A Search for correlations between turbulence and star formation in LITTLE THINGS dwarf irregular galaxies
Deidre A. Hunter, Bruce G. Elmegreen, Haylee Archer, Caroline E. Simpson, Phil Cigan
AA Search for correlations between turbulence and star formationin LITTLE THINGS dwarf irregular galaxies
Deidre A. Hunter , Bruce G. Elmegreen , Haylee Archer , Caroline E. Simpson , and PhilCigan ABSTRACT
Turbulence has the potential for creating gas density enhancements that ini-tiate cloud and star formation (SF), and it can be generated locally by SF. Tostudy the connection between turbulence and SF, we looked for relationships be-tween SF traced by FUV images, and gas turbulence traced by kinetic energydensity (KED) and velocity dispersion ( v disp ) in the LITTLE THINGS sample ofnearby dIrr galaxies. We performed 2D cross-correlations between FUV and KEDimages, measured cross-correlations in annuli to produce correlation coefficientsas a function of radius, and determined the cumulative distribution function ofthe cross correlation value. We also plotted on a pixel-by-pixel basis the locallyexcess KED, v disp , and H i mass surface density, Σ HI , as determined from therespective values with the radial profiles subtracted, versus the excess SF ratedensity Σ SFR , for all regions with positive excess Σ
SFR . We found that Σ
SFR and KED are poorly correlated. The excess KED associated with SF implies a ∼ .
5% efficiency for supernova energy to pump local H i turbulence on the scaleof resolution here, which is a factor of ∼ v disp in SF regions is also small, only ∼ .
37 km s − .The local excess in Σ HI corresponding to an excess in Σ SFR is consistent withan H i consumption time of ∼ . i in the inner disks. Subject headings: galaxies: irregular — galaxies: star formation — galaxies: ISM— galaxies: kinematics and dynamics Lowell Observatory, 1400 West Mars Hill Road, Flagstaff, Arizona 86001, USA IBM T ˙J. Watson Research Center, 1101 Kitchawan Road, Yorktown Heights, New York USA Department of Physics, Florida International University, CP 204, 11200 SW 8th St, Miami, Florida33199 USA George Mason University, 4400 University Dr., Fairfax, VA 22030-4444, USA a r X i v : . [ a s t r o - ph . GA ] J a n
1. Introduction
The gas in the inner parts of spiral galaxies is gravitationally unstable to the formationof clouds that can go on to form stars (Toomre 1964; Kennicutt 1989). However, in dwarfirregular (dIrr) galaxies, the atomic gas densities are much lower than in spirals and areapparently stable against this instability (Hunter & Plummer 1996; Meurer et al. 1996;van Zee et al. 1997; Hunter et al. 2011). Furthermore, in inner spiral disks star formationincreases as the gas density increases (Bigiel et al. 2008), while in dwarfs and outer spiraldisks the atomic gas density cannot predict star formation rates (SFRs, Bigiel et al. 2010).So, what drives star formation in dIrr galaxies?One process for creating clouds is compression of gas in a supersonically turbulentmedium (Elmegreen 1993; Mac Low & Klessen 2004). There is extensive evidence for inter-stellar turbulence in galaxies, and turbulence in typical dIrrs has been shown to be transonic(Burkhart et al. 2010; Maier et al. 2017) while that in spirals is generally supersonic (Maieret al. 2016). Furthermore, various distributions in stellar, cluster, and cloud properties indwarfs are consistent with sampling a fractal turbulent gas, including composite cumulativeH ii region luminosity functions (Youngblood & Hunter 1999; Kingsburgh & McCall 1998),stellar disk power spectra (Willett et al. 2005), mass functions of clouds and star clusters(Elmegreen & Efremov 1997; Hunter et al. 2003; Mac Low & Klessen 2004), H α probabil-ity distribution functions (Hunter & Elmegreen 2004), and the correlation between regionsize and the star formation time scale (Efremov & Elmegreen 1998). Dib & Burkert (2005)found evidence for scales in the interstellar medium (ISM) of Holmberg II less than 6 kpcin size that they interpret as due to a turbulence driver acting on that scale. And, Zhanget al. (2012) showed from H i spatial power spectra that either non-stellar power sourcesare playing a fundamental role in driving the ISM turbulence or the nonlinear developmentof turbulent structures has little to do with the driving sources. In addition, Hunter et al.(2001, 2011) have found regions of high velocity dispersion in the H i distribution of some dIrrgalaxies that correlate with a deficit of H i in a manner suggestive of long-range, turbulentpressure equilibrium (Piontek & Ostriker 2005).Turbulence can create density enhancements that initiate cloud formation (Krumholz& McKee 2005), but turbulence also heats gas, which can make it harder to form clouds(Struck & Smith 1999). So, how important is turbulence in driving star formation in dwarfs?It could be essential in outer disks where gas self-gravity is weak (Elmegreen & Hunter 2006).Also, a transition from subsonic to supersonic turbulence in the ISM could be the cause ofthe transition in the Schmidt-Kennicutt star formation rate-gas density relationship frominefficient star formation at low gas surface densities to star formation at higher densities(Kraljic et al. 2014). 3 –Conversely, how important is star formation in driving turbulence? Simulations suggestthat stellar feedback and supernovae drive turbulence on the scale of the galaxy thickness(Joung et al. 2009; Kim & Ostriker 2015), and it may drive turbulence in molecular clouds(Padoan et al. 2016), along with cloud self-gravity (Mac Low et al. 2017; Ib´a˜nez-Mej´ıa et al.2017). Feedback destroys molecular clouds as well (Kim et al. 2018). Models also suggestfeedback controls the SFR by adjusting the disk thickness and midplane density (Ostriker etal. 2010) or by compressing nearby clouds, causing them to collapse (Deharveng et al. 2012;Palmeirim et al. 2017; Egorov et al. 2017). On a galactic scale, feedback and self-gravityoperate together to drive turbulence (e.g., Goldbaum et al. 2016; Krumholz et al. 2018).These models are uncertain, however. Other simulations show no need for star formation todrive turbulence because they reproduce the velocity dispersion from self-gravity alone; theonly thing local feedback needs to do is destroy the clouds where young stars form, preventingthe SFR from getting too large (Bournaud et al. 2010; Combes et al. 2012; Hopkins et al.2011).Observations are usually not decisive about the connection between the SFR and turbu-lence. In a study of local dwarfs and low mass spirals, Stilp et al. (2013) found a correlationbetween the core velocity dispersion in H i line profiles and the H i surface density, sugges-tive of driving by gravitational instabilities, but they also found a correlation with SFR atΣ SFR > − M (cid:12) yr − kpc − . Stilp et al. (2013) show that both the H i velocity dispersionand Σ SFR decrease with radius in a galaxy; that makes correlations between these quantitiesambiguous as they both could depend on another parameter that varies with radius and noteach other.Zhou et al. (2017) studied 8 local galaxies with resolved spectroscopy and showed on apixel level that the velocity dispersion of ionized gas does not change over a factor of ∼
40 inSFR per unit area. Also for several hundred local galaxies in the same survey, Varidel et al.(2020) found a very small correlation between the galaxy-average vertical velocity dispersionof ionized gas and the total SFR, with the dispersion increasing by only 6 km s − for SFRsbetween 10 − and 10 M (cid:12) yr − . This contrasts with observations of high redshift galaxieswhere these authors show strong increases in dispersion with SFR density and total rate,respectively, for rate densities larger than ∼ . M (cid:12) yr − kpc − and rates larger than ∼ M (cid:12) yr − . This high-redshift correlation was earlier studied by several groups, including Lehnertet al. (2013) who observed that the velocity dispersion of ionized gas increases as the squareroot of the SFR per unit area. Lehnert et al. (2013) concluded that star formation was themain driver of turbulence and that it was also sufficient to maintain marginal stability ina disk. On the other hand, ¨Ubler et al. (2019) interpreted the increase in the ionized gasvelocity dispersion with SFR density for high redshift galaxies as the result of gravitationalinstabilities, following the theory in Krumholz et al. (2018). 4 –Bacchini et al. (2020) consider radial profiles of turbulent speeds and SFRs in local spiralgalaxies and account for all of the gas turbulence using supernovae from young massive stars.They get more effective turbulence driving than other studies because they include the radialincrease in disk thickness, which decreases the dissipation rate.In this paper we look for evidence of a spatial correlation between star formation andturbulence in the LITTLE THINGS sample of nearby dIrr galaxies. A spatial correlationcould be either a cause of star formation through the production of a gas cloud or a resultof star formation through mechanical energy input to the local ISM through feedback fromstars. We construct turbulent Kinetic Energy Density (KED) maps from the kinetic energyassociated with the bulk motions of the gas - velocity from H i velocity dispersion (moment 2)and mass from integrated column density (moment 0) maps, per unit area in the galaxy. Wecross-correlate the KED maps with far-ultraviolet (FUV) images that trace star formationover the past 200 Myr. Because we are using intensity-weighted velocity dispersions, the“turbulence” includes all bulk motions of the gas, including thermal and turbulent. This fol-lows the two-dimensional (2D) cross-correlation method used by Ioannis Bagetakos (privatecommunication) in analysis of the spiral galaxy NGC 2403.We also isolate turbulence in the vicinity of a SF region and determine the excessKED and velocity dispersion from that region alone. This method removes any backgroundturbulence that may be generated by other means, such as gravitational instabilities andcollapse.
2. Data
LITTLE THINGS is a multi-wavelength survey of nearby dwarf galaxies (Hunter et al.2012). The LITTLE THINGS sample is comprised of 37 dIrr galaxies and 4 Blue CompactDwarf (BCD) galaxies. The galaxies are relatively nearby ( ≤ (cid:48)(cid:48) is ≤
300 pc),contain gas so they have the potential for star formation, and are not companions to largergalaxies. The sample also covers a large range in dwarf galactic properties such as SFR andabsolute magnitude.We obtained H i observations of the LITTLE THINGS galaxies with the National Science Funded in part by the National Science Foundation through grants AST-0707563, AST-0707426, AST-0707468, and AST-0707835 to US-based LITTLE THINGS team members and with generous technical andlogistical support from the National Radio Astronomy Observatory. ). The H i -line data are characterizedby high sensitivity ( ≤ . − per channel), high spectral resolution (1.3 or 2.6 kms − ), and high angular resolution (typically 6 (cid:48)(cid:48) ).Ancillary data used here include far-ultraviolet (FUV) images obtained with the NASA Galaxy Evolution Explorer satellite (
GALEX ; Martin et al. 2005). These images trace starformation over the past 200 Myr. These data also yield integrated SFRs (Hunter et al. 2010)and the radius at which we found the furthest out FUV knot R FUVknot in each galaxy (Hunteret al. 2016). The SFRs are normalized to the area within one disk scale length R D , althoughstar formation is usually found beyond 1 R D . R D is measured from V -band surface brightnessprofiles (Herrmann et al. 2013). Several of the LITTLE THINGS galaxies without GALEX
FUV images are not included in this study (DDO 155, DDO 165, IC 10, UGC 8508). Pixelvalues of FUV and Σ
SFR are not corrected for extinction due to dust, which tends to be lowin these low metallicity galaxies.The galaxy sample and characteristics that we use here are given in Table 1. In someplots, we distinguish between those dIrrs that are classified as Magellanic irregulars (dIm)and those that are classified as BCDs (Haro 29, Haro 36, Mrk 178, VIIZw403).
3. Cross-correlations3.1. Two-dimensional
KED and FUV images were the inputs to the 2D cross-correlation. We geometricallytransformed the FUV image to match the orientation and field of view (FOV) of the H i mapusing OHGEO in the Astronomical Image Processing System (AIPS) and then smoothed itto the H i beam using SMOTH in AIPS. We also blanked the pixels outside of the galaxyFUV emission, replacing the blanked pixels with zeros, so that pure noise would not addto the correlation coefficient C coef . We constructed the KED maps as 0 . × N HI × v disp ,where N HI is the H i column density in hydrogen atoms per cm and v disp is the velocitydispersion in km s − . The conversion from counts in the KED maps to ergs pc − is given foreach galaxy in Table 2. Prior to executing the cross-correlations, we scaled both the FUV The VLA is a facility of the National Radio Astronomy Observatory. The National Radio Astron-omy Observatory is a facility of the National Science Foundation operated under cooperative agreement byAssociated Universities, Inc. GALEX was operated for NASA by the California Institute of Technology under NASA contract NAS5-98034. D a M V R H α b R FUVknotc R Dd R Bre log SFR
FUVD f
Galaxy (Mpc) (kpc) (kpc) (kpc) (kpc) (M (cid:12) yr − kpc − )CVnIdwA 3 . ± . − . ± .
09 0.69 0.49 ± ± ± − . ± . . ± . − . ± .
22 2.36 1.93 ± ± ± − . ± . . ± . − . ± .
16 1.51 3.02 ± ± ± − . ± . . ± . − . ± .
24 5.58 5.58 ± ± · · · − . ± . . ± . − . ± . · · · ± ± ± − . ± . . ± . − . ± .
17 3.69 3.39 ± ± ± − . ± . . ± . − . ± .
03 1.25 1.19 ± ± ± − . ± . . ± . − . ± .
03 2.26 2.89 ± ± ± − . ± . . ± . − . ± .
11 0.76 0.76 ± ± ± − . ± . . ± . − . ± .
12 1.23 1.34 ± ± ± − . ± . . ± . − . ± .
08 1.17 1.38 ± ± ± − . ± . . ± . − . ± .
15 3.18 4.23 ± ± ± − . ± . . ± . − . ± .
16 1.23 1.23 ± ± ± − . ± . . ± . − . ± .
24 2.84 3.37 ± ± ± − . ± . . ± . − . ± .
16 2.60 2.20 ± ± ± − . ± . . ± . − . ± .
16 1.73 2.65 ± ± ± − . ± . . ± . − . ± .
25 0.81 0.70 ± ± ± − . ± . . ± . − . ± .
25 2.24 2.25 ± ± ± − . ± . . ± . − . ± .
07 0.30 0.42 ± ± ± − . ± . . ± . − . ± . · · · ± ± · · · − . ± . . ± . − . ± .
10 0.42 0.59 ± ± ± − . ± . . ± . − . ± . · · · ± ± ± − . ± . . ± . − . ± . · · · ± ± ± − . ± . . ± . − . ± . · · · ± ± ± − . ± . . ± . − . ± . · · · ± ± ± − . ± . . ± . − . ± . · · · ± ± ± − . ± . . ± . − . ± .
20 5.58 6.79 ± ± ± − . ± . . ± . − . ± .
24 1.48 1.21 ± ± ± − . ± . . ± . − . ± .
03 0.88 0.47 ± ± ± − . ± . . ± . − . ± . · · · ± ± ± − . ± . . ± . − . ± .
14 0.51 0.65 ± ± ± − . ± . . ± . − . ± .
15 1.24 2.06 ± ± ± − . ± . . ± . − . ± .
11 0.96 0.86 ± ± ± − . ± . . ± . − . ± .
15 1.06 1.79 ± ± ± − . ± . . ± . − . ± .
26 1.17 1.45 ± ± ± − . ± . . ± . − . ± .
04 1.27 0.33 ± ± ± − . ± . a Distance to the galaxy. References are given by Hunter et al. (2012). b Radius of furthest out detected H ii region R H α in each galaxy from Hunter & Elmegreen (2004). Galaxies without H ii regions or with H ii regions extending beyond the area imaged do not have R Hα . c Radius of furthest out detected FUV knot R FUVknot in each galaxy from Hunter et al. (2016). Galaxies without
GALEX images have no value for this radius. d Disk scale length R D determined from the V -band image surface photometry from Herrmann et al. (2013). In the case ofgalaxies with breaks in their surface brightness profiles, we have chosen the scale length that describes the primary underlyingstellar disk. e Break radius R Br where the V -band surface brightness profile changes slope given by Herrmann et al. (2013). DDO 47 andDDO 210 do not have breaks in their surface brightness profiles. f SFR measured from the integrated FUV luminosity and normalized to the area within one R D from Hunter et al. (2010).The normalization is independent of the radial extent of the FUV emission in a galaxy. i column density have not been multiplied by 1.36 toinclude Helium and heavy elements. This factor will be used later when the efficiency ofKED generation is calculated.We decided not to remove the underlying exponential disks for the 2D cross-correlations.Although the SFR drops off with radius, the FUV image consists of knots of young starsand there can be large FUV knots in the outer disks. For the H i moment 0 and 2 maps,the H i surface density and velocity dispersion do, on average, change with radius too, butnot in a regular and homogenous fashion. Thus, in the 2D C coef maps exponential structurecould remain.Here, a C coef of 1 is perfectly correlated such that every bump and wiggle in one map isexactly reproduced in the other. A value of − correl images and corrmat analyze in IDL with a pythonwrapper. We used this command to shift one image relative to the other over and over againto yield a map of C coef . The peak pixel value in the C coef map is the adopted C coef . Forexample, for NGC 2366, we did a 150 ×
150 array of offsets. That is, we calculated the C coef for x,y offset of − −
150 to x,y offset of +150, +150. This produces a matrix of 301 × C coef of a piece of one of the galaxies “by hand” with a Fortran programwe wrote and we obtained the same peak C coef . The peak C coef and x,y shifts to that pixelare given in Table 2. The cross-correlation matrices are shown in Figure 1.The shift in x,y is also given in Table 2 relative to the disk scale-length R D , for bettercomparison to the size of the galaxy. The shifts vary between 0.02 R D (IC 1613) and 4.75 R D (Haro 36). 50% of the galaxies (18) have shifts less than 0.5 R D , 33% (12) have shifts between0.5 R D and 1 R D , and 17% (6) have shifts greater than 1 R D . 8 –Fig. 1.— Cross-correlation matrices for each galaxy. The images are displayed with the samecolor scale from C coef of zero to 0.8. 9 – 10 – 11 –Ioannis Bagetakos (private communication) examined the cross-correlation method onNGC 2403 as a function of image scale, focussing on scales of 0.23 to 3 kpc, and foundcorrelations on various scales for different images such as star formation tracers, dust, andH i . Thus, we divided our images up into square sub-regions 16 ×
16, 32 ×
32, 64 ×
64, and128 ×
128 pixels and computed the C coef in each box. The coefficient images constructedfrom this just look like noise and show no particular connection to the FUV image. So wedo not consider them further.We also applied alternate methods on one galaxy, NGC 2366 with a max C coef of 0.3, toexamine the robustness of our approach. This galaxy was chosen for initial and special testsbecause it has a giant H ii region and the H i velocity dispersion is high around this region,making it an interesting candidate for looking for a star formation-turbulence correlation.One problem with cross-correlations, in particular, can be caused by moderate signal-to-noise(S/N) pixels dampening the value of C coef . One simple diagnostic is a plot of the pixel valuesof the KED image against the pixel values of the FUV image, given that the FUV image hasbeen geometrically transformed and smoothed to the same pixel grid and beam size as theKED image. We normalized the pixel values in each image to range from 0 to 1, and thisplot is shown in Figure 2. If there were a notable correlation, we would expect a cluster ofpoints in the top right corner. If the images were anti-correlated, we would expect clusters ofpoints around the top left and bottom right. We do not see either of these extremes. Whilethere are some points in the top left corner, it is not a distinct cluster, rather it appears tobe consistent with a typical tail end of a simple distribution of points from 0 to 1.We also tried variations of weighted normalized cross-correlations and a wavelet analysisto NGC 2366. The zero-mean normalized cross correlation coefficient (ZNCC) is basicallythe standard Pearson Correlation measure ρ for a 2D image. Applied to NGC 2366, ZNCCis 0.26. Like ρ or r coefficients, +1 is perfect correlation, − C coef , implies a not very significant degree ofcorrelation. One way to deal with pixels with low S/N is to use a weighted normalized cross-correlation (WNCC). For this test, we weighted the pixels by the ratio of their signal to thestandard deviation of values in the map, which is effective at down-weighting backgroundnoise pixels. Using this method, we obtain a WNCC value of -0.023 – effectively zero,implying no significant correlation between the two images.For our final test on NGC 2366, we used a wavelet analysis to see if the degree of correla-tion depends on scale/resolution. In this process, each image is convolved with progressivelylarger 2D kernels or wavelets, in this case a Ricker or ‘Mexican Hat’ wavelet, and the cross-correlation is calculated at each of these scales or ’lags’. The result for NGC 2366 is shownin Figure 3. Strong correlations at a particular spatial scale would be evidenced by wavelet 12 –Fig. 2.— Values of pixels in KED image plotted against values of pixels in FUV image forNGC 2366. The images have each been normalized so that pixel values are between 0 and1. A notable correlation would be expected to appear as a cluster of points in the top rightcorner (or, in the other corners near values of 1.0 for anti-correlation), however this evidenceis not seen.cross correlation r w values of (cid:38) (cid:48)(cid:48) ), but r w isstill not significant.Thus, we conclude that no matter how we look at the FUV and KED images of NGC2366, the two images are mildly correlated at best and this does not change much with scale.The width of the peak signal in a cross-correlation matrix is expected to represent thescale of the correlation. However, in our matrices, the width is not well defined. The issueis demonstrated in Figure 4 where we show a radial plot and row and column cuts throughthe peak of the WLM C coef matrix. The peak is, of course, obvious, but the radial plot ismessy and the single row and column cuts show a complex background. The main featurein the cross correlation maps is the exponential disk because both the KED and the SFRdensity peak in the center with the exponential disk. The width of the C coef in Figure 4 13 –Fig. 3.— Cross-correlation coefficient r w for NGC 2366 KED and FUV images convolvedwith progressively larger ‘Mexican hat’ kernels. We find no significant correlation betweenthe two images at any resolved scale.is influenced more by the width of the disk than the scale of the 2D correlation. What totake as the baseline for a fit to the peak is also not clear. Therefore, we do not consider thewidths of the peaks further here. We also calculated the C coef in annuli from the center of the galaxy outward. Theimage was blanked outside of the target annulus, which were chosen to match those usedby Hunter et al. (2012) to produce the H i radial profiles. We normalized the pixel valuesin the annulus with respect to the average in the annulus, so in effect large-scale variations,such as the exponential fall-off with radius, are taken out. Then we measured the C coef forthe annulus. Figure 5 shows the C coef of the annuli as a function of annulus distance fromthe center of the galaxy. The annuli used galaxy center, ellipticity, and major axis positionangle determined from V -band images and given by Hunter et al. (2012).We see a wide variety of profiles. The central points in NGC 4163 and in VIIZw403reach a C coef of nearly 0.95 and a few other galaxies have peaks as high as 0.9. By contrastthe peak in DDO 210 occurs in the outermost annulus and only reaches a value of 0.14. Inmany galaxies the C coef drops in value with radius, but in many others it is relatively flat. 14 –Table 2. Correlation Coefficients and Offsets Galaxy Max C coef X shift a Y shift a Shift/ R D Offset (X × Y) b Calibration (10 ) c CVnIdwA 0.77 3 2 0.38 75 ×
75 7.67DDO 43 0.61 4 13 0.89 150 ×
150 18.33DDO 46 0.57 -5 -9 0.40 150 ×
150 26.77DDO 47 0.54 2 -4 0.13 150 ×
150 9.32DDO 50 0.41 -2 -8 0.14 300 ×
300 20.40DDO 52 0.56 6 -14 0.91 150 ×
150 24.86DDO 53 0.58 -4 5 0.36 150 ×
150 24.54DDO 63 0.41 8 -5 0.39 150 ×
150 18.81DDO 69 0.50 -15 -9 0.54 150 ×
150 28.29DDO 70 0.50 -23 18 0.63 300 ×
300 4.83DDO 75 0.52 -8 -2 0.43 300 ×
300 18.09DDO 87 0.48 -2 2 0.09 150 ×
150 18.73DDO 101 0.59 -15 5 0.76 150 ×
150 15.22DDO 126 0.62 2 3 0.15 150 ×
150 22.95DDO 133 0.65 1 2 0.05 150 ×
150 6.61DDO 154 0.50 10 -4 0.60 150 ×
150 17.69DDO 167 0.72 9 11 1.97 150 ×
150 22.95DDO 168 0.68 -2 0 0.08 150 ×
150 19.36DDO 187 0.74 -8 0 0.35 150 ×
150 25.58DDO 210 0.63 23 1 0.94 150 ×
150 8.77DDO 216 0.63 20 15 0.90 75 ×
75 3.54F564-V3 0.87 0 -4 0.40 150 ×
150 8.69IC 1613 0.33 1 2 0.02 300 ×
300 17.69LGS 3 0.46 13 -19 0.73 150 ×
150 8.05M81dwA 0.42 -11 -16 1.88 150 ×
150 18.01NGC 1569 0.40 30 6 1.65 300 ×
300 29.08NGC 2366 0.30 -6 -4 0.09 150 ×
150 21.51NGC 3738 0.48 -13 7 0.68 150 ×
150 25.50NGC 4163 0.62 4 12 0.83 150 ×
150 15.46NGC 4214 0.35 -88 8 2.57 300 ×
300 18.17SagDIG 0.51 -7 15 0.97 75 ×
75 1.84WLM 0.33 0 -32 0.20 150 ×
150 22.95Haro 29 0.63 -21 -1 2.69 150 ×
150 23.19Haro 36 0.34 -46 -54 4.75 150 ×
150 21.91Mrk 178 0.60 2 -1 0.33 150 ×
150 25.98VIIZw 403 0.68 1 -3 0.19 150 ×
150 12.11 a Offset of the pixel with the maximum C coef from the center of the array, in pixels. The pixel scale is1.5 (cid:48)(cid:48) except for DDO 216 and Sag DIG where it is 3.5 (cid:48)(cid:48) . b Offsets in pixels. c Constant by which to convert counts in KED maps to ergs pc − .
15 –Fig. 4.— Cuts through the peak in the C coef matrix of WLM. Top: Radial profile. Middle:Row plot. Bottom: Column plot.In a few galaxies the C coef drops precipitously from a relatively high value for the inner-mostannulus to near zero beyond that radius (DDO 167, F564-V3, Haro 36). 16 –Fig. 5.— Correlation coefficient between FUV and KED images in annuli as a function ofdistance from the center of the galaxy. The C coef profile is plotted from 0 to 1 for all galaxiesfor ease of comparison, and the radius is normalized by the disk scale length measured fromthe V -band image (Table 1). The pixel values in each annulus have been normalized by theaverage in the annulus, so large-scale trends with radius have been removed. 17 – 18 –
4. Results4.1. Cross-correlations
Generally, the 2D C coef indicate low levels of correlation between the FUV and KEDimages. In Figure 6 we plot the peak C coef against the integrated SFR for each galaxy tosee if a higher level of correlation is related to the overall SFR. There is no relationshipbetween the two values. In annuli, C coef can be as high as 0.9 in the center, indicating acorrelation, but the values tend to be low overall, and the radial profiles exhibit a wide rangeof shapes. From the images, visually most of the FUV is patchy and tends to be concentratedtowards the central regions of the galaxies while the H i often extends quite far outside theoptical/UV galaxy. So the birds-eye view of a dIrr might expect a higher correlation in thecentral regions where there is ample H i and FUV, with little correlation as you go fartherout where there are typically fewer FUV knots.By comparison, in the spiral NGC 2403 Ioannis Bagetakos (private communication)found that the FUV and H i surface mass density are uncorrelated with a C coef < .
20. Theydid, however, find correlations between dust and star formation ( C coef > .
55) and betweenPAHs and H i ( C coef ∼ . i data, as well as images at 8 microns,24 microns, H α , and FUV, and is nearby with an H i beam of 136 pc ×
119 pc. As an Scdspiral it is significantly larger and more massive than the dIrr galaxies in this study.
Since star formation is usually lumpy, we ask whether the lack of correlation betweenFUV and KED images is because KED is smooth compared to FUV or because lumpsin the two images do not correlate. Figure 7 shows the KED maps and FUV images atfull resolution. A contour of the FUV image is superposed on the KED map to facilitatecomparison. We see that FUV and KED maps are both generally lumpy although the lumpsare not necessarily located in the same place.To examine the degree of lumpiness, we looked at the fraction of pixels with raw valuesabove a given percentage of the maximum pixel value in the image. Specifically, we countedthe fraction of total pixels that have counts within 10%, 20%, 30%, 40%, and 50% of themaximum count value in each of the FUV and KED images. These data are shown in Figure8 as percentage of total pixels as a function of selected cut-off deviation from the maximumpixel value in the image. For example in CVnIdwA, the percentage of pixels with values 19 –Fig. 6.— Integrated SFR normalized to one disk scale length versus the maximum correlationcoefficient for each galaxy. The correlation coefficients are given in Table 2 and the SFRsare in Table 1.within 10% of the maximum value is 0.69% in the FUV image and 1.03% in the KED image,whereas the percentage of pixels with values within 50% of the maximum value is 5.65% inthe FUV image and 15.98% in the KED image.To understand what these plots mean we can compare the appearance of the galaxiesin Figure 7 with the plots in Figure 8. We see in the images that galaxies like LGS3, DDO87, DDO 133, and SagDIG have a few small FUV knots but more or bigger KED knots.The KED knots fill more of the area and so a higher fraction of the pixels are close to thepeak intensity. These galaxies have flat FUV profiles in Figure 8 because very few pixels areclose to the peak intensity, i.e., the FUV is spotty, but they have KED profiles that rise withpercentage of maximum pixel value because the KED is more uniform. DDO 43 is unique inthis sample because it is the only one with an approximately flat KED profile and an FUVprofile that rises with percentage of maximum pixel value. The reason is clear from Figure 7,which shows that the FUV image of DDO 43 is filled with bright spots, making most of theimage close to the peak pixel value, while the KED image has weaker peaks that are morespread out. DDO 167, on the other hand, has FUV and KED knots that are comparable insize, and FUV and KED profiles that rise together with percentage of maximum pixel value,as do DDO 47, DDO 101, F564-v3, and NGC 4163. Most of the galaxies have broader KEDdistributions than FUV emission, so their KED pixel percentages rise faster than their FUVpixel percentages as the top percentage of the maximum pixel value increases. 20 –Fig. 7.— Kinetic energy density (KED) maps and full-resolution FUV images for eachgalaxy. The major FUV knots are contoured and that white contour is shown on the KEDmap to facilitate comparison. The conversion of counts in the KED maps to physical unitsis given in Table 2. To convert FUV counts s − to flux in units of erg s − cm − ˚A − multiplyby 1 . × − . Figures for the rest of the galaxies in this study are available in the on-linematerials (72 images in 12 figures). 21 –The general rising trend of the curves in Figure 8 is mostly the result of the exponentialradial profile of the disk with the peaks in KED and FUV standing a nearly fixed fractionabove the mean profile. Figure 9 shows models for these curves assuming an exponentialdisk intensity profile I ( r ) = e − r , so the radius as a function of intensity is r ( I ) = − ln( I ).The radius at 10% of the peak is then r (10%) = − ln(1 − . πr (10%) . In general, for anintensity that is the fraction x down from the peak intensity, the fraction of pixels in thetotal disk is f ( x ) = π ( − ln[1 − x ]) / (cid:0) πr (cid:1) (1)where r max is the size of the disk measured in scale lengths. Figure 9 shows f ( x ) versus x inthree cases. The top curve is for an exponential profile with a scale length 1.5 times largerthan the middle curve and an overall galaxy size the same, r max = 2 scale lengths. The lowercurve also has a scale length 1.5 times larger than the middle curve but the overall galaxysize for the lower curve is 1.5 times larger ( r max = 3). Larger scale lengths for a given galaxysize make the percentage curves rise faster because more of the disk is close to the peakintensity at the center.The similarity of the model curves in Figure 9 to the observations in Figure 8 impliesthat the qualitative effect being captured by the fractional distribution is the result of theexponential disk. However, the percentage of pixels observed is much smaller than the modelpercentage, i.e., several percent or less for the observations compared to ∼
10% at the 50%top percentage of maximum pixel value. This difference implies that the peaks in the KEDand FUV distributions stand above the exponential disk, so their areas are a small fraction, ∼ Another way of looking at the data is to compare individual pixels in pairs of images.We have done that, examining KED, the velocity dispersion of the gas v disp , and H i surfacedensity Σ HI versus SFR surface density as determined from the FUV images, Σ SFR . Recallthat the FUV images were geometrically transformed and smoothed to match the pixel sizeand resolution of the H i images. For all galaxies but DDO 216 and Sag DIG, the pixelsize is 1 . (cid:48)(cid:48) and for these two it is 3 . (cid:48)(cid:48) . To compensate for radial trends, we determinedthe azimuthally-averaged Σ SFR , KED, H i and v disp in annuli from the center of the galaxyand subtracted that from the observations. We used optically-determined disk parametersof center, minor-to-major axis ratio b/a , and position angle of the major axis from Hunteret al. (2012). The widths of the annuli, constant in a given galaxy, were chosen to be thesame as those used to measure the H i surface density profiles of Hunter et al. (2012). Theazimuthally-averaged radial profiles of Σ SFR , KED, v disp , and Σ HI are shown for each galaxyin Figure 10. The pixel-pixel plots of excess KED, v disp and Σ HI versus excess Σ SFR are shownin Figures 11-13. All of these quantities except v disp were corrected to a face-on orientationby multiplying the fluxes by the cosine of the inclination. The KED units are erg pc − , v disp is in km s − , Σ HI is in M (cid:12) pc − and Σ SFR is in units of M (cid:12) yr − pc − . KED values in Figures10-13 have not been corrected for Helium and heavy elements. Note that only the regionsof relatively high Σ SFR are plotted, i.e., with positive excess above the annular average, andwe plot the logarithm of this excess. For the quantities on the ordinate, we consider bothpositive and negative excess values over the average, so they are not plotted in the log. Someregions of locally high Σ
SFR have locally low KED, v disp or Σ HI .In the radial averages shown in Figure 10, we see that KED, Σ SFR , and Σ HI generallydecline with radius. Tamburro et al. (2009) found this also for spiral galaxies. They alsofound that v disp declines with radius in their sample, but in our sample of dIrr the drop of v disp with radius is very minor, if any. They also find a clear correlation of KED with Σ SFR in pixel-pixel plots, whereas our Figure 11 does not show such a nice correlation. 26 –Fig. 10.— Azimuthally-averaged radial profiles of Σ
SFR determined from the FUV, KED(not corrected for He and heavy elements), v disp , and Σ HI . FUV emission is the limitingquantity in that it does not go out as far as the other quantities. Optical disk parameters(center, b/a , and major axis position angle) from Hunter et al. (2012) were used, and holesin the gas or FUV emission were not used in the averages. 27 – 28 –Fig. 11.— Pixel-pixel plots of the excess KED above the average value at each radius vs.the log of the excess Σ SFR . The density of points is color-coded. Figures for the rest of thegalaxies in this study are available in the on-line materials (there are 6 figures like this for36 galaxies). The KED has not been corrected for He and heavy elements. 29 –Fig. 12.— Pixel-pixel plots of the excess v disp above the average value at each radius vs.the log of the excess Σ SFR . The density of points is color-coded. Figures for the rest of thegalaxies in this study are available in the on-line materials (the remaining 36 galaxies areshown in 6 figures). 30 –Fig. 13.— Pixel-pixel plots of the excess Σ HI above the average value at each radius vs. logof the excess Σ SFR . The density of points is color-coded. Figures for the rest of the galaxiesin this study are available in the on-line materials (the remaining 36 galaxies are shown in6 figures).Figures 11 - 13 typically show concentrations of points at a low excess Σ
SFR and acontinuation of these points toward higher excess Σ
SFR . The low excess Σ
SFR are in the outer 31 –disks and the high excess Σ
SFR are in the inner disks. Some galaxies have two concentrationsof points in these figures.To quantify the pixel distributions, we determined the excess Σ
SFR and other quantitiesat the plotted concentrations. For each galaxy we made a histogram of the log of the excessΣ
SFR (the abscissa value) and found the peak at the low density concentration. The excess logΣ
SFR in that peak was determined from the average value in the three bins of the histogramcentered there. The bin width was 0.2 in the log of the excess Σ
SFR . Then for these threebins around the histogram peak for log excess Σ
SFR , we determined the mean value of thequantity plotted on the ordinate in the figures, i.e., the excess KED, v disp and Σ HI . Forthe higher excess Σ SFR , we took the mean value of the excess Σ
SFR and the other quantitiesfor all regions where the log of the excess Σ
SFR was larger than the high-SF edge of theconcentration of points, typically at − . − . − . v disp ,and Σ HI versus the mean of the log of the (positive) excess Σ SFR for all galaxies, with dotscorresponding to the low Σ
SFR concentrations in the outer disks and crosses correspondingto the high Σ
SFR in the inner disks. The curves in the KED plot show fitted relationshipsbetween the KED generated by supernovae and the Σ
SFR for the indicated efficiencies ofconverting SN energy into turbulence, and for galaxy scale heights of 850 pc and 540 pc.These theoretical KEDs come from equation 3.7 in Bacchini et al. (2020), which is
KED SN = η Σ SFR f cc E SN (2 H/v turb ) (2)where η is the efficiency of energy conversion from supernova to turbulence, f cc = 1 . × − M − (cid:12) is the number of core-collapse supernovae per solar mass of stars, E SN = 10 erg is the supernova energy, H is the disk thickness and v turb is the turbulent gas velocitydispersion (the ratio of these latter two quantities gives the turbulent dissipation time).Bacchini et al. (2020) compare the radial profiles of turbulent energies in 10 nearby galaxieswith the SFRs and derive an average efficiency of 1 . . . % if all of the turbulence comesfrom star formation. Because the required efficiency is relatively low, they concluded thatsupernovae related to star formation can drive most of the interstellar turbulence.For the dwarf galaxies studied here, we evaluate equation (2) using scale heights andvelocity dispersions from the average values for 20 dIrrs in Elmegreen & Hunter (2015), inTable 2 of that paper. For the concentrations of pixel values corresponding to the outerregions of the galaxies, we take the average scale height and v disp at 2 scale lengths, whichare H = 850 pc and v disp = 9 . − . For the inner regions, we take the values at 1scale length, which are 540 pc and 10.7 km s − . We also include Helium and heavy elements 32 –Fig. 14.— Mean excess KED corrected for He and heavy elements, v disp , and Σ HI versusthe mean of the log of the (positive) excess Σ SFR for all galaxies. Dots correspond to thelow Σ
SFR concentrations in the outer disks and crosses correspond to high Σ
SFR in the innerdisks.
Left:
The curves show fitted relationships between the KED generated by supernovaeand the Σ
SFR for the indicated efficiencies of converting SN energy into turbulence, and forgalaxy scale heights of 850 pc and 540 pc.
Middle:
Average v disp excess at each concentrationof excess Σ SFR in Figure 12. The excess velocity dispersion averages 0.34 km s − in the outerdisk and 0.17 km s − in the inner disk. Right:
Average Σ HI excess at each concentration ofexcess Σ SFR in Figure 13. There is a clear trend toward excess H i at local SF regions.in the KED by multiplying the H i mass surface density by 1.36. Then with f cc and E SN given above, equation (2) is fitted for the efficiency in the two cases, using for the moment v disp instead of v turb . The results are drawn as curves in the left panel of Figure 14. Theaverage local efficiencies for conversion of SN energy to KED are η = 0 . ± .
045 and0 . ± . η include thermal and turbulent motions, whichwere distinguished in several limiting cases by Bacchini et al. (2020) to get the desired v turb . If we assume Mach ∼ i ISM, then v turb = v disp / . andthe derived values of η decrease by the factor 0.7, preserving the ratio η/v used to matchthe KED. Alternatively, we could use thermal dispersions of 4.9 km s − and 6.1 km s − modeled for NGC 4736 and NGC 2403 respectively by Bacchini et al. (2020) to estimatethat v turb /v disp ∼ .
8, given that v disp ∼
10 km s − here. Then our derived η should decreaseby ∼ .
8. The Bacchini et al. (2020) galaxies were more massive than our dIrr galaxies,but the thermal contributions to v disp are not likely to be much different. These correctionschange the average value of η = 0 . η ∼ . η value is the average for the peak regions of star formation. It measures howefficiently star formation energy gets into H i turbulent motions locally in units of the super-nova energy per unit mass of young stars. When normalized this way, other types of energy 33 –related to star formation such as expanding HII regions and stellar winds are included in η too. What is not included as a source of turbulent motion is energy unrelated to star for-mation, such as gravitational energy from gas collapse on the scale of the ISM Jeans length,or collapse energy from transient spiral arms driven by combined gas and stellar masses, orshock energy from the relative motions of gas and stellar spiral density waves. If η ∼ . i gas, then the global turbulent energy pumped by all of the star formation in a galaxyshould equal our local η multiplied by the global star formation rate (along with the otherfactors in equation 2). Because Bacchini et al. (2020) found that the global turbulent energyis 1 . . . % of the energy derived from the star formation rate, there would seem to be moreenergy required than what star formation alone can provide. The excess energy needed is afactor of ∼ . / . − ∼ η ∼ . η = 0 .
009 assuming a pure warm phase H i . Also, ouraverage η for the inner disk regions in Figure 14 was higher than the average for the inner andouter disks combined (which gave the value 0.0048) by a factor of 1.2. But even within thisrange, the global energy from turbulence seems to be larger than what can be pumped fromstar formation alone, if we use local star formation rates as the basic means of calibrating η .Figure 14 for the KED excess has two high points for the inner disk which were notincluded in the efficiency fit. These are the galaxies Haro 29 with excess KED= 16 . × erg pc − , and NGC 1569 with excess KED= 141 × erg pc − . Correspondingly, Figure11 shows a scatter of individual pixel points to very high values of KED for these galaxies.The middle panel of Figure 14 shows the average v disp excess at each concentration ofexcess Σ SFR in Figure 12. The excess velocity dispersion is rarely larger than 1 km s − and averages 0.45 km s − in the outer disk, − .
34 km s − in the inner disk, and 0.37 kms − overall. Some local star formation regions have lower H i velocity dispersions than theaverage at that galactocentric radius, giving negative excesses in Figure 14. These typicallysmall excesses in the local H i velocity dispersion are consistent with the small feedbackefficiencies found above. There is relatively little generation of turbulence at the positionsof star-forming regions.The right-hand panel of Figure 14 shows the average Σ HI excess at each concentrationof excess Σ SFR in Figure 13. There is a clear trend toward excess H i at local SF regions,although in a few cases the H i is less than the azimuthal average. This general excesscorresponds to a ratio of Σ HI to Σ SFR that equals 6.5 Gyr in the outer disk, 1.2 Gyr in the 34 –inner disk and 1.6 Gyr overall, where this latter fit is shown by the curve in the figure. Forthis fit, the high point that is plotted in Figure 14 is excluded; that is for NGC 1569, wherethe ratio is 31 Gyr. This average ratio of ∼ . i and star formation in recentpapers (Hunter et al. 2019, 2020; Madden et al. 2020).
5. Discussion
Comparisons between the kinetic energy density or velocity dispersion and the localstar formation rate using cross correlations of several types and pixel-level excesses abovethe radial average quantities have shown virtually no connections between large-scale turbu-lence and star formation. Many of the galaxies have lumpy KED and FUV images but thelumps are not well correlated or anti-correlated spatially. This is contrary to some theoreticalexpectations and the simulations that have been designed to illustrate those expectationswhich suggest that feedback from star formation pumps a significant amount of interstellarturbulence, and thereby controls the interstellar scale height and average mid-plane density.While it is generally accepted that this mid-plane density controls the collapse rate of theISM and therefore the average star formation rate, the origin of the turbulence and othervertical forces which determine the scale height and density have been difficult to observedirectly. Most likely, the maintenance of a modest value for gravitational stability parameter Q controls the overall interstellar turbulent speed through pervasive and mild gravitationalinstabilities, which also feed the star formation process through cloud formation. This wasdemonstrated by Bournaud et al. (2010) and also underlies the Feedback in Realistic Envi-ronments (FIRE) simulations by Hopkins et al. (2014); the primary role of feedback is todestroy molecular clouds locally (Benincasa et al. 2020). Our data suggest that this feedbackdoes not extend far enough from molecular clouds to be visible in the H i at our resolution(from 26 pc at IC 1613 to 340 pc at DDO 52).
6. Summary
We have examined the relationship between star formation, as traced by FUV images,and turbulence in the gas, as traced by kinetic energy density images and velocity dispersionmaps in the LITTLE THINGS sample of nearby dIrr galaxies. We performed 2D cross-correlations between FUV and KED images, finding maximum C coef that indicate little 35 –correlation. A plot of integrated SFR against the maximum C coef also shows no correlation.We also performed cross-correlations in annuli centered on the optical center of the galaxyto produce C coef as a function of radius. In some galaxies the centers have C coef that arehigh enough to indicate a correlation, and in some galaxies the C coef drops off with radiusfrom the center.To look at the images a different way, we determined the fraction of pixels in the FUVand KED images with values above a given percentage of the maximum pixel value in theimage. Plots of these quantities show different behaviors for FUV and KED images in manyof the galaxies. Finally, we considered on a pixel-by-pixel basis the excess KED, v disp , andΣ HI above the average radial profiles of these quantities and plotted that versus the excessΣ SFR . There was a weak tendency to have a higher excess KED at a higher excess Σ
SFR ,corresponding to an efficiency of kinetic energy input to the local ISM from supernova relatedto star formation of about 0.5%. This is too small by a factor of about 2 for star formationto be the only source of global kinetic energy density. The excess v disp connected with starformation peaks is also small, only 0 .
37 km s − on average. The angular scale for these smallexcesses is typically 1 . (cid:48)(cid:48) , which, for a distance of 3 Mpc, corresponds to ∼
20 pc.We are grateful to Dr. C. Bacchini for comments on the manuscript. H.A. is grateful tothe Lowell Observatory Director’s Opportunity Network for funding to work on this project.Lowell Observatory sits at the base of mountains sacred to tribes throughout the region. Wehonor their past, present, and future generations, who have lived here for millennia and willforever call this place home.Facilities: VLA GALEX
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