Chapter 11. Main Sequence

Let us consider chemically homogeneous stars, that is, $\mu$ is a constant throughout the star.

Analytical Solutions

Energy Conservation

Assuming that

$$\varepsilon_\text{nuc}\simeq\varepsilon_0\rho^kT^s$$

Since the temperature is closely related to the total gravitational potential

$$\frac{k_BT}{\mu m_\text p}\sim\frac{GM}R$$

we have

$$L=\langle\varepsilon_\text{nuc}\rangle M\propto\left(\frac{M}{R^3}\right)^k\left(\frac{\mu M}{R}\right)^sM=M^{k+s+1}R^{-3k-s}\mu^s$$

Energy Transport

$$L\sim\frac{R}{\rho\kappa}T^4\propto M^{3+r-\alpha}R^{3\alpha-r}\mu^{4+r}$$

where

$$\kappa\propto \rho^\alpha T^{-r}$$

Finally

$$R\propto M^{1-\frac{2(k+\alpha+1)}l}\mu^{1-\frac{3k+3\alpha+4}l}$$

$$L\propto M^{3+2\alpha-\frac{2m(k+\alpha+1)}{l}}\mu^{4+3\alpha-\frac{m(3k+3\alpha+4)}{l}}$$

where

$$l=s+3k+3\alpha-r,\quad m=3\alpha-r$$

For the Sun, in pp-chain burning, $k=1,s=5$; the opacity mainly comes from electron scattering, so $\alpha=0,r=0$, then we have

$$R\propto M^{1/2}\mu^{1/8},\quad L\propto M^3\mu^4$$

More precisely, $R\propto M^{0.57-0.8}$, $L\propto M^{3.2-3.9}$.

For massive stars, the luminosity is asymptotically Eddington luminosity, which is proportional to $M$. For low-mass stars, free-free transition dominates the opacity, and $\kappa$ is given by Krammer's law $\kappa\propto \rho T^{-7/2}$, so that $\alpha=1$ and $r=7/2$. This gives $L\propto M^{5.4}$.

Fix $\mu$, $L\propto R^6$. On the other hand,

$$L\propto R^2T_\text{eff}^4\propto L^{1/3}T_\text{eff}^4\Rightarrow L\propto T_\text{eff}^6$$

Interior Stellar Structure

Radial Distribution

Subplot (a) - $\rho$ v.s. normalized $m$

$$\rho_\text c\propto\bar\rho\propto\frac{M}{R^3}\propto M^{-(0.8\sim1.4)}$$

So the central density has negative dependence on the stellar mass.

Subplot (b) - normalized $m$ v.s. normalized $r$

The normalized accumulated mass profile is simply given by the dimensionless Lane-Emden equation. So varying the mass only merely changes the profile.

Subplot (c) - $T$ v.s. normalized $m$

$$T_\text c\propto\frac MR\propto M^{0.2\sim0.4}$$

So the central temperature has negative dependence on the stellar mass.

Subplot (d) - $\varepsilon$ v.s. normalized $m$

The nuclear generation rate $\varepsilon$ is strongly dependent on the temperature. Meanwhile, the energy generation mechanism in the $10 M\odot$ star (CNO cycle) is much more efficient than that in the $1M\odot$ star (pp-chain).

Subplot (e) - $l$ v.s. normalized $m$

The luminosity generation is more concentrated in massive stars, since the temperature dependency for $\varepsilon$ in CNO cycle is much more dramatic than that in pp-chain.

$T\text c$ v.s. $\rho\text c$

In the $\rho-T$ plane, we know that at high $T$, low $\rho$ region, radiation pressure dominates; at low $T$, high $\rho$ region, electron degenerate pressure dominates. For all the other region, gas pressure dominates.

For a low-mass proto-star, as it evolves through Kelvin-Helmholtz contraction, $T\text c$ and $\rho\text c$ all increase. But chances are that the proto-star enters the electron degenerate pressure dominated region, when the contraction will stop, before $T\text c$ reaches $\sim10^7$ K and pp-chain is ignited. For $M<0.08M\odot$, pp-chain cannot be triggered at all, and the star, known as a brown dwarf, is dark, silent, and kind of boring to astronomers. For slightly more massive stars, hydrogen starts to burn, but there could still be a degenerate core. The nuclear fusion of degenrate materials is somehow unstable.

$\Psi$ denotes the fraction of the degenerate electrons. The $T\text c$-$\rho\text c$ curve has a turn-off at $\Psi=-2$, which should correspond to the existence of a degenerate core for $M\lesssim0.5M_\odot$.

Kippenhahn Diagram

• Blank region - convectively stable, radiation dominated

• Shaded region - convective unstable

  • Convection starts when the temperature gradient is unbearablily high and radiation itself cannot sufficiently take energy away. The temperature gradient is proportional to the opacity and the luminosity.

  • Low-mass stars tend to develop convective envelopes. Below certain temperature, free-free transition dominated the opacity, where $\kappa\propto \rho T^{-7/2}$. So for stars with lower mass, the opacity will increase dramatically, leading to a rapid grow in the size of convective envelope. Finally, extremely low-mass stars can be fully convective.

  • High-mass stars tend to develop convective cores because of the high energy generation rate in the center.

• Solid curves - given a stellar mass, describe how much mass is within a certain radius, say, $0.25/0.5 R$ - mass concentration.

• Dashed curves - given a stellar mass, describe where a certain fraction of total luminosity, say, $0.5/0.9 L$, is generated in the star, in mass coordinate - energy generation concentration.

Other Main Sequence Stars

For a helium main sequence star ($Y=1$), the critical temperature for ignation is much higher. The critical mass of a helium main sequence is $0.3M\odot$, below which the whole star would be degenerated at the termination of Kelvin-Helmholtz contraction, while no nulcear fusion is triggered. For a carbon main sequence, the critical mass reaches $0.8M\odot$.

Recall that luminosity strongly depends on the mean molecular weight,

$$L\propto \mu^4$$

so for a hydrogen main sequence, a helium main sequence, and a carbon main sequence of the same mass, the total luminosity $L$, as well as the effective temperature $T_\text{eff}$, increases. The HR diagram should look like below.

Helium and carbon stars are important progenitors for core-collapse supernovae. An evolved hydrogen main sequence star could harbor a steadily burning helium core. Once the hydrogen envelope is blown away, the helium core becomes a naked helium star, progenitors of Type Ib SNe (no hydrogen lines in spectra, helium rich). If the helium envelope is also blown away, a naked carbon star is probably born. They are progenitors of Type Ic SNe (neither hydrogen lines nor helium lines in spectra).