The Initial Mass Function

The initial mass function is the single number-spectrum from which almost every integrated property of a stellar system is built. When a molecular cloud collapses and fragments, it does not make one star of a definite mass — it makes a distribution of masses, thousands of feeble red dwarfs for every rare, blazing O star. The IMF, written $\xi(m)$ , is that distribution: the number of stars formed per unit mass interval in a single star-formation event, fixed at birth before stellar evolution has had time to rearrange it. Convolve the IMF with stellar-evolution tracks and you get a galaxy's luminosity, colour, and spectrum; weight it by mass and you get the stellar mass and the mass-to-light ratio; weight it by the yield of massive stars and you get the supernova rate and the chemical enrichment history. The IMF is the pivot on which star formation turns into everything astronomers actually observe. Two facts make it deceptively subtle. First, the same distribution tells three contradictory-sounding stories depending on how you weight it — count stars and it is overwhelmingly bottom-heavy, weigh the light and it is overwhelmingly top-heavy. Second, the IMF is written in two conventions, per unit mass $\xi(m)$ and per unit logarithmic mass $\phi(\log m)$ , whose power-law slopes differ by exactly one; getting that bookkeeping wrong corrupts every derived quantity. This article takes the physics-correct forms — Salpeter, Kroupa, Chabrier — and builds the full machinery: the conventions, the weighting integrals, the Jeans-mass origin of the characteristic mass, and why a distribution that is nearly universal across the Milky Way and its neighbours remains one of the deepest unsolved problems in star-formation theory.

Key takeaways

The initial mass function $\xi(m)=\mathrm{d}N/\mathrm{d}m$ gives the number of stars formed per unit mass in a single event, fixed at birth before any stellar evolution alters the population.
Salpeter (1955) showed that for $m\gtrsim 1\,M_\odot$ the IMF is a power law $\xi(m)\propto m^{-2.35}$ ; low-mass stars vastly outnumber high-mass stars.
Modern forms (Kroupa broken power law, Chabrier lognormal) flatten below $\sim 1\,M_\odot$ and turn over at a characteristic mass $m_c\approx 0.2$ – $0.3\,M_\odot$ instead of running away to zero.
The IMF exists in two conventions whose slopes differ by one: $\xi(m)\propto m^{-\alpha}$ and $\phi(\log m)\propto m^{-\Gamma}$ with $\alpha=\Gamma+1$ ; Salpeter is $\alpha=2.35\leftrightarrow\Gamma=1.35$ .
One distribution, three stories: number is dominated by the lowest masses, total mass by low-to-intermediate masses, but luminosity and metal production by the rare massive stars.
The characteristic mass traces the Jeans mass at the moment cooling fails and collapse turns adiabatic; the power-law tail traces scale-free supersonic turbulence.
Key idea: the IMF is a steeply declining function whose every observable is a mass-weighted integral over it — whether that integral looks top-heavy or bottom-heavy depends entirely on the power of $m$ you weight by.

The core insight

The IMF $\xi(m)$ is a steeply declining function of stellar mass: number is dominated by low-mass stars, total mass by low-to-intermediate stars, and total luminosity and metal production by the rare massive stars — so whether any integral over the IMF is effectively "top-heavy" or "bottom-heavy" is decided entirely by the mass-weighting of the observable.

Where this lives today

The IMF is where a great deal of modern astrophysics quietly cashes out. Every star-formation rate quoted for a galaxy — from an ultraviolet flux, an H $\alpha$ line, or a far-infrared luminosity — is an IMF-weighted conversion, because the light being measured comes from massive stars while the mass being inferred is dominated by dwarfs you cannot see; switching from Salpeter to Chabrier lowers inferred rates and stellar masses by a factor $\sim 1.6$ . The most active controversy is whether the IMF is truly universal: high-resolution spectroscopy of massive elliptical galaxies finds gravity-sensitive absorption features (the Na I and Ca II lines, sensitive to dwarf-rich atmospheres) pointing to a bottom-heavy IMF in their cores, while some starbursts and the earliest galaxies are argued to be top-heavy — either would break the assumption of a single universal $\xi(m)$ and reshape estimates of galaxy masses, dark-matter fractions, and chemical yields. Meanwhile ALMA resolves the core mass function in nearby molecular clouds — the prestellar cores that presumably set the IMF — and JWST is pushing constraints toward low-metallicity dwarfs and the first stars, whose possibly extreme top-heaviness governs reionization and the seeding of supermassive black holes. Seventy years on, the number $2.35$ is both indispensable and not fully understood.

The physical picture

Empty a single burst of star formation onto a mass axis and the pile is grotesquely lopsided. For every $60\,M_\odot$ O star there are tens of thousands of $0.2\,M_\odot$ red dwarfs; in raw counts the population is a swarm of dim, long-lived, low-mass stars with a few giants lost in the crowd. Yet each of those giants outshines a red dwarf by a factor of $10^5$ – $10^6$ , so if you re-weight the very same pile by luminosity instead of by number, it flips: now the handful of massive stars produce essentially all the light, and the swarm of dwarfs is invisible. The IMF is a lever, and which way it tips depends only on the power of $m$ you hang on it — $m^0$ counts stars, $m^1$ weighs mass, $m^{\sim 3.5}$ measures light. The first figure shows the distribution itself; the second shows the lever in action.

Figure

The stellar initial mass function φ(log m) = dN/dlog m on log–log axes. A Chabrier lognormal peaks at the characteristic mass m_c ≈ 0.2 M⊙, the high-mass tail follows the Salpeter power law φ ∝ m^−1.35, equivalently ξ ∝ m^−2.35, and the curve turns over near the hydrogen-burning limit 0.08 M⊙

The index changes nothing about the shape of the curve, but everything about what it means to integrate under it. The steepness of the Salpeter slope, $\alpha=2.35$ , is such that the number integral $\int\xi\,\mathrm{d}m$ is dominated by the lowest masses, the mass integral $\int m\,\xi\,\mathrm{d}m$ is only weakly bottom-heavy, and the light integral $\int L\,\xi\,\mathrm{d}m$ is dominated by the upper cutoff. That single fact — three integrals of one function pointing in opposite directions — is the second figure and the practical heart of the subject.

Figure

The same Salpeter IMF, three weightings. Per logarithmic mass bin the number of stars falls as m^−1.35, the mass falls only gently as m^−0.35, and the light rises steeply as m^+2.15 for a mass–luminosity slope a ≈ 3.5. All three curves cross at m = 1 M⊙, so number is dominated by the lowest masses and light by the highest

Physically, the peak of the distribution — the characteristic mass of a few tenths of a solar mass — is thought to mark the scale at which collapsing gas stops fragmenting. As long as the gas cools efficiently it stays cold and isothermal while its density climbs, the Jeans mass falls, and the cloud keeps breaking into ever-smaller pieces. Once a fragment becomes dense enough to trap its own radiation, compression heats it, the Jeans mass turns around and starts to rise, and fragmentation halts. The high-mass power-law tail, by contrast, plausibly inherits the scale-free character of the supersonic turbulence that stirs molecular clouds — which is why it looks so nearly the same everywhere.

The core equations

The IMF starts as a definition, not a law: it is simply the number of stars born per unit mass interval. Everything downstream is a choice of functional form or a weighted integral of this object.

\xi(m)\,\mathrm{d}m = \frac{\mathrm{d}N}{\mathrm{d}m}\,\mathrm{d}m \qquad (1)

where: $\xi(m)$ — number of stars formed per unit mass interval in a single event; $\mathrm{d}N$ — number of stars with mass in $[m,\,m+\mathrm{d}m]$ ; $m$ — stellar mass, conventionally in $M_\odot$ . Reading it: $\xi(m)$ is a density in mass, so it only becomes a number when multiplied by an interval and integrated. The same distribution is very often written per unit logarithmic mass, because star counts are naturally binned in decades of mass. The two forms describe identical physics but carry slopes that differ by one, and this is the most common source of error in the whole subject.

\phi(\log m) = \frac{\mathrm{d}N}{\mathrm{d}\log m} = m\ln 10\;\xi(m) \qquad (2)

where: $\phi(\log m)$ — number of stars per unit base-10 logarithmic mass interval; $\ln 10\approx 2.303$ — the factor converting $\mathrm{d}\log_{10}m$ to $\mathrm{d}\ln m$ ; the extra factor of $m$ is what shifts the slope. Reading it: if $\xi\propto m^{-\alpha}$ then $\phi\propto m^{-(\alpha-1)}$ , so a power law is one unit shallower in the log form. Salpeter's original result is the anchor of the whole field: a single power law that still describes the high-mass IMF beautifully across an enormous range of environments.

\xi(m) = \xi_0\left(\frac{m}{M_\odot}\right)^{-2.35} \qquad (3)

where: $\xi_0$ — normalization constant fixed by the total mass or number of the population; $-2.35$ — the Salpeter index $\alpha$ , equivalent to $\Gamma=1.35$ in the logarithmic form. Reading it: valid for $m\gtrsim 1\,M_\odot$ ; extrapolating it to arbitrarily low mass overpredicts dwarfs and makes the number integral diverge. Below a solar mass the IMF flattens, and Kroupa captured this with a multi-part power law whose breakpoints are pinned to the hydrogen-burning limit and the characteristic mass.

\xi(m)\propto\begin{cases} m^{-0.3} & 0.01\le m/M_\odot<0.08\\ m^{-1.3} & 0.08\le m/M_\odot<0.5\\ m^{-2.3} & m/M_\odot\ge 0.5\end{cases} \qquad (4)

where: the breakpoints at $0.08$ and $0.5\,M_\odot$ mark the hydrogen-burning limit and the flattening scale; the top segment $-2.3$ is Salpeter to within the quoted precision. Reading it: continuity of $\xi$ across each break fixes the relative normalizations. Chabrier showed the low-mass regime is equally well described by a smooth lognormal, which builds in a true peak rather than a kink.

\xi(\log m)\propto \exp\!\left[-\frac{(\log m - \log m_c)^2}{2\sigma^2}\right],\quad m\le 1\,M_\odot \qquad (5)

where: $m_c$ — characteristic mass, $\approx 0.2\,M_\odot$ for the single-star system IMF; $\sigma\approx 0.55$ – $0.69$ — the lognormal width in dex. Reading it: above $1\,M_\odot$ the lognormal is stitched to a Salpeter power-law tail, giving a form nearly indistinguishable from Kroupa's. The reason the IMF matters is that observables are weighted integrals of it, and every such integral needs two mass limits.

N_{\rm tot}=\int_{m_{\min}}^{m_{\max}}\!\xi(m)\,\mathrm{d}m,\qquad M_{\rm tot}=\int_{m_{\min}}^{m_{\max}}\! m\,\xi(m)\,\mathrm{d}m \qquad (6)

where: $m_{\min},m_{\max}$ — lower and upper mass cutoffs, typically $\sim 0.08\,M_\odot$ and $\sim 120\,M_\odot$ ; $N_{\rm tot},M_{\rm tot}$ — total number and total stellar mass formed. Reading it: $N_{\rm tot}$ is sensitive to $m_{\min}$ , while the light integral (below) is sensitive to $m_{\max}$ . The characteristic mass is not put in by hand — it emerges from the Jeans mass, the minimum mass a self-gravitating gas clump must exceed to collapse.

M_J \simeq \left(\frac{5k_BT}{G\mu m_H}\right)^{3/2}\left(\frac{3}{4\pi\rho}\right)^{1/2} \qquad (7)

where: $T$ — gas temperature; $\rho$ — mass density; $\mu$ — mean molecular weight ( $\approx 2.3$ for molecular gas); $k_B,G,m_H$ — Boltzmann's constant, Newton's constant, the hydrogen mass. Reading it: $M_J\propto T^{3/2}\rho^{-1/2}$ , so what happens to the temperature during collapse decides whether fragmentation continues or stops. Finally, converting the IMF into a light requires the main-sequence mass–luminosity relation, whose steepness is what makes the light integral top-heavy.

\frac{L}{L_\odot}\approx\left(\frac{m}{M_\odot}\right)^{a},\quad a\approx 3\text{–}4 \qquad (8)

where: $L$ — main-sequence luminosity; $a$ — the exponent, $\approx 4$ near $1\,M_\odot$ and flatter ( $\approx 3$ ) for the most massive stars. Reading it: with $a\approx 3.5$ and $\alpha=2.35$ , the light per unit mass rises as $m^{a-\alpha}\approx m^{1.15}$ , so light piles up at the top of the mass range.

Derivation AWhy the linear and logarithmic slopes differ by exactly one

The goal is to establish, cleanly and once, the relation $\alpha=\Gamma+1$ between the slope of the IMF written per unit mass and the slope written per unit logarithmic mass, because this single unit of slope is the most frequent silent error in IMF work: a paper that quotes "slope 1.35" and one that quotes "slope 2.35" can be describing the identical distribution. We change variables from $m$ to $\log_{10}m$ , track the Jacobian factor, and read off how it shifts a power-law exponent.

Step 1. Start from the definitions: the linear IMF is $\xi(m)=\mathrm{d}N/\mathrm{d}m$ and the logarithmic IMF is $\phi(\log m)=\mathrm{d}N/\mathrm{d}\log m$ . These count the same stars, so $\mathrm{d}N$ is common to both and the only difference is the measure, $\mathrm{d}m$ versus $\mathrm{d}\log m$ . Why: the physics lives in $\mathrm{d}N$ ; the slope difference is pure calculus of the binning variable.

Step 2. Differentiate the binning variable: $\mathrm{d}\log_{10}m = \mathrm{d}m/(m\ln 10)$ , so $\mathrm{d}m = m\ln 10\,\mathrm{d}\log m$ . Substituting, $\phi(\log m)=\mathrm{d}N/\mathrm{d}\log m = (\mathrm{d}N/\mathrm{d}m)(\mathrm{d}m/\mathrm{d}\log m)=\xi(m)\,m\ln 10$ . Why: the Jacobian $\mathrm{d}m/\mathrm{d}\log m = m\ln 10$ carries an explicit factor of $m$ , and that factor is the entire origin of the slope shift.

Step 3. Feed in a power law $\xi\propto m^{-\alpha}$ : then $\phi\propto m\cdot m^{-\alpha}=m^{1-\alpha}\equiv m^{-\Gamma}$ with $\Gamma=\alpha-1$ . For Salpeter, $\alpha=2.35$ gives $\Gamma=1.35$ , and a "flat in log" IMF ( $\Gamma=0$ ) corresponds to $\alpha=1$ , not $\alpha=0$ . Why: the extra power of $m$ from the Jacobian reduces the falling exponent by one, so the log form always looks shallower by unity.

Step 4. Sanity check with a limit: an IMF that is flat per logarithmic decade, $\phi=\text{const}$ , means equal numbers of stars in $[0.1,1]$ and $[1,10]\,M_\odot$ ; in linear form that is $\xi\propto m^{-1}$ , which indeed integrates to equal counts per decade. Why: reproducing the "equal per decade" intuition confirms the sign and size of the shift are right. *Verified: $\alpha=\Gamma+1$ ; Salpeter $2.35\leftrightarrow 1.35$ ; flat-in-log $\Rightarrow \xi\propto m^{-1}$ .*

Go deeper · Derivation B · 2 minNumber, mass, and light from a single Salpeter IMF

Derivation BNumber, mass, and light from a single Salpeter IMF

The goal is to compute, for a Salpeter IMF over a range $[m_1,m_2]$ , which end of the mass distribution dominates the total number, the total mass, and the total light — and thereby to prove that the same distribution is simultaneously "bottom-heavy" and "top-heavy" depending only on the observable. The whole result follows from a single elementary integral, $\int m^p\,\mathrm{d}m = m^{p+1}/(p+1)$ , applied with three different exponents, and reading off whether each is dominated by the lower or upper limit.

Step 1. Write $\xi(m)=\xi_0 m^{-\alpha}$ with $\alpha=2.35$ . The number is $N=\int_{m_1}^{m_2}\xi_0 m^{-\alpha}\,\mathrm{d}m=\dfrac{\xi_0}{1-\alpha}\big(m_2^{1-\alpha}-m_1^{1-\alpha}\big)$ , with exponent $1-\alpha=-1.35<0$ . A negative exponent means $m_1^{-1.35}\gg m_2^{-1.35}$ , so $N$ is set by the lower limit: most stars are the least massive ones. Why: a steeply negative exponent makes the antiderivative blow up toward small $m$ , which is the mathematical statement of "number is dominated by low-mass stars".

Step 2. The mass integral weights by an extra power of $m$ : $M=\int_{m_1}^{m_2}\xi_0 m^{1-\alpha}\,\mathrm{d}m=\dfrac{\xi_0}{2-\alpha}\big(m_2^{2-\alpha}-m_1^{2-\alpha}\big)$ , with exponent $2-\alpha=-0.35$ . This is still negative, so mass is still bottom-weighted, but only weakly — with an exponent of $-0.35$ the low and intermediate masses contribute comparably and the result is sensitive to $m_{\min}$ . Why: multiplying by $m$ lifts the exponent by one, moving it close to zero, so mass is spread far more evenly across the range than number is.

Step 3. The light integral weights by $L\propto m^a$ with $a\approx 3.5$ : $L\propto\int_{m_1}^{m_2} m^{a-\alpha}\,\mathrm{d}m=\dfrac{\xi_0}{1+a-\alpha}\big(m_2^{1+a-\alpha}-m_1^{1+a-\alpha}\big)$ , with exponent $1+a-\alpha\approx 2.15>0$ . Now the exponent is positive, so $m_2^{2.15}\gg m_1^{2.15}$ and the integral is dominated by the upper limit: essentially all the light comes from the most massive stars. Why: the mass–luminosity slope $a\approx 3.5$ overpowers the IMF slope $\alpha=2.35$ , flipping the sign of the exponent and hence the end of the range that dominates.

Step 4. Collect the punchline: number $\to m_1$ (low mass), mass $\to$ low-to-intermediate, light $\to m_2$ (high mass). A cluster is a swarm of dwarfs by headcount, weighs mostly in its ordinary stars, and shines almost entirely by its giants. Why: three integrals of one function, with exponents $-1.35$ , $-0.35$ , $+2.15$ , encode the single most important practical lesson of the IMF — always ask what power of $m$ your observable weights by. *Verified: exponents $1-\alpha=-1.35$ , $2-\alpha=-0.35$ , $1+a-\alpha=+2.15$ for $a=3.5$ ; matches Carroll & Ostlie §24.3.*

Go deeper · Derivation C · 2 minThe characteristic mass from the Jeans-mass turnaround

Derivation CThe characteristic mass from the Jeans-mass turnaround

The goal is to explain why the IMF has a peak at a few tenths of a solar mass rather than continuing to rise toward ever-smaller masses — that is, why fragmentation stops. The argument tracks the Jeans mass $M_J\propto T^{3/2}\rho^{-1/2}$ through the collapse of a molecular clump and identifies the moment its dependence on density reverses, which sets a minimum fragment mass. This is a scaling argument, not a first-principles theory of the IMF, but it captures the essential physics of the turnover.

Step 1. Begin in the isothermal phase. Molecular gas cools efficiently through line emission and dust, so as a clump contracts its temperature stays nearly fixed while its density $\rho$ climbs. With $T$ constant, $M_J\propto\rho^{-1/2}$ decreases as the clump densifies. A region whose mass exceeds the shrinking $M_J$ can itself break into pieces that each exceed the still-smaller $M_J$ — runaway hierarchical fragmentation into ever-smaller clumps. Why: a falling Jeans mass means every collapsing fragment is unstable to further sub-fragmentation, so the cascade drives toward low masses.

Step 2. The cascade cannot continue forever, because cooling eventually fails. Once a fragment's density is high enough to become optically thick to its own cooling radiation, the compressional work can no longer be radiated away and the gas heats adiabatically. For a monatomic-like adiabat, $T\propto\rho^{\gamma-1}=\rho^{2/3}$ . Why: trapping the radiation converts the isothermal collapse into an adiabatic one, which is the physical switch that ends fragmentation.

Step 3. Substitute the adiabatic law into the Jeans mass: $M_J\propto T^{3/2}\rho^{-1/2}\propto(\rho^{2/3})^{3/2}\rho^{-1/2}=\rho^{1/2}$ . The exponent has flipped sign — $M_J$ now increases with density. A fragment that keeps contracting only raises its own Jeans mass above its actual mass, so it becomes stable against further splitting. Why: the turnaround from $M_J\propto\rho^{-1/2}$ to $M_J\propto\rho^{+1/2}$ is a minimum in $M_J$ , and no fragment below that minimum can form.

Step 4. The Jeans mass at the density where cooling fails — of order tenths of a solar mass for typical molecular-cloud conditions — is therefore the smallest fragment that can be made, and is plausibly the origin of the characteristic mass $m_c\approx 0.2\,M_\odot$ . Why: identifying the opacity-limited minimum Jeans mass with the observed IMF turnover ties a scale in the distribution to a specific piece of thermodynamics. *Verified: $M_J\propto\rho^{-1/2}$ (isothermal) turns to $\propto\rho^{+1/2}$ (adiabatic, $\gamma=5/3$ ); minimum $M_J$ of order $0.1$ – $0.3\,M_\odot$ .*

Limiting cases

The IMF is piecewise by nature, and its behaviour — and the behaviour of integrals over it — changes character from regime to regime.

Case	Result	Meaning
High-mass tail ( $m\gtrsim 1\,M_\odot$ )	$\xi(m)\propto m^{-2.35}$	Salpeter power law; robust across environments. Number falls steeply, but light rises with mass.
Low-mass regime ( $0.1$ – $1\,M_\odot$ )	$\phi(\log m)\propto m^{-\Gamma},\ \Gamma\approx 0$	Flattens to near-constant per log decade; Kroupa uses $\alpha\approx 1.3$ from $0.08$ – $0.5\,M_\odot$ .
Turnover / characteristic mass	$m_c\approx 0.2$ – $0.3\,M_\odot$	Chabrier lognormal peak; below it the number per log interval declines.
Number integral, $m_{\min}\to 0$	$\int m^{-2.35}\mathrm{d}m\propto m^{-1.35}$ diverges	A physical lower cutoff near $0.08\,M_\odot$ is essential; number is sensitive to $m_{\min}$ .
Light integral	$\int m^{a-\alpha}\mathrm{d}m\propto m^{+1.15}$	With $a\approx 3.5$ the integrand rises; light is dominated by $m_{\max}$ , not $m_{\min}$ .
Pure top-heavy limit ( $\alpha<2$ )	mass integral dominated by $m_{\max}$	Hypothesized for the first stars; would invert the usual mass budget toward the giants.

Common trapSlope 1.35 and slope 2.35 are the same IMF — mixing the two conventions corrupts every integral

The linear IMF $\xi(m)=\mathrm{d}N/\mathrm{d}m\propto m^{-\alpha}$ and the logarithmic IMF $\phi(\log m)=\mathrm{d}N/\mathrm{d}\log m\propto m^{-\Gamma}$ satisfy $\alpha=\Gamma+1$ , so Salpeter is $\alpha=2.35$ or $\Gamma=1.35$ depending on which you write. These are not two different IMFs; they are one distribution in two coordinate systems. Before you integrate anything, pin down which convention an author means — a "slope of 1.35" plugged into a linear integral, or a "2.35" plugged into a logarithmic one, shifts every weighted exponent by a full unit and can flip whether an integral is dominated by the low- or high-mass end. The related trap: a mass integral $\int m\,\xi\,\mathrm{d}m$ can converge, diverge, or be pinned to a cutoff purely according to the slope, so never quote a total mass or light without stating $m_{\min}$ , $m_{\max}$ , and the convention in the same breath.

Worked examples

1 · The three faces of a Salpeter cluster (fully worked)

A star cluster forms with a pure Salpeter IMF, $\xi(m)=\xi_0(m/M_\odot)^{-2.35}$ , over the range $0.1$ to $100\,M_\odot$ . Find the mean stellar mass, the fraction of stars more massive than $1\,M_\odot$ , the fraction of the stellar mass in those stars, and how many core-collapse supernova progenitors ( $m>8\,M_\odot$ ) form per unit stellar mass.

Step 1. Total number. With $\int m^p\,\mathrm{d}m=m^{p+1}/(p+1)$ and $p=-2.35$ , $N=\xi_0\dfrac{m^{-1.35}}{-1.35}\Big|_{0.1}^{100}=\dfrac{\xi_0}{1.35}\big(0.1^{-1.35}-100^{-1.35}\big)=\dfrac{\xi_0}{1.35}(22.39-0.0020)=\mathbf{16.58\,\xi_0}$ . As expected the lower limit ( $0.1^{-1.35}=22.39$ ) dominates completely — the count is a swarm of dwarfs.

Step 2. Total mass. Now $p=-1.35$ : $M=\xi_0\dfrac{m^{-0.35}}{-0.35}\Big|_{0.1}^{100}=\dfrac{\xi_0}{0.35}\big(0.1^{-0.35}-100^{-0.35}\big)=\dfrac{\xi_0}{0.35}(2.239-0.1995)=\mathbf{5.83\,\xi_0}$ . The mean stellar mass is $\langle m\rangle=M/N=5.83/16.58=\mathbf{0.35\,M_\odot}$ — a red dwarf, even though the range runs to $100\,M_\odot$ .

Step 3. Fraction above $1\,M_\odot$ . Repeat over $[1,100]$ : $N_{>1}=\dfrac{\xi_0}{1.35}(1-0.0020)=0.739\,\xi_0$ , so $N_{>1}/N=0.739/16.58=\mathbf{4.5\%}$ . For mass, $M_{>1}=\dfrac{\xi_0}{0.35}(1-0.1995)=2.287\,\xi_0$ , so $M_{>1}/M=2.287/5.83=\mathbf{39\%}$ . One statement, two numbers: only $1$ star in $22$ exceeds a solar mass, yet those few hold nearly two-fifths of the mass.

Step 4. Supernova progenitors. Stars above $8\,M_\odot$ end as core-collapse supernovae: $N_{>8}=\dfrac{\xi_0}{1.35}(8^{-1.35}-100^{-1.35})=\dfrac{\xi_0}{1.35}(0.0604-0.0020)=0.0432\,\xi_0$ . Per unit stellar mass formed, $N_{>8}/M=0.0432/5.83=7.4\times10^{-3}\,M_\odot^{-1}$ , i.e. one core-collapse supernova per $\sim 135\,M_\odot$ of stars formed — a number that feeds directly into galactic feedback and chemical-enrichment models. (A Kroupa IMF, with fewer dwarfs, gives roughly one per $100\,M_\odot$ .)

2 · The Jeans mass of a dense core (fully worked)

A prestellar core has temperature $T=10\,\mathrm{K}$ , number density $n=10^{4}\,\mathrm{cm^{-3}}$ , and mean molecular weight $\mu=2.3$ . Estimate its Jeans mass and interpret the result for fragmentation.

Step 1. Mass density: $\rho=\mu m_H n = 2.3\times(1.67\times10^{-24}\,\mathrm{g})\times10^{4}\,\mathrm{cm^{-3}}=3.84\times10^{-20}\,\mathrm{g\,cm^{-3}}$ .

Step 2. The thermal factor: $\dfrac{5k_BT}{G\mu m_H}=\dfrac{5(1.38\times10^{-16})(10)}{(6.67\times10^{-8})(2.3)(1.67\times10^{-24})}=\dfrac{6.9\times10^{-15}}{2.56\times10^{-31}}=2.69\times10^{16}\,\mathrm{g\,cm^{-1}}$ , and raising to the $3/2$ power gives $4.42\times10^{24}$ in cgs.

Step 3. The density factor: $\left(\dfrac{3}{4\pi\rho}\right)^{1/2}=\left(\dfrac{3}{4\pi(3.84\times10^{-20})}\right)^{1/2}=(6.22\times10^{18})^{1/2}=2.49\times10^{9}$ .

Step 4. Multiply: $M_J\approx (4.42\times10^{24})(2.49\times10^{9})=1.10\times10^{34}\,\mathrm{g}=\mathbf{5.5\,M_\odot}$ . So a core at these conditions is unstable in chunks of several solar masses. As it collapses isothermally and $\rho$ climbs, $M_J\propto\rho^{-1/2}$ falls — by $n=10^{7}\,\mathrm{cm^{-3}}$ it has dropped to $\sim 0.2\,M_\odot$ — until opacity traps the heat and the turnaround of Derivation 3 halts the cascade near the characteristic mass. The single core seeds many low-mass stars, not one massive one.

3 · Reading a quoted slope correctly

A paper reports a cluster's high-mass "IMF slope" as $\Gamma=1.7$ in the logarithmic convention. Is this steeper or shallower than Salpeter, and what linear slope should you use to integrate the mass?

Step 1. Convert conventions with $\alpha=\Gamma+1$ : $\alpha=1.7+1=\mathbf{2.7}$ . Compared with Salpeter's $\alpha=2.35$ , this cluster's IMF is steeper — it has proportionally fewer massive stars.

Step 2. To integrate the mass, weight by one power of $m$ : the mass integrand is $m\,\xi\propto m^{1-\alpha}=m^{-1.7}$ , exponent $1-\alpha=-1.7<0$ , so the mass is even more strongly bottom-weighted than for Salpeter ( $-1.35$ ). Had you mistakenly used $\Gamma=1.7$ as the linear slope, you would have taken the mass integrand as $m^{-0.7}$ and overestimated the massive-star contribution — the exact error the warning box exists to prevent. Always convert first: $\alpha=\Gamma+1$ .

Physics of the portrait

The IMF is one of those rare quantities that carries a person's fingerprints for seventy years. Edwin Salpeter turned a messy pile of star counts into a single power law, and the number $2.35$ has appeared in essentially every stellar-population calculation since.

Origins · 1 minEdwin Salpeter and how this was discovered

Edwin Salpeter

1924–2008 · nuclear astrophysics and the statistics of star formation

Edwin Ernest Salpeter, born in Vienna in 1924 and driven from Austria by the Anschluss, became one of the twentieth century's most versatile astrophysicists — the "Sal" of the Bethe–Salpeter equation in quantum field theory and, independently, the discoverer of the triple-alpha process by which stars fuse helium into carbon. In 1955, at Cornell, he took the observed luminosity function of solar-neighbourhood stars, corrected it for the mass–luminosity relation, for the fact that massive stars die young and are underrepresented today, and for the larger scale height of low-mass stars, and fitted the result with a single power law: the number of stars per unit mass falls as $m^{-2.35}$ . He called it the "original mass function"; the name "initial mass function" and the symbol $\xi$ came later, but the slope stuck. Its durability is what makes the achievement remarkable — modern determinations with millions of stars have refined the low-mass end into the Kroupa and Chabrier forms, yet above a solar mass Salpeter's 1955 slope remains the default assumption in extragalactic astronomy, a quiet power law extracted from a few hundred nearby stars that still anchors the field.

History

1955

— Salpeter fits ξ(m) ∝ m^−2.35 to the solar-neighbourhood luminosity function

1979

— Miller and Scalo derive a lognormal-like disk mass function and confirm the low-mass flattening

1998

— turbulent-fragmentation models link the power-law tail to supersonic turbulence in molecular clouds

2001

— Kroupa codifies the multi-part broken-power-law IMF and argues for its near-universality

2003

— Chabrier recasts the low-mass IMF as a lognormal peaking at m_c ≈ 0.2 M⊙

2010

— gravity-sensitive spectral features reveal a bottom-heavy IMF in massive elliptical galaxies

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., §12.2 (star formation) and §24.3 (galactic evolution) · Binney & Tremaine, Galactic Dynamics, 2nd ed., §1.2 · Mo, van den Bosch & White, Galaxy Formation and Evolution, §9.6 (the IMF and star formation) · Dodelson, Modern Cosmology (star-formation inputs to structure formation). Primary papers: Salpeter, ApJ 121, 161 (1955) · Miller & Scalo, ApJS 41, 513 (1979) · Kroupa, MNRAS 322, 231 (2001) · Chabrier, PASP 115, 763 (2003).

Provenance. Slopes and functional forms are taken directly from Salpeter (1955), Kroupa (2001), and Chabrier (2003) as summarized in Carroll & Ostlie §12.2/§24.3 and Binney & Tremaine §1.2. A verifier should recompute: (1) the convention shift $\alpha=\Gamma+1$ from the chain rule; (2) the weighting exponents $1-\alpha=-1.35$ , $2-\alpha=-0.35$ , $1+a-\alpha=+2.15$ for $a=3.5$ ; (3) the worked Salpeter cluster over $[0.1,100]\,M_\odot$ : $\langle m\rangle=0.35\,M_\odot$ , $N_{>1}/N=4.5\%$ , $M_{>1}/M=39\%$ , one core-collapse supernova per $\sim 135\,M_\odot$ ; (4) the Jeans mass $M_J\approx 5.5\,M_\odot$ at $T=10\,\mathrm{K}$ , $n=10^4\,\mathrm{cm^{-3}}$ , $\mu=2.3$ ; (5) the isothermal-to-adiabatic Jeans-mass turnaround $M_J\propto\rho^{-1/2}\to\rho^{+1/2}$ . Numerical constants (Salpeter $2.35$ ; Kroupa breaks at $0.08$ and $0.5\,M_\odot$ ; Chabrier $m_c\approx 0.2\,M_\odot$ , $\sigma\approx 0.55$ ) are standard tabulated values.

Prerequisites. Molecular Clouds and Giant Molecular Clouds · The Jeans Instability and Jeans Mass · Virial Theorem for Molecular Clouds. Leads to. Stellar Population Synthesis · Galactic Chemical Evolution · Star Formation Rate Indicators.

Ask the tutor: Want to see how switching from a Salpeter to a Chabrier IMF changes a galaxy's inferred star-formation rate, or work through why massive ellipticals might be bottom-heavy? Ask me to walk through it.

Is the IMF really universal, or does it change from place to place?

Within the Milky Way and its nearby neighbours the IMF looks remarkably constant: the same Kroupa/Chabrier form fits star clusters, the field, and dwarf galaxies over a wide range of density and metallicity, and this near-universality is one of the strongest empirical facts in star formation. But "near" is doing work. There is growing evidence that the cores of massive elliptical galaxies are bottom-heavy (more dwarfs than Chabrier predicts), inferred from gravity-sensitive spectral lines and from dynamical and lensing masses, and theoretical arguments that the first stars and some intense starbursts were top-heavy. So the honest answer is: universal to excellent approximation in normal environments, with real and important departures in extreme ones — and the physical reason it should be universal at all is not fully understood.

What is the difference between the IMF and the present-day mass function?

The IMF is the mass distribution of stars at birth; the present-day mass function (PDMF) is what you observe now. They differ because stars die. Massive stars have main-sequence lifetimes far shorter than the age of the Galaxy, so a population that has been forming stars for billions of years has lost most of its high-mass stars — the PDMF is depleted at the top relative to the IMF. To recover the IMF from the PDMF you multiply the high-mass end by $T_{\rm gal}/\tau_{\rm MS}(m)$ , the ratio of the galactic age to the stellar lifetime, which corrects for the generations of massive stars that lived and died. Below about a solar mass, where $\tau_{\rm MS}$ exceeds the age of the Universe, no stars have died and the PDMF and IMF coincide.

Why does confusing α = 2.35 with Γ = 1.35 matter so much?

Because a slope error of one full unit propagates into every integral. If you take Salpeter's logarithmic slope $\Gamma=1.35$ and mistakenly treat it as the linear slope $\alpha$ , then your mass integrand becomes $m^{1-1.35}=m^{-0.35}$ instead of the correct $m^{-1.35}$ — you have made the IMF a full power of $m$ shallower, which dramatically overweights massive stars and can change a mass-to-light ratio or a supernova rate by large factors. The two conventions describe the identical physical distribution, but they are coordinates, not values, and mixing them is like mixing radians with degrees. The fix is a reflex: whenever you read a slope, immediately ask "per $\mathrm{d}m$ or per $\mathrm{d}\log m$ ?" and convert with $\alpha=\Gamma+1$ before doing anything else.

Why is there a characteristic mass at all — why not stars of any size?

There is a smallest star (below $\sim 0.08\,M_\odot$ hydrogen fusion never ignites) and a largest (above $\sim 100$ – $300\,M_\odot$ radiation pressure and instabilities tear the star apart), but the interesting scale is the peak around $0.2$ – $0.3\,M_\odot$ . That peak reflects when fragmentation stops. Collapsing molecular gas cools efficiently and stays cold, so its Jeans mass falls as it densifies and it keeps breaking into smaller pieces — until a fragment becomes opaque to its own cooling radiation, heats up, and its Jeans mass turns around and rises. The mass at that turnaround, of order a few tenths of a solar mass, is the smallest fragment nature can build, and it plausibly imprints the characteristic mass of the IMF. It is a scaling argument, not a complete theory, but it captures the essential thermodynamics.

How do you measure an IMF for stars you cannot individually resolve?

For nearby clusters you count stars directly: build a luminosity function, convert magnitudes to masses with a mass–luminosity relation and stellar-evolution models, correct for unresolved binaries and for completeness at the faint end, and read off $\xi(m)$ . For distant galaxies, where individual stars blur together, you instead work backwards from integrated light: stellar-population-synthesis models predict a galaxy's spectrum, colours, and gravity-sensitive absorption features for an assumed IMF, and you ask which IMF best reproduces the data. Dynamical and gravitational-lensing masses give an independent handle — a bottom-heavy IMF adds mass without light, so a galaxy heavier than its light suggests one. Each method has degeneracies (age, metallicity, dark matter), which is exactly why IMF variations in ellipticals remain contested.

What sets the upper mass limit of the IMF?

Two things. First, formation: as a very massive protostar accretes, its luminosity climbs so steeply that radiation pressure on infalling dust can, in the simplest picture, halt accretion — though disks, radiation beaming, and fragmentation let real stars grow past that naive limit. Second, stability: above roughly $100$ – $150\,M_\odot$ stars approach the Eddington limit and are shaken by pulsational and radiation-driven instabilities that shed mass violently, so stars much heavier than this are rare and short-lived. Observationally the most massive stars known sit around $200$ – $300\,M_\odot$ in the youngest, densest clusters, and integrated-light modelling usually adopts an upper cutoff near $100$ – $120\,M_\odot$ . Because the light integral is dominated by the top of the range, this cutoff matters far more for luminosities and ionizing budgets than the far more numerous dwarfs at the bottom.