Invitation to a Contemporary Physics (2004)

dinates are logarithmic, as is the case for the standard H–R plot, stars of di erent radii fall on di erent straight lines. If we move upward from the sun’s (or any star’s) location on the diagram, we encounter brighter, bigger objects. If we move downward, we find fainter, smaller stars. To the right of the sun’s location, stars with a similar luminosity are cooler and bigger; but those to its left are hotter and smaller. (But di erent masses may coexist in the same region, except along the main sequence band, as we’ll see below.)

Once the luminosities and temperatures for individual stars are determined, we can construct the H–R diagram for the observed luminous stars. The outstanding feature of the diagram so produced is that most of the stars fall in a narrow band, called the main sequence, which runs diagonally from the top left, where the most massive stars lie, to the bottom right, where the lowest-mass stars lie. Our sun, a type G star, occupies about the middle of the main sequence and is a very ordinary star.

Extending from about the sun’s position toward the top right is a region called the giant branch, inhabited by large, cool stars with very low mass density. Aldebaran in α Tauri — a spectral type K star, forty times bigger and a hundred times brighter than the sun — is a red giant; Betelgeuse in α Orionis — a spectral class M star, seven hundred times the size of the sun and radiating ten thousand times more energy — belongs to the category of supergiants. Extending across the diagram from the giant branch toward the left at constant luminosity is the so-called horizontal branch, where one finds very unstable configurations. Some pulse with a constant period; others flare up sporadically. The Cepheids belong to this region, although they are not usually plotted in observational H–R diagrams, precisely because of the variability in their apparent brightness.

In the lower left-hand corner of the H–R diagram, below the main sequence, we find a group of bluish-white, compact and very faint stars known as white dwarfs. Sirius B, the faint companion to the brightest star in the night sky, is such a star: five hundred times less luminous than the sun, yet only 2.7 pc away. Procyon B, also found in a binary system, radiates about ten times less than Sirius B and belongs to spectral type F. White dwarfs have typically the mass of the sun, but the size of the earth.

White dwarfs are one of the three forms of ‘dead stars.’ The other two are even more massive and compact. They emit no light at all, and so cannot be seen directly by optical means and would fall outside the standard H–R plot: they are neutron stars and black holes.

We see that the H–R diagram is more than just a plot of luminosity against temperature. It is a systematic representation of the correlations between the surface properties of the luminous stars. It separates stars into distinct groupings: the main-sequence stars, the giants, the supergiants and the white dwarfs. What causes these groupings? Are they somehow related to one another? Or more aptly: What happens in the interior of the stars to make them look the way they do?

It has been known for a long time that stars evolve from one form to another and that stars we see in di erent forms are stars at di erent ages. The English astrophysicist Arthur Eddington already understood in 1926 that stars exist because the two opposing forces at work in the systems — gravity which pulls inward on the stellar gas while gas pressure and radiation pressure which push outward — are just balanced. At about the same time, H. N. Russell and H. Vogt established an important result: the mass and chemical composition of a star determine uniquely how it looks at equilibrium. Simply put, give me a mass and a composition, and I shall tell you what the star looks like. It means that the main-sequence stars of similar masses must have similar pressures, radii, temperatures and other physical attributes. It also implies that, if some red giants and white dwarfs have a similar mass, it is just because they have evolved and acquired di erent compositions, and hence reached di erent equilibrium structures. And so, di erent forms of stars mean di erent internal compositions corresponding to di erent stages of evolution. How does the stellar composition change? The answer is, by nuclear reactions: these processes provide not only the mechanism for the star’s internal changes, but also the energy necessary for maintaining its stability. In the final analysis, the key to a good understanding of the stellar evolution lies in nuclear physics, particle physics and, eventually, general relativity.

8.2.1Summary

One of the most useful tools in astronomy is the Hertzsprung–Russell diagram, a two-dimensional plot of luminosity versus temperature of luminous stars. It gives a systematic representation of the correlations between the stars basic properties and hence of the stars at di erent stages of development. The path a star takes in the H–R diagram is the history of its evolution from its birth in an interstellar cloud to its end as a dead star.

8.3 Bright, Shining Stars

The space between stars is filled with rarefied and cold matter whose chemical composition reflects that of the universe as a whole. The most striking feature of our galaxy is its spiral pattern. The spirals are probably formed by matter-density waves that the Galaxy builds up as it rotates rapidly and di erentially about its own axis. Half of the interstellar matter is distributed very thinly throughout the Galaxy in the equatorial plane, with the rest found in the spiral arms as cloud formations of various sizes and densities. Of greatest interest for their roles in the theory of stellar evolution are the Small Molecular Clouds and the Giant Molecular Clouds, which can grow big and dense enough to contract through their own weight.

A clump of cloud becomes self-gravitating when it gains enough gravitational energy to overcome its dispersive internal turbulent motions. It can get gravitational energy either by adding mass from the surrounding medium or by being suddenly compressed by some powerful external force. Within a short time, the collapsing globule of gas will build up a dense central nucleus, which tends to fall away from the outer parts but which may continue to pick up infalling material from the envelopes to form a progressively opaque core having a typical stellar mass. The evolution of this object, or protostar, will take it through several successive phases (free fall, convection and radiation), characterized by increasingly higher internal temperatures, before it reaches its position on the main-sequence line of the Hertzsprung–Russell diagram.

8.3.1 Nuclear Sources of Stellar Energy

In the pre-main-sequence period, only two kinds of forces are involved — the gravitational and the electromagnetic force — but now, as the star reaches the main sequence, the other two fundamental forces9 of Nature enter the scene: the strong (nuclear) and the weak force, so called because the former is the strongest of all known forces, and the latter, much weaker than the electromagnetic force. In contrast to the first two, they have a very short range, 10−15 m or less — they will not act until the two interacting particles come that close together — and, for this reason, we cannot directly see their e ects in our everyday life. Yet, they are essential to the structure and the dynamics of matter at a deeper level and on the grander cosmic scale, and are, therefore, of primary importance to humans. These are precisely the two forces thought to be capable of producing the amount of energy necessary to sustain the long-lasting luminosity observed in most stars.

Except for the nucleus of hydrogen, which consists of a single proton, the nuclei of all other atoms, including isotopes of hydrogen, contain both protons and neutrons, collectively called nucleons.10 Nucleons interact with one another through a mainly attractive nuclear force and a repulsive electric force. Since the former is a hundred times stronger than the latter, it contributes the largest part to the potential energy in a nucleus.

The mass of a bound nucleon system always totals less than the sum of the constituent masses. This mass deficit represents11 the binding energy B of the

9Explanations on the notions of forces and particles we use here can be found in Chapter 9.

10The nucleus of hydrogen (H) consists of a single proton; the nuclei of its isotopes, deuterium (2H) and tritium (3H), contain, besides the requisite proton, one and two neutrons, respectively. Helium (4He) has two protons and two neutrons, and its isotope 3He two protons and a neutron.

Generally, a nucleus is designated by the symbol AZ X, where X stands for the name of the element, A the mass, or nucleon, number, and Z the charge number, which is often omitted. Thus, AZ X is the same as AX.

11Recall that a mass m is equivalent to energy mc2, according to the famous Einstein equation,

E = mc2.

uranium (235U) is often used as fuel; it breaks up into a variety of lighter nuclei, yielding about 0.9 MeV per nucleon. Needless-to-say, the total energy of the system is conserved throughout the reactions: there is no gain nor loss, only a transfer of one form of energy into another.

Hydrogen is by far the most abundant material in the universe: more than 90% of the atoms in the universe are hydrogen, and all but less than 1% of the remainder is helium. As the protostar is steadily contracting gravitationally, the density and the temperature near its center rise steeply (7.5 ×1031 protons/m3 and 15 × 106 K), and the conditions are set for hydrogen to ‘burn.’ Initially, the basic process is the fusion of two protons into deuteron. Since the two reacting particles carry positive charges, they tend to repel one another electrically. For them to come close enough together to allow the attractive part of the nuclear force to take over and create a bound nuclear state, they must have a su ciently high relative speed, which can arise naturally in large random thermal motions in a very hot gas (or it can be produced artificially in terrestrial accelerators). The proton–proton fusion reaction and all other common fusion reactions, which involve charged light nuclei, require a high temperature to proceed; for this reason, they are called thermonuclear reactions.

At the high temperatures prevailing in the core of the star at this stage, the particles have just about a thousand electron-volts in kinetic energy,12 which is insu cient for two protons to overcome their mutual electric repulsion (in excess of 1 MeV). However, they can still traverse the Coulomb barrier by ‘tunneling’ under it, a mechanism allowed by quantum mechanics (Fig. 8.4; see also Appendix B). The higher energy they have, the more vigorously they can tunnel and the more likely they are to complete the fusion process. And since there always exist some su ciently energetic particles in a thermal gas, the reaction is bound to occur, more or less frequently. The rate at which it actually occurs and hence the rate of energy production itself depend very critically on the temperature: fusion turns on and o quickly with the slightest changes in temperature. This is why nuclear fusion can ignite and maintain itself only in the deep interior of stars.

8.3.2 On the Main Sequence

With the onset of nuclear fusion, the protostar becomes a star — a temporarily stable object making a long pause on the main sequence as it traces its evolutionary path through the luminosity–temperature space. The star owes its exceptional stability to a constant interplay between the nuclear and gravitational forces (Fig. 8.5). As the rate of nuclear reactions is strongly influenced by the temperature, the production of nuclear energy might for an extremely short time exceed that of gravitational energy in the star’s central regions, and so the heated stellar

12Thermal energy is proportional to temperature. Particles in a thermal distribution at temperature 12 000 K have an average kinetic energy of 1 eV per particle.

material might expand locally to some infinitesimally small degree. Since the whole system is in quasi-equilibrium, the expansion will cause pressure and temperature to drop ever so slightly, which in turn immediately dampens the nuclear fusion rate. The gravitational pull will quickly take over as the dominant force, and the resulting compression instantly raises the temperature, restoring the system to its original equilibrium state which has been temporarily broken by local heating. This harmonious interplay between contraction and expansion, together with an abundant supply of hydrogen as fuel, contributes to maintain the star in an overall stable structural equilibrium, at constant radius and temperature, that can last for a considerable length of time. Most luminous stars we see in the sky belong to this period of their evolution, by far the longest in their active life.

The American physicist Hans Bethe (winner of the Nobel Prize for Physics in 1967) found that the most important mechanism for generating nuclear energy in stars, at this stage, is nuclear fusion, which produces, in net, helium from four hydrogen nuclei in two main reaction sequences — the proton–proton chain and the carbon–nitrogen cycle.

8.3.2.1The Proton–Proton Chain

The first step in this sequence, which gives the chain its name, is the combination of two protons into the only stable two-nucleon system, the deuteron, accompanied by a positron (e+) and a neutrino (νe):

Even at high temperatures, this reaction occurs very slowly because it is mediated in part by the weak interaction, which must occur to turn a proton into a neutron, in a process inverse to the more familiar β decay. Because of the Coulomb barrier, only protons in the high-energy tail of the thermal distribution can have su ciently high relative speeds to penetrate the barrier and come close enough together to interact and form a bound state. For these reasons the collision rate remains low, about 5 × 10−18 per second per proton. It is the slowest and least probable step (the bottleneck) in this chain of reactions. Nevertheless, there are so many protons around (like 1056) that the reaction can in fact take place at a su cient rate to get the chain rolling.

Once a deuteron is formed, it is most likely to combine with another proton to produce the helium isotope 3He. Once created, there are several ways for 3He to go, the most likely being to follow the ‘main branch’ and combine with another 3He to produce ordinary helium:

282 Bright Stars and Black Holes

Figure 8.6: Sequence of reactions in the main branch of the proton–proton chain of fusion.

The complete sequence, known as the proton–proton chain, is shown in Fig. 8.6. In net, we have four protons converted to helium:

Part of the energy generated is carried away as kinetic energy by the reaction products, leading to a net heating of the surrounding medium. The positrons will eventually annihilate with existing electrons, producing more gamma rays. All the gamma rays produced will interact with the medium to give additional heating; they will take some hundred thousand years to reach the surface. The neutrinos, in contrast, interact so weakly with the surrounding matter that they emerge directly from the core without contributing to the heating of the photosphere (the outer region of the star where the energy released in nuclear reactions is converted into light). Except for the small energy carried o by neutrinos, the mass deficit of the reaction product (26.7 MeV) becomes an e ective heat input for the star. In other words, of 1 kg of hydrogen, 7 g are converted into light, an amount equivalent to 180 GWh.

There are other ways for 3He to react: it could combine with protons, deuterons or α-particles. But reactions with protons are not possible because they would produce unstable systems which immediately break up on formation, while fusion with deuterons is highly improbable because deuterons are so scarce, being converted into 3He as soon as formed. The third alternative is the only one possible, when a 3He nucleus encounters a 4He nucleus to form the nucleus of 7Be, which is followed next either by the sequence,

or by the sequence,

The net reaction is the same for all three possible paths, but calculations identify the main branch (8.2) as the most important in solar-mass stars. We can test this

Figure 8.7: Sequence of reactions in the carbon–nitrogen fusion cycle.

prediction and discover the relative importance of the three alternatives by detecting the incoming neutrinos and analyzing their distinctive energy profiles. We have in the first case (8.2) a continuous distribution of neutrinos with a maximum energy of 0.42 MeV, and similarly in the second case (8.4) with a maximum of 14 MeV. In the third case (8.5), the two-body 7Be electron capture produces monoenergetic neutrinos mainly at 0.86 MeV.

8.3.2.2 The Carbon–Nitrogen Cycle

Bethe also found that higher-charge species can act as catalysts for alternative fusion schemes, involving heavier elements, that might compete with the proton–proton chain. The most important of these paths follows the carbon–nitrogen (CN ) cycle, which consists of a series of proton captures and beta decays, as shown schematically in Fig. 8.7. The net process is again the production of a helium nucleus from four hydrogen nuclei, exactly as in the p–p chain, with an associated release of energy. Carbon is needed to start the chain, but reappears at the end of the cycle to be used again in the first reaction of the subsequent cycle. Once initiated, the cycle can proceed more rapidly than the p–p chain because it does not have a bottleneck analogous to the deuteron formation. However, as the Coulomb barrier is six times higher for proton and carbon than for two protons, the CN cycle will not be competitive with the p–p chain until the central temperature rises well above 15 million K. Therefore, while the p–p chain is the main route for the synthesis of helium in stars less massive than 2 M , the CN cycle becomes the dominant source for energy production in more massive, much hotter stars, because the energy production in e ect then, with a rate that depends on temperature as T 16, will completely dominate the weaker source from the p–p chain, whose rate goes only as T 4.

8.3.2.3 Properties of Stars on the Main Sequence

The position of a star on the main sequence (MS) varies little with its age on the MS, or with the amount of hydrogen burned up, as long as the fuel in the core (containing 10% of the star’s mass) has not been completely exhausted.

The most important factor in determining a star’s position on the MS is its mass. The empirical fact that normal stars have masses in the range 0.1–60 M can be understood as follows. A protostar with a mass less than 0.1 M is called

a brown dwarf; it does not have enough self-gravity to compress its center to the high temperatures needed to complete the fusion process; after deuterium is burned out, the star will cool to a fully degenerate configuration without achieving the stellar status. On the other hand, an object with a mass much greater than 60 M will see its centers raised to temperatures so high that nuclear burning can proceed at furious rates, enabling radiation pressure (P T 4) to dominate over matter pressure (P T ). Eventually, the star ejects enough matter to fall below the critical mass. However, this does not mean that supermassive stars might not form and live, albeit for a short time. Stars with masses in the thousands or even millions of solar units may have an exceedingly short life, but, with their prodigious energy release, they would not pass without disturbing the universe.

Basically, a star shines because heat can leak out from its interior. The rate of this leakage determines the star luminosity L, which turns out (on general physical grounds) to vary with its mass as M4. This dependence means that an MS star of 10 M produces 10 000 times more power than the sun. And yet, with only 10 times more fuel to burn, it has an MS life-span 500 times shorter than the sun’s — 20 million years to the sun’s 9 billion years.

For a star to remain in equilibrium, it must adjust its configuration to have an exact balance between its kinetic and potential energy. This implies that its radius and internal temperature must be related, T (core) M/R. Since nuclear fusion is ignited at a specific threshold temperature, regardless of other properties, all stars reaching the MS should have roughly the same core temperature and hence the same mass-to-radius ratio. In other words, R M on the MS. Putting this result together with the two relations for luminosity (L M4 and L R2T 4),

we see that the surface temperature of stable stars increases (weakly) with mass

√

(T = constant× M ). Massive MS stars not only must have higher luminosities but also hotter surfaces than low-mass MS stars. This is the reason for the distribution of normal stars along the diagonal band in the H–R diagram.

Not all stars on the MS have the same internal structure, though. For one thing, small stars are denser than big stars, which can be understood if we just remember

that M/R is roughly constant on the MS, and so the mass density must vary as

ρ M−2.

Stars also di er in their internal make-up. Hot, massive stars have a convective inner zone surrounded by a radiative region, where energy is transported by radiation. For example, for a star ten times the mass of the sun and four to five times as hot, the convective core may occupy one fourth of the star’s volume and contains the same fraction of its mass (Fig. 8.8a). The temperature there may reach 25 × 106 K and the carbon–nitrogen cycle provides most of the star’s energy. As this process generates energy at a prodigious rate, radiative transfer alone becomes inadequate to remove the heat produced and must be helped out by convection.

But for red dwarfs, those stars on the lower part of the MS band, the reverse seems to hold. A red dwarf whose mass is 0.6 and luminosity 0.56 solar units has such a weak self-gravity that its center can reach just about 9 × 106 K, and so only