In this diagram the size of the cloud increases towards the right, and the force that is under discussion increases towards the top (those of you who love mathematics will note that the scale for each quantity is logarithmic, so that exponential relationships are shown as straight lines). As you go toward smaller sizes (on the left) gravity increases rapidly, but the pressure increases even more rapidly. For each 1% decrease in the size of the cloud, gravity will increase by 2%, but pressure will go up by around 4%, or about twice as fast. Because of this, unless something can be done to prevent the pressure from increasing as the cloud contracts, gravity will not be able to overcome the pressure, and the cloud will not contract to form a star.
Ways of Overcoming the Pressure
Despite the fact that it is difficult for clouds to contract to form stars, hundreds of billions of stars now exist in our Galaxy, showing that somehow any problems which prevent their formation must be overcome, so somehow the problem of overcoming the pressure must be solvable, and probably easily solvable.
(1)
One way in which this can occur has to do with the fact that in the spiral arms, there are hundreds of thousands or even millions of clouds clustered together in some areas. As they gradually move around they can collide and merge. There can be a number of effects associated with such collisions, but to keep things relatively simple let's just presume that two or more clouds very slowly pass through each other and for a while, are essentially a single cloud. The gas pressure inside the combined clouds is proportional to the overall mass, so it will go up by however many times the overall mass increases, but the gravity is proportional to the mass squared, so it can temporarily become much larger than the pressure, depending on how much mass becomes concentrated in one area or another. This would allow the combined clouds to, at least in some of their denser areas, start collapsing toward considerably smaller sizes and considerably higher densities, as the gravity pulls things in, closer and closer, denser and denser, as originally discussed a little while ago.
Of course as the size of a given part of the combined clouds decreases and the density increases, the pressure within that region will tend to increase, and as already discussed will in fact increase much faster than the gravitational force, but if the gravitational force starts out considerably larger, it may take a while for the pressure to catch up and as described below, by then things may have changed considerably. So this method of getting stars started could at least get us over the first hurdle.
(2)
There is also another way of getting things started, which depends upon the fact that massive stars tend to form very quickly, and when they do become extremely hot and extremely bright, lighting and heating up large areas all around themselves. If, in a region with many clouds (such as one of the spiral arms of the Galaxy) a number of massive stars can form and begin to light and heat up the surrounding gas, that gas can very quickly increase its temperature, from less than a hundred Kelvins to more than ten thousand Kelvins. This increase in temperature increases the pressure inside the gas by more than a hundred times, and such so-called HII regions begin to expand outward quite rapidly.
If there is nothing in the way of the expanding heated gas it will just expand into space and eventually become too thin to notice. But if it is expanding into a region filled with cooler clouds of gas it will press against them, eating away at their outsides (since the expanding gas is hot it will gradually heat up and eat away at the cooler gas), and at the same time compressing them with a force that is as much as a hundred times greater than the internal gravity and pressure of the denser clouds. We see many emission nebulae, as the hot glowing gas lit up by massive, bright stars are called, in which cold dark clouds of denser material are being compressed by the expanding heated gas. Depending upon the shape of the colder denser material, we call the darker region "dark globules" or "elephant-trunk" structures. The Horsehead Nebula, the Cone Nebula and the Eagle Nebula are all examples of elephant-trunk structures, and you can easily see, in many of the photographs of emission nebulae in textbooks (a favorite kind of illustration because they are so pretty) small dark blobs or dark globules which are surrounded by and being compressed by the heated gases surrounding them.
Since the heated gas may have a pressure which is as much as a hundred times the internal pressure of the colder darker gases that they are pressing on, they can force the colder clouds of gas to rapidly contract. As the clouds contract their own gravity will of course increase, and the external forces, by adding to the internal gravitational forces, can force the clouds to contract much faster and much further than a simple gravitational collapse would allow.
(3)
If these methods won't work there is also another, even more spectacular way of forces clouds to become stars. When the massive stars already referred to end their lives, they usually do so in spectacular explosions referred to as supernovae. In these fiery explosions huge amounts of gas can be ejected into space at tremendous speeds. When these gases collide with clouds of gas they violently compress the clouds, with forces far larger than any so far discussed. Because of this, it is not unusual to see, within arcs of gases which were ejected by supernovae, knots of hot bright stars which formed as a direct result of the impact of the expanding supernova gases with cooler, denser clouds.
The expansion of gases near young, hot stars of high mass, and the violent expansion of gases by the death of high-mass stars, can be so effective in helping clouds of gas turn into stars that as many as half of even the high-mass stars are formed in such a way, and almost all low-mass stars such as the Sun have some kind of help of this sort. (Note in passing: In discussions of the origin of the Solar System and the melting and differentiation of the planets it was mentioned that there must have been a large amount of short-lived radioactive elements in the gases which made up the cloud which became our Solar System. This suggests that not only was the Sun helped to form by the explosion of another, more massive star, but that a substantial minority of the heavy elements in our Solar System may have actually been created at the time of the death of that star, and mixed in with the materials which were in the already existing, cloud. So in several ways we may well owe our existence to the death of a massive star, and in a very real sense, are made of "stardust".)
Breaking the Relationship Between Temperature and Cloud Size
In order for a cloud of gas to form into a star, the tendency of pressure to rise faster than gravity and external forces must be overcome. Stars are millions of times smaller than interstellar clouds, and even if external forces and gravity are originally hundreds or thousands of times greater than the gas pressure, as the cloud contracts the pressure should rise dramatically, and catch up with the compressional forces when the cloud has contracted only a few dozens of times. So how can clouds turn into stars? We know that somehow this problem must be fairly easily solved, because there are hundreds of billions of stars. How is this accomplished?
One of the key problems in forming stars is that the temperature of the clouds tends to increase as they are compressed. If we could somehow stop this from happening, it would be much easier to form stars, because the pressure wouldn't rise as quickly, and the compression could continue for a longer period of time, pushing the cloud much closer to stellar size. As it turns out, the difference between the way in which the gas in the clouds interacts with light, and the way in which the "dust" (a mixture of silicate grains, carbon compounds, and ices) interacts with light, is the key to the problem.
When clouds are spread out, most of the material in any given region is gas. Typically, a cloud might contain a few dozen to a few hundred atoms (mostly of hydrogen and helium) per cubic inch, but only a few dust grains per cubic mile. The reason for the difference is party that the materials which make up the dust are rare -- amounting to only a tiny fraction of one percent of the total number of atoms -- and partly that each dust grain, although microscopic (studies discussed in the text show that the most common dust grains are about 1/50000th of an inch in size), consists of many billions of atoms, whereas the gas consists mostly of individual atoms or molecules containing only a few atoms. Since the dust grains require huge numbers of atoms each of which are relatively rare, the dust grains are far less common than the gas particles.
The gas particles only absorb certain types of light which pass through the cloud -- mostly ultraviolet light corresponding to wavelengths which hydrogen and helium atoms can absorb when in their "ground" states. Such light is quite rare in interstellar space, partly because all light is rare (the stars are mostly far away and pretty faint), and partly because only hot bright stars, which are not as common as ordinary stars because they don't last very long, can give off such ultraviolet radiation. As a result most of the light from the distant stars, which is visible or infrared light, passes right through the gas as if it weren't even there. There is a very small absorption of specific wavelengths by rarer atoms and molecules of gas (refer to the discussion of interstellar absorption lines in the text), but the vast majority of the light which passes through the clouds literally just passes through them, at least as far as the gas component of the clouds is concerned. In other words, the gas in the clouds does a pretty poor job of blocking or absorbing the light passing through it.
Now you might expect that if the gas doesn't absorb much light it would be relatively cold, as it is the light absorbed the gas that is primarily responsible for what little heat that the cloud has. But take the example of Venus. Venus reflects most of the sunlight which falls on it, so much in fact that if the light which did make it through the clouds to the surface were absorbed and emitted in a normal equilibrium, Venus wouldn't be much warmer than the Earth, despite being considerably closer to the Sun. However, Venus is much hotter than you'd expect, because as hard as it is for light to reach the surface, it is even harder for the infrared radiation emitted by the surface to get out, as a result of the runaway greenhouse effect caused by the atmosphere of Venus. In a similar way, if we want to understand the heat balance of interstellar clouds, we must consider not only how they absorb light, but also how they get rid of it.
It is very difficult for the clouds to absorb heat because they can only absorb tiny amounts of the visible light passing through them, and although they are pretty efficient at absorbing certain types of ultraviolet light, that kind of light is quite scarce in interstellar space. However, as hard as it is for the clouds to absorb light, it is even harder for them to get rid of it.
As you can read in the text, the hydrogen and helium atoms which make up the bulk of the gas are normally stuck in their so-called "ground" state. In this state they can only emit radiation by colliding with each other at very high energies, which would correspond to temperatures thousands of times higher than the actual temperature of the gas. As a result, they shouldn't be able to get rid of any of the heat that they've absorbed, and although they absorb very little at any given time, they should gradually heat up.
Eventually, as the gas increases in temperature, collisions of gas particles will put some of them into "excited" states which can emit radiation. For complex atoms such as iron or calcium, which have complicated structures and complicated energy level diagrams, small amounts of radiation can be given off once the gas is relatively hot. But these atoms are rare, and at the low temperatures in the interstellar clouds, give off radiation in only small amounts. Most of the radiation is given off by complex molecules, and by a particular radiation due to hydrogen atoms called the 21-centimeter radiation, because that is the wavelength of the radiation.
The 21-centimeter radiation represents a substantial fraction of the radiation by interstellar clouds of gas, and in fact is sufficiently strong that we use it to map the distribution of such clouds of gas in interstellar space. But the radiation is only given off once in a great while. To emit it, a hydrogen atom which is in its ground state must bump into another one, which doesn't happen very often, and end up in a very slightly energized version of the ground state. The ground state turns out to consist of two extremely close energy states differing by an incredibly tiny amount of energy, according to whether the electron going around the nucleus has a "spin" which is parallel to or anti-parallel to the "spin" of the nucleus. Switching from one of the two spin states to the other one involves the absorption (through collision) or the emission (through a 21-centimeter photon) of that very tiny amount of energy. Since the energy involved is so small, it takes over a million 21-centimeter radiations to get rid of the energy absorbed from a single ultraviolet photon, and the rate at which the 21-centimeter radiation is emitted is quite small, because the collisions which put the hydrogen atoms into the correct energy state to emit it are very rare.
To summarize,
the gas has a hard time absorbing light, and an even harder time getting rid of it, and as a result, is much warmer than you might expect given the tiny amount of light falling on a typical cloud. In fact, if the cloud absorbed and emitted heat in an easy, effective way it would probably have a temperature no more than a few degrees above absolute zero, whereas usually the gas temperature is around a hundred degrees above absolute zero, or even higher. This is very cold by our standards, since we live on a world where temperatures are more than 500 degrees above absolute zero, but still very hot compared to what we would expect if the clouds were in "equilibrium" with the light running through them.
In addition to the gas, however, we also have the small number of dust grains which are scattered throughout the cloud. The dust grains do not absorb light in the same way as the gas particles. They absorb or scatter it more like ordinary bits of solid material, and although because of their small sizes they have a much harder time interacting with light, especially longer wavelengths of light, they can interact with the light much more effectively than the gas particles. To see how this works, suppose that you were in the middle of one of these clouds. You would absorb starlight at a very small rate, because the starlight falling on you would be quite faint, and radiate heat, because you are warm, at a much greater rate. Both the absorption and emission would proceed at a rate comparable to that of a so-called "black body", because you are made of materials which are good at absorbing and emitting light. As long as you were relatively warm, the starlight that you were absorbing, although absorbed fairly well, would be far less than the heat your body was radiating, and you would cool off. However, as you got colder and colder the rate at which your body radiates heat would gradually decrease, and when you got cold enough it would be equal to the rate of absorption of starlight, and you would stop cooling off. This would happen, in most places in interstellar space, when your temperature was less than 10 degrees above absolute zero.
The dust grains in the interstellar clouds can't absorb and radiate heat as efficiently as you can, because they are microscopic and have trouble interacting with light waves which are bigger than they are (which is most of them). So they might not stabilize at quite the same temperature that you would, because their heat balance wouldn't be quite as efficient as yours. However, they would stabilize at a much lower temperature than the gas surrounding them, because they are much more efficient at absorbing and emitting light, and would be closer to the expected equilibrium temperature for objects in interstellar space.
The result of all this is that you can think of a typical interstellar cloud as consisting of a huge amount of relatively hot gas, and a very tiny amount of relatively cold dust grains. The dust grains, being made mostly of ices (because that is the most abundant materials making them up), are literally dirty pieces of ice, and compared to the gas, ice cold as well.
When the clouds are spread out over tens or hundreds of thousands of AUs, there are so few dust grains per cubic mile that any given gas particle is hardly ever going to run into an ice grain, and so the fact that the ice grains are much colder than the gas is of relatively little importance. But what would happen if the cloud were compressed, either by its own weight or some external force, to a size which is many times smaller -- say, ten times smaller than its original size? In that case, the density of the cloud would increase by a thousand times, meaning that each part of the cloud would have a thousand times as many gas particles,
and a thousand times as many dust grains per unit of volume. Under those circumstances, collisions between the gas and dust particles would occur a thousand thousand, or a million times more frequently. If, at the large size that it started out at, collisions with the dust grains were robbing the gas particles of only a tiny fraction of one percent of their heat, as the cloud contracted and became denser and denser, the dust particles would gradually have a better and better chance to steal heat from the gas, and radiate it away. Because of this, as a cloud compresses, although it should get hotter and hotter because the compression of the gas pumps heat energy into it, the ability of the dust grains to steal heat away from the gas can become so much more important that instead of heating up, the gas actually
decreases in temperature.
If this idea is correct, then clouds which have been compressed to a much smaller size, instead of being a hundred degrees above absolute zero, should be much closer to absolute zero. And in fact when we observe compressed clouds, referred to as dark globules because they look like dark little blobs, or as protostellar clouds because we believe that they are well on the way to becoming stars, we find that they are indeed, in most cases, much colder than typical interstellar clouds, and in some cases the outer parts of these dark globules are only a few degrees above absolute zero.
A Cloud Collapses, and a Protostellar Cloud Begins
The way in which stars start out is, therefore, like this. Either as a result of the merger of several clouds or some external force involving the life or death of a nearby massive star, a cloud begins to contract. As it does, the increasing density allows the dust inside the cloud to steal the heat energy of the compression from the gas and radiate it into interstellar space, preventing the cloud from heating up and in fact, more likely causing it to cool off. This allows the compressional forces to remain substantially higher than the internal pressure for a long period of time, and over a period of a few tens or hundreds of thousands of years (depending upon how violent and rapid the collapse is), the cloud shrinks from several tens or hundreds of thousands of AUs, to a much, much smaller size.
Throughout this collapse the gas in the cloud is, in a sense, in "free fall" or "accelerated free fall", pushed inward by forces much, much greater than the pressure. But this state of collapse cannot continue forever, as eventually the pressure will rise, catch up with the compressional forces, and stop the collapse at a stage of formation which, as mentioned above is referred to as a dark globule.
To see how this works, consider what the cloud would look like if we could speed up time by hundreds of thousands of times and watch it collapsing. At first it would be spread out over a vast space and be relatively transparent, because the gas allows most light to go right through it, and the dust is too spread out to affect the light much, either. But as the cloud collapses it would become denser and denser, and it would gradually become more and more opaque, meaning that the light of distant stars would have a harder and harder time getting through it.
Hopefully, you remember that we refer to idealized radiation such as that given off by stars as "black-body radiation", because, as discovered by Kirchhoff, if an object is a good absorber of heat and light it is also a good emitter of heat and light, and since nothing can absorb light energy better than a totally black object which absorbs all the light falling on it, such an object is a more efficient radiator than any other kind of object, as well.
When the gas cloud is spread out and most light just passes right through it, it is obviously not a very good absorber, so it is not a good emitter, either. The gas does a terrible job of absorbing and emitting light, and the dust hardly any better. But as the cloud shrinks it becomes harder and harder to see through. The gas still doesn't do a very good job of interacting with light, but as the dust becomes more and more thickly concentrated it does a better and better job, until it is so good at blocking light that you can't see stars through the cloud at all (refer to the discussion of interstellar extinction and reddening in the text for more about how the dust blocks the light of distant stars). But if the dust is doing a better and better job of blocking the light of stars -- that is, if it is acting more and more like a "black body" -- then it must also be doing a better and better job of radiating away the heat of the cloud -- that is, it must be stealing heat from the gas and radiating it into space, pushing the cloud to lower and lower temperatures more and more efficiently.
Because of this, as the cloud gets denser and denser and darker and darker, approaching the "dark globule" stage, it should gradually cool more and more, until it is close to absolute zero, as already discussed. But there is a limit to how far this can go, because as the cloud gets denser and denser there will come a time when the dust is so good at blocking light that it not only prevents the light of distant stars from passing through the cloud, but also prevents the heat which is being radiated away by dust near the center of the cloud from escaping. When this happens the cloud begins to alter its structure, becoming more and more like a star, with higher temperatures in the middle and lower temperatures near the outside, and as heat flows from the dense, opaque central regions toward the less dense, less opaque outer regions, we develop a structure which is hotter and denser in the middle, and less hot and less dense outside.
Quasi-Equilibrium Contraction: Differences Between Massive and Low-Mass Clouds
Once the temperature begins to rise in the core of the contracting cloud, meaning once it is a dark globule and heat radiated by the center cannot just escape to interstellar space, but must struggle to get through the intervening gas, the collapse of the cloud will begin to slow. For some clouds, which have large external forces compressing them, the cloud may already be about the size of our Solar System when this happens. For others, the cloud may be somewhat larger than our Solar System. But in all cases it will be substantially larger than a star, so there is still quite a bit of development left before the star that the cloud will eventually become has finally formed.
In the interval between the initial collapse and the final formation of the star the central part of the cloud will undergo a relatively slow, more-or-less steady contraction in which, at any given time the internal pressure caused by the extreme density increase which resulted from the initial collapse and the relatively high temperatures created when the dust became too thick for the internal heat to easily escape becomes essentially equal to the external forces and any intrinsic gravitation force, and if the heat of the core could be held onto the cloud would stop getting smaller, and would remain more or less stable at a relatively large size.
But despite this supposition, it is not possible for the cloud to remain constant in size at this point. For one thing there may be more gas further out which is still collapsing towards the central core and has not yet become so dense and opaque that its temperature is substantially increasing. In addition, the core must now be relatively hot and radiating energy (primarily infrared radiation) at a fairly substantial rate, and unless that radiation is somehow replaced, the gas in the core of the protostellar cloud or dark globule will cool off, reducing the pressure and allowing the compressional forces to force the cloud to still smaller sizes.
As it turns out, things don't work quite exactly that way. It is true that as heat is radiated away from the core there is a tendency for the gases to cool off, and for the compressional forces to push the cloud to still smaller sizes, but in the process energy is created and the gas, instead of cooling off, actually gets hotter. We refer to this stage of the star's formation as a gravitational contraction, and the heat generated by the contraction not only replaces the lost heat, but actually adds to the store of heat in the central core of the cloud, so that instead of cooling off it gets hotter and hotter.
The rate at which the core gets hotter strongly depends upon the mass of the cloud, because the more massive the cloud is, the greater the gravity it has and the greater the force that it exerts on itself as it contracts. But according to the laws which govern all heat engines (automobile engines, refrigeration units, air conditioning, etc), the compression of the gas will create an amount of heat which is proportional to the force which is compressing the gas. If the gas is compressed by only a small force only a small amount of heat will be generated and the gas will become only a little hotter, but if the gas is compressed by a large force, a much larger amount of heat will be generated and the gas will become considerably hotter.
The result of this is that a relatively massive cloud will, as it contracts, generate far more heat and become considerably hotter than a less massive cloud, which has a smaller gravity and a smaller compressional force as a result of its lesser mass, and because of that, as clouds of different mass contract the more massive ones will at a given stage of the contraction, be hotter at a given size or bigger at a given temperature, and in either case end up much brighter, radiating heat away into space at a much greater rate. A cloud which is going to become a low mass star might take a million years slowly radiating heat away to shrink from a given size and temperature and brightness to a particular smaller size and temperature and brightness, while a cloud which is going to become a high mass star, because it is much hotter at any given size, is much brighter at any given size and radiates heat away much more rapidly, causing it to shrink much more rapidly. This effect is so dramatic that if one cloud has as little as ten times the mass of another one, it can shrink as much as a hundred times faster, in this example becoming substantially smaller in just a few tens of thousands of years, instead of in a million years or so.
This tendency of massive clouds to generate larger amounts of heat from their greater gravitional compressions, to become brighter, and to shrink towards stellar status far faster than less massive stars, is one of the critical differences between stars of different masses, and causes them to form at different rates and to become different kinds of stars. Over a period of only a few tens of thousands of years a very massive cloud may collapse, undergo a relatively rapid contraction, and become a very hot, very bright upper Main Sequence star -- a star with a brightness of tens of thousands, hundreds of thousands, or even a million times the brightness of the Sun. Meanwhile a less massive object, like the cloud that became the Sun, may take several millions of years to slowly contract, always cooler, smaller and fainter at any given stage than the more massive object, and finally become a relatively ordinary Sun-like star. Still less massive clouds, which end up as stars with only a fraction of the mass of the Sun, would be still cooler, still fainter and shrink still more slowly, as they struggle to become the very small, relatively cool, very faint lower Main Sequence stars.
Please note that you can actually see the effects on this in the diagram in the text which shows the way in which stars approach the Main Sequence. That diagram, which shows evolutionary paths, as they are called, for stars which are approaching the Main Sequence, shows that the high mass stars form relatively quickly and are always relatively big, relatively bright and relatively hot compared to the lower mass stars, which form much more slowly.
Summary of the Preceding
To summarize what we have covered above, clouds are forced to contract by the merging of several clouds, or external forces created by massive stars which, because they form easily and quickly, form first in a region full of clouds of gas and dust, and as they live out their lives and die, trigger the formation of other stars, massive and not so massive.
The initial stage of the formation of stars is always relatively rapid, involving a decrease in the size of the original cloud of several dozens or even hundreds of times in size. During this collapse the dust in the cloud, as it becomes more and more thickly concentrated, steals heat from the gas, lowering its temperature and allowing the collapse to continue. Within a few tens or hundreds of thousands of years the cloud is much smaller, and doomed to become a star.
Towards the end of the initial collapse the dust becomes so dense that in the core of the collapsing cloud, heat and light can no longer escape and the temperature begins to rise, and pressure begins to rise quite rapidly, and the core of the cloud becomes more like a cool star than like a cloud, and the collapse begins to slow. At first the compression of the cloud by additional gases falling in from the outside may continue the collapse, but eventually the central core will become a more or less stable object, which continues to contract, but much more slowly, because it is slowly radiating heat away, through the surrounding layers of gas and dust.
This somewhat slower quasi-equilibrium contraction is relatively fast for massive objects because their weight, squeezing the gas of the cloud, generates large amounts of heat, making the cloud relatively hot and relatively bright for its size, causing heat to pour out of the central regions at a prodigious rate, and causing the cloud to contract to smaller sizes at a relatively rapid rate, and in fact in times as short as tens or hundreds of thousands of years.
For less massive objects, however, the lesser gravitational force contracting the object means that the gas doesn't get nearly as hot, and heat moves from one layer to another much more slowly, causing the contraction to be much slower. Depending upon how low the mass of the contracting cloud is, it may take millions or even tens of millions of years for the cloud to finally shrink enough to become a star.
Protostellar Clouds vs. Protostars
Up to this point we have been discussing objects which are relatively cool, even if considerably heated relative to their initial temperatures, and as a result are giving off mostly, if not exclusively infrared radiation. Such objects are sometimes referred to as protostellar clouds, to distinguish them from the next stage of stellar formation. That stage involves objects which have become so hot that they give off not only infrared but also visible radiation, and we refer to those objects as protostars. They are, if not still completely obscured by the gas and dust surrounding them, now visible as stars, but are still not stable objects. They are stars, visible objects shining by their own light, but not being stable objects, they are "proto"-stars. The next part of this "outline" (
part 2a) discusses how protostellar clouds become protostars.
