Estimating the Mass of the Galaxy From Star Counts
The Sun lies within the disk of the Milky Way Galaxy. In the region near us, stars are scattered about, in all directions, a half dozen or so light years apart. We can estimate the total number of stars in the disk from its overall size, which averages a couple of thousand light years thick, and the best part of a hundred thousand light years across, and the assumption that the stars are scattered about in a similar way throughout the disk. The result is a few tens of billions of stars. A similar calculation can be performed for the nucleus of the Galaxy. The nucleus is not as large as the disk, but stars are more thickly clustered inside it, from a few light months apart near the outside of the nucleus, to a few light weeks apart near its center, so the result comes out about the same: a few tens of billions of stars. It is more difficult to estimate the number of stars in the halo, as the vast majority of them are far too faint for us to see at all, but studies of the halos of other galaxies suggest that their stars are light decades apart, and estimates of the total numbers of stars in the halos come out in the same range, a few tens of billions. In other words, our Galaxy, and similar large galaxies, such as the Andromeda Galaxy, must contain many tens of billions of stars.
To estimate the mass of the galaxies, we examine the stars near us, to see how massive they are. Some, a small minority, are considerably heavier than the Sun, and quite a few are about the same mass as the Sun, but the vast majority are half the mass of the Sun, or less. As a result, the total mass of all the stars in the Galaxy, when compared to the mass of the Sun, should be a little less than their total number, or many tens of billions of Solar masses.
Estimating the Mass From Stellar Motions
There is, however, another way to measure the mass, namely by looking at the motions of the stars. To a first approximation, stars in our neighborhood are orbiting the center of the Galaxy in roughly circular orbits. If we knew how big those orbits are (about 25000 light years, or one and a half billion AUs, for the Sun, and the stars near it), and how fast they are moving around the center (about 200 miles per second), we could calculate the orbital period (a little less than 200 million years), and use Kepler's Third Law to obtain an approximate mass for the material which lies between us and the center of the Galaxy. Doing this gives a result of about 200 billion Solar masses, which is about ten times larger than the amount that we would estimate from star counts.
This surprising result is matched by star counts and velocity calculations in other galaxies, as well, and in clusters of galaxies, the velocities of the galaxies relative to each other also imply, at least if we assume that the clusters are gravitationally stable, masses around ten times what we would estimate from star counts.
The fact that in our Galaxy, in all other galaxies which have been well studied, and in every cluster of galaxies which we have observed, the mass estimated from star counts is only about one tenth of the mass estimated from stellar velocities, either means that all of these objects are unstable, and will fly apart, as a result of their high velocities, in a relatively short time, or that there are large amounts of mass which we are not taking into account. But unless our star counts are wrong, that mass can't be in the form of stars, or at least not in the form of the kind of stars that we can see, so what form is it in?
Dark matter. Most of what is, isn't visible or otherwise detectable.
The fact that huge masses, as much as ten times the mass of visible stars, seem to be hidden somewhere inside the galaxies, leads to the concept of dark matter. Even though dark matter's existence has been suspected for almost 70 years, its actual nature has remained a mystery, and still is not really known, but we can make a number of educated guesses about what it might be.
One possibility is that dark matter is some form of dead stars, such as white dwarfs, neutron stars, or black holes. White dwarfs are known to exist, but they are very faint, because they are only about the size of the Earth, and counts of white dwarfs in our neighborhood could easily be "off" by a factor of two or more, and in more distant regions, such as the halo and nucleus of the Galaxy, any estimate of the number of white dwarfs would be literally an educated guess, and perhaps more a guess, than educated.
Neutron stars are only a few miles across, and are therefore much smaller than the Earth, and white dwarfs, and as a result, it was thought that they should be completely unobservable, even if as close to us as the nearest known stars, and, of course, black holes are, by definition, completely unobservable, as nothing, including light, can possibly escape them. As it turns out, there are rare circumstances in which we can tell that neutron stars or black holes exist, such as in binary star systems which involve mass transfer from another star, but the vast majority of neutron stars and black holes should not be observable, so if there were large numbers of them scattered around the Galaxy, they could represent a considerable amount of mass. In fact, this was realized, right from the beginning, and for a long time astronomers hoped that such beasts didn't exist, and that somehow, massive stars always managed to eject enough mass that they could become white dwarfs, instead of neutron stars, or black holes, because if they couldn't, it would mean that most of the mass of the Galaxy, and of the Universe, might well remain forever unobservable, making the observations which could be made seem quite a bit less important, and perhaps even futile, in understanding the nature of the Universe.
Now that we know that neutron stars and black holes exist, and that we may be underestimating the numbers of white dwarfs by a considerable amount, it might seem reasonable to suppose that the dark matter is simply a very large number of "dead" stars, which just happen to be unobservable because they are too faint to see. The trouble with this idea is that if it is true, it will take a long, long time before we can prove that it is correct. In fact, if you had believed this to be the correct explanation say, 40 years ago, you would have had to conclude that dark matter was not a good thing to try to study, because you could spend your whole life looking for it, and never be able to find it. Even now, with far better tools and observing techniques, it is not at all certain that we could prove that dark matter is "just" dead stars and similar things, within our lifetimes. And, of course, if we could finally succeed in proving that this is the correct answer, it might seem more than a bit anti-climactic, and not terribly newsworthy, since, despite the difficulty of proving this to be the correct answer, it is a fairly reasonable, common-sense sort of idea.
As a result, for the last 40 years, efforts to look for dark matter have concentrated mostly on completely different ideas, ideas which, whether they are correct or not, at least have some chance of being proved right or wrong within a reasonable time.
Could Dark Matter Be Brown Dwarfs?
The trouble with assuming that dark matter is the hulks of dead stars is that, at least in our neighborhood, stars seem to be the best part of half a dozen light years apart, and if dark matter consists of relatively heavy things, the total number of them wouldn't have to be very many times greater than the number of visible stars, and so they would still be scattered pretty thinly around the Galaxy, certainly several light years apart. At those large distances, even white dwarfs are difficult to see, without knowing what to look for, and neutron stars and black holes are completely unobservable. But suppose that the dark matter is not in the form of relatively heavy things, but much, much less massive objects, such as brown dwarfs, the "stillborn" stars that result when a cloud of gas has too little mass to become a Main Sequence star. Such objects are less than a tenth the mass of the Sun, and in many cases, may be considerably less massive than that, so if their total mass was ten times the mass of the visible stars, there would have to be an extremely large number of them -- at least a few hundred times as many, in fact.
At first thought, the idea that there would have to be such huge numbers of brown dwarfs for them to be the dark matter makes the whole proposal seem absurd, but there is a happy side effect of this result, because if they do exist in such large numbers, they must be literally all over the place. The Galaxy would have to be practically crawling with the things, so that they were no more than a few light months apart, instead of light years, as normal stars seem to be.
Now, even at a distance of a few light months, brown dwarfs would be hard to observe, because they are too cool to give off any visible light worth speaking of, and even with infrared telescopes would be very, very faint. But the technology to observe them does exist, and in fact has existed for the best part of 40 years, so if the dark matter were brown dwarfs, then, as soon as the idea was proposed, it should be possible to start finding a few, and then some more, and finally, such huge numbers that it would be obvious that this is the correct answer. This makes the idea much more attractive, since, as pointed out above, a good subject for study should be something that has a good chance of being proven right, or wrong, sometime SOON, and the theory of brown dwarfs as dark matter was, as a result, quite eagerly accepted, when first proposed. For quite a while, magazine articles about dark matter would confidently proclaim that most of the mass of the Galaxy, and of the Universe, was made up of brown dwarfs, and "any day now," brown dwarfs would be found to exist in such large numbers that the idea would be known to be correct.
Unfortunately, despite nearly four decades of looking for them, brown dwarfs don't seem to exist in anywhere near the numbers required to explain dark matter. In fact, for most of that time, they didn't seem to exist at all. At any given time in the last 40 years, there were a few, to a few dozen, "candidates" for brown dwarfs, objects whose observable properties suggested that they might be brown dwarfs, but were not yet known to be brown dwarfs, for sure. In every case, however, it was soon shown that the observed objects were not brown dwarfs, at least until very recently. As technology has improved, it has finally, within the last few years, become possible to observe brown dwarfs, and we now suspect that they might be as common as normal stars, or at least, not too many times less common.
The trouble is, since brown dwarfs are much less massive than normal stars, to explain the dark matter, they can't be that few in number. They would have to be far more abundant than they seem to be, and as a result, although they do contribute a very small additional amount to the "observed" mass of the Galaxy, they don't in any way help explain dark matter.
Could Dark Matter Be Neutrinos?
Once it turned out that brown dwarfs wouldn't work, people working on the problem of dark matter turned their thoughts in another direction. Perhaps, instead of looking for a lot of relatively low mass objects, they should be looking for an incredible number of almost massless particles. Again, this sounds a little far-fetched, but there is actually a very good reason for hoping that this might be the answer -- the properties of the neutrino.
Neutrinos are massless, or nearly massless, particles which move through space at the speed of light, or very nearly the speed of light. They are known to exist in incredibly large numbers, as they are continually created inside stars, especially during the death-throes of massive stars, and were probably made in very large numbers during the Big Bang, in the Cosmic Fireball which began the Universe as we know it. As it turns out, it is extremely difficult to observe neutrinos, because they are practically "ghost" particles, capable of going through stars, or planets, as if they didn't even exist. Every second, it is estimated that several trillions of neutrinos pass through every square inch of the Earth's surface (and, therefore, your body, as well), but they don't notice that the Earth (or you) exists, and just go right through it, at least under normal circumstances.
Trying to observe such particles is obviously very difficult, but it can be done, using the right tools. You can't really look at all of them, but you can stop an occasional one, and from the theories which describe the physics of neutrinos, calculate how many there ought to be that we aren't managing to stop. This is, in fact, how we get the estimate of trillions per square inch per second, just above. Neutrino "telescopes" occasionally observe the effects of a neutrino interacting with material inside the "telescope," and by extrapolating to how many neutrinos didn't interact, we can estimate the total numbers.
The only trouble with this idea is that when their existence was first proposed, in the 1930's, as a way of explaining a troublesome physics experiment, neutrinos were thought to be massless. In the experiment, energy and momentum didn't seem to add up right, after the experiment, whereas mass did. So it was proposed that some kind of massless particle was carrying energy and momentum away, and the neutrino was "invented." A number of years later, particles having exactly the properties of the neutrino were observed, and were presumed to be the predicted particle. But if neutrinos are massless, then even in tens of trillions per square inch per second, scattered throughout the Galaxy, their mass-energy would be far, far less than the mass of known material, and would not explain the dark matter. So why was it proposed that neutrinos might be the dark matter?
As you have read in the book, neutrinos are created in the thermonuclear fusion reactions which power the Sun and other stars. Now that neutrino "telescopes" can observe neutrinos, it ought to be possible to observe the neutrinos coming from the Sun, and verify that they are indeed being created inside the Sun. But experiments to try to do just that have always shown that the number of neutrinos coming from the Sun is less than expected, in fact, at first, far less than expected.
To a certain extent, the fact that the number of neutrinos coming out of the Sun is less than expected could be due to errors in the theory of how the Sun works. We can't actually see what is happening inside the Sun, as there is far too much stuff in the way to see through, and in fact that is partly why people wanted to observe neutrinos in the first place. It is so hard for the light which is now being created in the center of the Sun to get out, that the light which is now leaving the surface of the Sun was actually created more than a million years ago, and in the long trip that it took from the center to the surface, it bounced back and forth, from one subatomic particle to another, trillions of trillions of times before finally reaching the surface, and during all those collisions, the light is completely changed, so that when it leaves the surface, it only tells us what conditions are like at the surface, and whatever happened to it during its long journey is completely "forgotten". For the neutrinos, however, the Sun might as well not exist, so when they arrive on the Earth, they are presumably completely unchanged from when they were created, only 500 seconds earlier. As a result, if we could properly observe them, we would gain valuable knowledge of the conditions in the center of the Sun.
Since the number of neutrinos coming from the Sun is lower than expected, a number of minor changes have been made in theoretical calculations of the Sun's structure, but there is a limit to how much those calculations can be changed, without making the results look quite different from what the Sun is known to be like. Most of the difference between the original estimates of how many neutrinos are coming from the Sun, and the observed numbers, can be explained by presuming that the core of the Sun is a little cooler than we originally thought, and a little denser, but there is still an intractable error of about 1/3 of the predicted amount, which can only be gotten rid of by changing the theory so much that it gives completely unreasonable estimates of what the Sun, and other stars, should look like.
This remaining error is sometimes referred to as the Solar neutrino "problem". For a long time, it was thought that the solution to the problem might lie in some astronomical factor not yet thought of, but in recent years, the opinion has been growing that the answer may lie, not in astronomy, but in the physics of the neutrino.
As mentioned above, it was originally thought that neutrinos have no mass. If this were true, then, like photons of light, they would travel at the speed of light, from the center of the Sun, to and through the Earth, without any change in their characteristics, because, for objects moving at the speed of light, time as we know it does not exist. For them, the Universe is an infinitely thin sheet, and they can travel all the way across it in no time at all (refer to the section on special relativity, if you are not familiar with this phenomenon). But suppose that neutrinos have a mass, although a very, very small one, so that it is difficult, if not impossible, to notice. Then they would not travel at the speed of light, but a tiny amount slower, and if they should happen to be "radioactive", and able to transform from one kind of neutrino into another kind, or into something completely different, some of the neutrinos emitted by the Sun might "disappear" before they reached us, producing the 1/3 error in the observed numbers. In fact, if such a thing can happen, the 1/3 error is just about the size of the effect that we would expect.
As a result, it is commonly believed, nowadays, that neutrinos probably do have some mass, and even if it is just a very tiny amount of mass, since trillions of trillions of them pass through any given part of the Earth at any given time, the total mass might be quite substantial, and for a while, it was hoped that perhaps this is what the dark matter in the Galaxy might be. After all, if this were the answer, then "any day now", when it becomes possible to correctly estimate the total numbers of neutrinos, and their individual masses, it would be possible to solve this problem, once and for all, and answers which can be obtained soon, if we are lucky, or smart enough, are far preferable to answers which might not be provable at all, or at least, any time soon.
For this reason, for a while, after the brown dwarf theory became an unlikely answer, the idea that neutrinos might make up most of the mass of the Universe was fairly popular. However, in 1987, a supernova was observed in the Large Magellanic Cloud, which is 170 thousand light years from us. In the death of the star which created that supernova, huge numbers of high energy neutrinos were created, and when they reached and passed through the Earth, neutrino detectors all over the Earth announced that something interesting had happened.
If neutrinos have mass, the more mass they have, the slower they will go. If their masses are very, very small, they would move at close to the speed of light, whereas if they have more mass, they would go a little slower (for a given amount of energy). By seeing how slow the neutrinos went, compared to light itself, it is possible to estimate the mass of the neutrinos from that supernova. As it turns out, however, the light from the supernova and the neutrinos from it arrived at the Earth, as far as we can tell, at just about the same time. Considering that they went 170 thousand light years, and took 170 thousand years to get here, the near dead heat between the light and the neutrinos means that the neutrinos must have been going at essentially the speed of light, and have practically no mass, after all. As a result of this observation, and subsequent laboratory experiments, it is now thought that neutrinos probably do have mass, but it is a very, very, very small amount -- at least billions, and perhaps trillions, of times less than the mass of the least massive particle known to have mass, the electron. As a result, even though there are large numbers of neutrinos passing throughout the Universe, it is not likely that they can have any substantial total mass, after all.
Could the Dark Matter be Some Sort of Strange Matter?
If the dark matter is not dead stars and such (which it could be, but will be practically impossible to prove, if it is), or brown dwarfs (which it cannot be, as they aren't numerous enough), or neutrinos (which just don't seem to weigh enough), then what could it be?
One possibility that has been raised, is that the dark matter is something, like neutrinos, that are scattered throughout all of space, but are very difficult to detect -- in fact, something that is so difficult to detect, and so different from any normal form of matter, that no one even knows that particles of this sort exist at all. This material is called Strange Dark Matter.
Of course, if we don't even know that the particles which make up strange dark matter exist, or anything at all about them, or how to observe them, why should we believe that they even exist, let alone make up most of the mass of the Galaxy, and perhaps, of the Universe? The answer is, perhaps we shouldn't do so. But wouldn't it be exciting if such material did exist, and Someday Soon, we could figure out what it is, and how to observe it, and found, all of a sudden, that this was indeed the answer? Of course it would! So, for quite some time now, this has been the preferred answer to what the dark matter might be.
Now, as to what this stuff would be, there are various sorts of answers, according to who is proposing them. Some theories of physics propose the existence of various types of particles which have never been observed, so that their properties are somewhat uncertain (although there are predictions as to some of their properties, as a result of the theory which predicts their existence). Among such particles are axions, Higgs bosons, vector mesons, gravitons, magnetic monopoles, and Weakly Interacting Massive Particles (WIMPs). Some of these particles are believed to definitely exist, but in unknown numbers, and with unknown masses, while others may or may not exist, outside the minds of the people who imagined them. So far, however, none of them have been shown to exist in even small numbers, let alone the staggering numbers that they would need to exist in, in order to explain dark matter. However, for various reasons, for the last 30 or so years, these particles, or something similar in characteristic, but otherwise different, from these, have been thought to make up most of the mass of the Universe. However, to explain why that is the case, we need to discuss dark matter in the Universe (to be written and posted ASAP).
Final Thoughts on Dark Matter in Galaxies
When you have read everything there is to read about dark matter in galaxies, and in the Universe, you should keep certain things in mind. Regardless of what the nature of dark matter in galaxies is, regardless of what it is made of, or why it is so difficult for us to observe, we know that it is there -- we just don't know what it is. It may be some strange form of ordinary matter, such as dead stars which are impossible, or nearly impossible, to observe, or some still stranger form of "strange" matter, which is like nothing that we have ever observed, at all. But regardless of what the answer to the question is, the masses of the galaxies are at least ten times greater than the masses of their visible stars, and so there is something there, something that we don't know about, something which we would dearly love to know about, which grips our imagination, and fuels the efforts to find it.
On the other hand, when you have read about dark matter in the Universe, you will find that if it exists, it cannot be dead stars, or any other kind of "normal" matter, no matter what form that matter might be in. If it exists, it must be made of some kind of "strange" matter -- perhaps not any kind of strange matter so far proposed, but definitely something which is not in any way like any kind of matter ever observed. However, that kind of dark matter, as you will read later, may not exist. In fact, there is absolutely no doubt that if it does exist, it is much rarer than was once believed. At one time, it was thought that between 90 and 99% of all the material in the Universe consisted of such strange dark matter, whereas now, we know that if it exists at all, no more than 20 to 30% of the mass of the Universe consists of such material. And, of course, since we have no observations that say what it is, or whether it even exists at all, perhaps it exists only in our imaginations.
So why have I even bothered to mention strange dark matter? The reason is, that for many years, it was thought that strange dark matter made up most of the mass of the Universe (as discussed in the yet-to-be-written dark matter in the Universe, and as discussed in your text), and, if so, that it ought to also make up most of the dark matter inside galaxies. So, when talking about dark matter inside our galaxy, which is known to exist, you will read that it must be mostly strange dark matter. That is not true. It is possible that strange dark matter exists, and if so, then it may make up a large part of the dark matter inside galaxies. But it is also possible that strange dark matter does not exist, and that all of the dark matter inside galaxies is just some form of ordinary matter in a very strange form. We just don't know. All we know is that there is something that makes up 90% of the mass of the galaxy, about which we know almost nothing save its existence.