Salpeter (1955)

Abstract (paraphrased)¶

Derives the birth-mass distribution of solar-neighbourhood stars by combining the observed present-day luminosity function with main-sequence evolution timescales. Salpeter defines the “original mass function” $\xi(\mathfrak{M})$ — the number of stars created per unit logarithmic mass interval per unit time — and finds it is a smooth power law over the fitted range. This single power law is the origin of “the Salpeter slope,” still used as the high-mass slope of essentially all later IMF parameterisations.

The original mass function (verified against the paper, §III–IV)¶

Salpeter defines the original mass function by (Eq. 2, p. 164)

and gives the central result (Eq. 5, p. 165)

i.e. the fit is calibrated over roughly $0.4 \lesssim \mathfrak{M}/\mathfrak{M}_\odot \lesssim 10$.

progenax adopts the linear convention $dN/dm \propto m^{-\alpha}$, so the canonical Salpeter value is $\alpha = 2.35$. (Kroupa 2001 and Chabrier 2003 both state Salpeter as $\alpha = 2.35$ in this convention and then adopt the rounded $\alpha = 2.3$ for their own high-mass tails.)

Salpeter explicitly flags that the steeper drop above $\sim 10\,\mathfrak{M}_\odot$
"is not yet clear … whether [it] is a real effect" (p. 165), so the −1.35 slope is best
supported over the fitted $\sim 0.4$–$10\,\mathfrak{M}_\odot$ window.

## Use in progenax

- [](../../10-theory/imfs/classic.md) — Salpeter as the high-mass slope of all IMFs.
- `progenax.imf.PowerLawIMF.salpeter()` — single-segment $\alpha = 2.35$ implementation
  (the only place progenax uses the *original* 2.35; the Chabrier/Maschberger/Kroupa
  classes use the modern rounded $\alpha = 2.3$).

## Notes

Observationally well supported above $\sim 1\,\mathfrak{M}_\odot$. Below that, the real
IMF turns over (lognormal/turnover forms — [](chabrier-2003.md), [](kroupa-2001.md),
[](maschberger-2013.md)). The −1.35 ↔ −2.35 convention is the single most common
source of confusion when comparing IMF papers; always check whether a quoted slope is
per $d\log m$ or per $dm$.