The Hayashi limit is a theoretical constraint upon the maximum radius of a star for a given mass.
When a star is fully within hydrostatic equilibrium—a condition where the inward force of gravity is matched by the outward pressure of the gas—the star can not exceed the radius defined by the Hayashi limit.
This has important implications for the evolution of a star, both during the formulative contraction period and later when the star has consumed most of its hydrogen supply through nuclear fusion.
[1] A Hertzsprung-Russell diagram displays a plot of a star's surface temperature against the luminosity.
On this diagram, the Hayashi limit forms a nearly vertical line at about 3,500 K. The outer layers of low temperature stars are always convective, and models of stellar structure for fully convective stars do not provide a solution to the right of this line.
Thus in theory, stars are constrained to remain to the left of this limit during all periods when they are in hydrostatic equilibrium, and the region to the right of the line forms a type of "forbidden zone".
These include collapsing protostars, as well as stars with magnetic fields that interfere with the internal transport of energy through convection.
[2] Red giants are stars that have expanded their outer envelope in order to support the nuclear fusion of helium.
However, they are constrained by the Hayashi limit not to expand beyond a certain radius.
Stars that find themselves across the Hayashi limit have large convection currents in their interior driven by massive temperature gradients.
Additionally, those stars states are unstable so the stars rapidly adjust their states, moving in the Hertzprung-Russel diagram until they reach the Hayashi limit.
[3] When lower mass stars in the main sequence start expanding and becoming a red giant the stars revisit the Hayashi track.
The Hayashi limit constrains the asymptotic giant branch evolution of stars which is important in the late evolution of stars and can be observed, for example, in the ascending branches of the Hertzsprung–Russell diagrams of globular clusters, which have stars of approximately the same age and composition.
This late recognition may be because the properties of the Hayashi track required numerical calculations that were not fully developed before.
[4] We can derive the relation between the luminosity, temperature and pressure for a simple model for a fully convective star and from the form of this relation we can infer the Hayashi limit.
We follow the derivation in Kippenhahn, Weigert, and Weiss in Stellar Structure and Evolution.
, which holds for an adiabatic expansion of an ideal gas.
Then, we use the hydrostatic equilibrium equation and integrate it with respect to the radius to give us
in the P-T relation and then eliminate pressure of this equation.
Luminosity is given by the Stephan-Boltzmann law applied to a perfect black body:
Finally, after some algebra this is the equation for the Hayashi limit in the Hertzsprung–Russell diagram:
for a cool hydrogen ion dominated atmosphere oppacity model (
): These predictions are supported by numerical simulations of stars.
[4] Until now we have made no claims on the stability of locale to the left, right or at the Hayashi limit in the Hertzsprung–Russell diagram.
The model is fully convective at the Hayashi limit with
If a star is formed such that some region in its deep interior has large
The convective fluxes of energy cooldown the interior rapidly until
In fact, it can be shown from the mixing length model that even a small excess can transport energy from the deep interior to the surface by convective fluxes.
This will happen within the short timescale for the adjustment of convection which is still larger than timescales for non-equilibrium processes in the star such as hydrodynamic adjustment associated with the thermal time scale.
Hence, the limit between an “allowed” stable region (left) and a “forbidden” unstable region (right) for stars of given M and composition that are in hydrostatic equilibrium and have a fully adjusted convection is the Hayashi limit.