I was asked to give a few lectures to some very clever secondary students. These lectures were to be held under the auspices of the Sutton Trust. The trust is doing a tremendous amount of work in bringing talented students from a wide range of backgrounds to have the “Oxford Experience”. They are exposed to lectures, tutorials and other activities that are typical of undergraduates here.
The idea behind my lectures was to have the students find out for themselves how to measure the size of the Universe. A tall order and not completely fair. They would have to rely on some sophisticated observations and could not practically do all the measuring themselves. The ideas, however should be simple to understand and with good data plots they should be able to work things out for themselves. It was a condensed version of these lectures that I gave to Jem over our weekly breakfast/lunch which takes place at our local greasy spoon. These lunches have become an important part of our weeks, where science meets art over large amounts of fat, starch and tea.
We want to measure the size of the Universe. What better place to start then our neighbourhood. Lets not just look at the streets or towns we live in but be more ambitious and look at the closest object in space we can discern: the Moon. The Moon is a round blob on the celestial sphere, an orb which mixes up distant and close objects onto a flat, spherical screen. We can only measure angles on this screen, we can’t measure the distance to any object directly by taking a ruler and stretching it all the way up. Our ability to see depth close by with stereoscopic vision can be developed to measure great distances. To remind you of what this is, stick your finger out in front of your eyes. Stretch your arm out and hold your finger upright. With both eyes open, you will see your finger in the foreground and the scenery in the background. Close one eye. Your finger will blend in with the background and you will lose your sense of depth. Now open that eye and close the other one. Your finger should shift slightly relative to the every thing else in the distance. As you can see the position of your finger relative to the background depends on which eye use. But we can do more than this. Move your finger closer to your face so that it is only a few inches away. Play the same game and you will find that your finger jumps much more across the background than before. The closer your finger is to your face, the more it moves when we switch our view point from one eye to the other.
We can use this idea do measure the distance to the Moon. By staring at the sky from one location on the surface of the Earth, we can chart out where the Moon lies relative to all the stars that are behind it on a given night. We can have someone else do that from another location some distance away from us. In these way we, and our collaborators, play the role of “eyes” on the Earth. The difference in the relative position of the Moon between the two viewpoints on the Earth can be used to gain a much larger sense of the depth than just opening and closing each eye. And clearly, the more distant the two viewpoints, the further we can measure distances. We can start to unpeel the night sky and discern the various layers which at first seemed superimposed and indistinguishable.
A good start is to map and measure the distances to the various planets in the Milky Way. For this we need two ingredients: a model for how the planets are laid in space and a method for measuring the distance to at least one of them. Copernicus, Kepler and Galileo gave us a brilliant working model which we will use here. In particular, Johannes Kepler gave us a rule of thumb which is all we need for setting up a simple model of the Solar System. This rule states that the speed at which planets revolve around the Sun is intimately tied to their distances to it. To be more specific, measure the number of days that that two planets, the Earth and Mercury for example, take to go around the Sun. Then cube each one of these numbers and divide one by the other. The resulting ratio will be exactly the same as the ratio of the squares of the distances of each planet to the Sun. In other words, if we measure the length of the year for each planet, and the distance of one of them to the Sun we can find the distance of the other. This simple relationship gives us a simple way of constructing a scale model of the Solar System. We now need the second ingredient to measure its true size.
What we have learnt about stereoscopic distance can be put to good use. Every once in a while, the planet Venus crosses in front of the Sun. It is sufficiently close to the Sun that it doesn’t eclipse it but it does appear as a small dot on the blazing yellow disk. Over a few hours it moves across, transiting from one edge to the other. Venus lies between us and the Sun and the distance is sufficiently close to us, that the tracks it makes will differ depending on the where we observe it from. If we choose two sufficiently distant view points, we can use the two tracks of Venus to measure our distance to it. By measuring our distance to it, and our knowledge of its orbital period, we can find the distance of the Earth to the Sun. With that knowledge we unlock the size of the Solar System.
The Solar System is immense but it is insignificant in the grand scheme of Universe. We need to be able to look further out. In getting a fix on the distance of the Earth to the Sun we can now, again, use stereoscopic vision to measure distances to some stars. Over a period of six months the Earth moves from one side of the Sun to the other. This is a phenomenal distance, about 300 million kilometres. And it gives us a fantastic baseline with which to see the depth of the cosmos. By observing how nearby stars are positioned relative to the distant backdrop of stars, we can measure our distances to them. This technique of using stereoscopic vision on these large scales is known as parallax and, with modern satellite observatories, can measure distances out to hundreds of light years, greatly transcending what we can do from the Earth. Yet this is still not far enough as we shall see.
To see further, we need to use a property of light that we are all familiar with. Suppose we have a 40 Watt light bulb shining at us from a few feet away. It will look bright and illuminate everything around us. If someone moves that light bulb away from us so that it is twenty or thirty feet away, it will look dimmer. And our surrounding won’t be as lit up as before. The further away it is, the dimmer it will look. If we call the amount of light that reaches us from the bulb, the brightness, then we easily see that the further away the bulb is, the smaller its brightness. There is, in fact, a very simple relationship between the brightness and the distance of the bulb. The brightness decreases with the square of the distance. In other words, if we move the bulb away from us so that it is twice the distance that it was before, then its brightness will fall by a factor of four. Thus, by measuring the brightness of the bulb, and knowing the amount of light it pumps out, also known as its luminosity, we can measure the distance to it.
How can we apply this idea to the cosmos? To start off with, we have stars which are beaming light out in all directions. The further away a star is, the dimmer it will be. So if we know the luminosity of a star and measure its brightness, we will know how far away it is. Of course, a star is not a light bulb. We don’t have a label telling us its luminosity. We could naively assume that all stars are the same. That is, exactly the same. Then they would all emit the same amount of light. The Sun could be used to measure the luminosity of our prototypical star. This measurement of the luminosity could then be used to find the distance to far away stars. Unfortunately it doesn’t work that way. Stars come in various sizes and luminosities and there has to be other ways of finding the luminosity of each individual star without assuming that it is the same as the Sun’s. In other words, one wants to find stars, or any luminous object for that matter, of which we can independently measure their luminosities and brightness’. Objects which have this convenient property are known as standard candles.
It is possible to use stars as standard candles in different ways. The simplest, but by no means most accurate way is to measure the colour of the star. Light can be split up into its various colours. That is what you do when you put a prism in front. If the light has more red, then the red part of the rainbow that emerges from the prism will shine more brightly. The same can be said if the light is bluer or more yellow. Now the dominant colour is related to the temperature of the star. The bluer the light, the hotter the star is. You may be familiar with this when you see piece of metal glowing in the heat. It starts off red and as it becomes hotter, it becomes bluer. By measuring the colour of a star we should be able infer its temperature and you wouldn’t be surprised to know that the hotter it is, the more luminous it should be. Yet this is not enough. The luminosity of the star also depends on its size. The bigger it is, the more surface it has emitting light and therefore the more luminous it is. So we can have big, cold stars emitting more light than small, hot ones. To resolve this we need to a way of measuring the size of the star. We can do this by looking at how fast the different bits of the star are jiggling about. By observing the way the light is distributed amongst the colours and how sharply focused its spectrum is in certain lines, we can infer how much motion there is in the star. Very small stars will have a lot of jitter and the spectral lines will be more smeared than larger, more placid stars. The amount of smearing in the spectra tells us how big the star is and, along with its colour, we can infer its luminosity. Measuring its brightness we can find the distance. This technique is known as spectroscopic parallax and can take us out to a few thousand light years, to the centre of our galaxies and out towards the edges.
Some stars have very peculiar properties. They swell and shrink periodically over time. When they swell they emit more light and when the shrink, their luminosity decreases. Some of these stars, known as Cepheids, do this over a period of days and they possess an intriguing property: the time it takes for them to go through a cycle, i.e. their period, is intimately tied to their luminosity. The longer the period, the more luminous they are. This gives us a very easy way to measure the luminosity of distant Cepheids. We measure their period and compare it to nearby ones of which we have been able to measure their luminosities. And once again, knowing their luminosities allows us to find their distances. Cepheid stars are very bright, about 100 to 1000 times brighter than the Sun. This allows us to use them to measure great distances, out to tens of millions of light years. Indeed Cepheid stars played a very important role in the beginning of the twentieth century. Edwin Hubble, the American Astronomer used them to find the distance to the Andromeda galaxy and prove that it wasn’t merely a cluster of stars within the confines of our own galaxy. In doing so he opened up the cosmos, proving it was seething mass of galaxies at tremendous distances from each other.
The furthest distances can be measured by using astounding explosive events known as supernovae. A supernova is the end product of stellar evolution in which the remains of a star are so dense that they collapse under their own gravitational pull. The result is an explosion which, for a brief period of time, emits as much light as a whole galaxy. This means it can be 10000000000 times as luminous as the Sun. Such luminous events can be seen at great distances. Some types of supernovae, known as Ia (pronounced “one ay”) seem to all be very similar. Which means that if we know what they look like close by, we should also know what they look like far away. Once again we have an object, or event, of which we know its luminosity and we can measure its brightness to estimate its distance. Supernovae can be used to measure distances out to tens of millions of light years.
We seem to have got quite far. So how big is the Universe? That may not be a fair question. The Universe may in fact be infinite in size. But that doesn’t mean we are able to see arbitrarily far. One of the facts unearthed by Edwin Hubble in the early twenties is that the Universe is expanding. This manifests itself in a tight relationship between the speed at which galaxies move away from us and their distances: the further away they are, the quicker they are moving away from us. This relationship is encapsulated in what has become known as Hubble’s law which says that the velocity is proportional to the distance where the proportionality constant is the Hubble constant. This proportionality has an interesting consequence. Suppose we pick any galaxy at a distance, d, away from us. Its velocity, moving away from us, is v and it should be equal to H times d, where H is the Hubble constant. It also means that at an early time, we would have been closer to that galaxy. And if we wind back the clock enough, the galaxy would have been right on top of ours. In other words, at some moment in the past, our galaxy and this other galaxy which is far away today, would have been completely coincident in space. We can work out how far back in time by dividing the distance they have to travel, d, by the speed at which they have been travelling. If we do that we find that this time is given by the inverse of the Hubble constant. And this will be true of any two galaxies we pick. In other words, every single galaxy would have started off at the same point on space at this special moment in time. This time is the beginning, also called the Big Bang and it was about 15 billion years in the past. It is the age of the Universe.
The age of the Universe gives us a fundamental limitation on how far we can see. Nothing can travel faster than the speed of light. So the distance that a light beam travels from the Big Bang until now is the furthest we will ever be able to see. This is, naturally 15 billion light years. So even if the Universe is infinitely large, we will never be able to see a portion of it which is 15 billion light years in scope. That is how big the Universe will seem to us.