Calibrating the relation of low-frequency radio continuum to star formation rate at 1 kpc scale with LOFAR

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A & A 622, A8 (2019)

https://doi.org/10.1051/0004-6361/201833905

LOFAR Surveys: a new window on the Universe

Calibrating the relation of low-frequency radio continuum to star formation rate at 1 kpc scale with LOFAR*

V. Heesen 1, E. Buie II2, C. J. Huff2, L. A. Perez2, J. G. Woolsey2, D. A. Rafferty 1, A. Basu3, R. Beck4, E. Brinks5, C. Horellou6, E. Scannapieco2, M. Bruggen1, R.-J. Dettmar7, K. Sendlinger7, B. Nikiel-Wroczynski8, K. T. Chyży8,

P. N. Best9, G. H. Heald10, and R. Paladino 11

1 U n iv e rs ity o f H am b u rg , H am b u rg er S ternw arte, G ojenbergsw eg 112, 2 1 0 2 9 H am bu rg , G erm an y e-m ail: v o lk e r .h e e s e n @ h s .u n i- h a m b u r g .d e

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4 M a x -P la n c k -In s titu te fu r R ad ioastrono m ie, A u f dem H u g e l 69 , 5 3 1 2 1 B on n, G erm an y

5 S chool o f Physics, A s tro n o m y and M a th e m atics, U n iv e rs ity o f H ertfo rd sh ire, H a tfie ld A L 1 0 9 A B , U K

6 C halm ers U n iv e rs ity o f T echnology, D ep t. o f Space, E arth and E nviro nm ent, O n sala Space O bservatory, 4 3 9 9 2 O nsala, Sw eden 7 A stronom isches In s titu t der R u h r-U n iv e rs ita t B ochum , 4 4 7 8 0 B ochum , G erm an y

8 A stro n o m ica l O bservatory, Jagiellonian U n iv e rs ity , u l. O r la 171, 3 0 -2 4 4 K ra k ó w , P oland 9 S U P A , In stitu te fo r A stron om y, R o y a l O b servatory, B la c k fo rd H ill, E din burgh E H 9 3 H J , U K 10 C S IR O A stro n o m y and Space Science, P O B o x 1130, B en tley , W A 6 1 0 2 , A u s tra lia

11 IN A F /Is titu to d i R ad ioastrono m ia, v ia G o b etti 101, 4 0 1 2 9 B olo g n a, Ita ly

R eceived 19 July 2 0 1 8 / A ccepted 15 O ctob er 2 0 1 8

ABSTRACT

C o n te x t. R ad io co ntinuu m (R C ) em ission in galaxies allow s us to m easure star fo rm a tio n rates (S F R s ) unaffected b y extinctio n due to dust, o f w h ic h the lo w -freq u en cy p a rt is uncontam inated fro m th erm al (fre e -fr e e ) em ission.

A im s . W e calibrate the conversion fro m the sp atially resolved 140 M H z R C em ission to the S F R surface density ( £ SFR) at 1 k p c scale.

R ad io spectral indices give us, b y m eans o f spectral ageing, a handle on the transport o f cosm ic rays using the electrons as a p ro x y fo r G e V nuclei.

M e th o d s . W e used recen t observations o f three galaxies (N G C 3 1 8 4 , 4 7 3 6 , and 5 0 5 5 ) fro m the L O F A R T w o -m e tre S ky S urvey (L o T S S ), and arch ival L O w -F re q u e n c y A R ra y (L O F A R ) data o f N G C 5 1 9 4 . M a p s w ere created w ith the facet ca lib ratio n technique and converted to radio E SFR m aps using the C ondon relation . W e com pared these m aps w ith h y b rid E SFR m aps fro m a co m b ination o f G A L E X fa r-u ltra v io le t and S p itz e r 2 4 p m data using plots tracing the relatio n a t the highest an gular resolution a llo w e d b y our data at 1.2 x 1.2 kp c2 resolution.

R e s u l t s . T h e R C em ission is sm oothed w ith respect to the h y b rid E SFR ow in g to the transport o f co sm ic-ray electrons (C R E s ) aw a y fro m star fo rm atio n sites. T h is results in a sublinear relatio n ( £ SFR)RC x [(E SFR)hyb]a, w h ere a = 0 .5 9 ± 0 .1 3 (1 4 0 M H z ) and a = 0 .7 5 ± 0 .1 0 (1 3 6 5 M H z ) . B o th relations have a scatter o f a = 0 .3 dex. I f w e restrict ourselves to areas o f young C R E s ( a > - 0 .6 5 ; Iv x va ), the re la tio n becom es alm ost lin e a r a t both frequencies w ith a « 0 .9 and a reduced scatter o f a = 0 .2 dex. W e then sim ulate the effect o f C R E transport by co nvo lvin g the hy b rid E SFR m aps w ith a G aussian k e rn e l u n til the R C - S F R relatio n is linearised; C R E transport lengths are I = 1 - 5 kp c. S olving the C R E diffu sion equation, assum ing dom inance o f the synchrotron and in verse-C o m p to n losses, w e find diffu sion coefficients o f D = ( 0 .1 3 - 1 .5 ) x 1028 cm 2 s-1 at 1 G eV .

C o n c lu s io n s . A R C - S F R relatio n at 1.4 G H z can be exp lo ited to m easure S FR s at red s h ift z ~ 10 using 140 M H z observations.

K ey w ords. rad ia tio n m echanism s: n o n -th e rm al - cosm ic rays - galaxies: m ag n etic fields - galaxies: star fo rm a tio n - radio continuum : galaxies

1. Introduction

R adio continuum (RC) em ission in galaxies em erges from tw o distinct processes: therm al fre e -fre e (brem sstrahlung) and non- therm al (synchrotron) radiation. B oth are related to the fo rm a

tion o f m assive stars. U ltraviolet (U V ) radiation from m assive stars ionises the interstellar m edium (ISM ), w hich gives rise to therm al brem sstrahlung em ission. Studies o f the origin o f n o n

* R ad io co ntinuu m flu x densities and fits files are o n ly a v a il

able at the C D S v ia anonym ous ftp to c d s a r c . u - s t r a s b g . f r ( 1 3 8 . 7 9 . 1 2 8 . 5 ) or v ia h t t p : / / c d s a r c . u - s t r a s b g . f r / v i z - b i n / q c a t ? J /A + A /6 2 2 /A 8

therm al R C em ission have found that supernova shocks from core collapse supernovae accelerate protons, heavier nuclei, and electrons; n on-therm al RC em ission dom inates at frequencies below 15 G H z; at higher frequencies, such as a few 10 GHz, fre e -fre e em ission dom inates w ith a possible contribution from spinning dust (S c a if e e ta l. 2010) . T he highly energetic elec

trons, know n as cosm ic-ray electrons (C R Es), spiral around interstellar m agnetic field lines, thereby em itting highly linearly p olarised synchrotron em ission. T he relation betw een the RC lum inosity o f a galaxy and its star form ation rate (SFR), h en ce

forth referred to as th e R C -S F R relation, is due to the interplay o f star form ation, CR Es, and m agnetic fields.

Astronomy

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Table 1. Properties of the sample galaxies.

A&A 622, A8 (2019)

G alaxy i d 25 D Mb Type N ucleus s f r r* log10(^SFR) log10(Mtot) vrot

(°) (arcm in) (M pc) (m ag) (M 0 y r- 1) (kpc) (M 0 y r-1 k p c-2 ) (M 0 ) ( k m s -1)

( 1) (2) (3) (4) (5) (6) (7) (8) (9) ( 10) ( 11) ( 12)

N G C 3 1 8 4 16 7.41 11.1 -1 9 .9 2 SBc H n 0.90 10.7 - 2 .6 2 11.09 210

N G C 4736 41 7.76 4.7 - 1 9 .8 0 Sab L IN E R 0.48 4.1 - 2 .0 7 10. 48 156

N G C 5055 59 11.75 10.1 - 21.12 Sbc T2 2.12 16.5 -2 .6 3 11.17 192

N G C 5 1 9 4 20 7.76 8.0 - 2 1 .0 4 SBc Hi i 3.13 15.1 - 2 .3 9 11.00 219

Notes. Data are from Walter etal. (2008) and Leroy etal. (2008) unless otherwise noted. Columns: (1) galaxy name; (2) inclination angle;

(3) optical diameter measured at the 25 mag arcsec-2 isophote in B-band; (4) distance; (5) absolute B-band magnitude; (6) galaxy morpholog

ical classification; (7) optical classification of the nuclear spectrum, from Ho et al. (1997) where Sy = Seyfert and T = transition object between Hii nuclei and LINERs; (8) SFR based on the hybrid FUV+24yum conversion that has an uncertainty of 0.3 dex; (9) radius of the actively star forming disc (within the last »100 Myr), estimated from the radial extent of the RC emission; (10) SFR surface density is ESfr = SFR/(nr*) with an uncertainty of 0.3 dex; (11) total mass Mtot = 0.233 x 1010r25v100 M0, where r25 is the radius of the galaxy estimated from d25 using the distance D, and v100 is the rotation speed in units of 100 km s-1, with an uncertainty of 0.1 dex; (12) maximum rotation speeds in the flat part of the rotation curve with typical uncertainties of ±3 km s-1.

T he global, integrated R C -S F R relation is very tight, as H e e s e n e ta l. (2014) have shown: using th e relation o f C ondon ( 1992) an d converting 1.4 G H z radio lum inosities into radio derived SFRs, these authors found agreem ent w ithin 50% w ith state-of-the-art star form ation tracers, such as far-ultraviolet (FU V ), H a , a n d m id - o r far-infrared (M IR ; F IR ) em ission. A n even tighter agreem ent can b e achieved if the radio spectrum is inte

grated over a w ide frequency ran g e (“b o lom etric rad io lum in o s

ity” ; T abatabaei et al. 2017) . M oreover, these authors found that the rad io lum inosity is a n on-linear function of the SFR , as p re dicted b y the m odel o f N iklas & B eck ( 1997) ; see Sect. 5.1 for details.

T hese findings highlight th at once properly calibrated, R C can be used as an unobscured star-form ation tracer in dusty, high- red sh ift galaxies (B esw ick e t al. 2015) if the relation also holds at low frequencies. This is the case if m agnetic fields at high red- shifts are sufficiently strong to ensure th at synchrotron losses o f CR Es dom inate over inverse-C om pton (IC) losses against the c o s

m ic m icrow ave b ackground (S chleicher & B eck 2013) . Studies o f the low -frequency R C -S F R relation have now becom e p o s

sible w ith the advent o f th e L O w -F requency A R ray (LOFA R;

van H aarlem et al. 2013) . G urkan e t al. (2018) found th at th e in te grated R C -S F R relation in the H-A TLAS field requires a b ro ken pow er law to b e described accurately. If fitted w ith a sin gle pow er law, th e relation betw een the 150 M H z R C lum in o s

ity and the S FR is L 150 ^ S F R 107, w hich is slightly super-linear.

C alistro R ivera et al. (2017) studied the red sh ift evolution o f the 150 M H z R C lum inosity as function o f F IR lum inosity (R C -F IR relation). T hey show ed the red sh ift evolution to b e a potentially im portant factor w hen calibrating the usefulness o f radio as a star form ation tracer. ChyZy et al. (2018) found th at the 150 M H z R C - F IR relation also holds in galaxies o f the local U niverse an d has a sim ilar scatter as for th e 1.4 G H z relation.

L ow -frequency RC spectra o f galaxies can b e m odified by additional m echanism s in com parison to G H z frequencies. F re e - free absorption by therm al electrons can cause Hii regions to becom e optically thick at low frequencies, leading to a spec

tral turnover (H e e s e n e ta l. 2018b). R elativistic brem sstrahlung and especially ionisation losses can also b e m uch m o re im por

tant, p articularly in starburst galaxies (M urphy 2009) . F u rth er

m ore, synchrotron self-absorption and R azin effects can fur

ther suppress th e radio em ission from dense ISM regions and produce spectral turnovers below 10 M H z (L acki 2013) . Star- burst galaxies, such as M 82, becom e optically thick a t 150 M H z

w hen using spatially resolved observations (C hyzy et al. 2018) . Since n uclear starbursts are sim ilar, alb eit on a sm aller scale, a spatially resolved study allow s us to separate these effects and the contribution from active galactic n uclei (A G N s) from our results.

C alibrating the R C -S F R relatio n requires a thorough u nder

standing o f the physical foundation th at gives rise to the relation in the first place. M easuring the n on-linearity of the sy n c h ro tro n -S F R relation and its p o tential dependence on galaxy type is crucial. L ow -frequency observations are p articu larly useful b ecause they allow us to study the dom inating syn

chrotron em ission in galaxies, w hich is largely free from the contribution of therm al em ission. Furtherm ore, low -frequency observations are ideally suited for large-area surveys, p ro v id ing us w ith statistically m eaningful sam ples. This w ork exploits the recently im proved im aging capabilities of LO FA R to build up such a sam ple and study the physics b ehind the R C -S F R relation. W ith spatially resolved observations w e can explore w hether w e see a flattening o f the R C -S F R relation in areas o f concentrated star form ation. Such a flattening w ould be hinting at a depression of R C intensities such as expected for fre e -fre e absorption.

F re e -fre e absorption is one o f the largest caveats in using low -frequency R C observations as a star form ation tracer, p artic

ularly in starburst galaxies. A debahr et al. (2013) found a spec

tral turnover for the integrated spectrum o f M 82 at 300 M H z, and even higher, at 600 M H z, fo r the starburst nucleus. Similarly, K apinska et al. (2017) found the 500 p c nuclear starburst region in N G C 253 b est d escribed b y an internally fre e -fre e absorbed synchrotron spectrum w ith a turnover frequency of 230 M H z.

Clearly, these tw o galaxies rep resen t th e m ore extrem e cases in the local U niverse and their starburst n uclei have SFR surface densities w ell in excess o f 1 M 0 y r-1 k p c-2 , hence are factor o f 10 o r higher than w hat is usually referred to as starburst galaxies.

L ow -frequency R C observations are p articularly appealing to study star form ation history across cosm ic tim es. A t redshift z ~ 10, the rest-fram e 1.4 G H z RC em ission can b e detected as 140 M H z em ission. S ince m o st deep large RC surveys w ith the n ex t generation o f radio telescopes w ill b e p erform ed at fre

quencies o f 1 -2 G H z, th e 1.4 G H z R C -S F R relation w ill be fur

ther established. This is for instance th e case for plan n ed sur

veys w ith th e Square K ilom etre A rray (SK A ) and its precursors, such as A ustralian Square K ilom etre A rray Pathfinder (A SK A P) E volutionary M ap o f th e U niverse (EM U ; N orris e t a l. 2011)

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V. Heesen et al.: Low-frequency radio to SFR relation at 1 kpc scale

and M eerK A T International G H z T iered E x tragalactic E x p lo ration (M IG H T E E; Jarvis & Taylor 2016) . Sim ilar surveys are p lanned w ith the n ex t generation Very L arge A rray (ngV LA ), the upgraded G iant M etrew ave T elescope (uGM RT), and M u lti

E lem ent R adio L inked Interferom eter N etw o rk (e-M ER LIN ).

This p aper is an exploratory study o f th e low -frequency R C -S F R relation in four nearby galaxies, three o f w hich (N G C 3 1 8 4 , 4736, an d 5055) w ere observed as p art o f the L O FA R Tw o-m etre Sky S urvey (LoTSS; S him w ell e t al. 2017) . T he L oT SS is a deep 120-168 M H z im aging survey th at w ill eventually cover the entire northern sky an d reach an rm s n oise o f 100 p J y b e a m -1 a t an angular resolution o f 5 arcsec.

In addition, w e added one galaxy that was observed p rev i

ously (N G C 5 1 9 4 (M 5 1 ); M u lc a h y e ta l. 2014), for w hich w e reduced the data using the latest strategy. T he galaxies w ere ch o sen from th e SIRTF N earby G alaxies Survey sam ple (SIN G S;

K ennicutt et al. 2003) , for w hich 1365 M H z m aps from the W esterbork Synthesis R adio T elescope exist (W SR T -SIN G S survey; B raun e t a l. 20 0 7 ). As in H eesen e ta l. (2014), w e use the com bined G A L E X 156 n m far-U V and S pitzer 24 p m M IR m aps from L eroy e t al. (2008) as our reference SFR surface d en sity m aps, in the follow ing designated as hybrid 2 SFR m aps. The reasoning b ehind this choice is that the F U V d ata trace O and B stars, so they can b e u sed as a star form ation tracer as long as the obscuration b y dust can be co rrected for. See Table 1 for a sum m ary o f th e properties o f our sam ple galaxies.

This p ap e r is organised as follow s. In Sect. 2 , w e present o ur observation strategy and data reduction techniques, including a com parison o f ou r new m ap o f N G C 5194 w ith th e previously p ublished m ap. Section 3 describes o ur results for the R C -S F R relation; subsections are devoted to th e m orphology (Sect. 3.1) , dependency on the rad io spectral index (Sect. 3.2), an d spatially resolved relation (Sect. 3.3) . In Sect. 4 , w e study th e transport o f cosm ic rays, conducting a sm oothing experim ent in Sect. 4.1 and applying a diffusion m odel in Sect. 4 .2 . W e discuss ou r results in Sect. 5, befo re w e conclude in Sect. 6 . In th e m ain p art o f the paper, w e present th e m aps o f N G C 5194 w ith th e rem aining galaxies p resented in A ppendix A .

2. Observations and data reduction

O ur new H igh B and A ntenna (HBA ) observations w ith LO FA R w ere taken w ith the LoTSS observing strategy (frequency and calibrator) set-up, observing th e L oT SS pointings, w hich w ere closest to our targets. O ur targets are w ithin 2° from the p o in t

ing centres, thus having a p rim ary beam attenuation o f less than 25% . In brief, w e u sed the H B A -dual inner m o d e to conduct 8 h observations; the 48 M H z b andw idth (1 2 0 -1 6 8 M H z) w as split equally over tw o target pointings, b ookended b y 1 0 m in flux- calibrator scans (i.e. 8.3 h scans); 50% o f the tim e o f each scan w as spent on ou r targets. W e stored the d ata at 16 channels per sub-band (12.2 kH z frequency resolution) and at 1 s tim e reso lution. T he archival observation o f N G C 5 194 was carried out slightly differently. T he bandw idth o f 48 M H z was equally d is

tributed betw een 116 and 176 M H z, w ith one pointing on the target and one pointing on the flux calibrator. See Table 2 fo r a jo urnal o f the observations.

T he d ata w ere red u ced w ith th e facet calibration technique, w hich m itigates the direction-dependent effects o f th e io n o sphere and b eam response that im pact low -frequency RC obser

vations w ith aperture arrays, such that im ages close to the therm al n oise level could b e obtained (van W eeren et al. 2 0 1 6 ; W illiam s e t al. 2016) . F irst, th e (u, v) data are calibrated w ith direction-independent m ethods using the PREFACTOR pipeline

Table 2. Journal of the observations.

N G C 3184

O bservation ID L 369724

O bservation date 2015 A ug 15

P roject L C 4 037

L oT SS pointing P 153+ 42

D istance to pointing centre 1=.1

Stations 60 (46 CS an d 14 RS)

P rim ary calibrator 3C 1 9 6 (L369720) N G C 4736 (observation 1)

O bservation date 2015 M ay 13/14

P roject LC 3_008

D istance to pointing centre 1°3

P rim ary calibrator 3C 1 9 6 (L343250) N G C 4736 (observation 2)

O bservation date 2015 Ju l 16

P roject L C 4_034

P rim ary calibrator 3C 1 9 6 (L350662) N G C 5055

O bservation date 2015 M a r 19/20

P roject LC 3_008

P rim ary calibrator 3C 1 9 6 (L280978) N G C 5194 (M 51)

O bservation date 2013 A pr 22/23

P roject LC 0_043

L oT SS pointing N /A

D istance to pointing centre 0 °

P rim ary calibrator 3C 2 9 5 (L127444) Notes. CS = core station; RS = remote station.

(de G asperin e t al. 2019) 1. This pip elin e first calibrates 3C 48 using the S caife & H eald (2012) flux densities, assum ing a p oint-like source. F rom th e resulting gain solutions the in stru m ental com ponents are extracted, nam ely the station gain am p li

tudes and the p hase variations due to the drift o f the clocks o f the L O FA R stations. T he latter are separated from the vari

ations due to the changing total electron content (TEC ) o f the ionosphere w ith the c lo ck -T E C separation. O nce deter

m ined, the instrum ental calibration solutions are applied to the target data, w hich are then averaged to 10 s tim e resolution and tw o channels p er sub-band frequency resolution (chan

n el w idth o f 9 7 .6 5 6 k H z ). T he d ata are calibrated in phase only using the G lobal Sky M odel (G SM ; Scheers 2011), w hich is a com pilation o f sources from th e V L A L ow -frequency Sky Survey R edux (V LSSr; L ane e ta l. 2014), the W esterbork N orthern Sky Survey (W E N SS; R engelink et al. 1997) , and the N R A O V L A Sky Survey (N V SS; C ondon e t al. 1998) . W ith the 1 h t t p s : / / g i t h u b . c o m / l o f a r - a s t r o n / p r e f a c t o r

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A&A 622, A8 (2019)

Table 3. Radio properties of sample galaxies.

G alaxy FW HM v 1 v2 ^1 S 1 S 2 a Int. area PA

(arcsec) (M Hz) (u Jy beam 1) (Jy) (arcm in2) (°)

( 1) (2) (3) (4) (5) (6) (7) (8) (9) ( 10) ( 11)

N G C 3 1 8 4 18.6 142 1365 130 35 0.389 ± 0.012 0.087 ± 0.002 - 0.66 ± 0.02 3.3 x 3.3 179 N G C 4 7 3 6 19.1 140 1365 200 42 0.800 ± 0.024 0.301 ± 0.006 - 0 .4 3 ± 0.02 3.0 x 2.5 296 N G C 5055 18.6 144 1365 110 36 2.082 ± 0.063 0.416 ± 0.008 - 0 .7 2 ± 0.02 5.6 x 3.7 102 N G C 5 1 9 4 17.1 145 1365 130 32 6.922 ± 0.208 1.402 ± 0.028 -0 .7 1 ± 0.02 6.5 x 5.8 172 Notes. Columns: (1) galaxy name; (2) angular resolution, referred to as the full width at half maximum (FWHM) of the circular synthesized beam;

(3,4) observing frequencies v1 and v2; (5,6) rms map noises ^ 1 and ^ 2, at v1 and v2, respectively; (7,8) integrated flux densities, S 1 and S 2, at v1 and v2, respectively; (9) integrated radio spectral index between v1 and v2; (10) major and minor axes dimensions of the elliptical integration area;

(11) position angle of the galaxy’s major axis from Walter et al. (2008).

Fig. 1. NGC5194. Panel a: RC emission at 151 MHz, as derived from LOFAR HBA observations (Mulcahy etal. 2014). The map has been calibrated with direction-independent phase calibration only. Panel b: RC emission at 145 MHz, obtained from the same data set as panel a but calibrated with direction-dependent phase and amplitude calibration. Both maps are presented with a logarithmic transfer function and have an angular resolution of 20 x 20 arcsec2, as indicated by the circle in the bottom left corner.

direction-independent calibration applied, the (u, v) data are inverted and deconvolved w ith a w ide-field CLEAN algorithm . A s a final step o f PREFACTOR, the CLEAN com ponents o f all the sources w ithin the 8° field o f view (FOV) are subtracted from the (u, v) data.

The residual, direction-independent calibrated (u, v) data w ith all sources subtracted, together w ith the subtracted m odel and solutions o f the phase-only calibration, are then used as the input for the direction-dependent facet calibration, for w hich w e used the FACTOR pipeline (Rafferty et al., in prep .)2. The FOV w as divided into approxim ately 20 facets around calibra

tor regions w ith integrated 167 M H z flux densities (of the full facet) in excess o f 0.3 Jy. O f those, facets in excess o f 0.8 Jy w ere processed one at a tim e, beginning w ith the brightest facet. The facet calibration technique allow s us to track and correct for the direction-dependent effects o f the ionosphere o f the E arth (effec

tively the “ seeing” at long radio w avelengths) and the station

h t t p s : / / g i t h u b . c o m / l o f a r - a s t r o n / f a c t o r

beam response by first self-calibrating on the calibrator region o f a facet and then using the solutions to update the m odel for the full facet, w hich in turn is used to update the residual (u,v) data. In the first step o f the calibration, fast, 10 s phase solutions are determ ined in sm all chunks o f » 2 M H z bandw idth to cor

rec t for the positional change and distortion o f sources. In the second step, slow, tens-of-m inutes am plitude solutions are used to track the variation o f the apparent flux density o f a source.

The target facets w ere corrected using the solution o f a nearby facet.

The direction-dependent calibrated (u, v) data w ere im ported into the C om m on A stronom y S oftw are A pplications (CASA;

M cM ullin et al. 2007) and inverted and deconvolved w ith the M S -M F S CLEAN algorithm (R au & C ornw ell 2011) . W e fitted for the frequency dependence o f the skym odel (nterm s = 2) and used angular scales o f up to the size o f the galaxy processed.

W e used B riggs w eighting, setting the robust param eter betw een 0.2 and 0.5 and the value w as adjusted to m atch the angular resolution o f the W SR T m aps. T his resulted in m aps w ith an 2

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effective central frequency ranging from 140 to 145 M H z w ith angular resolutions betw een 17 and 19 arcsec3. In the follow ing w e refer to these d ata as th e 140 M H z L O FA R data; see Table 3 for the m ap properties. W e fo u n d rm s n o ise levels betw een 110 and 200 p J y b ea m -1 , w hich is in approxim ate agreem ent w ith the expected sensitivity for our observations.

W e integrated flux densities in our m aps w ithin ellipses encom passing the 3 ^ contour lines. W e checked the flux d en sities and found them in good agreem ent w ith the 7C catalogue (w ithin 10%), except for N G C 3184 for w hich there is no entry.

H ence, w e did n o t apply any correction for th e w ell-know n sta

tion calibration b eam error (H ardcastle e t al. 2016) .

F or N G C 5194, w e have a m ap that w as p rocessed w ith direction-independent calibration b y M ulcahy e t al. (2014) . It is instructive to see the im provem ent th at com es from the direction- dependent calibration technique using the facet calibration technique. In Fig. 1, w e p rese n t the com parison o f the tw o m aps, w hich have been convolved to 20 x 2 0 a rc se c 2 reso lu tion. T he direction-independent m ap has an rm s n o ise o f 300

4 00 p J y b ea m -1 in the area surrounding the galaxy, w hereas the new m ap has a n o ise o f 1 8 0 -2 0 0 p J y b eam -1 , hence the im prove

m ent is alm ost a factor o f tw o. T he advantage o f the new m ap is also that the distribution o f the n o ise is m uch m ore uniform across th e m ap. F urtherm ore, sidelobes surrounding the b righter unresolved sources are significantly im proved. W e checked by fitting G aussians to these sources such th at the new m ap has a slightly im proved resolution by abo u t 1 -2 a rc se c , com pared to the old m ap. H ence, this com parison show s that im prove

m ent using the direction-dependent calibration is significant even w hen the ionosphere is fairly q uiescent as has been th e case for this observation.

3. RC-SFR relation 3.1. M orphology

R ather than w orking in observed flux density units, w e m ake the assum ption that the RC em ission is entirely due to recent star form ation. This was explored first b y C ondon ( 1992) , w ho, b ased on som e sim ple assum ptions, p redicted the follow ing relation (see H eesen e t al. 20 1 4 , for details):

This m ay change below 1 G H z, w here th e rad io spectral index o f galaxies m ay flatten to approxim ately - 0 .6 (C hyzy e t al. 2018), but in this p ap e r w e assum e a sim ple p ow er law w ithout this fur

ther com plication. Second, w e can convert C o n d o n ’s relation to a spatially resolved relation, thereby relating the SFR surface d en sity w ith the R C intensity rath er than lum inosity. U sing Eq. (3) in H eesen et al. (2014) , w e find

(2⁾

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C o n d o n ’s relation assum es a S alpeter initial m ass function (IM F) to extrapolate from the m assive stars (M > 5 M 0 ) that show up in the R C to th at o f all stars form ed (0.1 < M /M 0 < 100). The hybrid 2 SFR m aps o f L eroy et al. (2008) , w hich w e em ploy as our reference, are b ased on a broken pow er-law IM F as described in C alzetti et al. (2007), so th eir derived SFRs are a factor o f 1.59 low er than using the S alpeter IMF. W e have thus scaled C o n d o n ’s relation in this p ap e r accordingly.

C o n d o n ’s relation is close to th e resu lt o f the derivation o f M urphy e t al. (2011), w ho used the em pirical R C -F IR relation to derive SFRs th at are 15% lower. C o n d o n ’s relation can b e g en eralised in tw o w ays. F irst it can be scaled to any o ther frequency w hen one assum es a constant rad io spectral index. W e use a radio spectral index o f a = - 0 .8 (Iv k va ), w hich is the total (i.e. including non-therm al an d therm al em ission) rad io spectral index o f galaxies at G H z frequencies (T abatabaei et al. 2017) .

3 A n g u la r resolutions in this paper are referred to as the fu ll w id th at h a lf m a x im u m ( F W H M ) .

This quantity is defined in th e p lan e o f th e sky so that w e do n o t have to deal w ith p rojection effects. A lternatively, w e w ould have to correct the radio m aps to face-on and do the sam e cor

rection for the SFR m aps. T he correction factor is fairly sm all (0.5 < cos(i) < 1.0) for o ur sam ple and does n o t m ake m uch difference in th e analysis anyway, w hich is p erform ed in lo g -lo g plots. T here is also the subtle effect th at the b eam is elongated in the plane o f the galaxy along the m in o r axis. This is n o t corrected for w hen w e use rectangular regions, such as w e do, as they sam ple a larger dim ension along th e m inor axis. This w ould have to b e co rrected for by choosing regions w ith a sm aller dim ension along th e m in o r axis. W e chose to not do this. F irst, because the angular resolution o f ou r m aps is n o t sufficient to allow for it; second, H eesen et al. (2014) show ed th at results are n o t very sensitive to spatial resolution, com paring in that w o rk 0.7 and 1.2 kpc spatial resolutions.

T he resulting radio 2 SFR m aps both o f LO FA R and W SR T for N G C 5194 are p resen ted in F igs. 2 a and b, respectively. F or com parison, w e show in F ig. 2 c the h y brid 2 SFR-m ap. Clearly, the L O FA R m ap extends further than b oth the W S R T an d hybrid 2sfr m aps, p articularly along the m inor axis (m ajor axis position angle is PA = 172°). T he galaxy is inclined at i = 20°, such that w e m ay see a rad io halo in projection. This is at least suggested by th e m orphology, as the radio em ission does not extend very prom inently along the m ajo r axis.

W e n otice a nu m b er o f unresolved sources. M ost o f these sources are easily identified as b ackground rad io galaxies since they have no counterpart in the hybrid 2 SFR m ap. T here are a nu m b er o f com pact sources in th e hybrid 2 SFR m ap as w ell, but these are located in the spiral arm s; hence, w e assum e that they are m assive star form ation regions. W e m ask ed u n related b ac k ground sources and applied the m ask to all 2 SFR m aps. F urther

m ore, w e excluded com pact n uclear sources since they m ay be dom inated by A G Ns. W e stress th at these m ask ed sources co n tribute only a sm all am ount o f flux to th e radio m aps (at m ost

15%), therefore they are n o t influencing our results in a signifi

cant way. In Fig. 2d , w e show the ratio o f the L O FA R rad io to hybrid 2 SFR m ap w ith the m ask applied. T he m ap looks very sim ilar to the h ybrid 2 SFR m ap w ith the spiral arm s clearly visible.

T he heat colour scale is inverted, such that the m inim um ratio is found in th e spiral arm s and th e m axim um ratio is found in the outskirts o f the galaxy; th e inter-arm regions show interm ediate values for the ratio.

T he LO FA R 2 SFR m ap show s less variation betw een the sp i

ral arm s, the inter-arm regions, and th e outskirts o f the galaxy than the h ybrid 2 SFR m ap. T he radio m ap is in a w ay a sm oothed version o f the h ybrid m ap. This is usually ascribed to the d if

fusion o f th e CR Es aw ay from star form ation regions over their lifetim e (M urphy e t al. 2 0 0 8 ; H eesen et al. 2014) . T he diffusion length is expected to b e frequency dependent b ecause at low er frequencies C R Es have low er energies and longer lifetim es,

CsFR)gC 2 = 3.31 X 103 ( - ^ - )“

M 0 y r-1 kpc 2 V1 -4 G H z i Z F W H M \-2 Iv

\ arcsec / J y b e a m -1

^ F R r c 1 = 0.75 X 10 -21 ( ) ■

M 0 y r-1 'W H z - 1 '

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Fig. 2. NGC 5194. Panel a : RC emission at 145 MHz, as derived from LOFAR HBA observations. The intensities were converted into a map of the radio SFR surface density, ( I SFR)RC, using the 1.4 GHz relation of Condon scaled with a radio spectral index of -0.8. This map is shown at a logarithmic stretch ranging from 10-4 to 3 x 10-1 M0 yr-1 kpc-2. Panel b: same as panel a, but using a 1365 MHz map from WSRT-SINGS.

Panel c: hybrid SFR surface density map, ( I SFR)hyb, derived from a linear superposition of GALEX 156 nm FUV and Spitzer 24pm MIR emission, presented as inverted heat colour scale. Panel d: ratio, %, of the LOFAR ( I SFR)RC map divided by the hybrid ( I SFR)hyb map. The map is shown at logarithmic stretch using the heat colour scale, ranging from 10-11 to 1016. Areas that are light are radio bright, whereas dark areas are radio dim when compared with the hybrid I SFR-map. All maps have been convolved to a circular Gaussian beam with a resolution of 17.1 x 17.1 arcsec2.

The representation of the beam is shown in the bottom left corner of each panel. Panels a-c: unmasked maps, whereas panel d shows the area after masking background sources and the AGN-contaminated central area. In all panels, a 3<x cut-off has been applied.

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Fig. 3. NGC 5194. Panel a: radio spectral index distribution between 145 and 1365 MHz, presented in a cube-helix colour scale ranging from -1.2 to -0.4. Dashed contours are at -0.85 and -0.65, thus separating the galaxy in three zones. Areas with young CREs (a > -0.65) are predominantly found in spiral arms; areas with CREs of intermediate age (-0.85 < a < -0.65) are predominantly found in inter-arm regions; and areas with old CREs (a < -0.85) are found in the galaxy outskirts. Panel b: error of the radio spectral index distribution between 145 and 1365 MHz at logarithmic stretch, ranging from 0 to 0.2. As can be seen, the spectral index error only becomes larger than ±0.1 in areas with a < -0.85. In both panels, the maps were convolved to a circular synthesised beam of 17.1 x 17.1 arcsec2 resolution, which is outlined in the bottom left corner.

A mask has been applied to background sources and the central regions of NGC 5194 and its companion galaxy, NGC 5195. A 3<x cut-off was applied to both the 145 and 1365 MHz maps prior to combination.

assum ing th at synchrotron and IC radiation losses dom inate as they do outside o f the dense, gaseous spiral arm s (B asu et al.

2015). F igures 2 a and b hin t that this is indeed the case w ith the contrast o f the L O FA R m ap being even low er than that o f the W SR T m ap.

B efore w e conclude this section, w e briefly discuss the find

ings for the other three sam ple galaxies, the m aps o f w hich can be found in A ppendix A . N G C 3184 (Figs. A.1 and A .2) is only little inclined, and thus nearly in a face-on position. T he spiral arm s in the hybrid XS fr m ap has faint counterparts in the radio m aps. T he other tw o galaxies are m oderately inclined, N G C 4736 (Figs. A .3 and A .4) and 5055 (Figs. A.5 and A .6) , w ith i = 41° and i = 59°, respectively. A s in N G C 5194, w e found that the radio XSfr m aps extend further along the m inor axis than the hybrid m aps, m ore so than expected from thin inclined discs, suggesting the existence o f radio haloes. A ll three galax

ies show the sam e behaviour, nam ely that the radio em ission is a sm oothed version o f the hybrid XS fr m ap, the LO FA R m aps even m ore so than the W SR T m aps.

N otable features are a large (1 0 arcm in ) radio galaxy south

w est o f N G C 4736 (Figs. A .3 a and b), w here the northern lobe overlaps slightly w ith the em ission o f the galaxy. T he hybrid XSfr m ap (Fig. A .3c) shows a filam entary extension to the w est, o f w hich w e detect no counterpart in the radio. T his em ission is spatially coincident w ith a spiral arm visible in H i em is

sion (W alter et al. 2008) , and thus m ay be a tidal tail caused by p ast interaction. T his feature connects to a second outer ring visible in Hi, w hich m ay be caused by the L indblad re so nance (S chom m er & Sullivan 2016) . In nGc 5055, the W SRT

m ap (Fig. A .5b) show s tw o extensions south-east and north

w est o f the m ain body. W e find no counterpart in the LO FA R m ap (Fig. A .5 a) and neither is there one in either the hybrid XS fr m ap (Fig. A .5c) or in a m ap o f Hi em ission. T his galaxy has an extended, w arped Hi disc (B attaglia et al. 2006) and also an extended M 83-like F U V disc (T hilker et al. 2007) . However, their m orphology is different, w hich has m axim a north-east and south-w est o f the bright, inner disc, w hereas the W SR T exten

sions lie on the p erpendicular axis. H ence, w e conclude that this em ission m ay be an artefact o f the data reduction and w e exclude this part o f the galaxy from further analysis.

3.2. R adio sp e c tra l in dex

In Fig. 3 , w e p resent the 145-1365 M H z radio spectral index distribution in N G C 5194 w ith the corresponding error m ap; this is the total spectral index since w e do not correct for therm al em ission. T he m ap is highly structured, w here the galaxy can b e roughly divided into three areas: (i) the spiral arm s, w here a > - 0 .6 5 ; (ii) the inter-arm regions, w here - 0 .8 5 < a < - 0 .6 5 ; (iii) and the galaxies outskirts, w here a < -0 .8 5 . W e varied the boundaries in the spectral index selection in order to best separate these regions based on the spectral index alone. T he radio spectral index is now here flatter than - 0 .5 , w hich is the expected injection index o f young C R E s from the theory o f dif

fusive shock acceleration (B ell 1978) . T his is supported obser

vationally by the radio spectral index o f supernova rem nants, w hich is - 0 .5 ± 0.2 (R eynolds et al. 2012) . H owever, w ithout a fully sam pled radio spectrum fro m the M H z to the G H z regim e,

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F ig. 4. P lo t o f the in d iv id u a l galaxies, showing the sp atially resolved R C - S F R (( ESFR)RC- ( £SFR)hyb) relation . E ach data p o in t represents a 1.2 x 1 .2 k p c2 regio n that has been obtained fro m the hyb rid ESFR m ap (abscissa) and fro m the radio ESFR m ap (o rd inate). Shape and co lou r represent d ifferen t radio spectral indices betw een 140 and 1365 M H z . D o w n w a rd -p o in tin g red triangles represent regions w ith young C R E s (-0 .6 5 < a <

-0 .2 0 ); fille d green circles represent regions w ith C R E s o f in term ediate age (-0 .8 5 < a < -0 .6 5 ); and u p w ard -p o in tin g blue triangles represent regions w ith old C R E s ( a <-0 .8 5 ). S o lid b lack lines show the be st-fitting re la tio n and dashed lines show the C on do n relation . L e ft p a n e ls : results fo r L O F A R 140 M H z and r ig h t p a n e ls : fo r W S R T 1365 M H z . A 3 ix c u t-o ff was ap plied in a ll maps.

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Table 4. Spatially resolved RC-SFR relation, the (£SFR)RC- ( £ SFR)hyb relation, at 1.2 kpc spatial resolution.

G alaxy a 1 b 1 & 1,1200 a 2 b 2 &2,1200 F W H M1200

(dex) (dex) (arcsec)

( 1) (2) (3) (4) (5) (6 ) (7) (8)

N G C 3 1 8 4 0.46 ± 0 .02 - 1 .4 9 ± 0 .06 0 . 12 0 .60 ± 0 .02 - 0 .9 8 ± 0.05 0 . 10 22.29 N G C 4736 0.72 ± 0 .02 - 0 .7 4 ± 0.07 0 . 10 0.80 ± 0.03 - 0 .21 ± 0 .08 0 . 12 52.64 N G C 5055 0 .68 ± 0 .02 - 0 .5 3 ± 0.04 0.15 0.84 ± 0.02 - 0 .1 3 ± 0.05 0 .16 24.50 N G C 5194 0.51 ± 0 .02 - 0 .6 9 ± 0.04 0 .20 0.76 ± 0.02 - 0 .08 ± 0.05 0 .22 30.93 C om bined0 0.59 ± 0 .02 - 0 .7 7 ± 0.05 0.31 0.79 ± 0.02 - 0 .2 3 ± 0.04 0.25 - Young C R E b 0.93 ± 0.03 - 0 .21 ± 0 .06 0 .21 0.91 ± 0.03 1 o © 1+ 0 .06 0 .21 -

Notes. Columns: (1) data plotted; (2,3) best-fitting parameters for Eq. (3) for v1 « 140 MHz; (4) standard deviations for v1; (5,6) best-fitting parameters for Eq. (3) for v2 = 1365 MHz; (7) standard deviations for v2; (8) angular resolution given as FWHM of the map, which is equivalent to a projected spatial resolution of 1.2kpc. (a)Relation for the plot of the combined data points from the galaxy sample as presented in Fig. 5;

(b)relation for young CREs using only data points for which a > -0.65 as presented in Fig. 6.

w e cannot be too sure w hether th e C R Es are indeed young. A lter

natively, they cou ld b e old, so their intrinsic spectral index m ay be - 0 .8 , b u t fre e -fre e absorption suppresses the 140 M H z data point, turning it into an a o f - 0 .5 . Such a scenario is still possible and w e can only exclude therm al self-absorption w ith additional observing frequencies.

W e find a sim ilar situation in the o ther sam ple galaxies. The tw o galaxies w ith w ell-defined spiral arm s, N G C 3184 an d 5055, show good agreem ent betw een the radio spectral index distribu

tion an d the location o f the spiral arm s, even though the co n trast is n o t quite as p ronounced as in N G C 5194. This is because these tw o galaxies show less p ro m in en t spirals arm s in the hybrid 2 SFR m aps as w ell as in optical m aps. In N G C 3184, the spectral index is everyw here steeper than - 0 .4 and in N G C 5055 steeper than - 0 .5 , again in agreem ent w ith a C R E injection spectrum . N G C 3184 does n o t have any area, w here th e spectral index is steeper than - 0 .8 5 . This could be in p art caused by a sensi

tivity lim itation, since this galaxy has the jo in t low est hybrid 2 SFR value and thus low est RC surface brightness. N G C 5055 has em ission w ith a steep spectral index, in p articular along the m inor axis. It also shows som e flat spectral indices in the halo, b ut as discussed before, w e believe th at this is due to spurious em ission in the W S R T m ap; em ission th at w e d iscard for the fo l

low ing analysis. This is supported b y the fact th at ou r 1 4 0 M H z RC m ap shows a m o re extended halo than at 1365 M H z, but w ith a com pletely different m orphology hinting at a rad io halo that w e see in p rojection as discussed in Sect. 3.1 (see also B asu et al.

20 1 2 , fo r a 333 M H z m ap o f this galaxy).

W ith regards to th e spectral index distribution, N G C 4736 is different from the o ther galaxies since the spectral index is fairly flat. T he m axim um local spectral index is - 0 .2 , th ere

fore it becom es questionable w hether w e see the C R E injection spectral index. T he therm al fraction o f the R C em ission in the

“starburst rin g ” is so high that it flattens the spectral index even at these low frequencies (B asu et al. 2012). F urtherm ore, p o ssi

b le explanations fo r the flat spectral indices are ionisation and b rem sstrahlung losses, w hich both depend on neutral (atom ic and m olecular) gas density. S ince this galaxy has high gas sur

face densities o f 5 0 -1 0 0 M & p c -2 w ithin a radius o f 100arcsec (L eroy e t al. 2008) , w here w e observe m ainly th e flat spectral indices, such an explanation seem s likely. This is in fair ag ree

m en t w ith the > 200 M 0 p c -2 th at B a s u e t a l. (2015) suggested for areas o f a > - 0 .5 , b u t for spectral indices betw een 330 M H z and 1.4 G H z. Clearly, this is a strong function o f frequency, such that a threshold low er b y a factor o f tw o can b e expected fo r our low er observing frequency.

3.3. S patially r e s o lv e d R C -S F R relation

In this section, w e study the spatially resolved R C -S F R re la tion, the (2 SFR)RC- ( 2 SFR)hyb relation, at the resolution lim it o f our data. W e m easure 2 SFR values averaged in regions o f 1.2 x 1.2 k p c2 size in the radio 2 SFR m aps from L O FA R and W SR T and in the h ybrid 2 SFR m aps. P rior to this, w e convolved the m aps w ith a G aussian to a resolution (FW H M ) th at corresponds to a p rojected linear scale o f 1.2 kpc. W e applied 3& cut-offs in all m aps before creating the regions-by-regions plots. F o r each region, the rad io spectral index betw een 140 an d 1365 M H z was com puted. W e presen t the resulting plots fo r our four sam ple galaxies in Fig. 4 . In each p lo t the best-fitting least-squares lin ear relation (using the M arq u ard t-L ev en b erg algorithm ) is p re sented as w ell as the prediction from C o n d o n ’s relation. The least-squares fitting w as done in lo g -lo g space, fitting the fu n c

tion

log 10 [(^SFR)RC] = a log 10[(2 SFR)hyb] + b (3) H ence, in this notation one can w rite

(^SFr)rC = ^ K ^ F R ^ y b ^ (4)

w here a represents th e pow er-law slope o f the spatially resolved R C -S F R relation, w hen considering th at the rad io 2 SFR m ap is directly p roportional to the R C intensity, and b is a co nstant off

set. T he resulting best-fitting param eters can b e found in Table 4 . W e find the slope o f the relation is 0 .4 6 -0 .7 2 for L O FA R and 0 .6 0 -0 .8 4 for W SRT. This confirm s o ur earlier resu lt o f sublinear slopes (H eesen et al. 2014) . F o r each galaxy w e find th at the LO FA R slope is even flatter than for W SRT, such as w e already hinted in the study o f the m orphology (Sect. 3.1) .

A second resu lt is th at the offset from C o n d o n ’s relation is a function o f the rad io spectral index. A reas w ith steep spectral indices (green and blue data points) are relatively speaking “radio b right” , com pared w ith w h at is expected from C o n d o n ’s relation. D ata points representing areas w ith young C R Es (red d ata points) are in good agreem ent w ith the C ondon relation. Interestingly this is n o t a function o f th e hybrid 2 SFR.

In N G C 4736, w hich has th e hig h est spatially resolved R C -S F R slope, d ata p oints w ith young C R Es can b e found from 3 x 10- 4 to 6 x 10- 2 M 0 y r- 1 k p c - 2. T hese data points are in go o d ag ree

m en t w ith C o n d o n ’s relation (w ithin a factor o f 2), in p articu lar for L O FA R4. O n the other hand, w e find in N G C 5055 and 4 For WSRT the agreement can be improved, depending on the nor

malisation of the Condon relation.

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F ig. 5. P lo t o f the co m b ined data, showing the sp atially resolved R C - S F R ( (X SFR) RC- ( S SFR)hyb) relation . P a n e l a: results fro m L O F A R 140 M H z and p a n e l b: fro m W S R T 1365 M H z . E ach data p o in t represents a 1.2 x 1 .2 k p c 2 regio n th at has been obtained fro m the h y b rid E SFR m ap (abscissa) and fro m the radio E SFR m ap (o rd inate). Shape and co lou r represent d ifferen t radio spectral indices betw een 140 and 1365 M H z . D o w n w a rd - po inting red triangles represent regions w ith young C R E s ( - 0 . 6 5 < a < - 0 . 5 0 ) ; fille d green circles represent regions w ith C R E s o f in term ediate age ( - 0 . 8 5 < a < - 0 . 6 5 ) ; and u p w ard -p o in tin g blue triangles represent regions w ith old C R E s ( a < - 0 . 8 5 ) . S o lid b lack lines show the best-fitting relatio n and dashed lines show the C ondon relation . A 3 ix c u t-o ff was ap p lie d in a ll maps.

5194 young C R Es only in areas w here the h ybrid CSFR m ap exceeds approxim ately 10-2 M 0 y r- 1k p c- 2. T hese are also the areas w here th e agreem ent w ith C o n d o n ’s relation is best. In N G C 3184 the spectral index separation is n o t so clear w ith all points clustering around the C ondon relation.

In th e n ext step, w e com bined the data from all fo u r galaxies at a fixed linear scale o f 1.2 x 1.2 k p c2 in one p lo t fo r LO FA R and W SR T each, p resented in Fig. 5 . As for the individual g alax ies, w e find th at th e spatially resolved R C -S F R relation has a slope o f 0.59 for L OfAr, w hich is sm aller than for W S R T w ith

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Fig. 6. Plot of the combined data, showing the spatially resolved RC-SFR ((£SFR)RC- ( £ SFR)hyb) relation, for data points with spectral indices a > -0.65 only. Panel a: results from LOFAR 140 MHz and panel b: from WSRT 1365 MHz. Each data point represents a 1.2 x 1.2kpc2 region that has been obtained from the hybrid ESFR map (abscissa) and from the radio ESFR map (ordinate). Shape and colour represent the different galaxies. Solid black lines show the best-fitting relation and dashed lines show the Condon relation. A 3ix cut-off was applied in all maps.

0.79. W e also find th at the scatter o f th e LO FA R relation is w ith 0.3 dex larger than for W S R T w ith 0.25 dex. It now becom es even m o re strikingly apparent that the red d ata points are best in agreem ent w ith C o n d o n ’s relation regardless o f the hybrid 2 SFR, w hereas the green and blue data points lie p redom inantly above C o n d o n ’s relation.

As th e last step, w e investigated this further b y repeating a plot o f th e com bined data, but only o f d ata points w ith spectral indices a > - 0 .6 5 . This is show n in Figs. 6a and b for LO FA R and W SRT, respectively. Indeed, now all d ata points are in good agreem ent w ith C o n d o n ’s relation. M o st im portantly, the relation is now m uch m o re linear w ith a slope o f 0.93 for LO FA R and 0.91 for W SRT. This m eans that w e can find a n orm alisation factor for a linear spatially resolved R C -S F R relation. O ver three decades o f h ybrid 2 SFR, w e can find a relatio n th at strays only 0.15 dex from a linear relation. T he best-fitting relations are also fairly tight w ith scatters o f only 0.21 dex both fo r L O FA R and W SRT. H ence, this expands significantly on earlier results in H eesen e t al. (2 014), w here it w as suggested th at only areas w ith h ybrid 2 SFR values in excess o f approxim ately 10-2 M 0 y r-1 k p c -2 can b e recom m ended for th e use o f R C em ission as a reliable star form ation tracer. O ur new LO FA R observations suggest that it is instead the rad io spec

tral index that is the b etter d iscrim inant.

A nother approach w as used by D um as e t a l. (2011) and B asu e t a l. (2012) w ho m anually selected arm and inter-arm regions, w here th e fo rm er is dom inated b y young C R Es. A ccord

ing to B asu et al. (2 012) , th e arm regions o f N G C 5194 reveal a slope o f th e spatially resolved R C -F IR relation o f 1.0, w hile the slope in th e inter-arm regions is close to 2, probably ow ing to strong synchrotron losses o f C R Es diffusing from th e arm s.

B asu et al. (2012) show ed th at the arm regions in their 333 M H z m aps o f a sam ple o f galaxies display a higher spatially resolved R C -F IR relation slope o f 0.6 ± 0.1 com pared to th e inter-arm regions that have a slope o f 0.3 ± 0.1. Two o f their galaxies, N G C 4736 an d 5055, are also p art o f ou r sam ple. T hey find R C -F IR slopes that are flatter than ou r results even w hen w e consider only their arm regions w hereas w e fit the entire g alax ies. S ince their R C -F IR slopes at 1.4 G H z are also low er than the R C -S F R slopes w e m easure, w e conclude that the difference is due to th e intrinsic difference betw een ou r h ybrid 2 SFR m aps and the Spitzer 70 p m F IR m aps they used. Q ualitatively at least, our results are in good agreem ent.

T here is also som e rem aining scatter, for instance th e data points o f N G C 3184 are system atically below th at o f the other galaxies, m eaning that this galaxy is rad io w eak. This can be caused by w eak m agnetic fields and escape o f C R Es from the galaxy. T he latter can happen either through advection in a g alac

tic w ind o r diffusive transport, th e properties o f w hich w e inves

tigate in the next section.

4. Cosmic-ray transport 4.1. Sm oothing experiment

In this section, w e investigate th e properties o f th e cosm ic-ray transport. B ecause the C R Es are injected into the ISM at star for

m ation sites and are transported aw ay during their lifetim e, the resulting C R E distribution is a sm oothed version o f the 2 SFR m ap (B icay & H elou 1990; M urphy et al. 2006) . A s w e have seen in Sect. 3.3, the 1 k p c scale R C -S F R relation is sublinear, w here the slope o f th e relation is m ore shallow for L O FA R than for WSRT.

T he sublinear slopes have been reported before (Tabatabaei e t al.

2 0 1 3 ; H eesen et al. 2014), b u t the trend w ith frequency is new.

A lthough this has been suggested befo re from th e rad io spec

tral index as separate p aram eter in the 1 kpc R C -S F R relation (H eesen e t al. 2014) , w e can now m easure the cosm ic-ray tran s

port length as a function o f frequency and com pare this w ith theory.

F irst, w e m easure the transport length. This can b e carried out b y convolving th e h ybrid 2 SFR m ap w ith a suitable kernel to lin earise th e spatially resolved R C -S F R relation (T abatabaei et al.

2 0 1 3 ; H eesen et al. 2014) . T he first choice th at has to b e m ade is the shape o f the transport kernel. T here are tw o different m odes o f transport o f cosm ic rays in galaxies: advection in a galactic w ind and diffusion along m agnetic field lines. In an earlier w ork of M urphy et al. (2006) , it w as show n th at a exponential kernel is a m arginally b etter representation than a G aussian kernel, even though their approach w as slightly different; these authors stud

ied the correlation w ith only either M IR or F IR em ission using S pitzer 24 an d 7 0 p m em ission. W e expect advection to b e only o f im portance in the halo and since w e are studying em ission m ostly from the thin disc, w e expect diffusion to b e the d om inant process.

F or such a case w e expect that a G aussian diffusion kernel is the correct approach (H eesen et al. 2016) .

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A&A 622, A8 (2019)

Fig. 7. Smoothing experiment to measure the CRE diffusion length. Plotted is the slope of the (£SFR)RC- ( £ SFR)hyb relation, a, as function of the CRE diffusion length /CRE. LOFAR 140 MHz data are shown as open and WSRT 1365 MHz data are shown as filled symbols. The three galaxies are represented by red (NGC3184), green (NGC 4736), and blue (NGC5194).

In F ig. 7 , w e show the results o f this sm oothing experim ent.

We exclude N G C 5055 from the analysis, since the cosm ic-ray transport p rocess in this galaxy w ill b e investigated in the future (Sendlinger et al., in prep.). T he C R E diffusion w as sim ulated by convolving th e (2 SFR)hyb m ap w ith a G aussian kernel. In this way, the sublinear (2 SFR)RC- ( 2 SFR)hyb relation can b e lin earised, corresponding to a = 1 as show n b y the horizontal line in Fig. 7 , providing us w ith a m easurem ent o f th e C R E diffu

sion length. As w e can see, w e are indeed able to linearise the spatially reso lv ed R C -S F R relation in all studied galaxies; the resulting linearised plots are p resen ted in F ig. 8 . N ow w e define the length of the diffusion kernel, l , as the half w idth a t half m axim um (l = F W H M /2 ) o f the G aussian convolution kernel applied to th e hybrid 2 S fr m ap. T he C R E diffusion length can then b e derived as

l 2 = l 2 _ l 2 l CR E = l l beam -

W e can calculate th e diffusion coefficients using th e follow ing sim plified equation first:

(

6

⁾

w here t is the C R E lifetim e due to synchrotron and IC rad ia

tion losses. This is the 1D case for anisotropic diffusion along m agnetic field lines. F or a 3D case for isotropic diffusion, the diffusion coefficients w ould b e a factor o f four lower. T he CR E energy a t the observing frequency v can b e calculated from

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(5) w here w e correct for the sm oothing ow ing to th e lim ited re s olution o f our m aps w ith lbeam = 0.6 kpc. Thus derived d if

fusion lengths can b e found in Table 5 . F or LO FA R, w e find p rojected (in the p lan e o f th e sky) diffusion lengths betw een

1.3 and 4.8 kpc, and for W S R T betw een 1.0 an d 3.5 kpc. In each galaxy, the diffusion length for LO FA R is larger than for W SR T w ith a m inim um ratio o f 1.2 and a m ax im u m ratio o f 1.7. W e checked th at the intrinsic resolution o f th e LO FA R and W SR T im ages are the sam e, w hich w e expect, since w e cor

rected the L O FA R im age for the blurring effects o f the io n o sphere of the E arth w ith the d irection-dependent calibration (Sect. 2) . W e fitted u nresolved sources w ith a 2D G aussian fu n c

tion using I M F I T in A IP S and found agreem ent w ithin 5 -1 0 % ( 1 -2 arcsec) o f th e fitted F W H M in the LO FA R an d W SR T im ages.

w here Bg is the p erpendicular m agn etic field strength, w hich can b e approxim ated as Bg = V 2 /3 B0 for an isotropic turbulent m agnetic field, w here B0 is the total m agnetic field strength in the disc plane. T he C R E lifetim e is

(

8

⁾

A bove, w e have also included IC radiation losses, w here Urad is the radiation energy density including th e in terstellar radiation and cosm ic m icrow ave background. T he ratio of synchrotron to IC losses is equivalent to th e ratio o f th e m agnetic energy d en sity, UB = B2/(8 n ), to th e radiation energy density (see Table 5 for resulting C R E energies, lifetim es, diffusion lengths, and d if

fusion coefficients). W e find C R E energies betw een 1 an d 4 G eV and lifetim es o f approxim ately 100 and 3 0 M y r for LO FA R and W SRT, respectively. T he resulting diffusion coefficients are betw een 0.6 and 8.9 x 1028 cm 2 s-1 . T hese values are in b road D _ CRE /2

T

E (G eV ) _ A 1( V ) ( — ) ,

’ 1 16.1 M H z / \ juG /

t _ 8 3 5 2 x 109 ( g^v ) -1 (U G ) 2 ( 1 + ) y r '

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F ig. 8. L inearised ( £SFR)RC- ( £SFR)hyb re la tio n after convo lvin g the ( £SFR)hyb m ap w ith a G aussian ke rn el to sim ulate the diffu sion o f C R E s . L e ft p a n e ls : results fo r the 140 M H z L O F A R data; r ig h t p a n e ls : results fo r the 1365 M H z W S R T data. T h e length o f the G aussian kern el, l (= F W H M /2 ), is noted as w e ll. P a n e ls a and b: results fo r N G C 3 1 8 4 , p a n e ls c and d: fo r N G C 4 7 3 6 , and p a n e ls e and f : fo r N G C 5 1 9 4 . E ach data p o in t represents a 1.2 x 1.2 kp c2 regions th at has been obtained once fro m the hyb rid ESFR m ap (abscissa) and once fro m the radio ESFR m ap (o rd inate). Shape and co lou r represent d ifferen t rad io spectral indices betw een 145 and 1365 M H z . D o w n w a rd -p o in tin g red triangles represent regions w ith young C R E s ( - 0 . 6 5 < a < - 0 . 2 0 ) ; fille d green circles represent regions w ith C R E s o f in term ediate age ( - 0 . 8 5 < a < - 0 . 6 5 ) ; and u p w ard -p o in tin g blue triangles represent regions w ith old C R E s ( a < - 0 . 8 5 ) . S o lid b lack lines show the be st-fitting relatio n and dashed lines show the C ondon relation . A 3cx c u t-o ff was ap plied in a ll m aps.

agreem ent w ith w hat has been found in external galaxies before (B erkhuijsen e t al. 2 0 1 3 ; T abatabaei et al. 2013) and also w ith the M ilky W ay value o f 3 x 1028 cm 2 s-1 (S trong et al. 2007) . In one galaxy, N G C 5194, w e find th at the diffusion coefficient does n o t increase w ith energy, w hereas it m ay in the o ther tw o galaxies. W e return to discuss these findings in Sect. 5 .2 .

M urphy et al. (2008) studied the transport o f C R Es in the sam e sam ple as w e did. Interestingly, the resulting sm oothing lengths have the sam e trend that w e find w ith the sm allest length in N G C 4736 and the largest length in N G C 3184. H owever, their lengths are a factor o f tw o sm aller than w e find a t 1365 M H z. W e attribute this to the fact th at they sm oothed S pitzer 70 p m im ages, rath er than hybrid 2 SFR m aps as w e do. Since the F IR em ission is sensitive to less w arm dust than the 24 p m M IR em ission, the F IR em ission is m o re spread o u t over the disc and less concen

trated in spiral arm s and other areas o f rec en t star form ation.

This m eans that th e 70 p m m ap has to b e less sm oothed in order to resem ble the rad io m ap. This is in line w ith the findings o f B asu e t al. (2012) and our spatially resolved R C -S F R relation (Sect. 3.3) . F urtherm ore, M urphy e t a l. (2008) m in im ised the difference betw een the RC an d F IR m aps, w hereas w e linearised the R C -S F R relation such th at a difference can b e expected.

M u lc a h y e ta l. (2014) used scale-dependent w avelet tran s

form s o f the LO FA R 151 M H z an d S pitzer 70 p m im ages o f N G C 5194 and their cross-correlation spectra to m easure the d if

fusion lengths o f C R Es. T heir resu lt o f 1.45 kpc is low er by a factor o f 2.6 com pared to our resu lt at 145 M H z (Table 5) . This discrepancy indicates th at the cross-correlation coefficient rw o f 0.75 for m easuring the diffusion length w as too sm all. A choice o f rw = 0.85 w ould yield a diffusion length sim ilar to our result.