---
title: Protostars and Pre-Main-Sequence Evolution
draft: false
module: The Interstellar Medium
moduleNumber: 6
lessonNumber: 3
order: 603
summary: >
  A collapsing core becomes optically thick and forms a protostar that grows by
  accretion through a disk while driving bipolar outflows. The newborn star appears
  on the birthline and contracts down the fully convective Hayashi track, then
  crosses the radiative Henyey track to the zero-age main sequence, powered by
  gravitational contraction until hydrogen ignites. Below about 0.08 solar masses
  degeneracy halts contraction before ignition, dividing stars from brown dwarfs.
topics: [The Interstellar Medium]
sources:
  - book: Carroll & Ostlie
    ref: "Ch. 12 — The Interstellar Medium and Star Formation; §12.3 Pre-Main-Sequence Evolution"
  - book: Maoz
    ref: "Ch. 5 — Star Formation and Pre-Main-Sequence Evolution"
---

When a molecular cloud core collapses past the opacity limit, it stops radiating away
its compressional heat, becomes optically thick, and forms a small hydrostatic
**protostar** at its centre. The protostar is buried in the infalling envelope, grows
by accretion, and only later emerges as an optically visible **pre-main-sequence
star** that contracts toward the main sequence over millions of years. This lesson
follows that path: the accretion phase and the outflows and disks that accompany it,
the birthline where the star first becomes visible, the Hayashi and Henyey tracks
across the Hertzsprung–Russell diagram, arrival on the zero-age main sequence, and the
minimum mass below which hydrogen never ignites.

## Protostellar accretion and its luminosity

The collapsing core is not uniform: the centre reaches stellar density first and forms
a hydrostatic protostellar core, onto which the surrounding envelope rains down. The
**accretion rate** is set by the collapse of the cloud, roughly one Jeans mass per
free-fall time,

$$
\dot M \sim \frac{M_J}{t_{\text{ff}}} \sim \frac{c_s^3}{G}
\approx 2 \times 10^{-6}\ M_\odot\,\text{yr}^{-1}
$$

for a $10\ \text{K}$ core with sound speed $c_s \approx 0.2\ \text{km}\,\text{s}^{-1}$.
A solar mass assembles in a few times $10^5\ \text{yr}$. While the protostar is
accreting, its luminosity is dominated not by internal nuclear or contraction energy
but by the **accretion luminosity** — the gravitational energy released as each gram of
gas falls onto the stellar surface of radius $R$,

$$
L_{\text{acc}} = \frac{G M \dot M}{R}.
$$

> **Worked example.** A protostar of $M = 1\,M_\odot$ and radius $R \approx 3\,R_\odot$
> accreting at $\dot M = 10^{-5}\ M_\odot\,\text{yr}^{-1}$ radiates
> $$
> L_{\text{acc}} = \frac{G M \dot M}{R}
> \approx 4 \times 10^{28}\ \text{W} \approx 100\ L_\odot.
> $$
> Almost all of this emerges in the infrared, reprocessed by the dusty envelope, which
> is why the youngest protostars are luminous infrared sources with no optical
> counterpart. Their spectral energy distribution — a stellar photosphere buried under
> a cool dust envelope — defines the observational Class 0/I stages.

Accretion cannot proceed radially all the way to the surface because the infalling gas
carries angular momentum. It settles instead into a rotating **accretion disk**, and
matter spirals inward through the disk only as viscous stresses transport angular
momentum outward. A fraction of the accreting material is redirected along the
rotation axis and ejected as a pair of **bipolar outflows** — collimated jets that
plough into the surrounding cloud and shock it into the emission knots known as
Herbig–Haro objects. The disk and jets are the generic signature of a forming star.

$$
% caption: A protostar accretes through a rotating disk that carries angular
% momentum; part of the inflow is redirected into collimated bipolar jets.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% disk
\draw[very thick] (5.0,2.6) ellipse (3.0 and 0.65);
\node[anchor=west] at (8.1,2.6) {disk};
% central protostar
\fill (5.0,2.6) circle (4pt);
\node[anchor=north east] at (4.85,2.45) {protostar};
% bipolar jets
\draw[->, very thick] (5.0,3.1) -- (5.0,5.0);
\draw[->, very thick] (5.0,2.1) -- (5.0,0.2);
\node[anchor=west] at (5.15,4.6) {jet};
\node[anchor=west] at (5.15,0.6) {jet};
% infall arrows onto disk
\draw[->, black] (1.4,3.8) -- (2.6,3.0);
\draw[->, black] (8.6,3.8) -- (7.4,3.0);
\node[black, anchor=south] at (1.6,3.8) {infall};
\end{tikzpicture}
$$

## The birthline and the pre-main-sequence tracks

Once accretion ends and the envelope clears, the star becomes optically visible. Its
position when it first appears is not arbitrary: protostars of a given mass all emerge
at nearly the same place in the Hertzsprung–Russell diagram, along a locus called the
**birthline**, fixed by the accretion history. Stars are never observed above (younger
than) the birthline because at earlier stages they are still embedded and invisible.

Below the birthline the star is in hydrostatic equilibrium but not yet burning
hydrogen. It shines by **gravitational contraction**, releasing energy on the
Kelvin–Helmholtz timescale

$$
t_{\text{KH}} = \frac{G M^2}{R L} \approx 3 \times 10^{7}\ \text{yr}
$$

for solar values — the time to radiate away the gravitational binding energy at the
current luminosity. This is the duration of pre-main-sequence contraction, short
compared to the main-sequence lifetime but long compared to the accretion phase.

The newly revealed star is cool, extended, and **fully convective**: at low
temperature the opacity is high, radiation cannot carry the flux, and convection reaches
throughout. A fully convective star cannot be arbitrarily cool and remain in hydrostatic
equilibrium; there is a nearly vertical boundary in the H–R diagram, the **Hayashi
line**, to the cool (right) side of which no hydrostatic solution exists. The region
beyond it is the **Hayashi forbidden zone**. A contracting fully convective star sits
just to the warm side of this boundary and descends almost vertically — at nearly
constant effective temperature, with luminosity falling as the radius shrinks — along
its **Hayashi track**.[^co-pms]

$$
% caption: A contracting fully convective star cannot cross the Hayashi line into
% the forbidden zone; it descends the near-vertical Hayashi track at fixed color.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% axes: Teff decreasing to the right (astronomer convention)
\draw[->, black] (0,0) -- (10.4,0) node[right, black!70] {cooler};
\node[black!70, anchor=south west] at (0.1,0.05) {hotter};
\draw[->, black] (0,0) -- (0,5.6) node[above, black!70] {luminosity};
% forbidden zone shading to the right of the Hayashi line
\fill[black] (7.6,0.2) rectangle (10.2,5.4);
\node[black, anchor=north] at (8.9,5.2) {forbidden};
% Hayashi line (near vertical)
\draw[acc, very thick] (7.6,0.3) -- (7.2,5.3);
\node[acc, anchor=south, rotate=87] at (7.15,3.0) {Hayashi line};
% descending track just to its warm side
\draw[->, very thick] (7.0,4.9) -- (6.85,1.2);
\node[anchor=east] at (6.8,3.0) {track};
\end{tikzpicture}
$$

As the star contracts and its interior heats, the central opacity drops and a
**radiative core** grows outward from the centre. Once radiation carries the flux
through the interior, the star leaves the Hayashi track and evolves nearly horizontally
to the left — increasing effective temperature at almost constant luminosity — along
the **Henyey track**, until the central temperature reaches hydrogen ignition and it
settles onto the **zero-age main sequence** (ZAMS). Low-mass stars spend most of their
pre-main-sequence life on the near-vertical Hayashi track; high-mass stars develop
radiative cores early and cross almost entirely along the Henyey track.

$$
% caption: Pre-main-sequence tracks: low-mass stars descend the Hayashi track, then
% all cross leftward along Henyey tracks onto the zero-age main sequence.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (10.4,0) node[right, black!70] {cooler};
\node[black!70, anchor=south west] at (0.1,0.05) {hotter};
\draw[->, black] (0,0) -- (0,5.8) node[above, black!70] {luminosity};
% ZAMS diagonal
\draw[very thick] (1.2,5.2) -- (6.4,1.0);
\node[rotate=-38, anchor=south] at (3.6,3.3) {ZAMS};
% high-mass Hayashi descent then Henyey to upper ZAMS
\draw[thick, densely dashed] (7.6,5.2) .. controls (6.2,5.1) and (3.0,5.0) .. (1.5,5.0);
\node[anchor=west] at (7.7,5.2) {high mass};
% low-mass long Hayashi descent then short Henyey
\draw[thick] (9.2,4.6) -- (8.9,1.9);
\draw[thick] (8.9,1.9) .. controls (7.8,1.7) and (6.6,1.4) .. (5.9,1.25);
\node[anchor=west] at (9.3,4.6) {low mass};
\node[anchor=north, rotate=84] at (8.75,3.3) {Hayashi};
\node[anchor=north, font=\scriptsize] at (7.0,1.35) {Henyey};
\end{tikzpicture}
$$

## T Tauri stars and protoplanetary disks

The optically visible low-mass pre-main-sequence stars are the **T Tauri stars**:
irregularly variable, with strong chromospheric and X-ray activity, broad emission
lines from accreting gas, and infrared excesses from a surviving circumstellar disk.
The disk, now a **protoplanetary disk**, continues to feed the star at declining rates
($10^{-8}\ M_\odot\,\text{yr}^{-1}$ and below) while its outer regions cool and, over a
few million years, assemble planetesimals and planets. Angular-momentum transport
through the disk, driven by magnetorotational turbulence, is what allows the gas to
accrete inward while the disk as a whole spreads outward. The more massive analogues,
the Herbig Ae/Be stars, occupy the same evolutionary stage at higher mass. The
outflows persist in this phase as the jets that carve Herbig–Haro flows into the parent
cloud.

## The zero-age main sequence and the hydrogen-burning limit

Contraction ends when the central temperature reaches about $10^7\ \text{K}$ and
hydrogen fusion turns on. Nuclear energy generation then replaces gravitational
contraction as the luminosity source, the star stops shrinking, and it arrives on the
zero-age main sequence, where it will spend the bulk of its life. The pre-main-sequence
contraction time scales inversely with mass through $t_{\text{KH}}$: a $30\,M_\odot$
star reaches the main sequence in some $10^4\ \text{yr}$, a solar-mass star in about
$3\times10^7\ \text{yr}$, and a $0.1\,M_\odot$ star takes longer than $10^8\ \text{yr}$.

Whether hydrogen ignites at all depends on the mass. As a low-mass object contracts, its
core density rises, and electron **degeneracy pressure** — which does not depend on
temperature — begins to support it before it can heat further. If degeneracy halts the
contraction before the centre reaches the hydrogen ignition temperature, sustained
fusion never starts. The dividing mass is about

$$
M_{\min} \approx 0.08\ M_\odot,
$$

the **hydrogen-burning minimum mass**. Objects above it become stars; objects below it
are **brown dwarfs**, which fuse deuterium briefly (above roughly $13\ M_{\text{Jup}}$)
but never sustain hydrogen burning, and thereafter cool and fade indefinitely. The
threshold separates the stellar main sequence from the substellar regime.[^maoz-pms]

$$
% caption: The peak central temperature reached during contraction rises with mass;
% only above about 0.08 solar masses does it cross the hydrogen ignition threshold.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (10.4,0) node[right, black!70] {log mass};
\draw[->, black] (0,0) -- (0,5.4) node[above, black!70] {peak central T};
% rising central temperature with mass
\draw[very thick] (0.6,0.7) .. controls (3.0,1.8) and (5.0,3.6) .. (9.8,4.9);
% ignition threshold line
\draw[black, densely dashed] (0,3.0) -- (10.2,3.0);
\node[black!70, anchor=west, font=\scriptsize] at (7.0,3.25) {hydrogen ignition};
% crossing point
\fill[acc] (4.35,3.0) circle (2.4pt);
\draw[black, densely dotted] (4.35,0) -- (4.35,3.0);
\node[black, anchor=north, font=\scriptsize] at (2.3,0.9) {brown dwarfs};
\node[black, anchor=north, font=\scriptsize] at (7.4,0.9) {stars};
\end{tikzpicture}
$$

$$
% caption: The mass sequence from planets through deuterium-burning brown dwarfs to
% hydrogen-burning stars, divided at the deuterium and hydrogen ignition limits.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black, thick] (0.3,2.6) -- (10.1,2.6);
% dividers
\draw[acc, very thick] (3.4,2.1) -- (3.4,3.1);
\draw[acc, very thick] (6.6,2.1) -- (6.6,3.1);
% region labels
\node[black!70] at (1.8,3.5) {planets};
\node[black!70] at (5.0,3.5) {brown dwarfs};
\node[black!70] at (8.4,3.5) {stars};
% boundary captions below
\node[acc, anchor=north, font=\scriptsize] at (3.4,2.05) {deuterium limit};
\node[acc, anchor=north, font=\scriptsize] at (6.6,2.05) {hydrogen limit};
\node[black, anchor=north] at (10.1,2.05) {increasing mass};
\end{tikzpicture}
$$

The path from a collapsing core to a main-sequence star spans the accretion phase, the
Hayashi and Henyey contraction tracks, and the ignition of hydrogen — or, below the
minimum mass, the failure to ignite it. A star that reaches the zero-age main sequence
begins the long, stable hydrogen-burning phase whose structure and lifetime are the
subject of [the main
sequence](/astrophysics-cosmology/stellar-evolution/the-main-sequence-and-its-structure).

[^co-pms]: Carroll & Ostlie, §12.3 — Pre-Main-Sequence Evolution: the birthline, the Hayashi and Henyey tracks, T Tauri stars, and arrival on the main sequence.
[^maoz-pms]: Maoz, §5.4 — protostellar accretion, pre-main-sequence contraction, the Kelvin–Helmholtz timescale, and the hydrogen-burning minimum mass.