Hayashi limit

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. This is an extremely crude model of what occurs in convective stars, but it has good qualitative agreement with the full model with less complications. We follow the derivation in Kippenhahn, Weigert, and Weiss in Stellar Structure and Evolution.^[4]

Nearly all of the interior part of convective stars has an adiabatic stratification (corrections to this are small for fully convective regions), such that

${\displaystyle {\frac {\delta \ln T}{\delta \ln P}}=\nabla _{\text{adiabatic}}=0.4}$, which holds for an adiabatic expansion of an ideal gas.

We assume that this relation holds from the interior to the surface of the star—the surface is called photosphere. We assume ${\displaystyle \nabla _{\text{adiabatic}}}$ to be constant throughout the interior of the star with value 0.4. However, we obtain the correct distinctive behavior.

For the interior we consider a simple polytropic relation between P and T:

${\displaystyle P=CT^{(1+n)}}$

With the index ${\displaystyle n=3/2}$.

We assume the relation above to hold until the photosphere where we assume to have a simple absorption law

${\displaystyle \kappa =\kappa _{0}P^{a}T^{b}}$

Then, we use the hydrostatic equilibrium equation and integrate it with respect to the radius to give us

${\displaystyle P_{0}={\text{Constant}}*\left({\frac {M}{R^{2}}}T_{\text{eff}}^{-b}\right)^{\frac {1}{1+a}}}$

For the solution in the interior we set ${\displaystyle P=P_{0}}$ ; ${\displaystyle T=T_{\text{eff}}}$ 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:

${\displaystyle L=4\pi R^{2}\sigma \,T_{\text{eff}}^{4}}$.

Thus, any value of R corresponds to a certain point in the Hertzsprung–Russell diagram.

Finally, after some algebra this is the equation for the Hayashi limit in the Hertzsprung–Russell diagram:

${\displaystyle \log(T_{\text{eff}})=A\log(L)+B\log(M)+{\text{constant}}}$ ^[4]

With coefficients

${\displaystyle A={\frac {0.75a-0.25}{b-5.5a+1.5}}}$, ${\displaystyle B={\frac {0.5a-1.5}{b-5.5a+1.5}}}$

Takeaways from plugin in ${\displaystyle a\approx 1}$ and ${\displaystyle b\approx 3}$ for a cool hydrogen ion dominated atmosphere opacity model (${\displaystyle T<5000{\text{ K}}}$):

• The Hayashi limit must be far to the right in the Hertzsprung–Russell diagram which means temperatures have to be low.
• The Hayashi limit must be very steep. The gradient of Luminosity with respect to temperature has to be large.
• The Hayashi limit shifts slightly to the left in the Hertzsprung–Russell diagram for increasing M.

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. To the left of the Hayashi limit, we have ${\displaystyle \nabla <\nabla _{\text{adiabatic}}}$ and some part of the model is radiative. The model is fully convective at the Hayashi limit with ${\displaystyle \nabla =\nabla _{\text{adiabatic}}}$. Models to the right of the Hayashi limit should have ${\displaystyle \nabla >\nabla _{\text{adiabatic}}}$.

If a star is formed such that some region in its deep interior has large ${\displaystyle \nabla -\nabla _{\text{adiabatic}}>0}$ large convective fluxes with velocities ${\displaystyle v_{\text{convective}}\approx (\nabla -\nabla _{\text{adiabatic}})/2}$. The convective fluxes of energy cooldown the interior rapidly until ${\displaystyle \nabla =\nabla _{\text{adiabatic}}}$ and the star has moved to the Hayashi limit. 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. ^[4]

• Eddington limit