initial mass function
/ IMF /
When a cloud fragments and makes a fresh batch of stars, it does not make them all the same size. It makes a great many small, faint stars and only a precious few big, brilliant ones. The initial mass function is the recipe nature seems to follow: it describes how many stars of each mass are born out of a cloud, capturing the basic fact that low-mass stars vastly outnumber high-mass ones.
Concretely, the IMF is a graph of how the number of newborn stars falls off as you go to higher masses. The trend is steep: for every very massive star of, say, twenty solar masses, a star-forming region makes hundreds of Sun-like stars and thousands of small red dwarfs of a tenth of a solar mass. Written as a rough rule, the number of stars drops roughly as mass raised to a negative power above about a solar mass (the classic Salpeter slope), with a turnover so that very low masses, near the brown-dwarf boundary, become less common again. Remarkably, this distribution looks much the same in many different environments across the galaxy and its history, which is why it is treated as something close to a universal recipe.
The IMF matters because it underpins almost everything else in astrophysics. The few rare massive stars are short-lived but produce most of a galaxy's light, forge most of its heavy elements, and explode as supernovae, while the countless small stars hoard most of its long-lived mass. So the shape of the IMF controls how galaxies shine, how they enrich themselves with the elements of life, and how much invisible low-mass material they contain. Why the IMF takes the form it does, set by the interplay of fragmentation, turbulence, cooling, and feedback, is one of the deepest unsolved questions in star formation.
In a typical newborn cluster, for every star as massive as the Sun there might be a handful of half-Sun stars and dozens of tiny red dwarfs, but you may have to scan thousands of stars to find a single rare giant of ten or twenty solar masses.
Nature makes small stars by the thousand and giant stars only one in many thousands.
The IMF describes the masses stars are born with, not the masses surviving today; massive stars die fast, so an old population is even more dominated by small stars than the birth recipe alone implies.