Alan C. Pickwick
"EAAE Summerschools" Working Group
Manchester Grammar School (UK)
Abstract
It is almost a throw-away line that the Universe is expanding, but how do we know this for sure? In this workshop we will discover how to measure the distance to nearby and distant astronomical objects. We will find how to measure their velocities and we will see how their velocities and distances are linked together to predict the fate of the Universe in the far-distant future.
One hundred years ago we had no idea about how big the Universe actually is. We thought that the galaxies were 'little island universes' in the Milky Way. We thought that the stars were dotted through space and that the whole system was 'just hanging there'!
In this workshop you will have a chance to review the key measurements that totally changed our view of the Universe! By the end of the workshop you should be able to:
State the definitions of magnitude, the parsec and the light-year.
Calculate the distance of nearby stars given their annual parallax.
Calculate the relative distances of stars using the 'inverse square law'.
Explain how Henrietta Leavitt discovered that Cepheid variable stars can be used to measure the distances of stars and nearby galaxies.
Explain the importance of Type Ia Supernovae for measuring the distances of very distant galaxies.
Measure the velocity of a galaxy from its spectrum.
Understand that for distant galaxies, Edwin Hubble discovered the link between velocity and distance.
Explain how better observational techniques caused the value of the Hubble Constant to fall from 530 to 72 km/s Mpc in 80 years!
Explain how the value of the Hubble Constant at great distances could predict the fate of the Universe.
Distance Measurements
a) Definitions
The light-year
This is the distance light travels in a year. Using 3.0 x 108 m/s for the velocity of light, this is a distance of 9.46 x 1015 m (9 460 000 000 000 000 m).
Distance = velocity x time = 3.0 x 108 x Number of seconds in one year.
Use your calculator to check this here:
The astronomical unit
This is the mean radius of the Earth's orbit around the Sun.
It is equal to 1.496 x 108 km (often taken as 150 000 000 km).
The parsec
As the Earth travels round the Sun, the nearby stars appear to move relative to the distant 'fixed stars'. This motion is called the 'annual parallax'.
If a star appears to move by one second of arc ( 1/3600 of a degree) when the Earth moves by one astronomical unit the object is defined as being 1 parsec away.
If the star appears to move by 0.1 seconds of arc, the object is 10 parsecs away.
The name of this unit comes from the phrase 'second of parallax' being reversed to read 'parallax of second'.
One parsec is equal to 3.26 light-years.
b) Distance Measurement Methods
i ) Trigonometric Parallax
This method is useful for nearby stars, those out to a distance of about 100 parsec (about 326 light-years). In the diagram the Earth is shown at two measurement positions, six months apart. This arrangement is used to obtain maximum accuracy.
The nearby star would be many times more distant than the drawing implies. The angles a and b are measured and their average is angle p. The base of the triangle containing angle p is the Astronomical Unit (AU).
The great advantage of quoting distances in parsecs is that they are not affected by improvements in the accuracy of the AU. The AU was defined in 1963 to be 1.496 x 1011 m. However, more recent measurements give 1.49597892 x 1011 m (ą about 3 km !!).
Using trigonometry in figure 1 we see that: tan p = 1 AU / d
This formula may then be used to calculate the distance of 1 pc:
d = 1 AU / tan (1 second of arc)
d = 1.496 x 1011 / tan (1 / 3600)
d = 3.09 x 1016 m = 3.26 light-years.
Note that the nearest star, Proxima Centauri, is 4.22 light-years away. This means that the parallaxes of even the nearest stars are very small indeed. Therefore we can only use this method for nearby stars.
A Practical Parallax Exercise
Take a few sheets of A4 paper. Place two paper clips on the top edge of the papers. Hold the bottom edge of the paper against the bridge of your nose and adjust the height so that you can see both clips. Close your right eye and turn your head until the left hand paper clip lines up with an object about 2 metres away. Now open your right eye and close your left eye. Slide the right hand paper clip until it also lines up with the object that you have chosen. Check the adjustment by closing each eye in turn to make sure that both paper clips line with the chosen object.
Measure the distance between the paper clips and ask a friend to measure the distance between the centres of your eyes. The length of the A4 paper is 0.297 m.
This apparent movement of the object is caused by the different viewing positions of your left and right eyes. You can use 'similar triangles' to calculate the distance to the object your were looking at. This is using parallax to measure distance.
CQ / PR = Distance of Chosen Object / RQ
Distance of Chosen Object = 0.297 x CQ / PR = .....
Calculation of the distance of the star Vega
The annual parallax of Vega is 0.12892 seconds of arc.
Distance of Vega in parsecs = 1 / 0.12892 = ........... parsecs.
Using 1 parsec = 3.26 light-years.
Distance to Vega in light-years = ................ light-years.
The ESA Hipparcos Satellite has measured parallaxes of stars as small as 1/1000 second of arc. This gives the greatest measurable distance as 3250 light years. However, these stars are so close that they are 'in our back yard'!!
ii) Inverse Square Law
Now consider the effect of the Inverse Square Law. If one object is twice as far away as another, then its brightness will be one quarter ( 1 / 22 ) of the nearer object. To help to check that you have understood this, complete the table:
****** Demonstration using point source and squared paper at various distances.
****** Demonstration with string and cards at various distances.
iii) Magnitudes
Apparent Magnitude
Ever since the time of the ancient Greek astronomers, Hipparchus and Ptolemy, the visible stars have been divided into six magnitude groups as observed from the Earth. The brightest are called first magnitude and the ones that are just visible to the unaided eye are called sixth magnitude. The apparent magnitude depends on how brightly a star is shining and on how distant it is from the Earth; in other words, how bright it appears to the observer. In 1827 John Herschel deduced from his measurements that a five magnitude difference corresponded to a factor of 100 in the intensity (power per unit area) detected by the observer. In 1856 Norman Pogson proposed that all the magnitude steps should be the same ratio, one to another. This leads to the ratio being:
(100)1/5 : 1 = 2.51 : 1
Check this yourself now by multiplying 2.512 by itself five times.
Absolute Magnitude
To allow stars at different distances to be compared, it is convenient to adjust their Apparent Magnitudes to what they would have been if the stars were placed at the Standard Distance of 10 parsecs.
In the following, upper case letters refer to the star at 10 parsecs and lower case letters to the star at its true distance. The theory leads to the useful formula:
m - M = 5 log (d / 10)
where :
m is the apparent magnitude,
M is the absolute magnitude and
d is the distance in parsecs.
iv) Cepheid Variable Stars
What is a Cepheid?
Consider a star a little larger than the Sun. As it grows old it will become a Red Giant Star. During this process it passes through a phase where it 'breathes-in-and-out'. Heat is trying to escape from the core but it builds up inside the star. This pushes up its outer layers and the star increases in radius by a few percent. Now it has a larger surface area and so its looks brighter. With a larger surface area, the build-up of heat can radiate away into space. This allows the surface to return to normal and so the star's brightness also returns to normal. However the heat starts to build up again and so the process repeats. Each cycle takes some days to repeat.
Why are Cepheids important?
In 1912 Henrietta Swan Leavitt, working at Harvard, found that Cepheid variable stars could be used as 'standard candles'.
She worked on photographs of the Small Magellanic Cloud and found that:
The intrinsic brightness of a Cepheid Variable Star is related to the period of its light output variation.
The intrinsically brightest Cepheids vary with a period of perhaps 50 days. The intrinsically faintest Cepheids vary with a period of perhaps 2 days.
So if you find a Cepheid with a period of say 20 days, it will have the same intrinsic brightness as all the other Cepheids that have a period of 20 days.
However, if one looks fainter than another, it is because it is more distant.
In summary:
We can use a direct method to measure the distance to a few nearby Cepheids.
We can then use the Inverse Square Law to calculate the distance to Cepheids in quite distant Galaxies.
Recent Cepheid Results
To find the distance to the galaxies is not easy because they are too far away to show any parallax. The most methods involve 'standard candles'. We measure the distance to a nearby star and also measure its magnitude. If we then observe a similar star in a distant galaxy we can compare the magnitudes of the two stars and so find the distance to the galaxy.
These images show some of the most distant Cepheids observed. Dr Wendy Freedman of Carnegie Observatories has headed a team that has collected data from the Hubble Space Telescope. They observed 800 Cepheids, various Type Ia supernovae (see later) and two other distance indicators. They reported very good agreement between these methods for measuring distances to 18 distant galaxies. M100 is 56 million light years distant.
Exercise to Verify the Period-Luminosity Law
These four light curves are for Cepheids in M31, the Andromeda galaxy.
From the curves, read off:
The average apparent magnitude of the star and
The period of the variability.
Complete the table and plot the results on the graph.
Henrietta Swan Leavitt plotted similar graphs and realised that the mean apparent magnitudes of the stars were proportional to the logarithms of their periods. This was a major step forward. A year later, in 1913, Harlow Shapley was able to measure directly the distance to a few nearby Cepheids and so was able to link their absolute magnitudes to their periods. The 'standard candles' had been found!
This led to the final proof that the galaxies ('island universes') were outside our own Milky Way and that the scale of the Universe was truly massive.
Distance to Andromeda
The graph shows modern results for Cepheid variable stars. We shall use this data to estimate the distance to the Andromeda galaxy. Using the period of Star C from the previous exercise, look up its absolute magnitude on the graph. Be careful with the signs of the magnitudes!
M = Absolute magnitude of star C = .............
m = Apparent magnitude of star C = .............
Use the formula or the 'Nomogram Calculator':
m - M = 5 log (d / 10)
where
m is the apparent magnitude,
M is the absolute magnitude and
d is the distance in parsecs.
Hence d in parsec = ..................
Using 3.26 light-years equals one parsec, d in light-years = ..................
Although the Cepheids are super-giant stars, even the Hubble Space Telescope cannot observe them in the most distant galaxies. The next method solves this problem.
v) Type Ia Supernovae
Mechanism of the Explosion
Stars are formed when a cloud of gas and dust collapses under the attraction of gravity. Very often, several stars will be created from one cloud. Often these stars will attract each other and form pairs. Each will go through life with a companion! These are called binary star systems.
As the two stars of a binary system grow old, one of them will become a Red Giant star. Millions of years later the giant ball of gas will have moved away into space. All that will remain is the hot core of the star. This is called a White Dwarf star.
Now we have an interesting situation. Eventually, the second star will also become a Red Giant. It will grow in size and some of its hydrogen gas will be captured by its partner, the White Dwarf star. Eventually there will be enough hydrogen on the surface of the White Dwarf star for a massive 'Hydrogen Bomb' explosion. This is called a Type Ia supernova.
In detail, a Type Ia event occurs when a White Dwarf, composed mainly of carbon and oxygen, is one partner in a binary star system. As the other star goes into its Red Giant phase, gas is drawn down onto the surface of the White Dwarf. Some of this material burns to helium and then carbon and oxygen. When the White Dwarf passes the Chandrasekhar limit of 1.4 solar masses, the carbon core cannot resist the gravitational forces. The temperature rises, the core ignites and the star explodes. The resulting explosion consumes about one solar mass of carbon and oxygen. The product is nickel 56, which undergoes radioactive decay to cobalt 56 and finally to iron 56. This decay chain explains the exponential decay (60 day half-life) of the brightness of the Type Ia supernovae. This half-life allows Type Ia supernovae to be identified unambiguously.
Why are Type Ia Supernovae important?
We believe that all Type Ia Supernovae reach the same the peak intrinsic brightness. So we have another 'standard candle'. These explosions are so bright that we can use them to measure the distances of very distant galaxies.
Example Calculation
Observations show that a very distant supernova has a Peak Apparent Magnitude (m) = 20.5
From studies of nearby supernovae we know that the Peak Absolute Magnitude (M) = -19.5 (This is very very bright.)
Use the formula or the 'Nomogram Calculator':
m - M = 5 log (d / 10)
where
m is the apparent magnitude,
M is the absolute magnitude and
d is the distance in parsecs.
Hence the distance (d) =
Velocity Measurements
a) What is the Doppler Effect?
Imagine a star that is moving away from us. Atoms in it are emitting light waves at particular (rest) wavelengths. As the waves are being emitted, the atoms are moving away from us. This 'stretches out' the waves. It makes their wavelengths longer. It makes the light more red than it would be at rest.
Doppler Effect Formula
Change in wavelength / Original wavelength
= Relative velocity / Velocity of light
Doppler Calculation
A police radar speed trap emits microwaves of wavelength 0.030 m. A car moves away from it at 30 m/s. What is the wavelength of the microwaves as detected in the car.
Change in wavelength / 0.030 = 30 / 3 x 108
Wavelength as detected in the car = Change in wavelength + Original wavelength = .................
b) What is Redshift?
Be very careful with the definition of Redshift!
The Redshift that we see in the distant galaxies seems like the Doppler Shift:
Change in wavelength / Original wavelength
More recently however, astrophysicists have made a clear distinction between Redshift and Doppler shift. Redshift is caused by the expansion of the fabric of the space - the expansion of the 'graph paper' on which the Universe is drawn. Doppler Shift is caused by the relative motion of the source and the observer. Fortunately the simple mathematics are identical, but educators should be aware of this distinction.
Making Models of the Expansion of the Universe
Incorrect - Galaxies drawn on balloon.
Correct - Cotton wool on balloon. Clothes pegs on elastic. Paper clips on elastic bands.
3) Velocity and Distance
a) Hubble's Law Data Exercise
The light we receive from galaxies is very faint. With the naked eye we can only see one galaxy (Andromeda) in the Northern sky. To make a photograph of the spectrum of a galaxy requires a large telescope to collect the light. By the 1920s, spectra similar to the ones in this exercise had been collected. Work then progressed on finding the distance to the galaxies. Various methods were used and by 1929, Edwin Hubble had measured the distances to about twenty galaxies. With this information he was able to show a definite link between the distance to a galaxy and the velocity it is moving away from us.
You are provided with five spectra adapted from Hale Observatory originals. A spectral line of Calcium is marked. If the galaxy is stationary relative to us, the Calcium K-line is observed at its rest wavelength. If the galaxy is receding, the wavelength becomes longer - the Redshift.
In the table below, four of the galaxies have their distances given. Use the 'plate scale' on the diagram to find the recession velocities of the galaxies. You will have to judge the position of the K-line in the Hydra spectrum for yourself.
Fill in the table and plot the graph of recession velocity against distance.
Use the graph to find the distance to the galaxy in Hydra.
Find the gradient of the graph. This is the value of the Hubble Constant.
Expected value of the gradient is 72 km/s Mpc.
Note that this exercise uses modern distance values. Hubble would have originally obtained a value about 7 times greater (see later). Also, modern spectra are very, very much clearer than these! It is instructive to consider the difficulties that early workers in this field overcame.
b) The Recession of the Galaxies and the Age of the Universe
The spectral lines of all distant galaxies show red shift. The fabric of space-time is expanding, creating this red shift. Here is an idealised graph, extrapolated out to the velocity of light.
With a high degree of certainty, this graph can be plotted to about 10% of the distance to the edge of the observable Universe. Many galaxies have had their redshift and distance measured and so the lower part of the line is well defined.
Beyond that, there have only been about 100 very distant Type Ia supernovae observed. The line clearly extends approximately as shown but it will be several decades before its detailed shape is known.
The graph is extrapolated all the way up to the velocity of light (3 x 108 m/s).
Note that the velocity of light defines the most distant objects that we could ever see.
From the graph, read off the 'distance to the edge of the Universe'.
This distance in light-years gives the time directly.
The Greatest Distance Measured to Date
In March 2004, ESO announced that a newly discovered galaxy named Abell 1835 IR1916, has a redshift of 10. Using Hubble's Law, the galaxy is found to be 13 230 million light-years away. We see the galaxy as it was when the Universe was only 470 million years old. That is only 3 percent of its current age. These results were determined by spectral analysis of its light, not by observing a supernova in it.
c) Why did the value of the Hubble Constant fall so much over the last 80 years?
In 1929 Hubble measured the constant to be 530 km/s Mpc. At that tine he only had the velocities and distances of about 30 galaxies.
His velocity data was good but his distance data was in error. It turns out that all the galaxies are much further away than he thought.
Some of his distances were based on identifying bright stars in galaxies. It turns out that these 'stars' were compact ionized hydrogen gas clouds. Also, the brightest stars in galaxies are 25 times brighter than he originally thought.
Some of the Cepheid variable stars that he measured were of a slightly different class (W Virginis stars) and were about 1.5 magnitudes fainter than the classical Cepheids.
Some of the Cepheid variable stars that he measured were of a significantly different class (RR Lyrae stars) and were about 2 magnitudes fainter than the classical Cepheids.
All Cepheids are actually 1.8 magnitudes brighter than originally thought.
All these improvements in our knowledge of the Universe have allowed us to make the current estimate of the Hubble Constant as 72 km/s Mpc. This gives an age of the Universe of 13.7 billion years.
d) Fate of the Universe
When the Hubble graph is extended out to the most distant galaxies the data are so weak that it is not certain that the straight-line law holds. If the shape of the line/curve could be found with certainty, it would indicate the fate of the Universe. If the most distant galaxies are found to be moving away rapidly, then the universe will expand forever. If they are moving too slowly then gravity will pull them back to the Big Crunch. If the velocities are just right for the density of matter, then the Universe will come to rest at infinite size.
As the data improves it will be possible to plot a more precise Hubble graph. From its shape it will be possible to link the expansion of the galaxies to the amount of matter in the Universe. At present the amount of mass estimated from visible matter falls short by between 10 and 100 times that needed to close the Universe and so ensure a Big Crunch. Searches are continuing for black holes, brown dwarfs and molecular hydrogen clouds in particular, all of which are difficult to detect and are thought to hold a great deal of matter. Also 'dark matter' in the form of WIMPS (weakly interacting massive particles) is thought to be a significant factor. Finally, 'dark energy' is now thought to affect the expansion of the Universe at the greatest distances.
Books
Edwin Hubble the discoverer of the Big Bang Universe. Alexander S Sharov. Cambridge University Press 1993. ISBN 0-521-41617-5. The story of the discovery of the expanding universe and an excellent biography of Hubble. It ends with an excellent summary of the problems that face cosmologists.
Measuring the Universe. Stephen Webb. Springer-Praxis 1999. 1-85233-106-2. An excellent survey of the way in which the 'cosmological distance ladder' has been constructed.
Equipment
Project Star Spectroscope (PS-14/Plastic) $24 + p&p.
Sunspotter - The Safer Solar Telescope (LT-70) $335 + p&p.
