Protostellar Luminosity Function

Updated 20 September 2025

The Protostellar Luminosity Function (PLF) is the probability distribution of protostar luminosities, directly probing mass assembly through accretion processes.
It integrates the protostellar mass function and accretion history, with models like the isothermal sphere and turbulent core shaping its observed form.
Observations via SED integrations reveal environmental impacts and challenges such as sample biases and the luminosity problem in star-forming regions.

The Protostellar Luminosity Function (PLF) characterizes the present-day distribution of bolometric luminosities among protostars in active star-forming regions. Its detailed form reflects a combination of the instantaneous protostellar mass distribution, the mass accretion histories, and the underlying star formation physics. Because protostellar luminosity during the embedded phase is dominated by accretion energy and only partially reprocessed stellar emission, the PLF constitutes a direct, observable probe of the mass assembly process, the structure and evolution of molecular clouds, and the environmental dependence of star and cluster formation.

1. Definition and Formulation of the Protostellar Luminosity Function

The PLF, usually denoted ψ_p(L), is the luminosity probability distribution of protostars in a given cloud or cluster at a particular epoch. For a population of protostars, ψ_p(L) d ln L specifies the fraction with luminosities in the range [L, L + dL]. Its determination relies on two technical components: (i) the protostellar mass function (PMF), which captures the distribution of instantaneous protostellar masses, and (ii) the mapping between mass, accretion rate, and luminosity.

Mathematically, the foundational relations are:

The bivariate PMF, Ψ(m, m_f), expresses the number of protostars with current mass m and final mass m_f as

$Ψ(m, m_f) = \frac{ψ(m_f) \, τ_{acc}(m, m_f)}{⟨t_f⟩}$

where ψ(m_f) is the IMF for the final stellar mass, τ_acc is the characteristic time spent at mass m before the final mass m_f is reached, and ⟨t_f⟩ is the IMF-averaged formation time (McKee et al., 2010).

The PMF, ψ_p(m), is then

$ψ_p(m) = \int Ψ(m, m_f) \, d\ln m_f$

(McKee et al., 2010, Offner et al., 2011).

Accretion luminosity is parameterized as $L_{acc} \sim f_{acc} (G m \dot{m} / r)$ , where $f_{acc}$ is the radiative efficiency at the accretion shock (Offner et al., 2011, Bhandare et al., 18 Sep 2025). The evolution of $f_{acc}$ is nontrivial and crucial for mapping mass accretion to radiative output.

By convolving ψ_p(m) with the luminosity mapping determined by the accretion law and stellar structure model, the theoretical PLF is calculated. This synthesizes how protostars traverse mass and luminosity states as they build up stellar mass.

2. Physical Drivers and Theoretical Models Shaping the PLF

The shape and width of the PLF are set chiefly by the interplay between the mass accretion history and the origin of the protostellar luminosity. The main physical models considered in the literature are:

Model Type	Accretion Law	PLF Implications
Isothermal Sphere	$\dot{m} =$ constant (Shu 1977)	Narrow PLF; overpredicts high-L sources; slow for high-mass stars (McKee et al., 2010, Offner et al., 2011)
Turbulent Core	$\dot{m} \propto (m/m_f)^{1/2} m_f^{3/4}$	Broad PLF; matches observed spread and median; predicts shorter $t_f$ for massive stars
Competitive Accretion	$\dot{m} \propto (m/m_f)^{2/3} m_f$	Steep high-mass tail; gives roughly constant formation times for all $m_f$ (McKee et al., 2010)

Tapered variants introduce accretion rates that decline as m approaches m_f. Episodic accretion models further modulate time-averaged luminosities by invoking brief, intense accretion bursts (e.g., FU Ori events), which explain outliers and broaden the observed PLF (McKee et al., 2010, Dunham et al., 2011, Taquet et al., 2016).

Radiative efficiency at the shock, $f_{acc}$ , evolves from subcritical (where much energy is advected into the protostar) to supercritical (cold accretion, $f_{acc} \sim 1$ ) on short timescales (≈100 yrs post-formation), affecting the fraction of energy radiated (Bhandare et al., 18 Sep 2025). Thus, models with a time-dependent $f_{acc}$ capture the intrinsic luminosity spread more accurately than those with constant energy conversion.

3. Observational Construction and Environmental Dependence

The PLF is observationally mapped by integrating SEDs over broad wavelength ranges (typically 1–1000 μm), using photometry from instruments such as Spitzer, Herschel, and SOFIA/FORCAST, with luminosities derived via empirical relations involving mid-IR fluxes and SED slopes (Kryukova et al., 2012, Cheng et al., 2022). Detailed completeness analysis and contamination correction (e.g., edge-on disks, background galaxies) are essential to mitigate biases, especially at the low-luminosity end.

Systematic studies reveal strong environmental dependence:

High-mass star-forming clouds (e.g., Orion, Mon R2, Cygnus X) exhibit PLFs peaking near 1 $L_\odot$ with prominent high-luminosity tails (Kryukova et al., 2012, Cheng et al., 2022).
Low-mass clouds present broader or rising PLFs toward lower luminosities, often lacking a pronounced peak.
High protostellar surface or gas column densities correlate with an excess of luminous protostars, reflecting local enhancements in accretion rates (“collaborative” or competitive accretion) (Elmegreen et al., 2014, Kryukova et al., 2014, Cheng et al., 2022).

The measured PLF shows remarkable diversity not only between clouds but also within subregions, controlled by local clustering and gas column density (Kryukova et al., 2014). This is consistent with primordial mass segregation and/or environmentally regulated accretion physics.

4. Impact of Accretion Physics, Burst Phenomena, and the Luminosity Problem

The so-called luminosity problem—wherein median protostellar luminosities ( $\approx$ 1.5 $L_\odot$ ) are much lower than predicted by canonical, constant-accretion models—has driven refinement of accretion histories invoked for PLF modeling. Supported solutions include:

Extended protostellar lifetimes ( $\sim$ 0.3–0.5 Myr) with lower mean accretion rates (McKee et al., 2010, Offner et al., 2011).
Reduced radiative efficiency due to outflows carrying away mechanical energy ( $f_{acc} \approx 0.75$ ) (McKee et al., 2010).
Inclusion of episodic accretion, so that only a modest fraction (20–25%) of the final mass is gained in rare bursts; the remainder accrues during low-critical phases (Dunham et al., 2011, Taquet et al., 2016).

Hydrodynamic and MHD simulations incorporating infall-driven, variable accretion rates (including stochastic fluctuations from turbulent molecular gas dynamics and disk instability) both reproduce the observed PLF and resolve the luminosity problem. For typical $r_\star = 3 R_\odot (M_\star/M_\odot)^{0.5}$ , the predicted distribution and upper envelope of PLF align with observed spread without requiring extreme bursts for all sources (Padoan et al., 2014, Jensen et al., 2017).

5. Interpretive Power: Linking PLF, Star Formation Theories, and the IMF

The shape and statistical parameters of the PLF (median, mean, logarithmic standard deviation, fraction of very low-luminosity objects) provide rigorous constraints on star formation models:

Isothermal sphere models predict a too–narrow PLF and are strongly disfavored in active, clustered environments (Offner et al., 2011, Cheng et al., 2022).
Both turbulent core and competitive accretion models, especially with accretion tapering and acceleration, are preferentially favored when compared to large Spitzer, Herschel, and SOFIA/FORCAST samples (Offner et al., 2011, Kryukova et al., 2012, Cheng et al., 2022).
The two-component accretion (“2CA”) models, which blend core–like and clump–like accretion with random stopping, produce PMFs whose conversion to PLFs matches observations in known clusters, with a typical peak near $L_\odot$ (Myers, 2013).

The connection of PLF with the PMF and ultimately the IMF is made explicit: the PMF is the IMF weighted by the accretion time per mass bin; the PLF, in turn, reflects how protostars populate luminosity states given their mass accumulation trajectory. The high-luminosity tail of the PLF in dense environments encodes both mass assembly and feedback, as observed and as required from theoretical constraints.

6. Statistical and Empirical Methodologies

Several techniques are employed to estimate and analyze the PLF:

Empirical luminosity estimation relies on integrating SEDs and correlating mid-IR to bolometric luminosity via parameterizations involving SED slope α. For example,

$L_{bol} = L_{MIR} \times f(\log \alpha), \quad \alpha \geq 0.3$

(Kryukova et al., 2012, Cheng et al., 2022).

Statistical comparisons invoke Kolmogorov–Smirnov testing to assess the difference between PLFs in different environments, yielding very low $P$ –values for high vs. low stellar/gas density regions (Kryukova et al., 2012, Kryukova et al., 2014).
Luminosity–distance diagram methods offer an alternative path to estimating the PLF from incomplete samples, provided the spatial distribution is uniform. The number density n(L) can be inferred by $n(L) = D(L)^{-3}$ , where D(L) is the mean separation of objects at luminosity L (Zou, 2017). Caveats apply for clustered or heterogeneous regions.

7. Current Challenges and Future Directions

While progress has been made toward matching the observed PLF with theory, several limitations persist:

The degeneracy between mass and accretion rate, as well as uncertainties in radiative efficiency and protostellar radius, complicate the direct inference of accretion physics from PLF morphology alone (Hartmann et al., 24 Jul 2025, Bhandare et al., 18 Sep 2025).
Sample incompleteness at both the lowest and highest luminosities requires careful treatment of biases, especially as current dynamical mass estimates preferentially sample larger, resolved disks (Hartmann et al., 24 Jul 2025, Tobin et al., 2019).
The possible existence of bimodality in the accretion history and thus in the PLF (distinct photospheric-dominated vs. accretion-luminous populations) remains an open question requiring larger and better-characterized samples (Hartmann et al., 24 Jul 2025).
Evolving radiative efficiency at the shock necessitates time-dependent modeling, crucial for correctly linking observed luminosity distributions with evolutionary phase and accretion scenario (Bhandare et al., 18 Sep 2025).

Continued advances involving large high-resolution ALMA surveys, next-generation radiative transfer modeling, and more precise stellar evolutionary tracks are expected to refine the mapping between observed luminosity distributions and the fundamental processes governing protostellar mass and energy assembly.

In summary, the Protostellar Luminosity Function provides a quantitative, model-discriminating diagnostic of how protostars assemble their mass and radiate energy. Its form and environmental variation encode the physics of accretion, feedback, and cloud evolution, offering one of the most direct empirical tests of contemporary star formation models.