Figure 13.5 The Doppler effect in water waves. (A) A stationary bobber in a pond radiates concentric waves. (B) A bobber in motion chases its waves. An observer on the right will see short, frequent waves, on the left longer waves.
The way that Tully–Fisher use this information is to look at a well-known spectrum line that is found in the light from a distant galaxy. Ideally you would like to look at one edge of the galaxy and see blue-shift as it spins towards you and on the other edge red-shift as it spins away from you. But distant galaxies can appear point-like, in which case the shifts merge together; instead of separate blue- and red-shifts you just see a broadening of the line. But that still works. The broader the line, the faster the rotation, the brighter the galaxy. The Tully– Fisher method has been used to measure distances to galaxies 200 Mps (6 × 1024 m) away.
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There is one last candle technique we will describe: supernovae. Remember that the way a star works is that is balances the outward radiation pressure against the inward gravitational pressure. Near the end of its life it may just fizzle and collapse inward. However, in some cases, as it cools and collapses there may be enough pressure such that it can burn some of the heavier elements in it, for example carbon. “Burn” is really an understatement: it ignites—cataclysmically—into a massive explosion. The explosion may only take minutes, with the results fading in days (like the Crab nebula), but during that time the star may radiate as much energy as our Sun does in its whole lifetime. So much energy means that a supernova can briefly outshine its host galaxy!
There is one special type of supernova, known as the Type Ia supernova, that is of particular interest. The star must be of just the right size for this type of supernova to happen, which means it has a unique absolute magnitude. All Type Ia supernovae give up about the same amount of light and energy. They are also very bright, which makes them ideal candles. The only problem is that they are brief and uncommon. Fortunately there are a lot of candidates out there. In fact these supernovae are so bright that we can see and measure them out to about 1 Gpc (gigaparsec; 3 × 1025 m). With these distance scales we can map a vast section of space, well beyond the local group of galaxies.
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Space is filled with galactic groups and galactic clusters. These are clumps of galaxies that are gravitationally bound. Groups are generally smaller than clusters, but at this time there is no hard definition. Galactic groups generally have 50 or fewer galaxies and are only 1 to 2 Mpc across. Nearby Earth is the Canes Venatici I group, about 4 Mpc away, as well as several different Leo groups. The Leo groups are located beyond the constellation of Leo, but all at different distances.
Larger than the groups are clusters, like the Fornax cluster with almost 60 galaxies, located 19 Mpc (6 × 1023 m) away and the Eridanus cluster with more than 70 galaxies and found 23 Mpc (7 × 1023 m) away. But what dominates our part of the universe is the Virgo cluster. The Virgo cluster contains over 13,000 galaxies. Since it is about 5 Mps across, that means galaxies are spaced about every 100,000 pc, which we can compare to the diameter of our own galaxy—about 30,000 pc across—just to get an idea of the density of galaxies in a group or cluster.
Moving outward an order of magnitude we see that the local group and the Virgo cluster are part of a bigger collection of objects, which is called the Virgo supercluster, named after the dominant cluster in the region. This supercluster contains at least 100 galaxy groups or clusters and is about 33 Mpc (1024 m) across. We are now approaching 1% of the diameter of the universe (see Figure 13.6).
Figure 13.6 Comparison of the size of galaxies, group, superclusters and the universe.
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To measure the remaining 99% of the universe we must return to the work of Edwin Hubble. As we saw a few pages ago, in the 1920s Hubble used Cepheid variables to measure the distance to the Andromeda and Triangulum galaxies. But he did not stop there. He went on to measure the distance to a total of 46 galaxies and discovered a relationship between the distance to a galaxy and its red-shift. In 1929, only four years after showing that Andromeda and Triangulum were beyond the Milky Way, he published the correlation between red-shift and distance. We now call this Hubble’s law. The farther away a galaxy is, the faster it is moving away from us. A few near by galaxies are blue-shifted; that is, they are moving towards us. Most famously, Andromeda is on a collision course with us and is expected to collide in four billion years’ time, but this is just local motion. The trend for the vast majority of galaxies is that they are moving outward. This is solid evidence for the Big Bang and the expansion of the universe.
Einstein’s theory of general relativity, the most fundamental theory we have of gravity and space, makes a lot more sense in an expanding universe than in a static or steady-state universe. But in the early decades of the twentieth century the overwhelming opinion of astronomers was that space is the same as it has always been and will be. This prejudice was so strong that even Einstein accepted it. Einstein’s original formulation of general relativity predicted an expanding universe, but he added a term to his equations to cancel out the expansion. Later he would call this his “greatest blunder.” In fact in January 1931, he visited Hubble at the Mount Wilson observatory to thank him for finding the expanding universe and setting cosmology in the right direction.
The fact that distant galaxies are moving away from us faster than nearby galaxies, and so have a greater red-shift, can be used as a new technique for measuring vast distances. Hubble’s law has now been verified over thousands of galaxies, the distances of which we can independently measure with techniques such as Cepheid variables, Tully–Fisher, supernovae, as well as other methods. We have arrived at our last rung of the cosmic distance ladder. Hubble’s law states:
or the velocity (v) of a galaxy is Hubble’s constant (H0) times the distance to that galaxy(D). So, for example, if we observe a distant galaxy with a velocity of 7000 km/s (2% of the speed of light) then that galaxy is 100 Mpc (324 m) away. This is a fantastic tool. Now all an astronomer needs to do to measure the distance to a galaxy is to identify a line in the spectrum from that galaxy, perhaps a transition in hydrogen, and see how it has shifted compared to a non-moving source due to the Doppler effect. From that you know how far away the galaxy is. Now we can build a three-dimensional map of the universe with millions of galactic clusters, out to billions of parsecs.
What we see is called the large-scale structure of the universe, and it is beautiful and astonishing. When we plot the position of galaxies in three dimensions, not only do we find galaxies in groups, clusters and superclusters, but these collections of galaxies line up to form wispy filaments that reach across deep space (see Figure 13.7). The area of the universe with stuff in it seems to be gathered into sheets and lines. The areas with nothing in them are bubbles of inky black void. This structure has been described as foam, ironically the same word that has been used to describe the world at the Planck length. But to me what makes these structures so intriguing is that maps of the universe have names sprinkled across them. Perhaps the most famous is the Great Wall, a structure 60 Mpc away, 150 Mpc long, 90 Mpc wide, but only 5 Mpc thick.
Figure 13.7 The large-scale structure of the universe. Each point is a galaxy. Galaxies tend to bunch together into walls, and leave bubbles or voids. Right and left on this diagram are looking perpendicular to the disk of the Milky Way. We cannot see very far in the up and down direction because of the dust in gasses in our own galaxy. This data is from the Sloan Digital Sky Survey. Courtesy of Michael Blanton and the Sloan Digital Sky Survey (SDSS) Collaboration, http://www.sdss.org.
Areas without galaxies are also named, for example the Boötes void and the Capricornus void. It is odd to think of us humans, not even 2 m tall, naming regions of such magnitude. There is a bubble or void beyond the constellation Eridanus, which would take light about a billion years to cross. Originally it was named the WMAP cold spot, after the satellite that spotted it. This is like Beagle Bay in Australia being nam
ed after the ship that dropped anchor there and charted it. The Sloan Great Wall is named after the Sloan Digital Sky Survey, the project that charted it. This wall, at 420 Mpc (1.3 × 1025 m) long, is the largest known structure in the universe, although some recently spotted large quasar groups may challenge this title. Could Alfred P. Sloan have imagined that his name would be attached to such a vast region when he endowed the Sloan Foundation?
We are not quite at the end of the road. We are talking about structures that are as much as a billion light years across. But as we approach the edge of the universe, things get odd. First off, if you observe something that is a billion light years away you are also looking back in time a billion years because that is how long it took that light to reach us. The most distant thing we can see is the cosmic microwave background. These photons have been traveling since shortly after the Big Bang. Cosmologists tell us that for the first 380,000 years after the Big Bang the universe was a plasma and too hot for photons to penetrate. But since then the universe has cooled off and these microwave photons have been traveling towards us. This also forms the edge of what is referred to as the observable universe. Light from earlier, and so farther away, cannot be seen.
So how big is the observable universe? This is a complex problem because space and time get mixed together. However, let us start with something simple, like the Hubble constant, H 0 = 74 (km/s)/Mpc. It is a curious constant that has distance in both the numerator (km) and the denominator (Mpc), which we can divide to get H 0 = 74 (km/s)/Mpc = 2.4−18/s= 1/13.2 billion years. The Hubble constant is close to 1 over the age of the universe, which is 13.7 billion years. It is tempting to say that the size of the universe is the distance the fastest-moving galaxy has traveled since the Big Bang. Since nothing can move faster than light we might reason that the radius of the universe is about 13.7 billion light years (4.2 Gps or 1.3 × 1026 m). But we would be wrong. The universe is even bigger than that.
Light left that most distant object about 13.7 billion years ago (minus 380,000 years) and traveled at 3 × 108 m/s ever since then to get to our telescopes today. If that photon had an odometer it would read 13.7 Gly (measuring distance while sitting on a photon actually makes no sense). But the space that photon has traversed is now bigger than it was. The universe is expanding. In fact galaxies are not really moving away from us, but rather, the space between galaxies is growing.
The best analogy I know of to help us understand this is to look at the expansion of a rubber balloon as it inflates. If the balloon is covered with spots as it inflates, an ant sitting on any spot will see all other spots move away from it. This analogy is usually used to explain why all galaxies are moving away from us, yet we are not in a unique location. All galaxies see all other galaxies moving way from them.
Now let us take this analogy one step farther. Imagine that our spot-riding ant can walk 10 cm in 10 s. It starts from its home spot and arrives at its destination 10 s later. How far is it back to its home? The ant traveled 10 cm, but its home is farther away than 10 cm since the balloon was expanding during that trip. We see light that has traveled about 13 Gly, but that light shows us objects that are now three times farther away. The radius of the observed universe is about 14 Gps, 46 Gly or 4.3 × 1026 m in any direction.
Finally we can ask “What is the biggest structure in the universe?” The Great Sloan Wall is 420 Mpc across, whereas the whole universe is 28,000 Mpc. Are there any mega-macro-ultra-structures left? Are there collections that are 1000 Mpc across or larger? Apparently not, or at least we do not presently recognize them. This fact is sometimes known as “the end of greatness.” If I look at the universe with a resolution of 1025 mor1026 m you would see no structure. The universe at that scale appears to be uniform and homogenous.
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4.3 × 1026 meters from here to the edge of the universe and it continues to grow. In the few seconds it has taken to read this sentence the universe grew by over a billion meters, a million kilometers. The universe grows beneath our feet, like walking on a moving sidewalk in an airport or Alice running with the Red Queen in Through the Looking Glass,
“Well, in OUR country,” said Alice, still panting a little, “you’d generally get to somewhere else—if you ran very fast for a long time, as we’ve been doing.”
“A slow sort of country!” said the Queen. “Now, HERE, you see, it takes all the running YOU can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that!”
Through the Looking-Glass
Lewis Carroll
4.3 × 1026 meters. The universe is not only vast, but it continues to grow, and that growth affects light in the sort of ways Alice may have encountered through the looking-glass. And in that twisted way of light and space we may just be glimpsing the end of space—and the beginning of time.
14
A Little Chapter about Truly
Big Numbers
We have now talked about the range of time, from the lifetime of particles that decay in a yoctosecond (10−24 s) to the age of the universe, 13.7 billion years (4.3 × 1017 s), a span of 41 orders of magnitude. In length scales we have seen a range from the Planck length (1.6 × 10−35 m) to the width of the universe (8.6 × 1026 m), a span of 61 orders of magnitude. But to a mathematician these are small numbers. Even Archimedes was interested in bigger numbers. This book is primarily about the physical universe, but in this chapter I would like to digress once more and see the problems of “How Big is Big” from the viewpoint of numbers.
Archimedes told us how many grains of sands there might be if his ideal universe was filled with sand. I will instead address the question of how many particles there are. That is a big number, but physics needs even larger numbers when we talk about potential combinations, a cornerstone to Boltzmann’s statistical mechanics. We will briefly look at combinations as they have to do with entropy, as well as for the pure joy of numbers. And what numbers! They will break our notation scheme and we will have to find new ways of expressing even bigger numbers. Finally we will look at infinity, or rather at a few types of infinity. Some of this will seem like a deluge of abstract mathematics. Do not let the vastness overwhelm you. Do not treat it as something you have to know, but rather let this mathematics just flow. Let it excite and entertain you.
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As Archimedes explained, it takes 10,000 grains of extremely fine sand to fill the volume of a poppy seed; 40 poppy seeds side by side equal a finger’s width and 10,000 fingers’ width to a stade. Finally, since the diameter of the universe, according to Archimedes, was no more than 10 trillion stade, the whole volume could be filled with 1063 grains of sand. The radius that Archimedes used was 1016 m, or between one and two light years. That is a universe whose boundaries would be about half the distance to the next star. The real observable universe has a radius of about 4.3 × 1026 m; that is, ten orders of magnitude larger. If we filled this modern universe with Archimedes’ sand it would contain about 1095 grains.
Of course the universe is not filled with sand, but what it is filled with is still pretty impressive. The Sloan Digital Sky Survey estimates that the number of stars in the universe is about 6 × 1023. By coincidence that is about Avogadro’s number, the number of carbon atoms in 12 g, or the number of hydrogen atoms in 1 g. However it is only a coincidence, because chemist arbitrarily picked 12 g of carbon as a standard mole.
Sand may not fill the universe, but particles do. Amazingly enough, we can calculate the number of quarks in all of space. New deep-sky surveys tell us that the universe is “flat;” that is, it is not curving back on itself. That means that the universe maintains a unique balance between outward expansion, expressed in Hubble’s law, and inward gravitational collapse. So we can actually calculate the density of the universe by balancing Hubble’s constant and Newton’s gravitational constant and obtain the cosmological critical density:
Since we also know the size of the universe, the mass must be about 3 × 1054 kg. If that
is primarily protons and neutrons, that would be 2 × 1081 nucleons (mostly protons) and 6 × 1081 quarks. This is a number that pales compared to the number of grains of sand to fill the universe. It also implies that there are about 15 quarks or 5 atoms for every cubic meter of space. There is a big “if” on these numbers depending whether the world is made up primarily of protons and quarks. Presently we think 5% of the mass of the universe is “ordinary matter,” and 95% is dark matter. We do not know what dark matter is and I have just treated it as if it were made of quarks just cobbled together in some unusual way.
We can also look at the number of photons in the universe. The temperature of deep space is T = 2.72 K, a few degrees above absolute zero. Back in Chapter 5, we saw how Planck told us how to connect temperature to the number of photons. There are about 411 photons per cubic centimeter or 4 × 108 photons/m3. That adds up to 1.4 × 1089 photons. These are impressive numbers indeed, but to study the motion of the atoms in a teaspoon of tea, we would need bigger numbers.
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Back in Chapter 5 we also spent some time trying to understand Boltzmann’s description of heat, energy and entropy. We said that a well-ordered system was low in entropy and described the system in terms of six pennies. With six pennies in a line there are 26 = 64 combinations of heads and tails. Of those, only one is all heads. That combination (and the all tails) are the most orderly states. There are 6 ways of having one head, 15 ways of having two heads, and 20 ways to have 50% heads. These 20 arrangements: TTTHHH, THTHTH, TTHHTH,… are the most common states and the ones with the greatest disorder or highest entropy. But remember Boltzmann is thinking in terms of the number of atoms in a bottle of air.
If we have 100 pennies, there will be 1029 combinations that are 50% heads. If we had ten thousand pennies, the number increases to 103,008 combinations that are half heads. That is a number with 3,008 digits! Still, we are no where near Avogadro’s number of pennies, but fortunately, in Boltzmann’s equation for entropy it is the logarithm of combinations—effectively just the exponent—that matters.
How Big is Big and How Small is Small Page 25