Topic 5  Life and Death of a Star

In the exam you are expected to know about:

• The Hertzsprung-Russell diagram;

• General shape: main sequence, dwarfs and giants;

• Stellar evolution: path of a star similar to our Sun on the Hertzsprung-Russell diagram from formation to white dwarf;

•  General properties of supernovae, neutron stars and black holes;

• Calculation of the radius of the event horizon for a black hole.

• Schwarzchild radius R s

• 

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-Russel 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.

Notice that 

• the temperature scale is decreasing. 

• the classes of star are placed alongside the temperature scale;

• the luminosity scale is logarithmic to compress it;

• there is a spelling mistake.  Can you find it?

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.

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.

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 are common but hard to observe.

Question 1 Complete the table.  One has been done as an example:

ANSWER

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:

As the particles come together they collide increasingly frequently and the temperature begins to rise.  Star 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 picture shows the glow of very young stars.

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:

• gravity trying to make the star collapse in on itself;

• the outward force of the explosion (sometimes called the hydrostatic pressure).

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?  ANSWER

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.

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:

• the greater the mass, the longer it will before the hydrogen fuel runs out.

• the greater the luminosity, the sooner it will use up its supply of fuel.

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.

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 lump in space.

Question 3  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.

Data:

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.

ANSWER

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.

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

Novae and Supernovae

With more massive stars 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.  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 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.

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:

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 if there's a star nearby.  It gets sucked in:

The radius of the event horizon is called  the Schwarzchild Radius.  From gravity in Module 4, 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 Schwarzchild radius is given by:

[Rs - Schwarzchild radius (m); G - universal gravity constant 6.67 x 10^-11 N m² kg^-2; M - mass of the star (kg); c - speed of light (m/s)]

Worked Example

A star of mass 2 x 1031 kg forms a black hole.  What is the Schwarzchild 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 4

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

ANSWER

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