5.5: Post Main Sequence Evolution

The evolution of stars off the main sequence represents the response of the star to a depletion of the available fuel supply, and it can be qualitatively understood by examining the response of the core and envelope to the attempts of the nuclear burning regions to adjust to the diminution of the available hydrogen. In stars of the lower main sequence, the hydrogen-burning shell begins to move into a region of declining density resulting in a decrease in available hydrogen. For stars of the upper main sequence, the situation is somewhat different. The convective nature of the core ensures the existence of mass motions, which continue to bring hydrogen into the central regions for hydrogen burning until the entire core is depleted. These two rather different approaches to hydrogen exhaustion produce somewhat different evolutionary futures for the two kinds of stars, so we examine them separately.

a. Evolution off the Lower Main Sequence

The development of a helium core, which signals the onset of post main sequence evolution, is surrounded by a thin hydrogen-burning shell. The hydrogen burning continues in a shell around the helium core which steadily grows outward, in mass, through the star. However, the helium core must be supported against its own gravity as well as support the weight of the remaining star, and its energy sources are all on the outside. As a result, it is impossible for the hydrogen-burning shell to establish a temperature gradient within the helium core. Only gravitational contraction of the core will result in the release of energy inside the helium core, and except for this source of energy the helium core must be isothermal, with its temperature set by the burning of hydrogen surrounding it. But the rate of hydrogen burning is dictated largely by the mass of material lying above the burning zone, because this is the material that must kept in equilibrium. As the mass of the isothermal helium core increases, the equilibrium temperature of the core will also rise and this demand can be met only by a slow contraction of the helium core. The slow increase in the core temperature triggers a steady increase in the stellar luminosity.

As the isothermal core grows through the addition of He from the hydrogen-burning shell, the core temperature must rise in order for it to remain in equilibrium and support the outer layers of the star. Since an isothermal sphere is a unique polytropic configuration, it seems reasonable that there would be a limit to the amount of overlying material that such an isothermal core could support. This limit is known as the Chandrasekhar-Schönberg (C-S) limit. The limit will depend solely on the mass fraction of the isothermal core and the mean molecular weights of the core and envelope. Should the core exceed this limiting mass fraction, it must contract to provide the temperature and pressure gradients necessary to support the remainder of the star as well as itself.

Chandrasekhar-Schönberg Limit A detailed evaluation of the Chandrasekhar-Schönberg limit requires matching the isothermal core solution to the pressure required to support the overlying stellar mass. The maximum mass fraction that an isothermal core can have is \[q_{\mathrm{c}-\mathrm{s}} \approx 0.37\left(\frac{\mu_o}{\mu_i}\right)^2\label{5.4.1}\]

where \(\mu_\mathrm{o}\) and \(\mu_\mathrm{i}\) are the mean molecular weight of the outer region and core respectively. Although the specific calculation of \(q_\mathrm{c-s}\) requires detailed consideration of the isothermal core solution, we can provide an argument for the plausibility of such a limit by considering the Virial theorem for the core alone. \[3(\gamma-1) U_c+\Omega_c-3 P\left(\frac{4 \pi r_c^3}{3}\right)=0\label{5.4.2}\]

where \[\begin{aligned}
& U_c=\frac{k T M_c}{\mu_i m_h} \\
& \Omega_c \approx-\frac{G M_c^2}{r_c}
\end{aligned}\label{5.4.2a}\]

and \(\mathrm{r_c}\) is the radius of the isothermal core. The third term on the left-hand side arises because we cannot take the volume integrals, which yield the global theorem, over a surface where the pressure is zero. Thus, we must include a "surface" term which is effectively the surface pressure times the enclosed volume of the core. We may solve this expression for the pressure at the boundary of the core and obtain \[P\left(r_c\right)=\frac{1}{4 \pi r_c^3}\left[\frac{3(\gamma-1) k T M_c}{\mu_i m_h}-\frac{G M_c^2}{r_c}\right]\label{5.4.3}\]

Now we wish to find the maximum core radius which will provide sufficient pressure to support the remaining star. We can find a maximum pressure by differentiating equation \ref{5.4.3} with respect to \(\mathrm{r_c}\) and finding that value of \(\mathrm{r_c}\) for which the pressure gradient is zero. Certainly any core which yields a zero surface pressure gradient is the largest physically reasonable core. This calculation results in a maximum \(\mathrm{r_c}\) given by \[\frac{1}{r_c}=\frac{9(\gamma-1) k T}{4 \mu_i m_h G M_c}\label{5.4.4}\]

Substitution into equation \ref{5.4.3} yields the maximum surface pressure attainable at the surface of the core. \[P_m\left(r_c\right)=3\left[\frac{9(\gamma-1)}{4}\right]^4\left(\frac{k T}{\mu_i m_h}\right)^4 G^{-3} M_c^{-2}\label{5.4.5}\]

Remember that the homologous behavior of the temperature allows us to write \[T=\text { (const) }\left(\frac{\mu_o M}{R}\right)\label{5.4.6}\]

so we can eliminate the temperature from 5.4.5 and express the result with a term which is homologous to the pressure of the envelope. Thus, \[P_m\left(r_c\right)=\text { (const) }\left(\frac{T^4}{\mu_i^4 M_c^2}\right)=\text { (const) }\left(\frac{M^2}{R^4}\right)\left(\frac{\mu_o}{\mu_i}\right)^4\left(\frac{M}{M_c}\right)^2\label{5.4.8}\]

where the constant is the same as in equation \ref{5.4.1}. Thus, we see that the isothermal core can, at most, support about 37 percent of the mass of the star, but if the core is primarily helium, this limit is reduced to about 10 percent.

Degenerate Core Only for stars near the upper end of our range (i.e. M ≈ M_⊙) will the mass of the core approach the Chandrasekhar-Schönberg limit without becoming degenerate and the core undergoing further gravitational contraction. For stars with masses M ≤ 1.3M_⊙ the slowly developing isothermal core will be degenerate from a point in its development when the core mass is well below the Chandrasekhar-Schönberg limit. Under these conditions, that limit does not apply because the added pressure of the degenerate electron gas is sufficient to support nearly any additional mass. Thus, the isothermal helium cores of lower main sequence stars can increase to virtually any mass below the Chandrasekhar degeneracy limit. As mass is added to the core, we can expect the core to contract according to the mass-radius law for degenerate configurations that we derived in Chapter 1 [equation \ref{1.3.18}]. This law, following differentiation with respect to time, indicates that the core will shrink on the same time scale that mass is added to it, and that is the nuclear time scale.

Progress to the Red Giant Phase In terms of its physical size, this isothermal degenerate helium core is never very large. Thus the post main sequence evolution of a low-mass star can be viewed as the processing of stellar material through the burning zone, with the resultant helium being packed into a very small volume of systematically higher mean molecular weight. The declining density just above the helium core will lead to an increase in temperature, in order for the nuclear energy generation mechanisms to supply the energy required to support the star. However, an increase in the central temperature would lead to an increase in the temperature gradient and an increase in the luminosity. The increased luminosity, in turn, causes the outer envelope of the star to expand, decreasing the temperature gradient. Equilibrium is established at a higher shell temperature and somewhat greater luminosity and temperature gradient. The result is that the star moves upward and very slightly to the right on the H-R diagram. The process continues until the temperature gradient exceeds the adiabatic gradient. Then the entire outer envelope becomes convective. The increase in physical size of the envelope lowers the surface temperature and thereby increases the radiative opacity in the outer layers. This further decreases the efficiency of radiative transport and hastens the formation of the outer convection zone. The outer envelope is now well approximated by a polytrope of index n = 1.5, and the conditions for the Hayashi tracks become operative.

The star now approximately follows the track of a fully convective star only now in reverse. The continual decline of the available hydrogen supply in the shell burning region, which becomes extremely thin, leads to a steady increase in the shell temperature and accompanying rise in the luminosity. With the outer convection zone behaving as a good polytrope and efficiently carrying the energy to the surface, the energy loss is again limited by the photosphere and the star expands rapidly to accommodate the increased energy flow. The star now moves nearly vertically up the giant branch as a red giant.

Helium Flash As the temperature of the hydrogen-burning shell increases and the degenerate core builds in mass; the temperature eventually reaches approximately 10⁸ K. This is about the ignition temperature of helium via the triple-α processes. Under normal conditions, the burning of helium could begin in a measured way which would allow for an orderly transition of nuclear energy generation processes. However, the core is degenerate, so the electron pressure is only weakly dependent on temperature. Indeed, the limiting equation of state for total degeneracy does not contain the temperature at all. Thus helium burning sets in with unrelenting ferocity. With its extreme dependence on temperature, the triple-\(\alpha\) process initiates a thermal runaway which is limited only by the eventual removal of the degeneracy from the core. The complete equation of state for a partially degenerate gas does indeed, contain the temperature and at a sufficiently high temperature the equation of state will revert to the ideal-gas law. When this occurs, the core rapidly expands, cools, and reaches equilibrium, with helium continuing to burn to carbon in its center. The response of the core to this entire process is so swift that the total energy produced is a small fraction of the stored energy of the star. In addition, the site for the production of the energy is sufficiently far removed from the outer boundary that energy is diffused smoothly throughout the star and never makes a noticeable change in the stars appearance.

The duration of the flash, is so much shorter time than the dynamical time scale for the entire star that one would expect that all manifestation of the flash would be damped out by the overlying star and remain hidden from the observer. However, detailed hydrodynamical calculations⁷ indicate that the pressure pulse resulting from the rapid expansion of the core arrives at the surface with a velocity well in excess of the escape velocity. This may well result in a one-time mass loss of the order of 30 percent which would affect the subsequent evolution.

Terminal Phases of Low Mass Evolution Initially, after helium ignition, the hydrogen burning shell continues to supply about 90 percent of the required support energy. However, now an orderly transition of energy mechanisms can take place, resulting in the transfer from hydrogen burning to helium burning over an extended time. The star will move somewhat down the giant branch and out on the horizontal branch, from near the peak of the giant branch where the helium flash took place. Meanwhile the helium core is in convective equilibrium, with the convection zone extending almost to the hydrogen shell. The re-expansion of the core is responsible for the contraction of the outer envelope, causing the star to move out onto the horizontal branch. It appears likely, that after helium burning has ceased and the resultant carbon core is contracting, the outer envelope becomes unstable to radiation pressure and lifts off the star, forming a planetary nebula and leaving the hot core, which now relieved of its outer burden, simply cools. If the mass is below the Chandrasekhar limiting mass for carbon white dwarfs, the star continues to cool to the virtually immortal state of a white dwarf.

Structure and Evolution of White Dwarfs We have already discussed much that is relevant to the description of this abundant stellar component of the galaxy, and we will return to the subject in Section 6.4. In Section 1.3 we derived the equation of state appropriate for a relativistic and a nonrelativistic degenerate gas and found them to be polytropic. In Section 2.4 we developed the mass-radius relation for polytropes in general, which provides the approximate results appropriate for white dwarfs. In Chapter 6 we will see how the relativistic equation of state and the theory of general relativity lead to an upper limit of the mass that one can expect to find for white dwarfs. However, some description of the white dwarfs formed by the evolution of low-mass stars and their subsequent fate is appropriate.

There are basically two approaches to the theory of white dwarfs. The first is to observe that a relativistically degenerate gas will behave as a polytrope and to explore the implications of that result. The second is to investigate the detailed physics that specifies the equation of state and to create models based on the results. Cox and Giuli8 and references therein provide an excellent example of the latter. Our approach will be much nearer the former.

The ejection of a planetary nebula during the later phases of the evolution of a low-mass star leaves a hot degenerate core of carbon and oxygen exposed to the interstellar medium. While such a core may range in mass from about 0.1 of a solar mass to more than a solar mass, its future will be remarkably independent of its mass. While the actual run of the state variables will pass through regions of degeneracy through partial degeneracy to a nondegenerate surface layer, the basic properties of the star can be understood by treating the stars as polytropes.

From observation we know that the white dwarf remains of stellar evolution are about a solar mass confined to a volume of planetary dimensions and thus will have a density of the order of \(\rho=10^6 \mathrm{~g} / \mathrm{cm}^3\). If we assume that the gas is fully ionized, then the typical energy of an electron will be about 0.1 MeV for a fully degenerate gas. If the stellar core were at a temperature of 10⁷ΕK, the typical ion would have an energy of about 1 keV. Since energy densities are like pressures, even if the number densities were the same for the electrons and ions, the pressure of the electrons would dominate. In fact, since the typical ion produces many electrons, the dominance of the electron pressure, is even greater. Thus the structure will be largely determined by the electron pressure and the ions may be largely ignored. However, Hamada and Salpeter⁹ have shown that at densities around \(10^8 \mathrm{~g} / \mathrm{cm}^3\) the Fermi energy of the electron "sea" becomes so high that inverse beta decay becomes likely and some of the electrons disappear into the protons of the nuclei, causing the limiting mass to be somewhat reduced over what would be expected for a purely degenerate gas. Also, the thermal energy of the ions is lost, permitting the star to shine.

Since these stars are largely degenerate, most of the momentum states in phase space are full, and an electron that is perturbed from its place in phase space has to travel quite a distance before it can find an empty place. This implies that the mean free path of electrons will be very long in spite of the high densities. Such electrons play essentially the same role as the conduction electrons of a conductor so that the electrical and thermal conductivity will be very high in a degenerate gas. This crowding of the states in phase space also results in the reduction in radiative opacity since it is difficult for a photon to move an electron from one state to another. As a result, it is very difficult for temperature gradients to exist within a fully degenerate configuration. However, in the outer regions of the white dwarf where the gas becomes partially degenerate, the opacity rapidly rises and the conductivity drops, giving rise to a steep temperature gradient with the result that the energy flow to the surface is seriously impeded. Thus Aller has likened a white dwarf to a metal ball wrapped in an insulating blanket. Since the structure of a polytrope is stable and independent of the temperature, the evolutionary history of a white dwarf largely revolves on the details of its cooling.

In 1952, Leon Mestel¹⁰ took basically this classical approach to the cooling of white dwarfs and found that the cooling curve [dln(L)/dln(t)] was approximately constant and independent of time. Iben and Tutukov¹¹, using a much more detailed analysis and equation of state, found virtually the same result which they regarded as occurring through a series of accidents. Their results give \([\mathrm{d} \ln (\mathrm{~L}) / \mathrm{d} \ln (\mathrm{t}) \approx(-1.4,-1.6)]\) for \(5<\log \left(\mathrm{t}_{\mathrm{yrs} .}\right)<9.4\). It is true that a considerable number of effects complicate the simple picture of a polytrope wrapped in a blanket.

For example, while we may neglect the ions for an excellent approximation of the description of the white dwarf structure, the contribution of the ion pressure will make the star slightly larger than one would expect from the polytropic approximation. Because of the extreme concentration of the star, a small contraction produces a considerable release of gravitational energy, which is then to be radiated away. This extends the cooling time significantly over that which would be expected simply for a cooling polytrope. Toward the end of the cooling curve a series of odd things happen to the equation of state for the white dwarf. As the interior regions cool, they undergo a series of phase transitions first to a liquid state and then to a crystalline phase. Each of these transitions results in a "heat of liquefaction or crystallization" being released and increasing the luminosity temporarily. Problems of the final cooling remain in the understanding of the low-temperature high-density opacities that will determine the flow of radiation in this final descent of the white dwarf to a cool cinder, called a black dwarf in thermal equilibrium with the ambient radiation field of the galaxy. The question is of considerable interest since such objects could be detected only by their gravitational effect and could bear on the question of the "missing mass".

b. Evolution away from the Upper Main Sequence

The evolution of the more massive stars that inhabit the upper main sequence is driven by the same processes that govern the evolution of the lower main sequence, namely, the exhaustion of hydrogen fuel. However, the processes are quite different. The exhaustion of the convective core leads to the production of a helium center, as in the lower main sequence stars, but now the core will have to contend with the Chandrasekhar-Schönberg limit.

Nature of the Massive Helium Core In massive stars, as the hydrogen is depleted in the helium core, the temperature rises rapidly, to produce the energy necessary to accommodate the demands of stellar structure. For stars with masses greater than about 7 M_⊙, the resulting helium core will be greater than the Chandrasekhar-Schönberg limit; and to make up for the energy deficit caused by the failing hydrogen burning, the core will have to contract. Since the contraction must maintain a temperature gradient, the contraction will proceed rather faster than would be expected for an isothermal core. However, the steep temperature gradient established by the terminal phases of core hydrogen burning will be relaxed because the energy generated by gravitational contraction will not be as centrally concentrated as it was from hydrogen burning. This drop in temperature gradient will cause convection to cease, yielding a core in radiative equilibrium supplying the required stellar energy by contraction.

The contraction of virtually any polytrope will result in an increase in internal temperature. This is really a consequence of the Virial theorem. However, the cessation of hydrogen burning in the core and the resultant decrease in the temperature gradient imply a change in the overall structure of the core, and thus the polytropic analogy is somewhat strained. The decline in the temperature gradient actually implies that the core will suffer a reduction of its internal energy while the boundary temperature increases. This loss of internal energy goes into the support of the outer envelope. Thus, both this energy and the energy generated by contraction are available for the support of the outer layers of the star.

The increase in the boundary temperature of the core will result in a slow expansion and cooling of the radiative envelope for the same reason described for low-mass stars. After a suitable rise in the boundary temperature, hydrogen is reignited in a shell surrounding the helium core. This results in a marked change in the temperature gradient of the hydrogen-burning shell at the core boundary. The localization of the energy production in such a small region steepens the temperature gradient to the point where the temperature gradient exceeds the adiabatic gradient, driving the entire envelope into convection. The outer envelope now rapidly transfers the energy to the surface, which again becomes the limiting barrier to its escape. The star moves rapidly toward the giant branch as a star with a helium core surrounded by a hydrogen-burning shell and covered with a deep convective envelope. This envelope is so deep that it reaches into the region where nuclear processing has taken place, dredging up some of this material to the surface. The result of this process is evident in the atmospheric spectra of some late-type supergiants.

Stars with masses less than about 5M_⊙ will end their hydrogen burning with a helium core below that of the Chandrasekhar-Schönberg limit and will contend with a slowly contracting isothermal core right through the ignition of a hydrogen-burning shell. The future of such a star is mirrored in the behavior of lower-mass stars except that helium ignition takes place before the core becomes significantly degenerate. The result is that these stars experience no helium flash, and the transition to helium burning is orderly.

Ignition of the Massive Helium Core In both cases described above, the contraction of the helium core proceeds while the hydrogen shell is burning. In the more massive stars, where the core is above the Chandrasekhar-Schönberg limit, the star must do so to maintain a temperature gradient for its own support. In the less massive case, the core grows slowly as a result of the processing of the hydrogen in the energy-generating shell. The added mass results in a slow core contraction. In both instances, the decreasing density in the hydrogen-burning shell necessitates a rise in the temperature required for energy generation. After the increasing temperature gradient caused by this increasing shell temperature has forced the envelope to become fully convective, the convective envelope continues to expand, for the energy escape is again limited by the radiative efficiency of the photosphere.

Eventually the central temperature reaches 10⁸ K, and helium ignition takes place. The ignition has a dramatic effect on the core but does not exhibit the explosive nature of the helium flash. The core undergoes a rapid expansion and, because of the huge temperature dependence of the triple-a process, becomes unstable to convection. This produces an expanded convective helium core surrounded by a hydrogen-burning shell. This shell supplies more than 90 percent of the energy required to maintain the luminosity. The rapid core expansion is accompanied by a contraction of the outer envelope with a corresponding increase in the surface temperature. The star moves off to the left in the H-R diagram, maintaining about the same total luminosity. The hydrogen burning shell continues to supply the majority of the energy throughout the helium burning phase which proceeds in much the same manner as the main sequence core hydrogen-burning phase, but on a much shorter time scale.

Terminal Phases of Evolution of Massive Stars The end phase of the evolution of massive stars is still somewhat murky and the subject of active research. Initially, the helium burning continues in the core until the core becomes largely carbon. At the point of helium exhaustion, the core again gravitationally contracts, rising in temperature, until helium is ignited in a shell source around the core. For stars with a mass between 3 and 7 solar masses, there is some evidence that the carbon core which develops is degenerate and that ignition, when it occurs, occurs explosively, perhaps producing a supernova. For stars of more than 10 solar masses, carbon burning can take place in a nonviolent manner, producing cores of neon, oxygen, and finally silicon, each surrounded by a shell source of the previous core material which continues to provide some energy to the star. The results of silicon burning yield elements of the iron group for which further nuclear reactions are endothermic, and so this burning will not only fail to contribute energy for the support of the star but also rob it of energy. In addition, the densities become high enough that the electrons are forced into the protons of the nuclei by means of inverse b decay. This effect has been called neutronization of the core. Both mechanisms produce a significant number of neutrinos which also do not take part in the support of the star against gravity and can be viewed as "cooling" mechanisms for the core.

This rapid cooling precipitates a rapid collapse of the core followed by the entire star. The in-fall velocity soon exceeds the speed of sound, resulting in the formation of a shock wave, and interior densities may become large enough for the material to become opaque to neutrinos. Under some conditions the endothermic nuclear reactions may bring about the disintegration of the iron-group elements into 4He. Which process dominates for stars of which mass is not at all clear. The shock wave formed by the in-fall may "bounce", or the increased neutrino opacity may provide sufficient energy and momentum to ensure that a large fraction of the star will be blown into the interstellar medium.

The remnants, if any, could be a neutron star or a black hole. Although the details of the terminal phases of massive stars remain somewhat unclear, it is virtually certain that these phases are likely to end with the production of a supernova and the subsequent enrichment of the interstellar medium by heavy elements.

c. The Effect of Mass-loss on the Evolution of Stars

Throughout our discussion of stellar evolution we have assumed that the mass of the evolving star remains essentially constant. Certainly there is a reduction in mass from the nuclear production of energy, but this can never exceed 0.7 percent and therefore can be safely neglected. However, as we shall see in section 16.3, stars may exhibit rather large winds emanating from their atmospheres. In some cases these winds may result in mass loss rates exceeding 10^-5M_⊙ per year and so could lead to a significant reduction in the mass of the star during its nuclear lifetime.

Ever since the observation by Armin Deutch¹² that the red supergiant \(\alpha\) Herculis was losing mass faster than about 10^-8 M_⊙ per year, interest in the effects of mass loss on the evolution of a star has been high. Generally one would expect that a star slowly losing mass would follow the classical evolutionary track represented by models of stars having successively lower and lower mass. Thus for a main sequence star that is losing mass at a significant rate the evolutionary track on the HR diagram would not rise like the constant mass models, but move steadily to the right and perhaps down until the asymptotic giant branch is reached. In addition, the lifetime would be significantly enhanced as a result of the reduced stellar mass. Such were the conclusion reached by Massevitch¹³ and collaboraters¹⁴ in the late 1950's. However, the evolutionary tracks of 1-2M_⊙ failed to fit the HR diagrams of globular clusters and in the absence of evidence for mass loss from main sequence stars, their work was largely ignored. However, during the 1960s and 70s it became clear that virtually all early type stars exhibited significant stellar winds and it was likely that their evolution from the main sequence was affected^13,14. In general, the more massive the star, the greater the fractional mass loss rate will be. This would explain why stars in the range of a few solar masses where unaffected and the constant mass models fit the globular cluster HR diagrams relatively well. However, from studies of the ratio of blue to red supergiants in the Milky Way and other galaxies Humphreys and Davidson¹⁵ concluded that stars more massive than 50M_⊙ never made it to the red supergiant phase but remained confined to the left hand side of the HR diagram throughout their lives. Lamers¹⁶ found that this did not appear to be the case for stars in our galaxy concluding that a 100M_⊙ could only loose 15 percent of its mass during it main sequence lifetime.

The impact of such mass loss on the subsequent evolution of the star seems to depend on an accurate knowledge of the mass loss rate during the the evolution which in turn rests on the specifics of the origin of stellar winds. Using an empirically inspired mass loss rate, Brunish and Truran¹⁷ found that mass loss affected the evolution of stars less than 30M_⊙ more than the massive stars. However, Sreenivasan and Wilson^18,19 found including rotation and a theoretically motivated origin for the mass loss rate resulted in a much more complicated evolutionary history. By adjusting the amount of rotation present in the initial star they are able to match most of the observations. However, it is fair to say that as of the present much remains poorly understood about the specific role played by mass loss in the massive stars. It also appears that a proper understanding will require models that correctly couple the atmosphere to the interior and include rotation in a physically self consistent way.