Available energy sources

So far we have only derived that, because of the observed thermal equilibrium, the energy transport through the star must be independent of depth as long as there is no energy generation. The energy ultimately has to be generated somewhere in the star in order to keep up with the energy loss at the surface and to prevent the star from further contraction. The energy source ultimately determines the radius of the star.

Making use of the condition of hydrostatic equilibrium we estimated the internal temperature but we do not yet know what keeps the temperature at this level. In this chapter we will describe our present knowledge about the energy generation which prevents the star from shrinking further.

First we will see which energy sources are possible candidates. In Chapter 2 we talked about the gravitational energy which is released when the stars contract. We saw that the stars must lose half of the energy liberated by contraction before they can continue to contract. We might therefore suspect that this could be the energy source for the stars.

The first question we have to ask is how much energy is actually needed to keep the stars shining. Each second the sun loses an amount of energy which is given by its luminosity, L = 3.96 × 1033 ergs−1, as we discussed in Chapter 1. From the radioactive decay of uranium in meteorites we can find that the age of these meteorites is about 4.5 × 109 years. We also find signs that the solar wind has been present for about the same time.

In Volume 3 of Introduction to Stellar Astrophysics we will discuss the internal structure and the evolution of stars.

Many astronomers feel that stellar structure and evolution is now completely understood and that further studies will not contribute essential knowledge. It is felt that much more is to be gained by the study of extragalactic objects, particularly the study of cosmology. So why write this series of textbooks on stellar astrophysics?

We would like to emphasize that 97 per cent of the luminous matter in our Galaxy and in most other galaxies is in stars. Unless we understand thoroughly the light emission of the stars, as well as their evolution and their contribution to the chemical evolution of the galaxies, we cannot correctly interpret the light we receive from external galaxies. Without this knowledge our cosmological derivations will be without a solid foundation and might well be wrong. The ages currently derived for globular clusters are larger than the age of the universe derived from cosmological expansion. Which is wrong, the Hubble constant or the ages of the globular clusters? We only want to point out that there are still open problems which might well indicate that we are still missing some important physical processes in our stellar evolution theory. It is important to emphasize these problems so that we keep thinking about them instead of ignoring them. We might waste a lot of effort and money if we build a cosmological structure on uncertain foundations.

The light we receive from external galaxies has contributions from stars of all ages and masses and possibly very different chemical abundances.

Period–density relation

In Volume 1 we saw that there is a group of stars which periodically change their size and luminosities. They are actually pulsating (the pulsars are not). When Leavitt (1912) studied such pulsating stars, also called Cepheids, in the Large Magellanic Cloud she discovered that the brighter the stars, the lpnger their periods, independently of their amplitude of pulsation. In Volume 1 we discussed briefly how this can be understood. The pulsation frequencies are eigenfrequencies of the stars. They are similar to the eigenfrequencies of a rope of length 2l, which is fastened at both ends but free to oscillate in the center (see Fig. 18. la). If you pull the rope periodically down in the center, first slowly and then more rapidly, you find that for a given frequency ν0 a standing wave is generated in the rope. For this frequency you need to put in only a very small amount of energy, much less than for the other frequencies, for which running waves are generated which interfere with each other and are therefore damped rapidly. The frequency ν0, which generates the standing wave, is an eigenfrequency of the rope. If you increase the amplitude of the wave you still find the same eigenfrequency ν0. If you increase the frequency further you again find running waves until you reach another frequency ν2, three times as large as ν0, for which another standing wave is generated. This wave has two nodes and a wavelength which is a third of the wavelength for the eigenfrequency ν0 (Fig. 18.1c).

Evolution along the subgiant branch

Solar mass stars

From previous discussions we know that solar mass stars last about 1010 years on the main sequence. Lower mass stars last longer. Since the age of globular clusters seems to be around 1.2 × 1010 to 1.7 × 1010 years and the age of the universe does not seem to be much greater, we cannot expect stars with masses much smaller than that of the Sun to have evolved off the main sequence yet. We therefore restrict our discussion to stars with masses greater than about 0.8 solar masses, which we observe for globular cluster stars.

We discussed in Section 10.2 that for a homogeneous increase in μ through an entire star (due to an increase in helium abundance and complete mixing), the star would shrink, become hotter and more luminous. It would evolve to the left of the hydrogen star main sequence towards the main sequence position for stars with increasing helium abundance. In fact, we do not observe star clusters with stars along sequences consistent with such an evolution (except perhaps for the socalled blue stragglers seen in some globular clusters which are now believed to be binaries or merged binaries). Nor do we know any mechanism which would keep an entire star well mixed. We therefore expect that stars become helium rich only in their interiors, remaining hydrogen rich in their envelopes. Since nuclear fusion is most efficient in the center where the temperature is highest, hydrogen depletion proceeds fastest in the center. Hydrogen will therefore be exhausted first in the center.

Color magnitude diagrams for globular clusters

The best way to check stellar evolution calculations is, of course, to compare calculated and observed evolutionary tracks. Unfortunately we cannot follow the evolution of one star through its lifetime, because our lifetime is too short – not even the lifetime of scientifically interested humanity is long enough. Only in rare cases may we observe changes in the appearance of one star, for instance when it becomes a supernova. Another example occurred some decades ago when FG Sagittae suddenly became far bluer, a rare example of stellar changes which are too fast to fit into our present understanding of stellar evolution.

Generally evolutionary changes of stars are expected to take place over times of at least 104 years (except perhaps for stars on the Hayashi track, where massive stars may evolve somewhat faster). How then can we compare evolutionary tracks? Fortunately there are star clusters which contain up to 105 stars all of which are nearly the same age but of different masses. In such very populous clusters there are a large number of stars which have nearly the same masses.

In Fig. 17.1 we show schematically evolutionary tracks of stars with about one solar mass. They all originate near spectral types G0 or G2 on the main sequence. Their lifetime, t, on the main sequence is about 1010 years. The evolution to the red giant branch takes about 107 years.

Definition and consequences of thermal equilibrium

As we discussed in Chapter 2, we cannot directly see the stellar interior. We see only photons which are emitted very close to the surface of the star and which therefore can tell us only about the surface layers. But the mere fact that we see the star tells us that the star is losing energy by means of radiation. On the other hand, we also see that apparent magnitude, color, Teff, etc., of stars generally do not change in time. This tells us that, in spite of losing energy at the surface, the stars do not cool off. The stars must be in so-called thermal equilibrium. If you have a cup of coffee which loses energy by radiation, it cools unless you keep heating it. If the star's temperature does not change in time, the surface layers must be heated from below, which means that the same amount of energy must be supplied to the surface layer each second as is taken out each second by radiation.

If this were not the case, how soon would we expect to see any changes? Could we expect to observe it? In other words, how fast would the stellar atmosphere cool?

From the sun we receive photons emitted from a layer of about 100 km thickness (see Volume 2). The gas pressure Pg in this layer is about 0.1 of the pressure in the Earth's atmosphere, namely, Pg = nkT=105 dyn cm−2, where k = 1.38 × 10−16 erg deg−1 is the Boltzmann constant, T the temperature and n the number of particles per cm3.

Changes in radius, luminosity and effective temperature

In the previous chapter we considered only model stars in radiative equilibrium. We pointed out several mismatches between these models with real stars and attributed them in part to the influence of convection zones. Convection zones change stellar structure in two main ways:

1. (a) The radius of the star becomes smaller.

2. (b) The energy transport through the outer convection zones with the large absorption coefficients becomes easier due to the additional convective energy transport, so that the temperature gradient becomes smaller in comparison with radiative equilibrium. This may lead to an increased luminosity and Teff as well as energy generation.

If energy transport outwards due to convection is increased the star would tend to lose more energy than is generated, and so would tend to cool off. However, this does not actually happen, because it would reduce the internal gas pressure and the gravitational pull would then exceed the pressure force. The star actually contracts, the stellar interior temperature increases, thereby increasing the energy generation ∍ ∞ Tυ. With the larger energy generation the star is then able to balance the larger energy loss. The star is again in thermal equilibrium but with a smaller radius and a larger luminosity, which means with a larger effective temperature. As compared to radiative equilibrium the star moves to the left and up in the HR diagram (see Fig. 11.1). Convection decreases the equilibrium value for the radius.

Importance of Cepheid mass determinations

We know that Cepheids must be in an advanced state of evolution because the blue loops are the only way they can stay in the instability strip for any length of time. If we can determine mass and luminosity for a Cepheid we can check whether its luminosity agrees with what we expect without overshoot or additional mixing. A larger L might indicate additional mixing (see Fig. 15.3). In fact we could calibrate the amount of mixing for the Cepheid progenitor on the main sequence by determining mass and luminosity for a given Cepheid. Of course, we also have to know the chemical abundances and the correct κ. For a given L the derived masses of the Cepheid may differ by 50 per cent if for instance the assumed helium abundance is changed by a factor of 2.

We can also check the consistency of the stellar evolution and pulsation theories by determining masses of Cepheids in different ways, making use of either evolution or pulsation theory or of different aspects of those theories. If the theories are correct we should, of course, find the same mass, no matter how we determine it.

The period–luminosity relation

A number of Cepheids are found in galactic clusters. Their periods can be measured and their distances can be determined, for instance, by main sequence fitting or equivalent methods. We can thus find their absolute magnitudes averaged over one period. The first extensive study of distances for clusters with Cepheids was done by Sandage and Tammann (1968), and a more recent one was done by Schmidt (1984).

General discussion

So far we have talked about energy transport by radiation only. We may also have energy transport by mass motions. If these occur hot material may rise to the top, where it cools and then falls down as cold material. The net energy transport is given by the difference of the upward transported energy and the amount which is transported back down. Such mass motions are also called convection. Our first question is: when and where do these mass motions exist, or in other words where do we find instability to convection? When will a gas bubble which is accidentally displaced upwards continue to move upwards and when will a gas bubble which is accidentally displaced downwards continue to move downwards? Due to the buoyancy force a volume of gas will be carried upwards if its density is lower than the density of the surroundings and it will fall downwards if its density is larger than that of the surroundings.

From our daily experience we know that convection occurs at places of large temperature gradients, for instance over a hot asphalt street in the sunshine in the summer, or over a radiator in the winter. The hot air over the hot asphalt, heated by the absorption of solar radiation, has a lower density than the overlying or surrounding air. As soon as the hot air starts rising by an infinitesimal amount, it gets into cooler and therefore higher density surroundings and keeps rising due to the buoyancy force like a hot air balloon in the cooler surrounding air. This always occurs if a rising gas bubble is hotter than its surroundings.

The hydrostatic equilibrium equation

What information can we use to determine the interior structure of the stars? All we see is a faint dot of light from which we have to deduce everything. We saw in Volume 2 that the light we receive from main sequence stars comes from a surface layer which has a thickness of the order of 100 to 1000 km, while the radii of main sequence stars are of the order of 105 to 107 km. Any light emitted in the interior of the stars is absorbed and re-emitted in the star, very often before it gets close enough to the surface to escape without being absorbed again. For the sun it actually takes a photon 107 years to get from the interior to the surface, even though for a radius of 700 000 km a photon would need only 2.5 seconds to get out in a straight line. There is only one kind of radiation that can pass straight through the stars – these are the neutrinos whose absorption cross-sections are so small that the chances of being absorbed on the way out are essentially zero. Of course, the same property makes it very difficult to observe them because they hardly interact with any material on Earth either. We shall return to this problem later. Except for neutrinos we have no radiation telling us directly about the stellar interior. We have, however, a few basic observations which can inform us indirectly about stellar structure.

For most stars, we observe that neither their brightness nor their color changes measurably in centuries. This basic observation tells us essentially everything about the stellar interior.

Evolution along the giant branch

Just as for low mass stars, the evolution of high mass stars is caused by the change in chemical composition when hydrogen fuses to helium. These stars, however, have a convective core such that the newly formed helium is evenly mixed throughout the core. When hydrogen is consumed, the convective core contracts and also shrinks in mass (because the κ + σ per gram decreases and therefore ∇r decreases); the mixing then occurs over a smaller mass fraction, while some material, which was originally part of the convective region, is left in a stable region but with a slightly enriched helium abundance and also a slight increase in the N14/C12 and C13/C12 ratios. (See Figs. 13.2 and 13.4.) When the convective core mass reduces further, another region with still higher helium abundance and higher N14/C12 and C13/C12 is left outside the convection zone. The remaining convective core becomes hydrogen exhausted homogeneously while it contracts to a smaller volume and becomes hotter. The stars also develop hydrogen burning shell sources around the helium core. Again the core acts like a helium star with a very high temperature; the temperature at the bottom of the hydrogen envelope becomes too high to sustain hydrostatic equilibrium in the hydrogen envelope. The envelope expands and the stellar surface becomes cooler, moving the star in the HR diagram towards the red giant region. Again an outer hydrogen convection zone develops and reaches into deeper and deeper layers. Finally it dredges up some of the material which was originally in the convective core when it included a rather large mass-fraction of the star.

Schwarzschild's method

Before we can discuss the detailed structure of the stars on the main sequence we have to outline the methods by which it can be calculated. In Chapter 10 we have compared homologous stars on the main sequence. While we were able to see how temperatures and pressures in the stars change qualitatively with changing mass and chemical composition, we have never calculated what the radius and effective temperature of a star with a given mass really is. In order to do this we need to integrate the basic differential equations, which determine the stellar structure as outlined in Chapter 9. Two methods are in use: Schwarzschild's method and Henyey's method.

Schwarzschild's method is described in his book on stellar structure and evolution (1958). The basic differential equations are integrated both from the inside out and from the outside in. In the dimensionless form the differential equations for the integration from the outside in contain the unknown constant C (see Chapter 9), for the integration from the inside out the differential equations also contain the unknown constant D. A series of integrations from both sides of the star is performed for different values of these constants. The problem then is to find the correct values for the constants C and D and thereby the correct solutions for the stellar structure. At some fitting point Xf = (r/R)f we have to fit the exterior and the interior solutions together in order to get the solution for the whole star. At this fitting point we must of course require that pressure and temperature are continuous.