Invitation to a Contemporary Physics (2004)

C

Figure 8.8: Internal structure of stars in terms of prevailing modes of energy transfer, convection

(C) or radiation (R): (a) High-mass star; (b) Solar-mass star; (c) Low-mass star.

the p–p chain can be initiated. Since a lower temperature makes stellar matter more opaque, it is di cult for radiation alone to move energy produced at the center all the way to the surface; convection must do the job. A typical red dwarf has a radiative intermediate region enveloping a small hydrogen-burning core; it is surrounded, in turn, by a convective zone, which contains about one tenth of all the star’s mass (Fig. 8.8c).

How about the sun? This mid-MS star has a central temperature of about 14 × 106 K. It draws energy from fusion reactions of the p–p cycle and, to a much smaller extent, the CN cycle. It has a structure similar to that of a red dwarf, but its outer convective layer is considerably thinner, containing only about 2% of the sun’s mass (Fig. 8.8b). There is a high concentration of mass at the center, which arises from the nuclear ‘ashes’ accumulated ever since the sun reached the MS some 4.5 billion years ago.

8.3.3 Solar Neutrinos

The description given above of energy production processes in stars is part of the standard model of stellar structure and evolution. How can we test those ideas? The enhanced abundances of the expected products of the p–p and CN cycles over their initial values and the presence of the heavy elements, seen in spectral studies, provide very strong circumstantial evidence that the energy source inside ordinary stars is thermonuclear fusion. But it would be nice to have some more direct proof.

Evidently, the sun is a handy test-bed for the model. Nuclear reactions that take place in the center of this star produce photons and neutrinos, both detectable on earth. The photons interact with the solar material and take several hundredthousands of years to emerge into the surface layers, so that the light that reaches us from the sun comes from the energy released in processes occurring in the photosphere and is characteristic of the sun’s surface, not of its interior. Neutrinos, on the other hand, are weakly-interacting particles that rarely scatter o anything. A neutrino produced in the sun would go directly, according to the standard model, to

only by the gallium detectors, while the Kamiokande detector, with the highest threshold (7.5 MeV), can register only the most energetic neutrinos from the decay of 8B of the side branch (Eq. (8.4)). Davis’s chlorine detector, on the other hand, is sensitive to the 8B neutrinos as well as to monoenergetic neutrinos from electronassisted proton fusion (p + p + e− → 2H + νe) and from electron capture by 7Be as in Eq. (8.5).

All four groups have detected neutrinos, and Kamiokande has shown that the observed neutrinos do indeed come from the direction of the sun. However, in all cases, the rates measured are substantially smaller than the rate predicted by the standard solar model. The persistent disagreement between the calculated and the observed fluxes — the famous solar neutrino problem — was troubling, because it could mean that we understood solar physics or neutrino physics less well than we thought.

Neutrinos come in three flavors, associated with the three existing charged leptons (the electron, the muon and the tau). The solar p–p chain produces only electron neutrinos; they are the only ones assumed to emerge from the sun and, in fact, the only ones detected by the radiochemical detectors. But already in 1967, Bruno Pontecorvo suggested the possibility that neutrino flavors could become mixed in vacuo; he and V. Gribov interpreted the solar neutrino deficit first reported by Davis in 1968 as evidence for neutrino oscillation (the sinusoidal dependence on path length of the probability of the metamorphosis between two flavors; see Chapter 9 for further details). In 1985 S. Mikheyev and A. Smirnov suggested that such mixing might be su ciently amplified by interactions with matter on the way out of the sun to explain the missing neutrinos.

Recently, three new solar neutrino detectors, with vastly increased capacities, have come into operation, and are expected to clarify the situation. They are Super-Kamiokande, the successor to Kamiokande; the Sudbury Neutrino Observatory (SNO) in a northern Ontario mine; and Borexino in the Gran Sasso laboratory. The first two are sensitive to 8B neutrinos, whereas the third will concentrate its e orts on observing 863 keV 7Be neutrinos.

All these new detectors can register individual neutrino collision events and can detect neutrinos of any flavor through their elastic scattering o electrons in

ˇ

the detector’s liquid. But SNO, with its heavy-water Cerenkov detector, can also register signals from deuteron break-up:

and electro-production o deuterons:

While the former reaction is open equally to all three flavors, the latter is available only to the electron neutrino νe (because νµ and ντ in creating µ and τ would require energies far beyond the solar neutrino spectrum). Thus, the researchers at

288 Bright Stars and Black Holes

SNO can measure the flux of incoming neutrinos for all flavors from Eq. (8.6) and the flux of the incoming νe neutrinos from Eq. (8.7). In 2002 they reported that the νe component of the 8B solar neutrino flux is 1.76×1010 m−2 s−1 for a kinetic energy threshold of 5 MeV and the total flux, for all flavors, is 5.09×1010 m−2s−1, consistent with the standard solar model’s flux prediction for the electron neutrinos produced by 8B decays in the solar core (5.05 × 1010 m−2 s−1). The non-νe component measured, 3.41 × 1010 m−2 s−1, constitutes a strong evidence for solar neutrino oscillation.

8.3.4 Post-Main-Sequence Evolution

Although a star remains on the main sequence for a very long time with hardly any changes in its basic properties — deriving its exceptional stability from the abundant energy produced by fusing hydrogen into helium — this is only a temporary pause. When the hydrogen fuel in the central core (about 10% of the star’s mass) runs out, all transformed into helium, the star evolves o the main sequence, beginning a march towards its inevitable end. At this time, outside the core lies a thin shell in which hydrogen is still burning and, just above it, a thick envelope in which temperatures have never risen high enough to ignite hydrogen. What exactly happens next depends on whether we are talking about low-mass stars or highmass stars, but the issues common to both revolve around the burning of successive rounds of nuclear fuels and the flow of energy to the outside.

8.3.4.1 Evolution of Low-Mass Stars

Let us take a low-mass star, like our sun. The core, now depleted of its hydrogen fuel and filled with inert helium, continues to lose its heat and, in the absence of an adequate compensating nuclear energy production, contracts gravitationally, heating itself up as well as the surrounding shell. The energy generated in the hydrogen-burning shell contributes in part to the energy of the envelope above it and in part to the star’s surface luminosity. The star expands and its radius increases. As the luminosity remains constant or increases slightly, the e ective temperature decreases a little. Thus, immediately after leaving the main sequence, the star moves more or less horizontally to the right in the H–R diagram, turning itself into a subgiant (Fig. 8.10).

As the core keeps on contracting, heat continues to rush out into the overlying shell. The burning shell, with its ever increasing temperature, produces energy at rates so high that the pressure exerted on the outer layers causes them to expand greatly and the surface to shine brightly. As the outer envelope expands and cools, the surface temperature falls. But it cannot fall indefinitely. In regions just below the stellar surface, the temperature remains low and hydrogen exists as neutral atoms. But as we go deeper in, we find zones where temperatures are high enough to ionize hydrogen. Bubbles will form, rising quickly through the cooler material to

Figure 8.10: Post-main-sequence evolution of a 1 M star and a 10 M star.

the surface, and when the temperature at the surface drops below a certain limit, the whole envelope becomes convective. Hot gases can now flow rapidly upward, and the luminosity increases a thousandfold in about half a million years. As new energy now arrives at the surface, the temperature which has been falling there lately stabilizes around 3 500 K. This temperature barrier forces the evolutionary track to turn almost vertically upward for all but the most massive stars. We now have a very big red star, a red giant — 160 times bigger than the sun, 2 000 times brighter and somewhat cooler. This giant star has at its center a tiny core, a few thousand times smaller than the star itself but considerably heavier than initially, having gained mass from the helium ash of the surrounding shell. Meanwhile, outside, a strong hot wind has blown o a large fraction of the stellar material into space.

As the core continues to collapse, it becomes hotter and denser. The electrons that have long been pulled out of their atomic orbits are now so tightly packed that they evolve into a degenerate quantum state,13 while the more massive helium nuclei remain nondegenerate. When the central temperature reaches 2 × 108 K, triple-α fusion begins:

4He + 4He → 8Be + γ , 8Be + 4He → 12C + γ .

(An asterisk is used to indicate that 12C is formed in an excited nuclear state). Carbon combines with helium to form oxygen, 12C +4 He → 16O + γ. Or it can react with a proton to give 13N, an unstable isotope which quickly decays to 13C to

13See Appendix C. The key point is that the pressure of a degenerate gas depends on its particle density, not its temperature, so that compression does not lead to heating.

290 Bright Stars and Black Holes

undergo in turn the process 13C+α → 16O+n. This reaction is an abundant source of neutrons and will be called on to play a key role in nucleosynthesis in high-mass stars.

Triple-α fusion produces energy copiously at a rate that depends strongly on temperature (≈ T 40). Whereas in a normal star, the slightest local increase in the core temperature and pressure provokes a local expansion and a concomitant reduction of temperature, here the degenerate pressure is completely decoupled from thermal motions. So any increase in the core temperature leads to an accelerated energy production without the compensating e ects of a pressure increase and a stellar expansion. A runaway production of nuclear energy ensues: after ignition, helium burns in a flash. This sudden massive release of energy generates enough internal power to inflate the core, lowering both its density and temperature. The potential explosion, which would have catastrophically disruptive e ects, is thereby contained, and the electrons return to a normal, nondegenerate state. With a lowered temperature, the triple-α process shuts o , and even hydrogen burning in the outer shell slows down. Both temperature and pressure gradients drop throughout the star and, as the core itself expands, the outer envelope will contract. This period corresponds to the horizontal branch in the H–R diagram.

Eventually, the core expansion damps out and, abetted by the infall of the upper layers, the core starts contracting again, raising its temperature back to above 108 K. Helium burns anew but now in a nondegenerate environment. When the helium fuel in the core finally runs out and the nuclear energy production there comes to an end, a familiar sequence of events takes place. The outer layers collapse, releasing enough gravitational energy to ignite helium in a shell outside the now dormant carbon core. And in due course, convection sets in, sending the star up to the supergiant status, a red globe 180 R in radius. Now fed by the doubleshell sources of hydrogen and helium nuclear energy, the star shines brighter than 3 000 suns. Meanwhile, the core is compressed, and the electrons again become degenerate. The degenerate-electron pressure should be high enough to support the envelope of a low-mass star, so that the core, now a solid carbon–oxygen sphere, remains inert at a temperature below the next fusion point.

During its ascent on the asymptotic giant branch, as this period is called, the star loses so much mass in stellar winds that its outer envelope becomes very thin (assuming, as always, a low-mass star). At the top of the climb, successive sharp thermal pulses of extra energy production, interrupted by periods of quiet evolution, start in the thin, unstable helium-burning shell. Each thermal runaway ejects more matter into space until the last burst blows o what is left of the envelope, creating one of the most beautiful sights in the heavens: a planetary nebula, a diaphanous ring of ionized gases surrounding a very hot and very bright star, the size of the earth with half the mass of the sun.

The exposed hot core sheds its extended envelope, burns out its hydrogen and helium shells and, finally, with all inner fires extinguished, evolves into an inert carbon–oxygen star that slowly cools and dims to become a white dwarf.

Stars with masses of 1–6 M evolve in a similar way. In the H–R diagram, a typical low-mass star leaves the main sequence horizontally when all hydrogen in the core has been fused into helium, becoming a subgiant at the base of the giant branch. As hydrogen is burning in the shell surrounding the helium core, the star begins its ascent, and when it reaches the top of the climb, the core helium ignites in a flash, leading to its descent along the horizontal branch. When the core, now made up of carbon and oxygen, starts contracting, and while the helium shell and the hydrogen shell burn, the star begins its abrupt ascent on the asymptotic giant branch. Meanwhile, it is losing a considerable amount of material in stellar winds, making its outer shells spatially so thin that they become unstable to the nuclear inferno. When the last of the outlying layers have been blown o to form a planetary nebula, a bare core with a mass less than 1.4 M is exposed, and the star descends the H–R diagram to enter the region inhabited by the white dwarfs.

8.3.4.2Evolution of High-Mass Stars

We turn now to stars with masses greater than six suns. The evolution of these stars di ers from that of the low-mass stars we just described on two points. First, they evolve much more quickly because they lose energy through radiation at a much higher rate (remember the mass dependence of luminosity, L M4). Second, their large mass can keep the central core at a su ciently low density to allow the electron gas to remain essentially nondegenerate (recall ρ M−2), but still at a pressure and temperature high enough to ignite nuclear fuels all the way to iron.

In a high-mass star, as helium burning occurs under nondegenerate conditions, there is no helium flash, and the star moves leftward in the H–R diagram, towards higher temperatures (Fig. 8.10). Once all the hot helium fuel has been fused into carbon and oxygen, the newly formed inert core contracts by its own self-gravity and, with this released energy, ignites a ring of helium just outside. The evolutionary curve turns back to the right. There is little time for radiation to reach the surface, so the luminosity changes little and the evolutionary track progresses more or less horizontally in the H–R plot. When the central temperature reaches 5 × 108 K, carbon ignites, starting new chains of reactions, fusing two carbons into magnesium or neon. Although these reactions are extremely sensitive to temperature, they give rise to no disruptive events because carbon ignition has occurred before the electron degeneracy becomes important. The nondegenerate core can adapt itself to the rapidly changing conditions, e ciently releasing its extra pressure to the overlying shells until the carbon fuel is exhausted. So, very massive stars evolve quickly but smoothly — burning each successive round of fuel to exhaustion under normal-gas conditions, undergoing gravitational contraction, reaching quickly the temperature needed to ignite the newly-formed nuclear ash, and starting another fusion cycle. Carbon fuses into neon and magnesium; oxygen into silicon and sulfur; and silicon into nickel and iron. Each successive cycle proceeds at a faster pace: hydrogen burns for 10 Myr, helium for 1 Myr; the silicon layer is built in a few years,

H → He

He → C, O

C → Ne, Mg

O → Si, S

Si, S → Fe

Figure 8.11: Structure of a high-mass star at the end of the nuclear-fusion phase of its evolution.

and the iron core in a few days. Iron 56Fe is the most strongly bound of all nuclei, so no further energy can be extracted by nuclear fusion. The fusion process shuts o at this point, having played out its role in normal stellar evolution (Fig. 8.11).

When the star reaches the final phase, it has become a red supergiant, a thousand times larger than the sun. Its bloated envelope of unburned hydrogen is held tenuously by a central onion-like layered structure: a small inner core of hot iron surrounded by successive burning shells of silicon, carbon–oxygen, helium and hydrogen. Throughout all its post-main-sequence life and specially during the periods of evolutionary changes, the star loses gradually vast quantities of mass from its outer layers in hot winds and ejecta. Whether it will now die the quiet death of a white dwarf, or su er more violent convulsions before it goes depends on how much matter is left in the central core.

8.3.4.3Supernovae

Let us now consider a massive star on the edge, a thousand times larger than the sun, consisting of successive layers of burning hydrogen, helium, carbon–oxygen, neon–magnesium and silicon, and a central core of quiescent iron. The core may be some 1 000 km in radius and may hold 1.2–1.5 M at 10 billion K. As the source of nuclear energy at the center has dried up, the core is held up against its own weight only by the internal energy of a hot matter and the pressure of a degenerateelectron gas. But the support against the gravitational pull is being fast eroded, on the one hand, by energy-consuming processes in the interior (photodisintegration of nuclei, neutrino and neutron-rich nuclei production by electron capture) and, on the other hand, by the added weight of the infalling fusion products from the still-active upper layers.

When the core mass is nudged over 1.4 M , this fragile balance is broken, and the core collapses within a second to a structure 100 km in radius, squeezing matter within it to a density nearing that of nuclear matter. The implosion proper involves only the tiny inner core, while the bulk of the star shrinks at a relatively slower pace. Although nuclear matter is hard to compress because of the nuclear repulsive force at short distances (Fig. 8.4), it is not completely incompressible. The implosion, carried by momentum beyond the point of equilibrium, squeezes matter to a density slightly beyond that found in large atomic nuclei, before it finally grinds to a halt and vigorously bounces back. A pressure wave, built up at the center of the core, turns into a shock wave, carrying with it matter as it blasts its way out.

However, the shock wave fails to go all the way to the core limits — the material falling from the upper layers is crashing down on it while nuclear reactions saps its energy. In these processes, iron and other nuclei disintegrate into nucleons, costing 9 MeV per nucleon; or nucleons capture electrons and positrons to produce neutrinos and anti-neutrinos which carry away more energy. The shock wave stalls at some distance from the surface of the core.

What gets it started again is neutrinos. In the few seconds after the catastrophic collapse, the core is very dense and very hot, at several times 1011 K near its center. The only way it can cool down is by emitting a prodigious amount of neutrinos (while electromagnetic radiation is hopelessly trapped) and, thereby, transforming itself into a neutron-rich star. When the neutrinos move out through the star and are captured, they heat up the material there. Heat builds up a violent convection that breaks the tra c jam, re-energizes the shock wave and keeps it moving out of the core and into the surrounding envelope. Not only does the shock wave carry with it energy and matter at high speeds and violently heat up the outer layers, it also produces in its wake more heavy elements from the pre-existing materials.

In a few hours, the shock breaks out of the surface, moving at a speed of one tenth the speed of light. The star brightens within minutes to 10 billion times the solar luminosity, out-shining an entire galaxy: a supernova is born. Light appears first in the far ultraviolet, then shifts after a day to the visible; after a few months, most of the radiation is in the infrared, and the star fades out. Most of the outer envelope — made up of hydrogen, helium and heavy elements — is blown o , and, accelerated to a few 10 000 km/s, it expands while heating and stirring up the surrounding medium. The metal-enriched ejecta eventually cool o to mix with the interstellar gas and contribute to the composition of the next generation of stars. But it is not in the expanding gas nor in the electromagnetic radiation that the star releases its stupendous energy; it is rather the invisible and almost undetectable neutrinos that carry o 99% of the energy produced — 1046 joules, or a hundred times the energy radiated by the sun during its entire main-sequence lifetime.

The explosion throws o all the material of the parent star beyond a certain radius, enriching the interstellar medium with heavy elements, while the rest of the star settles down as a hot, neutron-rich ball, a hundred kilometers across, that quickly shrinks to one tenth of this size and slowly cools o . During this final period

of contraction, the remaining electrons are annihilated, and the matter that is left is primarily composed of tightly packed, degenerate neutrons: we have a neutron star. If the original star has a mass high enough, the relic core might exceed 3 M , and gravity will overcome the degenerate-neutron pressure to form, probably, a black hole.

Supernovae are spectacular and rare events. Among them, four have historical significance: SN1006 (supernova seen in the year 1006), described by medieval Chinese astronomers to be as bright as the half-moon; SN1054, also recorded by them and identified now as the event leading to the formation of the Crab Nebula; SN1572, sighted by Tycho Brahe, whose book about the event, ‘De Nova Stella,’ gave these objects a name; and finally SN1604, reported by Johannes Kepler, whose observations and explanations contributed to the final break with the Aristotelian traditional view that the realm of fixed stars was immutable.

But, of course, supernovae occur in other galaxies as well, as shown by Walter Baade and Fritz Zwicky, who initiated a systematic search in the 1930s. Since then, close to 150 supernova remnants have been detected in the Milky Way, and more than a hundred are being discovered every year in distant galaxies.

The more recent SN1987a is the nearest naked-eyed visible supernova seen since 1604. This event provides astrophysicists with a unique opportunity to test their current understanding of the most spectacular phenomenon in the heavens. Up to now, their knowledge about supernovae has been largely gained through theoretical modelings and computer simulations, but hard information about both historical supernovae and extragalactic supernovae is lacking. So it is fortunate that SN1987a is located only some 160 thousand light years away in one of the two nearest galaxies, the Large Magellanic Cloud. Its progenitor was a small blue supergiant having 18 M , and for the first time we detected the burst of high-energy neutrinos from the collapse of a massive star. Its evolution is practically unfolding before our eyes, and has provided us with a wealth of information about supernova physics, which helps to confirm, in particular, the basic predictions of the core-bounce picture.

8.3.5Summary

When a star heats up to ten million degrees at its center, the nuclear fusion of hydrogen (the most abundant of all elements) is ignited. Thermonuclear fusion is the only energy source that can sustain the radiation of stars for very long timespans. This process provides the most coherent explanation for a vast body of data, from the basic properties of stars on the main sequence to various phenomena marking the late stages of stellar evolution, in passing by the relative abundances of chemical elements observed in the universe.

According to the current stellar models, after 10% or so of hydrogen has been burned into helium, the star becomes structurally unstable. The relationships between its basic properties change drastically, and the star moves away from its