Star Formation
The Hayashi Track
Pre-main-sequence stars descending the H-R diagram — fully convective, constant Teff, shrinking radius
The Hayashi track is the near-vertical path on the H-R diagram of pre-main-sequence stars contracting toward the main sequence. Constant Teff ~4000 K, shrinking radius, decreasing luminosity. A 1 M⊙ star shrinks from 0.5 AU to 0.005 AU over 30 Myr.
- DiscovererChushiro Hayashi (1961)
- Effective temperature~3500-4500 K (pinned by H⁻ opacity)
- Interior structureFully convective
- 1 M⊙ duration~107 yr Hayashi + ~2-3 × 107 yr Henyey
- Radius change0.5 AU → 0.005 AU (factor ~100)
- Forbidden zoneTo the right of the track: no hydrostatic star
Interactive visualization
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Why a Sun-like star spends 30 million years before fusing hydrogen
A protostar reaches stellar density before its core can sustain hydrogen burning. Gravitational contraction continues, releasing energy at the rate of half the lost gravitational potential energy — the virial theorem. The star is in quasi-hydrostatic equilibrium but not in nuclear equilibrium: it shines, but with energy from gravity, not fusion. Hayashi (1961) showed that during this phase the star traces a very specific path on the Hertzsprung-Russell diagram — a path now bearing his name.
Two regimes dominate the path. While the star is fully convective — which it always is at the beginning — the photospheric temperature is locked by the H- opacity at ~4000 K, and the star contracts at fixed Teff, dropping L (because L = 4πR²σTeff4 and R is shrinking). This is the Hayashi track — a near-vertical descent on the log L vs log Teff diagram. Once a radiative core develops, the photospheric thermostat relaxes, Teff begins to rise, and the star moves leftward at roughly constant L — the Henyey track. Hydrogen fusion ignites only at the end of the Henyey segment, when central T crosses ~1.5 × 107 K. The star then settles onto the zero-age main sequence.
Why the photospheric temperature is locked
Negative hydrogen ions (H-) dominate the opacity in cool stellar atmospheres — an extra electron loosely bound to a neutral H atom. H- opacity has a violently steep temperature dependence:
κ_(H⁻) ∝ ρ^(1/2) T^9 (for T < 6000 K, partial ionisation)If the photosphere tries to cool, opacity plunges, radiation escapes more freely, and the photosphere is exposed to deeper hotter layers — Teff rises. If it tries to heat, opacity rockets, radiation is trapped, the photosphere puffs up, deeper layers are insulated, and Teff falls. The stiff opacity-temperature thermostat pins Teff in a narrow window (3500-4500 K for typical M-to-K-spectral-type pre-main-sequence stars). A fully convective star must sit on the Hayashi line; there is nowhere else to go.
To the right of the Hayashi line on the HR diagram lies the "forbidden zone": no hydrostatic star can have Teff below the Hayashi value without collapsing further. Pre-main-sequence stars descend the line; they cannot drift right.
Worked example — 1 M⊙ star, top to bottom
Take a 1 M⊙ protostar just emerged from its envelope, mass M = 1 M⊙, radius R = 100 R⊙ = 0.466 AU, Teff = 4200 K. We can compute its initial luminosity and the Kelvin-Helmholtz contraction timescale:
L = 4π R² σ T_eff⁴
= 4π · (100 R_⊙)² · (5.67 × 10⁻⁸ W/m²K⁴) · (4200 K)⁴
= 4π · (6.96 × 10¹⁰ m)² · 5.67 × 10⁻⁸ · 3.11 × 10¹⁶
≈ 1.35 × 10²⁸ W
≈ 35 L_⊙ (so L ≈ 35 L_⊙ at top of Hayashi track)
t_KH = G M² / (R L)
= (6.67 × 10⁻¹¹) · (1.989 × 10³⁰)² / [(6.96 × 10¹⁰) · (1.35 × 10²⁸)]
≈ 2.8 × 10¹³ s
≈ 0.9 Myr
Half-way down (R = 10 R_⊙):
L ≈ 35 · (10/100)² ≈ 0.35 L_⊙ (but Teff still ~4200 K, so L ∝ R²)
Actually Teff rises slightly to ~4400 K when radiative core appears.
At ZAMS (Sun's properties):
R = 0.9 R_⊙, L = 1.0 L_⊙, T_eff = 5777 K, age = 30 MyrSo the star drops in luminosity by a factor of ~35 (3.5 dex on a log plot) during the Hayashi phase as the radius shrinks by a factor of 10, then moves leftward at roughly constant L during the Henyey phase as Teff doubles from 4400 K to 5800 K. Total time from envelope shedding to ZAMS: ~30 Myr. The star never accretes a single fusion energy quantum during the trip — all of its 1043 J of radiated energy comes from gravitational potential.
How the track depends on stellar mass
| Mass (M⊙) | Hayashi Teff (K) | Top of track L (L⊙) | Top of track R (R⊙) | Time to ZAMS | Note |
|---|---|---|---|---|---|
| 0.1 | ~3000 | ~0.5 | ~3 | ~100 Myr+ | Fully convective throughout — no Henyey phase |
| 0.3 | ~3300 | ~2 | ~5 | ~150 Myr | Borderline; brief Henyey phase |
| 0.5 | ~3700 | ~5 | ~10 | ~100 Myr | Modest Henyey track present |
| 1.0 | ~4200 | ~35 | ~100 | ~30 Myr | Canonical Sun-like; clear Hayashi + Henyey |
| 2.0 | ~4500 | ~150 | ~200 | ~10 Myr | Intermediate mass; pronounced Henyey turn |
| 5.0 | ~4600 | ~1000 | ~400 | ~1 Myr | Massive: little Hayashi, mostly Henyey/radiative |
| 15 | ~4700 | ~10000 | ~600 | ~0.1 Myr | O-star; reaches main sequence as still embedded protostar |
The pattern: low-mass stars are mostly Hayashi (slow descent, no left-turn). Sun-like stars do both. Massive stars are mostly Henyey, reaching the main sequence so quickly they often emerge from their natal cloud already burning hydrogen.
A timeline of pre-main-sequence theory
- 1862. Kelvin estimates the Sun's gravitational-contraction lifetime at ~20 Myr — consistent with geological data of the time but in conflict with Darwin's estimate of much longer biological history.
- 1907. Helmholtz formulates the gravitational contraction mechanism rigorously.
- 1931. Henyey, LeLevier, Levée compute radiative-core models of contracting stars, predicting horizontal HR-diagram motion.
- 1955. Henyey track paper formalises the radiative-core pre-main-sequence segment.
- 1961. Chushiro Hayashi publishes Stellar Evolution in Early Phases of Gravitational Contraction in Pub. Astron. Soc. Japan, identifying the convective Hayashi track and the forbidden zone.
- 1962. Hayashi and collaborators (Hoshi, Sugimoto) extend the calculation to a grid of masses, producing the first complete pre-main-sequence isochrones.
- 1981. Iben's textbook calculations refine the tracks for modern stellar interiors codes.
- 1997. Baraffe, Chabrier, Allard models extend pre-main-sequence isochrones down to brown dwarfs and substellar masses.
- 2000s. PMS evolutionary tracks become standard tools for age-dating young clusters via HR-diagram fitting.
- 2018. Gaia DR2 parallaxes give 1% precision distances to nearby young clusters — observed HR diagrams match theoretical Hayashi tracks at the 5-10% level.
What the track determines
- Cluster ages. Plotting pre-main-sequence stars in a young cluster on the HR diagram and comparing with theoretical Hayashi isochrones gives ages with ~30% precision — the dominant clock for clusters younger than ~30 Myr.
- Initial mass function. Counting low-mass and intermediate-mass pre-main-sequence stars in young clusters and back-projecting to ZAMS positions reconstructs the IMF.
- Lithium depletion. Stars on the Hayashi track are fully convective, so they mix surface lithium into the hot interior where it burns. The depth of lithium destruction is a precise function of pre-main-sequence age — the "lithium clock."
- Stellar moment of inertia. Convective interiors are well-mixed; rotation is solid-body. As radiative cores form during the Henyey phase, rotation decouples and the surface spins down via magnetic winds.
- Deuterium burning. At ~106 K, deuterium fuses, providing a brief energy boost. Below 13 MJup, deuterium burning is the only nuclear reaction; this defines the planet / brown-dwarf boundary.
Common misconceptions
- "The Hayashi track is the only path for young stars." No — high-mass stars (> 5 M⊙) spend most of their pre-main-sequence time on the Henyey track. The Hayashi phase is dominant for solar-mass and lower stars.
- "Stars contract at constant luminosity." No — on the Hayashi track, Teff is constant but L drops as R². On the Henyey track L is roughly constant while Teff rises.
- "The track is a straight line." No — it is slightly inclined (a few percent slope on the log-log HR diagram) and curves into the Henyey track at the convective-to-radiative-core transition.
- "Hayashi was working alone in the 1960s in isolation." No — Hayashi had access to international stellar-evolution literature and built on Henyey's 1955 work. The Hayashi-Henyey two-phase picture is collaborative.
- "The forbidden zone is empty." Strictly, no hydrostatic star can lie there. But protostars in dynamical collapse (Class 0) can briefly cross into the forbidden zone before reaching hydrostatic equilibrium.
- "Brown dwarfs don't have Hayashi tracks." They do — just with a steeper slope, and they never leave to a Henyey phase, since they never become radiative enough.
Open questions
- Accretion luminosity offsets. Real pre-main-sequence stars are still accreting from their disks; the accretion shock adds 10-50% luminosity. This shifts observed HR-diagram positions and complicates age fitting.
- Birthline location. Where exactly do stars first become hydrostatic? Stahler-Palla "birthline" calculations from accretion-shock thermodynamics give predictions that observed protostars roughly follow.
- Magnetic effects. Strong surface magnetic fields modify the convective structure and shift the Hayashi track location at the 0.1 dex level — affecting age estimates.
- Brown-dwarf transition. Below ~0.08 M⊙, hydrogen never ignites and the Hayashi-track descent continues indefinitely.
Frequently asked questions
What is the Hayashi track?
The Hayashi track is the nearly-vertical path traced on the Hertzsprung-Russell diagram by a pre-main-sequence star as it gravitationally contracts toward the main sequence. Discovered by Chushiro Hayashi in 1961, it marks the locus of fully convective stars: their effective temperature is pinned near ~3500-4500 K by the steep temperature dependence of H⁻ opacity in cool stellar atmospheres. As the star contracts at constant Teff, luminosity decreases (L ∝ R²), so on a log L vs log Teff diagram the star moves nearly straight down. The Hayashi track is the right-hand 'forbidden zone' boundary on the HR diagram — no hydrostatic star can sit at lower Teff.
Why is Teff nearly constant during contraction?
The H⁻ opacity in cool stellar atmospheres rises steeply with temperature: κ ∝ ρ^0.5 T^9 or so. This means even a small drop in Teff causes opacity to plunge, the photosphere to recede, and Teff to rise back — a stiff thermostat. The result is that fully convective stars are locked at Teff between 3500 K (low-mass M dwarfs) and 4500 K (more massive protostars), regardless of luminosity. The star is free to change L (by changing R) but not Teff.
How long does the Hayashi phase last?
The Kelvin-Helmholtz timescale: t_KH = GM²/(R L). For a 1 M_⊙ star at the top of the Hayashi track (R ~ 0.5 AU = 100 R_⊙, L ~ 20 L_⊙), t_KH ~ 10⁶ yr. As the star contracts, R decreases faster than L, so the contraction accelerates. The total time on the Hayashi track for a Sun-like star is ~10⁷ yr; the subsequent Henyey track adds ~2-3 × 10⁷ yr; total pre-main-sequence time ~30 Myr. Massive stars (5+ M_⊙) contract in 10⁵-10⁶ yr; low-mass M dwarfs spend 100 Myr or more on the Hayashi track and may never leave it before reaching their hydrogen-burning main sequence.
What's the Henyey track?
The Henyey track (Henyey, LeLevier, Levée 1955) is the nearly-horizontal evolutionary segment on the HR diagram where a contracting pre-main-sequence star moves leftward (toward higher Teff) at roughly constant L. The transition from Hayashi to Henyey track happens when central temperature is high enough to make the core radiative (~10⁷ K). Once radiative cores form, the effective-temperature thermostat releases, and the star heats up as it continues contracting. Stars >0.5 M_⊙ all traverse a Henyey track; stars <0.35 M_⊙ remain fully convective and never have a Henyey phase.
When did Hayashi discover this?
Chushiro Hayashi published 'Stellar Evolution in Early Phases of Gravitational Contraction' in Publications of the Astronomical Society of Japan in 1961, demonstrating that fully convective stars must lie along a specific locus in the HR diagram set by the photospheric H⁻ opacity. Before Hayashi, theoretical models of pre-main-sequence evolution started lower on the HR diagram; he showed the right starting point is much higher and to the right — a fundamental correction. Hayashi (1920-2010) spent most of his career at Kyoto University; his other major contributions include big-bang nucleosynthesis (Hayashi 1950) and the minimum-mass solar nebula (Hayashi 1981).
How does the radius change?
Dramatically. A 1 M_⊙ star starts at the top of its Hayashi track with R ~ 100 R_⊙ ~ 0.5 AU and L ~ 20 L_⊙ at age 10⁵ yr. By age 10⁶ yr it has contracted to R ~ 30 R_⊙. By age 10⁷ yr R ~ 3 R_⊙. At the start of the Henyey turn (~10⁷ yr) R ~ 1.5 R_⊙. By the time it joins the zero-age main sequence at age 30 Myr, R ~ 0.9 R_⊙ = 0.0042 AU. So in 30 Myr the radius shrinks by a factor of ~100 (0.5 AU → 0.005 AU), while L drops from ~20 L_⊙ to 1 L_⊙ and core T rises from ~10⁵ K to 1.5 × 10⁷ K — enough to ignite p-p fusion.
How is the track tested observationally?
By dating pre-main-sequence stars in young clusters and plotting them on the HR diagram. Open clusters with known ages (Orion Nebula Cluster ~2 Myr, Taurus 1-3 Myr, ρ Oph ~1 Myr, NGC 2264 ~3 Myr, Upper Sco 5-10 Myr) provide samples where each member has identical age but a range of masses; their distribution on the HR diagram traces theoretical isochrones for various masses, all of which converge to a Hayashi track for low-mass members. Gaia parallaxes since 2018 give precise distances and luminosities; the agreement between observed and theoretical Hayashi tracks at 5-10% level validates the underlying convective/radiative physics.