The Main Sequence and Its Structure

A star spends most of its life fusing hydrogen into helium in its core. The locus this burning traces in the luminosity–temperature plane is the main sequence, a diagonal band that a randomly chosen star occupies about ninety percent of the time simply because core hydrogen burning is the longest-lived stable phase. The band is not a track a single star moves along; each point corresponds to a different mass. The structure at a given mass, the run of luminosity with mass, and the time spent burning all follow from the equations of stellar structure once the energy source and the transport mechanism are fixed.¹

The zero-age main sequence

A contracting protostar radiates gravitational energy on the Kelvin–Helmholtz timescale while its core heats. When the central temperature reaches $\sim 10^7\text{K}$, the pp chain switches on and nuclear energy generation rises until it balances the surface luminosity. Contraction stops: the star is now in both hydrostatic and thermal equilibrium, supported by an ideal-gas pressure maintained by nuclear burning. This chemically homogeneous, newly ignited configuration is the zero-age main sequence (ZAMS). Its position in the Hertzsprung–Russell diagram is fixed by one parameter, the mass, plus a weak dependence on composition through the mean molecular weight $\mu$. The uniqueness of the equilibrium for a given mass and composition is the content of the Vogt–Russell result used when integrating stellar models.²

Arrival on the ZAMS is the endpoint of pre-main-sequence contraction along the Hayashi and Henyey tracks, treated in protostars and the pre-main sequence. The lower end of the sequence terminates near $0.08M_{\odot }$, below which the core never reaches hydrogen ignition and the object becomes a brown dwarf; the upper end runs to $\sim 100M_{\odot }$, where radiation pressure and instabilities limit further mass.

Homology and the mass–luminosity relation

The dependence of luminosity on mass follows from a scaling argument. Suppose stars of different mass have the same dimensionless structure, so that any two are related by a uniform rescaling of radius and density. Such stars are called homologous, and each physical variable at fractional radius $x=r/R$ scales as a power of the total mass and radius. The governing relations are hydrostatic equilibrium, the ideal-gas equation of state, and radiative diffusion.

Hydrostatic equilibrium sets the pressure scale. From $\mathbf{d}P/\mathbf{d}r=-Gm\rho /r^2$, dimensional replacement of each factor gives

since $\rho \sim M/R^3$ and $m\sim M$, $r\sim R$. The ideal-gas law $P=\rho kT/(\mu m_{\mathrm{H}})$ then fixes the central temperature,

The central temperature rises with mass and falls with radius: more massive stars have hotter interiors. Radiative diffusion carries the luminosity outward,

using $\rho \sim M/R^3$ and $\mathbf{d}{T}^4/\mathbf{d}r\sim T_c^4/R$. Substituting $T_c\propto \mu M/R$ cancels the radius entirely when the opacity $\kappa$ is constant:

Constant opacity is the electron-scattering regime, $\kappa =0.2(1+X){\text{cm}}^2{\text{g}}^{-1}$, appropriate to the hot interiors of massive stars. The result $L\propto M^3$ is the mass–luminosity relation. When the dominant opacity is instead Kramers, $\kappa \propto \rho T^{-7/2}$, the same procedure gives a steeper $L\propto M^5$ with a mild radius dependence, so the observed exponent runs from about $4$ near a solar mass to about $3$ for the most massive stars. A single power law $L\propto M^{3.5}$ fits the ensemble to within the scatter set by composition and evolutionary state.

The mass–radius relation follows from a second constraint. The luminosity generated by nuclear burning is $L\sim Mϵ$ with $ϵ\propto \rho T_c^{\nu }$, and $\nu \approx 4$ for the pp chain, $\nu \approx 18$ for CNO. Equating this to the radiative luminosity above and eliminating $L$ gives $R\propto M^{(\nu -1)/(\nu +3)}$. For pp burning this predicts $R\propto M^{0.4\text{–}0.8}$, and for CNO burning $R\propto M^{0.6\text{–}0.8}$; the sequence is nearly $R\propto M^{0.7}$ on average. Radius grows more slowly than mass, so the mean density $\bar{\rho }\propto M/R^3\propto M^{-1.1}$ drops toward higher mass. Massive stars are large, hot, luminous, and tenuous; low-mass stars are small, cool, faint, and dense.

Upper and lower main sequence

The mode of energy transport changes across the sequence because the temperature sensitivity of the energy source changes. Two regimes divide at roughly $1.2M_{\odot }$.

• Upper main sequence ($M\gtrsim 1.2M_{\odot }$). The central temperature exceeds $1.8\times 10^7\text{K}$, so the CNO cycle dominates. Its rate $ϵ\propto T^{18}$ concentrates the entire luminosity in a small central volume. The radiative gradient there is far too steep for radiation to carry the flux, so the core is convective, mixing fresh fuel inward. The envelope, hotter and less opaque, stays radiative.
• Lower main sequence ($M\lesssim 1.2M_{\odot }$). The pp chain ($ϵ\propto T^4$) spreads energy generation over a larger central region, and the core is radiative. The cool outer layers have high opacity from the ${\text{H}}^{-}$ ion, which steepens the radiative gradient near the surface and drives an outer convective envelope. The Sun sits here: a radiative interior out to $0.71R_{\odot }$ with a convective zone above it.

The crossover in the burning mechanism is the same $\nu =(\tau -2)/3$ scaling derived from the Gamow peak in thermonuclear reaction rates and the Gamow peak; the two hydrogen networks and their crossover temperature are worked out in hydrogen burning: pp chains and the CNO cycle.

The main-sequence lifetime

The time a star burns hydrogen is the fuel available divided by the rate of consumption. A fraction $f\approx 0.1$ of the mass passes through the core and releases $0.007c^2$ per unit mass of hydrogen converted to helium, so the available energy is $E\approx 0.007fM{c}^2$. The burning rate is the luminosity, giving

For the Sun this evaluates to about $10^{10}\text{yr}$. Inserting the mass–luminosity relation $L\propto M^{3.5}$,

The lifetime falls steeply with mass: a $10M_{\odot }$ star burns out in about $3\times 10^7\text{yr}$, a $30M_{\odot }$ star in a few million years, while a $0.5M_{\odot }$ star would outlast the present age of the universe many times over. Massive stars are rare, luminous, and short-lived; low-mass stars are common, faint, and effectively eternal on cosmic timescales. This inverse scaling underlies the use of the main-sequence turnoff as an age indicator.

The band and the turnoff

The main sequence has a finite width for two reasons. Stars differ in composition, and higher metallicity shifts a star toward lower temperature at fixed mass. The larger effect is that a star evolves slightly during core hydrogen burning: as helium accumulates, the mean molecular weight rises, and $L\propto {\mu }^4$ forces the luminosity up. The star brightens and expands, drifting upward and to the right of the ZAMS until core hydrogen is exhausted at the terminal-age main sequence. The occupied region between ZAMS and terminal-age main sequence is a band roughly one magnitude wide, not a line.

A star cluster forms its members at nearly one time from one cloud, so they share an age and composition and differ only in mass. Plotting the cluster in the H-R diagram shows every member of mass above the turnoff already evolved off the main sequence, because for those masses $t_{\mathrm{MS}}$ is shorter than the cluster age. The main-sequence turnoff is the point where stars are just now exhausting core hydrogen: its mass satisfies $t_{\mathrm{MS}}(M)=t_{\mathrm{cluster}}$. Reading the turnoff luminosity and inverting the lifetime relation dates the cluster. As a cluster ages the turnoff slides down the sequence to lower mass and luminosity, so a young cluster turns off high on the diagram and an old globular cluster turns off just above the Sun's position.

The mass ordering established here propagates through everything that follows. Low- and intermediate-mass stars leave the main sequence to become red giants and end as white dwarfs, worked out in post-main-sequence evolution of low-mass stars. Massive stars run through the heavier burning stages in the evolution of massive stars and end in core collapse. The dividing mass, near $8M_{\odot }$, is set by whether the core ever reaches the temperatures required to burn carbon.