A Universal Luminosity Function for Radio Supernova Remnants
aa r X i v : . [ a s t r o - ph . C O ] J u l Draft version November 20, 2018
Preprint typeset using L A TEX style emulateapj v. 04/20/08
A UNIVERSAL LUMINOSITY FUNCTION FOR RADIO SUPERNOVA REMNANTS
Laura Chomiuk & Eric M. Wilcots Draft version November 20, 2018
ABSTRACTWe compile radio supernova remnant (SNR) samples from the literature for 19 nearby galaxiesranging from the SMC to Arp 220, and use this data to constrain the SNR luminosity function (LF)at 20 cm. We find that radio SNR populations are strikingly similar across galaxies. The LF can bedescribed as a power law with constant index and scaling proportional to a galaxy’s star formationrate (SFR). Unlike previous authors, we do not find any dependence of SNR luminosity on a galaxy’sglobal ISM density. The observed correlation between the luminosity of a galaxy’s brightest SNR anda galaxy’s SFR can be completely explained by statistical effects, wherein galaxies with higher SFRmore thoroughly sample the high-luminosity end of the SNR LF. The LF is well fit by a model of SNRsynchrotron emission which includes diffusive shock acceleration and magnetic field amplification, ifwe assume that all remnants are undergoing adiabatic expansion, the densities of star-forming regionsare similar across galaxies, and the efficiency of CR production is constant.
Subject headings: acceleration of particles — magnetic fields — radio continuum: galaxies — supernovaremnants INTRODUCTION
Surveys for supernova remnants (SNRs) in extragalac-tic systems have been ongoing for the past three decades,and we are now at the point where we can study SNRpopulations and use them to extract valuable insight intoSNR evolution and the interstellar medium (ISM). Thereare distinct benefits to extragalactic samples over stud-ies in the Milky Way. In other galaxies, all SNRs willbe at approximately the same distance, and therefore donot suffer the distance uncertainties that plague obser-vations of SNRs in our Galaxy. Also, comparing SNRsbetween galaxies provides a much larger dynamic rangein ISM conditions and a longer baseline for understand-ing how SNR characteristics and evolution depend onISM density, star formation rate (SFR), etc. As we standpoised for unprecedented depth in radio continuum imag-ing with the advent of the Expanded VLA and otherSquare Kilometer Array precursors, it is timely to lookback on the samples of extragalactic SNRs in the liter-ature, synthesize them, and attempt to draw physicalconclusions so we might know where to go with the nextgeneration of data.Extragalactic SNRs have been detected at many wave-lengths, but most commonly they are selected by theiroptical or radio emission. Optical surveys generallyuse narrow-band imaging of the [S II ] and H α emis-sion lines to detect SNRs and distinguish them fromH II regions. The first search of this kind was car-ried out by Mathewson & Clarke (1973) in the LMC,and these techniques have since been used to estab-lish large (5–100) samples of SNRs in the Local Group(Gordon et al. 1998; Braun & Walterbos 1993) and be-yond (e.g., Matonick & Fesen 1997; Matonick et al. 1997;Blair & Long 1997, 2004). Radio surveys use data atat least two frequencies to measure discrete sources’spectral indices and separate thermal H II regionsfrom synchrotron-emitting SNRs. These surveys have Electronic address: [email protected] University of Wisconsin–Madison, Madison, WI 53706 been successfully carried out with techniques rangingfrom single-dish observations in the Magellanic Clouds(Filipovi´c et al. 1998) to very long baseline interferom-etry in compact starburst galaxies (e.g., Lonsdale et al.2006). Our work here makes use of many radio SNRsurveys in the literature; see Section 2 for more de-tails. Multi-wavelength studies show that there is lim-ited overlap between optical- and radio-selected samplesof SNRs, because remnants will glow more brightly in theradio if they are expanding into dense ISM, whereas op-tical SNRs are more easily detected in less dense regionswhere there is less confusion from ongoing star formation(Pannuti et al. 2000).In this paper, we focus on studies of radio SNRs. Theradio luminosity of SNRs is due to synchrotron emissioncoming from cosmic rays (CRs) that have been acceler-ated by the SNR through first-order Fermi acceleration(Bell 1978) and are interacting with the SNR’s magneticfield. Due to its power-law spectrum, synchrotron emis-sion is brighter and also suffers less contamination fromthermal bremsstrahlung emission at lower frequencies;therefore, radio SNRs are typically selected by their 20cm emission. The 1.45 GHz spectral luminosities of SNRsdetected outside our Milky Way currently span five or-ders of magnitude, from ∼ erg s − Hz − in the SMCto ∼ erg s − Hz − in Arp 220 (For comparison, themost luminous Galactic SNR Cas A would have a spec-tral luminosity of 2.8 × erg s − Hz − , assuming a 1GHz flux density of 2723 Jy and a spectral index of α = − ∼ − and unaffected by the surroundingmedium. This phase ends when the SNR has swept upa mass of ISM or circumstellar material that is approx-imately equivalent to the mass initially ejected by theexplosion, implying a typical duration of 100–1000 years Chomiuk et al.(Berezhko & V¨olk 2004, hereafter BV04). Due to theshort duration of this phase, statistically very few SNRswill inhabit it at any one time. The end of free-expansionis called the Sedov time, and is important to studies ofradio SNRs because it marks a rapid period of very ef-ficient particle acceleration (BV04). At the Sedov time,a SNR has a diameter of ∼ v s ∝ t − / . BV04 show that a SNR’s radio luminositypeaks at the Sedov time and then declines throughoutthe Sedov phase. Most radio SNRs are thought to beobserved at approximately the Sedov time or in the Se-dov phase (Berkhuijsen 1986; Gordon et al. 1998, 1999,BV04). The Sedov phase lasts much longer than the free-expansion phase, and therefore it is statistically likelythat SNRs will be observed during their adiabatic evolu-tion.When the shock wave has slowed down to v s ≈ − , the SNR enters a temperature regime wherecooling become dynamically important. This transitionfrom the adiabatic phase to the radiative phase takesplace at a SNR age of ∼ × and a diameter of 20–50 pc (Blondin et al. 1998; Woltjer 1972). By this time,SNRs will have quite low luminosities in the radio andwill be difficult to detect, although they should start toglow brightly in UV, optical, and IR emission lines (e.g.,Weiler & Sramek 1988). As the SNR slows down evenmore, the cosmic rays which emit radio synchrotron willleak out of the SNR (BV04). Eventually, the shock speedbecomes equivalent to turbulent fluctuations in the ISM,and the SNR merges back into the ambient medium.Of course, there are many complications that affect anSNR’s evolution; not every SNR spends time in each ofthe four cartoon phases, and oftentimes the transitionsbetween the phases last longer than the phases them-selves (Jones et al. 1998; Reynolds 2005).Here, we study the populations of radio SNRs in 19nearby galaxies via their luminosity functions (LFs). TheLF can shed light on the evolutionary state of observedSNRs, particle acceleration physics, and magnetic fieldsin SNRs. Recently, Thompson et al. (2009) used the lu-minosities of extragalactic SNRs to place upper limits ongalaxies’ magnetic field strengths and test magnetic fieldamplification in SNRs. Their work does not fully treatthe SNR LF, but instead uses the most luminous SNRin a galaxy to constrain the behavior of the SNR popu-lation. With our statistical treatment of the LF we willexpand upon their work.In Section 2, we discuss the selection of SNRs and ourefforts to make the samples as homogeneous as possible,and in Section 3 we discuss the basic characteristics ofthe SNRs’ parent galaxies. In Section 4, we measurethe SNR luminosity function and compare it acrossgalaxies, and then we describe the LFs in terms ofphysical models of particle acceleration in Section 5. InSection 6, we investigate the possibility of truncation ofthe LF at the high-luminosity end. Finally, in Section 7we summarize our results. SNR SAMPLES
Selection Criteria
We have compiled samples of radio SNRs for 18 nearbygalaxies from the literature (listed in Table 1). It is im-portant to note that the SNRs presented here were allselected by their radio emission. The details of the SNRsearches at 20 cm can be found in Table 2. The data andthe selection criteria used by the original authors werelargely heterogeneous, so we homogenized the SNR sam-ples as much as possible by requiring a source to havethe following traits in order to be considered an SNR: • L-band flux density measurement (near 20 cm) atleast 3 σ above the local noise. • Non-thermal spectral index ( α ≤ − II re-gions from our sample. • A counterpart in a narrow-band H α image, in orderto distinguish an SNR from a background radiosource (any emission from the background sourcewould be redshifted out of the narrow-band filter).Here, we define α as S ν ∝ ν α where S ν is the source fluxdensity at frequency ν . In most cases, spectral indiceswere determined by comparing measurements at 20 cmand 6 cm. For sources where the 20 cm flux density is > σ but the 6 cm flux density is not, we place an upperlimit at 6 cm of three times the 1 σ error in the fluxdensity. We use this upper limit on the 6 cm flux densityto place an upper limit on the spectral index, and if thisallows us to say that the spectral index is less than − ; e.g.,Ulvestad 2000). The samples will be further winnoweddown in Section 2.4 using a luminosity criterion with theintention of creating complete samples of SNRs. SNRsin complete samples are marked with asterisks in Tables3–20.Whenever possible, we strictly applied our selectioncriteria, even though this forced us to exclude some ob-jects which the original authors considered to be SNRs.For example, Gordon et al. (1999) include sources intheir list of SNRs in M33 if they have optical counter-parts with [S II ]/H α > α > − II ] imaging for all18 galaxies, and so this optical emission line criterion The National Radio Astronomy Observatory is a facility ofthe National Science Foundation operated under cooperative agree-ment by Associated Universities, Inc.
NR LF 3can not be generalized to our entire sample. For the pur-pose of this paper, source selection based on [S II ]/H α isnot sufficient to be considered an SNR. However, thereare a few galaxies which make the standard observationsdifficult (see the below notes on the Magellanic Clouds,M82, and NGC 253), and our SNR selection criteria wereforced to be flexible in these cases.Additionally, we excluded recent radio SNe from oursamples; four radio SNe in M83, 41.95+57.5 in M82,SN1981K in NGC 4258, SN 1968D in NGC 6946, andSN 1994I in M51 are intentionally removed from the SNRsamples. Notes on Indvidual Galaxies
Below we discuss the SNR samples of each galaxy, inorder of increasing distance.
LMC:
The SNR sample in the LMC has significantlyworse angular (and spatial) resolution than any of theother sample galaxies because it is based on single-dishdata acquired with the Parkes telescope. We consideredall sources with an “SNR” or “SNR candidate” designa-tion in Filipovi´c et al. (1998); see their work for a de-scription of SNR selection criteria. The Parkes studyis a multi-frequency survey, but we required sources tohave 1.4 GHz flux densities and spectral indices between20 cm and 6 cm ≤ −
SMC:
The data for the SMC are significantly higherresolution than for the LMC because the radio mapshave ATCA interferometric data added to the Parkesdata. We considered sources with “SNR” (probably a bona fide
SNR) or “snr” (SNR candidate) designationin Payne et al. (2004). There were a few objects withSNR/snr designation but with α > − α values were taken fromthe Payne et al. catalog, where they use all available fluxdensities for the calculation). No flux density errors aregiven in Payne et al. (2004), but Filipovi´c et al. (2002)estimate that the calibration error can be up to 10% forextended sources. The images also have an r.m.s. noiseof 1.8 mJy/beam. We use the diameters as measured offthe 2.4 GHz images by Filipovi´c et al. (2005) to calculatethe number of beams each SNR subtends, and then usethe following equation to calculate conservative calibra-tion + measurement errors for the 20 cm flux densities: σ S . = p (0 . × S . ) + ( N bm × . . Here, S . is the SNR flux density at 1.45 GHz, N bm isthe number of beam areas the SNR extends across, and σ S . is the total error in the 1.45 GHz flux density. IC 10:
We applied our SNR criteria to the sourcesin Table 1 of Yang & Skillman (1993) and came awaywith three SNR candidates. Additionally, we include thenon-thermal superbubble that is the central subject oftheir paper. Although this source has an unusually largediameter ( ∼
130 pc) and probably results from multipleSN explosions, we include it here because similar sourcesin other galaxies would not be resolved (see Table 2)and would appear as typical SNRs. We compared theYang & Skillman 20 cm sources with an H α image fromGil de Paz et al. (2003). M33:
We applied the selection criteria describedabove to the list of radio sources in Gordon et al. (1999).Most sources were checked for H α counterparts by Gor-don et al. However, six radio sources fell out of therange of the Gordon et al. H α images, so we checkedthem against H α images from the UV/visible Sky Gallery(Cheng et al. 1997), downloaded from NED . We ex-cluded source 102 because it corresponds to the galacticcenter. NGC 1569, NGC 4214, NGC 2366, and NGC4449:
We include the SNR candidates fromChomiuk & Wilcots (2009), who used data at threefrequencies (20, 6, and 3.6 cm) to constrain spectralindices, and looked for H α counterparts to the radiosources. For sources with no 6 cm flux density measured,Chomiuk & Wilcots assumed an upper limit that was3 √ σ upper limit, and this tighterconstraint qualifies all five ambiguous SNR/H II regionsources as non-thermal; they are included here as SNRs. NGC 300:
We used the list of non-thermal radiosources in Pannuti et al. (2000), which were all checkedfor H α counterparts. M82:
We used different selection criteria for M82because, due to the compact and crowded nature ofthis galaxy, no 20 cm-selected SNR sample exists for it(higher resolutions can be achieved at higher frequen-cies). In addition, many SNRs in starburst galaxies suf-fer severe free-free absorption at 20 cm; objects whichare easily observable at 6 cm may become undetectableat 20 cm (Tingay 2004; Lenc & Tingay 2006). There-fore, selection by 20 cm data is unsuitable, and in-stead we use the list of 5- and 15-GHz selected SNRsfrom McDonald et al. (2002). All sources in their listhave spectral indices ≤ − α . We also use this linefit to determine the source’s flux density at 1.45 GHz,which allows us to correct for extinction in sources withsignificant free–free absorption by applying the power-law fit from higher frequencies to 1.45 GHz.Four of the SNRs are not measured by Allen & Kro-nberg, but are studied by McDonald et al. (2002). 20cm flux densities for these sources are found by fitting apower law between the 5 and 15 GHz data points at 200mas resolution, and extrapolating this fit to 1.45 GHz.Again, this method should correct for free–free absorp- The NASA/IPAC Extragalactic Database (NED) is operatedby the Jet Propulsion Laboratory, California Institute of Tech-nology, under contract with the National Aeronautics and SpaceAdministration.
Chomiuk et al.tion, but gives larger errors because there are only twodata points constraining the spectral index, rather thanthe 4–6 measurements typically provided by Allen & Kro-nberg.We did not require H α counterparts for this galaxydue to the extremely high dust extinction in the centerof M82, and the high density of sources implying thatbackground contamination is unlikely to be significant.Positions for sources come from McDonald et al. (2002). M81:
We used the list of radio sources inKaufman et al. (1987) and applied the criteria describedabove. All sources included in the Kaufman et al. cat-alog were checked by the original authors to have H α counterparts. NGC 7793:
We used the list of radio SNR candidatesin Pannuti et al. (2002). We eliminated one source with α > − α counterparts. NGC 253:
For the same reasons as in M82, we can-not select SNRs in NGC 253 by their 20 cm emission.Instead, in the central ∼
200 pc of NGC 253, we selectSNRs using the 6 cm source list in Ulvestad & Antonucci(1997); Lenc & Tingay (2006) show that free-free absorp-tion does not significantly affect 5 GHz flux densities. Wemeasure spectral indices for the sources using all avail-able data between 1.3 and 6 cm as presented in Table13 of Ulvestad & Antonucci. A source is considered anSNR if its spectral index is non-thermal and its 6 cm fluxdensity measurements is significant to > σ . We do notinclude 5.79-39.0 because it is assumed to be the nuclearregion of the galaxy by Lenc & Tingay.In addition, Ulvestad (2000) performed a search forSNRs at larger galactic radii, but this study is limitedin field of view and sensitivity at large radii due tobandwidth smearing. We did not include the “wide-fieldsources” listed by Ulvestad because the survey was veryincomplete at these radii. However, we did include the“compact circumnuclear sources” if they had α ≤ − α image for NGC 253. However, all ofthe radio sources are well within the optical disk of NGC253 where there are high levels of star formation, andit is unlikely these sources are background. Ulvestad(2000) estimates that perhaps one of the sources in the16-square-arcminute survey area may be a backgroundsource. Additionally, like M82, an H α image of NGC253 would be plagued by dust extinction and lead us tofalsely eliminate many SNRs. M83:
We used the radio flux densities from the 1990observations listed in Maddox et al. (2006), and appliedthe criteria listed above. The historical SNe 1923A,1950B, 1957D, and 1983N are detected in the radio byMaddox et al., but none of these sources fulfill our SNRcriteria (and even if they did, they would be intention-ally excluded from our sample here). We checked for H α counterparts to candidate SNRs using a SINGG image(Meurer et al. 2006) downloaded from NED. NGC 4736:
We included sources fromDuric & Dittmar (1988), who surveyed the circum-nuclear star-forming ring of this galaxy, if they fit our SNR criteria. We found H α counterparts in an imagefrom Knapen et al. (2004) downloaded from NED. NGC 6946:
We used the list of discrete radio sourcesfrom Lacey et al. (1997), and applied the criteria as de-scribed above. SN 1968D corresponds to source 82 inLacey et al., and is excluded from our sample. We uti-lized H α images from SINGS (Kennicutt et al. 2003) ac-cessed through NED. NGC 4258:
We include in our SNR sample the sourcesin Hyman et al. (2001) if they meet the above criteria.We did not include SN 1981K (van Dyk et al. 1992) eventhough it appeared in the observations of NGC 4258.
M51:
We used the list of discrete radio sources inMaddox et al. (2007) and imposed the selection criteriadescribe above. We used an ACS mosaic of M51 in H α ac-quired by the Hubble Heritage team to check that sourceshad an H α counterpart. We excluded source 58 whichcorresponds to SN 1994I. We note a remarkably brightsource (104) which, if it is an SNR, is unusually luminousfor a galaxy like M51. It is coincident with H α emissionand is therefore included here as an SNR. This source isdeserving of follow-up observations to investigate if it isindeed a super-luminous SNR. Arp 220
Recent VLBI studies have revealed a rich population ofdiscrete non-thermal radio sources in the ultraluminousinfrared galaxy (ULIRG) Arp 220 (Smith et al. 1998;Lonsdale et al. 2006). Despite the relatively large dis-tance to Arp 220 (77 Mpc), these studies achieve highspatial resolution of ∼ ± − . In addi-tion, Arp 220’s extreme characteristics make it difficult tocompare with other more quiescent nearby galaxies. Itsvery large SFR give it disproportionate influence whenwe investigate the composite SNR population of nearbygalaxies (as in Figure 5) or find linear fits to the data asa function of SFR (as in Section 4.2.) For these reasons,we do not include Arp 220 in our main sample of galax-ies, but throughout our analysis of the SNR LF, we willcompare our results from the 18 “normal” galaxies withSNR candidates in Arp 220.As an attempt at an SNR sample, we include here thenon-thermal “long-lived” sources (with 18 cm flux densi-ties which did not vary significantly over 11 years) fromParra et al. (2007). One “ambiguous” source, W15, isalso included because Parra et al. state that it is mostlikely a long-lived source. The flux densities listed inTable 21 were calculated by transforming the 18 cmoptically-thin synchrotron flux densities from Table 2 ofParra et al. (2007) to 1.45 GHz flux densities using theParra et al. spectral indices. Completeness was deter-mined by the same technique used for the other galaxiesas described in the next subsection. Defining Completeness
Completeness can be difficult to define for these sam-ples. Due to the shape of the primary beam, the noisein any interferometric radio image grows as a function ofNR LF 5
Fig. 1.—
Power-law fits (short-dashed lines) to cumulative SNR LFs (solid-line histograms). The true incompleteness limits (where thepower laws turn over) are marked with red long-dashed lines, while simple completeness limits defined as 3 times the image noise aredenoted by black long-dashed lines. radius from the image phase center. This was an issue forthe survey of M33 at 20 cm because of M33’s large an-gular size. At 6 cm, it affects practically every galaxy wepresent here due to the smaller primary beam at shorterwavelengths.In addition, sensitivity may drop off even more quicklywith radius due to chromatic aberration; if a galaxyis imaged with wide frequency channels (e.g., “contin-uum mode” with the current correlator at the VLA),sources at significant distances from the image center aresmeared radially, spreading their flux over a large areaand making them more difficult to detect and measure.This was a significant problem in NGC 253, NGC 4214,and NGC 4258. In these cases, SNR surveys were car-ried out over regions with less severe bandwidth smearing (typically out to a radius ≈ θ syn ν ∆ ν − , where θ syn is the half-power beam width of the synthesized beam, ν is the frequency of the observations, and ∆ ν is thefrequency channel width), and it is these regions thatare listed as the survey fields of view in Table 2. Werecognize that we are not surveying the entire galaxy forSNRs, and correct the relevant galaxy parameters likeSFR for this (see Section 3).SNRs are usually point sources at the distances sur-veyed here, but occasionally they are resolved. Whenthis is true, the surface brightness of the remnant, notjust the total flux, can determine if it is detected. Sur-face brightness limitations affect nearby high-resolutionSNR samples like the SMC and M33 most severely.Additionally, if an SNR is in an environment with vig- Chomiuk et al.orous star formation and high backgrounds, it will besignificantly more difficult to detect. With interferomet-ric observations, the brightness of the diffuse backgroundemission varies depending on the uv coverage of the map.For example, if a galaxy is only observed with the VLAin its most extended configuration, we will only be sensi-tive to objects with small angular scales, the backgroundwill be resolved out, and SNRs should be fairly easy todetect. The maps used in this study were made witha wide range of VLA configurations, and therefore theproblem of high backgrounds will vary quite a lot fromgalaxy to galaxy.The spatial resolutions of the SNR surveys vary from13 pc to 221 pc. To investigate the impact of reso-lution on the SNR catalogs, we produced a catalog ofdiscrete 20 cm sources in NGC 6946 observed with theVLA in its B configuration, and compared it with theLacey et al. (1997) catalog of radio sources detected inthe A-configuration. The B-configuration images haveapproximately three times lower resolution, and moresensitivity to diffuse emission. Of the 37 SNRs foundin the higher resolution images, only four were not de-tected in the B-configuration images because of poor res-olution and confusion with other sources. Another fourSNRs were not detected because of the higher diffusebackground in the B-configuration images. Therefore, itappears that resolution affects SNR samples only mildly.The observations of each galaxy in our sample havedistinct limitations, and therefore only a subset of theSNRs described in Section 2.2 can be considered to bein complete samples. We define completeness in termsof SNR spectral luminosity using a method developed byGordon et al. (1999) on the M33 SNR LF. We assumethat the cumulative LFs of SNRs can be described assingle power laws, and departures from them at the faintend are due to incompleteness. For each galaxy, we plotthe cumulative SNR LF as measured at 20 cm and fit apower law to it at the bright end. At fainter luminosi-ties, most galaxies’ LFs exhibit a break where they canno longer be described by this power law and must be fitwith a shallower one. We call this break in the cumula-tive LF the true completeness limit for the sample (seeAppendix A for more details on how the location of thisbreak is determined).Our cumulative LFs and power-law fits are shown inFigure 1 for our 18 sample galaxies. The low-luminositylimits where the LFs cease to be fit by power laws corre-spond to our completeness limits (red dashed line). Ad-ditionally, Figure 1 shows the flux density limit at whichone might expect to reliably measure point sources, at3 times the image noise. For most galaxies, the com-pleteness limit is significantly brighter than the surveyflux density limit, due to the reasons described above.Our definition of completeness appears to be conserva-tive, and unfortunately limits quite drastically the num-ber of SNRs in our sample for some galaxies (see Tables3–21, where SNRs in complete samples are marked withasterisks). In total, we have 259 SNRs in complete sam-ples across the 18 sample galaxies, and four SNRs in theArp 220 complete sample. PARENT GALAXY PARAMETERS
Some basic characteristics for our 18 sample galaxiesare listed in Table 1. All are nearby ( ≤ α and mid-infrared data as directed by Equation 7 in Calzetti et al.(2007):SFR(M ⊙ yr − ) = 5 . × − [ L ( Hα ) + 0 . L (24 µ m)](1)where L(H α ) and L(24 µ m) are luminosities in erg s − ,and the IR luminosity is converted from a spectral lu-minosity by multiplying by the frequency. This pre-scription corrects the H α luminosity for extinction usingthe 24 µ m emission (here we actually use 25 µ m emis-sion from the IRAS satellite). We use the H α luminosi-ties listed in Kennicutt et al. (2008), except for IC 10,whose H α luminosity comes from Hunter & Elmegreen(2004). The majority of 25 µ m flux densities come fromSanders et al. (2003), with a few measured by Rice et al.(1988), Lisenfeld et al. (2007), and Moshir et al. (1992).There are no IRAS measurements for IC 10, so we usedan H α -only SFR for this galaxy, assuming an inter-nal+foreground reddening of E(B-V) = 0.85, consistentwith values in the literature (Hunter 2001).Not all galaxies are completely covered by the SNRsurveys used here, and therefore it is not consistent tocompare SFRs measured from integrated luminositieswith the SNR populations derived from these surveys.In all cases except M33, M82, and NGC 253, we usearchival 24 µ m Spitzer images to measure the fractionof the SFR inside the SNR survey area. The disk ofM33 was not entirely covered by Spitzer observations, soin this case we used 25 µ m IRAS observations to tracethe star formation. In the cases of M82 and NGC 253,archival Spitzer/MIPS images were saturated, and wewere forced to use 20 cm radio continuum images fromCondon (1987) to trace the SFRs. For each galaxy, thefraction of its star-formation activity included in its SNRsurvey area is listed in Table 2.Previous studies of extragalactic SNRs have correlatedthe characteristics of SNRs with the global ISM den-sities of their parent galaxies (Hunt & Reynolds 2006;Thompson et al. 2009). For comparison with these stud-ies, we calculate rough estimates of the ISM density foreach galaxy ( ρ ) using the SFRs in the survey areas andthe Schmidt-Kennicutt law. We could use measurementsof the gas surface density (Σ g ) made directly from H I and CO measurements, but these quantities are usuallymeasured over the entire galaxy, and it would be verydifficult to calculate them in the SNR survey areas (thisis particularly important in cases like NGC 4736, whereNR LF 7the galaxy is only gas-rich and star-forming in a rathersmall area, and calculating the ISM density over the en-tire disk would dramatically dilute the true ISM densityin the region producing SNRs). Therefore, we insteadconvert SFRs into SFR surface densities (Σ SF R ), andthen use the relationship from Kennicutt (1998) to findΣ g : Σ g = 374 . . SF R (2)We assume that all galaxies have a constant gas scaleheight of 100 ±
50 pc (Thompson et al. 2009) althoughthis assumption probably contributes systematic er-rors, especially given evidence that the gas scale heightis a function of Hubble type (van den Bergh 1988;Brinks et al. 2002). We then use the inclinations in Table1 to find path lengths through the galaxies and divide Σ g by the path length to find an “average” volume density;these values of ρ can be found in Table 1. We recognizethat this method for calculating density is very rough,and should only be interpreted as an approximate diag-nostic.Basic data on Arp 220 are also listed in Table1. We use a SFR of 127 M ⊙ yr − (as found byAnantharamaiah et al. (2000), but converted to the IMFused by Calzetti et al. (2007)). We assume that 100% ofthe star formation activity is taking place in the SNRsurvey area, which is probably a slight overestimate. RADIO SNR LUMINOSITY FUNCTIONS
To convert our SNR flux densities to spectral luminosi-ties, we assume the distances to the galaxies in Table 1,and use the equation: L . = 1 . × S . × D (3)where D is the distance in Mpc, S . is the 1.45 GHz fluxdensity of an SNR in mJy, and L . is the SNR spectralluminosity in units of 10 erg s − Hz − . Errors in thespectral luminosities are calculated as: σ L . = p (1 . D σ S . ) + (2 . S . D σ D ) (4)where σ D is the error in the distance as described inthe previous section, and σ S . are the errors on the fluxdensity measurements as listed in Tables 3–21.Figure 2 shows cumulative luminosity functions for theSNRs in our 18 sample galaxies. The SNR spectral lu-minosities span almost four orders of magnitude, from ∼ erg s − Hz − in the SMC to almost 10 erg s − Hz − in M82 and NGC 253. Unfortunately many ofthe galaxies’ SNR samples do not overlap in luminos-ity space. For example, the most luminous SNR in M33is fainter than the least luminous SNR observed in M82.The galaxies which host the more luminous SNRs tendto be crowded and/or at relatively large distances, mean-ing that the SNR surveys of these galaxies will becomeincomplete at higher luminosities. Therefore it can bedifficult to directly compare SNR populations betweengalaxies. A similar problem was presented for high-massX-ray binaries in nearby galaxies by Grimm et al. (2003),and much of our analysis is inspired by theirs.The galaxies with the highest SFRs host the most lu-minous SNRs; in M82 and NGC 253, many of the SNRsare more luminous than Cas A. Can the differences be-tween the luminosities of SNR populations be completely explained by differences in SFR? Galaxies with higherSFRs will host SNe explosions more often, and thereforethe luminous end of the SNR LF will be more thoroughlypopulated. We also expect the total number of SNRs ina galaxy to scale with the SFR. In Figure 3 we againplot the cumulative LFs, but we scale each LF by theinverse of its parent galaxy’s SFR. Most SNRs adhere toa straight line in this diagram, implying that scaling bySFR removes most of the differences between LFs. Onenotable exception is the most luminous object in IC 10.It sits significantly above the line defined by the rest ofthe SNRs, implying that it is a much more luminous ob-ject than would be expected for a galaxy with the SFRof IC 10. This is not surprising, as Yang & Skillman(1993) identified the source as a superbubble and statethat it is probably powered by multiple SNe explosions.Because this object is so clearly aberrant, and becausethere is a recognized reason for its outlier status, we ex-clude it from further analysis. We also note that thehigh-luminosity end of this plot shows larger scatter thanthe low-luminosity end, implying that the SNR LF is notwell sampled at high luminosities in many galaxies. LF Power-Law Index ( β ) We can constrain the shapes of the differential SNRLFs if we make the simple assumption that all LFs canbe fit with a single power law. We write such an LF as: n ( L . ) = dNdL . = A L β . (5)where n ( L . ) is the number of SNRs with spectral lu-minosity L . , A is a scaling constant, and β is a neg-ative number. We determine the power-law index us-ing the maximum likelihood estimator as described inClauset et al. (2007):ˆ β ′ = − − n " n X i =1 ln L . ,i L . ,min − (6)For each galaxy, n is the number of SNRs in the com-plete sample, and L . ,min is the spectral luminosity ofthe least-luminous SNR in the complete sample. How-ever, for small sample sizes, ˆ β ′ will be biased low (morenegative) compared to the true β value. We can correctfor this bias with a simple analytic expression:ˆ β = (cid:18) n − n (cid:19) (cid:18) ˆ β ′ − n − (cid:19) (7)The standard error on ˆ β is also prescribed by Clauset etal. : σ ˆ β = ( − ˆ β − n ( n − √ n − β range from − − β can be found in Table 22. Figure 4 plotsthe power law index for each galaxy against the galaxy’sSFR. The ˆ β values are all consistent with one another,and there is no evidence for systematic trends in ˆ β withSFR (A Spearman rank correlation test gives a corre-lation coefficient of r s = 0.29 or a two-tailed p -value of Chomiuk et al. Fig. 2.—
Cumulative 20 cm luminosity functions for SNRs in the 18 sample galaxies. The spectral luminosity of Galactic SNR Cas A isalso marked as a solid vertical line.
Fig. 3.—
Cumulative SNR 20 cm luminosity functions scaled by the inverse of each galaxy’s SFR and then normalized by the SFR ofM82. Only SNRs in the complete samples are included. The spectral luminosity of Galactic SNR Cas A is also marked.
NR LF 9
Fig. 4.—
Best-fit values of β (the power-law index) plottedagainst SFR for each galaxy. Error bars represent 1 σ uncertain-ties in β determinations. The weighted-average β is marked witha dashed line. β = − . ± .
10. As can be seen in Figure 4, almostall of the galaxies fall within . σ of the mean, and noˆ β deviates by more than 1.7 σ . The galaxies with thelargest sample sizes (N >
20) and the best defined SNRLFs— M33, M82, NGC 6946, and M51— all have verysimilar LF indices in the range ˆ β = − − β = − . ± .
89 for this galaxy. All ofthe current data are consistent with being drawn from asingle power law.Additional evidence can be found for this assertion ifwe combine data from the 18 sample galaxies to make acomposite SNR LF, as can be seen in Figure 5. Becauseall of the surveys are sensitive to SNRs in the brightestbins, but only a few surveys are sensitive to SNRs in theleast luminous bins, we had to scale each bin to correctfor variable completeness. For a given bin, we summedup the SFRs for all galaxies whose SNR surveys are com-plete in that bin, and then we scaled the number countsin the bin by the inverse of this sum. This produces asmooth power law over almost four orders of magnitudein spectral luminosity. The power law is best fit with anindex β = − . ± .
07 (where the uncertainty is foundusing the true number of SNRs, n = 258). This is con-sistent within the uncertainties with ¯ β found by fittingeach galaxy individually and taking the weighted mean. LF Power-Law Scaling ( A ) As no maximum likelihood estimator is calculable for A , the normalization constant is found by binning thedata into spectral luminosity bins, and calculating thescaling that is needed to make L β . match the observednumber of sources in each bin. We impose the power-lawindex determined from the composite LF, β = − . A , we then takethe weighted average of the values found for each bin.To constrain the uncertainty on A , we run Monte Carlosimulations by randomly sampling the power law distri- bution function with the same number of points as are inthe complete SNR samples and determining A for thesesimulated data sets. After 10 runs per galaxy, we candetermine the uncertainty on A expected only due tosmall-number statistics. We then add this in quadraturewith the standard error of the weighted mean determinedfrom the data. Values for A are listed in Table 22. InNGC 4449, we excluded the brightest SNR from our cal-culation of A because it is a severe outlier in a smallsample, and therefore disproportionately affects A .We plot the differential LFs for our 18 sample galaxiesin Figure 6 and overplot the LF fits assuming β = − . A as described above. It is clear that many of theSNR samples suffer from small number statistics, but byand large the LFs are well-described by the power-lawformulation applied here.First and foremost, we would expect the 20 cm lumi-nosity of a SNR population to depend on a galaxy’s SFRif we assume that most radio SNRs are from the corecollapse of massive stars. In Figure 7, we plot A againstgalaxy SFR, and see that it correlates well with SFR.We fit log A as a function of log SFR by bootstrappinglinear least-squares fits to the data, excluding Arp 220from the fit because of the reasons discussed in Section2.3. We find: A = (cid:0) +18 − (cid:1) SF R . ± . , (9)which is marked by the solid line in Figure 7. A simplemodel for the SNR LF predicts that A should be linearlyproportional to the SFR (see Section 5.2), and our fit tothe data is consistent with this hypothesis. If we imposelinear proportionality between A and SFR, then the bestfit to the data is marked by a dashed line in Figure 7 andis expressed as: A = (cid:0) +17 − (cid:1) SF R. (10)Therefore, the current data are consistent with apower-law SNR LF with constant power-law index acrossgalaxies and scaling that is proportional to SFR. Moreattention will be given to the astrophysical implicationsof the LF in the next section. We also note that Arp220 (which was excluded from the fit) appears to be anoutlier, with unusually high A for its SFR, and this toowill be discussed in the next section. PHYSICAL MODELS FOR THE SNR LF
The synchrotron emission from radio SNRs is depen-dent on both the cosmic ray energy and the magneticfield energy in the remnant as described by Longair(1981): L ν ∝ a ( α ) V K B − α +1 ν α (11)where α is the spectral index of the synchrotron emissionas defined in Section 2.1; a is a constant that is dependenton α and can be found in Table 18.1 of Longair (1981); B is the magnetic field strength in the SNR; V is the volumeof the SNR; and K is the scaling factor of the CR electronenergy distribution, defined as N ( E ) dE = K E α − dE ,where N(E) is the number density of CR electrons of acertain energy in the remnant. Here, we will assumethe CR electron energy spectrum can be described as apower law E − which gives a synchrotron spectral indexof α = − . Fig. 5.—
Composite SNR LF including the SNR samples from all 18 sample galaxies. To correct for variable completeness between bins,the number of SNRs in each bin is scaled by the totaled SFR for all galaxies which are complete in that bin. Error bars represent simplePoissonian uncertainties. The dashed line is the best power law fit with β = − . In many systems, the minimum energy assumption isused to tease out the relative contributions of CRs andmagnetic fields to the synchrotron emission. However,there is really no physical reason for assuming equipar-tition in SNRs (Jones et al. 1998), and it is likely thatthe energy in the magnetic field is only a few percentof the energy in relativistic particles (Hillas 2005). Wetherefore do not make the minimum energy assumptionhere.BV04 model CR production and magnetic field am-plfication in SNRs with a full non-linear treatment ofdiffusive shock acceleration. They find that CR produc-tion peaks dramatically at the end of an SNR’s free ex-pansion phase, and during the Sedov phase the energyin CRs is approximately constant. Adiabatic losses tothe CR energy are presumably countered by low-levelongoing CR acceleration. The relatively low-energy CRelectrons which emit synchrotron in the radio (typicalenergies of ∼ ∼ years(Hillas 2005). We also note that synchrotron losses areunlikely to be important at 20 cm given the predictionsof BV04 that the amplified field strength is 10–100 µ Gin the Sedov phase. Thompson et al. (2006) state that the synchrotron cooling timescale is: τ syn = 8 . × B − / yr (12)at 1.45 GHz, where B = B /100 µ G. This is signifi-cantly longer than the duration of the Sedov phase (afew × years).The CR energy content of an SNR is only weakly de-pendent on the ambient ISM density according to BV04.In their models, increasing the ISM density by three or-ders of magnitude only increases the CR energy by a fac-tor of two in the Sedov phase (from 20% to 60% of theSN energy). This can also be seen if we evaluate the CRenergy using the simple test-particle assumption of Bell(1978, Equation 10) at the Sedov time (as the Sedov timeis when the vast majority of CRs are produced). CR en-ergy density is proportional to ISM density, but the SNRvolume at the Sedov time ∝ ρ − ; the two factors cancel,and the total CR energy is independent of density.Therefore, if we assume that most of the radio SNRsimaged in external galaxies are in their Sedov phase, wecan assume that their CR energy is roughly independentof time and ISM density. CR energy is simply a fractionof the SN explosion energy ( E SN ). If we in turn assumethat E SN is roughly constant, then the synchrotron emis-sion only depends on the magnetic field strength. Magnetic Field Compression Scenario
NR LF 11
Fig. 6.—
Differential SNR LFs for each of the 18 sample galaxies and power-law fits drawn with solid lines. All LFs are shown over twoorders of magnitude in spectral luminosity, except for NGC 4449 and M51 which have unusually large dynamic ranges. Power law indicesare held fixed at β = − .
07 and the scaling is allowed to vary. Vertical dotted lines represent the true completeness limits of the SNRsamples.
If, in a typical radio SNR in its Sedov phase, the mag-netic field is not amplified in the SNR but is insteadsimply compressed by the shock wave, then a compres-sion factor ( f ) describes the magnetic field in the SNRas a multiple of the magnetic field strength in the am-bient ISM ( B ). In the case of a strong shock passingthrough a randomly oriented magnetic field, f = 3.32(Reynolds & Chevalier 1981). When non-linear effectsare taken into account, the shock may become signifi-cantly modified and the compression factor may reach f ≈ dNdL . = dNdB (cid:18) dL . dB (cid:19) − . (13) dNdB describes the probability density (actually, the num-ber) of SNe exploding into an ISM with a given magnetic2 Chomiuk et al. Fig. 7.—
The LF scaling factor A as a function of SFR. The solidline is the best fit to the data (slope = 0.88), while the dotted linehas a slope of 1.0, representing a linear scaling of A with SFR. Arp220 is marked with a star and is excluded from the line fit. 30%uncertainties are plotted as error bars on SFR values. field strength. Let us assume a power law form for it: dNdB = D B η . (14)Using equation 11, we assume V K ∝ E SN as describedabove, and write dL . dB as: dL . dB ∝ E SN f . B . . (15)And finally, from equation 11, we know that B ∝ E − / SN f − L / . , so we find: dNdL . ∝ E ( − η − / . SN f ( − η − L ( η − . / . . . (16)Therefore, β = ( η − . / .
5. Assuming that β = − η = − ∼ SN and f constant). Of course, if there are many low-luminosity SNRs in M33 which are not currently observ-able, the implied dispersion in magnetic field strength could be significantly higher. The luminosities of SNRsin the 18 sample galaxies vary by almost four orders ofmagnitude, translating to a factor of a few hundred inmagnetic field strength.However, a simple B -field compression scenario is prob-ably not realistic. In young SNRs it is well establishedthat the magnetic field must be significantly amplifiedover the ambient ISM value to fit observations of X-ray synchrotron emission (V¨olk et al. 2005). In the Se-dov phase the X-ray synchrotron emission plummetsquickly (BV04), and therefore there is no direct testfor magnetic field amplification in more evolved SNRs.Thompson et al. (2009) used extragalactic radio SNRs asa test of magnetic field amplification by assuming that1% of the SN energy goes into CR electrons, and thenusing the 20 cm luminosities of SNRs to measure themagnetic field strength in SNRs. In normal star-forminggalaxies like the ones we study here, they find that themagnetic field strength in SNRs is greater (by factors ofa few to 10) than the strongest field obtainable by simplecompression of the ISM. They therefore claim that mod-est B field amplification is taking place in SNRs. Thesefindings only become stronger if we use the estimates ofBV04 which imply that only 0.2–0.6% of SN energy goesinto accelerating CR electrons. In this case, the SNRmagnetic fields will be stronger by a factor of 1.4–3 thanthose estimated by Thompson et al. Therefore, magneticfield amplification is likely shaping the SNR LF, and weuse it below to develop a physical interpretation of theSNR LF. Magnetic Field Amplification Scenario
BV04 assume that the magnetic field energy densityin an SNR is amplified to a fraction (1%) of the SNRpressure via the mechanism of Lucek & Bell (2000): B / (8 π ) = 0 . ρ v s . (17)This implies that the magnetic field is weakening as theSNR expands, and therefore the synchrotron luminositydecreases throughout the Sedov phase. We also assumethat the energy in CR electrons is a constant fractionof E SN and that v s can be described by the standardSedov similarity solution v s ∝ ( E SN /ρ ) / t − / . Thenthe spectral synchrotron luminosity scales as: L ν ∝ E . SN ρ . t − . . (18)This is consistent with the findings of BV04 despite theirmore detailed non-linear treatment of particle accelera-tion. Note that at a given SNR diameter, all SNRs shouldhave roughly the same spectral luminosity, with somespread due to E SN . However, remnants in denser mediareach the Sedov time (corresponding to their peak lumi-nosity) when they still have relatively small diameters.Therefore, their peak spectral luminosity is brighter thanthat of SNRs in lower density media, and they continueto be more luminous through much of the Sedov phase(see Figure 4 of BV04).In this case, we can write the SNR luminosity functionas: dNdL . = dNdt (cid:18) dL . dt (cid:19) − . (19) dNdt is the production rate of SNRs; if we assume thatNR LF 13most SNRs come from core-collapse SNe, then dNdt ∝ SFR. This is probably a fair assumption, as core-collapseSNe will preferentially occur in denser media than SNeType Ia and will therefore be more easily observable asremnants. dL . dt is simply the time derivative of equation18; we then use equation 18 to write t in terms of L . and find dNdL . ∝ SF R E . SN ρ . L − . . (20)Therefore, this simple model predicts β = − .
1, in verygood agreement with our observed β = − . B ∝ ρ v s , rather than B ∝ ρ v s (Bell 2004; see also Vink2008). This would predict a steeper time dependence forluminosity: L ν ∝ E . SN ρ . t − . (21)and a flatter power law index for the SNR LF: dNdL . ∝ SF R E . SN ρ . L − . . . (22)A SNR LF with a power law index of β = − B ∝ v s .Therefore, a magnetic field amplification model where B ∝ v s appears to best describe our data. We havealready stated that A appears to be proportional toSFR; can we exclude the possibility that A is actually ∝ SFR ρ . as predicted in Equation 20? In Figure 8,we visualize how the LF scaling factor A depends on ρ .We divide A by the SFR and plot it as a function of thedensity of the ISM for each galaxy; the model predictionof A /SFR ∝ ρ . is marked with a dashed line. Theredoes not appear to be any correlation of A /SFR with ρ for the 18 sample galaxies. A Spearman rank correlationtest gives a correlation coefficient of r s = − p value of 0.74, indicating no evidence of acorrelation. The data are very noisy, but we note thatM82 actually has a slightly lower A /SFR value than thedwarf irregular galaxies in our sample like IC 10 and theSMC, although the ISM density in these irregular galax-ies is approximately two orders of magnitude lower thanthe density in M82.“Average” ISM density may not be a good tracer of theISM density around SNRs, because, if all observed SNRsare from core-collapse SNe, they are exploding near theirstar formation sites. Therefore, the densities which arerelevant for the SNR LF are those in star-forming regions, not the global density of the galaxy. Perhaps the SNRLF is implying that the physical conditions inside star-forming regions do not vary much, even between dramat-ically different galaxies like the SMC and M82. This isconsistent with studies of the star cluster LF (e.g., Larsen2002), which imply that the masses of star clusters arerelatively invariant across galaxies and unaffected by theglobal ISM density.The 18 sample galaxies form a cloud at approximatelyconstant A /SFR, but Arp 220, with its extremely high Fig. 8.—
For each galaxy, the LF scaling factor A is divided bythe SFR and then plotted against the ISM density. Arp 220 ismarked with a star. Berezhko & V¨olk (2004) predict the depen-dence plotted as the dashed line: A /SFR ∝ ρ . . ISM density, displays an unusually high value of A /SFR.This is of questionable statistical significance, but may bean indication that A does indeed depend on density, andthat the conditions in the star-forming regions of Arp 220are fundamentally different from those in more quiescentgalaxies. More data on starburst galaxies are neededto better constrain the behavior of the SNR LF in thehigh ρ regime, as the SNR LF may have implications forhow global environment affects the physical conditions ofstar-forming regions.The lack of correlation between A /SFR and ρ alsosupports our assumption that a constant fraction of SNenergy goes into CR electrons. If the efficiency of cosmicray production did vary across galaxies, the most ba-sic expectation is that the efficiency would increase withincreasing ISM density (Bell 1978). If we assume thatthe energy in cosmic rays depends on the SN explosionenergy and ISM density as E CR ∝ E SN ρ γ , (23)where γ is a positive scaling index, then the SNR LF ofEquation 20 becomes modified to: dNdL . ∝ SF R E . SN ρ (0 . . γ )0 L − . . . (24)This implies that A should depend even more strongly on ρ than the A ∝ ρ . predicted above. Of course, there isno evidence for this in the data, as we have already seen4 Chomiuk et al.in Figure 8. The LF scaling is linearly proportional to theSFR, and there is no residual dependence on ISM density.We conclude that the density of gas surrounding SNRsdoes not vary much across galaxies, and therefore there islittle opportunity for variable efficiency in the productionof cosmic ray electrons across galaxies (regardless of thevalue of γ ). WHAT DETERMINES THE LUMINOSITY OF AGALAXY’S BRIGHTEST SNR?
Fig. 9.—
The 1.45 GHz spectral luminosity of the brightest SNRin each galaxy plotted against SFR. The solid line represents thebest linear fit in log-log space. Arp 220 is marked with a star andis excluded from the line fit. The dashed line is the mean fit toMonte Carlo simulations which represent the expected relation ifonly statistical sampling effects are taken into account.
We have seen that SFR can singlehandedly accountfor why some galaxies have higher total 1.45 GHz spec-tral luminosities in their SNRs than others, but can italso explain why some galaxies host much more lumi-nous individual remnants than others? There are twopossible explanations for why the brightest remnant inNGC 253 is ∼ ∝ ρ / . Therefore, perhaps SNRs in low- density galaxies are never as luminous as SNRs ingalaxies with dense ISM.2. The bright end of the SNR LF is dominated by asampling effect. In a galaxy with higher SFR, therewill be a larger population of SNRs, and one ismore likely to observe a SNR at its peak luminosity.In addition, there will be a larger population ofextremely massive stars, and, as we expect E SN tocorrelate with the mass of the progenitor, this willincrease the chance of a very luminous SNR ( L ν ∝ E . SN ). Therefore, in a galaxy with a higher SFR,one has a better chance of fleshing out the high-luminosity end of the LF (Lacey & Duric 2001).The Schmidt–Kennicutt Law implies that galaxies withhigher SFR will typically also have a denser ISM, so ascenario where the spectral luminosity of the brightestSNR (L max . ) ∝ SFR is not in direct conflict observation-ally with a scenario where L max . ∝ ρ . However, we candistinguish between these two possibilities by plottingeach galaxy’s L max . against its SFR (Figure 9). There isa solid near-linear correlation between these two quanti-ties which is best fit with the expression: L max . = (cid:0) +31 − (cid:1) SF R . ± . . (25)Again, Arp 220 is excluded from this fit.We can test if this relation is consistent with a simplestatistical sampling effect using Monte Carlo simulations.For each Monte Carlo run, we randomly sample the LFscaling A from a Gaussian distribution given log A =1.966 ± β = − × erg s − Hz − . We randomly choose this number of SNRs from apower law probability distribution with the same lowerlimit on luminosity and determine the spectral luminos-ity of the brightest remnant (L max,MC . ) in each galaxy.We then fit a line to log L max,MC . –log SFR just as wasdone for the real data. We perform 10 Monte Carlo runsin this fashion and finally calculate the mean slope andy-intercept from the 10000 individual line fits to find therelationship between L max and SFR which is purely dueto statistical sampling: L max,MC . = (cid:0) +46 − (cid:1) SF R . ± . . (26)The errors in the above equation are given by the stan-dard deviations of the slope and y-intercept from the 10 line fits; they represent the scatter one might expect inthe L max . –SFR relation due to random statistical sam-pling.The observed relation is consistent with the line de-rived from Monte Carlo simulations to within 1 σ ; seealso the similarity of the solid line and dashed line inFigure 9. This implies that truncation at the high lu-minosity end is not significantly affecting the observedLF; most galaxies simply do not have enough SNe explo-sions to thoroughly sample the SNR LF at the luminousend (right around the Sedov time). We can not excludethe possibility that SNR LFs are truncated at high lu-minosities, but if they are, their cut-off luminosities aresignificantly higher than the brightest observed SNRs.NR LF 15 CONCLUSIONS
We have analyzed 20 cm SNR samples in 19 galaxieswith SFRs ranging from 0.02 M ⊙ yr − to 127 M ⊙ yr − ,and we have reached the conclusion that the SNR LFis invariant across a wide range of host properties. AllLFs are consistent with a power law distribution with anindex β ≈ − B ∝ ρ v s , and are inconsistent with modelswhere B ∝ ρ v s . In applying the models, we assumethat the efficiency of CR production is constant, all SNRsare in the Sedov phase, and the densities of star-formingregions do not vary much between galaxies. These as-sumptions seem to describe all galaxies well, with thepossible exception of the ULIRG Arp 220. Its LF scal-ing may be too high to be explained by SFR alone, andmay imply that its star-forming regions are significantlydenser than those in the other 18 sample galaxies.In addition, we have shown that the correlation be-tween the luminosity of a galaxy’s brightest SNR and agalaxy’s SFR can be completely explained by statisticalsampling effects. The LF does not appear to be trun- cated at the high-luminosity end, and no physical justi-fication (e.g., variations in the ISM density) is needed toexplain why the brightest SNR in NGC 253 is three or-ders of magnitude more luminous than the brightest SNRin the SMC. Our findings support a scenario where theefficiency of CR production, the magnetic field strengthin SNRs, and the density of star-forming regions are alllargely independent of their host galaxies. ACKNOWLEDGMENTS
We are grateful to Ellen Zweibel, Todd Thompson,Brian Reville, John Everett and Mark Krumholz formany useful conversations. We also would like to thankJay Strader, Tommy Nelson, and Amanda Kepley fortheir insights. Finally, we acknowledge the work of ananonymous referee whose comments have improved thispaper. This material is based upon work supported un-der a National Science Foundation Graduate ResearchFellowship and was also supported by NSF grant num-ber AST-0708002.This research has made use of the NASA/IPACExtragalactic Database (NED) which is operated bythe Jet Propulsion Laboratory, California Institute ofTechnology, under contract with the National Aero-nautics and Space Administration. In addition, weacknowledge the usage of the HyperLeda database(http://leda.univ-lyon1.fr).
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NR LF 17
APPENDIX A: TURN-OVERS IN CUMULATIVE LFS AT LOW LUMINOSITY
As discussed in Section 2.4, we fit a power law to the luminous end of each galaxy’s cumulative LF, and the luminosityat which the cumulative LF turns over from the power law fit is considered the completeness limit. The location of thisturn-over is determined iteratively through a combination of line-fitting and visual inspection. First, we attempt to fita power law to a version of the cumulative LF which includes all SNRs in a given galaxy sample (see the left panel ofFigure 10). If the power law fit approximates the cumulative LF across the entire range of luminosity, we consider theSNR sample inherently complete. However, if at the lowest luminosities the power-law fit lies systematically above thecumulative LF, this implies that the SNR sample is incomplete at these low luminosities. This occurs in the LMC, ascan be seen in the left panel of Figure 10.Next, we try fitting a power law again, this time excluding the lowest-luminosity data point from the fit (equivalentto imposing a trial completeness limit on the data). If the cumulative LF (which now only includes data brighter thanthe trial completeness limit) still lies systematically below the power-law fit at low luminosities, the trial completenesslimit needs to be raised to higher luminosity, and the process repeated. The central panel of Figure 10 shows thefit to the LMC’s cumulative LF after the four least-luminous SNRs have been excluded; we now only consider datarightward of the vertical dashed line. However, the cumulative LF directly to the right of the dashed line still lies belowthe power law fit, so we can see that this trial completeness limit is still too low. We continue iterating and raisingthe trial completeness limit until our fit to the cumulative LF looks like that in the right panel of Figure 10. Thetrial completeness limit has been raised to exclude the eight lowest-luminosity SNRs in the LMC, and the cumulativeLF now adheres to a power law fit for luminosities brighter than the limit (rightward of the dashed line). Therefore,this is the “true” completeness limit for the LMC— 2.24 × erg s − Hz − . For all galaxies with the exception ofM51, this iterative process eventually converged to reveal a similar completeness limit. In the case of M51, the mostluminous remnant is so anomalously bright compared to the other SNRs that no reasonable power law fit could beachieved. We excluded the most luminous data point in M51’s cumulative LF so that good power law fits were found,and then proceeded to determine the completeness limit as described above. Fig. 10.—
To illustrate how the completeness limit is determined from cumulative LFs, the cumulative SNR LF for the LMC is picturedwith power law fits (short-dashed lines) over three different luminosity ranges. The vertical long-dashed lines mark the trial completenesslimit in each panel. In panel (A), all SNRs are included in the fit, while in panel (B), the four least-luminous SNRs (those to the left of thelong-dashed line) are excluded from the fit. The fit in panel (C) excludes the eight lowest-luminosity SNRs and corresponds to the truecompleteness limit.
TABLE 1Galaxy Parameters
Galaxy R.A. (2000) a Dec (2000) a Maj/Min Axis a i b Type a Distance c M B c SFR d ρ (hr:min:sec) ( ◦ : ′ : ′′ ) (arcmin) (deg) (Mpc) (mag) (M ⊙ yr − ) (M ⊙ pc − )LMC 05:23:34.5 − . ± . − − . ± . − f . ± . − . ± . − . ± . − − . ± . − . ± . − . ± . − . ± . − . ± . − − . ± . − − . ± . − . ± . − − . ± . − . ± . − . ± . − . ± . − . ± . − g − h i a From NED. b From HyperLeda (Paturel et al. 2003), except the values for the Magellanic Clouds which come from Groenewegen (2000) and theinclination for NGC 2366 from Hunter et al. (2001). c From Kennicutt et al. (2008). d SFRs are calculated using a combination of H α and 25 µ m fluxes as calibrated by Calzetti et al. (2007), and using H α luminosities fromKennicutt et al. (2008), and 25 µ m IRAS fluxes from the following references in order of preference: Sanders et al. (2003), Rice et al.(1988), Lisenfeld et al. (2007), Moshir et al. (1992). e Rough estimate of the average ISM densities in SNR survey areas, calculated as described in Section 3. A constant scale height isassumed for all galaxies, and therefore the uncertainty on these calculations is at least 50%. f From de Vaucouleurs & Freeman (1972). g Calculated assuming H = 75 km s − Mpc − . h Calculated using the RC3 B mag as provided by NED, the average of the foreground extinctions in Burstein & Heiles (1978) andSchlegel et al. (1998), and a distance of 77 Mpc. i As determined by Anantharamaiah et al. (2000), but converted to the IMF used in the Calzetti et al. (2007) calibration.
NR LF 19
TABLE 220 cm SNR Survey Parameters
Galaxy 20 cm Freq Telescope/ Synthesized Beam Spatial Res. a RMS Sensitivity F.O.V. SFR Frac. b (GHz) Config. (arcsec ) (pc) ( µ Jy/beam) (arcmin)LMC d e f g h c i h c h c j k l <
200 pc) m >
200 pc) n c h o p q r c s t a Geometric mean of the synthesized beam major and minor axes, taken at the distances listed in Table 1. b Fraction of the SFR (from Table 1) inside the SNR survey area. c Approximate: sensitivity at large radius is limited by chromatic aberration (bandwidth smearing). d Filipovi´c et al. (1998); e Filipovi´c et al. (2002) and Payne et al. (2004); f Yang & Skillman (1993); g Gordon et al. (1999); h Chomiuk & Wilcots (2009); i Pannuti et al. (2000); j Allen & Kronberg (1998) and McDonald et al. (2002); k Kaufman et al. (1987); l Pannuti et al. (2002); m Ulvestad & Antonucci (1997); n Ulvestad (2000); o Maddox et al. (2006); p Duric & Dittmar (1988); q Lacey et al.(1997); r Hyman et al. (2001); s Maddox et al. (2007); t Parra et al. (2007)
TABLE 3SNRs in the LMC taken from Filipovi´c et al. (1998) ID a R.A. (2000) Dec. (2000) S . α (hr:min:sec) ( ◦ : ′ : ′′ ) (mJy)B0450-7055* 04:49:48.85 -70:51:08.8 749 ± − . ± . ± − . ± . ± − . ± . ± − . ± . ± − . ± . ± − . ± . ± − . ± . ± − . ± . ± − . ± . ± − . ± . ± − . ± . ± − . ± . ± − . ± . ± − . ± . ± − . ± . ± − . ± . a An asterisk following the SNR ID denotes that the SNR is in the complete sample for this galaxy. See Section 2.4 for more explanation.
TABLE 4SNRs in the SMC taken from Payne et al. (2004)
ID R.A. (2000) Dec. (2000) S . α (hr:min:sec) ( ◦ : ′ : ′′ ) (mJy)J004100-733648 00:41:00.10 -73:36:48.6 77 . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . TABLE 5SNRs in IC 10 taken from Yang & Skillman (1993)
ID R.A. (2000) Dec. (2000) S . α (hr:min:sec) ( ◦ : ′ : ′′ ) (mJy)HL20a* 00:20:10.15 +59:19:13.6 1 . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − TABLE 6SNRs in M33 taken from Gordon et al. (1999)
ID R.A. (2000) Dec. (2000) S . α (hr:min:sec) ( ◦ : ′ : ′′ ) (mJy)2* 1:32:30.49 30:27:42.5 4 . ± . − . ± .
14* 1:32:39.83 30:38:17.9 0 . ± . − . ± .
35* 1:32:42.30 30:20:54.6 2 . ± . − . ± .
38* 1:32:53.29 30:38:10.2 0 . ± . − . ± .
89* 1:32:56.49 30:40:39.7 1 . ± . − . ± . . ± . − . ± .
316 1:33:02.59 30:29:38.5 0 . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± .
168 1:33:37.65 30:32:00.7 0 . ± . − . ± . . ± . − . ± . NR LF 21
TABLE 6 —
Continued
ID R.A. (2000) Dec. (2000) S . α (hr:min:sec) ( ◦ : ′ : ′′ ) (mJy)73* 1:33:40.49 30:45:57.8 1 . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± .
278 1:33:42.37 30:32:58.6 0 . ± . − . ± . . ± . − . ± .
184 1:33:45.00 30:36:00.5 0 . ± . − . ± .
885 1:33:45.42 30:36:26.8 0 . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± .
292 1:33:48.33 30:39:35.0 0 . ± . − . ± .
294 1:33:49.59 30:39:54.6 0 . ± . − . ± .
295 1:33:49.93 30:39:42.5 0 . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . TABLE 7SNRs in NGC 1569 taken from Chomiuk & Wilcots (2009)
ID R.A. (2000) Dec. (2000) S . α (hr:min:sec) ( ◦ : ′ : ′′ ) (mJy)N1569-04* 4:30:44.35 64:51:20.1 1 . ± . − . ± . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . TABLE 8SNRs in NGC 300 taken from Pannuti et al. (2000)
ID R.A. (2000) Dec. (2000) S . α (hr:min:sec) ( ◦ : ′ : ′′ ) (mJy)R1* 00:54:38.2 -37:41:47 0 . ± . < − . . ± . < − . . ± . − . ± . . ± . < − . . ± . < − . . ± . − . ± . . ± . < − . . ± . < − . . ± . < − . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . − . ± . . ± . < − . TABLE 9SNRs in NGC 4214 taken from Chomiuk & Wilcots (2009)
ID R.A. (2000) Dec. (2000) S . α (hr:min:sec) ( ◦ : ′ : ′′ ) (mJy)N4214-02* 12:15:34.74 36:20:17.1 0 . ± . − . ± . . ± . < − . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . − . ± . . ± . < − . . ± . < − . NR LF 23
TABLE 10SNRs in NGC 2366 taken from Chomiuk & Wilcots (2009)
ID R.A. (2000) Dec. (2000) S . α (hr:min:sec) ( ◦ : ′ : ′′ ) (mJy)N2366-07* 7:28:30.41 69:11:33.8 0 . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . TABLE 11SNRs in M82 taken from Allen & Kronberg (1998) andMcDonald et al. (2002)
ID R.A. (2000) Dec. (2000) S . α (hr:min:sec) ( ◦ : ′ : ′′ ) (mJy)39.10+57.4* 09:55:47.88 +69:40:43.6 7 . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . TABLE 12SNRs in M81 taken from Kaufman et al. (1987)
ID R.A. (2000) Dec. (2000) S . α (hr:min:sec) ( ◦ : ′ : ′′ ) (mJy)178* 09:55:47.74 +68:59:18.3 1 . ± . − . ± . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . < − . . ± . < − . . ± . − . ± . TABLE 13SNRs in NGC 7793 taken from Pannuti et al. (2002)
ID R.A. (2000) Dec. (2000) S . α (hr:min:sec) ( ◦ : ′ : ′′ ) (mJy)N7793-R1* 23:57:40.2 -32:37:38 0 . ± . < − . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . − . ± . . ± . − . ± . TABLE 14SNRs in NGC 253 taken from Ulvestad & Antonucci (1997) andUlvestad (2000)
ID R.A. (2000) Dec. (2000) S . α (hr:min:sec) ( ◦ : ′ : ′′ ) (mJy)1 00:47:26.81 -25:17:37.2 0 . ± . − . ± .
252 00:47:26.91 -25:17:33.0 0 . ± . − . ± .
313 00:47:27.53 -25:17:59.0 0 . ± . < − .
54 00:47:27.99 -25:17:15.4 2 . ± . − . ± .
257 00:47:28.42 -25:17:09.4 1 . ± . − . ± .
129 00:47:29.90 -25:17:38.6 3 . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . − . ± . TABLE 15SNRs in NGC 4449 taken from Chomiuk & Wilcots (2009)
ID R.A. (2000) Dec. (2000) S . α (hr:min:sec) ( ◦ : ′ : ′′ ) (mJy)N4449-07* 12:28:09.67 44:05:19.8 0 . ± . − . ± . . ± . < − . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . − . ± . NR LF 25
TABLE 16SNRs in M83 taken from Maddox et al. (2006)
ID R.A. (2000) Dec. (2000) S . α (hr:min:sec) ( ◦ : ′ : ′′ ) (mJy)2* 13:36:50.83 -29:51:59.6 0 . ± . < − .
73* 13:36:50.86 -29:52:38.5 0 . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . TABLE 17SNRs in NGC 4736 taken from Duric & Dittmar (1988)
ID R.A. (2000) Dec. (2000) S . α (hr:min:sec) ( ◦ : ′ : ′′ ) (mJy)6 12:50:49.2 +41:07:15 0 . ± . − . ± . . ± . − . ± .
269 12:50:49.5 +41:07:47 0 . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . TABLE 18SNRs in NGC 6946 taken from Lacey et al. (1997)
ID R.A. (2000) Dec. (2000) S . α (hr:min:sec) ( ◦ : ′ : ′′ ) (mJy)8* 20:34:31.05 +60:08:27.5 0 . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . < − .
444 20:34:49.69 +60:12:40.4 0 . ± . < − .
247 20:34:50.80 +60:07:47.9 0 . ± . < − . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . < − .
688 20:35:04.04 +60:09:54.3 0 . ± . < − . TABLE 18 —
Continued
ID R.A. (2000) Dec. (2000) S . α (hr:min:sec) ( ◦ : ′ : ′′ ) (mJy)89* 20:35:04.18 +60:10:54.7 0 . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . < − . . ± . < − . . ± . − . ± . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . − . ± . TABLE 19SNRs in NGC 4258 taken from Hyman et al. (2001)
ID R.A. (2000) Dec. (2000) S . α (hr:min:sec) ( ◦ : ′ : ′′ ) (mJy)12* 12:18:56.3 47:16:50 0 . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . TABLE 20SNRs in M51 taken from Maddox et al. (2007)
ID R.A. (2000) Dec. (2000) S . α (hr:min:sec) ( ◦ : ′ : ′′ ) (mJy)2 13:29:36.56 47:11:05.5 0 . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . < − . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . . ± . − . ± . NR LF 27
TABLE 21SNRs in Arp 220 taken from Parra et al. (2007)
ID R.A. (2000) Dec. (2000) S . α (hr:min:sec) ( ◦ : ′ : ′′ ) (mJy)W42 15:34:57.2123 22:30:11.482 0 . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . − TABLE 22SNR LF Parameters
Galaxy β A a LMC − . ± .
67 11 +5 − SMC − . ± .
47 4 . +1 . − . IC 10 − . ± .
52 2 . +2 . − . M33 − . ± .
17 13 +3 − NGC 1569 − . ± .
46 19 +5 − NGC 300 − . ± .
55 11 +3 − NGC 4214 − . ± .
24 18 +10 − NGC 2366 − . ± .
13 10 +5 − M82 − . ± .
26 670 +180 − M81 − . ± .
18 46 +28 − NGC 7793 − . ± .
70 24 +27 − NGC 253 − . ± .
48 870 +520 − NGC 4449 − . ± .
33 15 +14 − M83 − . ± .
48 98 +47 − NGC 4736 − . ± .
63 100 +70 − NGC 6946 − . ± .
29 140 +50 − NGC 4258 − . ± .
22 270 +130 − M51 − . ± .
25 140 +80 − Arp 220 − . ± .
89 57000 +39000 − a Calculated assuming β = − ..