Mr Toogood's Physics

Formation

All stars are formed from large cold clouds of gas and dust called nebulae. These clouds of mainly hydrogen can be huge, hundreds of light years across. In some areas of the nebula, with slightly higher densities, gravity will begin ‘clumping’ the gas together, which in turn attracts more and more matter. As this happens, any slight rotation that the cloud had will accelerate as the mass is concentrated at the centre.

This spinning cloud of gas will flatten out into a disc, which may go on to form planetesimals and create a solar system. The image below shows the disc of dust around several young stars.

As the gas at the centre of this collapsing cloud falls inwards its temperature increases rapidly a forms a protostar. If this protostar has enough mass it will continue to heat up until it reaches a critical temperature around $\quantity{10^{7}}{K}$ at which point thermonuclear reactions will start at the centre of the newly born star. When these reactions huge amounts of energy are released which causes the expansion the hot gas. The photons released from these reactions also produce an outwards pressure. Both of these factors are enough to prevent the further gravitational collapse of the protostar, and a hydrostatic equilibrium is created and the star moves onto the main sequence.

If the protostar does not have enough mass, these reactions will not start and the protostar will turn into a brown dwarf, which will remain hot, due to the contracting gases, but not get any hotter.

This process, from nebula to star typically takes around 1 million years.

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The main sequence

The main sequence in where stars spend 90% of their lifetimes. For the most massive stars, this may only be a few tens of millions of years, but for low mass stars such as the Sun, it may be tens of billions of years. Whilst the star is on the main sequence it is in hydrostatic equilibrium, and is dominated by nuclear fusion processes, the details of which you should also be studying in your A level nuclear module.

The main fusion process in a star is called the proton-proton chain, and you won’t be expected to recall the details of this in an exam. The proton-proton chain, which occurs mainly in smaller stars, brings together protons in three stages to form helium-4. Each stage more and more energy. The total energy released from this process is around $\quantity{25}{MeV}$.

This fusion process will only take place in the super hot core of the sun, which makes up only around 10% of the star itself. Photons released by this process will gradually make their way outwards, but they are constantly being absorbed, and re-emitted by atoms within the star. They have a very short mean free path, this results in photons taking millions of years to make their way through the dense plasma before being allowed to travel quickly through the star's photosphere. On average each re-emission of a photon takes place at a lower energy, so there is a temperature gradient from the core to the outer layers of the star, degrading photons created as high energy gamma photons to those within the visible spectrum. This process also releases ghostly neutrinos, which only react very weakly with matter, so they travel (almost at the speed of light) straight through the star’s dense interior. Therefore all stars also have a high neutrino flux. A solar mass star will continue this fusion process for around 10¹⁰ years.

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After the main sequence - Low mass stars

The fate of the Sun is similar to that of all low mass stars, under 6 solar masses. When the Sun runs out of its hydrogen fuel it will no longer be releasing enough heat to counteract the gravitational force trying to collapse it. As this happens the star’s core collapses and heats up even more, to $\quantity{10^{8}}{K}$, which begins the fusion of helium nuclei. This process brings together three helium nuclei together to make carbon in what is called the triple alpha process. This in turn releases so much heat that the outer layers of the Sun, which are mainly hydrogen and were never hot enough to be involved in fusion, to expand outwards a huge distance. As they do so, they appear redder, and the Sun will have become a red giant.

No one know just how much the Sun will expand when it becomes a red giant, and it could become large enough for its outer layers to reach all the way to Earth’s orbit. The Sun will remain in this red giant phase of its life for around one fifth of its life, or 2 billion years.

When it runs out of helium, it will contract again, but this time it cannot generate enough heat to start the next stage of fusion. It will lose gravitational control of its outer layers which will be shed as what is known as a planetary nebula.

The remaining hot core will form a burnt out, super hot core called a white dwarf. These are made from ionised carbon and oxygen surrounded by a sea of electrons which prevent further gravitational collapse. A white dwarf is around the size of earth, but have huge densities, around $\quantity{10^{9}}{Kg\,m^{-3}}$. White dwarfs are extremely hot, B class objects, and they will very slowly radiate their heat away to eventually become black dwarfs, but this process is thought to take several dozen billion years!

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After the main sequence - High mass stars

In stars with much more mass than the Sun, the fusion process can restart after the end of the red giant phase. When the helium runs out the core collapses, but this time there is enough mass to cause the temperature of the core to get high enough to start fusion between carbon atoms. Just as before the re-ignition of fusion prevents any further collapse. This third round of fusion creates nuclei of magnesium, neon, sodium, and aluminium. The star swells even more and becomes a red supergiant. This is followed by further collapse, re-ignition and fusion for a fourth round which creates silicon, sulfur, and argon. The star goes through several stages of collapse, heating, expansion, and fusion with each round of fusion continuing for less and less time until the star is layered up with different layers, rich in the elements that it has fused together, and with a core of solid iron.

When the core of the star is iron no further fusion can occur. You will have learnt in you A level module on nuclear physics why this is the case, but you will not be expected to explain this in the astrophysics exam.

Once the process of fusion stops in the core of the star there is nothing left to hold it up against the force of gravity, so catastrophic collapse is inevitable. The star falls in on itself and the falling gases rebound off the super dense iron core, creating a huge explosion called a type II supernova. From Earth, the supernova would appear as a star rapidly changing its magnitude as it became brighter, and it could potentially be as bright as the full moon, even from a distance of several parsecs. The shockwave from the supernova causes small areas of the expanding cloud of debris to fuse and create nuclei heavier than iron which are then scattered into space to form a nebula. The picture below show the remnants of a supernova viewed in 1987. The neutrinos released from an event like this escape the maelstrom much more easily than the photons, so are often detected before the light from the supernova, as was the case for the event below.

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Neutron stars

As the core of the giant star collapses after the initial supernova the nuclei and electrons present get squeezed into a smaller and smaller volume. Eventually the electron are absorbed by the protons and create a dense ball of neutrons.

Further collapse is prevented by neutron degeneracy pressure, which effectively prevents two fermions occupying the same space.

These neutron stars are very strange objects and are composed almost entirely of neutrons. They may have an atmosphere of hot plasma and a crust of heavy nuclei, and possibly a core of other unknown exotic matter, but their defining property is their neutron composition.

Their other defining property is their enormous density. A neutron star has a density similar to that of an atomic nucleus, $\quantity{10^{17}}{kg\,m^{3}}$. A neutron star with 1.4 solar masses would have a diameter of around $\quantity{10}{km}$, or to put it another way, a single teaspoon of neutron star would have a mass of 100 million tonnes!

Neutron stars also spin on their axis extremely quickly as a consequence of the the conservation of angular momentum. The slow rotation of the original star is sped up as its mass is concentrated into a smaller and smaller radius.They can also have extreme magnetic fields which accelerate charged particles to great speeds and create high intensity radio waves emitted from the star’s north and south magnetic poles. If these waves are emitted on the same plane as the Earth we can detect them as extremely regular radio pulses. These types of neutron stars are called pulsars and were first discovered by Jocelyn Bell Burnell in 1967. They are so regular that they are more accurate than atomic clocks and have periods ranging from 4 seconds to milliseconds.

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Gamma-ray bursts

Gamma-ray bursts are the most energetic electromagnetic events known in the universe. They appear as bright gamma sources at a great distance, and can produce, in a short period of time, as much energy as the total energy output of the Sun. To produce such high energy electromagnetic radiation, a very destructive process must be going on. They are thought to be caused by large supernova or hypernova events from the death of the very largest supergiant stars, or from the collision of two neutron stars in a binary system. This energy is usually highly focused, or collimated, in narrow beams from the poles of the exploding star.

Although thousands of gamma-ray burst have been observed, ranging in duration from a flash lasting a few milliseconds to several days of continuous output, no gamma-ray bursts have ever been observed in our galaxy. This is fortunate for us as it has been hypothesised that if a local gamma-ray burst directed at Earth it could cause a mass extinction of life!

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Black holes

When the core of a star with a mass greater than 10 solar masses collapses, not even the neutron degeneracy pressure the holds up neutron stars can prevent further collapse. The whole core of the star falls towards a point with infinite density. This creates a region of space with an infinite gravitational pull. This point is called a singularity. Although only very large stars form black holes, any amount of matter could be squeezed into a small enough volume of space to create a black hole.

The defining property of a black hole is its escape velocity, or the velocity an object would need to be projected at in order to move out of the gravitational well of the black hole. For a black hole the escape velocity is greater than the speed of light. As the speed of light is a sort of universal speed limit, and nothing can exceed it, it means that nothing can escape from the gravitational pull of a black hole, not even light itself.

This may sound odd, as we think of gravity only acting on mass, and light has no mass, but gravity actually warps the space around it, and light follows the shortest path through space. If that space is curved then the path that the light travels is also curved. If the curvature is infinite, as it is around a black hole, then light will be directed towards the singularity. This, however is well beyond the scope of an A level course.

However you do need to understand that although the gravitational force of a black hole drops off according to the inverse square law, there is a point, as discussed above, that the force is so great that the escape velocity is greater than $c$. This boundary, where the escape velocity is equal to $c$, is called the event horizon and it defines the edge of the black hole. The radius of a black hole’s event horizon is called the Schwarzschild radius and is given by the equation:

This can be derived from the equation for universal gravitation. In fact if any amount of mass is compressed within its Schwarzschild radius it will become a black hole. For a star with a mass of 10 solar masses ($10\,M_{\odot}$) the Schwarzschild radius would be:

Black holes are a fascinating state of exotic matter and are definitely worth reading about, but the their details, and weird and wonderful properties are not required in for exam.

This process would create what is called a stellar mass black hole, however, lurking at the centre of almost all galaxies are black holes that have the mass of millions of suns. These are called supermassive black holes. They have a Schwarzschild radius of hundreds or thousands of kilometres. The supermassive black hole at the centre of our galaxy is called Sagittarius A*. Although it is not visible, it is a strong radio source due to the radiation released from matter falling into it, and we are able to observe several stars local to it orbiting a very massive, but compact object.

Our supermassive black hole is relatively quiet, but some distant galaxies have huge black holes which are in the middle of feeding frenzies, as huge amounts of matter fall into their event horizons. This causes an accretion disc to form which in turn gets very hot. These become very bright objects known as quasars. All quasars are very distant, therefore very ancient, which suggests that quasars were common in an older epoch of the universe’s history, when galaxies were themselves younger, but are rare in today’s universe.

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Worked example

Centaurus A is the nearest example of an active galactic nucleus. Many astronomers believe a supermassive black hole at the centre of such a galaxy produces a quasar as it consumes the material of its nearby stars.

1. Explain what is meant by the event horizon of a black hole.

This is a simple question, but it is important to be clear. The event horizon is the boundary where the escape velocity = c. Many students write that the event horizon is a point, or a distance, but this is not correct. The question has not asked you explain any properties of a black hole, so it is wise to avoid doing so unnecessarily in case you confuse yourself and negate any mark that you have already got.

2. 1. The mass of the black hole is 60 million times the mass of the Sun. Calculate the radius of its event horizon.

   However it is important to notice that the mass of the black hole is 60 million time the Sun’s mass and you must copy the Sun’s mass correctly from the data booklet.

   $$R_{s}=\frac{2\times 6.67\times 10^{-11}\times 60\times 10^{6} \times 1.99 \times 10^{30}}{\left(3.00\times 10^{8}\right)}=\quantity{1.8\times 10^{11}}{m}$$
   2. Calculate the average density of the matter within its event horizon.

   We know the mass of the black hole, and we know the radius of its event horizon, so we can use the density equation, but we must remember that a black hole is a three dimensional object, so it would be spherical. Therefore we will need to calculate its volume as a sphere. This equation can be found on the data booklet also:

   $$V=\frac{4}{3}πr^{3}=\quantity{2.44290244743\times 10^{34}}{m^{3}}$$

   I have written out the whole calculator display here, so that I do not get any rounding errors on the next stage of the calculation.

   Density equals:

   $$ρ=\frac{m}{V}=\frac{60\times 10^{6} \times 1.99 \times 10^{30}}{\quantity{2.44290244743\times 10^{34}}{m^{3}}}=\quantity{4.8\times 10^{3}}{kg\,m^{-3}}$$

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