Astrophysics Tutorial 6 - Life and Death of a Star


Evolution of Stars

A Star is Born

Jeans Criterion

Stable Phase of a Star

Evolution of a Star
Lifetime of a Star

Cepheid Variables

Dying Star

Death of a Star

Novae and Supernovae

Classification of Supernovae

Type Ia Supernova

Nucleosynthesis in Supernovae

Neutron Stars
Black Holes

Schwarzschild Black Hole


Evolution of Stars

We can classify people by all sorts of different ways such as sex, race, creed, political beliefs and so on.  In the last topic we saw how we can classify stars according to their apparent and absolute magnitude, and their temperature and spectral analysis.  However these classifications do not tell us a great deal about the age of the star or how it has evolved.


A Danish astronomer Ejnar Hertzsprung recognised patterns within stars.  Independently an American, H N Russell, came up with the same sort of idea which gave rise to a useful tool called the Hertzsprung-Russell Diagram.  This is essentially a graph of temperature on the horizontal axis while on the vertical axis we can put the absolute magnitude or the luminosity compared with the Sun.  Sun = 1.


European Southern Observatory, Wikimedia Commons

Notice that 


Most stars lie along the main sequence, going from very bright blue stars to very dim red stars.  The Sun is somewhere in the middle of the main sequence.  They range from cool low power stars of absolute magnitude +15 to very hot high power stars of absolute magnitude -5.  The greater the mass of the star, the higher up the sequence it is.  Mass ranges from 0.1 to 30 solar masses.


To the top right there two distinct classes of star, the red giants and the red supergiants.  Although they are cool, they have to be big to achieve the luminosity.  The star Betelgeuse would engulf the orbit of Jupiter.  The magnitudes of these stars are between +2 and -2.  They are 10 to 100 times larger than the Sun.  They are more powerful than the Sun, but have lower surfaces temperatures.


To the bottom left we have dim stars.  Spectral line analysis suggests that they are very hot, but their low luminosity suggests that the stars are very small.  White Dwarfs are thought to be about the size of the Earth but with a mass similar to the Sun.  They have absolute magnitudes of +15 and +10.  They are common but hard to observe.


Worked Example

A red giant and a main sequence star both have an absolute magnitude of 0.  Their surface temperatures are 4000 K and 10 000 K respectively.  How any times larger is the red giant than the main sequence star?

Both stars have the same power output.

Use the Stefan's Law:

 P = 4pr2sT4


Let's call the red giant R and the mainstream M.  We can fiddle with the formula to write:

We can then put the numbers in:


Diameter of the red giant is the 39 = 6.25 times bigger.



Question 1

Complete the table.  One has been done as an example:



Luminosity (Sun = 1) 

Surface Temp (K)





Main sequence


20 000


Red Supergiant








Main Sequence




Main Sequence

Sirius B








A Star is Born

Space is not a complete vacuum.  There are about 10 atoms per cubic centimetre (compared to 1019 in a room).  As well as atoms there are molecules and specks of dust.  Many of these atoms and molecules come from stars that have exploded, which we will look at later.


Gravity is the driving force behind the birth of a new star.  From your studies in Module 4 you will remember that gravity is a very weak force but has an infinite range.  Gravity is always attractive, never repulsive.  It pulls the particles together, and they accelerate inwards.  The process is very slow indeed, but there is all the time in the universe for it to happen.  The picture below shows such as dust cloud:


Photo NASA


As the particles come together they collide increasingly frequently and the temperature begins to rise.  This process is called accretionStar formation tends to happen where the clouds are dense and have a mass about a hundred times that of the Sun.  There needs to be regions of non-uniform density.  Stars are usually born in clusters.


As the gas cloud collapses and heats up, it will emit significant amounts of infra-red radiation.  This is known as a protostar.  At this stage the temperature is still too low for nuclear fusion to happen.  If the mass is too low, the failed star ends up as a brown dwarf.  Some astronomers consider Jupiter to be a failed star.


As the material gets hotter, molecules are torn apart and atoms are ionised.  As the mixture gets hotter still a plasma is formed where atoms are stripped of most, if not all their electrons.  Finally an ignition temperature is reached and fusion starts.  The temperature is about 15 million oC.  The picture shows the glow of very young stars.


Photo NASA



The Jeans Criterion (IB students only)

The collapse of gases and material to form a star is only an occasional process.  We tend to assume that the cloud of gas is uniform and a regular shape.  However the process starts when there is some kind of instability.  This may be due to the action of a shock wave from the explosion of a supernova.  The Jeans Criterion is a model that enables astrophysicists to predict when a dust cloud will collapse to form a star.  It is named after the physicist and mathematician James Hopwood Jeans (1877 - 1946)


We have seen that accretion starts when the effect of gravity overcomes the outward force due to collisions of molecules.  We can also say that the potential energy is greater than the kinetic energy.  Once this is the case, the cloud will start to collapse.  For a cloud of given radius and temperature, there is a mass called the Jeans Mass (MJ), above which the cloud is liable to collapse.  The Jeans Criterion depends on:

The relationship is:


Worked Example:

A gas cloud of radius 1.0 109 m.  The temperature in that part of the universe is 3.5 K.  Assume the cloud is made up entirely of hydrogen (H2) molecules.

(a) Work out the mass of each molecule;

(b) Work out the Jeans Mass of the gas cloud.

(c) How does your answer to (b) compare with the mass of the Sun?

(d) Explain whether the accreted material will form a star?

(Avogadro's Number = 6.02 10-23 mol-1)

(Relative mass of hydrogen = 2.02 10-3 kg mol-1)

(Mass of the Sun = 1.99 1030 kg)


(a)  Mass of each molecule = 2.02 10-3 kg mol-1 6.02 1023 mol-1 = 3.36 10-27 kg

(b)  MJ = (3 1.38 10-23 J K-1 3.5 K 1.0 109 m) (2 6.67 10-11  N m2 kg-2 3.36 10-27 kg) = 3.23 1023 kg = 3.2 1023 kg (2 s.f.)

(c)  Fraction of the mass of this cloud compared to the Sun = 3.23 1023 kg 1.99 1030 kg = 1.62 10-7.

(d) The minimum size for a viable star is about 0.08 solar masses.  This cloud is far too small.  It may accrete to form a small brown dwarf.


The radius of this gas cloud is one million kilometres.  That is big.  Gas clouds that form stars are many times bigger.


We can use the density of the cloud to decide whether the cloud will collapse to form a star.  The density of the gas cloud that formed the Sun has been worked out to be about 1.0 10-19 kg m-3.  The current average density of the Sun is about  1400 kg m-3.



Stable Phase of a Star

Nuclear fusion releases a lot of energy.  Humans have achieved it but only in the context of an explosion that would make a thunderclap like a whisper.  The largest fusion bombs used deuterium (an isotope of hydrogen) to produce Helium.  The amount of fused gas used would fill little more than a large party balloon.  So why does a star not fly apart?


There are two opposing forces:

In the life time of a star, the two forces balance each other out and the star remains the same size.



Evolution of a Star

The Sun is a typical star.  It would have taken about 1000 years to coalesce (a very short time compared with the life time of stars) into a protostar of about 20 solar diameters, with a luminosity of about 100 times its present value.  The evolutionary path taken by more massive stars is shorter, because there is more gravity.


The diagram shows the evolution of stars of different masses.  The letter M refers to the solar mass, so 10 M is 10 solar masses.  The blue lines are the time spans in years.



So let's trace how the Sun evolved before it joined the main sequence (thick red line).   The luminosity would have been about 100 times what it is now for a period of about 10 000 years.  Over the 8 million years the luminosity reduced to its present value.  The temperature was relatively low, about 3000 K.  Finally over a period of about 10 million years the temperature gradually rose to about 6000 K.


Question 2

How would you describe the formation of a star that was much bigger than the Sun? 



Stars of small mass (less than about 0.8 M) can take 10 000 000 years to reach the main sequence.  Stars of mass 0.5 M may not even reach the main sequence at all.


A detailed account of what happens to stars that are off the main sequence can be found HERE.



Lifetime of a Star

Most stars spend most of their life on the main sequence of the HR diagram.  The life of a star is governed by its luminosity and mass:

Stars usually start off with 73% hydrogen, 25% helium, and 2% other elements.  Our Sun has used up most of its hydrogen and has more helium than hydrogen.  It is using about 4 million tonnes of hydrogen every second.  The Sun has been burning for about 4500 million years.  There is enough hydrogen to last the Sun another 4500 million years, so it won't go out tomorrow.


Stars of 25 solar masses last only a few million years, since they are extremely bright.  Stars of less than 1 solar mass burn themselves out very slowly; a star of 0.5 M can last up to 200 000 million years.



Cepheid Variables

Most stars show a constant luminosity in their lifetimes.  This means that:

Some stars pulsate, i.e. their luminosity changes periodically from a minimum to a maximum value.  They are called Cepheid Variables.  They were given this name as they were first discovered in the constellation of Cepheus.  The stars have masses of between 4 and 20 times that of the Sun.  They are also stars that are later on in their lives; they are using helium as a fusion fuel.  The apparent magnitude is shown on the graph:


The Cepheid Variable shown in the graph has a period of 5 days and the magnitude varies from +4.5 to +3.5.  The Sun has an absolute magnitude of +4.8.  Astronomers have found that there is a relationship between the period of a Cepheid and its luminosity.  There is a difference in the variation with metal rich and metal poor Cepheid Variables.


The best theory as to why there is this variation in luminosity is that the outer envelope of the star has a lot of helium in it.  We know that the helium atom is two protons, two neutrons and two electrons.  At high temperatures,  helium exists as helium He+ ions.  At even higher temperatures, the second electron is removed to form He2+ ions, a plasma.


The theory is that He2+ is more opaque, which means that less light gets through. The light interacts with the helium to heat it up.  As the star gets hotter, the helium expands.  As it expands, it becomes cooler.  The He2+ ions pick up electrons, to become He+ ions, which are more transparent.  The star gets brighter.  The cooler shell around the star contracts as it cools, and the process starts again.




Click here to see a video tutorial.



A Cepheid Variable with a period of 5 days has a luminosity that is about 1000 times that of the Sun.


From Tutorial 4 we saw that:


We can use this equation to give a difference in magnitude:

Dm = 2.5 lg 1000 = 7.5

Then we use:

Then putting numbers in and re-arranging:


log d - log 10 = 7.5 5 = 1.5


log d = 1.5 + 1 = 2.5


d = 102.5 = 320 pc



Question 3

How many light years is this?



Cepheid variables have well-defined luminosities and are referred by astronomers as standard candles.  Their absolute magnitudes do not vary with age or distance.




Dying Star

The centre of a star is where the fusion takes place.  The outer regions are not hot enough and there is still hydrogen in a shell about the core.  There is no convective circulation of gases into the core.


As fusion dies down, the expansive pressure reduces and gravity pulls the gases in.  They heat up and the pressure on the helium in the core rises.  Helium nuclei fuse to form heavier elements.  Hydrogen fusion increases in shells outside the core.  Therefore there is more helium and the core expands.


Meanwhile while the outer shell where there is hydrogen fusion moves outwards, and the star swells.  The star becomes a giant.  The core and the hydrogen fusion shell are relatively small, while the majority of the space of a red giant is taken up with a low density envelope.



When the Sun has reached this stage, all the inner planets would have been engulfed and fried to a crisp.  Jupiter and Saturn will lose their gas layers to reveal their rocky cores.


In the Sun, elements like carbon and oxygen will be formed in the core.  In more massive stars the conditions in the core will be sufficiently extreme further fusion will take place so that, for example, silicon nuclei will fuse to form iron (the most stable nuclide).



Death of a Star

Eventually gravity will overcome the expansive force.  In small stars, there is convection so that all the hydrogen fuses.  However the temperature never gets hot enough for helium fusion to happen.  The star collapses under gravity to become a white dwarf.  The volume is about the same as that of the Earth.  Eventually it cools to a black dwarf, a forlorn lump doing diddly-squat in space.


Question 4

A white dwarf has a mass of 0.2 solar masses.  Use the data below to calculate the density of a white dwarf, assuming it's the same size as the Earth.  Compare it with the density of the Earth and the Sun.


Mass of sun = 2.0 x 1030 kg;

Mass of earth = 6.0 x 1024 kg;

radius of Sun = 7.0 x 108 m;

radius of Earth = 6.4 x 106 m.



The answer you got shows the enormous density, about 1 million tonnes per cubic metre. 1 cubic centimetre would have the mass of 1 tonne; if you dropped it on your foot, it would bring tears to your eyes.


For Sun like stars, death is more spectacular.  The star expanding in the outer layers and contracting at the core.  The radiation pressure acting outwards pushes the outer layers away from the core to form a planetary nebula, a ring of gas that glows brightly because of the intense radiation from the core.    The material packs into an ever smaller volume until a limit is reached (determined by a quantum mechanical effect called electron degeneracy). The star becomes a white dwarf, gradually cooling down.


Photo NASA


The picture below shows a radio frequency image of a dying star.



Photo NASA

Novae and Supernovae

With more massive stars, where the core has a mass of more than 1.4 solar masses,  the limit is no longer observed and electrons start to combine with protons to form neutrons, releasing neutrinos and causing the core to collapse in on itself.  This is called the Chandrasekhar Limit, named after an Indian astrophysicist, Subrahmanyan Chandrasekhar (1910 - 1995).  The collapse takes less than 1 second, and the density rises to about 4 x 1017 kg m-3. We have a neutron star.  Like electrons, there is a limit to which neutrons are squashed together, determined by neutron degeneracy.  The interaction is:


p + e- n + ne


Just before a supernova explosion, a shower of neutrinos, X-ray, and gamma photons may be observed.  This is because the neutrinos released by this process and nuclear fusion propagate much faster than the shockwave passing through the body of the star.


The outer layers may also collapse in to collide with the dense core, which can no longer be compressed.  The material bounces back as a shockwave, which takes a few hours to propagate through the material.  The material is torn away in a titanic explosion called a supernova.  The absolute magnitude may be -15 or even -20.



Photo NASA


Often there is a cloud of gas, a nebula, which propagates away from the site of the explosion.  This can be seen in the picture below:


Photo NASA


Iron is the largest element that can be caused by fusion.  It is made in the largest of stars.  It requires the extreme conditions of a supernova to make elements that have nucleon numbers greater than 56.  So copper, with a nucleon number of 64, will have been made in a supernova explosion, as have many of the other rare elements in your computer.


It is thought that supernovae emit an audio-frequency hum just before they explode.



Classification of Supernovae

As with any star, we can analyse the light by looking at the absorption spectrum, and much can be learned about the explosions from the chemical compositions.  Astronomers classify the supernovae like this:


Type Ia Supernova

In binary star systems one of the stars becomes a white dwarf.  This is shown as Star B in the picture below. The white dwarf has a mass of about 1.4 times that of the Sun and has a large amount of carbon and oxygen as fusion products.  At this stage the force of gravity is not strong enough to cause cause collapse by electron degeneracy (where electrons are forced into the protons and turn the protons into neutrons).    Material from Star A is attracted by the gravity of Star B and is pulled away from Star B.  It lands on Star B and collects (accretes).  However, as material from the Star A accretes, the temperature rises high enough for carbon fusion to happen on Star B.  Carbon-carbon fusion results in magnesium.



Eventually the carbon fusion will result in a white dwarf that consists of magnesium, oxygen, and neon.  The white dwarf will collapse under gravity.  Electron degeneracy will cause a neutron star to be formed.


In rare cases the binary system consists of two white dwarfs like this:



In this case, the amount of material that accretes does not quite reach the critical value of 1.44 solar masses.  The temperature of the core rises as the pressure and density rises.  Strong convection currents occur in the atmosphere, a stage that is estimated to last about 1000 years.  Although it's not known how it starts, a carbon fusion flame ignites and spreads.  It sets off oxygen fusion. In a main sequence star, the process would be regulated by thermal expansion, but in a white dwarf, this does not happen, and the rise in temperature runs away.  The fusion events last only a few seconds, releasing vast amounts of energy (about 1044 J) to tear the star apart.  The shock wave propagates at speeds of up to 20 000 km s-1.  This blows away any remaining material of the companion star.  The process is different to normal supernovae in that the core does not collapse. 


The absolute magnitude of such an event is M = 19.3, with little variation.  This enables astronomers to use Type 1a supernovae explosions as standard candles, giving a good reading of the distance of the galaxies from the Sun.


Initial analysis of spectra reveal elements such as oxygen and calcium.  As time progresses, heavier elements are detected, such as iron, which is the largest element that can be formed by fusion.  Other elements of nucleon number 56 also arise, such as nickel-56.  This isotope decays by beta decay through cobalt-56 to iron-56, releasing high energy gamma photons, and photons released by the interactions of high speed electrons with matter.



All reactions in a star are fusion.  Although it might be tempting to assume that carbon and oxygen react to form carbon dioxide, this does not happen.  Chemical reactions involve the outer electron shells.  In the extreme conditions in a star, all electrons are removed, so that the elements exist as a plasma.  Once elements have cooled sufficiently to attract electrons into shells, only then can they form chemical bonds.


Type Ia Supernovae are rare events.  It is thought that they represent the first kind of supernova explosions that were the end of the earliest stars.  They are very distant objects.



Nucleosynthesis in Supernovae (IB students only)

In Nuclear Physics 7, we have seen that iron is the most stable nucleus.  Iron nuclei are the largest nuclei that can synthesised using fusion processes.  So we need to consider how larger nuclei like copper are formed.  These need to very high energy environment of a supernova explosion to form.  The production of such nuclei comes from neutron capture.  Neutron capture is where a neutron is absorbed into a nucleus as a result of a collision.  The neutron has to have sufficient energy to overcome the strong force.  This energy is lower than what would be required to capture a proton, which would have to overcome the electromagnetic force as well.


There are two kinds of neutron capture:

In a slow capture event, a seed nucleus, for example iron, absorbs a neutron to become a heavier isotope, for example:



This isotope is stable.  The iron-57 nucleus can capture another neutron to become iron-58.  This isotope is also stable.  It can capture a neutron to form iron-59.  This is unstable and decays by beta minus decay to cobalt-59.

The neutron capture chain is shown below:



The extent to which this process happens depends on the amount of neutrons are produced.  A typical reaction that produces neutrons is shown here:



The s-process is slow, taking thousands of years.  It depends of the neutron flux (rate at which neutrons are produced).  Neutron densities of between 109 and 1015 m-2 s-1 are observed. There can be long time periods between neutron capture events.  Iron-59 has a half-life of 45 days.  Much of it will have decayed to cobalt-59 (stable) before there is another neutron capture.


The kind of environment which can support the slow-process neutron capture would be the nebula of hot gases that results from a supernova explosion.  This can last for thousands of years.


The s-process can produce nuclides up to lead (Pb), but not heavy nuclei like uranium.  Such nuclides require a much more violent environment.



The rapid-process neutron capture events happen in a time period of a few seconds.  This is consistent with the fact that a supernova star collapses in about 45 seconds.  The environment we get is:


The process results in nuclides that are very neutron rich.  The neutron nuclei are very unstable, decaying by beta minus decay.  Iron-56 (stable) can absorb lots of neutrons in the high neutron density environment to produce nuclides like iron-64.  This decays by beta minus decay to cobalt-64, with a half life of 2.0 s.  The cobalt-64 isotope can capture neutrons rapidly to become cobalt-65.  Again this decays by beta minus to nickel-65 with a half-life of 1.2 s.  The neutron capture chain is shown:


Note that nickel-64 is stable, but can easily capture a neutron to become nickel-65 (which decays to copper-65 with a half-life of 2.5 h).  Copper-65 is stable.  By this time, the environment would have cooled somewhat and the neutron capture process will more likely be slow. 




Neutron Stars

The resulting neutron star is very small compared to the original star.  A sun-like star would give a neutron star of diameter 30 km.  The material in the Earth would be contained in a ball about 200 m across that would fit on top of your school (and squash it flat).


Question 5

 A ball of neutron star material has a mass of 6 1024 kg and a diameter of 400 m.  Calculate:

(a) the gravitational field strength on the surface;

(b) the escape velocity of a rocket trying to leave.



Some neutron stars rotate rapidly and give of beams off radiation with a regular period.  They are called pulsars.  The period can be as short as 33 ms, i.e. a frequency of 30 Hz. 




Such a star will be spinning at a rate of 900 times a minute.  When a pulsar was detected using a radio telescope in 1967, the astronomers thought initially that the regularity of the pulse suggested a civilisation.  They dubbed the pulsar LGM-1 (Little Green Men - 1).  The radio waves come from the interaction of charged dust particles with the magnetic field.


The maximum mass which produces a neutron star is considered to be 2.17 solar masses.  This is called the Oppenheimer-Volkoff limit.  This is named after the American Physicist Julius Robert Oppenheimer (1904 - 1967) and the Canadian physicist, George Michael Volkhoff (1914 - 2000).  Above this, the neutron star becomes a black hole.  The graph shows the idea:



It is possible for neutron stars to collapse into a black hole by:

Below the Oppenheimer-Volkhoff limit, the dead star is a neutron star.  This is the result of neutrons not being able to occupy the same energy level (Pauli Exclusion Principle).  Once the lowest energy levels are filled, the neutrons are forced into higher energy levels.  Up to the Oppenheimer-Volkhoff limit, the higher energy levels will provide an outward pressure to oppose the gravitational pressure.  There is space between the neutrons.


Above the Oppenheimer-Volkhoff limit, it is thought that the gravity overcomes the forces holding the neutrons apart.  They are packed ever closer, becoming very hot and dense.  The neutrons degenerate into quarks.  This state is usually a short-lived intermediate between a neutron star and a black hole.


Then a black hole will form.



Black Holes

In very massive stars the core can collapse so that even the limit determined by neutron degeneracy is broken.  The gravity is so great that the core keeps on collapsing, shrinking away, according to the theory, so that it is no more than a point in space.  This point is called a singularity, and the laws of Physics no longer apply.  The gravity field in a black hole is so strong that even light cannot escape.


The star is surrounded by an event horizon, inside which nothing can be seen.  It is the boundary at which light cannot escape.  Abandon hope all ye who enter here. 


You cannot see a black hole.  But you can tell where there's a black hole as jets of high energy particles are ejected


Picture from NASA, Wikimedia Commons


A passing star is gobbled up by a black hole:


Picture from NASA, Wikimedia Commons


Astronomers believe that at the centre of most galaxies is a super-massive black hole, a black hole of mass of 108 times the mass of the Sun.  Black holes have such high gravity that light can be bent.  This is called gravitational lensing.


Picture by Gallery of Space Time Travel, Wikimedia Commons


Schwarzschild Black Hole

The Schwarzschild black hole is the simplest black hole:


The diagram shows the structure:


The features are:


The radius of the event horizon is called  the Schwarzschild Radius, code RS.  From Fields Tutorial 3, the escape velocity of a rocket can be worked out by:


kinetic energy = gravitational potential energy


Rearranging gives:

The escape velocity of light can be similarly worked out:


So the Schwarzschild radius is given by:





Worked Example

A star of mass 2 x 1031 kg forms a black hole.  What is the Schwarzschild radius?  What is its density in the region bounded by the event horizon?



RS = (2 x 6.67 x 10-11 N m2 kg-2 x 2 x 1031kg) (3 x 108 m/s)2

RS = 29 600 m (29.6 km)


Density = mass/volume

Volume = 4/3 x p x (29600 m)3 = 1.09 x 1014 m3

Density = 2 x 1031 kg 1.09 x 1014 m3 = 1.8 x 1017 kg m-3.


Question 6

Much evidence is being produced suggesting that galaxies have black holes at their centres. For example, the spiral galaxy M51 may contain a black hole with a mass one million times greater than the Sun.

(i) Explain what is meant by the term event horizon.

(ii) Calculate the radius of the event horizon for the black hole in M51.

(AQA Past Question)



There are other types of Black Hole:

We will not consider these here.


The picture below summarises the life of a sun-like star:


And the picture here summarises the life of a star that is much bigger than the Sun:



The fate of the star following a supernova explosion depends on its size.  The largest stars end up as black holes.