Professor Ian Morison
What is a star?
If a large mass of gas is formed in the space between the stars, gravity will cause it to take on a spherical shape as it reduces in size. As the gas in the centre is compressed it heats up until the pressure produced is sufficient to halt the contraction. If the central temperature has then reached in excess of ~10 million degrees, the protons at it heart will be moving sufficiently fast (and hence have sufficient kinetic energy) so that a pair moving in opposite directions will be able to overcome their mutual repulsion and so get sufficiently close to allow the first stage of nuclear fusion to commence. The fusion process, which converts protons (hydrogen nuclei) to alpha particles (helium nuclei), generates energy which eventually reaches the surface and is radiated away as heat and light - the ball of gas has become a star.
This lecture will look at how stars in different mass ranges evolve during the later stages of their life, and describe the remnants left when they die: white dwarfs, neutron stars and black holes.
Low-mass stars: 0.05 to 0.5 solar masses
For a collapsing mass of gas to become a star, nuclear fusion has to initiate in its core. This requires a temperature of ~ 10 million K and this can only be reached when the contracting mass is greater than about 1029 kg, about one twentieth the mass of the Sun, or twenty times that of Jupiter.
In low-mass stars the conversion of hydrogen to helium by nuclear fusion is the same as in our Sun. However, whereas in stars of greater mass nuclear fusion only converts about 10% of the mass of the star (that residing in its core), in the lowest mass stars it is thought that convection currents mix the star's interior and so will allow much of the stars mass to undergo nuclear fusion so increasing the time during which they can carry out the fusion of hydrogen to helium - a period which is significantly longer than the present age of the universe. We thus have no direct observational evidence of what happens when fusion ceases in such stars and can only use computer modelling to investigate what might happen.
We will see that in order for helium to fuse into heavier elements, temperatures of order 100 million K are required and this requires sufficient mass in the star's envelope to provide the required pressure to enable such temperatures to be reached. In stars of mass less than 0.5 solar masses there is simply not enough pressure to give the temperatures that would allow helium fusion to begin. So when nuclear fusion, converting hydrogen to helium, finally ceases - and modelling of a 0.1 solar mass red dwarf suggests that this might be after six trillion years - the star will slowly collapse over a period of several hundred billion years to form what is called a white dwarf. Over many trillions of years, the white dwarf will cool until its surface temperature is below that at which significant light is emitted and the inert remnant will become ablack dwarf. No white dwarfs derived from low-mass stars yet exist, but they will be discussed in detail in the following section on mid-mass stars as their evolution also produces white dwarfs which can now be observed.
Mid-mass stars: 0.5 to ~8 solar masses
All stars in this range will have a common end state in the form of a white dwarf. There is, however, a difference in the process of nuclear fusion from hydrogen to helium in stars above and below ~2 solar masses.
For stars whose mass is less than ~2 solar mass stars, like our Sun, the bulk of its energy is produced by what is called the proton-proton cycle. However there is a more complex process called the Carbon-Nitrogen-Oxygen (CNO) cycle that provides 1 to 2% of the Sun's total energy output. In stars greater than about 2 solar masses this process, proposed independently by Carl von Weizsäcker and Hans Bethe in 1938 and 1939 respectively, becomes dominant. It provides a very efficient way of converting hydrogen to helium so the hydrogen in more massive stars burns more quickly so increasing the energy output of the core. As the greater energy output of the star must be balanced by radiation from its surface, the star becomes bluer and has a greater luminosity.
The reactions of the CNO cycle are:
12C + 1H → 13N + y
13N → 13C + e+ + ve
13C + 1H → 14N + y
14N + 1H → 15O + y
15O → 15N + e+ + ve
15N + 1H → 12C + 4He
The net result of the cycle is to fuse four protons into an alpha particle along with two positrons, two electron neutrinos (which carry some energy away from the star) and three gamma rays. The carbon acts as a catalyst and is regenerated. The two positrons annihilate electrons releasing energy in the form of gamma rays - each annihilation giving rise to two gamma rays.
Figure 1: The Carbon-Nitrogen-Oxygen Cycle.
When the CNO process reaches an equilibrium state, the reactions of each stage will proceed at the same rate. The slowest reaction within the cycle is that which converts 14N into 15O so, in order for this reaction to have an equal reaction rate, the number of nitrogen nuclei must be significantly larger that those of carbon or oxygen. Thus, over time, the relative amount of nitrogen increases until equilibrium is established. Whilst the total number of the carbon, nitrogen and oxygen nuclei is conserved, nitrogen becomes the most numerous nucleus, regardless of the initial composition.
This process produces essentially all of the nitrogen in the universe and thus has great significance for us as nitrogen is an essential element of all life-forms here on Earth.
An (perhaps surprising) increase in energy output as a star ages
As the proton-proton or CNO cycles convert hydrogen to helium in the core of the star, its mean molecular weight increases. The pressure of an ideal gas increases with the temperature of the gas, but decreases as the mean molecular weight of the gas rises. Assuming (correctly) that the matter in the core acts like an ideal gas, in order for the pressure to remain sufficient to support the overlying layers of the star as the amount of helium increases, the core's temperature must rise. The rate of, for example, the proton-proton chain is a function of the fourth power of the temperature, so that even though the percentage of hydrogen is reducing, the conversion rate of hydrogen to helium gradually increases over time. This, of course, increases the energy output of the star, so its luminosity will increase and, in order to radiate more energy, so will its surface temperature.
It is believed that, when formed, our Sun was ~30% less bright than at the present time and that over the next billion years its luminosity will increase by a further 10%. An obvious consequence is that our Earth will eventually become too hot for life (as we know it) to exist on its surface.
The Triple Alpha Process
Eventually, either by the proton-proton chain or the CNO cycle, the core of the star will be converted into 4He. At this point nuclear fusion stops so that the pressure in the core that prevents gravitational collapse drops. The core thus reduces in size but, as it does so, its temperature will rise. Finally when it reaches ~ 100 million K, a new reaction occurs - the triple alpha process (3), so called because it involves three helium nuclei which are also known as alpha particles. This is an extremely subtle process. The first obvious nuclear reaction that would happen in a core composed of helium is that two 4He nuclei fuse to form 8Be. But 8Be is very unstable - it has a lifetime of only 10-19 seconds - and virtually instantly decays into two 4He nuclei again. Only when the core temperature has increased to 100 million K, does it become likely that a further 4He nucleus can fuse with 8Be before it decays. The result is a12C nucleus. It is highly significant to our existence here on Earth that there is such a difference in temperature between that (~ 15 million K) at which the hydrogen fuses to helium and that (~ 100 million K) at which12C can be formed. If this were not the case, and the process could happen at the core temperatures close to that at which the proton-proton or CNO cycles operate, there would be no long period of stability whilst the star remains on the main sequence with a relatively constant luminosity. This, of course, has allowed stable temperatures to exist on Earth for billions of years and so enabled intelligent life to evolve.
Figure 2: The Triple Alpha Process.
But there is a further real problem in attempting to form 12C. A temperature of 100 million K is required to give the 4He nuclei a reasonable chance to fuse with a 8Be nucleus before it has a chance to decay. The 4He nuclei are thus moving very fast and so have appreciable kinetic energy. It would be expected that this energy would prevent a stable 12C nucleus arising as it would be sufficient to split the newly formed nucleus apart. (If a white billiard ball (4He) approached a red ball (8Be) very slowly they might just "kiss" and remain touching, but if it came in at high speed the energy of impact would split them apart).
So why is 12C so common? This problem was pursued with great vigour by the British astrophysicist, Fred Hoyle, in the early 1950's. As he then stated: "Since we are surrounded by carbon in the natural world and we ourselves are carbon-based life, the stars must have discovered a highly effective way of making it, and I am going to look for it".
He realised that the excess energy that was present in the reaction (and thus expected to break up the newly formed 12C nucleus) could be contained if there happened to be an excited state (called a "resonance" by particle physicists) of the carbon nucleus at just the right energy above its ground state. This is because, due to the quantum nature of matter, though atomic nuclei usually exist in their ground state, it is possible for them to absorb energy (such as an interaction with a gamma ray photon) and jump into an excited state. This will later decay back to the ground state with the emission of a gamma ray of the same energy. This is analogous to an atom absorbing a photon of energy which lifts an electron to a higher energy level. The electron will then, in one or more steps, drop back down the energy levels emitting photons as it does so.
Hoyle realised that a stable carbon nucleus could only result if it had an excited state that was very close in energy to that of the excess energy of the three 4He that came together in its formation. This would thus lift the resulting 12C nucleus into an excited state from which it could drop back to the ground state by the emission of a gamma ray photon and so reach a stable state.
Some experiments in the late 1940's had suggested that such an excited state might exist, but Hoyle had been told that these were in error. Hoyle argued that there must be an appropriate excited state otherwise we could not exist and pestered the particle physicists at the California Institute of Technology (Caltech), led by William Fowler, to repeat the experiments. Fowler did so (it is said, only so that Hoyle would go away) and found that there was indeed an excited state within 5% of the energy predicted by Hoyle! Hoyle was essentially using the "anthropic principle" - which says that our existence as observers puts constraints on the universe in which we live. William Fowler received the Nobel Prize in part for this work. Many believe that Hoyle should also have won the Nobel Prize for this incisive observation and his following work in showing how the elements are synthesized in stars.
For a given mass of gas, the 3 process only releases about 10% of the energy produced in forming helium nuclei from hydrogen, so the length of the helium burning phase will be about 10% of the star's life on the main sequence.
During its helium burning phase the core will be compressed to perhaps 1/50th of its original size and have a temperature of ~100 million K with, in addition, a shell of hydrogen burning surrounding the core. The energy so produced causes the outer parts of the star to also undergo significant changes. The radius of the star as a whole increases by a factor of ~ 10, but at the same time the surface cools to (in the case of a 1 solar mass star) a temperature of ~3,500 K. The star will then have an orange colour and the star becomes what is called (perhaps perversely) a "red giant".
For mid-mass stars less in mass than our Sun, this is about as far as nuclear fusion can take the formation of elements as there is not enough overlying mass above the core to allow its temperature to rise sufficiently for further nuclear fusion reactions to be carried out.
The stars in the upper part of this mass range are able to carry out one further nuclear reaction:
12C + 4He → 16O + y
This reaction and the 3 process are thought to be the main source of carbon and oxygen in the Universe today. But we could not exist if these elements stayed within the star - they must lose much of their material into space - this is the anthropic principle again - and this is exactly what is observed.
In the latter stages of their life, star become less stable and may well oscillate in size. As the stars size increases, its surface area will also increase, tending to increase its luminosity but, at the same time, the surface temperature will reduce so reddening its colour. As the emitted energy per unit area decreases as the fourth power of the temperature, the star's luminosity actually falls as the size increases. Conversely as the size of the star reduces, the colour will shift towards the blue and the luminosity will increase. The periodic changes in colour and luminosity result in what is called a "variable star". During this phase of their life the stars often have intense solar winds and so lose much of their outer envelopes into space.
Finally it appears that the star becomes so unstable that the outer parts of the star are blown off to form what is called a planetary nebulasurrounding the core remnant. Planetary nebulae are some of the most beautiful objects that we observe in the universe and many, such as the Ring and Dumbell Nebulas may be observed with a small telescope. Planetary nebulae are relatively common with over 1500 known, but it is expected that many more, perhaps over 50,000, will exist in the galaxy but are hidden by the dust lanes in our galaxy. The name "planetary nebula" is, of course, a misnomer as they have nothing to do with planets, but many do have a disk-like appearance. They are large tenuous shells of gas which are expanding outwards at velocities of a few tens of kilometres per second. They also contain some dust and have masses of typically one tenth to a fifth of a solar mass. Of order 10 planetary nebula are thought to be formed each year so the interstellar medium is being enriched by around a solar mass per year.
Figure 3: Planetary Nebula. In all cases, the stellar remnant can be seen at their centre
At the centre of a planetary nebula lies a white or blue-white star. They are not very bright so that relatively large telescopes are required to see them visually. (The author has once, using a 16 inch telescope under perfect conditions, observed the star at the centre of the Ring Nebula.) This star is approaching the final stage of its life when it will become a "white dwarf". Once nuclear reactions have ceased, what is left at the centre of the star will contract under gravity. It is composed mainly carbon and oxygen, and devoid of its outer layers through a combination of the intense stellar winds and the ejection of a planetary nebula. The fact that contraction finally ceases is due to a quantum-mechanical effect known as degeneracy pressure. In 1926, R.H. Fowler realised that, as a result of the Pauli exclusion principle, no more than two electrons (of opposite spin) could occupy a given energy state. As the allowed energy levels fill up, the electrons begin to provide a pressure - the electron degeneracy pressure - which finally halts the contraction. This pressure only depends on density, not temperature, and this has the interesting result that the greater the mass of the white dwarf, the less its radius!
A further consequence of being supported by electron degeneracy pressure is that there is a limiting mass which cannot be exceeded. This depends on the composition of the star; for a mix of carbon and oxygen, it turns out to be ~ 1.4 solar masses. This result was published in 1931 by Subramanyan Chandrasekhar when he was only 19! In 1983, Chandrasekhar rightly received the Nobel Prize for this and other work. We will see later what happens when the mass of the collapsing stellar remnant exceeds the Chandrasekhar Limit.
White dwarfs range in size from 0.008 up to 0.02 times the radius of the Sun. The largest (and thus least massive) being comparable to the size of our Earth whose radius is 0.009 times that of the Sun. The masses of observed white dwarfs lie in the range 0.17 up to 1.33 solar masses so it is thus obvious that they must have a very high density. As a mass comparable to our Sun is packed into a volume one million times less, its density must be of order one million time greater - about 1 million grams per cubic centimetre. (A ton of white dwarf material could fit into a matchbox!)
The Discovery of White Dwarfs
The first known white dwarf was discovered by William Herschel in 1783; it was part of the triple star system, 40 Eridani. What appeared surprising was that although its colour was white (which is normally indicative of bright stars) it had a very low luminosity. This is of course due to its small size so, although each square metre is highly luminous, there are far fewer square metres!
The second white dwarf to be discovered is called Sirius B, the companion to Sirius, the brightest star in the northern hemisphere. Friedrich Bessel made very accurate measurements of the position of Sirius as its proper motion carried it across the sky. The motion was not linear and Bessel was able to deduce that Sirius had a companion. Their combined centre of mass would have a straight path across the sky but both Sirius and its companion would orbit the centre of mass thus giving Sirius its wiggly path. Due to the close proximity with Sirius, Sirius B is exceedingly difficult to observe as it is usually obscured by light scattered from Sirius within the telescope optics. A very clean refractor has the least light scatter, and it was when Alvin Clark was testing a new 18-inch refracting telescope in 1862 that Sirius B was first observed visually.
The future of White Dwarfs
The observed surface temperatures of white dwarf stars range from 4000K up to 150,000K so they can range from orange to blue-white in colour. Their radiation can only come from stored heat unless matter is accreting onto it from a companion star. As their surface area is so small it takes a very long time for them to cool; the surface temperature reduces, the colour reddens and their luminosity decreases. The less the surface temperature the less the rate of energy loss, so a white dwarf will take a similar time to cool from 20,000K down to 5,000K as it will from 5,000K to 4,000K. In fact, the universe is not old enough for any white dwarfs to have cooled much below 4000K; the coolest observed so far, WD 0346+246, has a surface temperature of 3,900K.
Eventually the white dwarf will cool sufficiently so that there is no visible radiation and will then become a black dwarf. They could still, however, be detected in the infra-red, though will be very faint, and the presence of those in orbit around normal stars could still be deduced by the effect they have on the motion of a companion star.
High Mass Stars in the range >8 Solar Masses
Stars in this mass range have sufficient mass overlying the core so that the temperature of the core can increase beyond that in less massive stars. This allows the capture of alpha particles to proceed further. Having made carbon and oxygen, it is then possible to build up the heavier element having atomic numbers increasing by 4 - produced by the absorption of alpha particles. So that, in turn, the 16O fuses to 20Ne, the 20Ne fuses to 24Mg and then 24Mg fuses to 28Si producing a core dominate by silicon.
For each successive reaction to take place the temperature has to increase as there is a greater potential barrier for the incoming alpha particle to tunnel through. In the melee, protons can react with these elements to form nuclei of other atoms with intermediate atomic numbers such as 19Fl and 23Na, though these will be less common. A shell like structure results, with layers of the star containing differing elements, the heaviest nearest to the centre.
When the temperatures reach the order of 3 x 109 K, silicon can be transformed though a series of reactions passing through 32S, 36A and continuing up to 56N. The silicon burning produces a core composed mostly of iron (the majority) and nickel. Iron and its close neighbours in the atomic table have the most stable nuclei, and any further reactions to build up heavier nuclei are endothermic (they would absorb energy rather than provide it) so this is where nuclear fusion has to stop. The star is then said to have an iron core. This core is surrounded by shells in which the lighter elements are still burning giving an interior like that shown in figure 4.
Figure 4: The "onion like" shells of fusion burning during the later stages in the evolution of a giant star.
The energy released by each stage of burning is reduced and, as a result, the time spent carrying our each successive reaction becomes shorter: a star of mass 20 times that of our Sun will spend about 10 million years on the main sequence burning hydrogen to helium, then spend about 1 million years burning helium to carbon and about 300 years burning carbon to oxygen. The oxygen burning takes around 200 days and silicon burning is completed in just 2 days!
Once the core reaches its iron state, things progress very rapidly. At the temperatures that exist in the core (of order 8 x 109 K for a 15 solar mass star) the photons have sufficient energy to break up the heavy nuclei, a process known as photodisintegration. An iron nucleus may produce 13 helium nuclei in the reaction:
56Fe + y → 13 4He + 4n
These helium nuclei then break up to give protons and neutrons:
4He + y → 2 p+ + 2n
As energy is released when the heavy elements were produced, these inverse processes are highly endothermic (requiring energy to progress) and thus the temperature drops catastrophically. There is then not sufficient pressure to support the core of the star which begins to collapse to form what is called a neutron star.
In the forming neutron star, free electrons combine with the protons produced by the photodisintegration of helium to give neutrons, in the reaction:
p+ + e- → n + e
The electron neutrinos barely interact with the stellar material, so can immediately leave the star carrying away vast amounts of energy - the neutrino luminosity of a 20 solar mass star exceeds it photon luminosity by 7 orders of magnitude for a brief period of time! The outer parts of the core collapses at speeds up to 70,000 km per sec and, within about a second, the core, whose initial size was similar to the Earth, is compressed to a radius of about 40 km! This is so fast that the outer parts of the star, including the oxygen, carbon, and helium burning shells, are essentially left suspended in space and begin to infall towards the core.
The core collapse continues until the density of the inner core reaches about three times that of an atomic nucleus, ~ 8 x 1014 grams per cubic centimetre. At this density, the strong nuclear force, which in nuclei is attractive, becomes repulsive - an effect caused by the operation of the Pauli Exclusion Principle to neutrons and termed neutron degeneracy pressure. As a result of this pressure, the core rebounds and a shock wave is propagated outwards into the infalling outer core of the star. As the material above is now so dense, not all the neutrinos escape immediately and give the shock front further energy which then continues to work its way out to the surface of the star - there producing a peak luminosity of roughly 109 times that of our Sun. This is comparable to the total luminosity of the galaxy in which the star resides!
Type II supernova
This sequence of events is called a Type II supernova. The peak absolute magnitude of about -18 then drops by around six to eight magnitudes per year so that it gradually fades from view. We believe that such supernovae will occur in our galaxy on average about once every 44 years. Sadly, the dust in the plane of the galaxy only allows us to see about 10 to 20% of these and so they are not often seen.
The Crab Nebula
On July 4th, 1054 AD a court astrologer during the Sung dynasty, Yang Wei-T'e, observed a supernovae in the constellation Taurus. The gas shell thrown out in the supernova explosion was first discovered in modern times, by John Bevis in 1731 who included it in his sky atlas, Uranographia Britannica. Later, in 1758, it was independently discovered by Charles Messier whilst he was searching for the return of Halley's Comet. It became the first object in the Messier catalogue with the name M1. The Third Earl of Rosse, who drew its form using his 72 inch telescope in Ireland, thought that it appeared similar to a horseshoe crab and so he called it the "Crab Nebula", the name by which it is usually known.
Figure 5: The Crab Nebula.
The Crab Nebula is still, nearly 1000 years after it was first observed, expanding at rate of 1500 km per second and its luminosity is about 10,000 times brighter than our Sun. Much of this radiation appears to be the result of electrons, moving close to the speed of light (called relativistic electrons), spiralling around magnetic field lines in the nebula. The fact that the nebula still appears so energetic remained a puzzle until a neutron star (which is the remnant of the stellar core) was discovered in 1969 at the centre of the nebula. This will be described in detail below. The gas shell, now of order 6x4 arc minutes in size and shining at 8.4 magnitude, can still be observed with a small telescope.
The Crab Nebula is thought to be the remains of a Type II supernova. A supernova, (1987A) that was observed in the nearby galaxy, the Large Magellanic Cloud, in 1987 is also thought to have been a Type II supernova, but those observed by Tycho Brahe in 1572 and Johannes Kepler in 1604 are thought to have been caused by a different mechanism and are termed Type I Supernovae.
In Febuary 1987, a supernova was observed in the Large Magellanic Cloud, a galaxy close to our own Milky Way galaxy. Visible for a while to the unaided eye, it became the closest observable supernova since that of 1604.
Figure 6: Supernova 1987A in the Large Magellanic Cloud.
There is an aspect of its explosion that merits mention which was the result of a wonderful piece of serendipitous timing. In the late 1970's a particle physics model called the Grand Unified Theory (GUT) suggested that protons would decay with a half life of 1031 years. This means that if one observed a number of protons for 1031 years half would have decayed. This is obviously not an experiment that can be mounted, but the possible proton decay could be detected if one observed a very large number of protons for a relatively short period. The proposed decay process is:
The proton decays into a positron and neutral pion which then immediately decays into two gamma rays. The positron will annihilate with an electron to form two more gamma rays.
To this end, a number of detectors were built in the 1980's including that at the Kamioka Underground Observatory located 1,000 metres below ground in Japan. To provide the protons, 3,000 tons of pure water was contained in a cylinder 16 m tall and 15.6 m in diameter. The cylinder was surrounded by 1000 photo multiplier tubes attached to its inner surface which would be able to detect the gamma rays produced in the proton decay. It came into operation in 1983 and was given greater sensitivity in 1985. To date, even with a new detector containing 50,000 tons of water, no convincing proton decays have been detected and later versions of GUT suggest that the decay half life might be nearer to 1035 years. But what is critically important was that the detector, which came into full operation at the end of 1986 after its upgrade in 1985, could also detect neutrinos.
Relativity states that no particle can travel at the speed of light in a vacuum. However, in a dense media, like water, light travels at lower speeds. It is thus possible for a particle to travel through water faster than the speed of light. If the particle is charged, it will emit light radiation called Cherenkov radiation. The process is analagous to the formation of a sonic boom when an airplane exceeds the speed of sound. Neutrino interactions with the electrons in the water can transfer almost all the neutrino momentum to an electron which then moves at relativistic speeds in the same direction.
The relativistic electron produces Cherenkov radiation which can be detected by the photo multiplier tubes around the tank. The expanding light cone will trigger a ring of photomultiplier tubes whose position gives an indication of the direction from which the neutrino has travelled. This makes it more than just a detector - it forms a very crude telescope!
When SN1987A was seen to explode just a few months later (this being the serendipitous timing referred to earlier) the Kamiokande experiment detected 11 neutrinos within the space of 15 seconds. A similar facility in Ohio detected a further 8 neutrinos within just 6 seconds and a detector in Russia recorded a burst of 5 neutrinos within 5 seconds. These 24 neutrinos are the only ones ever to have been detected from a supernova explosion. Perhaps surprisingly, the neutrinos were detected some three hours before the supernova was detected optically. This is not because they had travelled faster than light! They had, of course, travelled out directly from the collapsing core of the star, whereas the visible light was not emitted until later when the shock wave reached the surface of the star. The detection of those 24 neutrinos was a perfect confirmation of the theoretical models that had been developed for the core collapse of a massive star and consistent with theoretical prediction that ~1058neutrinos would be produced in such an event. The Kamiokande observations also allowed an upper limit to be placed on the neutrino mass. If one assumes that the neutrinos began their trip somewhat ahead of the light from the supernova and given the fact that they arrived before the light having travelled through space for~ 169,000 years means that they must have been travelling very close (within one part in 108) to the speed of light. This, together with the fact that the higher and lower energy neutrinos arrived at the same time allows an upper limit to be put on the mass of a neutrino. It cannot be greater than about 3 millionths the mass of an electron.
Neutron Stars and Black Holes
What remains from this cataclysmic stellar explosion depends on the mass of the collapsing core. When stars, whose total mass is greater than ~ 8 solar masses, but less than ~ 12 solar masses, collapse the result is a Neutron Star - the core being supported by neutron degeneracy pressure as described above. The typical mass of such a neutron star would be ~ 1.4 solar masses so that it is, in effect, a giant nucleus containing ~ 1057 neutrons. It will have a radius between 10 to 15 km - the theoretical models are not all that precise. Assuming a radius of 10 km, the average density would be 6.65 x 1014 grams per cubic centimetre - more than that of an atomic nucleus!
Gravity at the surface would be intense; for a 1.4 solar mass star with a radius of 10km, the acceleration due to gravity at the surface would be 190 billion times that on the surface of the Earth and the speed of an object having falling from a height of 1 metre onto the surface would be 6.88 million kilometres per hour! A simple Newtonian calculation of the escape velocity from the surface gives a value of 0.643c. This implies that both special and general relativity need to be invoked when considering neutron stars. The structure of a neutron star is very complex; part may even be in the form of a superfluid sea of neutrons which will thus have no viscosity. This can give rise to an observable consequence as will be described later.
A neutron star may have an outer crust of heavy nuclei, the majority being of iron and nickel. Within this is an inner crust containing elements such as Krypton, superfluid neutrons and relativistic degenerate electrons. The inner crust overlays an interior of superfluid neutrons intermixed with superconducting protons and relativistic degenerate electrons. Finally there may be a core of pions or other elementary particles.
Figure 7: Cross Section of a Neutron Star:
Outer (solid) Crust - Nuclei of Iron and Nickel.
Inner Crust - Nuclei, superfluid electrons and electrons.
Outer Core - Superfluid neutrons, superconducting protons and electrons.
Inner Core - Condensed pions, kaons and quark matter ???
Like white dwarfs, neutron stars become smaller and denser with increasing mass, but there will become a point when the neutron degeneracy pressure can no longer support the mass of the star. So, in an analogous manner to the Chandrasekhar Limit for the maximum mass of a white dwarf, there is a limit, believed to be about 3 solar masses, beyond which the collapse continues to form a black hole as will be discussed later.
Stars rotate as, for example, our Sun which rotates once every ~ 25 days at its equator. The core of a star will thus have angular momentum. As the core collapses, much of this must be conserved (some is transferred to the surrounding material), so the neutron star that results will be spinning rapidly with rotational periods of perhaps a few milliseconds. The neutron star will also be expected to have a very intense magnetic field. This rotating field has observational consequences that have allowed us to discover neutron stars and investigate their properties.
When the neutron star is first born its surface temperature may approach 1011 K but rapidly falls to about 109 K. Neutrinos carry away much of the star's energy for about 1000 years whilst the surface temperature falls to a few million K. Photons - in the form of X-rays - then carry energy away from the surface which stays close to one million K for the next thousand years. Its luminosity will then be comparable to that of our Sun.
This explains why the Crab Nebula could be still visible. It was known that a star close to the centre of the nebula had a very strange spectrum. If this were the neutron star associated with the supernova explosion, its energy output would have kept the gas thrown out into the interstellar medium excited, so remaining visible. The way in which this was confirmed and how, to date, nearly 2000 neutron stars have been discovered is one of the most interesting stories of modern astronomy.
The Discovery of Pulsars
When stars are observed through the Earth's atmosphere they are seen to scintillate ("twinkle" is a rather nice if not scientific term that is often used). This is because irregularities in the atmosphere passing between the observer and the star act like alternate convex and concave lenses which sequentially converge the light from the star (so making it appear slightly brighter) and then diverge it (so reducing its brightness).
There is a similar effect related to radio sources caused by irregularities in the solar wind - bubbles of gas which stream out form the Sun expanding as they do so. It was realised that this could give a way of investigating the angular sizes of radio sources by studying the amount of scintillation observed when the source was at different angular distances from the Sun. It would also be a way of discovering radio sources with very small angular sizes - known as quasars. To carry out this experiment a very large antenna was required and Tony Hewish at the Mullard Radio Astronomy Laboratories at Cambridge recruited a Ph.D student called Jocelyn Bell to first help build the antenna - which was made up of an array of 2048 dipoles - and then carry out and analyse the observations. The array observed radio sources as they passed due south so a given radio source would be observed every sidereal day as it appeared on the meridian.
The signals from radio sources appeared on a roll of chart, about 400ft of which was produced each day. Soon Bell was able to distinguish between the scintillating signal of a radio source and interference, often from cars passing the observatory. In July 1967 she observed a "little bit of scruff" that did not look like a scintillating radio source but did not appear like interference either. A second intriguing feature was that it had been observed at night when a radio source would be seen away from the direction of the Sun and scintillation would not be expected to be seen. Looking through the charts, she discovered that a similar signal had been seen earlier from the same location in the sky. She observed that it reappeared again at always a precise number of sidereal days later which implied that the radio source, whatever it was, was amongst the stars rather that within the solar system. Hewish and Bell then equipped the receiver with a high speed chart recorder to observe the "scruff" in more detail and discovered to their amazement that it was not random, but a series of precisely spaced radio pulses having a period of 1.33724 seconds.
Figure: 8 Jocelyn Bell with the Cambridge array which discovered the first pulsar and the discovery record
Observations using a different telescope at Cambridge confirmed the presence of the signal and also that fact that the pulse arrived at slightly different times as the frequency of observation was changed. This effect is called dispersion, and is exactly similar to the fact that different wavelengths of light travel at different speeds in glass. The interstellar medium is not a perfect vacuum and so can cause this effect - but it would be only observed if the source of the pulses was far beyond the solar system.
At that time, no one in the radio astronomy group at Cambridge group could conceive of a natural phenomena that could give rise to such highly precise periodic signals - it seemed that no star, not even a white dwarf could pulsate at such a fast rate - and they wondered if it might be a signal from an extraterrestrial civilisation. Bell, who called the source LGM1 (Little Green Men 1), was somewhat annoyed about this as it was disrupting her real observations. When, later, a second source with similar characteristics but a slight faster period of 1.2 seconds was discovered she was somewhat relieved as "it was highly unlikely that two lots of Little Green Men could choose the same unusual frequency and unlikely technique to send a signal to the same inconspicuous planet Earth!"
A few days before the paper presenting these discoveries was published in Nature in February 1968, Hewish announced the discovery to a group of astronomers at Cambridge. Fred Hoyle was amongst them, and suggested that the signal might be pulsed emissions coming from an oscillating neutron star - the theoretical remnant of a supernova but never previously observed. After a press conference following the publication of the Nature paper announcing the discovery, the science correspondent of the Daily Telegraph coined the name Pulsar for these enigmatic objects.
Some three months later, in a paper also published in Nature, Thomas Gold at Cornell University in Ithaca, USA, gave a satisfying explanation for the pulsed signals. Gold suggested that the radio signals were indeed coming from neutron stars, but that the neutron star was not oscillating, but instead spinning rapidly around its axis. He surmised that the rotation, coupled with the expected intense magnetic field generates two steady beams of radio waves along the axis of the magnetic field lines, one beam above the north magnetic pole and one above the south magnetic pole. If (as in the case of the Earth) the magnetic field axis is not aligned with the neutron star's rotation axis, these two beans would sweep around the sky rather like the beam from a lighthouse. If then, by chance, one of the two beams crossed our location in space, our radio telescopes would detect a sequence of regular pulses - just as Bell had observed - whose period was simply the rotation rate of the neutron star.
Gold, in this paper, pointed out that a neutron star (due to the conservation of angular momentum when it was formed) could easily be spinning at such rates. He expected that most pulsars should be spinning even faster than the first two observed by Jocelyn Bell and suggested a maximum rate of around 100 pulses per second.
Figure 9: Twin beams emitted by a Pulsar.
Since then, nearly 2000 pulsars have been discovered. The majority have periods between 0.25 and 2 seconds. It is thought that as the pulsar rotation rate slows the emission mechanism breaks down and the slowest pulsar detected has a period of 4.308 seconds. There is a class of "millisecond" pulsars where the proximity of a companion star has enabled the neutron star to "pull" material from the outer envelope of the adjacent star onto itself. This also transfers angular momentum so spinning the pulsar up to give periods in the millisecond range - hence their name. The fastest known pulsar is spinning at just over 700 times per second - with a point on its equator moving at 20% of the speed of light and close the point where it is thought theoretically that the neutron star would break up!
Pulsars slowly radiate energy, which is derived from their angular momentum. This is so high that the rate of slowdown is exceptionally slow and so pulsars make highly accurate clocks and some may even be able to challenge the accuracy of the best atomic clocks. The periods of all pulsars slowly increase (except when being spun up to form a millisecond pulsar) and a typical pulsar would have a lifetime of a few tens of millions of years.
The linking of pulsars with supernova neutron star remnants was confirmed when the "odd" star close to the centre of the Crab Nebula was shown to be a pulsar with a period of 0.0333 seconds - rotating just over 30 times per second. A second pulsar was discovered within the Vela supernovae remnant and both this and the Crab pulsar also emit beams of radiation not just at radio waves, but across the whole electromagnetic spectrum including visible light, X-rays and gamma rays.
Most pulsars are seen along the plane of the galaxy, just as one would suspect as they are the remnants of stars but, perhaps surprisingly, a significant number are observed away from the plane. The 217km MERLIN array at Jodrell Bank Observatory is capable of making very precise measurements of the position of pulsars and has observed, from positional measurements made over a number of years, that many are moving at speed comparable to, and even exceeding, the escape velocity of the galaxy, which is ~ 500 km/sec. The highest pulsar speed so far measured (in this case by the USA's 5000 km VLBA array) is 1,100 km/sec - London to New York in 5 seconds!
These pulsars have obviously been ejected from the supernova explosion that gave rise to them with very great energies, enabling them to travel around the galaxy and, in some cases, to leave the galaxy into the depths of intergalactic space. It appears that, usually, the supernova explosion will be more intense on one side or the other of the central neutron star which is then ejected at high speeds rather like a bullet from a gun. In some cases it is even possible to track the course of a pulsar back to the gaseous remnant of the supernova. The situation where the resulting pulsar remains within the supernova gas shell, such as in the Crab Nebula, appears to be very rare.
If one projected a ball vertically from the equator of the Earth with increasing speed, there comes a point, when the speed reaches 11.2 km/sec, when the ball would not fall back to Earth but would escape the Earth's gravitational pull. This is the Earth's escape velocity. If either the density of the Earth was greater or its radius smaller (or both) then the escape velocity would increase as Newton's formula for escape velocity shows:
(v0 is the escape velocity, M the mass of the object, r0 its radius and G the universal constant of gravitation.)
If one naively used this formula in realms where relativistic formula would be needed, one could predict the mass and/or size of an object where the escape velocity would exceed the speed of light and thus nothing, not even light, could escape. The object would then be what is termed ablack hole.
Black holes have no specifically defined size or mass, but so far we have only found evidence for black holes in two circumstances. The first, with masses of up to a billion or more times that of our Sun, are found the heart of galaxies. The second are believed to result from the collapse of a stellar core whose mass exceeds ~ 3 solar masses - the point at which neutron degeneracy pressure can no longer prevent gravitational collapse.
The surface surrounding the remnant within which nothing can escape is called the event horizon. In the simplest case when the black hole is not rotating, the event horizon is surface of a sphere and has a radius, called the Schwarzschild radius, given by
RS = 2 GM/c2
The interior of an event horizon is forever hidden from us, but Einstein's theories predict that at the centre of a non-rotating black hole is a singularity, a point of zero volume and infinite density where all of the black hole's mass is located and where space-time is infinitely curved. This author does not like singularities; in his view they are where the laws of physics are inadequate to describe what is actually the case. We know that somehow, Einstein's classical theory of gravity must be combined with quantum theory and so, almost certainly, relativity cannot predict what happens at the heart a black hole.
Nucleons are thought to be composed of up quarks and down quarks. It is possible that at densities greater than those that can be supported by neutron degeneracy pressure, quark matter could occur - a degenerate gas of quarks. Quark-degenerate matter may occur in the cores of neutron stars and may also occur in hypothetical quark stars. Whether quark-degenerate matter can exist in these situations depends on the, poorly known, equations of state of both neutron-degenerate matter and quark-degenerate matter.
Some theoreticians even believe that quarks might themselves be composed of more fundamental particles called preons and if so, preon-degenerate matter might occur at densities greater than that which can be supported by quark-degenerate matter. Could it be that the matter at the heart of a black hole is of one of these forms?
The more massive a black hole, the greater the size of the Schwarzschild radius: a black hole with a mass 10 times greater than another will have a radius ten times as large. A black hole of one solar mass would have a radius of 3 kilometres, so a typical 10-solar-mass stellar black hole would have a radius of 30 kilometres.
The detection of stellar mass black holes
If a stellar black hole, formed when a massive star ends its life in a supernova explosion, existed in isolation, it would be very difficult to detect: gravitational micro-lensing, a method now being employed to detect planets, might just be able to do so. However, many stars exist in binary systems. In a binary system in which one of the components is a black hole, it appears that its gravity can pull matter off the companion star forming an accretion disk of gas swirling into the black hole. As gas spins up as it nears the black hole due to conservation of angular momentum, the differential rotation speeds give rise to friction and the matter in the accretion disk reaches temperatures of more that 1 million K. It thus emits radiation, mostly in the X-ray part of the spectrum.
X-Ray telescope have now detected many such X-ray binary systems, some of which are thought to contain a black hole. Observations of the orbital size and velocity of the normal star in the system enable one to estimate the mass of its companion. If this is both invisible and exceeds a calculated mass of ~ 3 solar masses, then it is likely to be a black hole. An excellent candidate in our own galaxy is Cygnus X-1 - so called because it was the first X-ray source to be discovered in the constellation Cygnus and is the brightest persistent source of high energy X-rays in the sky. Usually called Cyg X-1, it is a binary star system that contains a super-giant star with a surface temperature of 31,000 K (with its spectral type lying on the O and B boundary) together with a compact object. The mass of the super-giant is between 20 to 40 solar masses and observations of its orbital parameters imply a companion of 8.7 solar masses. This is well above the three solar mass limit of a neutron star, so it is thought to be a black hole.
In 1975, with colleagues, the author was able to pinpoint the precise location of a second black hole candidate Monoceros X-1 - an exciting day in his life!
Figure 10: Material accreting on to a black hole from a companion star.
Black holes are not entirely black
In the 1970's, Stephen Hawking showed that due to quantum-mechanical effects, black holes actually emit radiation - they are not entirely black! The energy that produces the radiation in the way described below comes from the mass of the black hole. Consequently, the black hole gradually looses mass and, perhaps surprisingly, the rate of radiation increases as the mass decreases, so the black hole continues to radiate with increasing intensity loosing mass as it does so until it finally evaporates.
The theory describing why this happens is highly complex and results from the quantum mechanical concept of virtual particles - mass and energy can arise spontaneously provided its disappears again very quickly and so does not violate the Heisenberg Uncertainty Principle. In what are called vacuum fluctuations, a particle and an antiparticle can appear out of nowhere, exist for a very short time, and then annihilate each other. Should this happen very close to the event of a black hole, it can sometimes happen that one particle falls across the horizon, while the other escapes. The particle that escapes carries energy away from the black hole and can, in principle, be detected so that it appears as if the black hole was emitting particles.
Black holes can be said to have an effective temperature, and unless this is less than the temperature of the universe the black hole cannot evaporate. This temperature is now ~ 2.7 degrees - the remnant of the radiation left over from the Big Bang. Eventually, in aeons, when the temperature of this relict radiation had fallen sufficiently and assuming Hawking's theory is correct, stellar mass black holes may finally begin to evaporate - on a time scales of 10100 years!
As yet, no black hole - perhaps formed at the time of the big-bang - has been seen to evaporate. A pity say's Stephen Hawking, as otherwise he might have received a Nobel Prize!
©Professor Ian Morison, Gresham College, 3 October 2008