5.4: The Structure and Evolution of Main Sequence Stars

When the track of a gravitationally contracting star intersects the main sequence, the star has reached a point in its life when it will be stable for an extended period of time. This is ensured observationally by the fact that about 90 percent of all stars reside on or very near the main sequence and so must be involved in the utilization of their most prolific and efficient source of energy − the fusion of hydrogen into helium. So we may take the intersection of the Henyey track with the main sequence as an indication that hydrogen ignition has begun in the stellar core.

Actually nuclear processes begin somewhat before the main sequence is encountered. The first constituents of the star to undergo nuclear fusion are deuterium and lithium which require conditions substantially below that of hydrogen for their ignition. However, their abundance is sufficiently low so that they provide little more than a stabilizing effect on the star as it proceeds along its Henyey track, causing the star to hook on to the main sequence.

For the sun, about a million years is required for the equilibrium abundances of the proton-proton cycle to be established with sufficient accuracy for their use in the energy generation schemes. At this point, the star can be said to have arrived at the zero-Age main sequence. As the name implies, this is generally taken as the beginning point of stellar evolution calculations, as the onset of nuclear burning makes the details of the prior evolution largely irrelevant to the subsequent evolution. In some real sense, the star forgets where it came from. Since it is fairly obvious that the effects of stellar evolution during the main sequence phase will result in little movement on the H-R diagram, we need to understand more of the structure of the interior to appreciate these effects. Therefore, we begin by describing the structure to be expected for the hydrogen burning models that describe the main sequence. The structure of main sequence stars can be readily broken into two distinct groups: those that occupy the upper half of the main sequence, and those that occupy the lower main sequence.

a. Lower Main Sequence Stars

We define the lower main sequence to be those stars with masses less than about 2 solar masses. For these stars, after the trace elements with low ignition temperatures have been exhausted and hydrogen fusion has begun, the equilibrium structure is established in about a million years. The mass of these stars is insufficient to produce a central temperature high enough to initiate the CNO cycle, so the primary source of energy is the proton-proton cycle. Models indicate that in the sun, 98 percent of the energy is supplied by the proton-proton cycle. The relatively low dependence on temperature of the proton-proton cycle implies that the energy generation will be less concentrated toward the center than would be the case with the CNO cycle. This and the modest central temperature imply that a temperature gradient less than the adiabatic gradient is all that is required to carry the energy produced by the p-p cycle. Thus, these stars have a central core which is in radiative equilibrium. However, in the sun, the adiabatic gradient is never far from the actual temperature gradient, and it would take a very little increase in the temperature gradient to cause the core to become unstable to convection. Indeed, the conditions for convective instability are met in the outer regions of these stars resulting in the formation of a convective envelope. In the sun, this point is reached at about 0.75R_⊙, so that about 98.8 percent of the mass is included in the radiative core. Ultimately, the situation is reversed near the surface, as it must be, for the energy leaves the surface of the star by radiation.

The existence of the radiative core in stars of the lower main sequence has a significant effect on the subsequent evolution of the star. The ⁴He, which is the end product of hydrogen burning, remains in the locale in which it is produced. However, since the production rate is strongly dependent on temperature, the helium abundance increases more rapidly as one approaches the center of the star. The helium must be supported against its own gravity while it contributes nothing to the support of the remainder of the star. As a result, the internal temperature will increase to maintain the luminosity in the face of decreasing hydrogen abundance and the increasing mass of the particles (i.e. the ⁴He). This is why the temperature scales with the mean molecular weight m, [see equation \ref{2.3.8}]. Thus, we should expect stars like the sun to slowly increase in brightness, as the internal temperature rises, during their main sequence lifetime. Indeed, the standard solar model indicates that the solar luminosity has increased by about 40 percent since its arrival on the zero age main sequence.

Toward the end of the star's main sequence life, the helium abundance will rise to the point where a core of helium, surrounded by a hydrogen burning shell, will form in the center of the star. The support of this isothermal helium core is eventually helped by the Pauli Exclusion Principle. In Chapter 1, we outlined the equation of state to be expected for a gas where all the available h³ volumes of phase space were filled. Because of their lower mass, this condition will be experienced first by the electrons. The degenerate equation of state does not contain the temperature and therefore permits the existence of an electron pressure capable of assisting in the support of the helium core; this equation is independent of the conditions existing in the hydrogen-burning shell. Thus as the core builds, we could expect its structure to shift from that of an isothermal sphere, described in Chapter 2, to that of a polytrope with a \(\gamma=5/3\), as would be dictated by the electron pressure of a fully degenerate gas. This change from an isothermal sphere to a polytrope will dictate the mass distribution, for the pressure of the ions becomes small compared to that of the electrons. However, because of the high conductivity of a degenerate gas, the configuration will remain isothermal since any energy surplus can immediately be transported to a region of energy deficit by electron conduction. Thus, the region is still known as the isothermal helium core, even though the pressure equilibrium is dictated by the electron pressure behaving as a polytropic gas with a \(\gamma=5/3\).

Therefore, the main sequence lifetime of a low mass star consists of a steady energy output from hydrogen burning in an environment of steadily increasing helium. On a nuclear time scale, the helium abundance increases preferentially in the most central regions causing the temperature to rise which results in a slow increase in the luminosity throughout the main sequence lifetime of the star. After about 10 percent of the radiative core mass has been consumed, an isothermal helium core begins to form and structural changes begin to occur very rapidly. This signals the end of the main sequence lifetime.

b. Upper Main Sequence Stars

The situation regarding the stellar structure for stars of more than 2 solar masses is nearly reversed from that of the lower main sequence. For stars on the main sequence, the observed mass-radius relation is approximately \[\mathrm{M \propto R^{4 / 3}}\label{5.3.1}\]

However, from the homology relations in Chapter 2 [equations \ref{2.3.8}], we know that \[\mathrm{T} \propto \mathrm{M} / \mathrm{R}\label{5.3.2}\]

Therefore, for stars along the main sequence, we expect the central temperature to increase slowly as we proceed up the main sequence in accord with \[\mathrm{T}_{\mathrm{c}} \propto \mathrm{R}^{1 / 3} \propto \mathrm{M}^{1 / 4}\label{5.3.3}\]

This slow rise in the central temperature will result in a greater fraction of the energy being produced by the more temperature-sensitive CNO cycle. Thus, by the time one reaches stars of greater than about 2 solar masses, the CNO cycle will be the dominant source of energy production. The much larger temperature sensitivity of the CNO cycle as compared to the p-p cycle means that the region of energy production will be rather more centrally concentrated than in stars of less mass. This, in turn, requires a steeper temperature gradient in order to transport the energy to the outer parts of the star. Since in the sun the radiative gradient was already quite close to the adiabatic gradient, this small increase is sufficient to cause the inner regions to become convectively unstable, and a substantial convective core will be established. However, in the outer parts of the star, the declining density causes the product of \(\overline{\kappa}\rho\), which appears in the radiative gradient [equation \ref{4.2.14}], to reduce the radiative gradient below that of the adiabatic gradient, and so convection stops. Thus, we have a star composed of a convective core surrounded by an envelope in radiative equilibrium. This role reversal for the core and envelope has a profound effect on the evolution of the star.

The presence of a convective core ensures that the inner regions of the star will be well mixed. As helium is produced from the burning of hydrogen, it is mixed thoroughly throughout the entire core. Thus, we do not have a buildup of a helium core that increases in helium abundance toward the center in these stars. Instead, the entire convective core is available as a fuel source for energy production at the center of the star. For this reason, energy production is remarkably steady in these stars until the entire convective core is nearly exhausted of hydrogen. Even as exhaustion approaches, the extreme temperature dependence of the CNO cycle implies that deficits produced by the declining availability of hydrogen fuel can be made up by modest increases in the temperature and hence minor changes in the structure of the star. Indeed, it is not until more than 99 percent of the convective core mass has been converted to helium that truly significant changes occur in the structure of the star and the star can be said to be leaving the main sequence.