Compact HII Regions as Clocks of Massive-Star Formation: Evidence for Long Formation Timescales

TL;DR

The paper addresses how massive stars assemble and how this affects the observed luminosity functions of OB stars and compact HII regions. By adopting the inertial–inflow model and performing a forward Monte Carlo analysis that couples a broken-power-law IMF with a mass-dependent growth law, the authors show that the OB and HII LFs encode the growth timescale $t_{ m form}(m_{ m f}) \propto m_{ m f}^{\alpha}$ and its normalization $\tau_0$, with a best-fit $\alpha\approx0.49$–$0.50$ and $\tau_0\approx1.9$–$2.0$ Myr. The observed knee luminosities, $L_{\rm k,OB}$ and $L_{\rm k,HII}$, arise from the fact that HII regions are powered by stars still growing, not by fully formed OB stars, which the analysis maps to a geometric relation between final masses and luminosities. Collectively, the results imply that massive stars in the Milky Way form over Myr timescales that increase with final mass, providing a natural resolution to the classical lifetime problem and highlighting the role of turbulent, mass-dependent accretion in star formation.

Abstract

We revisit the luminosity function (LF) of compact HII regions in the context of the inertial--inflow model, in which massive stars assemble over extended, mass-dependent timescales. The comparison of the compact-HII-region LF with that of OB stars is traditionally used to estimate the compact-HII-phase lifetime and is often cited as evidence for the classical ``lifetime problem" of HII regions. We show that once stellar growth during the ionizing phase is included, the LF comparison instead constrains massive-star formation timescales, so the lifetime problem turns into evidence for prolonged growth. We illustrate the principle with a simple analytic model and then forward-model the two LFs with Monte Carlo realizations. We also derive revised Galactic LFs for compact HII regions and OB stars from the Red MSX Source survey and the Alma Luminous Star catalogue. The joint LF constraints imply a growth law where the formation time is about 2 Myr for a $60\,M_\odot$ star, with a square-root dependence on mass. The revised OB-star LF exhibits a statistically significant knee at $\log_{10}(L_{\rm k}/L_\odot)=5.0$, while the HII-region LF knee occurs at lower luminosity, as expected in the interpretation that HII regions are powered by stars that are still growing in mass. We conclude that massive stars in the Milky Way form over Myr timescales that increase with their final mass.

Figures (5)

• Figure 1: Schematic illustration of the mass growth and luminosity mapping assumed in the classical interpretation of compact H II H2 regions (left) and in the IIM picture explored in this work (right). In both panels the stellar mass $m$ increases with time until it reaches the final mass $m_{\rm f}$, after which the star evolves on the main sequence (horizontal segment). Ionising emission turns on when the growing star crosses the threshold mass $m_{\rm ion}$ at $t_{\rm ion}$ (dot). Classical picture: the formation phase is short, and the compact-H II H2 stage is treated as a subsequent, comparatively brief phase of roughly fixed stellar mass, so that the embedded ionising source is effectively already at $m_{\rm f}$ and $L_{\mathrm \ion{H}{2}}(m_{\rm f})=L_{\rm OB}(m_{\rm f})$. This work: the compact-H II H2 phase occurs during continued accretion; the ionised emission is produced while $m<m_{\rm f}$, implying $L_{\mathrm \ion{H}{2}}(m)<L_{\rm OB}(m_{\rm f})$ and motivating a comparison between the compact-H II H2-region LF at luminosity $L$ and the OB-star LF at higher luminosities corresponding to the eventual $m_{\rm f}$. The coloured bars indicate approximate timescales for our reference $m_{\rm f}\simeq 60\,M_{\odot}$ case inferred in this work; they are not to scale and are intended to illustrate the ordering of phases, highlighting that in the growth scenario the compact-H II H2-region phase is concurrent with the stellar mass assembly.
• Figure 2: Schematic mapping between stellar growth tracks in the luminosity--time plane and the observed LFs in the IIM. Left: example evolutionary tracks for stars of different final masses $m_{\rm f}$. During the growth phase (blue), the luminosity increases as the star accretes; once the ionization threshold is reached (dotted horizontal line at $\log_{10}(L/L_\odot)=3$), the source is counted as a compact H II H2 region (blue shading) while it continues to brighten. After growth ends, the star enters the main-sequence phase (orange) at approximately fixed luminosity. Right: the compact-H II H2-region and OB-star LFs (shown here schematically, in the same style as Fig. \ref{['fig:OB_CHII_LF']}) can be viewed as projections of the track distribution: the H II H2-region LF counts the time spent in each luminosity bin along the shaded portions of all tracks, so a single luminosity bin receives contributions from a range of $m_{\rm f}$; in contrast, the OB-star LF counts the main-sequence lifetime at the luminosity corresponding to the final mass. This figure is intended purely to illustrate the geometric origin of the LF construction in the model (not to provide a quantitative fit).
• Figure 3: Left: OB-star LF from the ALS III catalog (blue circles with $1\sigma$ error bars) for $d\le 6$ kpc. The lower axis shows the bolometric luminosity, and the upper axis gives the corresponding main-sequence mass using the $L(m)$ relation (see Section \ref{['subsec:ALS_LF']} for details). Number densities assume an exponential vertical distribution with scale height $h=39$ pc. The M11 LF (gray squares) has been rescaled from $h=45$ pc used in Reed2005 to $h=39$ pc for comparison. The solid line shows a continuous broken power--law fit excluding the three lowest-luminosity bins (in light gray), with best-fit values given in the inset. The dashed black and gray segments indicate reference slopes corresponding to a Salpeter IMF with constant star--formation rate and to our IIM (Sect. \ref{['sec:OB_LF']}). Right: Compact H ii-region LF derived from the RMS survey (blue circles with $1\sigma$ error bars). The LF from M11 is also shown (open squares), as well as a broken power-law fit (red line) excluding the first two bins (in light grey). The black dashed lines show the slopes predicted by the IIM for Salpeter's slope $s=2.35$, growth-law index $\alpha = 0.5$, and average $L(m)$ relations at masses below and above the knee (see Sect. \ref{['sec:CHII_LF']} for details).
• Figure 4: Constraints on the massive-star growth–law parameters $\alpha$ and $\tau_{0}$. Top:$\Delta\chi^{2}$ map from the fit of the compact H ii-region LF using the best–fitting IMF. Filled contours show the joint 1, 2, and $3\sigma$ confidence regions for two parameters ($\Delta\chi^{2}=2.30,\,6.17,\,11.8$). The red star marks the best–fit point, with 1D profiled errors $\alpha = 0.48^{+0.07}_{-0.07}$ and $\tau_{0} = 1.90^{+0.18}_{-0.14}\,\mathrm{Myr}$. Bottom: effective $\Delta\chi^{2}_{\rm post}$ obtained after marginalising over the uncertainty in the broken–power–law IMF inferred from the observed OB-star LF. The red star indicates the posterior median, and the numbers in the top-left corner give the corresponding 68 per cent credible intervals, $\alpha = 0.49^{+0.12}_{-0.12}$ and $\tau_{0} = 1.94^{+0.29}_{-0.26}\,\mathrm{Myr}$.
• Figure 5: Comparison between the observed OB-star and compact H ii-region LFs (green squares and red circles respectively) and the model predictions (orange and blue lines). Both model LFs shown here are derived from the same broken power–law IMF giving the best fit to the observed OB-star LF. The blue curve is the best fit to the observed H ii-region LF corresponding to the best-fitting growth-law parameters $\alpha=0.49$ and $\tau_0=1.94$ Myr. The double-headed arrow marks the approximate knee-to-knee correspondence, from the H II H2-region knee at $m_{\rm k,\mathrm \ion{H}{2}}=18\,M_\odot$ to the OB-star knee at $m_{\rm k,OB}=25\,M_\odot$.