How Many Blue Stars Can You See?

Looking out at the night sky you might notice that most of the bright stars are blue or blue-white and relatively few are yellow like the Sun.  Part of this is because it is difficult to sense colour from faint objects but also most of the bright stars in the night sky are main-sequence stars (fusing hydrogen to helium in their cores) more massive than the Sun; therefore, they appear bluer. The most massive stars are inherently brighter so we can see them over a larger volume.   Their power increases as M⁴ so the volume increases as M⁶ so if there were the same density of ten-solar-mass stars as one-solar-mass stars you would see a million times more ten-solar-mass stars in the night sky than one-solar-mass stars.  The ratio is not this dramatic which tells us that massive stars are rare.

Massive stars don't live as long as solar-mass stars, in fact the lifetime goes as mass over power, so M^-3.  If you include this lifetime effect and assumed that the birthrate of ten-solar-mass stars equalled those of one-solar-mass stars then the density of ten-solar-mass stars would be a thousandth of that of one-solar-mass stars.   In this case the ratio of the number of ten-solar-mass stars to one-solar-mass stars visible in the night sky would be 1000 to 1.   In fact the ratio is not this dramatic, so the birthrate of stars must not be constant with mass.   Among the ten brightest main-sequence stars in the night sky, only one α Centauri is about the mass of the Sun.

This lets us know that the birthrate must decrease with mass. If it decreased as
M^-3 or more steeply with mass, the sky would not be dominated by stars more massive than the sky. If it decreases as M^-2, there would be ten times more ten-solar-mass stars in the sky, that solar-mass stars. In fact most are less than ten-solar masses and greater than one-solar mass, this tells us that the birthrate or initial mass function must decrease somewhere in between these two extremes. In fact the best guess is from more careful analysis is around M^-2.35.