Hayashi limit
Maximum radius constraint for stars in hydrostatic equilibrium.
The Hayashi limit is a theoretical constraint on the maximum radius a star of a given mass can have while in hydrostatic equilibrium. On a Hertzsprung–Russell diagram, it appears as a nearly vertical line near 2,500 K, marking a boundary beyond which fully convective stellar models have no solution. This limit governs key phases of stellar evolution, including the contraction of protostars and the expansion of red giants.
Quick Facts
- Named after
- Chūshirō Hayashi
- Temperature on h-r diagram
- about 2,500 K
- Recognized in paper
- 1961
Facts from the source article.
Derivation of the limit
A simple model for a fully convective star yields a relation between luminosity, temperature, and pressure from which the Hayashi limit can be derived. The interior of such a star is assumed to have an adiabatic stratification, with the relation holding from the interior to the photosphere. A polytropic relation between pressure and temperature is used, with the index set to a constant value of 0.4. At the photosphere, a simple absorption law is assumed. Hydrostatic equilibrium is integrated with respect to radius, and the pressure is eliminated from the resulting equation. Luminosity is given by the Stefan–Boltzmann law for a perfect black body. After algebra, the equation for the Hayashi limit in the Hertzsprung–Russell diagram is obtained, with specific coefficients. For a cool hydrogen ion dominated atmosphere opacity model, the limit lies far to the right (low temperatures), is very steep (large luminosity gradient with temperature), and shifts slightly leftward for increasing mass. These predictions agree with numerical simulations.
What happens when stars cross the limit
To the left of the Hayashi limit, a star has some radiative region; at the limit it is fully convective; to the right, models would require a condition that is not realized. If a star forms with a region where the temperature gradient exceeds the adiabatic gradient, large convective fluxes cool the interior rapidly until the gradient equals the adiabatic value and the star reaches the Hayashi limit. Even a small excess can transport energy from the deep interior to the surface within the convective adjustment timescale, which is longer than hydrodynamic timescales but shorter than the thermal timescale. Thus the Hayashi limit separates a stable allowed region (left) from an unstable forbidden region (right) for stars in hydrostatic equilibrium with fully adjusted convection.
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