SUBSTELLAR OBJECTS IN NEARBY YOUNG CLUSTERS. VII. THE SUBSTELLAR MASS FUNCTION REVISITED

We use the Spitzer-selected sample of young stellar/substellar objects presented by Gutermuth et al. (2009) which includes our two target regions. For each of the regions, a region of 25' × 25' centered on the core of the cluster was observed. The selection in Gutermuth et al. (2009) is based on colors and magnitudes in Spitzer and Two Micron All Sky Survey (2MASS) bands from 1 to 24 μm. Their multi-color selection process uses a series of criteria designed to exclude background stars, non-stellar emission features and extragalactic objects. According to their analysis, this process yields only minimal contamination (a few percent). An earlier version of this process was used in Gutermuth et al. (2008) in a survey of NGC 1333. Assuming that the distances to the clusters are similar, the identical selection procedure also ensures that the depth and completeness in terms of magnitudes in the two samples is comparable. This removes a major obstacle for an accurate comparison of the BD abundance.

Since the primary selection criterion is excess emission in the infrared, this sample only contains objects with emission from circumstellar material, i.e., either disks or envelopes. It does not include disk less young stars and BDs (Class III objects). Therefore, to infer star/BD ratios, we have to make the assumption that the fraction of objects with disks does not change with object mass in the low-mass regime. As recently shown in Dawson et al. (2013), this is a plausible assumption for many star-forming regions, including IC348. For NGC 1333, there might be a slight mass dependence, as the disk fraction in the total Spitzer-selected sample is found to be 83% ± 11% (Gutermuth et al. 2008), whereas the value for the very low mass objects is only 55%–66% (SONYC-IV). This indicates that the star/BD ratio in this cluster could be slightly overestimated.

The entire Perseus cloud, including the two target clusters have also been observed as part of the Spitzer Legacy program "From Cores to Disks" (C2D, PI: N. Evans). Their Perseus YSO catalog derived from IRAC photometry is discussed in detail in Jørgensen et al. (2006). We prefer to use the Gutermuth et al. (2009) selection, because it is slightly deeper, which is beneficial for our purposes. The downside of the Gutermuth et al. (2009) sample is the limited spatial coverage. Here the C2D catalog is useful to check the spatial completeness of our samples.

In Figure 1 we show the spatial distribution of the selected objects in NGC 1333 and IC348. With red squares we plot the samples from Gutermuth et al. (2009), with black crosses the C2D sample that covers the entire Perseus region. The figure shows that the number of YSO candidates outside the region covered by Gutermuth et al. (2009) is small compared with the total population. This is particularly true for NGC 1333 which shows a very compact profile and can be considered to be spatially complete (see also SONYC-IV). For IC348, there is a tail of a YSO population toward the south-west, which represents the transition region to another densely populated area in the Perseus star-forming region (Cambrésy et al. 2006). In addition, there are about 10 objects outside the coverage of the Gutermuth et al. (2009) survey. However, there is also a large number of additional YSO candidates only contained in the C2D sample within the cluster core. We checked the objects only in C2D and found that they do not show an obvious magnitude or extinction bias with respect to the sample we are using, thus, even in case we are missing members in the outskirts of the cluster, this is not going to affect our analysis in any significant way. While Muench et al. (2003) do find a difference in the IMF between the core and the halo in IC348, both regions (in their definitions) are within the survey area of Gutermuth et al. (2009). We conclude that the samples we are using are not affected by a spatial bias.

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Figure 1 also illustrates one major difference between IC348 and NGC 1333. The core of IC348 has about twice the diameter of the core of NGC 1333 (2.1 versus 1.2 pc, Gutermuth et al. 2009), i.e., the cluster volume in IC348 is about 8 times larger than in NGC 1333. On the other hand, IC348 has only about 20% more YSOs than NGC 1333, according to the Spitzer surveys (Jørgensen et al. 2008; Gutermuth et al. 2009). The fraction of diskless Class III objects is higher in IC348, taken into account that the total the YSO population in IC348 could be up to twice as large as in NGC 1333. This still implies that the object density in NGC 1333 is 4–7 times higher than in NGC 1333. Given the age, size, and number of members in these clusters, this difference is likely to be primordial and not caused by dynamical evolution (see Figure 1 in Gieles et al. 2012). Hence, these two clusters constitute an excellent test case to probe the effects of dynamical interactions and cluster potential on the formation of BDs.

For the overwhelming majority of young objects, masses can only be estimated indirectly by comparing an observed quantity with predictions from theoretical isochrones for a given age. For the observed quantity, there are two options, either the effective temperature or the luminosity (or a photometric proxy). The luminosity has the problem that model derivations are sensitive to the age for pre-main sequence objects that are still contracting. In addition, measurements can be affected by extinction as well as excess emission from disk and/or accretion. The effective temperature is problematic for other reasons; it depends on atmosphere models and can be altered by magnetic activity (Stassun et al. 2012). This can lead us to underestimate object masses by up to a factor of two.

For our chosen samples, accurate multi-band photometry is available, while the spectroscopic follow-up is not complete. Therefore, we will rely on photometry in the optical and near-infrared to estimate masses. We complement the 2MASS photometry provided by Gutermuth et al. (2009) with optical photometry from Luhman (1999) and Luhman et al. (2003) for IC348 (Landolt R and I band) and from SONYC-I for NGC 1333 (Sloan i and z band).⁷ For some objects without 2MASS near-infrared magnitudes, we were able to complement the dataset using the photometry from Muench et al. (2003) for IC348⁸ and SONYC-I for NGC 1333. In total, the samples contain 142 (for IC348) and 95 (for NGC 1333) objects with photometry in JHK, with smaller subsets of 86 (IC348) and 23 (NGC 1333) with additional optical magnitudes available.

For the objects with 2MASS photometry, we also obtain the errors as listed in the database (mostly between 0.03 and 0.05 mag). Similarly, the photometry from Muench et al. (2003) provides errors for all measurements. For the remaining photometry, errors for individual objects are not reported in the literature. For the optical magnitudes in IC348, we adopt a generic and conservative uncertainty of 0.1 mag. For the SONYC-I magnitudes in NGC 1333, we adopt errors of 0.1 mag for the optical bands and 0.05 mag for the near-infrared bands.

Based on the information given in the papers listed above, the error values adopted for the optical photometry should be typical for the samples. However, some objects might be affected by additional uncertainties introduced to calibration imperfections. Since the z and I bands are located at the long-wavelength edge of the sensitivity of the optical CCDs, they are highly susceptible to color terms in the calibration, which are difficult to measure with the usual photometric standard stars. This can introduce errors larger than 0.1 mag in individual sources, which cannot be quantified accurately. This issue is a particular problem for very red sources, since most of their optical flux is emitted in the part of the spectrum where the CCD sensitivity declines.

We derive masses using three different sets of isochrones from the Lyon group, BT-Settl (with AGSS2009 opacities), BT-Dusty (with AGSS2009 opacities), and BT-Nextgen (with GNC93 opacities).⁹ The latter two are updated versions from the standard AMES-Dusty and Nextgen models. The main difference among the three sets is the treatment of dust. In contrast to Nextgen, Dusty includes dust opacities. Settl includes a full dust cloud model. For more information on the isochrones, see Allard et al. (2011, 2001). Note that atmospheric dust becomes a major source of opacity for T_eff ≲ 2500 K (Helling et al. 2008), corresponding to M ≲ 0.02 M_☉ for young BDs, which is the low-mass limit in our analysis, thus the treatment of dust should not have a major effect on our results. The isochrones predict absolute magnitudes as a function of object mass in all photometric bands for which observations are available. They cover the range from 0.02 M_☉ or below to 1.4 M_☉ and are available for ages starting from 1 Myr, which is adequate for our purposes.

To compare the observed with the predicted magnitudes, we first shift the isochrones from absolute magnitude to the distance of our target regions, which enters here as a free parameter (see Section 2.3). We also re-bin the isochrones to a uniform stepsize of 0.01 M_☉, using a linear interpolation over a small portion of the isochrone. We then calculate a series of reddened isochrones for A_V = 1–20 mag in steps of 1 mag. For this step, the choice of the extinction law is important (see Section 2.3). The upper limit is chosen to be 20 mag, for two reasons. First, independent studies indicate that only a small fraction in the Perseus star-forming complex exceeds this extinction value (Lombardi et al. 2010). Second, beyond this value the samples are biased toward bright sources, thus, incompleteness becomes an issue.

After these preparations, the best fit for mass and A_V is determined with a χ² minimization. The number of degrees of freedom in this process is the number of photometric bands for which data is available (N = 3–5) minus the number of free parameters (mass and A_V, i.e., 2). For each object, we saved the combination of mass and A_V that results in the minimum value for χ², the corresponding reduced χ² ($\chi _r^2$, i.e., χ² divided by the number of degrees of freedom) and the number of available bands N. Objects with best fit value of A_V = 20 are discarded from the analysis—since this is the upper limit in our grid of isochrones, their mass estimate is not reliable. A typical example of the resulting $\chi _r^2$ plotted versus the best mass estimate is shown in Figure 2, left panel. This figure reveals that the procedure produces similar fitting results for high- and low-mass objects and for objects with and without optical photometry. We also show a typical A_V versus mass plot from this procedure (Figure 2, right panel). The upper limit in A_V is not changing significantly with mass, i.e., there is no evidence for an extinction bias in these samples (apart from the A_V < 20 cutoff).

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The distribution of reduced χ² can in principle be used to assess the goodness-of fit for our mass estimates. For a good fit, we expect $\chi _r^2$ to have an average of 1.0 and a standard deviation of ${\sim } \sqrt{2/N}$. However, as shown in Figure 2, our procedure yields significantly higher values in $\chi _r^2$. This indicates that either the model does not reflect the data well or that the errors are underestimated. In our case, the high values for $\chi _r^2$ are mostly explained by the fact that our model is discretely sampled in mass-A_V space. The stepsize in mass and A_V results in magnitude steps that are often larger than the typical photometric error. In addition, in some cases the errors of the optical photometry may be underestimated, see above. Therefore, we only use the procedure to select the best fit solution.

In our estimation of masses from photometry, the distance, age, as well as the extinction law, expressed in the quantity R_V = A_V/E_{(B − V)}, are considered free parameters. In addition, we have to choose the theoretical isochrone. What we call "scenarios" in the following are combinations of isochrone, distance, age, and extinction law for which we estimate masses. These scenarios have been chosen to cover the plausible range of these parameters and to give insight into the impact of the specific choice of a parameter or an isochrone on the mass estimates. In the following, we justify the choice of the range of the parameters.

For the extinction, we use the parameterized law by Cardelli et al. (1989), with R_V = 3.1, the canonical value used for the interstellar matter (ISM). This law yields extinction offsets that are consistent with the often-used extinction values published by Schlegel et al. (1998). In reality, R_V depends on the grain properties and is not the same for every line of sight; Cardelli et al. (1989) report values ranging from 2.6 to 5.6, with the overwhelming majority (22 out of 27 cases) below 4.5. We therefore use R_V = 4.5 as an alternative value to be able to assess the impact of the choice of R_V on the mass estimates.

The distances to the two clusters are not well constrained. The entire Perseus cloud is usually assumed to have an average distance of ∼300 pc, which we use as a default value. Based on the Hipparcos parallaxes for the early-type stars, de Zeeuw et al. (1999) estimate 318 ± 27 pc for the cloud. Based on a kinematical analysis of a much larger sample of A stars, Belikov et al. (2002) infer 300 pc (270–330 pc). However, there are indications that NGC 1333 is located at a shorter distance. Hirota et al. (2011) report a distance of 235 pc for NGC 1333 based on interferometry of the maser emission from a source that may be associated with the cluster. This is also consistent with an earlier photometric estimate of the distance of NGC 1333 (220 pc) by Cernis (1990). In addition, Lombardi et al. (2010) suggest, based on extinction map analysis, that the northern part of the Perseus region (where IC348 is located) is slightly more distant than the southern part (where NGC 1333 is located). To take this into account, we use a distance of 230 pc as an alternative value, noting that this is only a viable option for NGC 1333.

The ages that are typically quoted are 2–4 Myr for IC348 and 1–3 Myr for NGC 1333 (see Bally et al. 2008 and references therein). Judging from model-independent indicators of evolutionary state (fraction of objects with disks, fraction of objects in Class I stage, luminosity function), NGC 1333 is definitely younger than IC348, and both are clearly younger than star-forming regions with established ages of 5–10 Myr like Upper Scorpius and the TW Hydrae Association (e.g., Lada et al. 1996; Haisch et al. 2001; Gutermuth et al. 2008). In the context of our study, the relevant quantity is not the age of the cluster, but the average age of the objects contained in our samples, which may not include the youngest, embedded population because we require a near-infrared detection. In fact, we showed in SONYC-IV that most of the very low mass objects in NGC 1333 are consistent with an age of 1–5 Myr, based on their position in the Hertzsprung–Russell diagram (HRD). Therefore, we use a default value of 3 Myr, which is plausible for both clusters. To assess the influence of the age on the mass estimates, we additionally estimate masses for an age of 1 Myr.

To evaluate the impact of the choice of the parameters, we define six scenarios for which we estimate object masses. These scenarios are listed in Table 1. The default scenario 1 uses a distance of 300 pc, an age of 3 Myr, R_V of 3.1, and the BT-Settl model, which has the most recent opacities and the most advanced treatment of dust. In scenarios 2–4 we vary the cluster parameters. In scenario 2, we use the younger age of 1 Myr; in scenario 3, the alternative distance of 230 pc; and in scenario 4, the alternative value for R_V of 4.5. In scenarios 5 and 6, we switch to the Nextgen and DUSTY isochrones.

Table 1. Results from Mass Estimates for IC348 and NGC 1333

Cluster	No	D	t	Model	RV	R	Min–Maxa	Min–Maxb	α	$\overline{M}$
		(pc)	(Myr)
IC348	1	300	3	BT-Settl	3.1	3.6 (96/27)	2.8–4.4	2.6–4.1	0.72	0.27
IC348	2	300	1	BT-Settl	3.1	2.1 (88/42)	1.7–2.5	1.5–2.9	0.95	0.13
IC348	3	230	3	BT-Settl	3.1	1.9 (84/45)	1.5–2.2	1.3–2.1	1.00	0.16
IC348	4	300	3	BT-Settl	4.5	3.2 (94/29)	2.6–4.0	2.7–4.3	0.76	0.26
IC348	5	300	3	BT-Nextgen	3.1	4.0 (99/25)	3.1–4.9	3.4–5.5	0.67	0.19
IC348	6	300	3	BT-Dusty	3.1	2.9 (92/32)	2.3–3.5	2.4–3.4	0.81	0.22
N1333	1	300	3	BT-Settl	3.1	2.2 (43/20)	1.6–2.8	1.9–2.5	0.94	0.27
N1333	2	300	1	BT-Settl	3.1	2.1 (47/22)	1.6–2.7	2.0–2.5	0.94	0.13
N1333	3	230	3	BT-Settl	3.1	2.4 (47/20)	1.8–3.0	2.2–2.4	0.90	0.18
N1333	4	300	3	BT-Settl	4.5	2.0 (43/21)	1.6–2.6	1.9–2.5	0.96	0.30
N1333	5	300	3	BT-Nextgen	3.1	2.2 (44/20)	1.7–2.8	1.9–2.6	0.93	0.18
N1333	6	300	3	BT-Dusty	3.1	1.9 (42/22)	1.5–2.5	1.8–2.0	0.99	0.22

Notes. ^aStatistical uncertainties for R, only dependent on sample size. ^bMin–max range of R based on the number of objects close to the substellar limit, see Section 2.4.

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From the resulting mass distribution in a given scenario, we derive the cumulative distribution of object masses, i.e., the fraction of objects below a given mass, as a function of mass. These plots are shown in Figure 3. For each scenario, we determine the number of objects with 0.03 ⩽ M ⩽ 0.08 M_☉ (BDs) and with 0.08 < M < 1.0 M_☉ (stars) and calculate the star/BD ratio R. To be able to compare with the literature, we also calculate the slope of the mass function α for the power-law parameterization dN/dM∝M^−α, directly from the star/BD ratios, i.e., using two bins in mass, one for BDs and one for stars. In addition, we derive the median mass $\overline{M}$ for each scenario. The resulting parameters for the six scenarios are given in Table 1.

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An important part of this work is to evaluate the errors in the derived parameters. In our chosen experiment, five factors contribute to the uncertainties.

• 1.  
  Sample size. The part that is easiest to quantify is the statistical uncertainty, which is purely determined by the sample size. For this paper, we use the same approach as in SONYC-IV, which is based on the IDL scripts presented in Cameron (2011). In short, we calculate Bayesian confidence intervals from the beta distribution. We note that for large samples this procedure gives results that are very similar to binomial confidence intervals. The resulting values are listed in Table 1, Column 8. Typically, the sample size introduces an error in R of about ±0.3–0.9 for IC348 and ±0.5–0.6 for NGC 1333.
• 2.  
  Models. Masses are only defined in relation to evolutionary models, and at young ages the deficiencies of the available tracks are well documented (Baraffe et al. 2002; Wuchterl & Tscharnuter 2003). However, at the moment no tracks with self-consistent treatment of the collapse and infall are available to the community.¹⁰ Based on a dynamical mass estimate for the very low mass pre-main sequence object AB Dor C, which is older than our target regions, Close et al. (2005) claim that the existing mass-luminosity relations underestimate masses by a factor of about two for young objects, but this claim has been questioned (Luhman & Potter 2006). At young ages and very low masses, the only direct benchmark test for these tracks is the eclipsing BD binary 2M J05352184-0546085 (Stassun et al. 2007). The Baraffe et al. isochrones fail to reproduce the surprising temperature reversal in this object (i.e., the more massive object is cooler than the secondary), an effect that is likely related to the presence of strong magnetic fields on the primary (Stassun et al. 2012). The luminosities of the two components, however, are consistent with the isochrones. Irwin et al. (2007) have discovered and analyzed a very young eclipsing binary with component masses around ∼0.2 M_☉; that system confirms the isochrones within the errorbars as well. Thus, although more work is required to calibrate the isochrones, some preliminary trust in their validity seems warranted.
• 3.  
  Cluster parameters. As discussed in Section 2.2, several properties of the clusters affect the mass estimates, in particular the distance, the age, and the extinction law. From our set of scenarios documented in Table 1 (scenarios 1–4) we can assess how the uncertainties in these parameters propagate through the procedure. In general, changes in age and distance cause significant changes in the estimated mass distribution, while a change in the extinction does not. For NGC 1333, the induced variations in the star/BD ratios are small; R varies from 2.0 to 2.4, smaller than the statistical uncertainties. For IC348, the scatter is larger, from 1.9 to 3.6, but excluding the implausible scenarios with age of 1 Myr and distance of 230 pc this range shrinks to 3.2–3.6. We note that the star/BD ratio increases somewhat with assumed age. For example, for IC348 and an (unlikely) age of 5 Myr we obtain R = 6.7. This is easy to understand—as the objects evolve, they become fainter, i.e., the same magnitude corresponds to a larger mass. As a result, objects move from the substellar to the stellar domain, and the star/BD ratio increases.
• 4.  
  Completeness. The samples we are using have the same depth and completeness, in terms of magnitudes, which eliminates one major source of uncertainty. However, in terms of object masses the depth of the survey depends on the assumed distance and age. Assuming a shorter distance, as well as a younger age, will produce lower masses for the same magnitudes, i.e., the entire mass distribution would be shifted to lower masses, including the limits for completeness. For our default scenario with age of 3 Myr and distance of 300 pc, the magnitude limit of the samples corresponds to ∼0.02 M_☉. The alternative distance of 230 pc (scenario 3) implies a magnitude shift by 0.6 mag. According to BT-Settl isochrones, this translates to a mass limit of 0.015 M_☉. The alternative age of 1 Myr (scenario 2) yields a new mass limit of 0.012–0.015 M_☉. Thus, in these scenarios we would be sensitive to slightly lower mass objects, but since the object density is very low in this mass domain, this has only a minuscule effect on the mass distribution. Furthermore, it is not going to affect the star/BD ratios.
• 5.  
  Degeneracy. In many cases there are multiple mass/A_V combinations that fit the data with a similar χ². In particular, for an object with a best mass estimate around the hydrogen burning limit (0.07–0.09 M_☉) this method is unable to distinguish between a star and a BD. As an estimate of the introduced uncertainty in R we selected all objects in this mass regime and included them first in the BD count (for the lower limit of the star/BD ratio) and then in the star count (for the upper limit). The resulting ranges in R are listed in Table 1, Column 9. These intervals are ±0.1–0.3 for NGC 1333 and ±0.5–1.0 for IC348. We note that the degeneracy is less of a problem for the two other mass thresholds involved in the calculation of R (0.03 and 1.0 M_☉), simply because the number of objects around these limits is low.

Combining these error sources, the values of R are affected by an uncertainty of approximately ±1 for the two samples studied here. Currently this is about the best that can be done in terms of estimating this indicator for young star-forming regions. This translates to an uncertainty of about ±0.1 in the power-law slope α. The median of the mass function can be estimated with an accuracy of ±0.1 M_☉ for nearby star-forming regions.

Looking ahead, there are obvious ways to lower these uncertainties. First, future evolutionary tracks need to include realistic initial conditions and require more detailed calibration (e.g., with eclipsing binaries). Second, the extinction parameters need to be studied in more detail for individual regions. Third, independent estimates for the (relative or absolute) ages of young clusters are needed. Fourth, the accuracy in the distances of these clusters (and many other well-studied star-forming regions) should be improved. In this respect, the Gaia satellite will be a major opportunity, as it is anticipated to be able to measure distances with 1% accuracy for open clusters within 1 kpc (Prusti 2011), a huge improvement over the current estimates. Fifth, it is a worthwhile goal to obtain comprehensive sets of multi-filter photometry for star-forming regions. And sixth, additional survey work in rich star-forming regions with significantly more members (a factor of 10 or more) than the well-studied nearby regions can be used to minimize the statistical uncertainties. However, since these regions are only found at large distances of >2 kpc, such studies have to be postponed until larger facilities, namely James Webb Space Telescope or Extremely Large Telescopes (ELTs), are available.

Our procedure yields for each scenario and for each cluster a distribution of masses. Prior to calculating parameters for the IMF, we examine these distributions directly. In Figure 3, we show the cumulative distribution of object masses, i.e., the fraction of objects below a given mass, as a function of mass for all six scenarios. We compare the 36 combinations of the functions shown in Figure 3 (six for IC348 and six for NGC 1333) with a Kolmogorov–Smirnov test to search for differences in the mass distribution. By doing that, we test the null hypothesis that two given distributions are drawn from the same parent distribution. In Table 2, we list the probabilities that this hypothesis is valid.

For 14 out of 36 combinations, this probability is <5%, i.e., the null hypothesis should be rejected. Six of them are combinations that include scenario 5 for NGC 1333, i.e., the one that uses the Nextgen isochrone. This scenario produces an unusual large number of low-mass BDs (0.02–0.05 M_☉) for NGC 1333, which causes the discrepancy with other distributions. This shows that apparent differences in the mass distribution of young clusters can be introduced simply by the choice of the isochrone. We caution against comparing mass distributions derived with inconsistent isochrones.

Seven further combinations of scenarios with significant differences in the mass distribution include scenario 2 for one of the clusters, i.e., the scenario with an age of 1 Myr. As an example, we show in Figure 4 (left panel) a comparison between the default scenario for IC348 and scenario 2 for NGC 1333. From this figure the origin of the difference is clear—the mass distribution for NGC 1333 shows a pronounced "knee" at 0.15 M_☉. However, the mass distribution in IC348 gives almost exactly the same "knee" when assuming an age of 1 Myr (upper left panel in Figure 3). This effect is best explained by the "knee" in the mass-luminosity relation at this age, which is predicted to become weaker with age. The resulting differences in the mass distribution cannot be attributed to environmental differences.

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The combination of the default scenario for IC348 and scenario 3 for NGC 1333 produces a significant difference as well (see Figure 4, right panel). This combination assumes the shorter distance of 230 pc for NGC 1333. We note that three further combinations with this assumption give marginally significant differences with probabilities between 5% and 10%. As explained in Section 2.3, multiple independent studies indicate that NGC 1333 is located at a shorter distance than IC348, thus, this scenario is plausible.

In summary, his comparison shows that the choice of the cluster parameters and the choice of the isochrone can have noticeable effects on the estimated distribution of object masses. We find that there could be significant differences in the mass distributions between the two clusters, if NGC 1333 is at a shorter distance than IC348. In this case, our analysis indicates a larger proportion of very low mass objects with masses <0.3 M_☉ in NGC 1333.

From the mass distributions in the six scenarios we calculated the star/BD ratio, see Table 1 for the results. For the cluster NGC 1333, our six scenarios give star/BD ratios of 1.9–2.4, which is a range comparable to the statistical uncertainties (see Section 2.4). This means that our previous estimate for this cluster from SONYC-IV (R ∼ 2.3) is confirmed.

For IC348, we find a wider range of values between 1.9 and 4.0. These values imply that star/BD ratios reported in the literature for IC348 of ∼8 (Luhman et al. 2003; Andersen et al. 2008) are overestimated. A possible reason for these large values is survey incompleteness or low number statistics. As explained in Section 2.4, the star/BD ratio depends on age, therefore these large star/BD ratios in the literature would become viable if IC348 is, in fact, significantly older than 3 Myr. This is unlikely, as age-dependent observable quantities like the disk fraction are comparable to other 2–3 Myr old clusters (Dawson et al. 2013).

An age of 1 Myr and a distance of 230 pc are not plausible for IC348 (Section 2.3). Excluding the scenarios using these parameters gives a range for R between 2.9 and 4.0, which is our best estimate for this cluster. These values are somewhat larger than in NGC 1333, although still within the margin of error. For example, for the default case, the star/BD ratio is 3.6 for IC348 with a lower limit of 2.6, and 2.2 for NGC 1333 with an upper limit of 2.5. This is before taking into account the statistical uncertainties. Thus, based on the star/BD ratio the evidence for regional differences in the mass distribution of IC348 and NGC 1333 is tentative.

In Table 1 we also report two other quantities that are used in the literature to describe the IMF. The power-law slope of the mass function α is directly determined from the star/BD ratio and thus reflects the same trends reported in Section 3.2. For NGC 1333, α is 0.9–1.0; for IC349 0.7–1.0, or 0.7–0.8 after excluding the implausible scenarios. Note that the slope is an average value for the mass range 0.03–1.0 M_☉; therefore it is not unexpected to find values somewhere between the typical slope of 1.3 in the regime of low-mass stars (Kroupa 2001) and the typical value of ∼0.6 in the very low mass regime (see SONYC-IV).

Independent from the star/BD ratio, we determine the median mass for each scenario. These values vary between 0.13 and 0.30 M_☉, again indicating that the choice of the cluster parameters and the choice of the isochrone affect the results considerably. For a given scenario, the two clusters have very similar median masses (maximum difference is 0.04 M_☉ for scenario 4). Increasing age and distance will also increase the median mass. Note that Alves de Oliveira et al. (2013) have recently determined the mass function for IC348 from a different survey and find a characteristic mass (in the lognormal mass function) of 0.21–0.22 M_☉ consistent with the values derived here.