# The unexpectedly large proportion of high-mass star-forming cores in a Galactic mini-starburst

F. Motte, T. Nony, F. Louvet, K. A. Marsh, Sylvain Bontemps, A. P.

Whitworth, A. Men’Shchikov, Q. Nguyen Luong, T. Csengeri, A. J. Maury, et al.

► To cite this version:

F. Motte, T. Nony, F. Louvet, K. A. Marsh, Sylvain Bontemps, et al. The unexpectedly large proportion of high-mass star-forming cores in a Galactic mini-starburst. Nature Astronomy, 2018, 2, pp.478-482. 10.1038/s41550-018-0452-x. hal-01762416

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https://hal.science/hal-01762416v1

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# The unexpectedly large proportion of high-mass star-forming cores in a Galactic mini-starburst

F. Motte<sup>1,2*</sup>, T. Nony<sup>1*</sup>, F. Louvet<sup>3*</sup>, K. A. Marsh<sup>4</sup>, S. Bontemps<sup>5</sup>, A. P. Whitworth<sup>4</sup>, A. Men'shchikov<sup>2</sup>, Q. Nguyễn Lương<sup>6,7</sup>, T. Csengeri<sup>8</sup>, A. J. Maury<sup>2</sup>, A. Gusdorf<sup>9</sup>, E. Chapillon<sup>5,10</sup>, V. Könyves<sup>2</sup>, P. Schilke<sup>11</sup>, A. Duarte-Cabral<sup>4</sup>, P. Didelon<sup>2</sup> and M. Gaudel<sup>2</sup>

Understanding the processes that determine the stellar initial mass function (IMF) is a critical unsolved problem, with profound implications for many areas of astrophysics<fnref n="1" />. In molecular clouds, stars are formed in cores—gas condensations sufficiently dense that gravitational collapse converts a large fraction of their mass into a star or small clutch of stars. In nearby star-formation regions, the core mass function (CMF) is strikingly similar to the IMF, suggesting that the shape of the IMF may simply be inherited from the CMF²⁻⁵. Here, we present 1.3 mm observations, obtained with the Atacama Large Millimeter/submillimeter Array telescope, of the active star-formation region W43-MM1, which may be more representative of the Galactic-arm regions where most stars form⁶,⁷. The unprecedented resolution of these observations reveals a statistically robust CMF at high masses, with a slope that is markedly shallower than the IMF. This seriously challenges our understanding of the origin of the IMF.

The initial mass function (IMF), giving the relative numbers of stars born with different masses, between ~0.08 and ~100$M_\odot$, appears to be universal⁹,¹⁰,¹¹, although there are a few claims to the contrary, especially in young massive clusters¹⁰,¹¹. In nearby star-formation regions, stars with masses between ~0.08 and ~5$M_\odot$ are observed to form in cores, and the core mass function (CMF) in these regions has an approximately log-normal shape with a power-law tail at higher masses ($\gtrsim M_\odot$), very similar to the IMF, but with the peak shifted to higher masses by a factor $1/\epsilon \sim 2.5 \pm 1$ (refs ²⁻⁵). This suggests that the shape of the IMF is determined by the processes that determine the CMF, in which case there must be an approximately self-similar mapping from core mass to stellar mass, with a very small variance (see Methods). However, nearby star-formation regions are probably not representative of the regions where most stars are formed and they do not form massive stars (~8 to ~100$M_\odot$)¹². Previous attempts to resolve cores in high-mass star-formation regions have suffered from poor statistics¹³,¹⁴ and have therefore not allowed a critical comparison of the CMF and IMF¹⁴⁻¹⁶.

W43-MM1 is a massive cloud (~2×10⁴$M_\odot$) located at the tip of the Galactic bar—a very rich region in terms of cloud concentration and star-formation activity⁶,⁷,¹⁷ located ~5.5 kpc from the Sun¹⁸. Shock tracing lines such as SiO are detected throughout this

cloud, testifying to the prevalence of inflowing and colliding gas streams⁶,¹⁹. The large densities in W43-MM1, traced at high angular resolution by millimetre continuum radiation, suggest it will form a massive star cluster in the near future². We used the Atacama Large Millimeter/submillimeter Array telescope (ALMA) to image the complete W43-MM1 cloud (~3 pc²) at 1.3 mm (see Supplementary Fig. 1), with unprecedented resolution (~2,400 au), and identified a large sample of individual cores with a broad and previously unmatched range of masses.

Since the 1.3 mm map of W43-MM1 has a dynamic range of 30 in physical scale and 4,000 in flux, we used the *getsources* algorithm²⁰ to identify and characterize the cores. *getsources* decomposes cloud emission into a multi-resolution data-cube; two cube dimensions give the position on the sky, and the third dimension is the physical scale of cloud substructure. Cores were identified at small scales, and their positions, spatial extents (best-fitting ellipses at half-maximum), peak and integrated fluxes (corrected for local background), masses, and mean densities were computed (see Supplementary Table 1). Figure 1 shows the elliptical outlines of the 131 cores returned by *getsources*, of which 94 are deemed to be very robust (significance $\ge 7$). The cores have diameters ranging from ~1,300 to ~3,500 au once deconvolved with the 0.44" (or ~2,400 au) telescope beam; masses ranging from ~1 to ~100$M_\odot$; and mean H₂ densities ranging from ~10⁷ to ~10¹⁰ cm⁻³. Monte Carlo extraction simulations indicate that the sample is 90% complete down to ~1.6$M_\odot$ (see Methods).

The blue histogram in Fig. 2a displays the differential CMF of W43-MM1, $dN/d\log(M)$, giving the number of cores in uniformly distributed but variably sized logarithmic bins of mass²¹. The blue histogram in Fig. 2b displays the cumulative CMF of W43-MM1, $N(>\log(M))$, giving the number of cores with $M_{\text{core}}$ larger than M. If these histograms are truncated below the 90% completeness limit (~1.6$M_\odot$), thereby reducing the sample to 105 cores, they can both be fitted with power laws: $dN/d\log(M) \propto M^{\gamma_{\text{diff}}}$ with $\gamma_{\text{diff}} \sim -0.90 \pm 0.06$ and $N(>\log(M)) \propto M^{\gamma_{\text{cumul}}}$ with $\gamma_{\text{cumul}} \sim -0.96 \pm 0.02$ (red lines in Fig. 2a,b). For comparison, the magenta lines in Fig. 2a,b show the slope of the IMF at masses larger than ~1$M_\odot$, which for both the differential and cumulative forms of the IMF is -1.35 (refs ⁹,²²). The range of core masses from ~1.6 to ~100$M_\odot$ corresponds to a range of stellar masses smaller by a factor $1/\epsilon \sim 2.5$ (that is, stellar masses

<footnote n="1">¹Institut de Planétologie et d'Astrophysique de Grenoble, Université Grenoble Alpes, CNRS, Grenoble, France. ²AIM Paris-Saclay/Département d'Astrophysique, CEA, CNRS, Université Paris Diderot, CEA-Saclay, Gif-sur-Yvette, France. ³Department of Astronomy, Universidad de Chile, Santiago, Chile. ⁴School of Physics and Astronomy, Cardiff University, Cardiff, UK. ⁵Laboratoire d'Astrophysique de Bordeaux, OASU, Université Bordeaux, CNRS, Pessac, France. ⁶Korea Astronomy and Space Science Institute, Daejeon, Republic of Korea. ⁷NAOJ Chile Observatory, National Astronomical Observatory of Japan, Tokyo, Japan. ⁸Max-Planck-Institut für Radioastronomie, Bonn, Germany. ⁹LERMA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Université, Paris, France. ¹⁰Institut de Radioastronomie Millimétrique, Saint Martin d'Hères, France. ¹¹Physikalisches Institut, University of Cologne, Cologne, Germany. *e-mail: frederique.motte@univ-grenoble-alpes.fr; thomas.nony@univ-grenoble-alpes.fr; fabien.louvet@gmail.com</footnote>
<figure><img src="image_2.png" /><figcaption>**Fig. 1** | High-angular-resolution image of the W43-MM1 cloud, revealing a rich population of cores. 1.3 mm dust continuum emission, observed by the ALMA interferometer, is presumed to trace the column density of gas, revealing high-density filaments and embedded cores. The filled yellow ellipse on the right represents the angular resolution, and a scale bar is shown. Ellipses outline core boundaries (at half-maximum) as defined by the getsources²⁰ extraction algorithm. Core masses span the range ~1 to ~100M⊙ and can therefore be expected to spawn stars with masses from ~0.4 to >40M⊙ (ref. ⁵; see Supplementary Table 1). All cores are shown. Hashed ellipses indicate the most robust identifications.</figcaption></figure>

from ~0.6 to ~40M⊙), allowing a robust comparison with the higher-mass (≥M⊙) IMF. The result is stable against variations in the temperature model, dust emissivity, extraction algorithm and reduction technique (see Supplementary Table 2). We conclude that at masses larger than ~1.6M⊙, the CMF in W43-MM1 is markedly flatter than the IMF. This result seriously challenges the widespread assumption that the shape of the IMF is inherited directly from the CMF.

The main uncertainty in the CMF derives from the estimation of core masses using their measured 1.3 mm continuum fluxes (see Methods). For the most massive cores, we used a 1.3 mm image based on the signal in a narrow composite band (~65 MHz wide) that is not contaminated by line emission. For the lower-mass cores (the majority), we used an image based on the signal in the full band (~1.9 GHz wide), since line contamination for low-mass cores is expected to be negligible (see Cline in Supplementary Table 1). Variations in dust emissivity are presumed to be small among the W43-MM1 cores, since 90% of them have uniformly high mean densities (n̄H₂ ~ 3 × 10⁷–3 × 10⁸ cm⁻³) and warm temperatures (Tdust ~ 23 ± 2 K) (see Supplementary Table 1). The main source of uncertainty is the dust temperature, since this is critical for converting flux into mass. Figure 3 displays the mean dust temperatures used for estimating core masses. A mean line-of-sight dust temperature was estimated using Herschel 70–500 µm images⁶, Atacama Pathfinder Experiment 350 and 870 µm images¹⁸, and mosaics obtained with ALMA (present data) and Institute for Radio Astronomy in the Millimeter range (IRAM)⁷ interferometers at 1.3 and 3 mm, respectively. By applying the Bayesian point process mapping (PPMAP) procedure²³, we obtained column-density maps in different dust-temperature slices. The mean dust temperature in each pixel is then a column-density-weighted average along that line of sight. However, in the vicinity of 10 hot cores (of which 3 have been identified previously²⁴,²⁵), the local heating is not properly traced by the 2.5"-resolution PPMAP temperature image. Here, we divide the total luminosity of the W43-MM1 cloud (~2 × 10⁴ L⊙) between the cores in proportion to their associated line contamination in the wide 1.3 mm band. We then use the individual luminosities, L* < 10 to ~10⁴ L⊙, and PPMAP background temperatures to esti-

<figure><img src="image_3.png" /><figcaption>**Fig. 2** | W43-MM1 CMFs challenging the relationship between the CMF and IMF. a,b, Differential (a) and cumulative form (b). Above the sample 90% completeness limit, estimated to be Mcore = 1.6M⊙ (black vertical line), the W43-MM1 CMFs (blue histograms) are well fit by the single power laws dN/dlog(M) ∝ M⁻⁰.⁹⁰ in a and N(>log(M)) ∝ M⁻⁰.⁹⁶ in b (red lines and 1σ uncertainties). The error bars on the differential CMF correspond to √N counting statistics. The cumulative CMF in b is the more robust statistically: its 5σ global uncertainty (±0.13, hatched area) was estimated from Monte Carlo simulations. The W43-MM1 CMF is clearly flatter than the IMF⁹,²², which in the corresponding mass range has slopes dN/dlog(M) ∝ M⁻¹.³⁵ and N(>log(M)) ∝ M⁻¹.³⁵ (magenta lines).</figcaption></figure>

mate the mean core dust temperatures (T̄dust ~ 20 to ~90 K) and their uncertainties (see Supplementary Table 1) using an approximate radiation transport model²⁶. Monte Carlo simulations indicate that the mass uncertainties in Supplementary Table 1 correspond to a 5σ uncertainty of ±0.13 in the slope of the CMF (see Methods and black-hatched area in Fig. 2b).

The fidelity of the CMF also depends on whether we have correctly identified cores; that is, structures that (1) are gravitationally bound and are therefore destined to spawn stars and (2) already contain most of the mass that will eventually end up in those stars. Parenthetically, we note that core masses probably grow with time due to inflowing gas streams such as those observed towards many massive cores²⁷⁻²⁹. In respect of criterion (1), studies of gravitational boundedness using ¹³CS (5–4) lines from the present ALMA project to determine internal velocity dispersions indicate that W43-MM1 cores with Mcore > 12M⊙ are secure but the status of lower-mass cores would benefit from further investigation. In respect of criterion (2), the low luminosity-to-mass ratio of the whole region, Lbol/M ~ 5L⊙/M⊙ (ref. ¹⁷), and the low mean temperature, Tdust ~ 20 K (see Fig. 3), imply that the region is young. We note that the two most massive cores may assimilate further mass from their dense surroundings and eventually form stars of ~100M⊙, with L* ~ 10⁵ L⊙ on the main sequence. However, currently they are 10–50 times less luminous. We conclude that any protostars embedded within the W43-MM1 cores are at the very beginning of their accretion phase and therefore contain only a small fraction of their final stellar mass.

Finally, the fidelity of the CMF depends on the completeness of the core sample. Owing to increased source and background
<figure><img src="image_4.png" /><figcaption>**Fig. 3 |** Mean estimated dust temperatures for cores. Over most of the frame, T<sub>dust</sub> is the column-density-weighted mean obtained with PPMAP<sup>23</sup> using 2.5" (or 14,000 au) resolution (colour scale). In the vicinity of the 10 most luminous sources, the mean core temperature with 0.44" (or 2,400 au) resolution was calculated using an approximate radiation transport model based on their estimated luminosities<sup>26</sup>. The W43-MM1 filaments are located by the 4, 20 and 50 mJy beam<sup>-1</sup> contours of the 1.3 mm continuum map of Fig. 1. The T<sub>dust</sub> map shows cloud heating by the nearby cluster, screening by the W43-MM1 high-density filaments and strong local heating by the 10 most luminous (30 to 10<sup>4</sup>L<sub>☉</sub>) cores.</figcaption></figure>

confusion in the denser parts of the cloud, the core sample is 90% complete above ~1.6M<sub>☉</sub> outside the main filament and above ~4.5M<sub>☉</sub> within it (see Supplementary Fig. 2). As in previous CMF evaluations, we use the median of the detection thresholds (here ~1.6M<sub>☉</sub>), but even with the more conservative threshold of ~4.5M<sub>☉</sub>, the CMF is still markedly shallower than the IMF (see Supplementary Table 2). The 5σ uncertainty in the fit to the CMF on Fig. 2b (black-hatched area) reflects the above assumptions, along with contributions from the data-reduction method, extraction algorithm<sup>29</sup>, and effects of source and background confusion (see Methods).

Owing to the high sensitivity and resolution of our ALMA 1.3 mm image of the W43-MM1 cloud, we were able to obtain a robust CMF covering masses in the range ~1.6 to ~100M<sub>☉</sub>, and hence to conclude that the CMF in W43-MM1 is much shallower than the higher-mass (≥M<sub>☉</sub>) IMF; namely, N(>log(M)) ∝ M<sup>γ</sup> with γ ≈ −0.96 ± 0.13 instead of γ ≈ −1.35. This is in stark contrast with previous robust evaluations of the CMF in other regions and at lower masses, where the CMF appears to be very similar to the IMF<sup>2–5</sup>. Scenarios that might explain our result and still allow the shape of the IMF to be inherited from the shape of the CMF, with an approximately self-similar mapping, fall into two broad categories. In the first, W43-MM1 is not representative of the environments in which most low-mass stars form<sup>10,11</sup>; to obtain the complete IMF, stars formed in environments like W43-MM1 must be mixed with stars formed in environments that contain a higher proportion of low-mass cores. However, this scenario is hard to justify if the local IMF observed in extreme star-formation regions is similar to the universal IMF<sup>8</sup>. In the second, massive cores are over-represented in W43-MM1; either massive cores live significantly longer than lower-mass cores<sup>30</sup> (but current current lifetime derivations<sup>12</sup> suggest the opposite) or star formation in regions like W43-MM1 is prolonged, and massive cores and the stars they spawn only form during a short period (probably near the beginning), while lower-mass cores and low-mass stars form over a much longer period (which seems unlikely because it would require the low-mass period to be nearly ten times longer than the high-mass period, although this cannot

be ruled out). If none of the above scenarios appears to explain our results, that would imply that the mapping from the CMF to the IMF is not statistically self-similar. Either higher-mass cores must convert a smaller fraction of their mass into stars than lower-mass cores (which seems unlikely<sup>14</sup>) or they must spawn more stars with a wider logarithmic range of masses (which seems more likely), or some combination of these possibilities exists. The shape of the IMF—at least at masses greater than ~M<sub>☉</sub>—is then not simply inherited from the shape of the CMF and the processes that determine the IMF remain to be determined.

**Methods**

**Observations and data reduction.** The W43-MM1 ALMA observations were carried out between September 2014 and June 2015 (project number 2013.1.01365.S) using the 12 m interferometric array. With baselines ranging from 13 to 1,045 m, the 1.3 mm image is sensitive to emission on angular scales from the synthesized beam (0.37" × 0.53" (~2,400 AU at 5.5 kpc)) to the filtering scale of the interferometer for our observations (12"). The W43-MM1 cloud was imaged with a 33-field mosaic covering its 2.1 pc × 1.4 pc extent. The complete dataset contains eight spectral windows, of which the one most used here is dedicated to the 1.3 mm continuum centred at 233.45 GHz, with a bandwidth of 1.9 GHz and a channel step of 977 kHz. The complementary data taken with the 7 m interferometric array were not used because of their low signal-to-noise ratio.

Data were reduced using the CASA 4.3.1 software<sup>1</sup>, applying manual and SelfCalibration scripts. The SelfCalibration technique improved image quality provided that it started from the highest-sensitivity map integrated over the complete 1.9 GHz continuum band. We then applied the Clean algorithm with a robust weight of 0.5 and the MultiScale option<sup>25</sup>, which minimizes interferometric artefacts associated with missing short spacings. We did not use merged 12 and 7 m data because the MultiScale cleaning of 12 m-only data provides the optimum image sensitivity for extracting cores. From the residual map of the continuum map integrated over the complete 1.9 GHz band, we estimated noise levels from ~0.13 mJy beam<sup>-1</sup> in the outskirts to ~2.5 mJy beam<sup>-1</sup> towards the main filament. This variation was due to confusion caused by strong, extended and structured cloud emission<sup>13</sup>.

**Core extraction.** Compact sources were extracted using getsources (version 1.140127)—a multi-scale, multi-wavelength source-extraction algorithm developed for Herschel CMF studies<sup>20,34</sup>. The main advantage of this algorithm applied to interferometric images is that it can extract sources in varying backgrounds, from strong filaments to negative areas associated with missing short spacings. We also used the multi-wavelength design of getsources to simultaneously treat the highest-sensitivity (1.9 GHz-integrated) and line-free (65 MHz-integrated) 1.3 mm continuum images. During the detection step, getsources combines both images and defines a catalogue of sources with unique positions. At the measurement stage, getsources measures the background and flux of these sources on each map independently.

We post-processed the getsources catalogue to remove, through visual inspection, sources that were too extended, whose ellipticity was too large to correspond to cores or that were not centrally peaked. The final catalogue contains 131 sources with significance >5 (reliable), of which 94 cores have significance >7 (robust). Core characteristics were measured in the highest-sensitivity continuum image, except when core fluxes in the highest-sensitivity map were larger than those measured in the line-free image, due to line contamination. Sources with line contamination were massive cores harbouring hot cores. Supplementary Table 1 lists the cores detected in W43-MM1, and provides their significance, coordinates, diameters, line contamination ratio, and peak and total integrated flux (corrected for line contamination and local background). While the significance variable reflects the detection significance (equivalent to the signal-to-noise ratio of a source on the scale where it is best detected), flux uncertainties reflect the quality of the flux measurements.

**Core mass estimates.** The total mass of a core (gas plus dust), having uniform opacity throughout its solid angle, is given by

$$ 
\begin{align*} 
M_{\text{core}} &= -\frac{\Omega_{\text{beam}}}{\kappa_{1.3\text{mm}}} \frac{d^2}{\ln\left(1-\frac{S_{1.3\text{mm}}^{\text{peak}}}{\Omega_{\text{beam}} B_{1.3\text{mm}}(T_{\text{dust}})}\right)} \times \frac{S_{1.3\text{mm}}^{\text{int}}}{S_{1.3\text{mm}}^{\text{peak}}} \\ 
&= -M_{\text{core}}^{\text{opt thin}} \times \frac{\Omega_{\text{beam}} B_{1.3\text{mm}}(T_{\text{dust}})}{S_{1.3\text{mm}}^{\text{peak}}} \\ 
& \quad \ln\left(1-\frac{S_{1.3\text{mm}}^{\text{peak}}}{\Omega_{\text{beam}} B_{1.3\text{mm}}(T_{\text{dust}})}\right) 
\end{align*} 
$$

Here, Ω<sub>beam</sub> is the solid angle of the beam, d=5.5 kpc is the distance to the core, κ<sub>1.3mm</sub> is the dust opacity per unit mass of gas and dust at 1.3 mm, S<sub>1.3mm</sub><sup>peak</sup> and
$S_{1.3\text{mm}}^{\text{int}}$ are the peak and integrated monochromatic fluxes of the core at 1.3 mm, and $B_{1.3\text{mm}}(T_{\text{dust}})$ is the Planck function at dust temperature $T_{\text{dust}}$. $M_{\text{core}}^{\text{opt thin}} = S_{1.3\text{mm}}^{\text{int}} d^2/(\kappa_{1.3\text{mm}}B_{1.3\text{mm}}(T_{\text{dust}}))$ is the core mass in the optically thin limit. We have adopted $\kappa_{1.3\text{mm}} = 0.01 \text{ cm}^2 \text{g}^{-1}$ (ref. 35) as being most appropriate for high-density, cool to warm cores. Supplementary Table 1 lists for each core its mass and mean number density$^{36}$.

The 1.3 mm continuum flux of a core arises mostly from thermal dust emission, which is optically thin except towards the seven densest, most massive cores. Contamination by free-free emission is estimated to be low (<20%; ref. 17) for W43-MM1 cores, since none of the embedded protostars are sufficiently hot and luminous (<$10^3 \text{L}_\odot$ for all except 3 cores) to ionize significant H II regions. Line contamination was evaluated by comparing the fluxes measured in the total 1.9 GHz continuum band and in a selection of line-free channels summing up to 65 MHz. The mass uncertainties were derived from the flux uncertainties measured by getsources, and from the estimated dust-temperature error. We added the ±1 K errors of PPMAP temperatures$^{23}$ to the uncertainty derived from the self-shielding of 0.5" starless cores and/or the internal heating of 0.5" protostellar cores lying within the 2.5" PPMAP resolution$^{26,37}$. We estimated the absolute values of the core masses to be uncertain by a factor of a few, and the relative values between cores to be uncertain by ~50%.

To estimate the sample completeness, we performed 10 Monte Carlo simulations that placed 2,000 synthetic cores with masses in the range 1–10$M_\odot$ on the worst-sensitivity background image determined by getsources or on the original image containing real cores. Like the observed cores, the synthetic cores had small sizes, and this reduced confusion by nearby sources and gave an excellent measurement of the total core mass. We then simulated observation of these synthetic protoclusters and applied the same core extraction process as for the real data. Supplementary Fig. 2 shows that the 90% completeness level depends weakly on the background intensity. It is ~0.8±0.1$M_\odot$ in the outskirts of the protocluster, ~1.6±0.1$M_\odot$ where most cores lie and up to ~4.5±1$M_\odot$ on the main filament (see the areas outlined in Supplementary Fig. 1). The 90% completeness level also depends weakly on the source confusion, with an increase up to 2.4±0.2$M_\odot$ for overlapping sources, relative to the mean value, which remains at ~1.6$M_\odot$ (see Supplementary Fig. 2).

**CMFs.** We performed numerous tests and Monte Carlo simulations to prove the robustness of the CMF shape against interferometric artefacts, extraction methods (getsources$^{20}$ or MRE-GaussClumps$^{29}$), mass estimates and CMF representations. In developing the best strategy to reduce the W43-MM1 ALMA image, the core catalogue steadily improved, with progressively fewer false cores and more solar-type cores. The 94 robust cores (>7σ) were detected in almost all runs. Supplementary Table 2 lists the major tests performed to evaluate the uncertainty of the CMF fit. For statistical reasons, fits were more robust in cumulative form$^{38}$, with the complete sample of 131 reliable cores and the complete 1.6–100$M_\odot$ mass range. However, Supplementary Table 2 shows that the higher-mass part (>4.5$M_\odot$) of the W43-MM1 CMF is still always flatter at the IMF. Monte Carlo simulations (with 100,000 runs) showed that the mass uncertainties corresponded to the slope of the CMF, $\gamma_{\text{cumal}}$, having a 5σ uncertainty of ±0.13. Building separate mass functions for cores that do and do not contain accreting protostars was not feasible here. First, the most massive cores all contained young low-luminosity accreting protostars. Second, their outflows made it very hard to ascertain the precise nature of their lower-mass neighbours (that is, whether or not they too already contained accreting protostars).

**Self-similar mapping.** A statistically self-similar mapping means that, for example, the probability that a core with mass in the range 1.0–1.1$M_\odot$ spawns—possibly along with other stars—a star with mass in the range 0.40–0.44$M_\odot$ is the same as the probability that a core with mass in the range 10–11$M_\odot$ spawns a star with mass in the range 4.0–4.4$M_\odot$. A mapping with small variance is required because, if the variance were large, the peak of the IMF would be noticeably broader than the peak of the CMF, and this does not appear to be the case$^5$. (We note that incompleteness at low core masses can only make the peak of the CMF broader than current estimates and thereby strengthen this constraint).

To compute a notional conversion efficiency (the mean of the total mass of stars subsequently spawned by a core divided by the core's mass), we need to determine: (1) the mean factor by which a core's mass grows between the time when it is measured to determine the CMF and the time when it has finished forming stars, $\mu_{\text{map}}$ (>1); and (2) the mean fraction of this final mass that goes into stars, $\eta_{\text{map}}$ (<1). The notional efficiency is then $\mu_{\text{map}}\eta_{\text{map}}$ and can in principle exceed unity (which is why it is 'notional'). If we also know (3) the mean number of stars spawned by a single core, $N_{\text{map}}^*$, we can compute the upwards shift from the peak of the IMF to the peak of the CMF, $1/e = N_{\text{map}}^*/(\mu_{\text{map}}\eta_{\text{map}})$. Finally, if we know (4) the logarithmic variance of the mean distribution of stellar masses spawned by a single core, $\sigma_{\text{map}}^2$, this must be convolved with the logarithmic variance of the CMF to determine the logarithmic variance of the IMF, $\sigma_{\text{IMF}}^2 = \sigma_{\text{CMF}}^2 + \sigma_{\text{map}}^2$. Since $\sigma_{\text{map}}$ is necessarily finite, a self-similar mapping requires $\sigma_{\text{IMF}}^2 > \sigma_{\text{CMF}}^2$.

**Data availability.** The data that support the plots within this paper and other findings of this study are available from the corresponding authors (F.M., T.N. or F.L.) upon reasonable request. Requests for materials should be addressed to F.M., T.N. or F.L.

Received: 3 May 2017; Accepted: 26 March 2018;
Published online: 30 April 2018

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## Acknowledgements

This paper makes use of the following ALMA data: #2013.1.01365.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is

operated by ESO, AUI/NRAO and NAOJ. This project has received funding from the European Union's Horizon 2020 research and innovation programme StarFormMapper under grant agreement number 687528. This work was supported by the Programme National de Physique Stellaire and Physique et Chimie du Milieu Interstellaire of CNRS/INSU (with INC/INP/IN2P3), co-funded by CEA and CNES. A.P.W. gratefully acknowledges the support of a consolidated grant (ST/K00926/1) from the UK Science and Technology Funding Council. T.C. acknowledges support from the Deutsche Forschungsgemeinschaft via the SPP (priority programme) 1573 'Physics of the ISM'. A.J.M. has received funding from the European Research Council under the European Union's Horizon 2020 research and innovation programme (MagneticYSOs, grant agreement number 679937).

## Author contributions

F.M. and F.L. led the project. E.C., T.N., F.M. and A.J.M. reduced the ALMA data. F.L. ran getsources and the CASA simulator. T.C. ran MRE-GaussClumps. K.A.M. ran PPMAP. S.B. and A.M. performed the Monte Carlo simulations. F.M., T.N. and F.L. analysed the CMF results. F.M. and A.P.W. wrote the manuscript. F.M., S.B., F.L., Q.N.L., A.J.M. and P.S. contributed to the ALMA proposal. All authors discussed the results and implications and commented on the manuscript.

## Competing interests

The authors declare no competing interests.

## Additional information

**Supplementary information** is available for this paper at https://doi.org/10.1038/s41550-018-0452-x.

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