A Two-Component Power Law Covering Nearly Four Orders of Magnitude in the Power Spectrum of Spitzer Far-Infrared Emission from the Large Magellanic Cloud
aa r X i v : . [ a s t r o - ph . C O ] J un A Two-Component Power Law Covering Nearly Four Orders ofMagnitude in the Power Spectrum of Spitzer Far-InfraredEmission from the Large Magellanic Cloud
David L. Block
School of Computational and Applied Mathematics, University of the Witwatersrand,Private Bag 3, WITS 2050, South Africa
Ivˆanio Puerari
Instituto Nacional de Astrof´ısica, Optica y Electr´onica, Calle Luis Enrique Erro 1, 72840Santa Mar´ıa Tonantzintla, Puebla, M´exico.
Bruce G. Elmegreen
IBM T. J. Watson Research Center, 1101 Kitchawan Road, Yorktown Heights, New York10598 USA, [email protected]
Fr´ed´eric Bournaud
CEA, IRFU, SAp, F-91191 Gif-sur-Yvette, France.
ABSTRACT
Power spectra of Large Magellanic Cloud (LMC) emission at 24, 70 and 160 µ m observed with the Spitzer Space Telescope have a two-component power-lawstructure with a shallow slope of − . k , and a steep slopeof − . k . The break occurs at k − ∼ −
200 pc, which is interpretedas the line-of-sight thickness of the LMC disk. The slopes are slightly steeper forlonger wavelengths, suggesting the cooler dust emission is smoother than the hotemission. The power spectrum covers ∼ . k structure in what is effectively atwo-dimensional geometry. Small-scale driving from stellar processes and shockscause the high- k structure in a 3D geometry. This transition in dimensionalitycorresponds to the observed change in power spectrum slope. A companion papermodels the observed power-law with a self-gravitating hydrodynamics simulationof a galaxy like the LMC. Subject headings:
ISM: structure — galaxies: ISM — Magellanic Clouds — In-frared: ISM 2 –
1. Introduction
The Large Magellanic Cloud (LMC) is the first spiral galaxy in which an elongatedfeature of stars was identified. In a remarkable drawing of the LMC as seen with the naked eyemade by Sir John Herschel, a bar is clearly shown, which Sir John termed an “axis of light”.Also faithfully represented in that drawing is a prominent spiral arm to the north, as well astwo “embryonic” arms. The LMC belongs to the de Vaucouleurs classification bin SB(s)m,and is approximately 50 kpc distant (based on the distance modulus m − M = 18 . ± . ◦ –50 ◦ while the position angle of the lineof nodes lies between 120 ◦ and 150 ◦ ; an excellent review is provided by van der Marel et al.(2008). The study by van der Marel & Cioni (2001) using DENIS and 2MASS surveys yieldsan inclination of 35 ◦ and position angle of 125 ◦ , which are the values adopted in this study.The infrared emission from dust in the diffuse interstellar medium of our Galaxy hasbeen extensively studied from IRAS to the Herschel Space Observatory. The quantitativeGalactic model of Draine & Li (2007) consists of a mixture of amorphous silicate grains andcarbonaceous grains, each with a wide size distribution: sizes range from molecules containingtens of atoms to large grains greater than 1 µ m. Dust grain temperatures at the surfaces ofgiant molecular clouds in our Galaxy are hotter than in their cold interiors (Greenberg & Li1996), varying from ∼
15 K to ∼ ∼
23 K forsolar-type metallicities, increasing to 40 K at a low metallicity of 12 + log(
O/H ) ∼ ∼
25 K in the central portion of the stellar disk,declining to ∼
15 K in the outer domains (Figure 2 in Braine et al. 2010).Here we study the power spectrum (PS) of FIR dust emission from the LMC. Powerspectra of emission maps are an important but poorly understood diagnostic for interstellarstructures and the motions that cause them. PS of HI emission from sections of the MilkyWay are approximately power laws throughout the entire range of observed spatial frequen-cies (Crovisier & Dickey 1983; Green 1993; Dickey et al. 2001; Khalil et al. 2006). Similarslopes occur for PS of Milky Way CO emission (St¨utzki et al. 1998), IRAS and DIRBE 100 µ m emission (Gautier et al. 1992; Schlegel, Finkbeiner, & Davis 1998) and HI absorption(Deshpande, Dwarakanath, & Goss 2000). These power laws resemble theoretical expecta-tions from passive gas motions (Goldman 2000) and compressions (Lazarian & Pogosyan2000) in a turbulent fluid, so it is natural to interpret them in this way (Elmegreen & Scalo 3 –2004). The situation is much more complex for entire galaxies, however, as galaxies un-dergo rotation and contain supernovae, spiral waves, and other physical processes that arenot present in localized clouds. Still, Stanimirovic et al. (1999) found a power law covering2.5 orders of magnitude in the PS of HI emission from the Small Magellanic Cloud (SMC).Subsequent analysis of SMC dust emission (Stanimirovic et al. 2000) gave a similar result.In all of these cases, the two-dimensional PS is approximately a single power law with aslope between − . −
3, depending on the tracer and the depth of the velocity slice (e.g.,Dickey et al. 2001).The PS of HI emission from the Large Magellanic Cloud (LMC) is significantly differentfrom these others. Elmegreen, Kim, & Staveley-Smith (2001) found a two-component power-law with a break at a wavenumber that is consistent with the inverse of the disk thickness.They proposed that turbulent processes and other gas motions are approximately 2D onscales larger than the disk thickness, and similar processes are 3D on scales smaller than thedisk thickness. PS for integrated emission from turbulent gas has a slope that differs by 1 inthese two cases, with the steeper slope corresponding to 3D motions at large wavenumbers(Lazarian & Pogosyan 2000; Lazarian et al. 2001). In the LMC, the break scale increaseswith radius, indicative of a reasonable level of flare in the HI disk. Subsequent models ofdisk PS by Padoan et al. (2001) confirmed this interpretation.The same two-part PS was also found for discrete clouds in the Milky Way. Miville-Deschˆenes et al.(2003) found it in HI emission from the Ursa Major Galactic cirrus and modeled it with afractal Brownian motion cloud. Their model suggested that the break in the PS occurs at awavenumber k ∼ / (2 D ) for cloud depth D , and that the power-law slope changes by 1 atthe break. Ingalls et al. (2004) observed a two-component PS using Spitzer IR observationsof a molecular cloud in the GUM nebula. They found that the PS slope changed from − . × − arcsec − to − . × − arcsec − . The inferred cloud depth was 0.3 pc.A break in the PS of HI emission has not generally been observed in galaxies otherthan the LMC. Dutta et al. (2008) studied HI emission from the face-on galaxy NGC 628and derived a PS slope of − . − − . . ′′ , corresponding to 660 pc).PS of H α and HI emission from several dwarf galaxies showed power laws and no breaks(Willett, Elmegreen, & Hunter 2005; Begum, et al. 2006; Dutta et al. 2009b). In the SMC 4 –there is no break either. For the SMC, this lack of a break could be because the line-of-sightdepth is comparable to the transverse length (Westerlund 1997). This could be the case forthe dwarf galaxies also (Roychowdhury et al. 2010).PS miss the break if the high frequency part is dominated by the point spread func-tion of bright unresolved emission sources. These sources introduce a Gaussian-shaped dipat high frequency in the PS (e.g., Dickey et al. 2001). Models of this dip are shown inElmegreen, et al. (2003) and Block et al. (2009). To see a double power law, the part athigh frequency has to extend for a long wavenumber range from the inverse thickness of thedisk to the inverse spatial scale of the telescope resolution. If the thickness is smaller thanthe scale of the resolution, then the high frequency power law from 3D turbulence cannotbe observed. This problem is present for conventional ∼ ′′ HI resolution in galaxies atdistances greater than 2 Mpc. Our Spitzer infrared observations of the LMC have a resolu-tion scale at least 10 times smaller than the disk thickness, so we should be able to see anyhigh-frequency power law clearly.Optical observations of galaxy PS have sufficient resolution to see the thickness, butunless there are large numbers of stellar groupings with sizes spanning a wide range aroundthe disk thickness, the clarity of the PS break will be poor. A tentative PS break wasobserved optically in NGC 5055 (Elmegreen, Elmegreen & Leitner 2003), but not in NGC628 (Elmegreen et al. 2006). A break in the slope of the autocorrelation function for youngstars in several nearby galaxies was inferred to mark a change in geometric properties of thegas by Odekon (2008).Spitzer Space Telescope observations have such high angular resolution that 3.5-4 or-ders of magnitude in spatial scale can be covered for the LMC. This is much larger thanis possible for any other galaxy in any other gas tracer at the present time. The smallestresolvable scale is only a few parsecs, which is much smaller than the disk thickness, so thereis no contamination in the PS from the point spread function. Thus the FIR observationsof the LMC are ideal for an unprecedented view of correlated structures in a galaxy. In acompanion paper (Bournaud et al. 2010), we describe self-gravitating hydrodynamic simu-lations of galaxies like the LMC, with and without energy input from star formation. Weobtain a two component PS for projected density structure in the simulations too, and weexamine the velocities and energy sources that lead to this structure.In what follows, the observations are presented in Section 2.1, the PS are shown inSection 2.2, while Section 3 contains a discussion of the implications of our results. Theconclusions are in Section 4. 5 –
2. Observations2.1. Data
Our data for the LMC come from SAGE (“Surveying the Agents of a Galaxy’s Evolu-tion”; Meixner et al. 2006). The images at 24, 70 and 160 µ m were obtained with MIPS onthe Spitzer Space Telescope; they cover ∼ × λ/D at 24 and 160 µ m, 0.3 λ/D at 70 µ m). Thus, the instrumentallows the telescope-limited resolution of 6 ′′ , 18 ′′ , and 40 ′′ at 24, 70, and 160 µ m, respectively(Rieke et al. 2004). The first of these, 6 ′′ , corresponds to a spatial scale of 1.45 pc in theLMC. The theoretical Nyquist limit is 2 pixels, but in reality, the true Nyquist limit is 2resolution elements.At 24 µ m and 70 µ m, the images sizes we use are 8192 × . ′′ . ′′ µ m image was binned 2 × µ m, the image size is 2048 × . ′′ / px.As far as deprojecting images of the LMC are concerned, there is no unique “center” - thedynamical center of the HI is offset by almost a full degree from the photometric center ofthe bar (Westerlund 1997). We choose to conduct our deprojections about the dynamicalcentre of the LMC, which has right ascension α = 5 h . m and declination δ = − ◦ ′ (J2000.0), following section 7 in van der Marel et al. (2002). Figure 1 shows the MIPS image at 160 µ m and the 2D PS at the bottom right. The PSis the sum of the squares of the real and imaginary parts of the 2D Fourier transform (madewith fourn from Numerical Recipes ), averaged over each 2D wavenumber k = ( k x + k y ) / .We divided the PS into 15 wavenumber intervals and plotted the arithmetic mean for eachinterval as a dot in Figure 1. The PS has 2 distinct parts with different power law slopes.The steep slope at high k is − . ± .
13 and the shallow slope at low k is − . ± . /k = 200 parsecs. Double power laws also occur in the PSof the other MIPS channels (Figs. 2 and 3). The break in the middle of the PS on a logscale allows both power laws to be seen clearly; there are 1 . − µ m arises from cold dust in dark clouds. At 70 µ m, the emission 6 –comes from both warm and cold dust, and at 24 µ m the emission is from warm dust andPAH’s. The figures indicate that all of this dust has correlated structure spanning the entireresolvable disk of the LMC. The PS break is at 100 −
200 pc for all 3 MIPS bands, suggestingthat the disk thickness is about this value for each dust component. The least-square slopesare similar but with a slight progression. At high spatial frequency, they are − . ± . − . ± .
52 and − . ± .
34 for 160 µ m, 70 µ m and 24 µ m, respectively, and at low spatialfrequency, they are − . ± . − . ± .
36 and − . ± .
19 for these three passbands.The steepening of the slope with FIR passband indicates that there is relatively more spectralpower on large scales for longer wavelengths. This suggests that cool dust is more diffuseand less structured on small scales than warm dust. This conclusion is true regardless of thetotal fluxes and column densities of cool and warm dust, as these quantities contribute onlyto the k = 0 component of the PS.For comparison, the 2D PS of HI emission from the LMC has a slope of ∼ − . ∼ − . /k ∼
100 pc, increasing slightly with galactocentric radius.These slopes are both higher than the corresponding slopes for the dust emission. The HIPS is most like the 160 µ m PS, although even steeper, suggesting that HI is even moredominated by large scale structure than cool dust.In order to test the robustness of our analysis, we also deprojected the MIPS imagesabout the bar center as opposed to the dynamical center of the LMC; the resulting PS at 24 µ m remained unchanged, while the slopes of the two power laws at 70 µ m and 160 µ m wereonly marginally affected. We conclude that choice of center for deprojection purposes is notresponsible for the presence of two distinct power laws.
3. Discussion
The figures indicate that the relative thickness of the LMC dust disk compared to itsdiameter is only a few percent. This value comes from the ratio of the smallest wavenumberin the PS (the inverse of the largest scale) to the wavenumber at the break in the PS (theinverse of the disk thickness). The smallness of this ratio indicates that the LMC geometryis effectively 2D for some processes.Turbulence in a strictly 2D medium (infinitely thin) differs from turbulence in a 3Dmedium. For incompressible turbulence, an infinitely thin layer can have an inverse cascade ofenergy, i.e., energy moves from smaller to larger scales, and a direct cascade of mean-squaredvorticity, i.e., from larger to smaller scales (Kraichnan 1967; Leith 1968; Batchelor 1969). 7 –For 3D turbulence, the energy generally cascades to smaller scales. Thus 2D turbulence onlarge scales can be fed by small scale motions, while 3D turbulence on large scales can onlybe fed by larger scale motions (in the absence of a magnetic field or other physical processesthat force a connection between large and small scales).It would be interesting if an inverse cascade of turbulence occurred in a very thin galaxy.Then localized energy from gravitational instabilities on the scale of the Jeans length, whichis the disk scale height, and from supernovae, superbubbles, and so on, could power largescale motions and density correlations. A galactic disk is not strictly 2D even if it is thin,but the gas motions can still be highly anisotropic, making the disk effectively 2D. Forexample, density-wave and bar-driven streaming motions are often 5 to 10 times fasterthan the perpendicular motions that produce the disk thickness. Large scale motions maybe generated by disk self-gravity, tidal forces from companion galaxies (Mastropietro et al.2009), intergalactic ram pressure (Tonnesen & Bryan 2009; Dutta et al. 2010), and otherforcings. They contribute to large-scale structure and to the low- k part of the density PS.More localized energy input from stars and OB associations should drive 3D motions andstructures that contribute to the high- k part of the density PS (see also Hodge & Deshpande2006). Turbulence driven on large scales would have a shallow PS because of the 2D natureof the resulting flows, while turbulence driven on small scales would have a steep PS becauseof the 3D nature of its associated flows.Bournaud et al. (2010) ran simulations of galaxies like the LMC and found a two-component PS similar to what we observe here. This result occurred whether or not therewas input from supernova energy, suggesting that gravitational energy alone can drive galac-tic turbulence over a wide range of scales. The study also suggested that there may be somecontribution to the large-scale motions from an inverse cascade of energy in 2D. For example,they found a k − PS for the mean-squared vorticity, as in numerical experiments of strictly2D turbulence in protoplanetary disks (Peterson et al. 2007). Thus the LMC PS could bethe result of gravity-driven turbulence and other large-scale motions contributing to gasstructure in an effectively 2D medium wider than the disk thickness, supernovae, cloud-scalegravity and other small-scale motions contributing to turbulence in a 3D medium within thedisk thickness, and an inverse cascade of energy at the thickness scale up to larger scales,contributing to turbulence throughout the disk.
4. Conclusions
The LMC is the only disk galaxy where we can currently study interstellar motions ona range of scales covering a factor of 10 or more. An essential component of this study is 8 –the Spitzer Space Telescope, which provides the highest possible angular resolution for large-scale maps of interstellar emission. Using the SAGE images at 24 µ m, 70 µ m, and 160 µ m,we derived 2D power spectra of dust emission over the whole LMC disk. The results showedone power law covering a factor of 100 in scales at low spatial frequency and another, steeper,power law covering another factor of 100 in scales at high frequency. The low frequency powerlaw had been observed before with HI emission, but the high frequency power law was morelimited before, spanning only a factor of ∼
10 in range (Elmegreen, Kim, & Staveley-Smith2001). The current observations confirm the two-component nature of the interstellar mo-tions, suggesting that gas motions and turbulence are two-dimensional on scales larger thanthe disk thickness, and three-dimensional on scales smaller than the disk thickness. Theline-of-sight thickness is measured at the break point, and is between 100 pc and 200 pc,depending on the wavelength of observation.
Acknowledgments
We are grateful to G.G. Fazio and J. Hora for providing us with timelyaccess to the SAGE Survey images of the LMC after their public release. We are indebtedto S. Stanimirovic, L. Staveley-Smith and S. Kim for sending us their HI images of the SMCand LMC for testing our PS code. We are grateful to J. Scalo for comments on an earlyversion of this manuscript. A note of deep appreciation is expressed by DLB to AVENG andto Mr. F. Titi for their sponsorship of his research. DLB is indebted to Roger Jardine, KimHeller and to the AVENG Board of Trustees. This research is partially supported by theMexican Foundation CONACYT. This study is based on observations made with the SpitzerSpace Telescope, which is operated by the Jet Propulsion Laboratory, California Institute ofTechnology under a contract with NASA.
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12 – 13 –
Fig. 1.— Left: The LMC at 160 µ m (North is up, East is left). The brightest part of thisimage has an intensity of ∼ − . The bright cloud in the lower left is the giantmolecular and atomic cloud south of 30 Doradus. Upper Right: Two-dimensional Fouriertransform of the deprojected image. The rectangle betrays low level striping in the imageon the left. Lower Right: 2D PS with power law fits ∝ k α and ∝ k β at high and low k ,respectively; the slopes are indicated at the bottom left. Dots with error bars ( ± σ ) areaverages over a range in log k . The units of the PS are (MJy sr − ) for Spitzer calibratedimages. The smallest scale in the PS is 7.6 pc, which is 2 pixels (31 . ′′ ) and the theoreticalNyquist limit. The PS break at ∼
200 pc (top scale) is interpreted as the line-of-sight depthof the dust disk in the LMC. 14 –
LMC − 70 microns8.16 degrees Fourier Transform
Fig. 2.— Same as Figure 1, but at 70 µ m. The brightest region has an intensity of 4000MJy sr − . The smallest scale in the PS is 2.3 pc, which is 2 pixels (9 . ′′ ) and the theoreticalNyquist limit. The power spectrum spans 3.5 orders of magnitude. 15 – LMC − 24 microns8.16 degrees Fourier Transform
Fig. 3.— Image and PS of the LMC at 24 µ m. The brightest region has an intensity of 2800MJy sr − . The smallest scale in the PS is 2.4 pc, which is 2 pixels (10 . ′′′′