Online Astronomy eText: Stellar Evolution / The Sun
Review of Heat Flow Inside Stars

General Principles of Heat Flow Inside Stars
      Before discussing how protostars move to the Main Sequence, we need to discuss heat flow inside stars. Inside a star, the hot gases in each region radiate heat in all directions. Heat flows into each region from the hot gases above and below it, and energy may also be generated through nuclear reactions, if the region is close to the center of the star, or through gravitational contraction. All these energy flows must balance in a particular way, or the region would be heating up and expanding, or cooling off and contracting. To be specific, there must be a net flow of heat from the lower regions towards the higher ones, a net flow which is exactly equal to the rate at which heat is lost at the surface, after subtracting any energy generation in the layers lying closer to the surface.
      If you were inside a star, you could not tell which way is up, because you could only see a short distance through the gases, and conditions would be almost the same throughout the region you could see. Under these circumstances, light will flow upwards only if there happens to be more light coming from the lower layers than from the upper ones. Since there is no way to tell which way is up (and, even if there were, the pieces of atoms which make up the plasma inside a star wouldn't care which way is up), this requires that each layer must be brighter than the layer above it. In fact, if we define a "layer" as the distance that you can see through the gas, looking upwards or downwards, then at the bottom of each layer, the gas must be brighter than at the top by approximately the surface brightness of the star.
      As an example of how this works, consider the Sun. At the surface, it gives off over a million watts of power per square foot. Since there is no significant light coming into the Sun from outer space, this produces a net outward flow of radiation of over a million watts per square foot. Now, let's suppose that we were to move inwards by a distance equal to however far you can see, if you are at the surface (which is approximately the same as the bottom of the photosphere). This would require you to go downward several miles, because, near the surface, the gases are much thinner (more rarefied) than the air at the surface of the Earth. At this new location, there would be more than a million watts of power per square foot headed downwards, from the surface (remember, the gases don't know which way is up, so if there is a given amount of energy headed outwards, away from the Sun, at the surface, then there would be an equal amount headed towards the side, and downwards, as well). But the net outward flow of radiation in the new layer has to be the same million or so watts per square foot as at the surface. If there is a downward flow equal to this, then the outward flow at the bottom of this layer would have to be more than TWO million watts per square foot. Similarly, at the bottom of the layer below that, the outward flow would have to be over three million watts per square foot, and, if we continue to go downwards, defining each layer as the distance you can see from the previous layer above it, by the time you are a million "steps" below the surface, the brightness of the gas would have to be more than a million million watts per square foot.
      Of course, if each layer were several miles thick, as the layers near the surface are, then there wouldn't be room for a million layers of this sort inside the Sun. But as you go inwards, the gas becomes denser and denser, as a result of the compression by the gases which lie above it. At the surface, the Sun's gases are hundreds of times thinner than the air at the surface of the Earth, but by the time you are halfway into the Sun, the gases are hundreds of thousands of times denser than that, and in fact, are denser than water, despite the fact that they are primarily made of pieces of hydrogen and helium atoms, which would make a liquid of the same type of atoms considerably less dense than water, and in the center of the Sun, we estimate that the gases are another hundred times denser (more than 100 times the density of water), or nearly a billion times denser than at the surface. Because of this increase in density, although near the surface, a "layer" or "step" into the Sun would be several miles, near the center, each step is only a fraction of an inch, and the total number of "layers" is numbered in the trillions, which means that in the center, the gas must be giving off something of the order of several million trillion watts of power per square foot (this is assuming that the increase in brightness is always the same as near the surface, which isn't quite true, but is close enough for a rough approximation).
      Now, as it happens, at the surface of a star, and inside the star, the fact that you can't see all the way through the gas, as you can in the atmosphere (which is, in fact, defined as the region that you can look through), causes the gas to absorb and radiate light in the same way that is discussed in your text as so-called black-body radiation. For black-body radiation, one thing, and one thing only, determines the brightness of the gas, namely its temperature. So, to have a continual increase in brightness, you have to have a continual increase in temperature, as well. To see how much of a temperature increase you need, you use a relationship which is called Stefan's Law:

The brightness per square foot is proportional to the fourth power of the temperature.

      Since the brightness per square foot in the center of the Sun is trillions of times greater than at the surface, the temperature must be thousands of times greater (and, in fact, is about 2500 times greater, being close to 15 million Kelvins, versus only 6000 Kelvins at the surface). The same sort of relationship would be true for all stars, except that in bright stars, as you go downwards, the number of steps is larger, and the temperatures are even higher, and in fainter stars, as you go downwards, the number of steps is smaller, and the temperatures are not as high.

Heat Flow in the Interior of a Star -- the Radiative Core
      In the inner parts of a star (the radiative core), heat flows outward by the diffusion of photons of light (using the term "light" to mean any kind of electromagnetic radiation, whether visible light, or invisble radiation such ultraviolet radiation and X-rays). Diffusion, in this case, refers to the fact that the photons of light can only go through the gas for a short distance before they are scattered (bounced off of) or absorbed by the gas particles. Mostly, in the deep interior, they are scattered by electrons, so the process of light scattering is sometimes referred to as electron scattering.
      After a photon is scattered, or bounced off of an electron, so to speak, it will probably be heading in a completely different direction from its original direction, and may (although this is fairly rare) even have a slightly different energy, as a result of sharing some of its energy with the electron (or vice-versa). Because the photons don't get to go very far in their original direction before scattering (a distance equal to the thickness of the "layer" that they are passing through, which is less than an inch, near the center of the Sun), they only gradually manage to end up in a different place from where they started out, in what is sometimes described as a random walk. On the average, this process slows down the flow of radiation by an amount proportional to the number of steps involved. If light can move all the way through a region without being scattered, it will take a very small amount of time to do so (namely, the distance involved divided by the speed of light, which is a very short time, even for the widest layers involved). However, if the distance is, say, a hundred times the distance that light can travel in a single step, then, on the average, the light will undergo a hundred squared, or ten thousand, scatterings before it passes through the region, and will be slowed down by a hundred times. Since the number of steps from the surface of the star to the center is, as discussed above, a few trillion, the light takes, on the average, a few trillion times longer to get out of the star than it would if it could just zip right from the core, where it is created through nuclear fusion, to the surface. In the case of the Sun, it would take less than three seconds for light to make that journey, at the speed of light, but since it is slowed down by more than a trillion times, it actually takes more than a million years for light to move from the core to the surface. This means that at any given time, there is more than a million years worth of heat and light trapped inside the Sun, and it is this huge amount of trapped radiation that allows the gas to maintain the huge temperatures that it has, and provides it with the presure required to hold up its weight (which is several tens of millions of times the weight of the Earth, taking into account both the mass of the Sun, and its gravity).
      Now if it takes such a long time for the light to leak out of the star, and there is even the slightest chance that, in some of the collisions, there might be some exchange of energy between the gas particles and the light photons, you might expect that, after some time (and the corresponding distance within the star), the average energy of the gas particles would become equal to the average energy of the photons, and the range of particle energies (given by the Maxwell-Boltzmann Distribution) would be comparable to the range of photon energies (given by the spectrum of the light). However, the energy of the gas particles (or, more accurately, the average random energy of motion per particle) is directly proportional to the temperature of the gas that the light is leaking through, so if the temperature of the gas is higher, which means that the gas particles have a greater energy per particle, then the average energy of the photons leaking through and escaping from one region to another will be higher, as well. But as described the Planck's Law (see chapter 4 in your text), the energy of the photons is inversely proportional to the wavenlength of the light.
      Summarizing the above, if the average energy per photon is equal to the average energy per gas particle (providing the photons have had to struggle through the gas for a while), the average energy per gas particle is proportional to the temperature, and the energy of each photon is inversely proportional to its wavelength, then the result is:

The temperature of the gas is inversely proportional to the average wavelength of its spectrum

This relationship happens to be exactly the same as Wien's Law. It is one of three relationships which describe the way in which black-body radiation is emitted by glowing liquids and solids, and, as it turns out, as discussed here, by gases which are so thick that light has to struggle through them for some time before finally escaping from them. It is much harder to show, without complex mathematics, that the other two relationships are also obeyed, but as in the case of Wien's Law, the following result can be verified. If light has to struggle through a gas for some time in order to get through it, then, by the time that it has finally passed through the gas, it will no longer look like it did when it first entered the gas. Instead, it will be identical to black-body radiation, corresponding to the temperature of the gas. It is for this reason that, as stated above, the relationship between temperature and brightness inside a star is given by Stefan's Law, which is one of those black-body radiation laws. Everywhere inside a star, all the way from the center to the surface, the radiation which flows through the gas is black-body radiation corresponding to the temperature of the gas. When light interacts with gas in this way, we say that the light and the gas are in Local Thermodynamic Equilibrium, meaning that the heat properties (or thermodynamics) of the gas, and of the light, are in balance (or equilibrium).
      Summarizing what has been stated so far, as you go down into a star, each "layer" must be brighter than the layer above it, by an amount approximately equal to the surface brightness of the star, and to accomplish this, as you go down into the star, each layer must be HOTTER than the layer above, by an amount which would be calculated from Stefan's Law.
      There is, however, a notable exception to the above rule, when you are not inside the star, but outside it. As you should already know, in the atmosphere of the Sun, the temperatures increase as you go upwards, not downwards. This seems to violate the above statement, but there is a reason for that. In low-density gases (gases so rarefied that you can look right through them), the light does not have to struggle to get through the gas. In fact, the light emitted by the surface of the Sun goes through most of the atmosphere without even noticing that it is in the way. As a result, it is neither slowed nor altered by its passage through the gas. Instead, it just goes right through it. In this circumstance, the rule that the layers have to be brighter as you go downwards is obeyed, but the rule that temperature has to increase is not obeyed. The contradiction is due to the fact (explained in class, but, so far, not in this supplement) that for optically thin gases (gases thin enough to look through), the gas and the light are not in Local Thermodynamic Equilibrium, and temperature is not related to brightness by the black-body radiation laws, but instead, depends upon temperature and density. It is only in optically thick gases (gases so thick that you cannot see through them, and light has to struggle through the gas for some time to get out of it) that the black-body relationships are true. In the atmosphere of the Sun, since the gases are rarefied, and are optically "thin", density is more important than temperature in determining how bright the gases are. The corona is very hot, but there is practically nothing there, and the large brightness that you would expect from its temperature, multiplied by practically nothing, comes out millions of times fainter than the surface of the Sun. In the chromosphere and photosphere, the gases are cooler, but are denser than in the corona, and as you go downwards, and the density increases, the brightness increases, as well. It is only when you reach the "surface" of the Sun, and the gases become optically thick, that the black-body relationship begins to apply. But from there, all the way to the center of the Sun, that relationship is the only one that has any effect on the brightness, so the temperature has to increase in lockstep with the brightness.

Heat Flow in the Outer Regions of a Star -- Convective Envelopes
      So far, we have restricted our discussion to the regions, deep in the interior of a star, where radiation moves from one region to another through the diffusion of individual photons. This region is called, as mentioned above, the radiative core of the star (there is also a smaller region, where the nuclear fusion of the star occurs, which is sometimes referred to as the nuclear core). However, near the outside of the star, things may work a little differently, because another factor may come into play.
      As mentioned early on, the photons may be either scattered by (bounced off of), or absorbed by the gas particles. The electrons, which make up more than half of the particles in the stellar interior, and any bare nuclei, which make up all the other particles deep within the star, where temperatures are many millions of degrees, cannot actually absorb photons. All they can do is bounce the photons in random directions, and (occasionally) exchange energy with them. However, in the cooler regions near the surface of the star, there may be atoms or ions which have one or more electrons attached to their nuclei, despite the relatively high temperatures. This is especially true of many-electron atoms such as iron, silicon, or oxygen. In the outer regions, near the surface of the star, these particles may have a chance to actually absorb the energy of a photon, and in one way or another, hold onto it for some short period of time. Eventually, they will give up the energy they absorbed, either as a similar photon, or as several photons of lower energy, with a total energy approximately equal to the energy of the absorbed photon. The time involved between the absorption and subsequent emission(s) is very short, but since light travels so fast, even if the energy is held onto for even a few millionths of a second, it may "block" or "slow" the outward flow of the light as much as several millions of individual scatterings. As a result, even if the amount of such absorption is relatively small, it may enhance the opacity (the amount of blockage of the light by the gas), and slow down the light, by a substantial amount.
      Whether this is true or not depends upon the density of the gas in the region where the temperature is low enough for absorption to occur. If the star is huge, and relatively hot, the region near the surface which is cool enough for such absorption is so spread out, and so low in density, that the overal opacity of the gas (its ability to block gas) is very low, and so any additional opacity added by absorption is not very important, and nothing happens, except that the light has a slight additional amount of slowing. However, if the star is smaller, or cooler, so that the region cool enough for absorption reaches deeper into the star, and the density of the regions where absorption is significant is somewhat larger, the additional absorption may become quite significant, and cause an interesting side effect, namely a vertical mixing, or convection of the gases, in a region referred to as the convective envelope (so-called because it is in convective motion, and is on the outside of the star).
      Whether convection occurs or not depends upon how fast the temperature has to increase, as you go inwards, in order to cause the light from the interior of the star to flow through the gas at a rate equal to the surface brightness of the star. In the upper Main Sequence stars referred to as blue supergiants, the star is as much as 20 times the size of the Sun, which gives those stars nearly ten thousand times the volume of the Sun. Such stars are several tens of times as massive as the Sun, but since they have such huge volumes, they are as much as a hundred times less dense than the Sun. Near the surface, instead of being hundreds of times less dense than the air at the surface of the Earth, they are hundreds of thousands of times less dense. Halfway down from the surface to the center, instead of being denser than water, they are hardly any denser than ordinary air. And even in the center, where the Sun is more than a hundred times as dense as water, such stars are only about the same density as water.
      Since such stars have such low densities, it is easy for the light which is made inside them to leak from layer to layer, and as you go inwards, the temperature increases relatively slowly (as little as three Kelvins per mile). Even though, in the center, they require huge temperatures (as much as 30 or 40 million Kelvins) in order to produce the immense amount of thermonuclear fusion which sustains their brightness, they are so large that the temperature can go up very slowly, and still attain very high temperatures in the center.
      In the case of the Sun, however, the gas is much denser, and so it is much harder for the light inside the Sun to struggle from layer to layer. The effect of this is to increase the rate at which temperature increases (in the same way that, at the surface of Venus, the greenhouse gases in its atmosphere, by making it hard for the heat of the planet to escape, raise the surface temperature by nearly a factor of three, compared to what it would otherwise be). Whereas a blue giant may have a temperature gradient (the rate of temperature increase) of only three degrees per mile, the temperature gradient of the Sun is around 30 Kelvins per mile (about 15 million Kelvins, in less than half a million miles).
      Because of the fact that, in the outer regions where the light is blocked by both scattering and absorption, the denser gas within the Sun makes it much harder for light to escape, and causes a much higher temperature gradient than in larger, hotter Upper Main Sequence stars, the rate of temperature rise can become so large that the gas is not stable against vertical mixing, or convection, and vertical mixing of regions as large as the Earth, or larger, moving upwards (and downwards) at speeds of several miles to a few tens of miles per second, helps move heat outwards. In very large, hot stars, with much lower densities in the regions where absorption is significant, the temperature gradient required to create convection is higher than the actual temperature gradient, and there is no convection. Heat flows outwards only through the diffusion of photons (radiation) from one place to another. But as you move from hotter stars to cooler, denser stars, the actual temperature gradient in the outer regions increases, and a convective envelope forms in the regions near the surface where density and opacity are high enough to force such mixing. As you move to still cooler, or denser, stars, the convective envelope grows, reaching further and further into the star, and for the coolest, and densest, stars, the envelope actually reaches down into the core of the star.