The Core Mass Function across Galactic Environments. III. Massive Protoclusters

We created individual raw, flux-corrected, and number-corrected "true" CMFs for each clump. We then combined all sources together to create final CMFs. All three final CMFs are shown in Figure 3, with the raw CMF in black, the flux-corrected CMF in blue, and the true CMF in red. Masses were binned using the same methods as Papers I and II, with all bins equal in size in ${\rm{\Delta }}{{\rm{log}}}_{10}M$ with five bins per dex, and a bin centered on 1 M_⊙. We assumed N^1/2 Poisson counting uncertainties for each bin. Uncertainties on the final, "true," CMF are the same fractional size as those of the flux-corrected CMF's bins. However, note that these underestimate the true uncertainties, as they do not allow for additional uncertainty in f_num.

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From inspection of Figure 3 we see that the raw CMF exhibits a broad peak between about 2 and 5 M_⊙, while for the flux-corrected CMF this peak is wider, extending from about 2 to 8 M_⊙. The number correction yields a significantly increased population of low-mass (≲1 M_⊙) cores in the true CMF.

We fit a power law of the form of Equation (1) to the binned raw, flux-corrected, and true CMFs. First, we follow the fiducial weighted least-squares (WLS) method of Papers I and II, which involves minimizing differences in the log of ${dN}/d\mathrm{log}M$, normalized using the average of the asymmetric Poisson errors. For the one empty bin at the high-mass end, we again follow Papers I and II and treat it as an effective upper limit by assuming that the point is a factor of 10 lower than the level if the bin had one data point and set the error bar such that it reaches up to the level if there were one data point in the bin. Note that the results are insensitive to these details of how the empty bin is treated.

The fiducial method of Papers I and II was to fit the CMF over a standard range from the mass bin centered at 1 M_⊙ (i.e., for cores down to a minimum mass ${M}_{\min }=0.79\ {M}_{\odot }$) up to the highest-mass bin that is populated. These fits are shown in Figure 3 (left) and listed in Table 2. We see that the values of the power-law index, α, are relatively low, e.g., α = 0.50 ± 0.07 for the true CMF, compared to the Salpeter index of 1.35. We also note that a single power-law fit over this mass range is not a very accurate description of the CMFs, which is indicated by the fits having very low p values (<0.001).

Table 2. Best-fit Power Laws

CMF	Type	${M}_{\min }$	α	Δα	p
Raw	WLS	0.79*	0.44	0.07	<.001
⋯	MLE	0.79*	0.55	0.04	<.001
⋯	MLE-B	0.79*	0.67	0.06	<.001
⋯	MLE	3.6	0.95	0.09	.15
Flux-corrected	WLS	0.79*	0.30	0.06	<.001
⋯	MLE	0.79*	0.48	0.03	<.001
⋯	MLE-B	0.79*	0.56	0.05	<.001
⋯	MLE	6.0	1.04	0.11	.11
True	WLS	0.79*	0.50	0.07	<.001
⋯	MLE-B	0.79*	0.60	0.05	<.001
⋯	MLE-B	5.0	0.94	0.08	.12

Note. ${M}_{\min }$marked with a * were assigned, rather than being found via the fitting process. The MLE type is a fit to the unbinned sample of cores, while MLE-B type is a fit to a binned histogram of the data.

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We also consider maximum likelihood estimation (MLE) methods for fitting a power law to the CMF, which have been shown to outperform least-squares-based methods (Clark et al. 1999; White et al. 2008). In particular, Clauset et al. (2009) proposed methods to reliably fit power laws to distributions that are not binned into histograms through direct numerical maximization of the likelihood and this is the method we implement here for fits to the raw and flux-corrected CMFs. We note that several other CMF studies have also adopted aspects of these methods (e.g., Swift & Beaumont 2010; Olmi et al. 2013; Lu et al. 2020).

We assume a power-law form of $f(M)={{CM}}^{-(\alpha +1)}\,={(\alpha /{M}_{\min })(M/{M}_{\min })}^{-(\alpha +1)}$. To estimate α for an unbinned sample of cores above some minimum mass M_min we apply an MLE (Newman 2005) in which

$$\begin{eqnarray}&&\alpha =n{\left[\sum _{i=1}^{n}\mathrm{ln}\displaystyle \frac{{M}_{i}}{{M}_{\min }}\right]}^{-1},\end{eqnarray} \tag{ 4 }$$

where M_i are the masses of the cores that have masses $\geqslant {M}_{\min }$ and n is the number of cores in the sample selected for the fit. We first adopt M_min = 0.79 M_⊙ (corresponding to the bin centered at 1 M_⊙). The standard error on α is ${\rm{\Delta }}\alpha =\alpha /\sqrt{n}$. We only fit to the raw and flux-corrected CMFs using this method, as the number-corrected CMF is defined by bin only. We also use a second MLE method (MLE-B) to fit to binned data, i.e., fitting power laws directly to the histograms of the raw, flux-corrected, and true CMFs. This method is based on that proposed by Virkar & Clauset (2014), with a log-likelihood function above some minimum bin M_min of the form

$$\begin{eqnarray}&&{ \mathcal L }=n\ \alpha \ \mathrm{ln}\ {M}_{\min }+\sum _{i=\min }^{k}{h}_{i}\ \mathrm{ln}\ [{M}_{i}^{-\alpha }-{M}_{i+1}^{-\alpha }].\end{eqnarray} \tag{ 5 }$$

The results of employing these MLE and MLE-B fitting methods with M_min =0.79 M_⊙ are presented in Table 2. We derived moderately steeper power-law indices than those found through WLS-based fitting, e.g., for the true CMF we derived α = 0.60 ± 0.05 (via MLE-B), while the WLS method found α = 0.50 ± 0.07. As illustrated for the raw and flux-corrected CMFs, the differences in the results of MLE and MLE-B fitting are at a similar level, with the latter yielding slightly steeper power-law indices.

We next consider single power-law fits to the CMF, but with the minimum mass of the fitting range, M_min, allowed to vary to obtain the "best" description of the data, as described below. This analysis is important as a first step in identifying potential break points in the CMF, which may have a physical origin that is important for setting the characteristic masses of cores and, eventually, stars.

We first estimated the optimal M_min for each CMF through fitting power laws beginning at every unique core mass (or mass bin) within the sample and calculating the Kolmogorov–Smirnov (K-S) distance D between each cumulative distribution and their corresponding fit:

$$\begin{eqnarray}&&D=\max \left|F(M)-{F}_{n}(M)\right|,\end{eqnarray} \tag{ 6 }$$

where F(M) is the cumulative distribution function (CDF) of the power law model and F_n (M) is the CDF of the sample. Identifying the mass M_min that minimizes a distance-based statistic provides an estimate of where power-law behavior is most likely to begin (e.g., Reiss & Thomas 2007; Clauset et al. 2009).

However, the K-S statistic is not especially sensitive to differences in the tails of distributions and it performs best when the compared distributions diverge near their medians. This presents a particular problem when assessing asymmetrical distributions like power laws. As an alternative distance metric that is more sensitive to differences in the tails of distributions, we also consider the Anderson–Darling (A-D) statistic A²:

$$\begin{eqnarray}&&{A}^{2}=n{\int }_{-\infty }^{\infty }\displaystyle \frac{{\left({F}_{n}(M)-F(M)\right)}^{2}}{F(M)(1-F(M))}{dF}(M),\end{eqnarray} \tag{ 7 }$$

where n is the number of cores or bins included in the fit. Other distance metrics such as Pearson's χ² cumulative test statistic are also potentially applicable for this goal, but have been found to be less powerful than K-S- or A-D-based approaches (Virkar & Clauset 2014).

We show the values of D and A² as a function of M_min for the raw, flux-corrected, and true CMFs in Figure 4. There is a broad minimum of D values between about 3 and 15 M_⊙ for the raw and flux-corrected CMFs, with absolute minima at ∼7 and ∼12 M_⊙, respectively. The true CMF shows a broad minimum between 0.2 and 50 M_⊙, with an absolute minimum of D at 12.5 M_⊙. In contrast, A² values decrease as M_min increases, and then plateau to near-constant minimum values above ∼3 M_⊙ for the raw and flux-corrected CMFs. Values of A² for the true CMF remain roughly constant from about 8 to 80 M_⊙. The absolute minimum values of A² for the raw, flux-corrected, and true CMFs occur at 7, 8, and 50 M_⊙.

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These profiles of D and A², which have broad ranges of M_min where the values are at a similar low level, indicate that a single power-law description of the CMF begins to become reasonable somewhere around M_min ∼ 3 M_⊙. To quantify further, we calculated the goodness of fit of a power law as a function of M_min. However, typical hypothesis testing is not applicable here, since if the parameters of a model are derived from the data to which the model is then compared, then the standard critical values used to generate p-values from the K-S or A-D statistics are no longer valid because the tests' requirements of independence are not met (Lilliefors 1967). To circumvent this issue, correct p-values can instead be estimated via bootstrap resampling (Babu & Rao 2004; Clauset et al. 2009).

To carry out this analysis, we first fit a power-law to the observed CMF using the fiducial MLE method, but with a fixed M_min. We recorded the values of the K-S statistic, D, and A-D statistic, A², and counted how many cores were below M_min, n_low, and how many were above, n_high. Next, we generated a synthetic CMF through a two-part, semi-parametric method based on the procedure proposed by Clauset et al. (2009). To create the low-mass end of the synthetic CMF ($M\lt {M}_{\min }$), we randomly selected n_low cores from the pool of the n_low real cores. Note, this selection was done with replacement, i.e., any core could be chosen more than once. This ensured that the low-mass end of the synthetic sample would follow roughly the same distribution as the real CMF (which is not guaranteed to follow a power law). To create the high-mass end of the synthetic CMF ($M\gt {M}_{\min }$), we generated n_high synthetic cores more massive than M_min that were distributed according to the power-law fit to the observed CMF. This ensured that the high-mass end of the synthetic sample is guaranteed to follow a power law (although apparent deviations can arise due to random sampling of a finite population). We then combined the low-mass and high-mass samples into a full synthetic CMF covering the entire mass range of the observed CMF.

We note that to create a synthetic CMF for the true CMF, which by definition does not possess any information about the distribution of cores within each mass bin, we generated the corresponding number of masses uniformly distributed across each mass bin. We then performed the sampling procedure described above and re-binned the resulting synthetic sample according to the binning scheme originally used to create the binned CMFs described in Section 3.2.1.

Once a given synthetic sample was created, we fitted a power law to it using the fiducial MLE method, including fitting an optimal M_min. By creating a sample which is guaranteed to follow a power law above a certain mass, but fitting a power law without knowledge of what that mass was, we replicate the conditions inherent to our fitting process for the real CMFs. Due to the random fluctuations in the shapes of the synthetic CMFs this allows their fits to yield ${M}_{\min }{\rm{s}}$ that are greater or less than the M_min actually used to create that CMF. This preserves the integrity of the goodness-of-fit calculation. We then repeated this procedure N = 3000 times, i.e., with 3000 different synthetic CMFs for each M_min, recording the value of $\hat{D}$ or ${\hat{A}}^{2}$ resulting from the power-law fit to each synthetic CMF.

We calculated the fraction of synthetic values of $\hat{D}$ or ${\hat{A}}^{2}$ that were larger than the original D or A² fit to the real CMF and we estimated a p-value via $p=N(\hat{D}\geqslant D)/N$ and $p=N({\hat{A}}^{2}\geqslant {A}^{2})/N$. Since these statistics are meant to be minimized, this represents the percentage of synthetic fits which, although known to actually follow a power law above M_min, resulted in a "worse" fit, i.e., due to random sampling effects. This percentage is equivalent to the p-value for the fit to the real CMF; if many synthetic CMFs randomly appear to be a worse fit, it indicates that the real CMF is likely to be well described by said power law. Alternatively, if very few of the synthetic CMFs (e.g., <10%, i.e., p < 0.1) result in fits which are worse than the fit to the real CMF, this indicates that the real CMF is unlikely to be described by that distribution.

For the raw and flux-corrected CMFs, we performed this bootstrapping procedure for 100 values of M_min evenly distributed between 0.1 and 10 M_⊙. For the true CMF, we performed this procedure for 11 mass bins between 0.1 and 10 M_⊙. We note that the power of this test decreases as M_min is raised and n_high decreases, which effectively increases uncertainties on individual p-values at higher M_min.

Figures 4 (top right) and (bottom right) show the p-values of power-law fits to the CMFs as a function of M_min for the D and A² statistics, respectively. The black and blue curves, corresponding to the raw and flux-corrected CMFs, respectively, were smoothed using LOWESS regression (Locally Weighted Scatter-plot Smoothing; Cleveland 1979) with a smoothing parameter of 0.05 determined by cross-validation. This method fits polynomials to many small subsets of the data through WLS regression to reduce noise in the data and make visual assessment of trends easier.

We find that a power law becomes a consistently plausible fit (p ≥ 0.1) to the raw CMF above ∼3 M_⊙ as assessed by the K-S statistic and above ∼3.6 M_⊙ as measured by the A-D statistic, and is not adequate at describing the CMF below these masses. A power law becomes appropriate for the flux-corrected CMF above ∼6 M_⊙ as indicated by both the K-S and A-D statistics. We find no significant evidence that power-law behavior holds below this range. A power law becomes an adequate descriptor of the histogram of the true CMF above ∼5 M_⊙ as assessed by the K-S statistic and also above ∼5 M_⊙ as measured by the A-D statistic.

Considering the various results derived from the K-S- and A-D-based methods, we report final "best" fits that begin at the lowest significant (p ≥ 0.1) M_min value as assessed by A². We prefer the A-D statistic due to the known issue of the K-S statistic being insensitive to differences in the tails of compared distributions. The single power-law best fit to the raw CMF is then α = 0.95 ± 0.09 with M_min ∼ 3.6 M_⊙ (p = .15). The best fit to the flux-corrected CMF is α = 1.04 ± 0.11 above M_min ∼ 6.0 M_⊙ (p = .11). The best fit to the true CMF is α = 0.94 ± 0.08 above the bin with a lower limit of M_min ∼ 5.0 M_⊙ (p = .12). These results are presented in Table 2, and in Figure 3 (right).

The results of the last section already indicate that a single power-law description of the true CMF is only valid at the high-mass end above about 5 M_⊙. Here we compare the goodness of fit of single and multi-component continuous power laws, i.e., piecewise power laws (PPLs), to the CMF to further investigate if a break in the single power law is present and what are the best PPL descriptions. PPL descriptions of the CMF and IMF have been made previously in a number of studies (e.g., Johnstone et al. 2000; Kroupa 2001; Reid & Wilson 2006).

For the mass ranges defined by minimum-mass bins M_min centered from 1 and 16 M_⊙ we fit a single power law, a double PPL, and a triple PPL by performing non-linear WLS regression. The double PPL can be expressed as

$$\begin{eqnarray}p(M)=\left\{\begin{array}{ll}{{AM}}^{-{\alpha }_{1}} & M\lt {M}_{1}\\ A({M}_{1}^{-{\alpha }_{1}+{\alpha }_{2}}){M}^{-{\alpha }_{2}} & \mathrm{otherwise}\end{array}\right.\end{eqnarray} \tag{ 8 }$$

and the triple PPL as

$$\begin{eqnarray}p(M)=\left\{\begin{array}{ll}{{AM}}^{-{\alpha }_{1}} & M\lt {M}_{1}\\ A({M}_{1}^{-{\alpha }_{1}+{\alpha }_{2}}){M}^{-{\alpha }_{2}} & {M}_{1}\leqslant M\leqslant {M}_{2}\\ A({M}_{1}^{-{\alpha }_{1}+{\alpha }_{2}})({M}_{2}^{-{\alpha }_{2}+{\alpha }_{3}}){M}^{-{\alpha }_{3}} & {M}_{2}\lt M\end{array}\right..\end{eqnarray} \tag{ 9 }$$

The different PPL fits resulting from this method for ${M}_{\min }=0.79\ {M}_{\odot }$ are shown in Figure 5. Note, all fits include the entirety of the high-mass end of the CMF up to the highest-mass core M = 252 M_⊙, with the empty bin at 125 M_⊙ being assigned a count as in the original WLS fitting in Section 3.2.2.

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The Akaike Information Criterion (AIC; Akaike 1973) can be used to choose a best-fit model for a given data set by measuring the "distance" between the observed and modeled distributions. For WLS estimation, this takes the form

$$\begin{eqnarray}\begin{array}{rcl}{\mathrm{AIC}}_{\mathrm{WLS}} & = & N\ \mathrm{ln}\left(\displaystyle \frac{\sum _{j=1}^{N}{w}_{j}^{-2}{\left({y}_{i}-f({t}_{j},{q}_{\mathrm{WLS}})\right)}^{2}}{N}\right)\\ & & +2({\kappa }_{q}+1),\end{array}\end{eqnarray} \tag{ 10 }$$

where N is the number of bins being fit and κ_q is the number of parameters being fit (Banks & Joyner 2017). For a single, double, and triple PPL κ_q = 2, 4, and 6, respectively. N ranges between 7 and 13 depending on the ${M}_{\min }$ selected. However, for sample sizes N that are small in comparison to κ_q , as occurs here, the AIC is known to perform poorly. The corrected AIC_c (Sugiura 1978) accounts for this effect by adding an additional penalty term to the AIC formula:

$$\begin{eqnarray}&&{\mathrm{AIC}}_{\mathrm{WLS},{\rm{c}}}={\mathrm{AIC}}_{\mathrm{WLS}}+\displaystyle \frac{2({\kappa }_{q}+1)({\kappa }_{q}+2)}{N-{\kappa }_{q}}.\end{eqnarray} \tag{ 11 }$$

We calculate the single, double, and triple PPL fit AIC_WLS,c values as a function of ${M}_{\min }$ for each CMF. This allows for analysis of the relative likelihood of these forms as a function of ${M}_{\min }$. We define the AIC differences Δ_i (AIC) (Wagenmakers & Farrell 2004) as

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{i}(\mathrm{AIC})={\mathrm{AIC}}_{i}-{\mathrm{AIC}}_{\min },\end{eqnarray} \tag{ 12 }$$

where ${\mathrm{AIC}}_{\min }$ is the minimum AIC_WLS,c of the three fits. The likelihood of a model is proportional to exp $(-\tfrac{1}{2}{{\rm{\Delta }}}_{i})$, which allows for the calculation of Akaike weights w_i (AIC),

$$\begin{eqnarray}&&{w}_{i}(\mathrm{AIC})=\displaystyle \frac{\exp \left(-\tfrac{1}{2}{{\rm{\Delta }}}_{i}\right)}{\sum _{k=1}^{K}\exp \left(-\tfrac{1}{2}{{\rm{\Delta }}}_{k}\right)},\end{eqnarray} \tag{ 13 }$$

which are the normalized relative likelihoods of that model and where K is the number of candidate models. This is equivalent to the probability that each model is the "best" model. The w_i (AIC) values for all model types fit to the three CMFs as a function of ${M}_{\min }$ are presented in Figure 6.

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We find that for low values of ${M}_{\min }\sim 1\ {M}_{\odot }$ all CMFs prefer a double PPL. However, they eventually transition to strongly favor a single power law as M_min is raised. This turnover occurs at about 4 M_⊙ for the raw CMF, 5 M_⊙ for the flux-corrected CMF, and also 5 M_⊙ for the true CMF. A triple PPL is never indicated as being more appropriate than the other forms.

When fitting over the "standard" range of the CMF with M_min = 0.79 M_⊙, the best double PPL fit to the raw CMF is where M₁ = 4 M_⊙, α₁ = −0.43 ± 0.13, and α₂ = 0.85 ± 0.07. The flux-corrected CMF fitting results are similar. For the true CMF the best-fit results are M₁ = 15 M_⊙, α₁ = 0.11 ± 0.12, and α₂ = 1.29 ± 0.26. Note, the break mass bins M₁ for the raw and flux-corrected CMFs are identical to the bins selected by fitting a single power law to the binned CMF through performing an MLE and minimizing A². This reinforces that a break in the CMF exists in this mass range. The break mass for the true CMF is higher due to the effects of number corrections inflating the number of cores in the lower-mass bins near 1 M_⊙.

We conclude that this consistent transition from a double to single power law as M_min increases indicates the presence of a break in the CMF between about 4 and 15 M_⊙, depending on the overall range of the CMF that is fit and being moderately sensitive to whether number corrections are applied to the CMFs. These results are consistent with those derived from the K-S- and A-D-based methods, i.e., which also support a break in power-law behavior of the CMF around 4 to 6 M_⊙.

We next consider log-normal fits to the CMF. Such log-normal descriptions of the IMF and/or CMF have been discussed by, e.g., Larson (1973), Zinnecker (1984), and Adams & Fatuzzo (1996). The log-normal probability density function takes the form

$$\begin{eqnarray}&&p(M)=\displaystyle \frac{1}{\sigma M\sqrt{2\pi }}\exp \left(-\displaystyle \frac{{\left(\mathrm{ln}\ M-\mu \right)}^{2}}{2{\sigma }^{2}}\right),\end{eqnarray} \tag{ 14 }$$

where $\mu =\mathrm{ln}\bar{M}$ with $\bar{M}$ being the mean mass and σ² is the variance in ln M, i.e., σ describes the width of the distribution in ln M.

Chabrier (2003) found σ ∼ 0.7 for the IMF. This was later revised to σ ∼ 0.55 (Chabrier 2005). Subsequently, Bochanski et al. (2010) derived σ ∼ 0.3 for the IMF. Previous studies of the CMF have derived values of σ between 0.6 and 2 (see, e.g., Reid & Wilson 2006; Swift & Beaumont 2010; Olmi et al. 2013; Könyves et al. 2015).

Fitting only for M > 0.79 M_⊙, we derive the log-normal fits shown in Figure 7 (left). The results of fitting using the next-higher-mass bin as the minimum are shown in Figure 7 (right). We find that the global log-normal fit is quite sensitive to the number corrections applied to transform the flux-corrected CMF into the "true" CMF. These corrections become relatively large for the bin centered at 1 M_⊙ and so the fit parameters are also quite sensitive to whether this bin is used in the fitting. For example, the mean core mass is $\bar{M}=0.90\ {M}_{\odot }$ when the fit is carried out down to this bin and $\bar{M}=0.55\ {M}_{\odot }$ (and with a somewhat broader log-normal) when going down to the next-higher-mass bin.

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We explored the goodness of fit of log-normal distributions and single power laws by calculating the ratio R of their log-likelihoods (Vuong 1989; Clauset et al. 2009):

$$\begin{eqnarray}&&R=\sum _{i=1}^{n}[\mathrm{ln}\ {p}_{\mathrm{pl}}({M}_{i})-\mathrm{ln}\ {p}_{\mathrm{ln}}({M}_{i})].\end{eqnarray} \tag{ 15 }$$

The significance levels of this test were calculated through the powerlaw Python package (Alstott et al. 2014). We report values of R normalized by its standard deviation, $R/({\sigma }_{R}\sqrt{n})$. To assess the binned CMFs, we generated the corresponding number of random masses uniformly distributed in each mass bin and calculated the median R over 2000 iterations of the same process. In our implementation of this test, positive values of R indicate that a power law is preferred, while negative values indicate that a log-normal distribution is preferred. p-values larger than p = 0.1 indicate that there is no significant difference between the goodness of fit of the two distributions.

For ${M}_{\min }=0.79\ {M}_{\odot }$, a log-normal distribution is a significantly better fit to the raw and flux-corrected CMFs than a single power law (R = −4.6, p < 0.001 for the raw CMF; R = −5.7, p < 0.001 for the flux-corrected CMF). However, there was no significant evidence that a log-normal was a better descriptor of the true CMF than a single power law (R = −1.2, p = 0.25). When fitting with the next bin in mass as the minimum, i.e., M_min = 1.26 M_⊙, a log-normal distribution remained more appropriate than a single power law for the raw and flux-corrected CMFs (R = −3.2, p = 0.002 for the raw CMF; R = −5.4, p < 0.001 for the flux-corrected CMF). However, there continued to be no significant evidence that a log-normal fit was more appropriate for the true CMF than a single power law (R = −1.03, p = 0.30).

We also compared a log-normal fit to PPLs using the methods of Section 3.2.3. We calculated the AIC_c values and Akaike weights w_i (AIC) for a log-normal fit and for single, double, and triple power-law fits to the CMF. With ${M}_{\min }=0.79\ {M}_{\odot }$, the raw and flux-corrected CMFs were both best described by a log-normal fit, with w_i (AIC)_ln = 0.97 and w_i (AIC)_ln = 0.99. However, for the true CMF a log-normal fit was not the best description (w_i (AIC)${}_{\mathrm{ln}}=0.03$) and a double PPL was preferred (w_i (AIC)_double = 0.85). Above ${M}_{\min }\,=1.26\ {M}_{\odot }$, a PPL was still preferred for the true CMF (w_i (AIC)_double = 0.97).

In summary, these results indicate that a single log-normal is not a better fit of the true CMF compared to a double PPL, at least over the range of masses explored in this study, down to about 1 M_⊙. We do note, however, that single log-normals are relatively good fits to both the raw and flux-corrected CMFs, so this result for the true CMF is dependent on the accuracy of our estimated incompleteness corrections. Other models, such as hybrid models with a log-normal at low masses plus a high-mass power-law tail are likely to be possible good descriptions of the true CMF. However, to advance further a CMF measurement with larger numbers of cores (to reduce Poisson errors) and probing to lower masses (to more accurately define the expected peak) are needed.

We varied the parameters used to identify cores with the dendrogram to test the effects on our results. Our fiducial method is with a minimum intensity F_min = 4σ, a minimum increase in intensity δ = 1σ, and a minimum area S_min = 0.5 of the size of the synthesized beam. We kept δ constant and tested combinations of F_min = 3, 4, or 5σ and S_min = 0.5 beam or 1 beam. We found a minimum of 128 cores with F_min = 5σ and S_min = 1 beam, and a maximum of 293 cores with F_min = 3σ and S_min = 0.5 beam. For the true CMF, M_min typically ranged between 3 and 8 M_⊙ with values of α between 0.94 and 1.10. This shows that our results using the fiducial method are not very sensitive to the choice of dendrogram parameters.

Fitting to histograms of the CMF can create a bias in the derived values of α through the information loss created by binning. The fiducial binning method involves five bins per dex. We varied the number of bins per dex and fit a power law to the resultant histogram using the fiducial binned MLE method (MLE-B). As expected, we found that the α derived for each histogram converged on the values derived from fitting to the unbinned sample of cores as the number of bins per dex increased. To be consistent with the methods of Papers I and II we report values derived from fitting to histograms with five bins per dex, but note the potential bias in these binned MLE based results. The results shown in Table 2 comparing MLE and MLE-B results indicate that these biases are present at a level corresponding to Δα ≃ 0.1, which is about the uncertainty level caused by Poisson sampling uncertainties of the CMFs.

We also examined the variation in the CMF as a function of clump distance. Clumps distances ranged between 0.79 kpc and 3.5 kpc. Figure 8 compares the CMFs of the 15 clumps (containing 127 cores) within 3 kpc and the 13 clumps (containing 95 cores) farther than 3 kpc. Unsurprisingly, we are able to probe to lower-mass cores in the near sample. The far sample contains a handful of cores that are very close to the noise limit, subject to very large number correction factors ≳20, and that are mainly responsible for the dramatically boosted "true" CMF at ≲1 M_⊙ in the far sample. This gives us further reason to be cautious about the reliability of the derived CMF in this regime. Conversely, the shape of the CMF of the near sample, which is subject to smaller number corrections in the ∼1 M_⊙ regime, strengthens our confidence in the result that a break is seen in the CMF at masses ≳5 M_⊙.

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We performed K-S tests comparing the raw and flux-corrected CMFs of the near and far clumps, and found no significant evidence that the samples are drawn from different populations (p = 0.09 and p = 0.07, respectively). We did find significant differences between the true CMFs (p < 0.001), which is likely the result of the increased effect of number corrections in the far sample.

We fit single power laws to the near and far CMFs using the fiducial MLE fitting process and found good agreement with the fits to the complete CMFs. The fitted values of α are largely within the estimated errors of α in the original fits. The values of M_min ranged between ∼2–4 M_⊙ for the near sample and ∼3–5 M_⊙ for the far sample. Thus, note that both samples independently show evidence for a break in the CMF via this method.