Caption: A log-log plot of initial mass function (IMF or ξ) as estimated by various authors (in some cases maybe a bit approximated???) with the normalization ξ = 1 at stellar mass = 1 M_☉.

Features:

1. The IMF is the mass distribution of stars at the time of star formation when the stars are on the zero-age main sequence (ZAMS). It is a fundamental ingredient in the galaxy evolution, and thus in cosmic evolution.

2. The horizontal axis is stellar mass in units of solar masses (i.e., units of M_☉).

3. The vertical axis is IMF (or ξ) itself and NOT ξ(m)Δm as the label claims (see Wikimedia Commons: File:Plot of various initial mass functions.svg).

4. The Chabrier03system IMF (i.e., the violet curve) may be the best (see Ivan Baldry, 2008, "Note on the use of stellar IMFs in cosmology").

   However, the IMF probably varies with star formation and cosmic time. So no one IMF can apply everywhere at all times.

5. All the IMF curves in the plot approximately converge for stellar mass >∼ 1 M_☉ to the Salpeter55 IMF which was originally proposed by Edwin Salpeter (1924--2008) in 1955 (see Wikipedia: Initial mass function: Salpeter (1955)). The Salpeter55 IMF has a simple functional form:

     ξ(m) = ξ_{0}*(m/M_☉)**(-α)  , 

   where m is stellar mass, α = 2.35, and for this plot ξ_{0} = 1.

   The Salpeter55 IMF is unrealistic for m → 0, since that gives ξ(m) → ∞---which CANNOT be since there are NOT infinitely many zero stellar mass stars. The explanation is that in 1955, there was inadequate evidence about the lower end of the IMF and Salpeter could only give a simple large stellar mass asymptotic formula.

   Actually, yours truly wonders why Salpeter did NOT set α = 7/3 = 2.333 ... since that must give virtually the same fit to the observations at the α = 2.35 value and may be a clue to why the large stellar mass IMF is what it is. The clue is α = 7/3 suggest that some simple relation dictates ideal star formation for large stellar mass. But if it is a clue, no one has deciphered it it seems.

6. Note that the Chabrier03system IMF shows a maximum IMF at ∼ 0.04 M_☉. This maximum is actually in the brown dwarf region of the IMF.

7. The dividing line between stars and brown dwarfs is thought to be ∼ 0.08 M_☉ ≅ 80 Jupiter masses (see Wikipedia: Red dwarf: Description and characteristics; Wikipedia: Brown dwarf).

   Below the line, objects never have hydrogen burning to helium-4 (He-4) though they can do a little nuclear burning: deuterium fusion where deuterium (D, H-2) is burnt to helium (He-3) and those ∼> 65 Jupiter masses have lithium burning where lithium-7 (Li-7) is burnt to helium-4 (He-4) (see Wikipedia: Brown dwarf).

8. The upper limit to stellar mass is currently ∼ 225 M_☉. Stars near and above 100 M_☉ are extremely rare and have very unusual evolutions that are NOT well understood (see Wikipedia: List of most massive stars). Stellar winds may usually limit the growth of protostars to ∼< 100 M_☉ on the main sequence (CK-305--306).

9. The Chabrier03system IMF is thought to be correct at least that there is maximum IMF and there is NOT a quasi-infinity of brown dwarfs as m → 0, but it is hard to be sure where the maximum is exactly. Brown dwarfs are small and dim, and so it is hard to do an accurate count of them and obtain their mass distribution. They may be as numerous as stars in the observable universe (see John Wenz, 2017, Astronomy), but because of their low mass, they make only a small or negligible contribution to the mass-energy of the observable universe.

10. Brown dwarfs were theorized to exist in the 1960s (see Wikipedia: Brown dwarf: Early theorizing) and the first one was discovered was in 1995 (see Wikipedia: Brown dwarfs: Milestones). As of 2021, ∼ 3000 brown dwarfs have been discovered (see Katelyn Allers, 2021, Scientific American, scroll to ∼ 50%).

11. Yours truly thinks brown dwarfs are rather boring astronomical object, except as forming part of the mass continuum from stars to planets.