 9, and 12. At these radio frequenciesthe extent of NGC 4631 on the sky is larger than the primarybeam of the VLA antennas. Therefore, the λ ff elsbergmaps, obtaining a value of (430 ˘
20) mJy at λ ˘
16) mJy at λ α emission (Fig.5). The dominant features in the total intensity maps are the ex-tended central emission and the huge and asymmetric radio halo.The bright central emission extends „ λ µ m, represented by the “x”, lies within ourcentral radio peak of the three; the westernmost coincides withthe huge star forming region CM 67.The λ above the major axis.The southern part of the halo is also asymmetric in shape withthe largest extent in the southeast. Even the single dish E ff elsbergmap at λ HPBW. The northern asymmetric dis-tribution looks similar also at λ
20 cm by Hummel et al. (1991)and in the L-band regime by Heald et al. (2009). However, a sec-ond larger extent of the radio emission is visible at λ Fig. 4.
Total radio intensity map at 4.85 GHz (E ff+ VLA) in contours with apparent magnetic field orientation. The length of the vectors isproportional to the I pol (1 fl . µ Jy { beam). It is overlayed on a colorscale optical DSS image (blue band). The half-power beam width is12 . ˆ .
08 and the rms noise ( σ ) is 23 µ Jy { beam. Contour levels are given by σ ¨ p´ , , , , , , , , q . The “x” indicates thedynamical center of NGC 4631. (Fig. 1) in the northwestern halo which is already indicated inthe λ With the integrated total flux-densities of NGC 4631 shown inTable 4, we fit a power law ( S tot ν α tot ) to calculate the in-tegrated total spectral index of this galaxy. The flux density at1.365 GHz was taken from Braun et al. (2007) and at 10.55 GHzfrom Dumke et al. (1995). These values were incorporated be-cause they are the most recent total flux-densities publishedof NGC 4631. The integrated total spectral index obtained is α tot “ ´ . ˘ .
04; it is in agreement with that publishedby Hummel & Dettmar (1990). This spectral index accounts forboth the thermal and the nonthermal components of the radioemission.However, the distinct spectral behavior of each of the twocomponents can be used to distinguish one from the other. Thethermal emission is considered to have a spectral index of α th “´ . S th ν ´ . ). The thermal fraction can be estimated withrespect to that measured at another frequency as follows: f th “ f th ¨ ˆ ν ν ˙ ´ . ´ α tot . (1)By adding the approximate thermal contributions of the ra-dio and H α emission, Golla (1999) estimated that NGC 4631 hasa total thermal flux density at C-band (4.5-5.0 GHz) of 56 mJy.With this value we deduced a thermal fraction at 6.2 cm of 13%.In Table 4 we show our derived thermal fractions for each ofthe given frequencies. With the thermal fractions we derived thenonthermal flux-densities to estimate α nth . In Fig. 6 the solid redline represents the best fit to the power law resulting in a non-thermal spectral index of α nth “ ´ . ˘ . Table 4.
Total flux-densities and thermal fractions ( f th ) of NGC 4631at four di ff erent frequencies. Frequency [GHz] Total flux density[mJy] f th ˘ a ˘
20 13%8.35 310 ˘
16 19%10.55 265 ˘ b N on t h e r m a l f l ux - d e n s it y [ m J y ] Fig. 6.
Integrated radio spectrum of the nonthermal emission ofNGC 4631. The red solid line shows the best-fit power law, with a non-thermal spectral index of α nth “ ´ . ˘ .
03. 5ilvia Carolina Mora and Marita Krause: Magnetic field structure and halo in NGC 4631 D ec li n a t i o n ( J ) Right ascension (J2000)12 42 30 15 00 41 4532 36353433323130 0 100 200 300 400
Fig. 5.
Total radio intensity map at 4.85 GHz (E ff+ VLA) in contours with apparent magnetic field vectors. It is overlayed on a colorscale H α image. The length of the vectors is proportional to the I pol (1 fl . µ Jy { beam). The angular resolution is 12 . ˆ .
08 HPBW and the rmsnoise ( σ ) is 23 µ Jy { beam. Contour levels correspond to σ ¨ p´ , , , , , , , , q . While the observed vertical extent of the radio emission of anedge-on galaxy depends on the sensitivity of the radio map (i.e.,the signal-to-noise ratio), the vertical scale height of the emis-sion is a more appropriate parameter to describe and comparethe vertical emission profile of edge-on galaxies. Of course, thefitting procedure has to take into account the telescope beam sizeand the inclination of the galaxy. This was done by introducingan e ff ective beam size in the following way. We determined theintensity distribution in NGC 4631 along the major axis of themerged total intensity map at λ ˝ . This distribution was convolvedwith a Gaussian function with 12 HPBW to simulate the e ff ectof the telescope beam. The HPBW of the resulting distributionis the e ff ective beam size. For the merged λ .We determined the vertical scale heights at λ each, centered on the nucleus. These profiles were fittedwith model distributions (for details see Dumke et al. 1995) con-sisting of an intrinsic one- or two-component exponential, or aGaussian profile convolved with the e ff ective beam size. The fitswere made separately for the emission above (north = n) andbelow (south = s) the disk midplane (from east to west).As the merged λ Table 5.
Vertical scale heights for the thin and thick disk. n = northand s = south. The eastern (1), central (2) and western (3) strips have awidth of 150 each. Range along 8.46 GHz 4.86 GHzStrip major axis h thin h thick h thin h thick [ ] [pc] [kpc] [pc] [kpc] n1 -225 to -75 420 2.9 440 3.2n2 -75 to 75 410 2.2 490 3.4n3 75 to 225 160 1.1s1 -225 to -75 30 1.8s2 -75 to 75 290 1.7s3 75 to 225 810 2.9 Mean ˘
280 2 . ˘ . The fitted values for the scale heights vary widely within thedi ff erent strips in NGC 4631 at λ ˘
50 pc for the thin disk and 1 . ˘ . I n t e n s it y [ m i c r o J y / b ea m ] n1 -225" < x < -75" 0 50 100 150 200 250Distance from major axis ["] s2 I n t e n s it y [ m i c r o J y / b ea m ] s1 s3n2 -75" < x < 75" n3
75" < x < 225"
Fig. 7.
Total radio intensity profiles of the merged radio maps in µ Jy { beam perpendicular to the major axis of NGC 4631. The measured pointsare averaged in strips of 150 width along the major axis. The x-axe of the plots give the distance from the major axis in [ ]. The upper panels referto the northern part (i.e., above the major axis) and the lower panels to the southern part (i.e., below the major axis) of NGC 4631. The thick redlines represent the two-component exponential fit to the data at 4.85 GHz; the thin black lines in n1 and n2 represent the fit to the 8.35 GHz data. the typical dumbbell structure visible in NGC 253, for exam-ple (Heesen et al. 2009), and in NGC 4565 (Krause 2009). Thedumbbell structure with its smaller extent of the emission aboveand below the central region of the disk and larger z-extents atlarger radii seems to reflect dominating synchrotron losses in amagnetic field that is strongest along the central galactic planeof the galaxy (Heesen et al. 2009). The galaxy NGC 4631 is dif-ferent in this respect. This will be further discussed in Sect. 4. The linearly polarized emission of NGC 4631 at 8.35 GHz and4.85 GHz is plotted as a colorscale in Figs. 8 and 9 with a reso-lution of 12 . ˆ .
08 HPBW.There is evidently more polarized emission above than be-low the plane of NGC 4631. At λ „ . „ . λ ff ects atshorter wavelengths.The polarized emission of NGC 4631 can be separated intofour extra-planar quadrants with respect to the eastern peakof the triple radio source (see Sect. 3.1) at position α “ h m s ; δ “ ˝ . The northwestern quadrant hasthe brightest polarized intensity at both wavelengths, followedby the southeastern quadrant. At λ λ D ec li n a t i o n ( J ) Right ascension (J2000)12 42 20 15 10 0532 34 0033 300032 300031 30 0 100 200
Fig. 8.
Colorscale of the polarized emission over total radio in-tensity contours at λ ff+ VLA), with apparent magneticfield vectors. The length of the vectors is proportional to the I pol (1 fl . µ Jy { beam). The colorscale gives the polarized inten-sities in µ Jy { beam. The half-power beam width is 12 . ˆ . p µ Jy { beam q¨p´ , , , , , , , , q . (1994), who state that it is a small tributary of the northeasternspur visible in total power (see Fig. 1) and that it seems to origi-nate from the eastern peak of the triple source (CM67).In order to analyze the large-scale structure and increase thesignal-to-noise ratio, the Stokes Q- and U-maps were smoothedto a resolution of 25 and polarized intensities were determinedfrom these (see Fig. 10). Again, there are four peaks of polarizedemission distributed around the dynamical center of NGC 4631,in each of the four quadrants previously mentioned. D ec li n a t i o n ( J ) Right ascension (J2000)12 42 25 20 15 10 05 00 41 55 50 4532 35 0034 300033 300032 300031 3000 0 100 200
Fig. 9.
Colorscale of the polarized emission over total radio inten-sity contours at λ ff+ VLA), with apparent magnetic fieldorientation. The length of the vectors is proportional to the I pol (1 fl . µ Jy { beam). The colorscale gives the polarized inten-sities in µ Jy { beam. The half-power beam width is 12 . ˆ . p µ Jy { beam q ¨p´ , , , , , , , , q . The polarized intensity shows a minimum along the galac-tic midplane at both wavelengths. This is expected as Faradaydepolarization e ff ects are usually strongest in this area becausethe thermal electron density, the LOS, and the magnetic fieldstrength are largest. A depolarization along the midplane has al-ready been observed in several other spiral edge-on galaxies, forexample NGC 5775 (Soida et al. 2011). Furthermore, if there isa magnetic field parallel to the disk accompanied by strong ver-tical magnetic fields above and below the disk, we also expectstrong beam polarization along the region of the projected tran-sition between both magnetic field components. To minimize thebeam depolarization, observations with high resolution are re-quired, while high frequency observations reduce the Faradaydepolarization. We even found that the merging of single-dishdata to the interferometer data increases the zone of depolar-ization along the midplane of NGC 4631 (see Fig. 9). This isexpected along the transition zone of two large-scale magneticfield patterns with di ff erent orientations: the beam depolarizationin the single-dish maps is much larger and reduces the observedU- and Q-signal at positions where the interferometer beam stilldetects a signal. In these cases the merged map shows less po-larized intensity than the original interferometer map (see Fig.9). The polarization angles can only be determined with an n ¨ π am-biguity which leads to an uncertainty in the RM derivation. For n “
1, at wavelengths of 3.6 and 6.2 cm, we would have am-biguous rotation measure values of: RM “ π {p λ ´ λ q «˘ { m , which is high compared to the expected valuesfor a galactic disk.The obtained rotation measures are biased by the Galacticforeground component ( RM fg ), which is quite small in the di-rection of the sky in which NGC 4631 is located. Nevertheless,we corrected for this by subtracting the value of RM fg “ p´ ˘ q rad { m (Heald et al. 2009) from our rotation measures.The overall RM distribution obtained does not present val-ues that exceed ˘
615 rad { m . Even without being a ff ected bythe n ¨ π ambiguity, a rotation of the polarization angle by 90 ˝ along the LOS through the emitting source leads to strong dif- D ec li n a t i o n ( J ) Right ascension (J2000)12 42 15 00 41 4532 3736353433323130
Fig. 10.
Linearly polarized emission at λ ff+ VLA), with ap-parent magnetic field orientation. The length of the vectors is propor-tional to the polarization degree (1 fl and the rms noise ( σ ) is 20 µ Jy { beam. Contour levels are givenby σ ¨ p´ , , , , , , , q . The “x” indicates the dynamical cen-ter. In the electronic version the angles are color coded as shown in thecolorfan in the upper-right corner. ferential Faraday depolarization within the galaxy. This is ex-pected at λ | RM | «|p˘ π { q{p . ¨ ´ q | «
400 rad { m and means that the galaxyis no longer transparent in polarized emission. Hence, this im-plies that the derived values for the rotation measure may beincorrect. To be on the safe side we set our rotation measurethreshold to | RM | ď
350 rad { m and clipped both the rota-tion measure distribution and the polarization angles to satisfythis condition. We determined the rotation measure distributionat 25 with the original VLA maps at λ ff elsberg λ λ resolution. In Fig. 11 we show these rotationmeasures with respect to the E ff elsberg polarized emission at λ HPBW). Overall, the rotation measure distributionis smooth and regular. The concentration of the thermal emis-sion towards the disk, due to the elevated electron densities inthe high starforming regions, may explain why | RM | increasesclose to the plane and especially around the dynamical center.We note that the rotation measure in the northeast quadrant isnegative with increasing | RM | values from the midplane in a re-gion around the northeastern spur (see Sect. 3.4). This region,which we call the Northeastern extension , will be discussed fur-ther in Sect. 4.
The apparent magnetic field orientation (which are theobserved polarization angles rotated by 90 ˝ ) are shown as vec-tors in the merged maps in Figs. 4, 5, 9, and 10 at λ λ λ ˝ yields the orientation of the magnetic field compo- D ec li n a t i o n ( J ) Right ascension (J2000)12 42 30 15 00 41 45 3032 38363432302826 -200 0 200
Fig. 11.
Rotation measure distribution colorscale (6.2cm-Merged + ff ) over E ff elsberg λ pol contours, withintrinsic magnetic field. Length of the vectors is proportional to theE ff elsberg λ fl σ level of the polarized intensity maps (at bothfrequencies) and where | RM | ě
350 rad { m . The colorscale gives therotation measures in rad { m . All values are from the data at an angularresolution of 85 HPBW. Contour levels of the E ff elsberg λ p µ Jy { beam q ¨ p´ , , , , q . nent perpendicular to the LOS (B K -vectors). It is worth notingthat the polarized intensity is only sensitive to the field orien-tation but not to the field direction, hence it cannot distinguishbetween parallel and antiparallel field directions in the plane ofthe sky.We determined the vectors at those locations where the po-larized intensity at both wavelengths is above 2 σ and where therotation measures are below | RM |
350 rad { m , which iswhere the derived values for the rotation measures are reliable,as discussed in Sect. 3.5. By error propagation the error in the in-trinsic polarization angles is σ ψ int “ σ ψ ` p λ ¨ σ RM q . Hence,in areas where the polarized emission is twice the rms noise( σ I pol « σ Q , U ), the error in the intrinsic polarization angles isat 6.2 cm „ ˝ and at 3.6 cm „ ˝ . However, this error scalesdown linearly with increasing polarized intensity.Merging with E ff elsberg data is essential for recovering thelarge-scale emission of the total intensity and the linear polar-ization in NGC 4631. In special cases, however, the polarizedsignal may even be weakened by the merging process, e.g.,along a transition zone of large-scale magnetic field patternswith abruptly changing field orientations (as discussed in Sect.3.4). At λ ff ects along the midplane we de-termined the intrinsic magnetic field orientation derived solelywith the VLA data at both frequencies. To increase the signal-to-noise ratio we smoothed these data to 25 and corrected themfor Faraday rotation with the corresponding RM map (see Sect.3.5). The map of the intrinsic magnetic field in presented in Fig.12. We can clearly see that the halo magnetic field orientation inthe eastern side of the galaxy is almost symmetric with respect D ec li n a t i o n ( J ) Right ascension (J2000)12 42 25 20 15 10 0532 34 0033 4530150032 4530150031 4530
Fig. 12.
Intrinsic magnetic field orientation perpendicular to the LOS(B K -vectors) derived from the 25 resolution data (VLA only). Thecontours show the VLA λ λ fl µ Jy { beam). These vectors were clipped at the 2 σ level of the polarized intensity maps (at both frequencies) and where | RM | ě
350 rad { m . The contour levels correspond to p µ Jy { beam q¨p´ , , , , , , , , , q . In the electronic version the an-gles are color coded as shown in the colorfan in the upper-right corner. to the major axis of NGC 4631 and that it curves radially out-wards. At the edges of the western side of the maps we can alsodistinguish that the halo magnetic field begins to curve outwards.This forms part of the X-shaped halo magnetic field seen in otheredge-on galaxies. As claimed by Golla & Hummel (1994), alongthe spur-like feature observed in polarized emission (see Sect.3.4) the magnetic field orientation seems to follow this spur. Inaddition, above the dynamical center the halo magnetic field isorientated perpendicular to the major axis of the galaxy.Along the central region of the disk the B K -vectors are ori-ented parallel to the galaxy’s plane extending over several beamsizes. We corrected the λ ff elsberg map (Fig. 1) forFaraday rotation with the RM map determined between this mapand the merged λ (Fig. 11). Themap is shown in Fig. 13 and presents a plane-parallel magneticfield in the outer regions along the midplane that had previouslybeen indicated by Krause (2009). With the plane-parallel vectorsin Fig. 12 we can now see for the first time the missing link of adisk-parallel magnetic field in the central 5-7 kpc in NGC 4631. The observed Faraday depolarization (DP) (wavelength-dependent) in NGC 4631 was determined by the ratio of the po-larization degree at λ ff elsberg data set) with a resolution of 85 . Since the mapsof the polarization degree at both frequencies have the same an-gular resolution, the beam depolarization is equal in both mapsand cancels out, except for the e ff ects along the midplane as dis-cussed in Sect. 3.4. The observed DP is presented in Fig. 14. Itwas determined from data truncated at 2 σ of the I pol maps andat 3 σ of the total intensity maps. The depolarization values insome regions are DP ą
1; this may be due to additional errorsin the maps caused by the large separation between the pointingpositions of the di ff erent observations.As expected, the depolarization is higher in the plane of thegalaxy which could be the result of several factors. The H α dis-tribution of NGC 4631 (Fig. 5) indicates that the thermal elec- D ec li n a t i o n ( J ) Right ascension (J2000)12 42 30 15 00 41 45 3032 363432302826
Fig. 13.
Intrinsic magnetic field orientation derived from the 85 resolution data (6.2cm-Merged + ff ). Contours correspond tothe E ff elsberg λ ff elsberg λ fl µ Jy { beam). These vectors were clipped at the2 σ level of the polarized intensity maps (at both frequencies) andwhere | RM | ě
350 rad { m . The contour levels correspond to p µ Jy { beam q ¨ p´ , , , , , , , , q . In the electronicversion the angles are color coded as shown in the colorfan in the upper-right corner. trons are concentrated in the disk of the galaxy in the star form-ing regions. The galaxy NGC 4631 is known to have strong starformation activity (see Sect. 1). Since star formation causes themagnetic field to become more turbulent, the interstellar mediumin the disk may be considerably more turbulent producing in-ternal Faraday dispersion. Furthermore, the long LOS throughthe disk may also cause depolarization by di ff erential Faradayrotation within NGC 4631. In addition, rotation measure gradi-ents due to the decrease of thermal electrons in z-direction canalso produce depolarization, as estimated by Golla & Hummel(1994) for our frequency range.The depolarization seems to increase towards the extent ofthe northeastern halo of NGC 4631. In the uppermost regionof the northeastern halo, we also have high rotation measures(about ´
150 rad { m ) as seen in Fig. 11. This is surprising sincein general the depolarization e ff ects decrease towards the halobecause of the decreasing thermal electrons and the decreasingLOS through the emission. Another possibility is that the B q -field in this region is stronger. This would be the case if the mag-netic field lines are highly ordered and bend away from us alongthe spur. This would cause an increase in the B q -field and hencestrong depolarization and high rotation measures. If one assumes equipartition between the total energy densitiesof cosmic rays and that of the magnetic field, the magnetic fieldstrength can be derived from the nonthermal radio emission asdescribed in Beck & Krause (2005). In Sect. 3.6 we describedthe configuration of the B K -field orientation of NGC 4631 to becomposed of an X-shaped field in the halo, an extra-planar ver-tical field above the dynamical center and a field along the disk.We estimated the magnetic field strengths with the total and po-larized emission of the E ff elsberg λ ff erent re- D ec li n a t i o n ( J ) Right ascension (J2000)12 42 30 15 00 41 45 3032 383634323028260.0 0.5 1.0 1.5
Fig. 14.
Colorscale of the depolarization distribution (6.2cm-Merged + ff ) over E ff elsberg λ pol contours, with in-trinsic magnetic field. Length of the vectors is proportional to theE ff elsberg λ fl σ level of the polarized intensity maps (at bothfrequencies) and where | RM | ě
350 rad { m . The contour levels cor-respond to p µ Jy { beam q ¨ p´ , , , , , , , , q . All val-ues are from the data at an angular resolution of 85 HPBW. Contourlevels of the E ff elsberg λ p µ Jy { beam q ¨ p´ , , , , q . gions within the halo and the disk of more than a beamsize each,according to the di ff erent field geometries in the disk and halo,assuming a constant nonthermal spectral index of α nth “ ´ . ff erent regions . Thewidths of the two regions along the disk are about 2 kpc each.The region with the vertical field above the dynamical centerextends from about 1 ´ | z | » | r | » Northeastern extension onlythe upper part of the northeastern halo was considered (betweenabout 4 ´ α nth , the assumedLOS, and inclination of the field. Table 6.
Magnetic field strengths in di ff erent areas of NGC 4631. B t -Total field strength. B ord - Ordered field strength. Region Area B t r µ G s B ord r µ G s Halo N-E;S-E;N-W;S-W 10 ˘ ˘ Northeasternextension a ˘ ˘ ˘ ˘ ˘ ˘ ˘ ˘ Within the range of uncertainties, the total and ordered fieldstrengths seem to be very similar in all of the considered regionsof the X-shaped field. Furthermore, the strength of the total fieldin the central region of the disk is comparable to the strength ofthe perpendicular field above it within the margin of uncertain-ties. In addition, the strength of the total magnetic field in theentire disk seems to be similar to the strengths of the total fieldin the halo. The strengths of the ordered field in the halo seemsto be at least as strong as that of the ordered field in the disk. Thedegree of uniformity (the ratio of ordered to total field strength)in the halo is generally higher than in the disk, especially alongthe
Northeastern extension .
4. Discussion
In Sect. 3.6 we described our results for the magnetic field ori-entation of NGC 4631 on the sky plane (B K ). In the halo it ischaracterized by an overall X-shaped configuration and a strongperpendicular field above the galactic center. The edge-on spiralgalaxies NGC 5775 (Soida et al. 2011) and NGC 4666 (Soida2005) also have strong perpendicular components above and be-low their galactic centers. Dynamo simulations by Hanasz et al.(2009) may be able to explain not only this X-shaped morphol-ogy but also these vertical fields.The Faraday-corrected single-dish E ff elsberg map at λ α Ω dynamo.The galaxy NGC 4631 was regarded as an exception, but ourresearch indicates this is not the case.The total magnetic field strength in the disk of NGC 4631, B t « ˘ µ G, is comparable to the total field strength of thestar forming edge-on galaxy NGC 5775 (Soida et al. 2011), butit is not as strong as the strength in the starburst galaxy NGC 253(Heesen et al. 2009) which also has strong star formation alongits disk. However, the average strength of the total magnetic fieldin the halo, B t « ˘ µ G, is higher than in other the edge-on galaxies studied so far. In addition, the strengths of the or-dered field in the halo B ord « p ´ q µ G seem to be larger thanthe strength of the ordered field in the disk B ord « ˘ µ G(see Table 6). Hence, strong di ff erential Faraday rotation in thedisk as well as in the halo is expected. This, together with strongbeam depolarization along the transition zone between the hori-zontal disk field and the mainly vertical halo field in the centralregion of NGC 4631, might explain why the plane-parallel fieldin this disk was so di ffi cult to detect when compared to othergalaxies.Our rotation measure distribution between λ q ) in the halo ofNGC 4631. As the RM distribution presents no type of sym-metric pattern with respect to the major or minor axis, we can-not associate it with a specific parity of the presumably dynamogenerated magnetic field in the disk. Close to the galactic planethe rotation measures are high ( | RM | P r , ě s rad { m ), as one would expect because of the long path-length through theentire galactic disk and the concentration of thermal emission.In the northeastern halo of NGC 4631 the rotation measuresincrease farther upwards, reaching values of up to ´
150 rad { m .Since the thermal electron density decreases with increasing z-values (Hummel et al. 1991), this indicates that the strength ofthe B q in the uppermost region is considerable. Assuming abending of the field by about 40 ˝ in this Northeastern extension we calculated the total and ordered field strength and obtainedvalues of B t « ˘ µ G and B ord « ˘ µ G, while thetotal field strength without a bending is only B t « µ G. Theratio of the ordered to the total field strength is highest here, i.e.,the degree of uniformity of the magnetic field structure along the
Northeastern extension is highest compared to other regions inthe halo (and the disk). In addition, the increase of depolarizationand the decrease of the polarized emission can also be explainedby a bending of the magnetic field lines away from us.The orientation of the halo B K -field in the northeastern quad-rant of NGC 4631 seems to be aligned with the HI-spur locatedin this region (see Sect. 1). Since this is also about parallel tothe X-shaped halo field, it is di ffi cult to distinguish which partof the B K -vectors might be related to the HI-spur and hence todetermine if these features are actually related. This HI spur (la-beled 4 in Rand 1994) is part of an extended HI structure aroundNGC 4631 which is related to tidal interaction mainly betweenthe three galaxies NGC 4631, NGC 4656, and NGC 4627. Ithas been argued already by Hummel et al. (1988) that this tidalinteraction of NGC 4631 is responsible for the large extent ofits radio halo (e.g., Hummel & Dettmar 1990). According toHummel et al. (1988) the magnetic field in the disk may havebeen pulled out of the disk together with the gas leading to strongordered magnetic fields in the halo of NGC 4631. The large or-dered magnetic field strength in the halo and their strong ver-tical components could now be verified by our observations athigher frequency. The smooth RM-pattern even indicates a co-herent magnetic field structure in the halo.Other huge HI tails and bridges have been observed aroundNGC 4631 (labeled 1 to 5 in Fig. 4 of Rand 1994). The inter-action of NGC 4631 with NGC 4656 and NGC 4627 was simu-lated many years ago by Combes (1978) indicating that the ma-terial in features 1 and 4 comes from NGC 4631, whereas that offeature 2 in the southeast and feature 3 in the northwest orig-inate from the now dwarf elliptical galaxy NGC 4627. If so,material may also stream onto NGC 4631. This may explainthe di ff erent scale heights in the di ff erent strips in NGC 4631(see Sect. 3.3) which are indeed smallest in the southeast andnorthwest (s1 and n3 in Table 5). An optical CCD study ofthe vertical structure in NGC 4631 also showed an east-westasymmetry of the thick disk of stellar emission in the north-ern half (Ann et al. 2011). Opposite the radio emission they de-tected the highest values of more than 2 kpc in the northwesternpart of the halo. They interpret this as di ff use stellar emissionand speculate that tidal debris of the satellite galaxies accret-ing to NGC 4631 contribute to this di ff use stellar emission. Thisfits with the simulations by Combes (1978) mentioned above.Furthermore, Rand & van der Hulst (1993) detected two super-shells in HI of 2-3 kpc in diameter of which the largest has beenmodeled to be formed by an HVC impact (Rand & Stone 1996).Even extraplanar cold dust has been found at distances up toabout 10 kpc in NGC 4631 which may be related to this HI su-pershell (Neininger & Dumke 1999).As mentioned in Sect. 3.3 the radio halo in NGC 4631 doesnot show the characteristic dumbbell structure which is typicalof dominating synchrotron losses. Strong star formation in the disk alone cannot be responsible for the deviating vertical scaleheights in NGC 4631, as other galaxies with even higher SFRor starbursts, like NGC 253, show the usual values published byKrause (2012). All this and the fact that the scale heights aresignificantly larger than usual, supports the view that convec-tion may be strong in NGC 4631. Rand et al. (1992) draw thesame conclusion from their H α observations. The H α emissionis very irregular and disturbed with a patchy high-z structure.They could not detect a smooth, extended, di ff use H α halo andsuspected that convection is strong and matter is transported tolarge distances in the halo which is then responsible for the X-ray emission.Tidal interaction or minor mergers may also induce starformation in the disk (Arshakian et al. 2009). In fact, anothergalaxy with strong tidal interaction is the starburst Galaxy M 82,which had already been compared with NGC 4631 several timesin the literature. Recently, Adebahr et al. (2013) was able todetermine scale heights at four di ff erent wavelengths for thisgalaxy. All scale heights were smaller than the usual values anddi ff ered systematically between the northern and southern halvesof M 82. The lower values in the south of M 82 where the IGMmedium is expected to be denser than in the north because ofthe interaction, may lead to higher cosmic ray losses and hencesmaller scale heights there. Hence tidal interaction may influ-ence the vertical scale heights either directly by dragging outmaterial and magnetic field from the disk into the halo or in-directly by inducing a higher star formation in the disk and / ormodifying the loss processes of the cosmic rays in the halo as aresult of intergalactic material produced by the tidal interaction.
5. Conclusions
We merged radio data from the E ff elsberg and the VLA tele-scopes at λ λ – We fit a single power law with the total flux-densities ofNGC 4631 at four di ff erent frequencies and obtain a totalintegrated spectral index of α tot “ ´ . ˘ .
04. By sub-tracting the thermal fraction of the emission from the totalflux-densities, we obtain a nonthermal integrated spectral in-dex of α nth “ ´ . ˘ . – The vertical scale heights in NGC 4631 vary significantlyin di ff erent regions within the galaxy. Their mean valuesat 4.85 GHz are with 2.3 kpc (370 pc) for the thick (thin)disk higher than the mean values found in six other edge-onspiral galaxies studied so far (Krause 2012). There are sev-eral indications that convection may be strong in NGC 4631,which may originally be related to its tidal interaction withits neighbouring galaxies. Because of the tidal interaction,material may even stream onto NGC 4631. Hence, this in-teraction may be responsible for the large spread of the val-ues of the vertical scale heights within NGC 4631 and itslarger mean values compared to those of the other six edge-on galaxies which are not strongly tidally interacting. A sim- ilar result has recently been found for M 82 (Adebahr et al.2013). – The rotation measures between λ – We detected in NGC 4631 a plane-parallel magnetic fieldstructure along its entire disk, similar to all the other edge-ongalaxies observed so far. This plane-parallel magnetic fieldin the disk is the expected edge-on projection of the spiralmagnetic field structure observed in face-on galaxies. – We estimated a total magnetic field strength in the disk ofNGC 4631 of B t « ˘ µ G and an ordered field strengthof B ord « ˘ µ G. The total magnetic field strength in thehalo is comparable to that in the disk. However, the degreeof uniformity in the halo is generally higher than in the disk,especially along the
Northeastern extension .Future research on NGC 4631 will include more sensitiveobservations with higher angular resolution in a wider wave-length range taken with the Extended Very Large Array (EVLAor Karl G. Jansky VLA). This will allow a point by point spec-tral index analysis and a quantitative analysis of cosmic ray lossprocesses in NGC 4631. In addition, rotation measure synthe-sis (Brentjens & de Bruyn 2005) will be applied to these multi-channel polarization observations to separate the contributionsfrom the disk and halo in this galaxy.
Acknowledgements.
We thank Krzysztof Chyzy and Michael Dumke for theirhelp with the data reduction. We appreciate the helpful comments from RainerBeck and the anonymous referee.
References