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An analysis of the FIR/RADIO Continuum Correlation in the Small Magellanic Cloud

The local correlation between far-infrared (FIR) emission and radio-continuum (RC) emission for the Small Magellanic Cloud (SMC) is investigated over scales from 3 kpc to 0.01 kpc. Here, we report good FIR/RC correlation down to ~15 pc. The reciprocal slope of the FIR/RC emission correlation (RC/FIR) in the SMC is shown to be greatest in the most active star forming regions with a power law slope of ~1.14 indicating that the RC emission increases faster than the FIR emission. The slope of the other regions and the SMC are much flatter and in the range of 0.63-0.85. The slopes tend to follow the thermal fractions of the regions which range from 0.5 to 0.95. The thermal fraction of the RC emission alone can provide the expected FIR/RC correlation. The results are consistent with a common source for ultraviolet (UV) photons heating dust and Cosmic Ray electrons (CRe-s) diffusing away from the star forming regions. Since the CRe-s appear to escape the SMC so readily, the results here may not provide support for coupling between the local gas density and the magnetic field intensity.


Introduction
One of the most puzzling and robust relationships in extragalactic astronomy is the virtually ubiquitous correlation between far-infrared (FIR) and radiocontinuum (RC) measurements of star-forming galaxies (Yun, Reddy, and Condon 2001) (hereafter referred to as Y01). The essentially linear relationship extends for over five orders of magnitude (Condon 1992) exhibiting the amazingly small scatter of ∼ 0.2 dex (z dex = 10 z ). Virtually all star forming galaxies are included: normal barred and unbarred spiral galaxies, irregular and dwarf galaxies, and Seyferts including radio-quiet quasars. This correlation has been found to apply to galaxies even beyond z=1 (Appleton et al. 2004).
The standard explanation for these observations is found in models that use the conversion of energy resulting from the formation of young massive stars (> 20 M ⊙ ). FIR emission is posited to be the result of dust heated by UV radiation from these massive young stars. These stars also provide energy for the thermal radio emission found in star forming regions. Massive stars quickly evolve to the terminus of their life cycle and explode into supernovae. The supernova remnants (SNRs) expand rapidly into the local interstellar medium (ISM) and the shock wave provides the mechanism for accelerating Cosmic Ray electrons (CRe − s) (Helder et al. 2009;Schure et al. 2009). Those electrons interact with the galactic magnetic field and are thus responsible for the non-thermal (synchrotron) component of the RC emission (Condon 1992). They also interact with particles in the ISM and contribute to the thermal (free-free) component of the RC emission along with UV radiation from hot massive stars.
This simple picture cannot explain the small dispersion in the FIR/RC ratio measured among the diversity of galaxies which span the observed ranges of star formation rates (SFRs), magnetic field strengths, metallicity, dust grain chemistry, and ISM mass values.
The scale at which models break down provides an important constraint on the physical mechanisms detailed in that model. For galaxy-wide scales some models work well on spiral galaxies. For very small scales, for example, within several kpc of the Sun, the FIR/RC ratios of star forming regions are not consistent with expected FIR/RC values. Instead, the radio emission appears to be primarily thermal (Haslam and Osborne 1987). Hughes et al. (2006) examined the Large Magellanic Cloud (LMC) for 60 µm FIR / 21 cm RC correlations on spatial scales from 1.5 to 0.05 kpc and found a tight correlation on spatial scales above ∼ 50 pc.
The SMC can be described as a gas-rich late-type dwarf galaxy (Bolatto et al. 2007). It has a gas-todust ratio 17 times higher than the Milky Way Galaxy (Koornneef and de Boer 1984). It is a member of the local group and is classified as irregular (ImIV-V) (Sandage et al. 1994). It may also be a satellite galaxy of the Milky Way (Westerlund 1997). The SMC is centered at 60.6 ± 3.8 kpc (Hilditch, Howarth, and Harries 2005) from the Galaxy. The spatial scale of the SMC is 0.3pc/ ′′ . The line of sight to the SMC has an extinction of only AV∼0.2 magnitudes and a reddening of E(B-V)∼0.04 (Bolatto et al. 2007;Schlegel, Finkbeiner, and Davis 1998).
The SMC appears to have been tidally disrupted by a recent close encounter with the LMC (Murai and Fujimoto 1980). There is no trace of any spiral structure in the SMC (Sandage and Tammann 1981) but both a ′′ bar-like ′′ structure and a ′′ wing ′′ have been reported (Westerlund 1997). The bar structure is probably the body of the SMC and contains most of its gas and star formation activity. The wing feature continues into the bridge which is an H i region extending from the SMC to the LMC (Bolatto et al. 2007). The SMC may be more complex than the LMC considering that various population centroids vary by ∼ 20 ′ in R.A. and by ∼ 1 • of declination (Westerlund 1997). The SMC has an absolute luminosity of M B = 2.9m (Westerlund 1997).
The SMC has the lowest metallicity of any gasrich galaxy in close proximity (Bolatto et al. 2007). This makes it one of the best galaxies to provide understanding of processes in very high redshift galaxies. Measurements suggest that the dust grains are mostly silicates rather than carbonaceous grains which distinguish the SMC's ISM from that of the Galaxy (Weingartner and Draine 2001).
The stars resolved in the SMC are blue and red supergiants with M B = −7m. These Population I stars are accompanied by a resolved region of Population II stars with M V = −3m. Schwering and Israel (1993) identified 249 infrared sources in the SMC. The morphology of the Population II stars has been measured and appears to suggest a spheroidal distribution without the prominent irregular features seen in the younger stars (Bolatto et al. 2007;Cioni, Habing, and Israel 2000;Maragoudaki et al. 2001;Zaritsky et al. 2000). Earlier studies of the motion of H i clouds were interpreted as representing rotation (Westerlund 1997). Recent observations, however, find no evidence for rotation in the SMC (Piatek, Pryor, and Olszewski 2008), (Costa et al. 2011).
In this paper, Section 2 discusses the origins and characteristics of the data used in terms of the frequencies of each data set, characteristics of the data, and source instruments. Section 3 discusses the analysis methods applied to the data sets. Section 4 is a discussion of results of the analysis and considers some FIR/RC galaxy models. Data processing and analysis for this paper was done with Miriad (Sault, Teuben, and Wright 1995), Karma (Gooch and Barnes 1996), Matlab (Matlab 2010), DS9 (Joye et al. 2003) and Ftools (Blackburn, Payne, and Hayes 1995).
Also, included are data sets for H i (Staveley- Smith et al. 1997;Stanimirovic et al. 1999), Hα (Smith, Leiton, and Pizarro), and CO (Mizuno et al. 2001) detailed below. In order to make comparisons between the data sets, all of the data are regridded to a common pixel size of 5 ′′ using a gnomonic projection centered at 15 • RA and −73 • DEC (equatorial J2000 coordinates). The whole SMC is defined as the data shown in Figure 1. The irregular coverage of the SMC by the Spitzer data suggested that a division of the data sets into several regions would be appropriate. The 160 µm Spitzer data is used as an approximate template for the division into the five regions. Figure 1 shows the regional definitions superimposed on all of the data sets as defined by the 160 µm Spitzer data. It is clear that the bar of the SMC is represented by regions 1-3 and the wing by regions 4 and 5. Table 2 lists the coordinates of the boundaries and centers of the regions as well as their areas. Table 3 contains the flux densities from each region and from the whole SMC.

Infrared Data
The IRAS data used in this paper are the improved reprocessed data from the IRAS satellite survey (Miville-Deschnes and Lagache 2005). The data are in two bands: 60 µm and 100 µm and have been corrected for zodiacal light, calibration, zero levels and striping problems (see Figure 1).
The data used from the Spitzer satellite are from the MIPS instrument. They are from the Spitzer Survey of the SMC (S3MC) covering most of the bar and wing (Bolatto et al. 2007) in two bands, 70 µm and 160 µm. The data are corrected for zodiacal light, Galactic foreground and Cosmic Infrared background using Leroy's published data ) to adjust the final calibration.

Radio-Continuum Data
Data from two radio telescopes are used in this study; the Parkes Radio Telescope and the Australian Telescope Compact Array (ATCA). For high-resolution correlation investigations, the combined ATCA+Parkes data is used and for the low resolution thermal fraction investigations, the Parkes data from Filipovic et al. (1997) is used exclusively due to its superior overall flux-density accuracy.
The high resolution RC data at 21 cm and 13 cm are from observations made with the ATCA telescope in 1992 from October 6 th to the 8 th . The observations were made using five antennae in the 375 m east-west array configuration (Wong et al. 2012;Filipovic et al. 2005) at 1.42 GHz and at 2.37 GHz. The angular resolution is ∼98 ′′ (FWHM) and ∼40 ′′ (FWHM). There were 320 pointings used to create a mosaic of data covering an area of (∼ 20 • ) 2 . The ′′ short spacing ′′ low spatial frequency data fill is accomplished using Parkes survey data (Filipovic et al. 1997). The 6 cm and 3 cm high resolution continuum data are from (Dickel et al. 2010), at 4.8 GHz and 8.6 GHz, using the EW352 (February 2005) and EW367 (February and March 2006) configurations in 3564 individual pointing positions achieving resolutions of 35 ′′ and 22 ′′ respectively. The data were reprocessed to include data from a sixth antenna approximately 6 km from the center of the ATCA as well as including the low spatial frequency data from the Parkes survey (Haynes et al. 1991). The addition of this incomplete, 2 ′′ -3 ′′ interference pattern, provides greater resolution to the image for the evaluation of previously unresolved sources and will be used here to enhance morphological studies using wavelet cross correlations. The additional data does not cover the full extent of the SMC but covers all of the area studied in this paper. See Haynes et al. (1991); Xu et al. (1992); Klein et al. (1993);Filipovic et al. (1995Filipovic et al. ( , 1997Filipovic et al. ( , 1998, Filipovic et al. (2002); Payne et al. (2004);Filipovic et al. (2005); Reid et al. (2006); Payne et al. (2007), and Crawford et al. (2011);Wong et al. (2011Wong et al. ( ,a, 2012).

H i Data
The H i 21 cm data are from observations made with the ATCA telescope in 1992 from October 6 th to the 14 th and August 2 nd and 3 rd . The observations were made using five antennas in the 375 m east-west array configuration , 1997 at 1.42 GHz. The angular resolution is ∼90 ′′ (FWHM). There were 320 pointings used to create a mosaic of data covering an area of (∼ 20 • ) 2 . The ′′ short spacing ′′ low spatial frequency data fill is accomplished using Parkes survey data (Stanimirovic et al. 1999) from 1996 March 14 th to 17 th .
The column densities are corrected for self-absorption using the correction factors from Stanimirovic et al. (1999). The H i mass is calculated for each region and the whole SMC and is shown in Table 3. Results reported here of 2.8 · 10 8 HI M ⊙ from a 3 • × 3 • reduced data set compare favorably to a published value of 3.8 · 10 8 HI M ⊙ from Stanimirovic et al. (1999) for a 5 • × 5 • image.

Hα Data
The Hα data is from the Magellanic Cloud Emission Line Survey (MCELS) program using the UM/CTIO Curtis Schmidt telescope (Smith, Leiton, and Pizarro). The central 3.5 • × 4.5 • of the SMC are imaged in overlapping 1 • × 1 • fields with 2048 2 pixels per field in such a way as to provide at least two samples of each pixel. The Hα filter is centered on 500.7 nm with a width of 3 nm. Two continuum channels are also imaged, 685 nm, δ = 9.5 nm, and 513 nm, δ = 3 nm for stellar contribution subtraction.
2.5 12 CO j = (1 − 0) Data (115.271 GHz) CO data is from the NANTEN sub-millimeter observatory on Pampa la Bola in the Atacama Desert, Chile a Also see Haynes et al. (1991); Xu et al. (1992); Klein et al. (1993); Filipovic et al. (1995Filipovic et al. ( , 1997Filipovic et al. ( , 1998 Table 1. Data are displayed with North to the top and East to the left and J2000 coordinates in degrees. These plots are smoothed with a 25 ′′ Gaussian kernel and scaled by the square root of the intensity. (Mizuno et al. 2001). The NANTEN telescope is used at 2.6 mm wavelength and the observations have a resolution of 2.6 ′ . Regions 1-3 are well covered by this data. Region 4 has coverage primarily around N84 and there is only incidental coverage of Region 5. The conversion factors for the CO data to H 2 mass used in this paper are from Leroy et al. (2011). Leroy calculated 3 different conversion factors for the SMC for different parts of the SMC which varied by 40%. (Leroy's West → Region 1; North → Region 3; East → Region 4; Average of Leroy's West and North → Region 2 and Region 5). The results of this conversion is expected to reveal only the most dense H 2 clouds. The conversion factors applied to the 5 regions and the overall average value applied to the entire SMC are shown in Table 4.

Background sources
Since the SMC is essentially transparent to 1.42 GHz RC radiation and it presents a cross section of several deg 2 , many background sources are expected to be visible in the SMC RC data. These background sources were catalogued by Payne et al. (2004) Crawford et al. (2011);Wong et al. (2011Wong et al. ( ,a, 2012. This data consist of 717 RC sources of which 616 were identified as background sources. They are plotted in Figure 1 along with the numbers assigned to each of the five regions on an image of the 21 cm RC data.

Analysis
The analysis presented here uses cross correlations between several different wavelengths using traditional pixel-by-pixel correlations as well as wavelet correlations. The pixel-by-pixel correlation studies provide intensity correlations as a function of position. The wavelet cross correlation technique probes morphological correlations which are sensitive to changes over a range of spatial scales. FIR/RC correlation is studied over the entire SMC as well as the regions defined previously.
All of the correlation analysis with RC data is done excluding data at the coordinates of the background sources. The mask consisted of Gaussian function with a FWHM of 2 ′ applied to the coordinates of each background source. The data sets also had a median filter of 15 ′′ × 15 ′′ applied to remove zero pixels and provide a slight smoothing.
3.1 SMC and Regional FIR/RC ratios An examination of the FIR/RC correlation in the SMC compared to measurements from more distant galaxies is shown in Figure 2. This correlation is measured down to scale factors of less than 1 kpc by considering each of the five regions in the SMC (see Figure 1). Three of these regions cover the bar structure and two cover the portion of the wing structure that is closest to the bar. The structure of the correlations within the regions is shown in images which display the organization of the FIR and RC emissions giving rise to the measured values.
Using the 60 µm IRAS IR flux and the 21 cm RC flux, a FIR/RC calculation is made for each of the five SMC regions and the whole SMC (defined as a 3 • × 3 • region centered at RA 15 • Dec −73 • ). These results are directly compared to the survey done by Y01. Y01's catalog identified radio counterparts to the IRAS Redshift Survey galaxies with 60 µm IR fluxes of 2 Jy. The catalog includes 1750 galaxies and contains radio positions and redshifts. Also tabulated are 1.4 GHz radio fluxes from the selected galaxies and their IRAS fluxes.
An ordinary linear least squares fit is performed on this data to determine the FIR/RC ratio as defined by the data from Y01. The fit equation calculated in this paper is Equation 1:

Hz
(1) For each 5 ′′ × 5 ′′ pixel of the SMC data sets, the value of the FIR/RC ratio is calculated. The sums of this ratio for each region is plotted on the graph with Y01's data set ( Figure 2). The line described by the fit equation is extended through the SMC data sets presented in this paper to show how the fitted ratio of the Y01 data corresponds with this SMC data. The correlation is evident in all five regions and for the whole SMC. The distance from Y01's regression fit line to the regional and complete SMC data are very small: from ∆=0.017 dex to ∆=0.115 dex. These differences changed slightly depending on whether the background sources are included or not. The differences for each region and the whole SMC are shown in Table 5 including the change due to background sources.
For comparison, Figure 2 also plots the LMC correlation calculated from data published on the LMC by Hughes et al. (2006). The distance to Y01's regression line for the LMC is calculated here to be ∆=0.308 dex.

Pixel by Pixel analysis
Pearson's linear correlation coefficient calculation is performed between several pairs of data sets, Equation 2. For each pair of data sets, the data set with the smallest FWHM beamwidth is convolved with a Gaussian kernel to smooth it to the same FWHM as the lower resolution data. The sampling is then done at a spacing corresponding to the beamwidth of the lower resolution data to guarantee statistical independence of the samples. In this equation f N refers to the N th image, f N i refers to the i th pixel of the N th image and r p is the Pearson's linear correlation coefficient.
The error in this correlation is calculated as follows: The formal errors for these data sets are ≃10% since n, the total number of points in a data set, is ∼10 3 when the resolution of the data sets are considered. Of course, the real error is possibly larger due to calibration errors and other systematic errors in the data. Table 5 Comparison of the difference between the data fit to data from Yun (Yun, Reddy, and Condon 2001) and the current measurements. The contribution to the fit from the background sources is shown to contribute less than 10 percent to the fit discrepancy. Perfectly correlated images would have a coefficient of 1 and perfectly anti-correlated images would have a coefficient of -1. Table 6 shows various pairs of data sets for the 5 regions and the correlations therein. Values above 0.7 or 0.8 in Table 6 indicates moderate to strong correlation. Those values are printed in bold face type.
Region 1 and Region 5 have the highest correlations with the FIR-H i data sets and includes N 19 and NGC 267. SMC SNR position data from Filipovic et al. (2005) is used to determine the SNR population of each region. Region 1 has the largest concentration of H i for the SMC (see Table 3) and a complex of 5 SNRs around SNR B0045-734. There are 4 other SNRs in Region 1. Region 5 contains no identified SNRs. Region 2 shows very little correlation in any of the data set pairs. There are two identified SNRs in Region 2. Region 3 is dominated by the H ii region N 66 and NGC 346, and shows good correlations in the FIR-RC data sets as well as the Hα correlations with the RC and FIR data. Region 3 contains 12 SNRs. Region 4 shows moderate to strong correlations of the IR-CO data sets for the coolest dust. Region 4 is not well covered by the 70 µm or the CO data. There are no known SNRs in Region 4. It does, however, contain NGC 456. Figure 3 shows the scatter plots of the 21 cm continuum and 60 µm IR data for the pixels of all five regions and the whole SMC. For each pair of data sets, the data set with the smallest FWHM beamwidth is convolved with a Gaussian kernel to smooth it to the same FWHM as the lower resolution data. The sampling is then done at a spacing corresponding to the beamwidth of the lower resolution data to guarantee statistical independence. It also shows two different Least Squares Fits to each data set. The blue line is the Ordinary Least Squares Fit, OLS and the green line is the Weighted Least Squares Fit (WLS) calculated using the Matlab function lscov (Matlab 2010). The OLS assumes the error of each data point is the same and consequently weights every data point equally. The WLS uses a weighting for each data point which here is calculated as the square root of the sum of the squares of the values of each axis linearly weighted from 0.01 to 1 scaled to the value of the data point from the minimum value to the maximum value of each data set.
The difference in the slope and offset of these fit lines and the averages are shown in Table 7. The last row contains the differences in percent of the OLS slope and the WLS slope of the fitted lines. It is expected that the OLS fit would be a better representation of the entire data set whereas the WLS fit is weighted to lower the contribution of data with larger calculated errors and would describe the strongest data without the effect of outlier points.   Yun, Reddy, and Condon (2001). Regions 1-5 and whole SMC are plotted and labeled. (1-plus, 2-circle, 3-asterisk, 4-x, 5-square, and SMC-diamond). Additionally, the correlation calculated from data published on the LMC by Hughes et al. (2006) is plotted using the star marker.  For each pair of data sets, the data set with the smallest FWHM beamwidth is convolved with a Gaussian kernel to smooth it to the same FWHM as the lower resolution data. The sampling is then done at a spacing corresponding to the beamwidth of the lower resolution data. The two lines in each plot are the ordinary least squares and weighted least squares fits to the data. The OLS is in green, the WLS in blue. The contours in the data plot show the density of data points and the Gaussian distribution of the background noise. Those contours are at 5, 30, 60, and 90% of the peak data density. The position of the square is the average of the data set.
Strong to moderate Pearson correlations in FIR-RC, FIR-H i, and IR-Hα in Regions 1 and 3 seen in Table 6 appear to show the effect of two populations of data in those regions. The range of OLS slopes in the regions is 0.594 to 0.978 while the WLS slopes range from 0.638 to 1.144 over the regions. In every region the OLS slope is flatter that the WLS slope. The set of contour lines to the left of the figures is from the background noise using measurements from the north east corner of the whole SMC data set which are not considered to be part of the SMC. The measured average value and the RMS noise are used to calculate a background Gaussian noise distribution and the results of this distribution are used to create the noise contours for the background. Also shown are four contours which are created by binning the luminosity data into a 100×100 bin array. The contours shown are at 5, 30, 60 and 90% of the maximum bin count. Visually examining these contours indicates that 90% of the data is found with values quite symmetric to the fitted lines, both OLS and WLS. This suggests that the WLS is a better representative of the data in the regions. The Gaussian noise distribution calculated for the measured background noise uses the same scaling. It is clear that there is not a significant contribution by noise to the scatter plot for any of the five regions. However, the whole SMC plot includes portions of sky which are not part of the SMC and contain only noise so that there is considerable overlap seen in the lower luminosity values of the scatter plot with the noise contours. The plotted square in each data set is the location of the average pixel value.

SMC FIR/RC ratio map
In order to relate particular physical structures in the SMC with variations in the FIR/RC ratio, Figure 4 is calculated showing the log ratio of FIR and 21 cm RC data. The q value map is calculated using the formula from Helou et al. (1988) with the IRAS 60 µm and 100 µm data sets. The FIR value is calculated in equation 4. F IR W m −2 = 1.26 · 10 −14 2.58S 60µm + S 100µm Jy The q values of the pixels are calculated with Equation 5.
The average value of the ratio is q=2.65. This suggests that the SMC has a slight FIR excess when compared to Y01's average figure of q=2.34. Approximately Table 7 For each pair of data sets, the data set with the smallest FWHM beamwidth is convolved with a Gaussian kernel to smooth it to the same FWHM as the lower resolution data. The sampling is then done at a spacing corresponding to the beamwidth of the lower resolution data. Data set averages for RC 21 cm, Log(Luminosity 1.4 GHz)(W/Hz), and FIR, Log(luminosity 60 µm) sols, and fit parameters to y=ax + b using Ordinary Least Squares fit, OLS, and Weighted Least Squares fit, WLS. These numerical values are shown graphically in Figure 3. The differences in slope values are also shown with the WLS always larger than the OLS value. For comparison, Y01's data as fitted here using OLS gives a = 0.995 and b = 12.38. 98% of Y01's galaxies are within a factor of 5 of 2.65, ∼1.6 to ∼3.0. Figure 4 shows the distribution of q values across the SMC from q <2.15 to q >2.95. Region 1, with an average q of 2.80, has an a large portion of its area with q ≥2.65. This suggests that there is abundant dust in the area. Relatively high Pearson correlations of FIR with H i are also seen in Table 6 for Region 1. There are 9 SNRs in this region. The strongest RC feature in this region is N 19 containing SNR B0045-734 (Payne et al. 2004) Wong et al. 2011Wong et al. ,a, 2012. The other regions have extended areas where the RC excess is present.

Regions
Region 2, with an average q of 2.58, has a large portion of its area with q ≤2.65. With two SNRs in this region, it could be inferred that there is not much starforming activity in this region so that the dust is cooler.
Region 3, with an average q of 2.70, has 12 SNRs with similar areas with q <2.65 and q >2.65. The high star formation rate leads to the very high Pearson correlation seen in all of the IR data maps with the 21 cm data. Hα correlates also very well with the RC data and the FIR data.
Region 4, with an average q of 2.75, is dominated by background radio source SMC B0109-7330. It has a portion of its area with significant dust and gas in the eastern part of the region. It is the only region that has strong correlation of CO with the IR emission in the Pearson correlation results.
Region 5, with q of 2.73, has very small total energy emission, 0.2 -0.3 of the average of the other 5 regions. Scaled for area, it is ∼0.4 -∼0.6 of the average.

Wavelet analysis
Wavelets can be used as a tool for scaling analysis (Frick et al. 2001). It is a technique that convolves the data with a family of self-similar analyzing functions that depend on scale and location. The family of analyzing functions is defined by dilations and translations of the analyzing function, also called the mother function. In two dimensions, the continuous wavelet transformation can be written as follows: In this expression, x = (x, y) and f ( x) is the two dimensional data set for which the Fourier transform exists, Ψ * ( x) is the complex conjugate of the analyzing function, a is the scale factor, and κ is the normalizing parameter.
The analyzing function used here is the Pet Hat (PH) function (Frick et al. 2001). This function is defined in terms of its Fourier transform as follows: Ψ k = cos 2 π 2 log 2 k 2π : π < k < 4π = 0 : otherwise (7) This function specifies an annulus in Fourier space with a median radius of 2π and is non-zero only for the specified annulus.

Wavelet cross correlation
Given a two dimensional data set, the energy of the transformation as a function of scale factor a over the entire plane is: The wavelet cross correlation coefficient as a function of a is calculated as follows: where the subscripts refer to the source data sets used in the cross correlation.
The error in the cross correlation is calculated as follows: where L is the linear size of the data set. For the data used here ∆r w (a) is 15%.
For each pair of data sets used in wavelet cross correlation, the data set with the smallest FWHM beamwidth is convolved with a Gaussian kernel to smooth it to the same FWHM as the lower resolution data. The sampling is then done at a spacing corresponding to the beamwidth of the lower resolution data. Figure 5 shows wavelet correlations of several data sets with the 21 cm RC in all 5 regions. The smallest scale that can be used to analyze the data is the resolution of the data. The largest scale is dependent on the overall size of the data set. The smallest linear dimension of the data set should be no less than four times the largest scale factor used in the analysis. The resolution of the 21 cm data is 90 ′′ (27 pc) and the finest resolution calculated is 80 ′′ (24 pc). The correlation is performed with IRAS 60 µm and 100 µm data with 120 ′′ (36 pc) resolution, Spitzer 70 µm and 160 µm data with 18 ′′ (5.4 pc) and 40 ′′ (12 pc) resolution respectively, H i data with 98 ′′ (30 pc) resolution, Hα with 3 ′′ (1 pc) resolution, and CO data with 156 ′′ (46 pc) resolution. For the data sets used here, the maximum valid scale value for the wavelet cross correlation is 794 ′′ (238 pc) for Regions 1-4 and 501 ′′ (150 pc) for Region 5 except for the CO data.
The CO map is much more irregular than the other data sets. Region 1 is close to being fully covered with the cross correlation results beginning at 125 ′′ . Region 2 is not well covered and has a large hole in the coverage in the center of that region. The only scale factors that satisfy validity requirements are at 125 ′′ and 200 ′′ , both of which are shown. Region 3 is also close to being fully covered with the cross correlation results beginning at 125 ′′ . Region 4 is split into two smaller areas. Interpreting the wavelet cross correlation in the region is problematic. Region 5 is just a sliver of data that is too small and irregular to use in a wavelet correlation. Wavelet correlations of Regions 4 and 5 are not calculated. Table 8 shows the smallest scale factors for which excellent wavelet cross correlation is measured. Frick et al. (2001) regarded 0.75 or above as representing an excellent correlation between images and will be referred to here as the correlation threshold. The scale factors are extracted from the data shown in Figure 5. The data clearly shows strong morphological correlations of dust emission in the FIR with the 21 cm continuous emission from the defined regions of the SMC down to beamwidth limited minimum scale factor, ∼ 35 pc. In Regions 2 and 3, the limit of the correlations is primarily due to the data resolution. Table 9 shows the wavelet correlations extracted from Figure 6 which correlates the Spitzer IR data with the higher resolution 3 cm and 6 cm data, (Dickel et al. 2010). The wavelet cross correlation with the Spitzer data shows that the FIR/RC wavelet correlation scale factors extend down to ∼15 pc. Within each region, all of the FIR bands show similar correlation patterns with RC data with respect to scale factor. The coverage of the regions is quite good for the Spitzer 160 µm data but less so for the Spitzer 70 µm data particularly in Regions 1, 3, and 4. There are also some unexpected differences between the 3 cm and 6 cm correlations with the FIR data.

Wavelet cross correlation for other bands
H I cross correlation is seen in Figure 5 with 21 cm RC in Region 1. The correlation initially dips before beginning a slow rise to the correlation threshold at ∼130 pc. Region 2 never achieves correlation threshold and Region 3 doesn't until ∼170 pc. Regions 4 and 5 have approximately the same scale factor of ∼85 pc. Hα correlations from Figure 5 in Table 8 with 21 cm RC are essentially at the resolution of the H i data at ∼25 pc for Regions 1 through 3. Regions 4 and 5 have correlation threshold at larger scale factors of ∼85 pc. High-resolution Hα correlations from Figure 6 in Table  9 with 3 cm and 6 cm shows that the Hα wavelet cross correlation extends down to much smaller scale factors from 9 pc to 33 pc for Regions 1 through 4.
The CO correlations are very sparse since the map of the data is rather incomplete. Region 1 had correlation threshold at 97 pc and above. No significant correlation is detected in Region 2. Region 3 had correlation threshold at 229 pc. No other correlations are possible.

Discussion
The FIR/RC correlation coefficient, q, determined in this study is very close to the expected value for nor-  mal galaxies even though the SMC is structurally different from the spiral galaxies which constitute most of the population on which the correlation is based. Spirals generally have a thick underlying non-thermal disk. The expected thermal fraction for the disk of a spiral galaxy is ∼0.1 (Condon 1992). The three regions that cover the SMC bar measured here all have high thermal ratios of ∼0.54 to ∼0.95 (see Figure 7). The existence of the bar, however, is questioned by Zaritsky et al. (2000). Their argument centers around the hydrodynamic interaction between gaseous components providing the physical mechanism for the SMC morphology. They further suggest that the SMC is actually spheroidal with highly irregular recent star formation. The overall thermal fraction of the SMC is measured here using 1.4 GHz and 4.8 GHz data to have a value of ∼0.43 which is higher than previously published values of 0.15-0.40 for the SMC by Loiseau et al. (1987) using 1.4 GHz and 2.3 GHz data but lower than the value of 0.71 measured by Israel et al. (2010) using 5 GHz and 10 GHz data. However, each of the SMC regions measured here except Region 2 has a thermal fraction of >0.71. The triggers for star formation events in massive spirals are believed to be density waves (Beck 2000). In the SMC, there is not enough mass to support such a mechanism so that the trigger in the SMC is likely to be stochastic which leads to a discontinuous formation history with bursts of star formation lasting short periods of time (Klein and Graeve 1986). With only two SNRs detected in Region 2, there is clearly no evidence of a galactic nucleus with its traditional high star formation rate. Figure 4 shows the distribution of q values in the SMC and there is no indication of the q value decreasing from a ′′ nucleus ′′ to the edge of the galaxy as Helou and Bicay (1993) expect in their ′′ leaky box ′′ FIR/RC model. Figure 4 does show that most of the SMC bar is consistent with q values within Y01's expected limits. The depth of the SMC extends 5-10 kpc along the line of sight, (Martin, Maurice, and Lequeux 1989). The measured thermal fraction leads to the conclusion that there are not enough CRe − s to generate higher non-thermal emissions and/or that the magnetic field strength is too low in most regions. The published values of the magnetic field are 5 − 10 µG along the bar (Ye and Turtle 1991). Sreekumar and Fichtel (1991) concluded that the cosmic ray energy density level in the SMC is three to five times too low to be in equilibrium with the cosmic ray sources. This low cosmic ray density is described by Klein, Grave, and Beck (1984) to be a result of the internal pressure from the CRe − s. This causes rapid diffusion of the CRe − s out of the star-forming regions. This is consistent with the observation that the bar has primarily thermal RC emission in Region 1 and Region 3 where there is high density H i and star formation activity. Region 2 may be populated primarily with CRe − s which have diffused out of the two star-forming regions in Region 1 and Region 3.

FIR/RC Correlation Models
The FIR/RC correlation models examined here start with the assumption that the source of relativistic electrons and ionizing photons is the supernova rate. UV photons from massive stars are absorbed by dust which re-emit the energy in the FIR. The relativistic electrons from the SNRs interact with magnetic fields or interact with ionized gas to produce RC radiation. Voelk (1989) proposed a calorimeter theory which assumes that all of the ionizing photons are absorbed by dust and all of the energy of the CRe − s are dissipated by RC emission within the galaxy. This does not appear to be consistent with the SMC since Sreekumar and Fichtel (1991) determined that most of the CRe − s escape the SMC.
Applying the ′′ leaky box ′′ model of Helou and Bicay (1993) most of the ionizing photons and CRe − s would escape the SMC. The observational prediction of the model is that q decreases radially from the center of the galaxy to the edges of the galaxy. Regions 1 and 3 are the main star-forming regions in the SMC at the present time. Star formation in Region 2 must be suppressed since the H 2 content is the lowest of any region except for Region 5, see Table 3. Due to the irregular nature of the SMC, different position angles from these starforming areas point towards very different environments as seen in Figure 4. This does not appear to match the gradual decrease in q modeled by Helou and Bicay particularly since neither a nucleus nor a disk can be identified in the SMC. Hoernes, Berkhuijsen, and Xu (1998)  Of these criteria, the SMC is also close enough for well resolved FIR and RC measurements. But the SMC is not dominated by the non-thermal RC component with the overall SMC non-thermal emission at 57% and a minimum non-thermal emission of only 5% in Region 3, see Figure 7.
The supernova rate (Rs) for the SMC is about 1/350 years, Rs SMC = 2.8 · 10 −3 SN · yr −1 (Filipovic et al. 1998) whereas the Galaxy has a supernova rate of about 1/50 years, Rs Galaxy = 20 · 10 −3 SN · yr −1 (Diehl et al. 2006). Dividing these rates by the galaxies' respective volumes shows that the supernova rate per unit volume is higher in the SMC than in the Galaxy: Whereas M 31 has a star formation rate of an order of magnitude below the Galaxy with those two galaxies approximately the same volume, the SMC has a volumetric supernova rate an order of magnitude above the Galaxy or two orders of magnitude above M 31.
The SMC does not appear to be consistent with the criteria needed to apply H01's decomposition model.

Decomposition of the SMC RC into thermal and non-thermal emission
Data from the Parkes single dish telescope are used for the RC thermal fraction calculations (Filipovic et al. 1997) due to its superior overall flux-density accuracy. The Parkes data were converted to a data map with 5 ′′ resolution using the Miriad regrid program to match the 6 cm data set. The thermal fraction of the emissions at 21 cm are calculated using equation 13 from Niklas, Klein, and Wielebinski (1997). S 6cm and S 21cm are the measured flux densities of the regions.
In equation 13, f th is the thermal fraction, f nth is the non-thermal fraction, f nth = (1 − f th ). The thermal spectral index used is α th = 0.1. The non-thermal spectral index is assigned a value of α nth = 1.09 after Bot et al. (2010). The thermal fractions calculated are 0.81, 0.54, 0.95, 0.84, and 0.76 for Regions 1 through 5 and 0.43 for the SMC, see Figure 7. The thermal fraction calculation assumes that the synchrotron emission spectrum can be described by a single power law and that the ISM is optically thin. This assumption can introduce a bias into the calculated thermal fraction if the spectral index is not constant. The index is expected to steepen in locations that are increasingly remote from the area where the CRe − s were initially accelerated since the energy loss of CRe − s is ∼ E 2 . Conversely, in star-forming regions the thermal fraction will likely be overestimated as the spectral index is flatter than in other regions. However, the entire SMC as well as each region shows approximately 4 to 9 times the expected thermal fraction of 0.1 (Condon 1992) which suggests that CRe − s are not necessarily the source of the RC energy that makes the FIR/RC correlation so consistent.

Regional Decomposition of the SMC
The top panel of Figure 7 shows the relationship of the slope of the Weighted Least Squares Fit (WLS) of 21 cm RC verses 60 µm emissions from Table 7 plotted against the estimated thermal fractions. The regions show a strong tendency towards increasing WLS slope with increasing thermal fraction.
Region 3 has the highest WLS slope, the highest thermal fraction, and the highest 21 cm flux density of the five regions. It also has the best Pearson correlation, Table 6, of any region for RC 21 cm with all of the IR data sets, excellent Pearson correlation of 13 cm RC with the higher temperature IR emissions of 60 µm and 70 µm and excellent Pearson correlation of Hα with RC 21 cm, 13 cm, IR 60 µm, 70 µm and 160 µm covering cold dust to warm dust emitters. The locations of the high H i concentration correlates with the IR emissions by way of the high Pearson correlation indicating that the H i and dust are colocated. It has excellent wavelet cross correlation of the 21 cm with 60 µm down to the resolution limit of 35 pc. Figure 6 shows the wavelet cross correlation of the higher resolution FIR and RC data (70 µm and 160 µm Spitzer data with 6 cm and 3 cm RC data), indicating that the correlation extends down to 15pc. This region includes N 66 and NGC 346 which is an H ii region with embedded stellar association (Bica and Dutra 2000) and the strongest star-forming region in the SMC (Cignoni et al. 2011)  RC with 60 µm extends down to 50 pc. Region 4 has excellent Pearson correlation of CO with IR emissions at 60 µm, 100 µm and 160 µm with the correlation increasing as the temperature of the dust emission decreases. Interestingly, the 21 cm flux for Region 1 is only slightly below Region 3 but Region 4 has the second lowest flux at about 1/2 of the flux of Region 1. The CO data in Region 4 contains NGC 456 but much of the remainder of the region is not covered in the CO data. Region 5 has a WLS slope of 0.72 and a thermal fraction of 0.76. The region contains the open cluster NGC 419 which has been examined by Rubele, Kerber, and Girardi (2010) who suggests that the star formation history of the cluster displays continuous star formation over 1.2 to 1.9 Gyr. This region exhibits no strong Pearson correlations in any data set pairs investigated here (Table 6). There is excellent wavelet correlation down to 130 pc which is the largest measured by this study of the SMC. Oddly, the Spitzer 70 µm wavelet correlation with 6 cm RC gives a much smaller correlation threshold, 50 pc, but the Spitzer 70 µm is consistent with the 60 µm correlation threshold when correlated to 6 cm and 3 cm RC data.
Region 2 has the smallest WLS slope of the regions with a value of 0.62. It also displays the smallest thermal fraction of 0.54. It contains NGC 330 which is an open cluster and two SNRs. It contains no strong Pearson correlations. The wavelet correlation threshold is at the resolution limit of 35 pc. The higher resolution wavelet correlation graphs, Figure 6, show the the correlation actually goes down to 19 pc.
The thermal fraction of the SMC is estimated here to be 0.43. This is higher than the typical value of ∼ 0.1 given by Condon (1992) for spiral galaxies. It is very close to the value of 0.45 for the LMC measured by Hughes et al. (2006). The average value of the thermal fraction of the 5 regions is 0.78 over an area of 2.9 deg 2 whereas the SMC measurement is over 9 deg 2 . This suggests that the regions, primarily in the bar, emit mostly thermal RC energy but that the larger area contains significant non-thermal emission from a large population of CRe − s that have diffused out of the bar. The star formation activity is in the bar where the H i has high density. The non-thermal emission is taking place where the H i has low density outside of the bar. The small size of the SMC suggests that the cosmic ray leakage into the nearby areas would be substantial. The total 21 cm flux from the regions is 26.5 Jy and the total flux in the 3 • × 3 • data set used for these thermal fraction calculations is 36 Jy.
It is interesting to note that the q values that are calculated including only the thermal portion of the RC emission satisfies Y01's range of q values for normal galaxies for the five regions: 1) 2.90, 2) 2.85, 3) 2.72, 4) 2.83, 5) 2.85. The ′′ greater ′′ SMC has a q value of 3.02 which is just at the 98% q value consistent with Y01's correlation study.

Conclusions
We show in this study that the FIR/RC correlations for the SMC and for each of the five regions defined here, are consistent with Y01's regression from 1750 galaxies. The study presented here considers the ′′ greater ′′ SMC and the sub-regions of the bar and wing as well.
Pearson correlation coefficients of RC 21 cm emissions with 60 µm and 100 µm emissions have positive correlation only in Region 3 which is the region with the highest star formation rate. Not surprisingly, Region 3 also has positive Pearson correlation of Hα with RC and FIR emissions. Interestingly, Region 1 has twice as much H i and H 2 as Region 3 and has the second highest star formation rate in the SMC. Its strongest Pearson correlations are in terms of H i with FIR emissions. Since the star formation rate in the SMC is predicted to be stochastic, Region 1 may have a suppressed supernova rate at this time.
Pixel by pixel scatter plots indicate two populations of correlations in terms of 21 cm RC and 60 µm emissions. The highest slope is from Region 3 which also has the largest difference between the WLS and OLS slope values, 1.14-0.98, 14.5%. Region 1 has the second largest slope and the second largest difference of slopes of 0.85-0.73, 13.7%. This correlates to the regions with the strongest star formation rate having the strongest indication of two distinct emission populations. The other regions and the SMC have smaller slopes and less significant differences of 7.8% to 3.6%.
The wavelet cross correlation scale factors for the FIR with the RC for the regions are all within 20 to 60 pc except for Region 5 which is ≈130 pc and may have few features at small scale factors. There is no clear trend displayed in this correlation. It is interesting to note that the region with the highest thermal fraction, Region 3, and the region with the lowest thermal fraction, Region 2, have the the smallest wavelet correlation scale factors. Hα has wavelet cross correlation scale factors with the 6 cm RC emission of between 9 and 14 pc for all of the regions except for Region 5 which is 87 pc.
The regions have q values of: 1) 2.80, 2) 2.58, 3) 2.70, 4) 2.75, 5) 2.73. The measured value of q for the SMC is 2.65 which includes portions of the greater SMC that are not included in regions comprising the bar and wing.
The SMC and all of the regions have q values above Y01's correlation value of 2.34 but well within the values of 1.64 to 3.04 between which 98% of the q values for the galaxies measured by Y01 were found. This suggests that the RC emission in the SMC is slightly deficient compared to the FIR emission. It could be due to CRe − s escaping the bar or wing of the SMC without losing much of their energy. Since the non-thermal fraction of the RC emission is so small, the non-thermal RC emission is significantly under-luminous considering that the non-thermal fraction is expected to be dominant. This may be due to the small physical size of the SMC and a low magnetic field density.
The SMC displays a thermal fraction that is much higher than is common for disk galaxies. Disk galaxies typically have a thermal fraction of 0.1 (Condon 1992) while the SMC is measured here to have a value of 0.43. The regions have even higher measured values of the thermal fraction, 1) 0.81, 2) 0.54, 3) 0.95, 4) 0.84, and 5) 0.76. This suggests that galaxies can have most of their RC radiation derived from thermal sources powered by the radiation field from high mass stars in H ii regions rather than by SNRs. There appears to be a weak correlation between the thermal fraction and the q values. The thermal fraction values above 0.5 correspond to q values above 2.7 and thermal fraction values below about 0.5 have q values of below 2.7. The SMC and all five of the defined regions have FIR/RC correlations consistent with Y01 q values when considering only thermal RC radiation. Thermal radiation can be the major source of RC emission from a galaxy suggesting that measurements of the RC emission of galaxies could lead to over estimating the super nova rate by assuming the 0.1 ratio of thermal to non thermal radiation.