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masses of 50 M&, 10 M&, 5 M&, 1 M&, 0.5 M&and 0.1 M&does the Salpeter

mass function predict? ■

As the Exercise 8.7 shows, if the Salpeter mass function is correct, there should be over 10 million low-mass stars (just above the brown-dwarf limit) for every star formed at the upper mass limit for stars.

Checking this has proved difficult however. Low-mass stars are necessarily very faint and very difficult to detect. In recent years though, advances in infrared astronomy have enabled censuses to be carried out of low-mass stars in the solar neighbourhood. It turns out that there are rather fewer low-mass stars than predicted by the Salpeter mass function. Instead, the mass function seems to have a peak around a mass of 0.5 M&, and falls off either side of this with roughly the

same slope. The most common stars in the Universe are therefore M-type stars with about half the mass of the Sun.

8.5.2 The single-star fraction

Observationally it is clear that many stars exist in binary systems, or in clusters. For instance, the nearest star system to the Sun, Alpha Centauri, is actually a triple system consisting of a close binary pair (α Cen A and α Cen B) in orbit with a more distant third star (known as Proxima Centauri).

Historically, in the 18th century it had been recognized that the fraction of visual double stars was too high to be due to chance alignments, and by 1802 William Herschel had catalogued hundreds of visual pairs and computed the first orbits of binary stars. In 1983 Helmut Abt had claimed that 70% – 80% of all solar-type stars were in binaries, and this led to a long-held belief that most stars in the galaxy were in fact formed as part of binary systems.

However, in recent years it has been realized that the binary-star fraction is a function of stellar type (and therefore stellar mass). Recent surveys have concluded that, whilst the binary-star fraction is indeed high for massive stars, it is somewhat less for lower-mass stars. Furthermore, since most stars in the Universe are low-mass (∼ 0.5 M&) stars, the binary-star fraction is not as high as had been

supposed.

It is actually easier to talk about the single-star fraction, since that avoids complications concerning binary, triple, quadruple, etc., systems containing more than one star. The fraction of stars that are single appears to be about 25% for

massive stars (M > 2 M&), about 45% for solar-type stars, and about 75% for

low-mass (M ∼ 0.5 M&) stars. When the fraction of single stars is weighted by

the initial or present day stellar mass function, it turns out that about 65%–70% of stars are actually single, and most of these are low-mass M-dwarfs.

8.6 Stellar life cycles

Having examined the process by which stars are born, we conclude the book by revisiting the idea of stellar life cycles. In Chapters 6 and 7 you read about planetary nebulae leading to white dwarfs, and supernovae leading to neutron stars and black holes. You might have thought of these as the final link in the chain of the life and death of stars. But the process does not end. Rather, mass-loss via stellar winds, planetary nebulae and supernovae is a vital link in a cycle of activity that explains the chemical evolution of entire galaxies and the Universe.

As you have just learnt in this Chapter, stars form in gravitationally contracting gas clouds, and gravitational contraction continues to drive stellar evolution throughout a star’s life (Chapter 2). Stars ignite hydrogen in their cores if their masses exceed ≈ 0.08 M&, and produce helium and possibly nitrogen (via

the CNO cycle) (Chapters 1, 3 and 4). If their masses exceed ≈ 0.5 M&,

later they ignite helium to produce carbon and oxygen (Chapter 5), and if their main-sequence masses exceed ≈ 10 M&they produce elements up to

the iron peak (Chapter 6). Carbon and nitrogen synthesized in low-mass and intermediate-mass stars (M ≤ 2 M&and 2 M&≤ M ≤ 8 M&respectively) are

returned to the interstellar medium via mass-loss late in the life of AGB stars (where s-process nuclei are also made) and planetary nebulae (Chapter 6). Most heavier elements are produced in supernovae or their progenitors (parent stars). Type II supernovae (SN II) eject the products of nucleosynthesis that range from helium in the outer layers of the star, down to iron-peak elements on the edge of the iron core, and even elements beyond the iron peak that are produced by the r-process during the explosion (Chapters 6 and 7). Type Ia supernovae (SN Ia), whilst being lower-mass objects not possessing an iron core prior to the explosion, are prolific producers of iron during the explosion.

The nucleosynthesis products ejected during (though not necessarily produced during) supernova explosions enrich the interstellar medium, and accompanying shock waves from the explosion are believed to compress interstellar clouds and trigger the next episode of star formation. Stars that form from gas enriched by these events have a higher metal content (i.e. elements other than hydrogen and helium) than previous generations of stars. Their stellar structure also differs, which in turn affects the ways they evolve.

The oldest stars seen in the Galaxy today are observed to have very low

abundances of metals; they are referred to as Population II stars. These stars are found within the bulge near the centre of the Galaxy and in the halo of the Galaxy, including the globular clusters. In contrast, younger stars (such as the Sun) have higher abundances of metals; they are referred to as Population I stars. These are typically found in the spiral arms of the Galaxy. Of course, there is a gradual transition from metal-poor Population II stars to metal-rich Population I stars, and the disc of the Galaxy contains a population of stars with a range of metallicities.

Summary of Chapter 8

The metallicity of a star essentially depends on how many generations of stars preceded its formation.

An important point to note is that even Population II stars contain some metals. It is therefore suggested that an even earlier generation of metal-free stars once existed in the Galaxy. These hypothetical Population III stars must have been formed from primordial hydrogen and helium only, and lived their lives before the observed Population II stars were born. Such stars are postulated to have been extremely massive (perhaps several hundred solar masses) and consequently would have had extremely short lives (less than a million years).

The physics of extrasolar planets is discussed in the companion volume to this book: Transiting

Exoplanets by C. A. Haswell.

By observing the different chemical compositions, ages, and space motions of the stars that make up galaxies, astronomers can piece together the history of entire galaxies, and ultimately the Universe. Such investigations also have a very direct bearing on our presence, here on planet Earth. By studying extrasolar planets (planet orbiting other stars), a strong correlation has been found between the presence of planets and the metallicity of their host star – a star with a higher abundance of heavy elements is more likely to possess planets. The possibility of life in the Universe is therefore intimately linked with the processes of stellar evolution and nucleosynthesis.

Summary of Chapter 8

1. The Jeans criterion for collapse, ETOT < 0, can be expressed in several

ways. A cloud of radius R collapses if its mass M exceeds the Jeans mass:

M > MJ= _2Gm3kT R. (Eqn 8.1)

A cloud of mass M condenses if its average density exceeds the Jeans density: ρ > ρJ= , 3kT 2Gm -₃ 3 4πM2. (Eqn 8.3)

2. The changing parameters of collapsing gas clouds may be tracked on a log(temperature)–log(density) diagram. The Jeans criterion indicates that the critical boundary for collapse has the form

log₁₀T = 1₃log₁₀ρ + constant.

Adiabatic collapse for an ideal gas with 3 degrees of freedom would have a slope 2/3, and would always lead back to the Jeans line, thus failing to form stars. Rather, collapse is initially non-adiabatic. Initially clouds cool, until heating and cooling timescales become comparable. High-density clouds then collapse almost isothermally until they become optically thick, after which they contract almost adiabatically.

3. The Jeans equations show that if the cloud collapses (R decreases and ρ increases) without heating up, then progressively smaller submasses satisfy the Jeans criterion. This leads to fragmentation of the collapsing cloud. 4. A 1 M&molecular cloud fragment that has just become unstable to collapse

at a temperature of 20 K would have a Jean’s density

5. A cloud collapses almost isothermally and in free fall (τff ≈ 105yr) until the

liberated gravitational energy has completely dissociated molecular hydrogen (H2) and then ionized atomic hydrogen. For 1 M!of hydrogen,

the necessary amount of energy (∼ 3 × 1039_{J) is liberated when the}

protostar collapses to R ≈ 150 R!.

6. Ionization of hydrogen creates a plasma of protons and electrons and hence also increases the opacity. The protostar becomes opaque to its own radiation and heats up, and the pressure slows the collapse. The protostar approaches hydrostatic equilibrium. By the time hydrostatic equilibrium has been established, the internal temperature has risen to ∼ 30 000 K.

7. In hydrostatic equilibrium, the star undergoes the long Kelvin–Helmholtz phase in which it slowly collapses, remaining in hydrostatic equilibrium. The loss of gravitational potential energy due to collapse is at a rate sufficient to replenish the energy radiated from the surface.

8. The Kelvin–Helmholtz contraction comprises two parts: the Hayashi track (where the protostar is fully convective) and the Henyey phase (where the protostar is dominated increasingly by radiative transport).

9. The Hayashi convective boundary exists because cooler stellar material has higher opacity, according to the Kramers opacity law κ ∝ ρT−3.5_{. High}

opacity impedes the radiative heating of layers further out and hence leads to a steep radiative temperature gradient; a steep radiative temperature gradient promotes convection.

10. The Hayashi boundary is at Teff ≈ 3000–5000 K (depending on mass,

composition and luminosity), and protostars descend along the boundary from high luminosity to low luminosity at roughly constant effective surface temperature. Stars below 0.5 M!reach the main sequence at the base of

their Hayashi tracks, whilst still fully convective.

11. Stars more massive than 0.5 M!enter the subsequent Henyey contraction

phase before reaching the main sequence. During this phase the luminosity goes as L ∝ T_eff4/5.

12. Protostars are generally still cloaked in dust, but can be observed in radio, infrared, and X-ray wavelengths. Bipolar outflows (along the rotation axis) are often seen, along with evidence for circumstellar discs. As the dust is blown away from the luminous protostar, a T Tauri pre-main-sequence star appears in the H–R diagram near the fully convective (Hayashi) boundary. 13. The initial stellar mass function seems to peak at around 0.5 M!and fall off

either side with a power law dependence roughly ∝ M±2.35_{. Most stars in}

the Universe are therefore low-mass M-dwarfs.

14. The binary-star fraction is a function of stellar mass. Only about a quarter of high-mass stars are single, but around half of solar-type stars are single. The single-star fraction for stars around 0.5 M!is about 75%.

15. Stellar evolution is a cyclic process, with the nuclear-processed products of one generation seeding interstellar space to be incorporated into subsequent generations of stars.