What sets the typical stellar mass near 0.2 solar masses?
In plain words
The most common birth mass of stars is about a fifth of the Sun's mass, and this peak is found almost everywhere we look. Why nature prefers this mass, and whether it changes in very different environments, is not explained.
Precise statement
The initial mass function $dN/d\operatorname{log} m$ peaks at $m_c \sim 0.2-0.3\,M_{\mathrm{sun}}$ and falls as dN/dlog m ~ m^-1.35 (Salpeter slope, 2.35 in $dN/dm$) above $\sim 1\,M_{\mathrm{sun}}$. Derive $m_c$ from first principles, distinguishing between the thermal Jeans mass at the transition to dust-dominated cooling, the mass of the first hydrostatic core, protostellar radiative heating, and turbulent fragmentation, and predict how $m_c$ scales with metallicity, gas pressure and radiation field.
What would settle it
Simulations converged in resolution that predict $m_c$ and its environmental scaling, tested against measured IMFs in clusters of different metallicity and density.
Status in the literature
Unverified note
The 2024 review by Hennebelle and Grudic argues that dust opacity and the first hydrostatic core set the peak, which is not yet confirmed by tests across environments.