5.6: Summary and Recapitulation

In this chapter we have sketched the evolution of normal stars from their contraction out of the interstellar medium to their probable fate. We have not discussed many topics and details which are important to the detailed understanding of stellar evolution and some important problems remain unsolved. For the evolution of individual stars, an area of singular importance that was acknowledged only in passing concerns the origin of the elements. The production of the elements through nuculosynthesis was suggested by Burbidge, Burbidge, Fowler and Hoyle²⁰ and the early view of the important processes are reviewed by Bashkin²₁. The manifestation of these elements in the outer layers of the star and the internal processes by which they got there are covered by Wallerstein²². In addition, we have said nothing about the fascinating topic of the evolution of close binary stars where the futures of the components are linked through the process of mass exchange. We have said nothing about the mass loss from massive stars that may alter the evolution of these stars. Nor have we touched on the tricky processes by which a white dwarf cools off. We also avoided the effects of rotation and magnetic fields on the evolution of stars along with the details of the dynamic collapse of stars, and these should be regarded as fertile areas for research. However, we did delineate major events in the lives of normal stars. Specifically, we used the efficiency of energy transport, the temperature sensitivity of the nuclear reactions, and the radiative ability of the photosphere to indicate the probable direction that the evolution of stars will take. Simple arguments of efficiency lead to a remarkably accurate description of the early phases of stellar evolution to the main sequence. Post main sequence evolution is made more complicated by the zonal nature of the star, complications to the equation of state, and the existence of multiple energy sources. Nevertheless, we can see the basic conservation laws of physics at work during the latter phases of stellar evolution and can get a feel for the important processes at work. We close this discussion with another view of the interplay between the core and outer envelope along with a detailed look at the evolution of a 5M_⊙ star.

a. Core Contraction - Envelope Expansion: Simple Reasons

For years there has been some debate over why the envelope of an evolving star expands when the core contracts, for many people find the result counterintuitive. A number of explanations have been suggested and objections have been made to almost all, of being simplistic or incomplete. Some have regarded the question as being so complicated that it is not useful to search for a single cause, and in response to the question of envelope expansion they simply say, "It happens because my computer tells me it does." This is no answer at all, for it offers no insight into the physical phenomena that result in the particular behavior exhibited by the star. Certainly the physical situation which leads to the expansion of the envelope during core contraction is complicated and simple answers, in some sense, will always be incomplete. However, we should make the effort to identify the important processes at work which dominate the result.

It would be useful if we could clarify the question a little. A star goes through several different phases as it evolves from the main sequence to the giant branch of the H-R diagram, and all result in an expansion of the envelope accompanying some contraction of the core. However, the structure of the core and that of the envelope differ widely in these various phases as do the magnitude and time scale for the resulting core contraction-envelope expansion. We attempted to make plausible the expansion of the convective envelope which accompanies the temperature increase of the hydrogen-burning shell, resulting from the contraction of the radiative helium-rich core, by appealing to the behavior of convective polytropes. Although such envelopes will not be the complete polytropes of the Hayashi tracks, it seems reasonable that the stars will approach the tracks in their general behavior, for the same principles that result in the Hayashi tracks are operative in the expansion of the convective envelope. However, in envelope expansion, the processes are reversed and less perfectly followed, since only the envelope is involved. The general expansion of the radiative zone overlying the helium core which initiates the departure of the star from the main sequence cannot be described in the same manner. The expansion would be represented by a partial polytrope having an index that varies in time. Indeed, for the lower main sequence stars, that zone is surrounded by a convective region that, together with the radiative zone, makes up the envelope outside the hydrogen-burning shell. This compound envelope undergoes the expansion.

With such a great variety of situations leading to envelope expansion, does it even make sense to look for a common cause? If we found one, it would be necessarily vague about details since it must apply in a wide variety of circumstances. Should this common cause exist, it must result from a very general principle in order for it to apply in these many diverse circumstances. Let us consider two very general principles to see if they can provide a qualitative indication as to how the entire star will behave should the central regions, embodying the majority of the mass, contract. First, the conservation of energy will require that the total energy of the star be written as \[\mathrm{E}=<\Omega>+<\mathrm{U}>-\int_0^{\mathrm{t}} \mathrm{Ldt}+\int_{\mathrm{V}} \int_0^{\mathrm{t}} \varepsilon \mathrm{dtdV}\label{5.5.1}\]

For any time scale that is less than the Kelvin-Helmholtz time, the magnitude of the integrals will be less than either the gravitational or internal energy, because the Kelvin-Helmholtz time is essentially the time required for the star's luminosity to lose an amount of energy equal to the internal energy. Since the integrals appear with opposite sign and are approximately equal, and since we have included all sources of energy available to the star explicitly, we may write \[\text { E. }<\Omega>+<\mathrm{U}>\text {. constant }\label{5.5.2}\]

The contractions and expansions of interest occur on time scales very much longer than the dynamic time scale, so we can be certain that the time-averaged form of the Virial theorem will apply. Thus \[\mathrm{<\Omega>+2<U>=0}\label{5.5.3}\]

The combination of equations \ref{5.5.2} and (5.5.3) requires that the gravitational and potential energy, separately, be constant. Now as long as we average over a dynamical time, equation \ref{5.5.3} will be valid, while equation \ref{5.5.2} becomes more exact for shorter times t since the integrals of equation \ref{5.5.1} will contribute less. Thus we may drop the averages and expect that for any time greater than the dynamical time but shorter than the Kelvin-Helmholtz time \[\Omega \text {. constant . } \Omega_{\mathrm{c}}+\Omega_{\mathrm{e}}\label{5.5.4}\]

Now, for upper main sequence stars, the mass of the core substantially exceeds that of the envelope, \[\Omega . \mathrm{GM}_{\mathrm{c}}^2 / \mathrm{R}_{\mathrm{c}}+\mathrm{GM}_{\mathrm{c}} \mathrm{M}_{\mathrm{e}} / \mathrm{R}_*\label{5.5.5}\]

If, for simplicity, we further hold the masses of the core and envelope constant during the core contraction, we have \[\frac{d R_*}{d R_c} \approx-\left(\frac{M_c}{M_e}\right)\left(\frac{R_*}{R_c}\right)^2 \ll-1\label{5.5.6}\]

The sign of equation \ref{5.5.6} indicates that we should expect the observed radius of the star to increase for any decrease in the core radius, and the magnitude of the right-hand side implies that a very large amplification of the change in core size would be seen in the stellar radius. One can argue that the assumptions are only approximately true or that the time scales involved occasionally approach the Kelvin-Helmholtz time, but that will affect only the degree of the change, not the sign. Indeed, detailed evolutionary model calculations throughout the period of evolution from the main sequence to the giant branch indicate that the total gravitational and internal energy is indeed constant to about 10 percent. The accuracy for shorter times is considerably better. For lower main sequence stars, the mass of the core is less than that of the envelope. Nevertheless, a result similar to equation \ref{5.5.6}, with the same sign, is obtained although the magnitude of the derivative is not as large.

The nature of this argument is so general that we may expect any action of the core to be oppositely reflected in the behavior of the envelope regardless of the relative structure. Thus, we can understand the global response of the star to the initial contraction of the core when the overlying layers are in radiative equilibrium as well as the subsequent rapid expansion to and up the giant branch when the outer envelope is fully convective. In addition, contraction of the stellar envelope following the core expansion accompanying the helium flash, which leads to its position on the horizontal branch, can also be qualitatively understood. In general, whenever the core contracts, we may expect the envelope to expand and vice versa. Detailed model calculations confirm that this is the case.

Table 5.1 History of 5M_⊙ Star

Point	Duration	Elapsed Time	Primary Physical Activity
Location	yr	yr
(1-2)	6.4 x 10^7	6.40 x 10^7	H burning core
(2-3)	2.2 x 10^6	6.62 x 10^7	Core exhaustion and contraction
(3-4)	1.4 x 10^5	6.63 x 10^7	Establishment of hydrogen-burning shell
(4-5)	1.2 x 10^6	6.75 x 10^7	H shell thickens
(5-6)	8.0 x 10^5	6.83 x 10^7	H exhaustion, envelope expansion to convection
(6-7)	5.0 x 10^5	6.88 x 10^7	Core contraction, envelope expansion
(7-8)	6.0 x 10^6	7.48 x 10^7	He ignition and burn, envelope contraction, and core expansion
(8-9)	1.0 x 10^7	8.48 x 10^7	Primary He burning phase
(9-10)	1.0 x 10^6	8.58 x 10^7	He core grows, envelope expands
(10-11)	<10^5	8.58 x 10^7	Core contraction, He shell ignition
(11-12)	<10^4	8.58 x 10^7	He exhaustion before C ignition

b. Calculated Evolution of a 5 M_⊙ star

In this final section we look at the specific track on the H-R diagram made by a 5M_⊙ star as determined by models made by Icko Iben. This is best presented in the form of a figure and is therefore shown in Figure 5.2 above. Similar calculations have been done for representative stellar masses all along the main sequence so the evolutionary tracks of all stars on the main sequence are well known. The basic nature of the theory of stellar evolution can be confirmed by comparing the location of a collection of stars of differing mass but similar physical age with the H-R diagrams of clusters of stars formed about the same time. A reasonable picture is obtained for a large variety of clusters with widely ranging ages. It would be presumptuous to attribute this picture to chance. While much remains to be done to illuminate the details of certain aspects of the theory of stellar evolution, the basic picture seems secure.

Figure 5.2 delineates the evolution of a 5M_⊙ star from its arrival on the main sequence through its demise at the onset on carbon burning²³. The labeled points are points of interest discussed in the chapter and their duration, place in the stellar lifetime, and the significant physical process taking place are given in Table 5.1.