The K-band luminosity functions of super star clusters in luminous infrared galaxies, their slopes, and the effects of blending
Zara Randriamanakoto, Petri Vaisanen, Stuart Ryder, Erkki Kankare, Jari Kotilainen, Seppo Mattila
MMon. Not. R. Astron. Soc. , 1–18 (2012) Printed 6 October 2018 (MN L A TEX style file v2.2)
The K-band luminosity functions of super star clusters inluminous infrared galaxies, their slopes, and the effects ofblending
Z. Randriamanakoto , (cid:63) , P. V¨ais¨anen , , S. Ryder , E. Kankare , J. Kotilainen ,S. Mattila , South African Astronomical Observatory, P.O. Box 9 Observatory, Cape Town, South Africa University of Cape Town, Astronomy Department, Private Bag X3, Rondebosch 7701, South Africa Southern African Large Telescope, P.O. Box 9 Observatory, Cape Town, South Africa Australian Astronomical Observatory, P.O. Box 915, North Ryde, NSW 1670, Australia Tuorla Observatory, Department of Physics and Astronomy, University of Turku, V¨ais¨al¨antie 20, FI-21500 Piikki¨o, Finland Finnish Centre for Astronomy with ESO (FINCA), University of Turku, V¨ais¨al¨antie 20, FI-21500 Piikki¨o, Finland
Accepted 2013 January 29. Received 2013 January 29; in original form 2012 September 6
ABSTRACT
Super star clusters (SSCs) are typically found in interacting galaxies and tracean extreme form of star-formation. We present a K -band study of SSC candidates ina sample of local luminous infrared galaxies (LIRGs) using two adaptive optics in-struments (VLT/NACO and Gemini/ALTAIR/NIRI). In addition to facilitating SSCdetections in obscured environments, this work introduces SSC studies in hosts withhigher star-formation rates (SFRs) than most previous studies. We find that the lumi-nosity functions (LFs) of the clusters are reasonably well-fitted by a single power-lawwith the values of the index α ranging between 1.5 to 2.4 with an average value of α ≈ .
9. This value appears to be less steep than the average α ≈ . D L ∼
70 Mpc) blend-ing effects have to be taken into account, and are investigated using Monte Carlosimulations of blending effects for LFs and a photometric SSC analysis of the well-studied Antennae system which is artificially redshifted to distances of our sample.While blending tends to flatten LFs our analyses show that ∆ α is less than ∼ M K < − Key words: galaxies: interactions - galaxies: star clusters: general - infrared: galaxies
Young massive star clusters are related to triggers of starformation (SF) in galaxies and contain clues to the physi-cal conditions under which extremely strong SF happens.These clusters, often called super star clusters ˝ (SSCs)when masses are in the 10 - 10 M (cid:12) range, are typicallyfound in interacting and merging gas-rich galaxies. Theirbirth, evolution and disruption are not well understood whilethese issues are very interesting in the context of clustered (cid:63) E-mail: [email protected]
SF in general (e.g. Lada & Lada 2003) and in particular sincethey might be the progenitors of globular clusters (e.g. Ash-man & Zepf 1992; Holtzman et al. 1992; Ho & Filippenko1996; Elmegreen & Efremov 1997). For a recent review ofyoung and massive star clusters see Portegies Zwart et al.(2010).SSCs can be used to trace the history of bursts of SFin galaxies (e.g Bastian 2008; Escala & Larson 2008; Escala2011; Adamo et al. 2011b) since SSCs are bright and rel-atively simple to model as single stellar populations. It isimportant to understand how wide-spread starbursts can bein interactions and mergers, particularly in the context of © a r X i v : . [ a s t r o - ph . C O ] F e b Z. Randriamanakoto et al. high-redshift galaxy formation, since these conditions couldbe similar to what is happening at z >
1. Local LIRGs (lumi-nous IR galaxies, log L IR /L (cid:12) >
11) exhibit extreme star for-mation and large populations of SSCs and they are believedto be good analogs for higher- z star formation in general(e.g. Alonso-Herrero et al. 2009; Elbaz et al. 2011; Tekolaet al. 2012).Although the young and compact SSCs (age ∼
10 -100 Myr, r eff ∼ D L (cid:46)
25 Mpc) non-LIRG starburst systems, simplybecause it is much easier (e.g. Whitmore et al. 1993, 1999;Bik et al. 2003; Haas et al. 2008). SSCs were, in fact, oneof the first discoveries of the
Hubble Space Telescope (HST) while imaging the central regions of NGC 1275 (Holtzmanet al. 1992). In particular the closest major merger system,the Antennae (NGC 4038/4039, log L IR = 11 .
0) has beenamongst the most studied SSC hosts, providing a sampleof thousands of these clusters. There have been some in-dividual cases studied further away, and at higher SF lev-els, such as Haro 11 with log L IR = 11 .
22 at 81.2 Mpc andthe Bird galaxy (IRAS 19115-2124) with log L IR = 11 .
87 at206 Mpc, by Adamo et al. (2010) and V¨ais¨anen et al. (2008),respectively. The only published work thus far characterisingSSCs from a significant sample of galaxies is that of Miralles-Caballero et al. (2011), who analysed optically-selected starforming knots, SSCs, or complexes of clusters in 32 LIRGsand ultraluminous IR galaxies (ULIRGs, log L IR /L (cid:12) > N ( L ) dL ∼ L − α dL, (1)where α ∼ α ∼ − . α ≈ . − .
0, though at distances >
200 Mpc where res-olution and blending effects may play a role, they find thatthe LF flattens to α ∼ I -band CLF in their verystrongly star forming blue compact dwarf galaxies (54 to82 Mpc), while Surace et al. (1998) and Inami et al. (2010)find similar slopes between 1.1 and 1.8 in the B -band forthree (U)LIRGs.In our ongoing study we are expanding the SFR rangeof SSC host galaxies to include LIRGs from their lowestlimit to ULIRGs, and including galaxies in various stages ofinteractions, from isolated and paired galaxies, to interact-ing, merging and merger remnant stages with a sample of(ultimately) dozens of targets (V¨ais¨anen et al. 2012). Fur-thermore, most of the studies thus far have been done inthe optical, making extinction effects potentially difficult inthe notoriously complex dusty environments of gas-rich in-teractions. We use K S -band adaptive optics (AO) observa-tions which match the spatial resolution of the HST opticalstudies. Crucially, the use of near infrared (NIR) has greatpotential in opening a new angle into the SSC populationsin that they probe deeper into the dusty birth regions ofthe SSCs and the very obscured regions in the inner partsof the galaxies. There is also an interesting time-window atages of ∼
10 Myr when SSCs are expected to be very NIR-luminous due to their high-mass stars entering the red su-pergiant (RSG) phase. There are very few NIR studies ofSSCs compared with the optical ones, including Lai et al.(1999), Mengel et al. (2005), Alonso-Herrero et al. (2006), Hereafter, we will refer to both the Johnson K -band and2MASS K S -band observations as K -band. ©000
10 Myr when SSCs are expected to be very NIR-luminous due to their high-mass stars entering the red su-pergiant (RSG) phase. There are very few NIR studies ofSSCs compared with the optical ones, including Lai et al.(1999), Mengel et al. (2005), Alonso-Herrero et al. (2006), Hereafter, we will refer to both the Johnson K -band and2MASS K S -band observations as K -band. ©000 , 1–18 SC LFs in LIRGs Pollack et al. (2007), V¨ais¨anen et al. (2008), and Adamoet al. (2010).The main specific objectives of this work are to derivethe K -band LFs of the massive star clusters and to evaluatethe effect of blending on the LFs. In forthcoming papers wewill study the mass functions and age distributions of thetargets by combining optical data with the NIR photometry.The paper is organised as follows: we describe the data andits reduction in §
2. The cluster analysis is presented in § § §
5; finally, we summarise our findings thensuggest our future work in § H = 73 km s − Mpc − , Ω M = 0 .
27, and Ω Λ = 0 . We analyse a representative sample of ten local LIRGs froman ongoing NIR AO survey which is mainly intended tosearch for dust-obscured core-collapse supernovae (see e.g.Mattila et al. 2007; Kankare et al. 2008, 2012; V¨ais¨anenet al. 2009). The targets analysed here were drawn fromthe
IRAS Revised Bright Galaxy Sample (Sanders et al.2003). They lie at redshifts of 0.01 < z < L IR /L (cid:12) ) = 11 . f /f < K -band using two different ground-based instruments with AOimaging: the NAOS-CONICA on the ESO Very Large Tele-scope (VLT/NACO) and the ALTAIR/NIRI on the Gemini-North telescope (ALTAIR/NIRI). For AO correction, natu-ral guide stars were used for the NACO data, whereas laserguide stars, with a tip/tilt reference star, were used with theALTAIR/NIRI data – this requirement of suitably brightreference stars is a further sample selection constraint, butdoes not bias the LIRG characteristics in any way. Our present sample includes two NACO galaxies,IRAS 18293-3413 and IRAS 19115-2124, the latter also be-ing the subject of our pilot SSC study in relatively distantgalaxies compared to most SSC works (V¨ais¨anen et al. 2008).The data taken with the VLT UT4 and NACO S27 cam-era have a plate scale of 0.027 ˝ pix − and a field of view(FOV) of 27 ˝ . The AO correction worked well resulting in aFWHM ∼ ˝ for the point sources. Frames were taken withexposure times of 30 sec in dithering mode with an integra-tion time per pointing of 90 sec , and Table 1 lists the totalintegration times. More details of the NACO observations,as well as data reduction, are given in Mattila et al. (2007)and V¨ais¨anen et al. (2008) for the two galaxies, respectively. The other 8 targets come from a recent multi-epoch sur-vey using Gemini-North during 2008-2012. The pixel scaleis 0.022 ˝ pix − , yielding a FOV of 22 ˝ . These data also havea final resolution of ∼ ˝ . Each individual frame has an exposure time of ∼ sec , and in this case separate sky-frames were taken for sky subtraction in addition to on-target dithering. Refer to Table 1 for details of observations.Our Gemini data were reduced using IRAF -based tasksincluding flat-fielding and sky subtraction. Individual frameswith significantly lower quality PSFs were excluded, as wellas some with abnormal electronic noise. A weaker, horizon-tal stripe pattern was still evident and was removed by acustom-made de-striping algorithm. The final images wereproduced by co-adding by average-combining the individualframes from different observing runs after shifting them toa common reference. Table 1 again lists the effective integra-tion times per target. The astrometry calibration was per-formed with the IRAF task
CCMAP : we downloaded archival
HST /ACS data of the fields and accurately re-calibratedthese using the Guide Star Catalog II, and then added theWorld Coordinate System (WCS) into the FITS headers ofour NIR images using the larger FOV
HST frames as refer-ence images.
We obtained K -band images of all the Gemini targets usingthe Nordic Optical Telescope (NOT) NIR Camera and spec-trograph (NOTCam). In the wide field imaging mode, NOT-Cam has a pixel scale of 0.234 ˝ pix − and a FOV of 4’. Theimages were flat-field corrected, sky subtracted and com-bined using IRAF -based tasks. The NOTCam images wereused as intermediate images to determine photometric zero-points for the smaller FOV NIRI images.
For object detection we ran
SExtractor v2.5.0 (Bertin& Arnouts 1996) on unsharp-masked versions of the im-ages (see the top-right panel of Figure 1 in the case ofIRAS 18293-3413); unsharp masking was done in order tomake point-source detection more uniform in varying back-ground conditions, though we stress that photometry (seebelow) was done always on original images. Critical param-eters for detection were tuned to minimise spurious sourcesand to include faint and also extended sources in the out-put catalogues. A threshold of ∼ σ above the backgroundRMS noise combined with an upper value of the minimumnumber of ∼
10 adjacent pixels above threshold were even-tually chosen.We performed aperture photometry on the combinedimages using the task
IRAF/PHOT with 3 and 5 pixel aper-ture radii (approximately equal to a FWHM ∼ ˝ of apoint source) and sky annuli from 5 to 7 and 7 to 10 pixels( ∼ ˝ wide) for the NACO and the ALTAIR/NIRI data,respectively. A small sky annulus is necessary for a goodsky sampling in the strongly varying background in betweenlarger-scale features of the galaxy, while small apertures are IRAF is distributed by the National Optical Astronomy Obser-vatories, which are operated by the Association of Universities forResearch in Astronomy, Inc., under cooperative agreement withthe National Science Foundation. © , 1–18 Z. Randriamanakoto et al.
Galaxy name Exp time RA DEC l b log L IR SFR m − M D L ( sec ) (J2000) (J2000) ( degrees ) ( degrees ) ( L (cid:12) ) ( M (cid:12) yr − ) (Mpc)IC 694 1260 11 28 33.5 +58 33 45 141.9 55.4 11.60 a a −
04 00 53 23.4 9.4 11.35 38.1 33.79 57.3IRAS F17138-1017 990 17 16 35.8 −
10 20 39 12.2 15.6 11.42 44.7 34.29 72.2IRAS 18293-3413 1230 18 32 41.1 −
34 11 27 0.1 -11.3 11.81 109.8 34.37 74.6MGC +08-11-002 1140 05 40 43.7 +49 41 41 161.6 9.9 11.41 43.7 34.51 79.9IRAS F16516-0948 900 16 54 24.0 −
09 53 21 9.5 20.5 11.24 29.5 34.88 94.8IC 883 1440 13 20 35.3 +34 08 22 82.9 80.6 11.67 79.5 35.02 101IRAS 19115-2124 1410 19 14 30.9 −
21 19 07 16.1 -14.4 11.87 126 36.56 206
Table 1.
Total exposure times, equatorial (RA, DEC) and galactic ( l, b ) coordinates of the observed sample are shown in this tableordered by distance. The luminosity distance, D L , and the distance modulus, m − M , are retrieved from NASA/IPAC ExtragalacticDatabase (NED) while the logarithmic value of the IR luminosity, L IR , is as estimated by Sanders et al. (2003) with a slightly differentsetting of the cosmological parameters. a These two galaxies are components of an interacting system Arp 299; the total L IR isdivided between IC 694 (known as nucleus A) and NGC 3690 (nuclei B+C) following Mattila et al. (2012) the circum-nuclear SF portionbeing evenly split. An empirical relation using L IR by Kennicutt (1998) is used to derive the SFR. Figure 1.
The NACO field of IRAS 18293-3413.
Top left : The field of the whole interacting system with contours demarcating thefour selected background regions for completeness analysis.
Top-right : A slightly smaller field around the primary galaxy is shown afterunsharp masking. The SSC detections are made from this image.
Lower left : All SSC candidate detections are overlaid as white on theoriginal image (where the photometry is performed) while those that meet all the SSC candidate selection criteria presented in Section 3.2are shown as red.
Lower-right : Same as previous, but only for a zoomed-in region around the nucleus. Small tick marks are in 1 ˝ units,except in the last panel where they are in 0.1 ˝ units. © , 1–18 SC LFs in LIRGs needed to minimise blending effects, especially in the case ofthe crowded SSC populations detected in the NACO data ofIRAS 18293-3413 (hence the smaller apertures with NACOdata).Since the apertures are small, aperture corrections areessential. In addition, the PSF shape is expected to varyacross the frames as a function of distance from the star usedas the AO-reference. Aperture corrections were determinedbased on the curve-of-growth (out to 1.0”) of sufficientlybright and isolated stars at various locations throughout theimages. In the case of NACO data the aperture correctionswere found to be dependent on the distance of the AO ref-erence star, while in the Gemini data, where the laser guidestar is located in the centre of the field within the targetgalaxy, no such systematics were found. These aperture cor-rections a c , as a function of radial position from the AOstar if required, were then applied to measurements of can-didate clusters using an aperture of 0.1” radius. The NACOdata a c values applied to individual candidate SSCs rangefrom − .
23 to − .
07 mag and for the ALTAIR/NIRI datawe adopted an averaged constant a c = − .
23 mag. We esti-mate the uncertainty of the aperture correction in a singleframe to be typically ∼ K S -band point source catalogue using 2MASS com-mon stars in the FOV, making our magnitude system thesame as that of the 2MASS Vega-based K S filter. In caseswhere the 2MASS point-sources were outside the small FOVof the Gemini data, NOTCam K -band images were usedas intermediate images to estimate the zero-points. ForMCG +08-11-002 and CGCG 049-057, comparing the totalintegrated flux of the galaxy with the corresponding 2MASSmagnitude was the only option to estimate their zero-points.We estimate the absolute calibration to be accurate to ∼ Deciding whether a detected source is a SSC candidate ornot is challenging when working with a single filter and deal-ing with a sample of distant targets where both stars andstar clusters have similar PSF sizes. In our case, the follow-ing steps were carried out to generate the final K -band SSCcatalogue for each target.(i) First of all, only objects falling on detectable opticalor NIR emission from the galaxy within the frames wereconsidered. The edges of frames where the noise is higherwere also excluded in case the host galaxy extended there.After this process, we assumed that the catalogue is mainlycomposed of real sources since we had already tuned thedetection parameters to avoid false detections in the innerregions of the frames.(ii) Foreground contamination by Milky Way stars isa real possibility, especially for targets with low galacticlatitude ( b < | ◦ | , see Table 1). Therefore, we estimatedthe potential effect of contamination in our data by using Name N(SSC) Comp.limit ( K -band)( σ m (cid:54) Table 2.
After imposing our selection criteria, the final numberof SSC candidates for each target is given. We tabulate also theapparent and absolute magnitudes of the 50 % completeness levelcorresponding to the background region where more than ≈
50 %of the data points are below the contour level limiting that region(see text). The brighter absolute magnitude completeness limitfor IRAS 19115-2124 is due to its much larger distance comparedto the other targets. the Besan¸con model (Robin et al. 2003). IRAS 19115-2124,IRAS 18293-3413, and IRAS 17578-0400 yield the largestnumbers of expected K -band stars according to the model,but even in these fields the maximum numbers of contam-inating stars at K ∼
16 mag corresponding to the brightestmagnitude bins of SSCs, are at most at ∼
10 % of the totalSSC candidate numbers within a typical 20 ˝ × ˝ area ofthe galaxies. At fainter magnitudes star counts are at negli-gible levels, as are counts at all magnitude bins for the restof the fields, according to the models. However, as a consis-tency check, we also counted obvious point sources in thesky areas outside of the galaxies in our fields, derived theirsurface densities in magnitude bins, and compared to ob-ject numbers within the host galaxy area. As an example weshow the low galactic latitude IRAS 18293-3413 case in theleft panel of Fig. 2. These numbers are consistent with theBesan¸con model predictions at bright magnitudes, but tendto be somewhat larger at K >
16: it could, for example, bethat there are some real SSCs well outside of the projectedgalaxy area of our targets, or that the models do not probeMilky Way stars at faint enough levels. Though there mightbe some contamination at the brightest K ∼ −
16 bins,these would be ”obvious stars” that are removed in the fi-nal selection step (v) below and at fainter levels the starnumbers still are not large enough to make any significantdifference to the ultimately derived luminosity functions ofSSCs. We hence conclude that foreground contamination isnot significant for our results, and no further star vs. SSCseparation was attempted for our dataset.(iii) Photometric uncertainties were taken into accountfairly conservatively to have a robust list of SSC candi-dates: we excluded all detections having errors σ m > © , 1–18 Z. Randriamanakoto et al.
Figure 2.
Left:
Raw surface densities of detected objects within the galaxy area of frame IRAS 18293-3413 are shown in blue, whilethose of objects found outside the galaxy area in the same frame are shown in red. As the latter are likely foreground stars, it is seenthat stellar foreground contamination is insignificant even in this low galactic latitude case. Apparent K -band magnitude scale is shownat the top. Right:
An example of SSC candidate selection for the same field of IRAS 18293-3413. A FHWM vs concentration index C plot of the detected objects from SExtractor with the cutoffs for candidate selection indicated. The limits are checked with simulationsto be consistent with the parameter space of extracted intrinsic point sources ( ∼ σ m < C , defined as the difference in magnitudes betweenthe 3 and 5 pixel radius apertures ( m px − m px ), as well ascuts on the FWHM. A small value of the index ( C ∼ C ∼ C isless accurate for fainter sources, we also used the FWHM ofthe detection, as measured by the task RADPROF in pixels tomake the selections more robust. Plots such as that shownin the right panel of Figure 2, were generated for each targetfield to help us distinguish between truly fuzzy objects andunresolved SSC candidates at different S/N levels.The cutoff values were decided with the help of simula-tions and also by-eye checks interactively for consistency. Aspart of Monte Carlo simulations to define completeness cor-rections (described in Section 3.3) simulated intrinsic pointsource PSFs were extracted from the real data frames, andthe cutoffs in FWHM and C were adjusted to encompass the output parameter space of the detections of the input pointsources in the simulations made on the real data frames.The differing AO corrections on different datasets introducessome variation, but in most of the cases SSC candidate se-lection included objects having values of 0 . (cid:46) C (cid:46) . . (cid:46) FWHM (cid:46) . SExtractor are marked on the image, but only those whichmet all the selection criteria above are in red (most are ex-cluded because of their magnitude uncertainty in this case).
The raw SSC catalogues obtained in the previous sectionare affected by incompleteness due to photometric detec-tion limits of the observations; the varying and complicatedbackground resulting from the different diffuse componentsof the host galaxies; and potential crowding of sources. Tocorrect the data for incompleteness bias, we ran Monte Carlo(MC) completeness simulations with each science image. APSF model for each target field was first created. The LIRGsMCG+08-11-002 and CGCG 049-057 did not have isolatedstars in their fields, and for them we used a representativePSF model from other fields having a similar distance fromtheir tip-tilt reference star, and hence similar AO-correctionbased from the reference star. Using the model, we created ©000
The raw SSC catalogues obtained in the previous sectionare affected by incompleteness due to photometric detec-tion limits of the observations; the varying and complicatedbackground resulting from the different diffuse componentsof the host galaxies; and potential crowding of sources. Tocorrect the data for incompleteness bias, we ran Monte Carlo(MC) completeness simulations with each science image. APSF model for each target field was first created. The LIRGsMCG+08-11-002 and CGCG 049-057 did not have isolatedstars in their fields, and for them we used a representativePSF model from other fields having a similar distance fromtheir tip-tilt reference star, and hence similar AO-correctionbased from the reference star. Using the model, we created ©000 , 1–18 SC LFs in LIRGs Figure 3.
The results of Monte Carlo completeness simulationsfor IRAS 18293-3413 within regions of different background levels.The green dashed line corresponds to the innermost region withthe highest background.
Upper panel:
The fraction of simulatedpoint sources recovered as SSC candidates in the simulation asa function of apparent and absolute K magnitude in the lowerand upper axes, respectively. The 50 % and 80 % completenesslimits corresponding to the middle2 ˝ region are a reasonableapproximation of overall completeness limits of point sources inIRAS 18293-3413; they are shown as the horizontal dashed lines. Lower panel:
The y-axis plots the input minus output magnitudeas a function of the input magnitude in the same MC simulation.The error bars reflect the scatter of this difference in the simu-lation and is plotted for one curve only for clarity. The verticaldashed lines show the two completeness levels determined abovefrom the middle2 ˝ region. artificial stars with IRAF/DAOPHOT which were used in thesimulation.The main idea of the simulation is to record how manyof the input objects, added randomly to the science images,are detected by
SExtractor at a given magnitude range withexactly the same manner and configuration parameters asthe ones chosen for optimal detection in § K -mag in steps of 0.5mag. We generated 1000 random positions of artificial starcentroids. To avoid systematic errors, we used a new subsetof random positions for each magnitude step. In addition weran the simulation separately at different background levelsof the field. These levels were determined by defining three(four in cases of IRAS 19115-2124 and IRAS 18293-3413, thelatter having the largest dynamical range between the back-ground and point source brightness) approximately equalranges in the pixel values of a smoothed background map ina logarithmic scale, ranging from an essentially empty sky tothe LIRG’s core, but excluding the nucleus itself. Figure 3,upper panel, shows as an example the resulting complete-ness curves for the case of IRAS 18293-3413. The simulationswere also used to test the accuracy of the photometry forsystematic effects. The lower panel of Fig. 3 shows the differ-ence of input and output magnitudes of the detected sources.In the fainter bins, magnitudes cannot be measured reliablyanymore due to very variable background, crowding, andinaccurate aperture corrections. However, the ∆mag valuesbecome larger than our typical overall photometric uncer-tainties only at or below the completeness limits relevant toeach galaxy and background region used for subsequent LFanalysis, and any corrections for these possible systematicsare not attempted.Each individual SSC photometric data point was thencorrected for the incompleteness bias with respect to its ob-served magnitude as well as its location in the complex back-ground field, as long as its magnitude value was above the50 % completeness limit of the region in question. Represen-tative completeness values are shown in Table 2. After thiscorrection, we are ready to construct the luminosity func-tions of the SSCs in each target field. K -BAND SSCLUMINOSITY FUNCTION Once the SSC candidates have been selected for each tar-get, we can construct a binned LF and then fit a functionto its shape. LFs are constructed as a function of absolute M K magnitude; the distance modulus m − M of the ob-served sample listed in Table 1 was used in the conversion.Completeness corrections were applied to the counts of SSCcandidates as a function of observed magnitude and back-ground region of the detections, as discussed above. Figure 4shows all the observed LFs of our targets; both raw andcompleteness-corrected counts are shown.After this, we fit a power-law of the form N ( L ) dL ∼ L − α dL in log-log space to the LF shape. The fitted datapoints are weighted using their respective Poisson-noise un-certainties √ N . In a mag-logN plot, a linear fit is expressedas follows: log N ( M K ) = β M K + con (2)where the relation between the power-law index α (see Eq. 1)and the linear slope β is (Elmegreen & Efremov 1997): α = 2 . β + 1 (3) © , 1–18 Z. Randriamanakoto et al.
Figure 4. K -band LFs of the sample using a constant bin size. The black solid line is the incompleteness-corrected LF while the dashedgrey line is the original one. The single power-law fit of the data points is represented by the cyan solid line while, in the case ofIRAS 18293-3413, the grey solid one and the dashed blue line result from broken power-law fits, and the curved green line is a Schechterfunction fit. The vertical lines mark the 50 % (grey) and the 80 % (cyan) completeness levels. The y-axes scales are not the same, becausethe number of SSC candidates differs from one target to another, making the optimal constant bin size different in each case. © , 1–18 SC LFs in LIRGs Name α con α var χ red IC 694 2.29 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Table 3.
Power-law indices from weighted linear fitting of theLFs in Fig. 4 using the relation in Eq. 3. α con and α var are, re-spectively, the indices derived from binning with a constant and avariable bin width. χ red show the reduced Chi Square values forthe single power-law fits using the constant binning. The uncer-tainties in the slopes α are derived from the rules of propagationof errors in Eq. 3, after calculating the uncertainty in β which isthe weighted linear slope shown in Eq. 2. Note that in the case ofa small dataset, e.g. IRAS F16516-0948 and IC 883, the value of χ red may not be a good representation of the goodness of the fit. Due to the very small number of SSC candidates inMCG+08-11-002 and CGCG 049-057, N=12 and 10 respec-tively, we did not fit their LFs.
As a first step we fitted the SSC candidate LF distribu-tions with a single power-law shape using a constant binsize. Since the targets do not have the same number of SSCcandidates, each galaxy has its own constant bin size to bal-ance having statistically enough sources per bin while try-ing to maximise the number of bins overall. The LF binswere fitted from the brightest bin down to the last binabove the 80% completeness limit, the data points plot-ted as squares with error bars in Fig. 4 indicating the binswhich were fit. The resulting power-law indices estimatedusing Eq. 3 are shown in Table 3 as α con , and the fittedLFs are plotted in Fig. 4 as the cyan line. The values of α con range from 1.5 to 2.4. The average over the sample is α con = 1 . ± .
30, or α con = 1 . ± .
28 if the most distantLIRG, IRAS 19115-2124, is excluded. The quoted errors arethe formal uncertainty of the fits. Fitting a single power-law function to the LF of the combined dataset down to a M K = − . α con = 1 . ± .
25 (Fig. 5). This is quite similar to the av-erage slope.To check whether a single power-law is a good approx-imation of the LF, we estimated the reduced chi-squarestatistic χ red (the ratio of chi-square χ and the degreesof freedom of the dataset) in each case. In most cases χ red values (see Table 3) indicate that a single power-law appearsto be a reasonable fit to the data. Note that the value of χ red may not reflect the goodness of the fit when dealing with asmall number of data points. Figure 5. K -band SSC LF of the sample, except the dataset fromIRAS 19115-2124. The single power-law fit of the data points isrepresented by the cyan solid line while the vertical lines markthe average values of the 50 % (grey) and the 80 % (cyan) com-pleteness levels. The shape of the LF may be affected by sample binning:for example, Ma´ız Apell´aniz & ´Ubeda (2005) argued thatthe difference in the value of α can be as large as 0.3 forsmall datasets. To check for consistency of our results, weconstructed the LFs using two different methods: first usinga constant bin size as above, and secondly using a variablebin size and assigning an equal number of objects to each bin,as proposed by Ma´ız Apell´aniz & ´Ubeda (2005). Table 3 liststhe values from both methods: α con is the slope of the LFfor a constant bin size, while α var is the one that resultsfrom using a variable bin size. Though the values derivedfrom the two binning methods are slightly different, theyare consistent within the error estimates of each fit.In addition, we ran systematic tests on the effect ofusing a particular size of the constant-size bin. The scatterof the fitted index values α with a wide range of bin sizesis of the same order, ∼ .
15, as the uncertainties of theslope fits presented in Table 3, and we conclude that binsize does not influence our final results. The characteristicsof the resulting LF slopes will be discussed in § Figure 4 suggests that in the case of IRAS 18293-3413 a two-component power-law would be a better fit to the CLF.Hence we also fitted this LF of the target with a brokenpower-law with both slopes as free parameters, but thebreak, or bend point, M bend fixed at a location chosen byeye. The form of the fit is expressed as follows in a mag-logNplot: logN ( M K ) ∼ (cid:26) β M K for M K (cid:62) M bend β M K for M K < M bend (4) © , 1–18 Z. Randriamanakoto et al. where β and β are related, respectively, to the bright andfaint slopes α = 2 . ± .
15 and α = 1 . ± .
47 of the CLFdouble power-law fits using Eq. 3, also plotted and labelledin Fig. 4 as α and α .We also performed a Schechter fit to the CLF of thetarget with the form of: φ ( M ) dM = con × X α S +1 e − X dM (5)where X = 10 . M (cid:63)K − M ) (6)which resulted in a characteristic magnitude of M (cid:63)K ≈− . α S = 1 . ± .
19. Notethat the values of α and α S are similar for the two typesof fit.A bent LF for IRAS 18293-3413 fits better than a singlepower-law slope at the ∼ . σ level. Though the differenceis not very large, it is interesting, especially given that thisgalaxy has the largest statistics of SSCs in our present sam-ple. We will discuss the case further in Section 5.2.2. The distances of our sample galaxies, 45 < D L (Mpc) < ∼ ˝ corresponds to aphysical size of ∼
20 to 40 pc depending on the distance, andnearly 100 pc in the case of IRAS 19115-2124, and our pho-tometric apertures are of comparable size. Given that theeffective radii of SSCs from the literature are in the range of3 to 5 pc (e.g. Whitmore et al. 1999), such aperture sizes maycontain more than a single SSC candidate. Hence, blendingof SSCs and complexes of SSCs will most probably contami-nate our SSC counts despite the use of AO imaging. Is it thenreasonable to even refer to these as SSCs, or should we rathertalk, for example, about knots of star-formation ˝ (Miralles-Caballero et al. 2011)? To address this we next attempt toestimate how much blending and crowding affect our anal-ysis. We first perform a simulation estimating the effect ofblending on the LF slopes, then examine what happens tophotometry of SSCs of a nearby system when it is moved toa larger distance. We also investigate the relation betweenthe SSC surface density and confusion limits. In the end weconclude that blending effects are not significant within theSSC luminosity range considered, and in our case the term SSC candidate ˝ is a perfectly reasonable one for sources inour photometric catalogues. We performed a MC blending simulation to quantify the ef-fects of crowding on the values of the power-law index α .A random population of N artificial sources was createdwithin the same magnitude range as our observational data( − (cid:46) M K (cid:46) −
12) drawn from a LF with an initial index α init = 2. We randomly selected two artificial sources fromthis population then blended them together. This processwas repeated x times until the original population had x blended sources in total, corresponding to a blending rate y = x/N %. The blending was done step-by-step, so thata new ˝ blended source entered back into the catalogue asa new source and may be randomly blended with a thirdsource, etc. A new power-law index, α new was determinedfrom a fit to the blended source luminosity distribution. Thedifference in the slopes, ∆ α = α init − α new was then de-termined. We ran the simulation with y ranging from 5 to60 per cent in steps of 5 %. A particular blending realisationwas repeated 1000 times for each blending rate. We also ranthe MC simulation for α init = { } . We found thatblends of 3 or more sources are rare below ∼
20 % blendingrates.Figure 6, left panel, plots ∆ α against the blending ratewith different values of α init . The sense is always that of flattening of the slope. A blending rate of 10 % for examplewould lead to a deviation of ∆ α ∼ α init = 2. Thefigure indicates that to get significant flattening of α by 0.3or more, one would need blending rates of ∼
25 % or higherfor an initial slope of 2, or nearly 40 % for the slope to flat-ten from 2 to 1.5. The blending rates needed are higher fora steeper initial slope and lower for a flatter initial slope asexpected. The blending rate is obviously related to the sur-face density of objects in the field, and we will return to thelikely real blending rates of our targets in Section 5.1.4.Figure 6, middle panel, plots the change in magnitudeof the brightest object; with more blending there is a greaterchance that the brightest object is in fact a blend. Again wesee that significant changes, ∆mag > .
5, in the magnituderequire blending rates of 40 % or higher.The simulation does not take into account varying de-grees of blending, nor changes in completeness limit. Mostsignificantly, however, the simulation implicitly assumes arandom surface distribution of targets, whereas SSCs areclearly clustered in (most) galaxies. While the simulationgives a good feel for the expected levels of changes in LFslopes due to blending, a more realistic estimate should bebased on real data.
The Antennae system, at D L ∼
20 Mpc, is a popular labo-ratory for SSC studies (e.g. Whitmore et al. 1999; Zhanget al. 2001), and we use its well-studied star cluster pop-ulation for a blending estimate. In particular, we retrieved
HST /WFC3 images (PI: Whitmore) in the UVIS/F814W( I -band) and IR/F160W ( H -band) filters from the Hub-ble Legacy Archives. Adopting a distance of 22 Mpc the0.04 ˝ pixels in the I -band translate to a 4.3 pc physical sizeand the PSF to ∼ H -band dataset the pixel sizeis equivalent to 9.6 pc and the PSF to ∼
16 pc.We first convolved and rebinned the I -band Anten-nae image to correspond to being 4 times further away. At88 Mpc it is well within the distance range of our sample (Ta-ble 1), and the PSF resolution element of 32 pc is also simi-lar to that in our NACO and ALTAIR/NIRI images. SSCswere detected both from the original and convolved images.We do not take into account the change in detection limits,so as to test only the effects of blending. The photometryand LF constructions were done similarly to our AO datadescribed above, except that we do not attempt complete-ness corrections here as they are irrelevant to the main goal.Following other HST studies of the Antennae (e.g. Whit- © , 1–18 SC LFs in LIRGs Figure 6.
Results from our MC blending simulation.
Left : the difference in the indices ∆ α = α init − α new plotted against the blendingrate considering different values of α init . The figure indicates for instance that a blending rate of 8 % would lead to a deviation of ∼ α init = 2. Middle:
The same simulations, but plotting the magnitude difference of the brightest source. Notethat intrinsic clustering characteristics are not taken into account in this simple simulation.
Right:
Surface density expressed as beamsper source ˝ vs. blending rate; see Section 5.1.3 for details. A figure of 20 beams per source ˝ is traditional confusion limit rule-of-thumb,and it is seen that at this surface density the blending rate is ∼
10 %, resulting in ∆ α ∼ . Figure 7.
LFs of SSCs derived from
HST images of the Antennae galaxies.
Left:
The black histogram shows the LF from the original I -band (F814W) image, and the blue after convolving and rebinning the image to correspond to a 4 times larger distance ( ≈
84 Mpc).The best-fit power-law slopes are indicated.
Right:
The same test for the H -band (F160W) image, but now the comparison is made sothat the convolved image matches both the PSF size and the pixel size of our IRAS 18293-3413 NACO image. In both tests only mildflattening of the LF slope is observed. more et al. 2010) an aperture radius of 0.1 ˝ was used on theoriginal (unbinned) image, corresponding to 10.8 pc. Nom-inal photometric zero-points and aperture corrections weretaken from the WFC3 on-line manuals. The resulting LF isshown as the black points in Fig. 7, left panel, and a singlepower-law slope of α = 2 .
22 was found; the fit is performeddown to M I ≈ − . α = 2 .
26 in the V -band.The source extractions and photometry were done inthe same way for the redshifted ˝ convolved image. An ad-ditional aperture correction is needed however since the PSFand pixel characteristics change, and the new correction wassimply determined by matching the extracted magnitudesof a handful of foreground stars in the different images andapertures. The LF shown in blue in Fig. 7 results from ex- tractions from the convolved image when using a small 0.75pixel aperture, corresponding to 13 pc radius. As expected,the completeness limit is some 2 magnitudes brighter dueto blending only, while the LF slope becomes only slightlyflatter at α = 2 .
09. The aperture size used does not have aneffect on the slope, but we will return to this aspect in moredetail below.We also examined what happens to the 50 and 200brightest original SSCs in the convolved image. Of the 50brightest ones 7 were not detected as individual objects af-ter the redshifting ˝ (14 % blending rate) while 53 of the 200brightest ones were not detected (27 % blending). In the MCsimulation of Sec. 5.1.1 these blending rates would have re-sulted in a flattening of the slope by ∆ α ≈ .
20 and ≈ © , 1–18 Z. Randriamanakoto et al. magnitudes of the brightest 50 SSCs are a mere ∼ . H -band image of the Antennae. This time wemodified the original HST image to match both the physicalpixel scale and PSF size of the NACO IRAS 18293-3413 im-age at D L ≈
75 Mpc. Rebinning was not necessary becauseof the smaller pixels of the NACO instrument, while a fac-tor of 2.5 widening of the PSF was performed. Magnitudeswere measured in 20 pc radii in the original image, matchingthe aperture used in the Whitmore et al. (e.g. 2010) NIC-MOS data, and 30 pc radii in the redshifted case, as donefor the NACO IRAS 18293-3413 images. The resulting LFsare shown in Fig. 7, right panel, indicating a flattening ofthe slope by 0.15, very similar to what was found for the I -band HST image. The magnitudes do not change signifi-cantly this time either, the average difference being less than0.1 mag for the 200 brightest SSCs.To understand and differentiate between the effects ofspatial resolution and photometric aperture used with agiven resolution, we redid the tests above with different con-volutions using PSFs ranging from 10 to 100 pc, as well asusing numerous aperture sizes in the same range. Some re-sults become clear. First of all, the photometric apertureused at a given resolution does not change the slope signifi-cantly . Variations of ∼ . α . Secondly, the LF does however shift to brighter mag-nitudes as the apertures grow. To recover as closely as pos-sible the intrinsic SSC counts the smallest possible apertureshould be used , assuming a reliable aperture correction canstill be determined. With the largest tested apertures thebright SSCs brighten by nearly a magnitude, while usingapertures smaller than about 20 −
30 pc radius, the bright-ening stays within typical photometric errors of ∼ . α = 0 . H and I -band respectively) until sizesof about 40 pc are reached, after which the slopes flattenrapidly reaching α ∼ . HST I -band which illustrates these effects. While numer-ous faint SSCs disappear or are blended into bright SSCs,the latter are generally recovered in the convolved imagewith close to their proper magnitudes, unless located in verycrowded regions. For example the more isolated bright SSCson the edges of the image are recovered within ∼ . Distance [Mpc]10 20 40 80 120 200PSF [ ˝ ]0.05 ˝
440 (1900) 440 (480) 220 130 58 220.075 ˝
440 (850) 220 98 59 26 100.10 ˝
440 (480) 120 55 33 15 5.60.20 ˝
120 30 14 8.3 3.7 1.40.30 ˝
53 13 6.1 3.7 1.6 0.620.50 ˝
19 4.8 2.2 1.3 0.59 0.221.00 ˝ Table 4.
Surface densities, in units of kpc − , which would re-sult in confusion-limited observations of SSCs, when the limit isdefined as 20 beams per source ˝ . For small distances/PSFs theassumed 10 pc physical size of SSCs is resolved and the value cor-responds to this confusion limit; the value corresponding to theactual resolution element is given in parentheses. Historically, blending and crowding as discussed above arereferred to as confusion ˝ especially in the radio and far-IRstudies, and in particular bright source confusion ˝ . Whena confusion limit is reached depends, in addition to theshape and size of the spatial resolution element, on theslope of the source counts. With very steep slopes confu-sion noise dominates, i.e. the many undetected sources atand below a detection limit give rise to an effective noise-level (see e.g. V¨ais¨anen et al. 2001 and references thereinfor full discussions) that deeper observations cannot pene-trate. With much flatter slopes the bright sources lying tooclose together for satisfactory extraction at a given spatialresolution tend to dominate. The SSC source count slopesare closer to this latter regime, but it is prudent to searchfor this surface density-related confusion limit also in caseswhere the images are not truly confusion-limited yet.A 20 to 40 beams per source ˝ confusion limit is oftenused as a rule-of-thumb. For example, for a resolution ele-ment FWHM = 0.1 ˝ , setting the beam size asΩ = π F W HM , then 20 beams per source corresponds to having one SSC perevery 0.47 ˝ × ˝ region. Hence, having over 500 sourceswithin the area of IRAS 18293-3413 (see Fig. 1) would meanreaching the limit, while we detected ∼ − . If SSCs are seen more denselypacked than this then better resolution is not likely tohelp extend SSC detection. Over the approximately 9 × size of the Antennae, for example, this would meansome 40000 sources; extrapolating the counts we extractedfrom the HST image, this level would be reached around M I ∼ − . regions of thegalaxy system, the confusion limit must be reached at muchbrighter magnitudes since the SSCs are clustered. This isreflected in the fact that significant incompleteness startsappearing already at M I ∼ − ©000
Surface densities, in units of kpc − , which would re-sult in confusion-limited observations of SSCs, when the limit isdefined as 20 beams per source ˝ . For small distances/PSFs theassumed 10 pc physical size of SSCs is resolved and the value cor-responds to this confusion limit; the value corresponding to theactual resolution element is given in parentheses. Historically, blending and crowding as discussed above arereferred to as confusion ˝ especially in the radio and far-IRstudies, and in particular bright source confusion ˝ . Whena confusion limit is reached depends, in addition to theshape and size of the spatial resolution element, on theslope of the source counts. With very steep slopes confu-sion noise dominates, i.e. the many undetected sources atand below a detection limit give rise to an effective noise-level (see e.g. V¨ais¨anen et al. 2001 and references thereinfor full discussions) that deeper observations cannot pene-trate. With much flatter slopes the bright sources lying tooclose together for satisfactory extraction at a given spatialresolution tend to dominate. The SSC source count slopesare closer to this latter regime, but it is prudent to searchfor this surface density-related confusion limit also in caseswhere the images are not truly confusion-limited yet.A 20 to 40 beams per source ˝ confusion limit is oftenused as a rule-of-thumb. For example, for a resolution ele-ment FWHM = 0.1 ˝ , setting the beam size asΩ = π F W HM , then 20 beams per source corresponds to having one SSC perevery 0.47 ˝ × ˝ region. Hence, having over 500 sourceswithin the area of IRAS 18293-3413 (see Fig. 1) would meanreaching the limit, while we detected ∼ − . If SSCs are seen more denselypacked than this then better resolution is not likely tohelp extend SSC detection. Over the approximately 9 × size of the Antennae, for example, this would meansome 40000 sources; extrapolating the counts we extractedfrom the HST image, this level would be reached around M I ∼ − . regions of thegalaxy system, the confusion limit must be reached at muchbrighter magnitudes since the SSCs are clustered. This isreflected in the fact that significant incompleteness startsappearing already at M I ∼ − ©000 , 1–18 SC LFs in LIRGs Figure 8.
A 10 ˝ by 8 ˝ (1.1 by 0.9 kpc) region in the HST I -band image within the Antennae (region E in Whitmore et al. 2010).
Left:
The original image with ∼
160 SSC candidates detected.
Right:
The image after taking the galaxy four times further away. Only ∼ panel). If the HST image was truly confusion-limited, andassuming a LF with α = 2, the completeness limit wouldbe expected to brighten by 3 magnitudes after the factor 4convolution, whereas a change of 2 mag was observed.We list in Table 4 physical surface densities per kpc at which the confusion limit is likely to be reached at agiven distance and spatial resolution, given the definitionsabove. With the typical PSF of ∼ . ˝ , the distances ofour sample, and the number of detected SSCs, only thecore regions of IRAS 18293-3413 come anywhere close tothese confusion limits, as well as IRAS 19115-2124 due toits distance. This is not to say that individual SSCs will notblend of course, especially in clustered star-forming regionsof the galaxies as shown in the tests based on the Anten-nae. For example, the surface density of detected objects inthe left panel of Figure 8 is ∼
160 kpc − . At a distance of ∼
20 Mpc and for a resolution element of 0.075 ˝ this is notquite yet confusion-limited according to Table 4, though it isapproaching it. This is also seen from the corresponding 78beams-per-source surface density. However, within the clus-tered sub-region the equivalent values would be ∼
500 kpc − and ∼
25 beams-per-source, respectively, i.e. that region isconfusion-limited.To see how the surface densities quantitatively relate tothe LF slope changes, we ran another MC simulation addingincreasing amounts of randomly distributed equally brightsources in an otherwise empty frame and extracting themwith
SExtractor using typical parameters. This was donehundreds of times at several surface densities, and the frac-tion of unrecovered sources is simply the blending rate ateach surface density. The change in LF slope with a givenintrinsic LF shape was already simulated in Section 5.1.1 asa function of this blending rate. The right panel of Fig. 6connects the two by plotting the beams per source ˝ surfacedensity against the blending rate. The black curve is the in-trinsic surface density of the simulation, and the red curveis calculated from the extracted surface density. An ob-served surface density of objects can be used to get an ex-pected blending rate using the red curve (for completeness-corrected counts the black curve is more appropriate) and this can then be converted to a likely ∆ α value of the LF.While the vertical displacement of the surface density vs.blending rate curves will depend on the source detection al-gorithm and clustering of objects, our tests show that therelations do serve as a realistic approximation of the quan-titative effects involved. The average surface density of detected SSC candidates inour target galaxies ranges from a low of 0.3 kpc − in IC 694,or 168 beams per source, to a high of 0.8 kpc − in the caseof IRAS 18293-3413, or 87 beams-per-source. According toFig. 6, the corresponding blending rates are well below 5 %,meaning that ∆ α < . < . α values for each individual target,but rather stress that these must be less than 0.1. However,in our most extreme case of the central ∼ × α ∼ .
15 would be expected in thatregion.As a further check, Fig. 9 shows separately the LFs forSSCs out to distances D L (cid:54)
60 Mpc, and for D L >
60 Mpc(excluding IRAS 19115-2124 since it is so much furtheraway). If blending was a serious issue in our sample, themore distant subsample would be expected to show a flatterslope. The single power-law indices of the LFs below andabove 60 Mpc are α = 1 . ± .
11 and α = 1 . ± .
23 re-spectively, i.e. there is no significant difference in the LFslope, nor in the numbers of detected SSCs, with distance.In summary, we are confident that in the luminosityrange of interest, M K < −
14 mag, any flattening of LFslopes due to blending is below ∆ α ∼ .
1. With the possibleexception of IRAS 19115-2124 where the spatial resolutionis just a physical size of 90 pc, the luminosities of detected K -band point sources will be dominated by a single brightSSC rather than whole knots of star formation. In the case ofAntennae-like SSC populations, blending and crowding do © , 1–18 Z. Randriamanakoto et al.
Figure 9.
SSC LFs of two subsamples generated from our observational data but segregated by distance. The dataset from IRAS 19115-2124 is not included to avoid bias in the analysis.
Left:
SSC LF of the closer subsample where the targets have distances (cid:54)
60 Mpc.
Right:
SSC LF of the more distant targets, excluding IRAS 19115-2124. The values of the slopes appear to be consistent within theuncertainties. flatten the LF slope, but significantly so only at resolutionspoorer than a 40 pc physical size. The photometric aperturesused should be as small as possible to recover intrinsic lu-minosities, though the aperture does not have a significanteffect on the LF shape. Assuming a 10 pc scale for SSCs,the confusion limit is reached at a surface density of 440SSCs per kpc , or less when clustering and larger resolutionelements are a factor (Table 4). α The results from our fitting procedure in § α ∼ α con = 1 . ± .
30 for a constant binsize, and α var = 1 . ± .
23 for a variable bin size. Val-ues of α reported in the literature vary widely from 1.7 to2.4 (e.g. Elmegreen & Efremov 1997, Whitmore et al. 1999,Elmegreen et al. 2002), with slopes in normal spirals oftenin the upper part of this range. Our α values tend to bein the less steep part of the range, and in at least one case(IRAS 18293-3413) either a broken power-law or a Schechterfunction would yield a better fit. These two issues are dis-cussed next. Improper incompleteness corrections can produce artificiallyflattened slopes. We note however that the slopes are fittedto a range where at most 20 % of SSCs are missed, and where the corrections are still reliable. The most significant changeis likely to be caused by blending of SSCs but as discussedin the previous section, this is expected to be ∆ α ≈ . α values with distance tothe galaxies, which would have been expected if significantblending was present (Fig. 9). Therefore, we conclude thatthe intrinsic SSC LF slopes, the combined and averaged LFshaving α ≈ . ± .
25 and 1 . ± .
30, respectively, are notaffected by these observational effects, with any systematiceffects being outweighed by the statistical uncertainties.Normal spiral galaxies tend to have power-law indices inthe range of α ∼ α ≈ . ± . α ≈ . ± . V - and I -bands; if several bandswere used, we chose the reddest band. As discussed earlierthe well-studied interacting Antennae system also has α ∼ . α ≈ .
9, appears flatter than the slope in normal galaxies.The large sample of LIRGs studied in Vavilkin (2011) showsan average α ∼ .
8, though blending effects there mightstill need to be explored. Similarly, Miralles-Caballero et al.(2011) find α ∼ . <
100 Mpc, andeven flatter values at certain interaction stages of LIRGs.Moreover, Adamo et al. (2010, 2011b) recently found flatterpower-law slopes while probing the star cluster properties inHaro 11 and Mrk 930. Both of the targets are blue compactdwarfs with intense star formation, and the former can also ©000
100 Mpc, andeven flatter values at certain interaction stages of LIRGs.Moreover, Adamo et al. (2010, 2011b) recently found flatterpower-law slopes while probing the star cluster properties inHaro 11 and Mrk 930. Both of the targets are blue compactdwarfs with intense star formation, and the former can also ©000 , 1–18 SC LFs in LIRGs Figure 10.
SSC LFs of two subsamples generated from our observational data, excluding IRAS 19115-2124, but broken by the averageSFR.
Left:
SSC LF of the subsample where the targets have SFR (cid:54) M (cid:12) yr − . Right:
SSC LF of the other set of targets with highSFRs. A power-law fit until the 80 % completeness level generates a quite similar value of α in both cases. be classified as a LIRG. Is it the case therefore that LFs aresystematically flatter in extreme SF cases?To see if there is a trend with SFR within our sample,Fig. 10 shows LFs separately for SSCs with host galaxy SFRsless than, or greater than the average SFR ∼ M (cid:12) yr − ,again with IRAS 19115-2124 excluded. The values of theslopes are α = 1 . ± .
16 for the subsample below theaverage and α = 1 . ± .
19 for those above. Thus, while wedo not find a correlation between the LF slope and the SFRwithin our present small sample, our results together withother recent studies suggest that there is a real differencebetween the SSC LF slopes of LIRGs and those of more qui-escent galaxies. This needs to be verified with larger samplesfor better statistics, and with careful blending analysis sincelarger samples will by necessity involve more distant targets.Apart from effects related to the observations them-selves, perhaps the most fundamental cause for flattenedLFs, or breaks/bends in the LF for that matter, would comefrom mass- and/or age-dependent cluster disruption, as wellas from differences in the cluster formation with differentenvironments. Young star clusters are most vulnerable todisruption, leading to variation of the integrated LF as timepasses. Since LFs are the integrated sum of the distribu-tions of individual initial LFs of SSCs of different ages andmasses, any selective (e.g. mass-dependent) disruption ofthem would be seen as changes in the (integral) total SSCLFs (e.g. Mengel et al. 2005; Gieles et al. 2006a; de Grijs& Parmentier 2007). Kruijssen et al. (2012) argue that clus-ter formation efficiencies decrease in galaxies with higherSFRs due to increased tidal disruption in the extreme envi-ronments ( cruel cradle effect ˝ ), which might lead to flatterLF slopes if lower mass SSC formation is more affected. Al-ternatively, high SFRs may favour the formation of moremassive GMCs (e.g. Wei et al. 2012). However it is impossi-ble to conclude whether disruption is the main cause of theflatter slopes until we have analysed data from other filtersto be able to derive LFs and MFs in different mass and ageregimes. In the case of IRAS 18293-3413 fitting two independentslopes produces better Chi Square values ( χ red = 0 . M K ≈− . M (cid:63)K ≈ − . χ red = 0 . M ∼ − −
11 mag in
V RI filters (e.g. Whitmoreet al. 1999; Gieles et al. 2006a,b; Santiago-Cort´es et al. 2010;Adamo et al. 2010, 2011a). Assuming typical un-extincted (and age dependent) optical-to- K -band colour indices of 2to 3 mag, one would expect a corresponding bend to appearat M K ∼ −
12 to −
13 mag. Mengel et al. (2005) find a bendat a significantly brighter level M K ∼ − . © , 1–18 Z. Randriamanakoto et al.
Figure 11.
Starburst99 model of 2 × M (cid:12) SSP cluster witha Kroupa IMF. The three curves show the absolute magnitudeof the star cluster in B -(blue), I -(green) and K -(red) bands. Theinset highlights that the K -band luminosity of star clusters islikely to peak close to age 10 Myr while they peak somewhatearlier in the optical. tennae (but see Fall et al. 2009) not unlike our result forIRAS 18293-3413, suggesting significant extinction effects.The bend-point of a double power-law at M K ∼ − . M (cid:63)K ∼ − . minimum characteristic mass of M (cid:63) ≈ − × M (cid:12) , or perhaps double this mass range ifwe adopt an average mass-to-light ratio over the first 30 Myrof age (see next Section). These masses are higher by fac-tors of 5 to 10 than the characteristic mass found for SSCsin normal spirals (Gieles et al. 2006a; Larsen 2009) whilethey are very similar to those suggested for galaxies withmuch higher SFRs, such as LIRGs (see e.g. Bastian 2008).However until there is clearer evidence for real bends in SSCLFs from a larger dataset it is premature to read too muchinto these masses. K -bandSSCs An interesting characteristic of SSCs observed at NIR wave-lenghts is a relatively narrow time-frame at ages of ∼
10 Myrwhen the high mass stars in the SSCs enter the RSG phasemaking them very NIR-luminous and suddenly ˝ red in theoptical-to-NIR colours compared to earlier blue stages. Thisis seen in Fig. 11 which plots as an example the evolutionof cluster brightness with time in the BIK filters, derivedfrom a
Starburst99 (Leitherer et al. 1999) model assumingan instantaneous SF with a fixed mass of 2 × M (cid:12) , and aKroupa IMF. Hence, a lower limit on SSC masses detectedin this work can be obtained by assuming the mass-to-lightratio at that ∼
10 Myr age. In many of our galaxies the mostmassive detected SSC has M K ∼ −
18 mag correspondingto a mass of 4 × M (cid:12) at 10 Myr of age or more than2 × M (cid:12) at 30 Myr; clearly we are sampling very massiveclusters in LIRGs. SSCs with masses in excess of 10 M (cid:12) have been foundbefore (e.g. Bastian et al. 2006; Portegies Zwart et al. 2010).The faintest-detected SSCs in our sample have M K ∼ − ≈ × M (cid:12) , while the pho-tometric 80 % completeness limits in different targets corre-spond to lower limits of 1 − × M (cid:12) . We note that withsuch massive clusters, the luminosities of individual brightRSG stars are negligible compared to the total integratedflux of the star cluster. Therefore, stochastic effects shouldnot introduce significant scatter in the mass-to-light ratiosand inferred ages of our clusters (Fouesneau et al. 2012). In this work we studied the characteristics of massive starclusters in the extreme environments of local interactingLIRGs. Observations were performed in the K -band filterusing two different NIR AO instruments with pixel scalesof ∼ ˝ pix − and a FWHM ∼ ˝ for point sources.The galaxy sample consists of LIRGs in the redshift range0.01 < z < K -band luminosity functions ofSSC candidates in our targets, and because of the distancesinvolved, we also carefully evaluated the effect of blendingon the power-law index α of the LFs. Extensive photomet-ric completeness simulations were done, as well as checkingthe effects of sample binning and foreground contamination,which turned out to not be significant. The main results fromthis work can be summarised as:(i) The SSC luminosity function is probed at high com-pletion down to M K ∼ −
14 or −
15 mag in our sample. Inthis range all the LFs are reasonably well fitted by a singlepower-law, though in the case of IRAS 18293-3413 a doublepower-law or a Schechter function is a better approxima-tion. The values of the best-fit slopes vary with a wide rangefrom α = 1 . . α ≈ .
9, andthe combined SSC LF, excluding the most distant target,at α ≈ .
8. The slopes appear slightly flatter than those innormal spirals which typically have α ≈ .
2. The sampleis still too small to find definitive trends of the slope withSFR or interaction stage. One or more of age, extinction,and mass-dependent cluster disruption effects can all leadto small α values at the faint end, but cannot be unambigu-ously separated from the present dataset alone.(ii) We carefully examined the possibility of blending ofSSCs in our target LIRGs. Though blending does happenwith the typical resolutions of 30 to 40 pc physical sizes ofour sample, we showed that it is not enough to change theLF slopes by more than ∆ α ≈ . − .
1, nor change drasti-cally the measured luminosities of SSCs. Hence we concludethat out to ∼
100 Mpc it is quite possible to accurately mea-sure SSC properties with 8-m class telescope adaptive op-tics and
HST imaging. In addition to deriving some generalblending/confusion properties, we found that the photomet-ric apertures used do not affect the LF slope, but small aper-tures are necessary to recover the luminosities as correctlyas possible. Worsening spatial resolution tends to flatten themeasured LF slopes through increasing blending. However, ©000
HST imaging. In addition to deriving some generalblending/confusion properties, we found that the photomet-ric apertures used do not affect the LF slope, but small aper-tures are necessary to recover the luminosities as correctlyas possible. Worsening spatial resolution tends to flatten themeasured LF slopes through increasing blending. However, ©000 , 1–18 SC LFs in LIRGs the effect becomes pronounced only when close to the con-fusion limit. In the case of SSCs distributed as in the An-tennae, we determined that LF slopes at M H < −
12 are notreliable if the spatial resolution corresponds to a physicalsize larger than ∼
50 pc.All our findings are based on observations with a singlefilter and thus the estimation of ages of the star cluster can-didates was beyond the scope of this work. However, we arein the process of estimating the physical characteristics ofthe selected star cluster candidates with the help of archival
HST data and by using J and H imaging from Gemini to beable to constrain the SSC evolution models more robustly.In addition, with a larger sample of southern LIRGs andstarbursts with the VLT/NACO currently available, and thenext generation of multi-conjugate AO systems promising todeliver much more stable PSFs across a larger field of viewwe expect to grow both the number and quality of SSC LFsto probe correlations with LIRG host galaxy characteristicsand environments. ACKNOWLEDGMENTS
We thank the two referees for their thoughtful and very use-ful comments. ZR acknowledges financial support from theSouth African Square Kilometre Array and PV from theNational Research Foundation. SM and EK acknowledgefunding from the Academy of Finland (project: 8120503).Based on observations obtained at the Gemini Observa-tory, which is operated by the Association of Universi-ties for Research in Astronomy, Inc., under a cooperativeagreement with the NSF on behalf of the Gemini partner-ship: the National Science Foundation (United States), theNational Research Council (Canada), CONICYT (Chile),the Australian Research Council (Australia), Minist´erio daCiˆencia, Tecnologia e Inova¸c˜ao (Brazil) and Ministerio deCiencia, Tecnolog´ıa e Innovaci´on Productiva (Argentina).Observations were taken as part of programs GN-2008A-Q-38, GN-2008B-Q-32, GN-2009A-Q-12, GN-2009B-Q-23, andGN-2010A-Q-40 (PI: S. Ryder). And based in part on ob-servations made with the European Southern Observatorytelescopes, Paranal, Chile, under programmes 072.D-0433and 073.D-0406. Based in part on observations made withthe NASA/ESA
HST , obtained from the data archive atthe Space Telescope Science Institute, which is operated bythe Association of Universities for Research in Astronomy,Inc., under NASA contract NAS 5-26555. Based in part onobservations made with the Nordic Optical Telescope, oper-ated on the island of La Palma jointly by Denmark, Finland,Iceland, Norway, and Sweden, in the Spanish Observatoriodel Roque de los Muchachos of the Instituto de Astrofisicade Canarias.
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