Statistical Analysis of Supernova Remnants in the Large Magellanic Cloud

MCSNR J0447-6918 (Figure 1; top left)—A large optical shell (245'' × 245'') was present with an enhanced [S ii]/Hα ratio of >0.4. Also, some weak 20 cm emission was detected in the NE part but no reliable flux density estimate was possible. No sensitive X-ray coverage is available in this field.

MCSNR J0449-6903 (Figure 1; top right)—Turtle & Mills (1984) originally proposed this source to be an SNR exemplifying the typical evolved shell-type SNR morphology. Although there was no obvious optical identification in MCELS, we estimated a radio spectral index of $\alpha =-0.50\pm 0.01$ using measured flux densities of 108 mJy, 83 mJy, and 45 mJy at 36 cm (843 MHz), 20 cm (1377 MHz), and 6 cm (4800 MHz), respectively. Unfortunately, no sensitive X-ray coverage is available at this point.

MCSNR J0456-6950 (Figure 1; middle left)—This source is a potential radio SNR based on a shell-like radio structure. We have no definite optical confirmation, and X-ray surveys have not covered this region.

MCSNR J0457-6923 (Figure 1; middle right)—This source was classified as a potential optical SNR based on an [S ii]/Hα ratio of >0.4 as well as an evident radio emission. No sensitive X-ray coverage is available.

MCSNR J0457-6739 (Figure 1; bottom left)—This object exhibits a shell-like optical nebula with a somewhat enhanced [S ii]/Hα ratio of ∼0.4 and a shell-like radio-continuum morphology. However, we were only able to measure a flux density at 20 cm of 25.9 mJy. No sensitive X-ray coverage is available.

MCSNR J0506-6815 = [HP99] 635 (Figure 1; bottom right)—HP99 recorded an object at this position, giving it the name [HP99] 635.¹¹ They listed an extent of 313 (very low likelihood), and estimated HR1 and HR2 values of 1.00 ± 0.76 and −0.05 ± 0.19, respectively. Therefore, the X-ray source could be a point source unrelated to the SNR candidate. Although there was no optical (MCELS) signature, we detected extended radio-continuum emission and estimated flux densities of 37 mJy at both frequencies (36 cm and 20 cm), pointing to a flat spectral index, which is indicative of a PWN. This object is classified as an SNR candidate primarily based on its radio and optical morphology as the X-ray emission association is not clear at this point.

MCSNR J0507-6847—Chu et al. (2000) observed this source, consisting of a large ring of diffuse X-ray emission, and proposed it as an SNR candidate. They found an X-ray luminosity within the range expected for SNRs, and predicted an age of ∼5 × 10⁴ yr based on the Sedov solution. Blair et al. (2006) did not detect this object in their far-UV survey of the MCs. We did not detect associated optical (MCELS) or radio-continuum features in this large object, and we propose that this object might represent a superbubble similar to 30 Dor C (Sano et al. 2017).

MCSNR J0507-7110=DEM L81 (Figure 2; top left)—Davies et al. (1976) listed this object as DEM L81, describing the source as a faint semicircular arc extending $4\buildrel{\,\prime}\over{.} 5$. We identified a southern arc in the MCELS images ahead of the extended radio-continuum emission. As the whole source is very complex, we could not measure any flux density at our radio frequencies. However, the source appears stronger at lower frequencies and is indicative of a steep non-thermal radio-continuum spectrum. Also, there is no sensitive X-ray coverage in this field.

MCSNR J0510-6708 (Figure 2; top right)—We determined an enhanced [S ii]/Hα ratio of >0.5 in the shell. The radio-continuum emission is centrally located and very weak (S${}_{20{\rm{cm}}}=2.5$ mJy). At present, there is no sensitive X-ray coverage in the direction of this object.

MCSNR J0512-6716 (Figure 2; middle left)—A prominent X-ray ring from XMM-Newton images can be seen with some very weak radio emission overlapping. We could not confirm optical identification of this object.

MCSNR J0513-6731 = [HP99] 544 (Figure 2; middle right)—HP99 named this object [HP99] 544, recording an extent of 274, in addition to an HR1 measurement of 1.00 ± 0.29. A low likelihood for the extent leaves the possibility of an X-ray point source unrelated to the SNR candidate. We detected a weak but distinctive [O iii] emission surrounding a distinct radio source. The spectral index was determined to be α = −0.56, based on measured flux densities of 28.7 mJy at 36 cm and 21.5 mJy at 20 cm.

MCSNR J0513-6724 = [HP99] 530 (Figure 2; bottom left)—HP99 gave this object the name [HP99] 530, recording an extent of 175, in addition to HR1 and HR2 ratios of 1.00 ± 0.21 and 0.15 ± 0.17, respectively. Because these hardness ratios suggest a "hard" source and also given the low source extent likelihood, HP99 suggest the source could be unrelated to the SNR candidate. We found a strong radio point source with flux densities of 31 mJy (at 36 cm) and 23 mJy (at 20 cm), implying a non-thermal spectral index of α = −0.61. There is also a weak [S ii] ring, although somewhat smaller than the X-ray extent.

MCSNR J0527-7134 (Figure 2; bottom right)—We classified this object as an SNR candidate based on an enhanced [S ii]/Hα ratio of >0.4 as well as a shell-like radio-continuum morphology. We estimated flux densities of 31 mJy (at 36 cm) and 24 mJy (at 20 cm), implying a non-thermal spectral index of α = −0.52. This candidate was observed very recently with XMM-Newton (2016 October, PI: P. Kavanagh). The detection of soft X-ray emission, correlated with the optical and radio shells, suggests that this source is a bona fide SNR. A detailed study of MCSNR J0527-7134 will be presented elsewhere.

MCSNR J0538-6921 (Figure 3; left)—Turtle & Mills (1984) originally proposed this source to be an SNR. This is the only strong LMC SNR candidate to date that has been detected in radio frequencies alone. The estimated spectral index is $\alpha =-0.59\pm 0.04$, based on measured flux densities at various frequencies.

MCSNR J0539-7001 = [HP99] 1063 (Figure 3; right)—HP99 assigned this source the name [HP99] 1063, recording an extent of 182, in addition to listing HR1 and HR2 values of 1.00 ± 0.17 and −0.17 ± 0.10, respectively. They classified the X-ray source, which was constant in flux during the ROSAT observations, as an SNR candidate. We found a weak radio point source in the center of this remnant with measured flux densities at 36 cm of 5.8 mJy and at 20 cm of 4.6 mJy, giving a spectral index of α = −0.47.

Using the rule-of-thumb bandwidth, we performed smooth bootstrap resamplings and estimated f(x) from the original data points. To minimize BIMSE(h), we used a golden section search algorithm (Kiefer 1953), which narrows down the interval that contains the minimum value by comparing the BIMSE(h) values at four points. To reduce the influence of the Monte Carlo error on the minimizing procedure, we allowed the h values within the algorithm to change only in increments of ${h}_{{\rm{s}}}={10}^{\mu ({h}_{0})-2}$, where μ presents the order of magnitude of its argument. The search was stopped when the size of the h interval containing the minimum fell below h_s. At each given h value, the ${f}_{h}^{* }(x)$ was calculated for all resamplings and the ${({f}_{h}^{* }(x)-f(x))}^{2}$ term was averaged to obtain BIMSE(h).

In the work of Faraway & Jhun (1990), the authors stated that their procedure can be used in an iterative manner and hence improve the values obtained for an optimal smoothing bandwidth. We iterated the above procedure until the difference between the input and output h values fell below h_s or until this difference h_dif changes sign, in which case we took the input value h_in of the final iteration as the outcome of the procedure. Even in such a case, h is calculated with a $\approx \mu ({h}_{\mathrm{in}})-1$ order of magnitude accuracy. The change of sign in h_dif can be avoided and the accuracy improved simply by applying a larger number of bootstrap resamplings. However, Faraway & Jhun (1990) noted that, at the time, their procedure was very computationally intensive and that they obtained satisfactory results using only 100 resamplings. The calculations described in this work took up to $\approx 1$ day of computing per examined data sample on a standard desktop PC using 500 bootstrap resamplings. However, this computing time was dominated by repeating the procedure for computing confidence bands (described below), which required a much greater computation intensity. To increase the speed of the computations, we calculated the Gaussian distribution values only within the five standard deviations from the mean, considering the values outside this interval to be zero.

Faraway & Jhun (1990) noted that a by-product of their procedure is the estimation of confidence intervals as desired quantiles of the ${f}_{\hat{h}}^{* }(x)$ distributions, where $\hat{h}$ is the output optimal bandwidth of their smoothed bootstrap procedure. However, the confidence band derived in this manner depends on $\hat{h}$, which depends on h₀. To avoid this we applied a common bootstrap resampling (without the addition of a smoothing term θ) to the original data sample. For each of the resamples obtained in this way, we applied the described smoothed bootstrap procedure to obtain ${f}_{\hat{{h}_{\bullet }}}^{* }(x)$, where the bullet sign designates that an optimal smoothing bandwidth was obtained for the common bootstrap resample of the original data sample. At each selected point x_j along the x-axis, we then calculated the desired confidence fraction f_ci of the ${f}_{\hat{{h}_{\bullet }}}^{* }({x}_{j})$ values. First, we calculated the median of the given ${f}_{\hat{{h}_{\bullet }}}^{* }({x}_{j})$ array of values and then the  upper and lower limits of confidence bands as the ${f}_{\mathrm{ci}}/2$ fraction of the total number of array elements from the median value. The median value was also used as the value of the smoothed density distribution $\hat{f}({x}_{j})$ at a point x_j.

If the testing x_min and x_max are the lowest and highest values of the data sample, respectively, the density distributions were calculated in the $[{x}_{\min }-5\ast {h}_{0},{x}_{\max }+5\ast {h}_{0}]$ interval at the centers of 10³ bins of equal width. The confidence bands are calculated as an ${f}_{\mathrm{ci}}=0.75$ fraction of the total number of 500 common bootstrap resamplings (•) of the original data sample. We also used 500 smooth bootstrap resamplings (∗) for each (•) resampling. For each (•) distribution, we calculated the mean, mode, and median. The uncertainties of the mean, mode, and median were calculated similarly to the confidence bands for the smooth density distribution, and we give the confidence band with the higher discrepancy from the median as the uncertainty. Confidence bands were also obtained at ${f}_{\mathrm{ci}}=0.75$, similar to $\hat{f}({x}_{j})$. The smooth density functions, with their parameters obtained as described above, for diameter, radio flux, spectral index, and ovality, are presented in Section 5.

A similar procedure to that described in this section can be generalized to a 2D case. We applied the procedure in 2D to the LMC and SMC radio surface brightness and diameter data to check if there were any significant emergent data features (Figure 20). The generalization of the procedure can be done in two ways, either by using a 2D Gaussian kernel (where a kernel is a function of two variables and two smoothing bandwidths), or as a product kernel of separate Gaussian kernels in each dimension. We adopted the latter. As described in Feigelson & Babu (2012), for a 2D case, the density estimate in (x, y) space is calculated as

$$\begin{eqnarray}&&{f}_{{h}^{x}{h}^{y}}(x,y)=\displaystyle \frac{1}{{{nh}}^{x}{h}^{y}}\displaystyle \sum _{i=1}^{n}\,K\left(\displaystyle \frac{x-{x}_{i}}{{h}^{x}}\right)K\left(\displaystyle \frac{y-{y}_{i}}{{h}^{y}}\right),\end{eqnarray} \tag{ 8 }$$

where h^x and h^y are the bandwidths for the kernel components in x and y, respectively. For the 2D density estimate procedure, only the smoothed bootstrap was used to minimize over BIMSE. Due to higher computation requirements, we did not use common bootstrap resamplings to estimate confidence intervals. For the input values to the procedure h^x₀ and h^y₀ (similar to the 1D case), we used rule-of-thumb bandwidths calculated separately in each dimension. For the range of h^x and h^y we calculated and minimized BIMSE. The values of h^x and h^y where BIMSE reached a minimum were used as the next iteration input values. The procedure was repeated until the output values for h^x and h^y equal the input values at the given level of accuracy.

To investigate the spatial distribution of SNRs in the LMC, annotations containing the size and position angle of the 59 confirmed and 15 candidate remnants were superimposed on the H i peak temperature map from Kim et al. (1998). There is an indication of a connection between the higher H i density (also including Hα and radio continuum) and the location of SNRs, which seems to follow a spiral structure (Filipovic et al. 1998). The mean foreground H i column density in the direction to the 59 confirmed and 15 candidate remnants was estimated to be $\sim 2\times {10}^{21}$ atoms cm⁻² (with a Standard Deviation SD = 1 × 10²¹ atoms cm⁻²), while the "empty" LMC 4 supershell (Bozzetto et al. 2012b) exhibits a mean H i density of 5 × 10²⁰ atoms cm⁻² (SD = 1 × 10²⁰ atoms cm⁻²). Curiously, we found only one SNR (MCSNR J0529-6653 inside the LMC4 supershell) to be located outside the apparent spiral structure of the H i distribution. Also, MCSNR J0527-6549 expands in a very rarified environment with a mean H i column density of 6 × 10²⁰ atoms cm⁻². We note that our 16 type Ia SNR sample might be expanding in a somewhat lower density environment (mean = 1.9 × 10²¹ atoms cm⁻²), while the CC sample of 23 LMC SNRs exhibits a mean H i column density of 2.4 × 10²¹ atoms cm⁻². However, SDs of both samples are quite large (SD = 1.1 × 10²¹ atoms cm⁻²). Therefore, this is indicative of a different molecular environment in which type Ia and CC LMC SNRs are expanding, although it should be taken with caution.

To compare the multifrequency emission from the 59 known SNRs in the LMC, we plotted a Venn diagram (Figure 6) that summarizes the number of SNRs exhibiting emission in the different EM domains. It is important to note that the lack of detected emission does not mean that the remnant does not emit such a radiation. However, it may indicate that the emission is under the sensitivity level of current surveys. As for the candidate remnants, many of them were entangled in unrelated emission or not part of current surveys, making it difficult to construct a worthwhile Venn diagram for these sources. We also note that there are examples of SNRs such as the SMC SNR HFPK 334 (Crawford et al. 2014) or the Galactic Vela Jr SNR (Filipović et al. 2001; Stupar et al. 2005) that could not be identified in optical frequencies despite extensive searches.

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For comparison to our LMC results, Venn diagrams were constructed for various other nearby galaxies, shown in Figure 7. We note that some of these galaxies do not have deep X-ray and/or radio coverage. Still, our results are closely comparable to those found for the SMC (Filipović et al. 2005), which is to be expected as it is the most similar to the LMC. The obvious common trait between the LMC and SMC SNR populations is that they are ubiquitous in X-rays because of low foreground absorption toward the MCs. We concluded from Figure 6 that this sample is not under severe influence from observational biases. Although the LMC and SMC XMM-Newton surveys reach similar depth, the LMC X-ray field coverage is somewhat incomplete and therefore our present LMC X-ray SNR sample is likely incomplete as well.

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Comparing Figures 6 and 7, we note that all other galaxies have high numbers of detected SNRs only in optical frequencies, with small numbers of SNRs in cross-sections of the Venn diagrams. This is expected, as Pannuti et al. (2000) argued that X-ray and radio SNRs are mixed/embedded and therefore confused with H ii regions in distant galaxies. Also, we argue that the detection of optical SNRs in distant galaxies is biased toward lower densities. Therefore, we detect only smaller numbers of X-ray/radio SNRs in the more distant samples—ones that are brighter and in denser, star-forming regions.

Interestingly, in a revised catalog of 294 Galactic SNRs by Green (2014), 93% were detected in radio, ∼40% in X-ray, and only ∼30% at optical wavelengths. This points to a clear selection bias, which limits the optical and X-ray detection. This is most likely due to obscuration from dust and clouds, as well as the lack of deeper observations at various frequencies. Also, we note that NGC 55 is an edge-on spiral galaxy and therefore, only a small number of SNRs can be detected due to obscuration.

To measure the extent of the SNRs in the LMC, an ellipse was fitted to delineate the bounds of emission for all confirmed and candidate SNRs in this study (Bozzetto et al. 2014b; see Figure 7). A multiwavelength approach was used and the given size takes into account the optical, radio, and X-ray emission. Such an approach was taken as some shells may appear incomplete at optical wavelengths, although complete at radio or X-ray wavelengths, and vice versa. A somewhat typical example of this is where emission in one band (e.g., X-rays) is located in the center and encased by an optical/radio shell, e.g., MCSNR J0508−6902 presented in Bozzetto et al. (2014b, Figure 2), where the radio (5500 MHz) and optical (Hα) emission form the "ring" of emission, inside which the X-ray (0.7–1.1 keV) emission resides. The major and minor axes, in addition to the position angle of these measurements, are listed in Tables 1 and 2. The resulting PDF showcasing the distribution of these data is displayed in Figure 8, where the diameter was taken as the geometric average of the major and minor axes (Tables 1 and 2).

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The mean value of the reconstructed distribution for the 59 confirmed and 15 candidate remnants was found to be 39 ± 4 pc for confirmed SNRs, and 41 ± 3 pc for the entire sample (Figure 8). This increased mean value for the entire sample is most likely due to many of the candidate remnants being larger and weaker, only being recently detected by newer and more sensitive instruments. The majority (36 out of 59 or 61%) of the remnants were found to exhibit diameters in a range of 15–50 pc. These values are moderately larger than the value found in the study of M83 SNRs by Dopita et al. (2010), where a mean diameter of 22.7 pc (with standard deviation of SD = 10.3 pc) was found for a sample of 47 remnants. Also, in a study of the SMC, Filipović et al. (2005) found a mean diameter of 30 pc, and argued that such a value indicates that most of the remnants are in the adiabatic (Sedov) evolutionary stage. The results of this study are more in line with those found by Long et al. (2010) in their study of M33, finding a median of 44 pc, and by Lee & Lee (2014) of M31, which showed a strong peak at D = 48 pc.

The spherical symmetry of the LMC SNR population was measured to investigate whether or not a trend exists between the type of SNR and how circular their morphology appeared. Lopez et al. (2011) suggested a link between the spherical thermal X-ray morphology and the remnant type, where those SNRs resulting from a type Ia SN explosion were more spherical than those from an CC SN. In this study, we define SNR spherical symmetry via

$$\begin{eqnarray}&&\mathrm{Ovality}\,( \% )=\displaystyle \frac{2({D}_{\mathrm{maj}}-{D}_{\min })}{{D}_{\mathrm{maj}}+{D}_{\min }},\end{eqnarray} \tag{ 9 }$$

where D_maj and D_min are the major and minor axes, respectively. This was done for all 59 confirmed remnants using the diameter measurements shown in Table 1 (Col. 5).

We plot the smoothed density distribution for the ovality data (as explained in Section 4) in Figure 9. The large uncertainty for the mode parameter and shape of the confidence bands strongly suggest the existence of statistically significant bi-modality. To test whether this bi-modality is related to progenitor type, we plotted the smoothed density distributions for the subsamples selected by progenitor type: type Ia, CC, and unknown (with progenitor type being undetermined).

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In Figure 10 the mode values for all examined subsamples overlap within the estimated uncertainties at the 75% confidence level, so it is unlikely that the examined subsamples are actually coming from statistically distinct populations of SNRs. Also, from the shapes and parameters of the distributions for type Ia and CC progenitor types, it appears that no distinction based on the progenitor type can be made and that the expected ovality of ∼0.16 should be even smaller since the progenitors from the unknown group are more spherical in shape. In total, ∼27% (16 out of 59) of the LMC SNRs exhibit ovality up to 5% (0–0.05) and only 2 out of 16 (∼13%) type Ia SNRs' ovality are within this range, which does not support the hypothesis that type Ia SNRs are more spherical in shape.

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To further test if there is a statistically significant independence for ovality subsamples based on SNR progenitor type, we conducted an Anderson–Darling test (Anderson & Darling 1954), studied in more detail for a two-sample case by Pettitt (1976). The test calculates the parameter

$$\begin{eqnarray}&&{A}_{\mathrm{nm}}^{2}=\displaystyle \frac{{nm}}{N}{\int }_{-\infty }^{\infty }\displaystyle \frac{{({F}_{n}(x)-{G}_{m}(x))}^{2}}{{H}_{N}(x)(1.0-{H}_{N}(x))}{{dH}}_{N}(x),\end{eqnarray} \tag{ 10 }$$

where F_n(x) and G_n(x) are the empirical cumulative distribution functions (ECDF) for sample 1 with n points and sample 2 with m points, while H_N(x) is the ECDF of a joint sample with $N=m+n$ points. The integral of the squared deviations weighted with the joint ECDF factor gives a robust measure of the difference, even at the tails of the distribution where, by definition, all ECDFs converge toward 0 and 1. The resulting value of A²_nm is then compared against the critical value of A^2'_nm at a specific level of confidence (α) to test the null hypothesis that the two samples represented by F_n(x) and G_n(x) are sampled from the same underlying distribution. The results of the test are shown in Table 3. It is apparent that even for the selected confidence limit of 0.1, ${A}_{\mathrm{nm}}^{2}\lt {A}_{\mathrm{nm}}^{2^{\prime} }$, meaning that the calculated ${A}_{\mathrm{nm}}^{2}$ is within the 90% confidence interval from the expected value for ${A}_{\mathrm{nm}}^{2^{\prime} }$. This indicates that the null hypothesis cannot be rejected at the α=0.1 level and that the tested subsamples could plausibly be sampled from the same distribution. This argues against the earlier stated existence of a correlation between the type of SNR and their spherical symmetry.

Table 3.  Results of the Anderson–Darling Two-sample Test

S1 versus S2	n	m	Anm2	${A}_{\mathrm{nm}}^{2}{^{\prime}}$
				$\alpha =0.1$
Unknown versus type Ia	20	16	1.150	1.933
Unknown versus CC	20	23	0.331	1.933
Type Ia versus CC	16	23	1.556	1.933

Note. The comparison samples (S1 and S2) are given in the first column while the rest of the columns present the number of objects in S1, number of objects in S2, calculated value of the Anderson–Darling variable for the two-sample test (A_nm²), and value for the specific confidence level α (${A}_{\mathrm{nm}}^{2}{^{\prime}}$) at $\alpha =0.1$ (10%) taken from Pettitt (1976), respectively.

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In addition, we make a scatter plot in Figure 11 of the geometric mean diameter versus ovality, color-coded by the ages of the remnants. For this population, there does not appear to be any conclusive evidence that the type of SN explosion (Table 1; Col. 11) correlates with the ovality (as defined here) of the resulting SNR or its known age, estimated from the various multifrequency measurements (Table 1; Col. 12).

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For the 28 LMC SNRs older than 10,000 yr (Figure 11), we found a suggestive progenitor type. This is somewhat surprising, as for older remnants, signatures of progenitor type are likely to be faint and hence harder to determine. Also, in Figure 10, the distribution of objects with undetermined progenitor type appears to be skewed toward smaller ovality values. This implies that older remnants might be more spherical in shape.

We expect that future surveys with high accuracy ovality measurements, better age determinations, and more complete samples are of crucial importance for a better assessment of this subject.

The cumulative number–diameter relation, also known as $N(\lt D)$ − D, shows the number of SNRs smaller than a given diameter. Assuming a rough uniformity in SN explosion and environment, it is possible to estimate the evolution of SNRs and the rate of SN occurrence via

$$\begin{eqnarray}&&N(\lt D)=\displaystyle \frac{t(D)}{\tau },\end{eqnarray} \tag{ 11 }$$

where t(D) is the age of the remnant and τ is the average time between the SN events, i.e., ${\tau }^{-1}$ is the average supernova rate. As a remnant evolves, it is expected to pass through different phases of hydrodynamical evolution. The turnover from the free-expansion phase to the Sedov phase, i.e., when the mass of the swept-up material is higher than the ejecta, was approximated using the equation from Spitzer (1978):

$$\begin{eqnarray}&&{R}_{\mathrm{pc}}\approx 2.13{\left(\displaystyle \frac{{M}_{E}}{{n}_{o}}\right)}^{1/3},\end{eqnarray} \tag{ 12 }$$

where R_pc is the transition radius in parsecs, n₀ is the number density of the ISM per cubic centimeters, and M_E is the ejecta mass in solar units. If we assume progenitor masses between the Chandrasekhar limit (1.4 ${M}_{\odot }$) and that of a large star (40 ${M}_{\odot }$) expanding in ambient densities between 0.1 and 1.0 cm⁻³, the expected turnover from free expansion to Sedov expansion would fall between ∼4.8 pc and ∼31.4 pc. It should be noted, however, that a progenitor star's mass may exceed the given 40 ${M}_{\odot }$(Gal-Yam et al. 2009), and that densities can vary significantly (e.g., cold atomic regions of the ISM have densities of 20–50 cm⁻³, while molecular regions may be as high as ${10}^{2}\mbox{--}{10}^{6}$ cm⁻³ (Ferrière 2001 and references therein). Therefore, such cutoff levels are far from being well-established. The upper limit of Sedov expansion (i.e., the turnover to snowplow evolution) is estimated by Woltjer (1972) to be 50 kyr, or $D\,\sim$ 48 pc. These estimated turnover diameters are in moderate agreement with Berkhuijsen (1987), who found evidence that SNRs are radiative between ${D}_{R}({n}_{0}=1)\simeq 20$ pc to 40 pc.

If we assume that the expansion is in the form of a power law $D\propto {t}^{m}$, then $N(\lt D)\propto {D}^{a}$, where $a=1/m$. This means that for the free expansion, $a\approx 1$ is expected, while in the Sedov phase, the slope in log–log space would be a = 2.5.

Early studies of the $N(\lt D)$ relation for SNRs in the Galaxy and the LMC found exponents of $\approx 1$ (Mathewson et al. 1983; Mills 1983; Mills & Turtle 1984), implying that the majority of the remnants are in free expansion. However, Berkhuijsen (1987) made the point that such an exponent is readily mimicked by any influence tending to randomize diameters. Later studies by Long et al. (1990), Smith et al. (1993), and Gordon et al. (1998) found steeper slopes, common to later phases such as Sedov or snowplow expansion. For example, in their study of SNRs in M33, Gordon et al. (1998) found an $N(\lt D)$ relation consistent with Sedov expansion and inconsistent with free expansion, inferring that if the ISM is similar to our own Galaxy and the LMC, then these galaxy surveys are seriously incomplete. In their study of SNRs in M83, Dopita et al. (2010) found a slope consistent with free expansion for nuclear remnants, whereas remnants residing in the disk generally followed the expected value of the radiative phase ($N(\lt D)\propto {D}^{7/2}$). In M31, Lee & Lee (2014) found two breaking points in the data: one at 17 pc and another at 50 pc. The first component (SNRs with $D\lt 17$ pc) showed a power-law slope of $a=1.65\pm 0.02$, while the second component ($17\,\mathrm{pc}\lt D\lt 50\,\mathrm{pc}$) was in line with Sedov expansion, with a slope of $a=2.53\pm 0.04$.

Here, we make distinctions between the types of SNRs based on their morphologies. Lee & Lee (2014) defined so-called A-type remnants as those with well-defined shells. The slope for this subclass of M31 SNRs was found to be $a=2.15\pm 0.09$ for 25 pc $\lt \,D\lt 45$ pc. In their study of M33, Gordon et al. (1998), using a maximum likelihood estimate fit to the $N(\lt D)$ relation, found slopes for D_max = 30 pc and D_min = 8 pc and 10 pc of $a=2.2\pm 0.05$ and $a=2.3\pm 0.06$, respectively, while D_max = 35 pc and D_min = 8, 10, and 15 pc showed slopes of $a=2.0\pm 0.04$, $a=2.0\pm 0.05$, and $a=1.1\pm 0.06$, respectively. For the SMC, Filipović et al. (2005) found an overall slope for the galaxy of $a=1.7\pm 0.2$.

In Figure 12, we present our results for the LMC. The top of Figure 12 shows cumulative counts versus diameter for all 59 confirmed LMC SNRs in log–log scale. The red line shows the best fit with a slope $a=1/m=0.96$. Since the sample above D > 40 pc seems to be somewhat incomplete, only the first part of the curve was fitted (for an alternative view, see, e.g., Badenes et al. 2010). The population exponent $a=0.96$ is close to 1, even given the larger population and more complete sample size of this study. Therefore, the earlier suggestion by Berkhuijsen (1987) regarding randomized diameters readily mimicking such an exponent is probably the case in our LMC sample, and not that the relation is indicative of the SNR population in the galaxy to be in free expansion. The exponent $1\lt a\lt 2.5$ may also indicate that the population is somewhere between free expansion and the Sedov phase. Although this is unlikely in the case of the LMC sample, we have used a simple model for the SNRs' expansion velocity (Arbutina 2005; Finke & Dermer 2012),

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}{v}^{2}=\displaystyle \frac{{k}_{1}{E}_{o}}{{k}_{2}{M}_{E}+4\pi {R}^{3}{\rho }_{o}/3},\ \ v=\displaystyle \frac{{dR}}{{dt}},\end{eqnarray} \tag{ 13 }$$

and combined it with ${dN}/{dD}=1/(2\tau v)$ in order to perform a non-linear fit to the data (again, only the first part of the curve was fitted). In Equation (13). E_o is the explosion energy, M_E mass of the ejecta, and ${\rho }_{o}$ is the ISM density. The constants k₁ and k₂ are determined in such a way that when E_o = 1 foe¹² and M_E = 1.4 ${M}_{\odot }$, v ≈ 20,000 km s⁻¹, and when $R\gg {R}_{\mathrm{pc}}$, the velocity tends toward the Sedov solution. In Figure 12 (bottom) we show the differential distribution for the number of LMC SNRs versus diameter. The red curve represents the best-fit model obtained by applying Equation (13), which is still far from good. For a fixed energy of 1 foe and ${M}_{E}\sim 10$ ${M}_{\odot }$ (although the sample could contain few SNe Ia), the fit gives a supernova¹³ rate of $0.55$/cy (twice as high as rates found in the literature, e.g., 0.23/cy; van den Bergh & Tammann 1991), while the density is quite low, ∼0.03 cm⁻³, which is not surprising for SNRs still close to the free-expansion phase. As explained above, this density is unrealistically low.

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The mean spectral index of the reconstructed distribution is $\alpha =-0.52\pm 0.03$ (with sample SD = 0.13), calculated from the 41 LMC SNRs where a reliable spectral energy distribution could be estimated. Individual remnant indices as well as associated properties can be found in Table 1, while a reconstructed differential distribution (similar to the reconstructed distributions from Figure 8) of these values can be seen in Figure 13. The mean value of −0.52 is in line with the theoretically expected spectral index of $\alpha =-0.5$ (as discussed above). The left side of the distribution is predominately composed of young SNRs, while older remnants or those harboring a pulsar are found toward the right end of the PDF.

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Comparing to other galaxies, Filipović et al. (2005) found a mean spectral index of −0.63 (SD = 0.43) for confirmed and candidate SNRs in the SMC. In the MW, Clark & Caswell (1976) found a mean spectral index of −0.45 with an SD of ∼0.15.

To investigate the relationship between a remnant's age and its radio spectral index, we plotted these two properties against each other (Figure 14; top), resulting in a power-law fit for the 29 SNRs with an established age in the LMC:

$$\begin{eqnarray}&&\alpha =-0.97\pm 0.09\times {{\rm{t}}}_{\mathrm{yr}}^{-0.06\pm 0.02},\end{eqnarray} \tag{ 17 }$$

where t_yr represents the remnant's age in years. The fit parameter values are calculated as mean values from an array of 10³ values for each parameter obtained by fitting (non-weighted fit) the resamples of the original data sample (for more details on the bootstrap procedure, see Efron & Tibshirani 1993). The parameter uncertainties are calculated as standard deviations of the corresponding arrays. Some of the earlier studies (e.g., Clark & Caswell 1976) indicated that a remnant's spectral index did not appear to be correlated with any other parameters. However, this conclusion was generally formed when smaller samples of remnants were available or used. In our study, we found the trend that younger remnants exhibit significantly steeper spectral indices, while mid-to-older remnants show flatter indices. Also, we found that the average spectral index for this sample of 29 LMC SNRs with known age (and spectral index) is α = −0.55, which is fractionally steeper than that for the whole sample (α = −0.52). It is likely that this is because it is easier to obtain an age for younger, and therefore brighter, SNRs, which have steeper spectral indices.

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The notion that older SNRs exhibit flatter indices was first recognized by Harris (1962) based on observational evidence. Onić (2013) and Urošević (2014) suggested that SNR diminishing may be explained by the contribution of the second-order Fermi mechanism, higher shock compressions, or/and thermal bremsstrahlung. Conversely, the steeper spectra found for younger SNRs may be explained by optically thin synchrotron emission produced from accelerated electrons and compressed magnetic field produced at the shock front (Staveley-Smith et al. 2005). Further, Bell et al. (2011) suggested that expansion into a Parker spiral may produce a geometry favoring quasi-perpendicular shocks and spectral steepening.

One should also not forget that there is evidence that the SNR radio spectral index significantly flattens when the SNR shell interacts with surrounding molecular clouds (Ingallinera et al. 2014). The linear fits in Figure 14 are not likely to capture all of the features of the data as we do not know precisely what causes the spread. It could be a wide variety of factors spanning from the excess in N_H, differing supernova energies and types, to differing number densities.

To compare our results with those from other galaxies, we re-plotted all values in Figure 14 (bottom) alongside spectral index and age data from SNRs in the MW (Green 2014) and SMC (Filipović et al. 2005, 2008). As can be seen, there is a good alignment between the galaxies. SNR ages, except for the few historical SNe, are derived assuming a hydrodynamical model, usually Sedov, linking age $t\,\propto \,R/{v}_{{\rm{s}}}$. The shock speed ${v}_{{\rm{s}}}$ is derived from X-ray fitting ($\propto \sqrt{{kT}}$) and is independent of distance (D), so $t\,\propto \,R\,{({kT})}^{-1/2}$ $\,\sim \,\theta D{({kT})}^{-1/2}$, where θ is the observed angular diameter. Thus, distance uncertainties contribute most of the uncertainty in age measurements. As noted in Section 1, D is most poorly known for Galactic objects, leading to large age error bars. However, for the LMC objects, a common distance of 50 kpc is assumed so the age measurement does not suffer from such large uncertainties.

In a similar fashion to the age–spectral index relation, a diameter–spectral index relation was created, which gives a better view of the entire sample of remnants, resulting in

$$\begin{eqnarray}&&\alpha =(-0.8\pm 0.1)\times {D}_{\mathrm{pc}}^{-0.12\pm 0.03}.\end{eqnarray} \tag{ 18 }$$

As expected from Figure 14, smaller (which would generally imply younger) remnants tend to exhibit steeper spectral indices, while their larger counterparts show flatter indices. Similarly to the age–spectral index relation, we compare these results with those from the SMC and MW sample (see Figure 15, bottom). There is a loose alignment in the results, although remnants in the MW generally appear to exhibit flatter indices at the same diameter compared to the LMC sample.

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Glushak (1985) studied the evolution of young shell SNRs using eight Galactic remnants, in addition to the one located in the LMC. He found that the remnants' spectral indices flattened as they got larger and older, in line with the results in this study. Glushak (1996) used positive α-values expressed in the form

$$\begin{eqnarray}&&\alpha =P\mathrm{log}\left(\displaystyle \frac{{D}_{\mathrm{pc}}}{\mathrm{pc}}\right)+{\alpha }_{0}\end{eqnarray} \tag{ 19 }$$

to find values of $P=-0.58\pm 0.07,{\alpha }_{0}=0.62\pm 0.03$ for M82 and NGC 253 ($0.3\lt D\lt 4$ pc), and $P=-0.54\ \pm 0.03,{\alpha }_{0}=1.03\pm 0.02$ for the Galaxy and M31 ($3\ \lt D\leqslant 21$ pc). He concluded that young shell SNRs evolve differently in galaxies with and without starbursts. Using the same method for the sample of LMC SNRs listed in Table 1, we found $P=0.18\pm 0.04,{\alpha }_{0}=-0.79\pm 0.07$ (see Figure 15; bottom), which shows that the spectral index flattening is much less severe as the remnant evolves, in line with the age–spectral index relation.

Similar to the SNR spherical symmetry tests (Section 5.4), the independence of the flux density subsamples presented in Figure 16 was tested with the Anderson–Darling two-sample test. For the CC versus type Ia SNRs, the Anderson–Darling variable test value is 6.4, which is significantly higher than 3.857 (a critical value for 0.01 level from Pettitt 1976), indicating (with a probability of $\gt 99 \%$) that CC and type Ia remnants belong to vastly separate populations of objects. Similar results are found for CC versus Unknown SN type with even higher confidence since the value of the Anderson–Darling variable is 7.2. This implies that SNRs with an undetermined progenitor type are more likely to originate from type Ia than CC progenitors. This is in agreement with the remaining case (type Ia versus Unknown) where the Anderson–Darling variable is 0.73, which is well below 1.933 (a critical value at 0.1 level value as also used in Table 3 for testing ovality based subsamples), indicating that the tested samples are likely sampled from the same underlying distribution. We also emphasize that the second youngest (310 yr old; Bozzetto et al. 2014a; Hovey et al. 2015) SNR in the LMC—MCSNR J0509-6731—is a well-known type Ia and a relatively weak radio emitter with ${S}_{1\mathrm{GHz}}$ of 97.4 mJy.

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The flux density distribution for the 40 LMC SNRs that have estimated flux densities are shown in Figures 16 and 17.¹⁴ The flux density variable is, by its own nature, heavily biased with sensitivity selection. The faint objects are not usually detected in surveys alongside the majority of the objects but rather with specialized high sensitivity observations. This is reflected in the fact that only ∼70% (40 out of 59) of known LMC SNRs have radio flux density measurements. When candidate SNRs are added, this number remains similar—∼67% (49 out of 74). The object DEM L71, which has the lowest flux value, is a well-observed and studied object, unlike much of the sample (see also Section 6.4). Compared with other objects in the vicinity of the PDF mode (which is close to DEM L71), this object and its immediate neighbor [HP99] 460 appear to be rather well-separated. This implies that the highest concentration of detected objects is very close to the sensitivity limits of the related observations and that surveys with sensitivities below 50 mJy should give a large number of new detections.

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Although there are known LMC SNRs with 1 GHz flux densities less than 50 mJy, most of the sub-50 mJy sample is lacking reliable flux density estimates. This is because of confusion due to unassociated nearby emission, e.g., an H ii region or a nearby and strong background source. Also, some of the remnants are lacking flux density estimates because they are either too weak to be accurately measured or fall below the detection limit of the present generation of surveys. Future radio telescopes (such as Australia Square Kilometre Array Pathfinder, ASKAP) with higher sensitivity and resolution will be able to account for these remnants and provide a more complete sample.

We compared our estimated radio flux densities at 1 GHz (Table 1) and broadband X-ray flux in the 0.3–8 keV range from Maggi et al. (2016). There are 58 known LMC SNRs with X-ray flux and/or radio flux density measurements (see Figure 18). Only one confirmed LMC SNR (J0521-6542) has no measurements at either frequency.

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At present, the SNR with the faintest measured radio flux density is DEM L71. As it is a well-studied SNR (and one of the brightest X-ray SNRs in the LMC), there are many deep radio observations available for DEM L71 that make its detection and radio flux density measurement easier than for most of the LMC sample. The color-coded symbols in Figure 18 at ∼6 mJy, indicating radio non-detections, are not representative of true flux density limits. While the rms noise will vary significantly across the LMC, the average radio sensitivity limit for the non-detected sample is >10 mJy. Likewise, the three X-ray non-detections (ticks at ∼10⁻¹⁴ erg s⁻¹ cm⁻²) stem from a lack of proper coverage (e.g., observations with XMM-Newton or Chandra), and it is premature to conclude that these sources are intrinsically X-ray fainter.

The two flux-bright sources (N49B and N63A) with ${F}_{0.3-8\mathrm{keV}}\gt 1\times {10}^{-11}$ erg s⁻¹ cm⁻² have ages of 3500 and 10,000 years old. The other outlier is N157B (also known as 30 Dor B), which has a bright/young cometary PWN and is 5000 yr old. The total X-ray flux (SNR+PWN, ${F}_{0.3-8\mathrm{keV}}\gt 5\,\times {10}^{-12}$ erg s⁻¹ cm⁻²) is given, although the thermal component (SNR only) is only $\sim 2\times {10}^{-13}$ erg s⁻¹ cm⁻² as for the bulk of the sample. The radio flux density for this object (∼3 Jy) also includes a significant fraction of PWN emission, but is harder to separate as both have a non-thermal spectrum.

It appears that SNRs younger than 10,000 years with higher flux and flux density values in X-rays and radio, respectively, show some correlation between these values.  Although it is difficult to quantify, we point to the possible correlation between young type Ia and CC SNRs, though the latter appears somewhat brighter (in both X-rays and radio) than the former. Also, the data points near the plotted 1-1 correspondence line in Figure 18 imply that Type Ia objects might be somewhat younger than CC SNRs. For the 43 objects that have age data (Table 1), the results of the AD two sample test (see sections 5.4 and 6.3) do not justify the assumption that CC (20 objects) and Type Ia (15 objects) age data subsamples are independent. However, the distribution of objects with an undetermined progenitor type (eight objects) is distinct from the distributions of Type Ia (at $\alpha =0.05$ level) and CC objects (at $\alpha =0.1$ level). This is consistent with Figure 18. The objects with undetermined progenitor types, unknown ages, and ages $\gt 10\,000\,\mathrm{yr}$ have, in general, ${F}_{0.3-8\mathrm{keV}}\lt \times {10}^{-12}$ ergs s⁻¹ cm⁻². This implies that objects with unknown ages are thus likely to be older remnants that have lost the signatures of their progenitors over time.

There is also a number of radio non-detections that fall into the X-ray flux range of ${F}_{0.3-8\mathrm{keV}}\,\lt \,1\times {10}^{-12}$ erg s⁻¹ cm⁻², which confirms the need for more sensitive observations of the LMC SNR population in radio.

Following the theoretical work done initially by Shklovskii (1960), the important relation connecting the radio surface brightness ${{\rm{\Sigma }}}_{\nu }$ of a particular SNR at frequency ν and its diameter D can be written in general form as

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\nu }(D)={{AD}}^{-\beta }.\end{eqnarray} \tag{ 20 }$$

The parameter A depends on the properties of both the SN explosion and the ISM (e.g., SN energy of explosion, ejecta mass, the density of the ISM, the magnetic field strength, etc.), while β is thought to be independent of these properties (Arbutina & Urošević 2005) but explicitly depends on the spectral index α of the integrated radio emission from an SNR (Shklovskii 1960). Parameters A and β are obtained by fitting the data from the sample of SNRs with known distances. Despite all of the criticism of the ${\rm{\Sigma }}\mbox{--}D$ relation (e.g., Green 2005), it remains an important statistical tool in estimating distances to an SNR from its observed, distance- independent, radio surface brightness.

The best-fit correlation (Equation (20)) is a straight line in the $\mathrm{log}D\mbox{--}\mathrm{log}{\rm{\Sigma }}$ plane. However, explicit care has to be taken to use the appropriate form of regression. As concluded in Pavlović et al. (2013), the slopes of the empirical ${\rm{\Sigma }}\mbox{--}D$ relation should be determined by using orthogonal regression because of its robust nature and equal statistical treatment of both variables. Both variables suffer from significant scatter and it is not statistically justified to treat one of them as independent. Nevertheless, this is the usual practice, and regressions that minimize over offsets along one variable while the other one is considered as independent, such as ${\rm{\Sigma }}=f(D)$ or $D=f({\rm{\Sigma }})$, are very often used due to their simplicity. In this work we used the more robust orthogonal regression since it minimizes over the orthogonal distance of the data points from the fit line and consequently, both variables have the same significance without one of them being considered the function of the other and measured with infinite accuracy. The orthogonal fitting performs well, regardless of the regression slope for data sets with severe scatters in both coordinates. However, for the extragalactic samples of SNRs, distance (and hence SNR diameters) can be obtained with higher accuracy than for the Galactic remnants, since they can be approximated to reside at the distance of the host galaxy. Even in such a case, the depth effect and intrinsic effects can cause significant scatter in D and orthogonal fitting should be preferred.

We can see from extragalactic SNR samples that the intrinsic scatter still dominates the ${\rm{\Sigma }}\mbox{--}D$ relation and errors in distance determination are not crucial, for both Galactic and extragalactic SNRs. Here, the intrinsic scattering originates from the modeling of diverse phenomena with just two parameters (Σ and D) and not taking into account individual characteristics of SNRs such as different SN explosion energies, densities of ISM into which they expand, evolutionary stages, etc. Also, data sets made up of extragalactic SNRs do not suffer from the Malmquist bias because all SNRs are at the same distance, while they still suffer from other selection effects caused by limitations in sensitivity and resolution (Urošević et al. 2005).

For the 40 LMC SNRs with measured flux densities, we estimated the radio surface brightness via

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\nu }=5.418\times {10}^{-16}\displaystyle \frac{{S}_{\nu }}{{\theta }^{2}},\end{eqnarray} \tag{ 21 }$$

where ${S}_{\nu }$ is the integrated flux density in Jansky (Jy) and θ is the diameter in arcseconds. Our data sample consisting of SNRs from the LMC is displayed in Figure 19, where there appears to be a correlation between Σ and D, in confirmation of the theoretical models. Significant scattering is still present despite having more precise SNR diameters in comparison to the Galactic sample. This is expected due to an intrinsic scatter which dominates any errors arising from the measurement process. Furthermore, SNRs formed from type Ia explosion have lower surface brightnesses than those arising from CC SN events. This is in agreement with the theoretical prediction that the larger ISM density produces greater synchrotron emission from an SNR, given in the form ${\rm{\Sigma }}\propto {\rho }_{0}^{\eta }\propto {n}_{H}^{\eta }$ (Duric & Seaquist 1986; Berezhko & Völk 2004, 2010), where ${\rho }_{0}$ and n_H are the average ambient density and hydrogen number density, respectively. As the distance to the LMC of 50.0 ± 1.3 kpc is determined to a very high accuracy (Pietrzyński et al. 2013), the ${\rm{\Sigma }}\mbox{--}D$ relation is in our case more important from a theoretical point of view, for comparison with Galactic and other extragalactic relations and also for comparing the environments into which SNRs expand.

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Surface brightnesses were calculated for the sample of 40 LMC SNRs at 5 GHz and orthogonal fitting was applied. The resulting fit shows a ${\rm{\Sigma }}\mbox{--}D$ slope of $\beta =3.78\pm 1.20$, which is very close to the slope $\beta =3.9\pm 0.9$ obtained for the sample of 31 compact SNRs from the starburst galaxy M82, at the same frequency (Urošević et al. 2010). The slope errors were calculated by using the bootstrap method. We have done 10⁵ bootstrap data resamplings for each fit.

For the purpose of proving the universality of the ${\rm{\Sigma }}\mbox{--}D$ law for SNRs, regardless of which samples are used, we constructed a composite sample containing available extragalactic SNR populations at the same frequency of 5 GHz (214 SNRs in total), which is shown in Figure 21. Properties of the 10 extragalactic samples considered in this paper are listed in Table 4. The entire sample has a compact appearance for the presented variable range, with an overall slope of $\beta =3.60\pm 0.15$, very close to that of the LMC sample in this study. The resulting slope indicates that the observed extragalactic SNRs are mainly in the Sedov phase of evolution, as it was predicted by the values of slopes that are theoretically derived (Duric & Seaquist 1986; Berezhko & Völk 2004).

Table 4.  Parameters of Available Nearby Galaxy Data Samples Using Their 5 GHz Radio Fluxes

Name	Number of SNRs	β	${\rm{\Delta }}\beta$	Reference
Entire data sample	214	3.60	0.15	This work
LMC	40	3.78	1.25	This work
SMC	19	5.16	5.76a	Filipović et al. (2005)
M82	31	3.88	0.91	Fenech et al. (2008)
M31	30	2.58	0.68	Urošević et al. (2005)
M33	51	2.66	0.85	Urošević et al. (2005)
Arp 220b	6	⋯	⋯	Batejat et al. (2011)
NGC 4449, NGC 1569	37	4.68c	2.58	Chomiuk & Wilcots (2009)
NGC 4214, NGC 2366

Notes.

^aA large bootstrap error for slope β is expected for small samples like the SMC, containing only 19 SNRs. ^bThis sample contains only six SNRs and therefore calculating the ${\rm{\Sigma }}\mbox{--}D$ slope does not make physical sense. ^cThe ${\rm{\Sigma }}\mbox{--}D$ slope was calculated for the composite sample containing 37 SNRs in four galaxies: NGC 4449, NGC 1569, NGC 4214, and NGC 2366.

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In Figure 22 we show the theoretical "equipartition" models for radio evolution similar to the ones given by Reynolds & Chevalier (1981) and Berezhko & Völk (2004). The emissivity is defined as

$$\begin{eqnarray}&&{\varepsilon }_{\nu }={c}_{5}K{(B\sin {\rm{\Theta }})}^{(\gamma +1)/2}{\left(\displaystyle \frac{\nu }{2{c}_{1}}\right)}^{(1-\gamma )/2},\end{eqnarray} \tag{ 22 }$$

where ${c}_{1},{c}_{3}$ and ${c}_{5}={c}_{3}{\rm{\Gamma }}(\tfrac{3\gamma -1}{12}){\rm{\Gamma }}(\tfrac{3\gamma +19}{12})/(\gamma +1)$ are from Pacholczyk (1970). The model assumes $K\propto {\epsilon }_{\mathrm{CR}}$, ${\epsilon }_{\mathrm{CR}}\ \approx \tfrac{4}{\gamma +1}{\epsilon }_{B}\sim \delta \rho {v}^{2},\ \ {\epsilon }_{B}=\tfrac{1}{8\pi }{B}^{2}$ (Arbutina et al. 2012, 2013), where we applied Equation (13), and assumed a CR proton-to-electron number of 100:1. The slope of ${\rm{\Sigma }}\mbox{--}D$ in the adiabatic phase is then given approximately as $\beta =\tfrac{-3\alpha +7}{2}$, where α is the spectral index. From Figure 22, we also observe that, as expected, the type Ia events could have overall lower surface brightness and smaller diameters than the known CC population. However, no definite conclusion can be drawn as there are large population of yet unknown SN types.

In addition to dependence on SNR size, the radio surface brightness of an SNR might also depend on the properties of the ambient medium into which the SNR expands. Arbutina & Urošević (2005) argued that based on the progenitor type, SNRs are likely to be associated with an ambient medium of different densities. Consequently, depending on the progenitor type (ambient density), SNRs should form a broad band corresponding to evolutionary tracks in the ${\rm{\Sigma }}\mbox{--}D$ plane. The dependence of ${\rm{\Sigma }}\mbox{--}D$ evolution on ambient density is further explored in Kostić et al. (2016). Under the assumption that SNRs expanding in an ambient medium of higher density have higher Σ, they argued in favor of the dependence of the slope of the ${\rm{\Sigma }}\mbox{--}D$ relation on the fractal properties of the ISM density distribution in the areas crowded with molecular clouds. These clouds are denser than the surrounding ISM and therefore SNRs emit more synchrotron radiation while expanding within it. After reaching the edge of the cloud, SNR evolution continues outside the cloud, and it emits less radiation due to the lower density of the ISM. This effect might result in different slopes of the ${\rm{\Sigma }}\mbox{--}D$ relation.

To further analyze the density dependence, the PDF of the data in the ${\rm{\Sigma }}\mbox{--}D$ plane should be obtained. As described in Vukotić et al. (2014) this has many advantages compared to the standard fit parameter-based analysis since all the information from the data sample is preserved and not just projected onto the parameters of the fit line. We calculated the ${\rm{\Sigma }}\mbox{--}D$ PDF using the kernel density smoothing described in Section 4.3. To test if there are statistically significant data density features in the ${\rm{\Sigma }}\mbox{--}D$ plane, in relation to the ${\rm{\Sigma }}(\rho )$ dependence, 2D kernel smoothing was performed on the ${\rm{\Sigma }}\mbox{--}D$ LMC and SMC data sample (containing 40 + 19 = 59 SNRs; SN 1987A not included in this sample). One hundred smooth bootstrap resamplings were applied in each step of the iterative procedure initialized with ${h}_{0}^{\mathrm{log}D}=0.129$ and ${h}_{0}^{\mathrm{log}{\rm{\Sigma }}}=0.389$. The ranges over which the BIMSE was calculated were ${h}^{\mathrm{log}D}=[0.01,0.3]$ and ${h}^{\mathrm{log}{\rm{\Sigma }}}=[0.1,0.7]$, with steps of 0.01 in both dimensions. We obtained optimal smoothing bandwidths at ${h}^{\mathrm{log}D}=0.12$ and ${h}^{\mathrm{log}{\rm{\Sigma }}}=0.34$. The resulting ${f}_{{h}^{\mathrm{log}D}{h}^{\mathrm{log}{\rm{\Sigma }}}}(\mathrm{log}D,\mathrm{log}{\rm{\Sigma }})$ in Figure 20 was calculated on a regular 100 × 100 grid mapped on $\mathrm{log}D=(0.5,2.5)$ and $\mathrm{log}{\rm{\Sigma }}\ =(-22.5,-17.5)$ ranges.

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From the given contour plot (Figure 20), it is evident that there are no ${\rm{\Sigma }}\mbox{--}D$ data groupings in parallel tracks that are emergent, or any other features possibly indicative of SNRs expanding out of molecular clouds or outgrowing the relevant density scale of the molecular clouds (as analyzed in Kostić et al. 2016). Further analysis and theoretical work are required on this matter in addition to thorough surveys.