Topic 6 Cosmology

In the exam you are expected to know about:

Doppler effect

for v << c applied to optical and radio frequencies

Calculations on binary stars viewed in the plane of orbit

Hubble’s law 

Red shift

v = Hd 

Simple interpretation as expansion of universe; estimation of age of universe, assuming H is constant

Qualitative treatment of Big Bang theory

Quasars as the most distant measurable objects

Discovery as bright radio sources

Controversy concerning distance and power – use of inverse square law

Quasars show large optical red shifts; estimation of distance.

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); c - speed of light (m/s)]

For wavelength the equation is similar:

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.

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

Worked example

Question 1  A star is moving away from the Earth at 5000 km/s.  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?  ANSWER

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:

We can see that the pattern is the same, but the colours are different.

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.  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:

If the stars are of different masses we see orbits 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.

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.

ANSWER

The same principle as in the question is used to determine the rotation of a star.

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.  The graph shows the idea:

The equation from this graph is:

• 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 Ho 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.

Question 3  A distant galaxy has a red-shift of 15 %.

(a) What is its speed of recession?

(b) If Ho has a value of 100 km s^-1 Mpc^-1, what is its distance?

ANSWER

The Age of the Universe

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 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.  By converting Mega parsecs 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 Ho has a value between 40 and 100 km s^-1 Mpc^-1

Question 4  What is the age of the universe using these extremes of value?  1pc = 3.086 x 10¹⁶ m  ANSWER

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)?

Quasars

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.  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.

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