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

The Large Magellanic Cloud (LMC) is the first spiral galaxy in which an elongated feature of stars was identified. In a remarkable drawing of the LMC as seen with the naked eye made 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 as two "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 ± 0.10, following Freedman et al. 2001). Studies of the LMC concur that the inclination to the line of sight of the (warped) disk lies in the range 30°–50° while the position angle of the line of nodes lies between 120° and 150°; an excellent review is provided by van der Marel et al. (2009). The study by van der Marel & Cioni (2001) using DENIS and 2MASS surveys yields an inclination of 35° and a position angle of 125°, which are the values adopted in this study.

The infrared emission from dust in the diffuse interstellar medium (ISM) of our Galaxy has been extensively studied from IRAS to the Herschel Space Observatory. The quantitative Galactic model of Draine & Li (2007) consists of a mixture of amorphous silicate grains and carbonaceous grains, each with a wide size distribution: sizes range from molecules containing tens of atoms to large grains greater than 1 μm. Dust grain temperatures at the surfaces of giant molecular clouds in our Galaxy are hotter than in their cold interiors (Greenberg & Li 1996), varying from ∼15 K to ∼7 K, respectively. Cold dust grains are a crucial component in the ISM of galaxies (e.g., Block et al. 1994). With regard to the dust temperature–metallicity relations in external systems, Engelbracht et al. (2008) find an anticorrelation between dust temperature and metallicity. Equilibrium dust temperatures are ∼23 K for solar-type metallicities, increasing to 40 K at a low metallicity of 12 + log(O/H) ∼ 8 (see their Figure 5). First insights from a Herschel study of Messier 33, with its strong metallicity gradient, show that dust temperatures are ∼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. Power spectra (PS) of emission maps are an important but poorly understood diagnostic for interstellar structures and the motions that cause them. PS of H i emission from sections of the Milky Way are approximately power laws throughout the entire range of observed spatial frequencies (Crovisier & Dickey 1983; Green 1993; Dickey et al. 2001; Khalil et al. 2006). Similar slopes occur for PS of Milky Way CO emission (Stützki et al. 1998), IRAS and DIRBE 100 μm emission (Gautier et al. 1992; Schlegel et al. 1998), and H i absorption (Deshpande et al. 2000). These power laws resemble theoretical expectations from passive gas motions (Goldman 2000) and compressions (Lazarian & Pogosyan 2000) in a turbulent fluid, so it is natural to interpret them in this way (Elmegreen & Scalo 2004). The situation is much more complex for entire galaxies, however, as galaxies undergo rotation and contain supernovae, spiral waves, and other physical processes that are not present in localized clouds. Still, Stanimirovic et al. (1999) found a power law covering 2.5 orders of magnitude in the PS of H i 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 a slope between −2.8 and −3, depending on the tracer and the depth of the velocity slice (e.g., Dickey et al. 2001).

The PS of H i emission from the LMC is significantly different from these others. Elmegreen et al. (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 two dimensional on scales larger than the disk thickness, and similar processes are three dimensional on scales smaller than the disk thickness. PS for integrated emission from turbulent gas has a slope that differs by 1 in these two cases, with the steeper slope corresponding to three-dimensional motions at large wavenumbers (Lazarian & Pogosyan 2000; Lazarian et al. 2001). In the LMC, the break scale increases with radius, indicative of a reasonable level of flare in the H i disk. Subsequent models of disk 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-Deschênes et al. (2003) found it in H i emission from the Ursa Major Galactic cirrus and modeled it with a fractal Brownian motion cloud. Their model suggested that the break in the PS occurs at a wavenumber k ∼ 1/(2D) for cloud depth D and that the power-law slope changes by 1 at the break. Ingalls et al. (2004) observed a two-component PS using Spitzer IR observations of a molecular cloud in the Gum Nebula. They found that the PS slope changed from −2.6 at wavenumbers less than 2 × 10⁻³ arcsec⁻¹ to −3.5 at wavenumbers greater than 4 × 10⁻³ arcsec⁻¹. The inferred cloud depth was 0.3 pc.

A break in the PS of H i emission has not generally been observed in galaxies other than the LMC. Dutta et al. (2008) studied H i emission from the face-on galaxy NGC 628 and derived a PS slope of −1.7 without a break. Their PS covered only a factor of 10 in scale, which may be too small to resolve out two separate components. Dutta et al. (2009a) saw a break in NGC 1058 with a steepening of the slope from −1 to −2.5 at an inverse wavenumber of 1.5 kpc, which they converted to a disk thickness of 490 pc. Their PS also spanned a factor of 10 in scale, but the galaxy may be too distant (10 Mpc) to resolve the disk thickness with H i (their angular resolution was 136, corresponding to 660 pc). PS of Hα and H i emission from several dwarf galaxies showed power laws and no breaks (Willett et al. 2005; Begum et al. 2006; Dutta et al. 2009b). In the SMC there is no break either. For the SMC, this lack of a break could be because the line-of-sight depth is comparable to the transverse length (Westerlund 1997). This could be the case for the dwarf galaxies also (Roychowdhury et al. 2010).

PS miss the break if the high-frequency part is dominated by the point-spread function of bright unresolved emission sources. These sources introduce a Gaussian-shaped dip at high frequency in the PS (e.g., Dickey et al. 2001). Models of this dip are shown in Elmegreen et al. (2003b) and Block et al. (2009). To see a double power law, the part at high frequency has to extend for a long wavenumber range from the inverse thickness of the disk to the inverse spatial scale of the telescope resolution. If the thickness is smaller than the scale of the resolution, then the high-frequency power law from three-dimensional turbulence cannot be observed. This problem is present for conventional ∼10'' H i resolution in galaxies at distances greater than 2 Mpc. Our Spitzer infrared observations of the LMC have a resolution scale at least 10 times smaller than the disk thickness, so we should be able to see any high-frequency power law clearly.

Optical observations of galaxy PS have sufficient resolution to see the thickness, but unless there are large numbers of stellar groupings with sizes spanning a wide range around the disk thickness, the clarity of the PS break will be poor. A tentative PS break was observed optically in NGC 5055 (Elmegreen et al. 2003a), but not in NGC 628 (Elmegreen et al. 2006). A break in the slope of the autocorrelation function for young stars in several nearby galaxies was inferred to mark a change in geometric properties of the gas by Odekon (2008).

Spitzer Space Telescope observations have such high angular resolution that 3.5–4 orders of magnitude in spatial scale can be covered for the LMC. This is much larger than is possible for any other galaxy in any other gas tracer at the present time. The smallest resolvable scale is only a few parsecs, which is much smaller than the disk thickness, so there is no contamination in the PS from the point-spread function. Thus, the FIR observations of the LMC are ideal for an unprecedented view of correlated structures in a galaxy. In a companion paper (Bournaud et al. 2010), we describe self-gravitating hydrodynamic simulations of galaxies like the LMC, with and without energy input from star formation. We obtain a two-component PS for projected density structure in the simulations too, and we examine 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 in Section 2.2, while Section 3 contains a discussion of the implications of our results. The conclusions are given in Section 4.

2.1. Data

2.2. Power Spectra

Figure 1 shows the MIPS image at 160 μm and the two-dimensional PS at the bottom right. The PS is the sum of the squares of the real and imaginary parts of the two-dimensional Fourier transform (made with fourn from Numerical Recipes), averaged over each two-dimensional wavenumber k = (k²_x + k²_y)^1/2. We divided the PS into 15 wavenumber intervals and plotted the arithmetic mean for each interval as a dot in Figure 1. The PS has two distinct parts with different power-law slopes. The steep slope at high k is −3.08 ± 0.13 and the shallow slope at low k is −2.15 ± 0.48. The break is at approximately 1/k = 200 parsecs. Double power laws also occur in the PS of the other MIPS channels (Figures 2 and 3). The break in the middle of the PS on a log scale allows both power laws to be seen clearly; there are 1.5–2 orders of magnitude of wavenumber on each side for the shorter wavelength bands.

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Emission at 160 μm arises from cold dust in dark clouds. At 70 μm, the emission comes from both warm and cold dust, and at 24 μm the emission is from warm dust and polycyclic aromatic hydrocarbons. The figures indicate that all of this dust has correlated structure spanning the entire resolvable disk of the LMC. The PS break is at 100–200 pc for all three MIPS bands, suggesting that the disk thickness is about this value for each dust component. The least-squares slopes are similar but with a slight progression. At high spatial frequency, they are −3.08 ± 0.13, −2.97 ± 0.52, and −2.55 ± 0.34 for 160 μm, 70 μm, and 24 μm, respectively, and at low spatial frequency, they are −2.15 ± 0.48, −1.83 ± 0.36, and −0.78 ± 0.19 for these three passbands. The steepening of the slope with FIR passband indicates that there is relatively more spectral power on large scales for longer wavelengths. This suggests that cool dust is more diffuse and less structured on small scales than warm dust. This conclusion is true regardless of the total fluxes and column densities of cool and warm dust, as these quantities contribute only to the k = 0 component of the PS.

For comparison, the two-dimensional PS of H i emission from the LMC has a slope of ∼−3.7 at high spatial frequency and a slope of ∼−2.7 at low spatial frequency (Elmegreen et al. 2001). The break is at a scale of 1/k ∼ 100 pc, increasing slightly with galactocentric radius. These slopes are both higher than the corresponding slopes for the dust emission. The H i PS is most like the 160 μm PS, although even steeper, suggesting that H i is even more dominated by large-scale structure than cool dust.

In order to test the robustness of our analysis, we also deprojected the MIPS images about the bar center as opposed to the dynamical center of the LMC; the resulting PS at 24 μm remained unchanged, while the slopes of the two power laws at 70 μm and 160 μm were only marginally affected. We conclude that choice of center for deprojection purposes is not responsible for the presence of two distinct power laws.

The figures indicate that the relative thickness of the LMC dust disk compared to its diameter is only a few percent. This value comes from the ratio of the smallest wavenumber in the PS (the inverse of the largest scale) to the wavenumber at the break in the PS (the inverse of the disk thickness). The smallness of this ratio indicates that the LMC geometry is effectively two dimensional for some processes.

Turbulence in a strictly two-dimensional medium (infinitely thin) differs from turbulence in a three-dimensional medium. For incompressible turbulence, an infinitely thin layer can have an inverse cascade of energy, i.e., energy moves from smaller to larger scales, and a direct cascade of mean-squared vorticity, i.e., from larger to smaller scales (Kraichnan 1967; Leith 1968; Batchelor 1969). For three-dimensional turbulence, the energy generally cascades to smaller scales. Thus two-dimensional turbulence on large scales can be fed by small-scale motions, while three-dimensional turbulence on large scales can only be fed by larger-scale motions (in the absence of a magnetic field or other physical processes that 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, which is the disk scale height, and from supernovae, superbubbles, and so on, could power large scale motions and density correlations. A galactic disk is not strictly two dimensional even if it is thin, but the gas motions can still be highly anisotropic, making the disk effectively two dimensional. For example, density-wave and bar-driven streaming motions are often 5 to 10 times faster than the perpendicular motions that produce the disk thickness. Large-scale motions may be 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 other forcings. 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 three-dimensional motions and structures that contribute to the high-k part of the density PS (see also Hodge & Deshpande 2006). Turbulence driven on large scales would have a shallow PS because of the two-dimensional nature of the resulting flows, while turbulence driven on small scales would have a steep PS because of the three-dimensional 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 there was input from supernova energy, suggesting that gravitational energy alone can drive galactic turbulence over a wide range of scales. The study also suggested that there may be some contribution to the large-scale motions from an inverse cascade of energy in two dimensions. For example, they found a k⁻¹ PS for the mean-squared vorticity, as in numerical experiments of strictly two-dimensional turbulence in protoplanetary disks (Petersen et al. 2006). Thus, the LMC PS could be the result of gravity-driven turbulence and other large-scale motions contributing to gas structure in an effectively two-dimensional medium wider than the disk thickness, supernovae, cloud-scale gravity and other small-scale motions contributing to turbulence in a three-dimensional medium within the disk thickness, and an inverse cascade of energy at the thickness scale up to larger scales, contributing to turbulence throughout the disk.

The LMC is the only disk galaxy where we can currently study interstellar motions on a range of scales covering a factor of 10⁴ or more. An essential component of this study is the Spitzer Space Telescope, which provides the highest possible angular resolution for large-scale maps of interstellar emission. Using the SAGE images at 24 μm, 70 μm, and 160 μm, we derived two-dimensional power spectra of dust emission over the whole LMC disk. The results showed one 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 power law had been observed before with H i emission, but the high-frequency power law was more limited before, spanning only a factor of ∼10 in range (Elmegreen et al. 2001). The current observations confirm the two-component nature of the interstellar motions, suggesting that gas motions and turbulence are two dimensional on scales larger than the disk thickness, and three dimensional on scales smaller than the disk thickness. The line-of-sight thickness is measured at the break point, and is between 100 pc and 200 pc, depending on the wavelength of observation.

We are grateful to G. G. Fazio and J. Hora for providing us with timely access to the SAGE Survey images of the LMC after their public release. We are indebted to S. Stanimirovic, L. Staveley-Smith, and S. Kim for sending us their H i images of the SMC and LMC for testing our PS code. We are grateful to J. Scalo for comments on an early version of this manuscript. A note of deep appreciation is expressed by D.L.B. to AVENG and to Mr. F. Titi for their sponsorship of his research. D.L.B. is indebted to Roger Jardine, Kim Heller, and to the AVENG Board of Trustees. This research is partially supported by the Mexican Foundation CONACYT. This study is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA.