Astrophysics Tutorial 7 - Cosmology

 Contents

The Doppler Effect

You will know the Doppler effect as the falling note of a car or train horn as it approaches, passes, and then goes away from you. The importance of the Doppler effect is that it is seen with light waves and radio waves.  For any object that is moving with a speed much less than that of light, it can be shown that the change in frequency is given by: [Df - change in frequency (Hz); f - original frequency (Hz); v - speed of object (m s-1); c - speed of light (m s-1)]

Sometimes we use the term Doppler Shift, code z, which is the fractional change in frequency. For wavelength the equation is similar: In a similar way we use the Doppler Shift, code z, which is the fractional change in wavelength. For these equations:

• Objects moving towards the observer have a positive speed;  moving away from the observer the speed is negative.

• If the object is moving away, the frequency is lower so that Df is negative.  The wavelength will be longer.  Astronomers call this red shift.

• If the object is coming towards the observer, the frequency is higher, so Df will be positive.  The wavelength will be shorter.  Astronomers call this blue shift.

The Doppler shift can be summed up in this table:

 Doppler Shift, z Moving towards Moving Away Frequency Positive (higher) Negative (Lower) Wavelength Negative (shorter) Positive (Longer)

While we will focus on the Doppler Effect in Astrophysics, it can be used to measure blood flow with ultrasound (Medical Physics Tutorial 5), or detect the movement of vehicles using radar.  A radar gun used by the police emits microwaves and measures the Doppler shift as a car approaches.  The traffic cop gets a direct readout of the speed of the car.  Fixed speed cameras work the same way.  If you are above the tolerance on the speed limit (about 10 %), the camera will take a picture, and you will get a speeding ticket (3 points on your licence).

 Worked example The wavelength of a pale blue line in the hydrogen spectrum is 486.27 nm as measured in the lab.  When the same spectral line is looked at in a star, the wavelength is now 486.94 nm.  What is the speed of recession? Answer Find out the change in wavelength: Dl = 486.94 - 486.27 = 0.67 nm = 0.67 x 10-9 m.   Use: Substitute into the equation:   0.67 x 10-9  m ÷ 486.27 x 10-9 m = -v ÷ 3 x 108 m s-1   v = -4.1 x 105 m s-1    (Negative means away from us.  Strictly speaking, speed should be velocity.)

 A star is moving away from the Earth at 5000 km s-1.  A certain wavelength has been detected in its spectrum which corresponds to a line of wavelength 350 nm as measured in a laboratory.  What is the wavelength of this line?

The minus sign tells us that the star is receding from us.  The longer wavelength is called red shift, i.e. it has been shifted towards the red end of the spectrum.  This is shown below.  The top spectrum is what we would expect for an element in the lab.  The bottom spectrum shows the same pattern, but shifted towards the red. We can see that the pattern is the same, but the colours are different.

 Question 2 Venus has a diameter of 12 200 km and a rotational period of 243 days. (a) What is its angular velocity and its linear speed of rotation at the equator. (b) Radio waves of wavelength 1.0 m are used to determine the speed of rotation.  What is the expected shift in wavelength reflected at opposite edges of the equator? The same principle as in the question is used to determine the rotation of a star.

The Doppler effect is used in other ways:

• looking at the rotational period of stars

• rotational periods of planets.

• Orbital period of binary stars.

60 % of the stars are actually pairs, making our Sun as a single star in the minority.  (It is thought that Jupiter is a failed star.)  Binary stars consist of two stars orbiting about their common centre of mass.  If the stars are of equal mass the orbits are like this: We can use the Doppler shift to tell us how the stars are orbiting.  Let's look at a single line which we know is yellow in the lab: That line for Star A is blue shifted, which means it has a shorter wavelength, so is approaching us.  That for B is red shifted which means that Star B is going away. Notice that now the stars have moved around in their orbit, the blue shift and red shift are less. When the stars are in this position we only get the one spectral line as both stars are neither moving towards us nor away from us.

If binary stars are a long way away, it may not be possible to resolve them.  However we can still see a change in their spectroscopy.  These are called spectroscopic binaries.  These two stars are the same mass and are in the same orbit. Star A is coming towards the observer, so its light is blue-shifted.  Star B is moving away, so its light is red-shifted.  When the two stars are in line with the observer, the line on the absorption spectrum is as expected.  Over a period of time, the two absorption lines would move between the middle position and the extremities.  So half an orbit later, we would see: Question 3 A binary star system is studied where it is concluded that both stars are of the same mass.  Their orbital period is 2.4 years.  A certain element is known to give a spectral line of 460 nm.  And this is observed at time zero.  0.6 years later, the same line is observed to be at 459.92 nm.  At the same time another spectral line is seen. (a) Where is the second spectral line seen? (b) Explain the observation. (c) What is the orbital speed of the star? (d) What is the radius of the orbit? (1 year = 365.25 days) If the stars are of different masses we see orbits like this: Both stars will have the same orbital period.  The smaller star has a larger orbital radius, hence larger linear speed.  The change in wavelength will be greater for the smaller star.

 Worked Example Two stars are orbiting each other as a spectroscopic binary.  One star has more mass than the other.  A spectral line of an element merges once every 1.5 years.  It has a laboratory wavelength of 486 nm.  The line then splits so that there is a difference of 0.042 nm and 0.024 nm.  Calculate: (a) the orbital speed of each star; (b) the radius of each orbit. Answer (a) Work out the Doppler Shift, z for each star:   z = 0.042 nm ÷ 486 nm = 8.64 × 10-5   z = 0.024 nm ÷ 486 nm = 4.94 × 10-5   Work out the speed for each star: v = zc = 8.64 × 10-5 × 3.0 × 108 m s-1 = 2.59 × 104 m s-1   Similarly: v = 1.48 × 104 m s-1   (b) Use: T = 1.5 y × 365.25 dy-1 × 86400 s d-1 = 4.73 × 107 s                    For the faster star: r  = 2.59 × 104 m s-1 × 4.73 × 107 s = 1.95 × 1011 m = 2.0 × 1011 m                                  2p   For the slow star:   r = 1.48 × 104 m s-1 × 4.73 × 107s = 7.0 × 1011 m                               2p

For centuries we have wondered if there are other planets out there.  A lot of science fiction is based on such planets that can support carbon-based life.  It is only recently that we have seen evidence for the existence of planets outside the Solar System.  Evidence was observed as early as 1917, but was not interpreted as such.  The first planet that was conclusively proved to exist outside the Solar System was discovered in 1988.  Now there are reckoned to be about 3700 extrasolar planets or exoplanets.  Exoplanets orbit around a star (or a binary system), in the same way as the planets of the Solar System orbit the Sun.

Direct detection of exoplanets is very difficult.  All planets reflect the light of their parent star, but the light is usually too faint to be observed from even the best telescopes.  Also the faint light is often swamped by the light of the parent star.  Direct observation is possible if the planet is very big (bigger than Jupiter) as such planets can give out large amounts of radiation and can be detected. Image by NASA - Wikimedia Commons

There are a number of ways of indirect detection of exoplanets.  The most common way is to study the radial velocity of a planet-star system.  We tend to think of planets orbiting a fixed point.  However a planet will exert a gravitational force on its parent star, so that both will orbit around a single point of the system's centre of mass.  The idea is shown in the picture below: We don't see the planet, but instead we see the star moving from side to side.  It happens to the Sun under the influence of Jupiter.  The movement can be detected by Doppler Shift. When the star is approaching us, spectral analysis reveals that the star is blue shifted.  When it goes away from us there is red-shift.  The difference in speed is about 13 m s-1 for Jupiter and the Sun.  (The Earth exerts a force on the Sun such that the difference in speed is 9 cm s-1).  Equipment has been made that can detect a Doppler Shift that is equivalent to a speed of as little as 1 m s-1.

The variation of the speed with time is shown on a graph like this: This plot was retrieved from the Exoplanet Orbit Database and the Exoplanet Data Explorer at exoplanets.org, maintained by Dr. Jason Wright, Dr. Geoff Marcy, and the California Planet Survey consortium.

In this case, the speed of approach is about 100 m s-1, while the recession speed is about 140 m s-1.  This could show that the star is moving away from us about 20 m s-1.  The orbital period of the planet is about 3 years.

This method has been used to detect most exoplanets observed so far.  It is a method that is suitable for planets that are relatively close to Earth, about 160 light years.  Also the stars need to be relatively low mass.  Jupiter sized planets have been detected to a distance of about 10 Astronomical Units from their parent stars.

Another way of detecting the presence of an exoplanet is to observe its transit of its parent star, provided of course that its transit is in a suitable plane for us to observe it.  The intensity of the light received from the star is reduced as the planet crosses its face.  The diagram shows the idea: There are other methods that have detected exoplanets, but these are beyond the scope of this discussion.  Here are a link you may find interesting:

The hunt is on for exoplanets that have the possibility of life, i.e. have atmospheres that have a temperature of about 270 - 300 K, and show signs of water.  Several of these have been detected.  There is every possibility that there are life forms on other planets.

It is worth remembering that the laws of physics as we know them here on Earth would apply to all planets, wherever they are in the Universe.  All the knowledge gleaned from our observations of the Universe are underpinned by the Newtonian Laws of Physics (or classical Physics).  Newton's Laws were written three hundred and fifty years ago, but we use them today to send space probes to far-off places.

Hubble's Law

An American astronomer E P Hubble established a relationship between the velocity of recession of a galaxy and its distance from the Earth.  He chose a galaxy that could be resolved into individual stars and analysed spectra against known spectra.  He studied:

• the red shift;

• the speed of recession (moving away);

• the distance from the Earth by measurements of Cepheid Variables.

He found that the further the galaxy was from the Earth, the faster it was moving away.

The graph shows the idea: The equation from this graph is:

v = H0d

• We can work out the recession velocity by using the red shift;

• The distance is worked out by other means which are not needed here.  You can look these up in any astrophysics text book.

The constant H0 is called Hubble's Constant, and has the value 70 ± 30 km s-1 Mpc-1.  There is a lot of discussion on the precise value of Hubble's Constant, and there is a lot of uncertainty.  The value you are given in the AQA A-level exam is 65 km s-1 Mpc-1.  In SI units, H ≈ 2.4 × 10-18 s-1

More local galaxies do not fit this pattern, because of gravitational interactions.  The Andromeda galaxy is actually approaching the Milky Way, and the two will collide.  This will not happen for several thousands of millions of years, so don't lose sleep over it.

 Worked Example The wavelength of a spectral line in the spectrum of light from a distant galaxy was measured at 398.6 nm. The same line measured in the laboratory has a wavelength of 393.3 nm. Calculate: (a) the speed of recession of the galaxy; (b) the distance to the galaxy.   (c = 3.0 × 108 m s−1, H = 65 km s−1 Mpc−1) Answer (a) Calculate the red shift:    Dl = 398.6 - 393.3 = 5.3 nm   z = 5.3 ÷ 398.6 = 0.0133   v = 0.0133 × 3.0 × 108 m s-1 = 3.99 × 106 m s-1 = 4.0 × 106 m s-1.  This is 4000 km s-1.   (b) d = v/H = 4000 ÷ 65 = 61 Mpc When using Hubble's constant, remember to convert m s-1 to km s-1.  You do know how to do that, don't you?

 Question 4 A distant galaxy has a red-shift of 15 %. (a) What is its speed of recession? (b) If H0 has a value of 100 km s-1 Mpc-1, what is its distance?

There are two possible reasons for red-shift in light:

• Stars are moving away from us.

• The universe itself is expanding.

If the second reason were correct, it would seem that the Universe has been much smaller in the past, and that at one point the universe was entirely concentrated in one point.  This led to the theory about how the universe began, in a stupendous explosion, the Big Bang.  (The latter term was first used in a radio broadcast by the Astronomer Royal, Sir Fred Hoyle, in 1953.  He believed that the Universe was in a steady state, and his use of the term "Big Bang" was dismissive in a sarcastic way.)  The Steady State Theory says that the Universe was as it is, is, and always will be.  The Big Bang Theory has considerable credibility among today's astronomers.

If we look at Hubble's graph, we can say that a galaxy at 1000 Mpc is receding at 7000 km s-1.  By converting Megaparsecs into kilometres we can work out the age of the universe as 4.4 x 1017 s, about 14 000 million years.  You can try it for yourself.

The age of the universe is given as: We know that H0 has a value between 40 and 100 km s-1 Mpc-1

 Question 5 What is the age of the universe using these extremes of value?  1 pc = 3.086 x 1016 m At the upper limit of the Hubble constant, the universe is about 10 000 million years old.  Studies suggest that the Earth is about half this age.

The simplest model to use to explain how the universe is expanding is one you can easily try for yourself.  Mark with a felt tip some spots on a balloon.  Then blow the balloon up.  As it inflates the spots get further apart.  As the universe expands, the galaxies move further apart.

There is considerable debate about the fate of the universe:

• Will it keep on expanding and we will be left abandoned in the middle of nowhere?

• Will it bounce back, with everything eventually coming back to where it started from (the Big Crunch)?

Evidence for the Big Bang

There are a number of pieces of evidence that provide convincing support for the Big Bang:

• Relative abundance of hydrogen and helium;

• Dark Energy

The Cosmic Background Radiation (or Cosmic Microwave Background) was discovered in 1964 by two American radio astronomers, Arno Penzias (1933 - ), and Robert Wilson (1936 - ).  They had constructed a giant horn antenna for use in radio astronomy and satellite communication experiments.  The instrument, a Hogg Antenna is shown in the picture: NASA, Wikimedia Commons

Their initial experiments kept on bringing up a microwave signal of wavelength 1.87 mm that corresponded to a temperature of 3 K.  Their thoughts were at first that the instrument was not working properly.  They checked it over thoroughly and even removed a pigeon's nest.  But the signal was still there.  After much thought and discussion, the conclusion was that the signal was due to cosmic background radiation, predicted in 1948.

This background radiation is universal and gives the Universe a temperature of 2.7 K (very cold).  Things get colder as they expand.  If you run a butane (Camping Gaz) stove for about 30 minutes, you will find that it is cold.  Energy is taken in from the surroundings to allow the butane to boil and expand as a gas.  If you allow compressed air to escape rapidly, the container gets cold.  Using the same kind of argument, we can say that the Universe has expanded and was a lot hotter than it is now.

Although the Universe is a very cold place, a steady state theory would suggest that deep space would have a temperature of 0 K.

Stars and galaxies contain hydrogen and helium in a ratio of 3:1 by mass.  So what?

Helium nuclei have a mass 4 u.  A helium nucleus consists of 2 protons and 2 neutrons.  There must be 12 hydrogen nuclei, each of 1 u, to fit in with the ratio. A hydrogen nucleus is 1 proton.  Therefore, for every two neutrons, there are 14 protons.  In other words, the proton to neutron ratio is 7:1.  The argument goes that when the universe cooled sufficiently to allow quarks to form in triplets to make baryons,  protons formed more readily than neutrons.  Calculations involving rest energies of the baryons predict the observed ratio of protons to neutrons of 7:1.

In 1998, supernovae were observed that were much further away than expected.  They concluded that the stars must have been accelerating away, and this had been going on for at least 5000 million years.  The expectation had been that distant objects would be decelerating as gravitational forces were acting.  Many more observations have supported this acceleration.  Force has to be applied to get acceleration.  Work has to be done to accelerate an object.  The energy supplied to do this work is called dark energy, but the nature of this is very unclear.

It is thought that most of the material of Universe consists of dark matter.  Its key property is that it does NOT interact with the electromagnetic force.  Therefore does not emit, reflect, or absorb electromagnetic radiation.  Its presence has been detected by the gravitational effect it has on visible matter.  Figures from CERN suggest that 27 % of the universe is dark matter, while only 5 % is matter as we know it.

The evidence for dark matter is that the orbital speeds of stars and galaxies should diminish the further away they are from the centre of a galaxy.  Instead they remain roughly constant. The speeds of galaxies within a cluster suggest much more mass than that which is observed.

Gravitational lensing provides evidence. Image from Hubble - Wikimedia Commons

Models have been used to work out the mass required to give a lensing effect like this.  The mass is considerably more than what is observed.

Theoretical physicists are working on the theory of supersymmetry in which each known particle has a companion particle, so far undiscovered.  For example, quarks have particles called squarks which are the super-symmetrical partners of the quarks.  Similarly the W boson has a partner called a wino. Particle physicists are now trying to detect such particles at CERN.  The key to detecting them is unexpected results with momentum. The particles in supersymmetry are NOT antiparticles.

Some theoretical physicists think that there is an association between the dark matter and the Higgs Boson.  The rest energy of dark matter is considered to be between 100 GeV (1 × 1011 eV) and 1 TeV (1 × 1012 eV).  The Higgs Boson has a rest energy of 125 GeV, and is an elementary particle.  When the Universe immediately after the Big Bang was still extremely hot, theoretical physicists suggest that matter and dark matter were balanced in thermal equilibrium.  Since it is believed that matter is derived originally from the Higgs Boson, it makes sense that dark matter is as well.   It gave rise to a range of particles known as WIMPS (weakly interacting massive particles).  The weak part refers to the weak interaction.

Some physicists think that Dark Energy is connected with the Higgs Boson, but others disagree.

Dark energy makes up 68 % of the universe.  It is associated with the vacuum of space.  It is described as being evenly distributed in both space and time, and is considered to provide the repulsive force that is causing the universe to expand.

Much of what went on in the Big Bang happened in the first second - rather less than the time you took to read this.   The timeline is presented in the table below:

 Time Name Size of the universe / m Temperature / K What happened 0 The Big Bang 0 ∞ All matter in the Universe was confined to an infinitesimally small space. 0 - 10-43 s Pre-Planck Era <10-35 >1032 Nobody is sure.  All fundamental forces are unified. 10-43 - 10-36 s Grand Unification Epoch 10-35 1032 Gravity separates from the other fundamental forces.  Earliest elementary particles and antiparticles. 10-36 - 10-32 s Inflationary epoch 0.1 1027 The strong force separates.  This causes massive expansion of the Universe, which consists of a hot quark-gluon plasma. 10-32 - 10-12 s Electroweak epoch 1010 1022 Exotic particles come into existence, for example W, Z, and Higgs Boson. 10-12 - 10-6 s Quark epoch 1015 1021 Quarks start to form.  Fundamental forces take their present form.  Matter and antimatter particles annihilate.  There is a small surplus of matter.  Some sources suggest that the Universe temperature increased markedly (by about 100 000 times) 10-6 - 1 s Hadron Epoch 1017 1018 Protons form.  Electrons colliding with protons are captured to form neutrons.  Neutrinos are formed, travelling at the speed of light.  Protons combine with electrons to form hydrogen atoms. 1 s - 3 min Lepton epoch 1018 1010 With the annihilation of protons and antiprotons, there are many high energy photons, which collide to cause pair production of electrons and positrons. 3 - 20 min Nucleosynthesis 1020 109 In this period, the temperature is sufficiently high to allow the fusion of protons and neutrons to form the first very simple nuclei, like hydrogen, helium, and lithium.  After 20 minutes, the temperature has fallen to too low a value to allow fusion to happen. 20 min - 240000 y Photon Epoch 1024 109 The Universe is a hot opaque soup of plasma.  Leptons and antileptons are annihilating to produce photons.  These interact frequently with protons, electrons, and nuclei. 240000 - 300000 y Recombination and decoupling 1025 3000 The nuclei are now combining with electrons to form atoms.   Small clumps of material start to gather.  As electrons combine with nuclei, the universe starts to become transparent to light.  Photons are no longer interacting with protons and electrons.  These are released (decoupled) to form the Cosmic Background Radiation. 300000 - 150 ×106 y Dark Ages 1026 60 The Universe is dark, as the first stars are only just starting to be formed.  There is little activity and the Universe is dominated by dark matter. 150 - 1000 × 106 y Reionisation 1026 19 The first quasars are formed by gravitational collapse.  This causes ionisation in much of the surrounding universe.  Most of the universe is ionised again to form plasma, 1000 - 5000 × 106 y Star and Galaxy formation 1027 4 Small clumps of cosmic gas coalesce under the influence of gravity.  As the material gathers, its temperature rises sufficiently to cause fusion.  Stars a born.  These stars a metal-free, and are usually large and short-lived.  They form supernovae that produce many heavy elements from their explosions.  Large volumes of matter tend to gather together to form galaxies 13.7 × 109 y Present 1027 2.7 The expansion of the universe continues and stars are recycled.  The furthest objects that can be observed are the cosmic background radiation photons.

When the Big Bang occurred, it is assumed that all space time was crammed into that infinitesimally small space.  The explosion did not propagate into a pre-existing vacuum.  Nor did the space time lead the explosion.  Space-time expanded with everything else in the universe at the same rate.

Quasi-stellar objects (quasars) were discovered in 1960.  They are very luminous objects at immense distances.  They appear to light telescopes as stars but are not typical:

• They outshine complete galaxies;

• Spectra  show lines that correspond to no known elements.

• However the lines were in fact considerably red-shifted.

• Some are intense radio sources.

The red shift suggests that the objects are moving away from us at 15 % of the speed of light.  According to Hubble's Law, that means that they are a very long way away.  So we can say that Quasars are:

• very distant;

• very bright;

• smaller than a galaxy.

Here is a picture of one: Nobody is quite sure what they are, but the predominant belief that :

• They are super massive black holes about 106 to 109 times the solar mass.

• They gobble up stars at about 10 solar masses every year, i.e. a giant cosmic Hoover.

• The dense flow of matter can force jets of matter to stream away from the disc.

Computer modelling suggests a structure like this: A stylised painting shows what a quasar would look like from a relatively safe few hundred light years. What you see will depend on your viewing point:

• from the side you see a radio galaxy;

• closer to the line of the jet you see a quasar;

• in the line of the jet you see a blasar (an extremely luminous galactic object.)

Some astronomers believe that there are quasars closer to home, in nearby galaxies.  However the accepted belief is that they are distant objects.  Therefore they were around very early in life of the university.  The furthest yet observed is quasar 0051 279 discovered in 1987.  It is receding at 93 % of the speed of light and is 13 100 million light years away.

Being gobbled up by a super massive black hole is not yet on the agenda, as both the Milky Way and Andromeda are relatively inactive galaxies, with few stars near the supermassive black holes at the centre.

Critical Density  (Welsh Board)

We have considered the origins of the Universe in the Big Bang, but we have not thought about the fate of the Universe.  We know that the Universe is currently expanding. What is going to happen to it?  Will it eventually stop expanding, and fall back on itself (the Big Crunch)?  Will the expansion eventually be halted and the Universe stays where it is?  Or will it continue to expand (even if at an ever decreasing rate)?

In the Theory of General Relativity, Einstein argued that gravity has the effect of curving the surrounding space.  This is described in Physics 6 Tutorial 1.  The fate of the Universe is determined by the density of the matter that is in it.

• If the density of the matter in the Universe is high, we have a closed Universe that can be modelled as a sphere.  The gravity of the Universe will slow the expansion down, and eventually start to pull all the material back together.  In the closed universe, parallel rays of light will eventually meet at some extremely distant point.

• If the density of the matter in the Universe is low, we have an open Universe, the gravitational forces are insufficient to stop the expansion.  The Universe will continue to expand.  Parallel rays of light will diverge.  This can be modelled as the hyperbolic geometrical shape.

• If the density is just right, we have a flat Universe.  Here the parallel rays of light remain parallel.  The expansion stops at an infinite time.

We can show this as a schematic: The density to give rise to the flat Universe is called the critical density.  The critical density is given by the equation: where:

• H0 is Hubble constant.  In SI units, H0 = 2.2 × 10-18 s-1.

• G is the gravitational constant = 6.67 × 10-11 N m2 kg-2.

 Show that the critical density is about 10-26 kg m-3. You must use the SI units for Hubble's Constant in this case.   Use of H0 as 70 km s-1 Mpc-1 will give an answer of 9.0 × 1012 kg m-3.  This is the density of a black hole.

Derivation

Consider the Universe as a uniform hollow sphere of mass m in which every particle is evenly spread throughout, whether it is from dust, stars, planets, or dark matter.  It forms a material of uniform density of a very low value, and has a mass, M.  The sphere is expanding at a speed v The material experiences a gravitational force inwards.  We can ignore the gravitational effects of the material outside the shell (if there is any) as its extent is infinite, and any pull to one side is cancelled out by the pull to the other side.

We can write an expression for the mass (= density × volume): From our studies of Newton's Laws concerning gravitational fields, we know that the gravitational potential energy, Eg, is given by: Strictly speaking we should use the minus sign, but we will keep Eg positive in this argument.

We substitute for M: Notice that the r term downstairs has cancelled with the r3 term upstairs to leave r2 upstairs.

The shell is expanding.  The total kinetic energy is given by the usual: From our studies of the Hubble constant, we know that:

v = H0r

So we can substitute for v and write: The definition of potential energy is the work done to move the object to infinity.  At this point the potential energy is zero.  By the Law of Conservation of Energy, the kinetic energy is also zero, so we can equate the two equations for potential and kinetic energy to give: We can see that the r2 and the m terms cancel: The density term r is the critical density, which we will call rc.  Now we can rearrange to make rc the subject to give us: The value of H0 is in the range of  1.6 × 10-18 s-1 to 3.2 × 10-18 s-1.

 Use your answer to question 6 to work out how many hydrogen atoms there are in every cubic metre. Mass of Hydrogen atom = 1.67 × 10-27 kg.