Review of Stellar Evolution Prior to the Protostar Stage|
The Contraction of Protostellar Clouds
As previously discussed, most of the formation time between the initial collapse of an interstellar cloud and the formation of the star that results from that collapse is spent in a slow contraction. During the first stage of this slow contraction all stars, regardless of their mass, contract in a fairly similar way. Heat generated by their contraction leaks from layer to layer and escapes from their surfaces, tending to reduce the internal temperature of the protostellar cloud. But the temperature and density of the cloud provide the pressure which supports it against its weight or whatever external forces is helping its weight compress it, so the loss of heat causes the cloud to contract. As it does so, its density and weight automatically increase, and the pressure must also increase, to match the increased weight; which as it turns out, requires the temperature to increase, by an amount approximately equal to the temperature decrease which would have occurred if the cloud didn't contract; as a result, at any given time, the heat contained within the cloud is approximately equal to all the heat that it radiated since the time it began to contract.
How Different Mass Stars Approach the Main Sequence
At any given time, the forces striving to compress the cloud are more or less in balance with its internal pressure, and if there were no further radiation, it would maintain a constant size and structure; but since it continually radiates heat into space, it is continually contracting, albeit relatively slowly, from one state in which pressure and weight are in balance, to a smaller state in which they are still in balance, but with all the quantities describing its structure -- density, temperature, pressure, and gravity -- gradually increasing, save for the reduction in size, in a state of quasi-equilibrium. At any moment, pressure and weight are in balance (or equilibrium), but gravity is slowly winning the battle, as the balance is only maintained, as heat is lost, by gradually reducing the size of the cloud.
More massive clouds generate more heat from a given contraction than less massive clouds, because the heat generated by compression of a gas is proportional to the force compressing the gas, which in this case is the weight of the cloud. As an example, a cloud ten times more massive than a less massive cloud of the same size would have a hundred times more weight, and generate a hundred times more heat from a given contraction. This would make it ten times hotter (temperature being proportional to the heat generated by the compression, divided by the mass of the cloud), and ten thousand times brighter (luminosity being proportional to the fourth power of the temperature). Because the rate of the cloud's contraction depends upon how fast it radiates heat into space, the higher luminosity of more massive clouds causes them to contract much faster than less massive clouds (in the example given, one hundred times faster, because heat is generated a hundred times faster, but radiated away ten thousand times faster than in the less massive cloud); and as a result, massive clouds can become stars in only tens or hundreds of thousands of years, while less massive clouds become stars in millions or tens of millions of years.
Toward the end of the protostellar cloud contraction, the central temperatures reach ten thousand Kelvins, and hydrogen begins to ionize, turning from a gas to a plasma, or totally ionized gas consisting of bare nuclei and free electrons. This is a phase change, similar to the melting of ice or boiling of water, and during phase changes, temperature remains constant, regardless of how much heat is poured into the material. As a result, further contraction of the cloud rapidly increases its density, pressure and weight, but without any change in the temperature, the weight increase outraces the pressure increase, and the cloud goes into a rapidly accelerating free fall, which reduces its size by a factor of ten or more times, decreases its volume by a thousand or more times, increases its density by a thousand or more times, and increases its weight by ten thousand or more times, in less than a decade. Within those few years, the clouds changes from a large, cool cloud of gas, radiating only infrared radiation, to a much smaller, denser, hotter cloud of ionized gas, radiating much larger amounts of infrared and visible light -- a protostar.
The Contraction of Protostars
Once all the hydrogen in the protostellar cloud is ionized, temperature rapidly rises, until helium begins to ionize, at which point the temperature becomes temporarily stuck once again; but once all the helium is ionized and then doubly ionized. During the multiple ionizations of hydrogen and helium, enough heat is used to ionize those gases to -- if it were somehow possible -- raise the temperature of neutral gases to well over a hundred thousand degrees. As a result, the pressure can't catch up with the weight of the protostellar object until the temperature has gone well over a hundred thousand degrees; and in fact, depending upon the rapidity and violence of the ionization collapse, which depends upon the mass of the object, equilibrium isn't achieved until central temperatures rise to a quarter to half of a million degrees.
So, in just a few years, we go from a cloud with central temperatures of only a few thousand degrees, to an object with central temperatures of hundreds of thousands of degrees; from an atomic hydrogen cloud radiating only infrared heat, to an ionized plasma radiating visible light; and hence, from a protostellar cloud, to a visible star. However, despite the huge rise in temperature, and the tremendous change in the internal conditions of the object, it is is one way very similar to the protostellar cloud it used to be -- namely, it is still radiating heat, in fact far greater amounts of heat, and it still has no way to replace that heat, other than gravitational contraction. So, once the ionization collapse is over, the protostar resumes the relatively slow, quasi-equilibrium contraction that it had, as a protostellar cloud.
The diagram below (similar to one in your textbook) shows the way in which stars change, as they contract. Massive objects are, as before, far brighter than less massive ones, and as a result, contract much more rapidly, just as in the case of protostellar cloud contraction. But in addition to the difference in brightness, and in contraction rate, there are marked differences in the way that different mass objects contract. Massive objects (1) become hotter and brighter as they contract, while low massive objects (2) remain relatively cool, and get fainter and fainter, and intermediate mass objects (3) behave like low mass objects at first, and like higher mass objects later on. Why these different mass objects behave so differently is the main topic of the material immediately following.
1. High-mass stars start off bright, and stay bright, despite shrinking, by getting hotter on the outside as well as on the inside. They take only a few tens of thousands or hundreds of thousands of years to form, because they throw away their energy so fast.
2. Low-mass stars start off faint, and get even fainter as they shrink, by getting hotter on the inside but staying cool on the outside. They take tens of millions of years to form, because they use their energy so slowly.
3. Medium-mass stars, like the Sun, start off with medium brightness, get fainter at first, like low-mass stars, then, as their central temperatures rise, begin to behave more like high-mass stars, and get hotter and brighter. They take a few millions of years to form, since they use their energy at middling rates.
Why Do Different Mass Protostars Evolve So Differently?
Next: The Main Sequence
At the end of the ionization collapse which converts large, cool, atomic and molecular infrared-radiating protostellar clouds into small, hot, plasma visible-light protostars, the formation of the stars is not complete. Far from it. For they are now radiating more heat and light than ever, and still have no way to replace it, save by contracting, compressing their gases to higher density, temperature, pressure and gravity, and in other words, resuming the quasi-equilibrium contraction which preceded the ionization collapse. In other words, although the ionization collapse very quickly converts an invisible, infrared object into a visible protostar, it doesn't change the basic thermodynamics which control its contraction. And as a result, the protostars immediately resume their inexorable march toward smaller, denser, hotter structures.
As the HR Diagram above shows, however, there are substantial differences between the way in which the higher-mass stars evolve, compared to the lower-mass stars. As before, the higher-mass stars generate far more heat from a given contraction, and as a result, are much brighter, and contract far faster; so they move across the HR Diagram from the upper right, to the upper left, in just a few tens or hundreds of thousands of years. And the lower-mass stars, which generate far less heat, and are much fainter, and contract much slower, move from the middle right, to the lower right, over a period of tens of millions of years. But aside from the difference in time of contraction/formation and brightness, there is another difference, which is very significant, not so much for the lives of the stars, but for their deaths. Namely, the high-mass stars move to the left in the Diagram, becoming hotter and brighter as they contract, despite getting smaller and smaller, which the low-mass stars move downwards in the Diagram, becoming fainter as they contract. What causes this difference, why does the Sun do the one thing at first, and the other later, and what effect does it have on the lives and deaths of stars?
The Fight Between Density and Temperature
The cause of the different evolutionary paths (as the curves drawn by showing what the stars are like at different times during their formation are called) of high and low mass stars is the different way in which their internal density and temperature change, as they get smaller. Remember that high-mass stars generate a lot more heat during their contraction, because they have a larger gravity at any given size, which squeezes their gases harder, and makes them hotter. Thus, a relatively small reduction in size can produce a relatively large increase in temperature.
Now, the reduction in size increases the density inside the protostar, which makes it harder for heat and light to pass through the gases, reach the surface, and escape. But it also increases the temperature inside the protostar, which means that there is more light trying to get out. And what happens as a result, depends upon how fast the density and temperature increase, relative to each other.
In a high-mass star, the relatively large amount of heat generated by a given contraction means that temperature and hence luminosity go up faster than density, so even though there is more gas trying to block the outward flow of radiation, there is so much more radiation trying to get out that it is able to overcome the increased density, and flow outwards just as fast or faster than before. But since the protostar is decreasing in size as it contracts, to maintain an equal or faster flow of radiation means that the heat radiated by the surface per square foot must be increasing, which can only occur if the surface temperature increases. In other words, if the increase in temperature due to a decrease in size produces an increase in luminosity greater than the increased opacity of the gas due to an increase in density, then the star will increase in brightness despite its contraction, and must move to the left in the HR Diagram, to reflect its increasing surface temperature.
But for a low-mass star, it takes a much larger decrease in size to produce a given increase in temperature, because the relatively lower gravity of the lower-mass star produces less heat for a given contraction. So in smaller, less massive stars, as the star gets smaller, denser and hotter, the increase in opacity caused by the increase in density overcomes the less intense radiation of the slowly increasing internal temperature, and as the star gets smaller and smaller, it becomes fainter and fainter, which means that there is no reason for it to increase its surface temperature. So, if the increase in density due to a decrease in size produces an increase in opacity greater than the excess radiation produced by the increased temperature, the star will get fainter and fainter as it contracts, with little change in surface temperature.
In other words, the more massive stars move to the left because they get hotter faster than they get denser, while the less massive stars move downward because they get denser faster than they get hotter. But what about stars like the Sun? To understand what happens there, we must discuss an unseen effect of the differing ways in which the high and low mass stars contract.
Convection in Stars of Different Masses
The ionization collapse of the protostellar clouds is a rapid, violent event, and at its end, all protostars are undergoing violent vertical mixing, or convection, throughout their interiors (note that one result of this is that they are relatively uniformly mixed at this point, regardless of whether the clouds they formed from were uniformly mixed or not). Whether they maintain this convective motion, however, depends upon how the temperature gradient within the protostars, which is what drives the vertical mixing, changes as they shrink and evolve toward the Main Sequence.
Think of the situation in the current-day Sun. In the deep interior, only scattering impedes the outward flow of radiation. Because the gas is extremely dense, it is very difficult for the light to move outward, and it takes more than a million years for it to move from the core, where it is produced, into the surrounding layers. But as it nears the surface, there is less and less stuff in the way, and the radiation moves outward faster and faster, through the less and less dense gas. However, about a third of the way from the surface, the temperature drops to a low enough value that in addition to scattering, there is a significant amount of absorption by individual atoms or ions, which makes it harder for the radiation to get through the gas than might be expected.
As in the case of the greenhouse effect on Venus, if it is harder for radiation to escape from a region, the temperatures increase in the lower portions of the region, so that enough extra radiation is trying to get out, to balance the extra difficulty of getting out. But if the temperatures increase too much, so that the temperature gradient (the rate at which temperature changes with depth, compared to the distance and/or density of the gas) is too large, then the gas becomes unstable against vertical mixing, and becomes convective.
In the Sun, the outer third of its interior is not only relatively opaque thanks to the additional opacity provided by absorption, but also due to the relatively high density in the outer layers of the Sun, caused by its relatively small size. If we were dealing with a much larger, brighter, hotter star, such as those at the top of the HR Diagram, the outer regions which are cool enough to have significant absorption would have much lower densities (typically, about 1% that of the Sun's similar temperature regions), and despite the higher opacity due to absorption, the low density would result in a relatively low overall opacity, and radiation would just pour out of the gas, without any help from vertical mixing, or convection. So in the Sun, thanks to its relatively high density in its cooler outer layers, the temperature gradient is relatively high, and we have convective mixing; but in much bigger, hotter stars, the relatively low density in their cooler outer layers causes a lower temperature gradient, and convective mixing doesn't occur.
To get an idea of how this works, in very massive stars, central temperatures may be three to four times those inside the Sun; but the sizes of the stars are ten or twenty times larger than the Sun; so the increase in temperature per mile is three to five times higher in the Sun, than in the larger stars -- about 30 Kelvins per mile in the Sun, and only 6 to 10 Kelvins per mile in the larger stars. This is of course an average, and there are areas in the Sun where the temperature gradient is even larger; and because of that, convective mixing occurs. But with their lower overall temperature gradient, convective mixing does not occur in the outer regions of more massive stars, at all.
Conversely, lower-mass stars are not as hot as the Sun, so the regions which have both absorption and scattering reach much deeper into the stars, and since those stars are smaller and denser than the Sun, anyway, the regions that have large opacities also have very high densities, making it even harder for heat and light to escape; and temperature gradients average closer to 100 Kelvins per mile, and convective zones can reach all the way down into the core of the star, despite the high temperatures in the core.
Convection in Contracting Protostars
Now, let's revisit the contraction of protostars. The higher-mass protostars get much hotter as they contract, than they get dense. This means that they are relatively spread out, and although relatively cool compared to their later, Main Sequence structures, so that the opacity of their gases per pound of material is fairly high, being relatively spread out means that it is relatively easy for heat and light to escape through their gases (as already stated), so as they contract, they get hotter and brighter on the outside, and move to the left in the HR Diagram. As they do so, the higher temperatures inside and outside, combined with their relatively large size and low density, cause the convective zone, which extended throughout the star at the start of the protostar stage, to rapidly shrink toward the surface of the star, and long before reaching the Main Sequence, there is no convective zone on the outside of these stars, at all. In other words, as we "see" high-mass stars move to the left in the HR Diagram, not only are they getting hotter and brighter on the outside (as well as on the inside, as all protostars do), but they are changing from a totally convective structure, to a less convective structure, to a structure with no external convection. That is, as a protostar moves to the left in the HR Diagram, its outer convective zone shrinks toward the surface, and may even disappear.
For low-mass protostars, however, the contraction to smaller sizes results in rapidly increasing density, which makes it harder and harder for light to escape, and increases the temperature gradient, so that instead of contracting, the convective zone continues to fill the whole structure of the star. That is, as a protostar moves downward in the HR Diagram, it remains totally convective. This was discovered by an astrophysicist named Hayashi, so we say that the protostar slides down the Hayashi slope (as in a ski slope), toward the lower Main Sequence.
Now, in the case of the Sun, early on, it is relatively cool and dense, and both absorption and scattering are important in determining its internal opacity, so as it contracts, it slides down the Hayashi slope, just like the lower mass stars, getting fainter and fainter, but hardly any hotter (at the surface), as it gets smaller and smaller. But since the Sun is a middling mass star, when its central temperatures rise to several million degress, and absorption ceases in its core, the scattering in the core is not adequate to maintain the previously high temperature gradient, and the convective zone shrinks toward the surface of the star; and as it does so, and heat moves outward more and more easily (with less and less absorption to help fight its outward movement), the Sun moves to the left in the HR Diagram, getting hotter and brighter, just like the higher mass stars.
Now, in some ways, the details just discussed -- the difference in the rate of density and temperature increase, and the continuation of total convection, or movement toward lesser convection -- are relatively unimportant, in discussing the formation and life of stars. For regardless of the details of how the protostars shrink, the more massive ones will always be bigger, hotter and brighter than the less massive ones, and form faster. But when we discuss stellar death, later on, these details, and most particularly, just how convective (or not) different mass stars are, during their Main Sequence lifetimes, will become very, very important; and it is because of that, that I have discussed this topic in such detail.