D.2 Stellar characteristics and stellar evolution

Spectral classes

1) A classification of stars based on their surface temperature and colour. 2) The spectral classes are called O, B, A, F, G, K, and M (remembered as Oh Be A Fine Girl Kiss Me!).

How surface temperature may be obtained from a star's spectrum

1) The wavelength at which most radiation is emitted is measured from an intensity-wavelength graph. 2) Using Wien's law, the surface temperature is determined.

Black hole

A point in space where the curvature is infinite; an end product in the stellar evolution of very massive stars (mass of core greater than the Oppenheimer-Volkoff limit).

Birth of a star

Stars are formed out of contracting gases and dust in the interstellar medium, which has hydrogen as its main constituent. Initially the star has a low surface temperature and so its position is somewhere to the right of the main sequence on the HR diagram. As the star contracts under its own weight, gravitational potential energy is converted into thermal energy and the star heats up; it begins to move towards the main sequence. The time taken to reach the main sequence depends on the mass of the star; heavier stars take less time.

Stability of a white dwarf (electron degeneracy pressure)

The conditions in the core mean that the electrons behave as a gas, and the pressure they generate is what keeps the core from collapsing under its weight. This pressure is called electron degeneracy pressure and is the result of a quantum mechanical effect, referred to as the Pauli Exclusion Principle, which states that no two electrons may occupy the same quantum state.

Evolution of a low-mass star after the main sequence

The evolutionary path takes the star off the main sequence and into the red giant region. The star gets bigger and cooler on the surface and hence becomes red in colour. The time taken to leave the main sequence and reach the planetary nebula stage is short compared with the time spent on the main sequence: it takes from a few tens to a few hundreds of million years. The path then takes the star to the white dwarf region. The star is now a stable but dead star. No nuclear reactions take place in the core.

Planetary nebula

The huge release of energy from the nuclear reactions blows away the outer layers of the red giant star in an explosion called a planetary nubeula; mass is thrown into space, leaving behind the carbon core (and some oxygen).

Nature of science: Evidence from starlight

The light from a star is the best source of information about it. The distribution of frequencies tells us its surface temperature, and the actual frequencies present tell us its composition, as each element has a characteristic spectrum. The luminosity and temperature of a star are related, and together give us information about the evolution of stars of different masses. Using this evidence, Chandrasekhar predicted a limit to the mass of a star that would become a white dwarf, while Oppenheimer and Volkoff predicted the mass above which it would become a black hole. The development of theories of stellar evolution illustrates how, starting from simple observations of the natural world, science can build up a detailed picture of how the universe works. Further observations are then needed to confirm or reject hypotheses.

Oppenheimer-Volkoff limit

The maximum mass of a neutron star, about 3 solar masses.

Evolution of a massive star after the main sequence

The star moves off the main sequence and into the red supergiant area. As the path moves to the right, ever-heavier elements are produced. Eventually, the star undergoes a supernova explosion and ends up as either a neutron star or a black hole.

Exiting the main sequence

When about 12% of the hydrogen in the star has been used up in nuclear fusion, a series of instabilities develops in the star, upsetting the delicate balance between radiation pressure and gravitational pressure. The star will then begin to move away from the main sequence. What happens next is determined mainly by the mass of the star. Other types of nuclear fusion reactions will take place and the star will change in size and surface temperature (and hence colour).

Hertzsprung-Russel (HR) diagram

1) A plot of luminosity versus temperature for stars. The vertical axis represents luminosity in units of the Sun's luminosity. The horizontal axis shows the surface temperature of the star. The temperature decreases to the right. Note that the scales are not linear. The slanted dotted lines represent stars with the same radius. 2) Once we know the temperature of a star (e.g. through its spectrum), the HR diagram can tell us its luminosity with an acceptable degree of accuracy, provided it is a main-sequence star.

HR region: Red supergiants

1) Above the red giants are the red supergiants (very large and cool). 2) Even larger and brighter than red giants.

White dwarf

1) An end product in stellar evolution in which electron degeneracy pressure is in equilibrium with gravitational pressure. 2) Exposed, and with no further energy source, the star is doomed to cool down to practically zero temperature and will then become a black dwarf.

Neutron star

1) An end product in stellar evolution in which neutron degeneracy pressure is in equilibrium with gravitational pressure. 2) Neutron degeneracy pressure keeps the neutron star stable, provided the mass of the core is not more than about 2-3 solar masses (Oppenheimer-Volkoff limit)

Cepheid variable

1) Cepheids variable stars are stars whose luminosity is not constant in time but varies periodically from a minimum to a maximum, the periods being typically from a couple of days to a couple of months. 2) The brightness of the star increases sharply and then fades off more gradually, as shown in the light curve.

Determining distance using data on Cepheid variables

1) Measure the period (e.g. from light curve). 2) Find corresponding luminosity from a luminosity-period graph. 3) Apparent brightness should be given (or calculated from given information). 4) Calculate distance using the formula for apparent brightness. 5) The Cepheid method can be used to find distances up to a few megaparsecs.

HR region: Main sequence

1) Most stars fall on a strip extending diagonally across the diagram from top left to bottom right. This is called the main sequence. 2) The higher the luminosity of a main-sequence star, the higher its mass. So as we move along the main sequence towards hotter stars, the masses of the stars increase. Thus, the right end of the main sequence is occupied by red dwarfs and the left by blue giants. 3) The common characteristic of all main-sequence stars is the fusion of hydrogen into helium. 4) Main-sequence stars produce enough energy in their core, from the nuclear fusion of hydrogen into helium, to exactly counterbalance the tendency of the star to collapse under its own weight.

HR region: Instability strip (variable stars)

1) Region to the left of the red giants. 2) Cepheids occupy a strip between the main sequence and the red giants on an HR diagram.

Stellar spectra

1) Sets of wavelengths that can be emitted by stars. 2) The energy radiated by a star is in the form of electromagnetic radiation and is distributed over an infinite range of wavelengths. 3) A star is assumed to radiate like a black body. 4) Surface temperature and chemical composition can be determined from the spectrum.

HR region: Red giants

1) Some large stars, reddish in colour, occupy the top right. These are the red giants (large and cool). 2) Bright, large, cool, reddish, tenuous. 3) The luminosity of red giants is considerably greater than that of main-sequence stars of the same temperature. 4) Treating them as black bodies radiating according to the Stefan-Boltzmann law means that luminosity which is 10(3) times greater than that of the Sun corresponds to a surface area which is 10(3) times that of the Sun, and thus a radius about 30 times greater. 5) The mass of a red giant can be as much as 100 times the mass of the Sun, but their huge size also implies small densities. 6) A red giant will have a central hot core surrounded by an enormous envelope of extremely tenuous gas.

HR region: White dwarfs

1) The bottom left is a region of small stars known as white dwarfs (small and hot). 2) Dim, small, hot, whitish, dense.

Mass-luminosity relation

1) The luminosity of main sequence stars is proportional to a power of their mass. 2) Main-sequence stars in the upper left-hand corner of the HR diagram have a very high luminosity and therefore are very massive. 3) The mass-luminosity relation can only be used for main-sequence stars.

Reason for the variation of Cepheid variables

1) The reason for a Cepheid star's periodic variation in luminosity is the periodic expansion and contraction of the outer layers of the star. 2) As radiation rushes outwards, it ionizes helium atoms in the atmosphere of the star. The freed electrons, through collisions, heat up the star's atmosphere. This increases the pressure, which forces the outer layers of the star to expand. 3) When most of the helium is ionised, radiation now manages to leave the star, and the star cools down and begins to contract under its own weight. 4) This makes helium nuclei recombine with electrons, and so the cycle repeats as helium can again be ionised. 5) The star is brightest when the surface is expanding outwards at maximum speed.

How the chemical composition of a star may be determined from the star's spectrum

1) We obtain an absorption spectrum in which dark lines are superimposed on a background of continuous colour. Each dark line represents the absorption of light of a specific wavelength by a specific chemical element in the star's atmosphere. 2) Most stars have essentially the same chemical composition, yet show different absorption spectra. The reason for this difference is that different stars have different temperatures (hence electrons in different energy states).

Wien's displacement law

1) Wien's displacement law relates the peak wavelength to the surface temperature. 2) It implies that the higher the temperature, the lower the wavelength at which most of the energy is radiated.

Main-sequence star

As a star is compressed more and more (under the action of gravity), its temperature rises and so does its pressure. Eventually, nuclear fusion reactions commence, resulting in the release of enormous amounts of energy. The energy released can account for the sustained luminosity of stars. Thus, nuclear fusion provides the energy that is needed to keep the star hot, so that its pressure is high enough to oppose further contraction, and at the same time to provide the energy that the star is radiating into space. On the main sequence, the main nuclear fusion reactions are those of the proton-proton cycle, in which the net effect is to turn four hydrogen nuclei into one helium-4 nucleus.

Nuclear reactions in low-mass stars after the main sequence

Helium collects in the core of the star, surrounded by a thin shell of hydrogen and a bigger hydrogen envelope. Only hydrogen in the thin inner shell undergoes nuclear fusion to helium. The temperature and pressure of the helium build up and eventually helium itself begins to fuse (helium flash), with helium in a thin inner shell producing carbon in the core. In the core, some carbon nuclei fuse with helium to form oxygen. Oxygen is the heaviest elements that can be produced in low-mass stars; the temperature never rises enough for production of heavier elements. The hydrogen in the thin shell is still fusing, so the star now has nuclear fusion in two shells, the H and He shells.

Fate of stars depending on mass

The mass of a star is the main factor that determines its evolution off the main sequence. 1) If the mass of the core of a star is less than the Chandrasekhar limit of about 1.4 solar masses, it will become a stable white dwarf, in which electron degeneracy pressure keeps the star from collapsing further. 2) If the core is more massive that the Chandrasekhar limit but less than the Oppenheimer-Volkoff limit of about 2-3 solar masses, the core will collapse further until electrons are driven into protons, forming neutrons. Neutron degeneracy pressure now keeps the star from collapsing further, and the star becomes a neutron star. 3) If the Oppenheimer-Volkoff limit is exceeded, the star will become a black hole.

Chandrasekhar limit

The maximum mass of a white dwarf star, about 1.4 solar masses.

Nuclear reactions in a massive star after the main sequence

The process begins much the same way as for low-mass stars, but differences begin to show when carbon fuses with helium in the core to form oxygen. If the mass of the star is large enough, the pressure caused by gravity is enough to raise the temperature sufficiently to allow the formation of ever-heavier elements: neon, more oxygen, magnesium and then silicon; eventually iron is produced in the most massive stars, and that is where the process stops, since iron is near the peak of the binding-energy curve. It would require additional energy to be supplied for iron to fuse.

Nuclear reactions leading to a supernova

The star is very hot in the core. Photons have enough energy at these temperatures to split nuclei; in about one second millions of years worth of fusion is undone. Nuclei are in turn ripped apart into individual protons, electrons, neutrons and photons. Because of the high densities involved, the electrons are forced into the protons, turning them into neutrons and producing neutrinos that escape from the star. The star's core is now made up almost entirely of neutrons, and is still contracting rapidly. The Pauli Exclusion Principle may now be applied to the neutrons: if they get too close to one another, a pressure develops to prevent them from getting any closer. But they have already done so, and so the entire core now rebounds to a larger equilibrium size. This rebound is catastrophic for the star, creating an enormous shock wave trailing outwards that tears the outer layers of the star apart. The resulting explosion, called a supernova, is much more violent that a planetary nebula. The energy loss from this explosion leads to a drastic drop in the temperature of the star, and it begins to collapse.

Cepheid variables as standard candles

There is a precise relationship between the average luminosity of Cepheids and their period. The longer the period, the larger the luminosity. This makes Cepheid stars standard candles - i.e. stars of a known luminosity, obtained by measuring their period.