Chapter 14. Protostar

Hayashi's Line (Track)

All types of stars have a radiative layer at the surface, even including low-mass stars, of which the outermost region is super-adiabatic. As a result, the following conditions are satisfied.

For the hydrostatic equation,

$$\int_R^\infty\text dr\frac{\text dP}{\text dr}=-\int_R^\infty\rho\frac{GM}{r^2}\text dr$$

Here the lower limit of the integral is taken as a certain radius $R$ close to the stellar surface, while the upper limit is formalistically taken to be infinity. The integrant in the RHS of the equation, however, contributes mainly in a small region around $R$. As a result,

$$P_\infty-P_R=-\frac{GM}{R^2}\int_R^\infty\rho\text dr=-\frac{GM}{R^2}\frac1{\langle\kappa\rangle}\int_R^\infty\rho\kappa\text dr$$

where

$$\langle\kappa\rangle\equiv\frac{\int_R^\infty\rho\kappa\text dr}{\int_R^\infty\rho\text dr}$$

Now we consider a special $R$ so that

$$\tau\equiv\int_R^\infty\rho\kappa\text dr=\frac23$$

This is the so-called photospheric condition and gives the definition of the photosphere radius $R$. For simplicity we set $P_\infty\sim0$, so we have

$$P_R=\frac23\frac{GM}{R^2}\frac1{\langle\kappa\rangle}\simeq\frac23\frac{GM}{R^2}\frac1{\kappa_0\rho_R^aT_\text{eff}^b}$$

Let's consider a fully convective star within radius $R$, of which the main atmospheric opacity comes from $\ce{H-}$, and $a\sim0.5, b\sim9$. This is usually the case for protostars and red giants with huge, cool, convective envelope. Now that $\nabla\simeq\nabla_\text{ad}$, the EoS is

$$P_R=K\rho_R^{5/3}\equiv EGM^{1/3}R\rho_R^{5/3}$$

where $E\equiv0.4242$ is a numerical factor by solving Lane-Emben equation of $n=3/2$ numerically. And for ideal gas, this EoS is equivalent to

$$P_R=\frac{k_B}{\mu m_\text p}\rho_RT_\text{eff}$$

We have substituted $TR$ with $T\text{eff}$ because the effective temperature also denotes the temperature at the photoshpere.

Finally, the luminosity is given by

$$L=4\pi\sigma_{SB}R^2T_\text{eff}^4$$

For given $M$, we have four equations and five unknown quantities ($L, R, T_\text{eff}, \rho_R, P_R$), so they can be reduced to ONE final relation,

$$\log T_\text{eff}=A\log L+B\log M+C$$

$$A=\frac12\cdot\frac{3a-1}{9a+2b+3},\quad B=\frac{a+3}{9a+2b+3}$$

Recall that $a\ll b$ for $\ce{H-}$ opacity, thus $A, B\simeq0$. Precisely, $A\simeq 0.01, B\simeq 0.14$, so $T_\text{eff}$ does not change a lot when $L$ and $M$ vary.

Notes

• For given mass, the Hayashi's line is a very steep, almost vertical line in the HR diagram.

• The dependency of position of Hayashi's line on $M$ is also weak, though $T_\text{eff}$ would slightly increase for higher mass.

• The solution is realized when $E=E_\text{cr}=0.4242$. For fixed $M$ and $L$, if we allow $E$ to vary,

  $$\log T_\text{eff}=3B\log E+C'$$

  It seems that if $E$ goes down, so will $T\text{eff}$, and the stars goes to the right hand side, known as the forbidden region. But if $E<E\text{cr}$, the Lane-Emden solution becomes collapse-type, which is unphysical (Hayashi 1961). On the contrary, $E$ could be larger than $E_\text{cr}$, when the actual stellar structure is given by a nested polytrope (radiative core + convective envelope).

  
• In Hayashi's original paper, the definition of $E$ is different from ours.

  $$E'\equiv(2.5)^{3/2}\xi_1^{2.5}\theta_1'^{0.4}=4\pi E^{-3/2}=45.48$$

  and

  $$\Rightarrow T_\text{eff}=-2B\log E'+C''$$