Summary of Previous Discussion
In From Protostars to the Main Sequence, we discussed how different mass stars "move" across the Hertzsprung-Russell Diagram during their contraction to the Main Sequence (the curve from upper left to lower right found in some partial form in the HR Diagram of any group of stars): (1) massive stars move directly to the left, becoming hotter, brighter and non-convective, (2) low mass stars move downward, becoming fainter and fainter while hardly changing their temperature and remaining totally convective, and (3) middling mass stars like the Sun act like low mass stars at first and high mass stars later on, and are only partially convective by the time they reach the Main Sequence.
The approach of stars to the Main Sequence
The Approach to the Main Sequence
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As the stars approach the Main Sequence, they are at any given time, in the same sort of quasi-equilibrium that they have been in for most of the time since their initial collapse, with pressure and compressive forces in balance at any given time, but with the continual radiation of heat and light into space causing the stars to gradually contract to smaller, denser, hotter, higher pressure/gravity objects. As a result, their interiors, and most particularly their cores, soon rise from the hundreds of thousands of degrees that they were at the end of the ionization collapse, to millions of degrees; and when the temperatures near ten million Kelvins, things begin to change.
Up to this point, any heat loss by the protostars can only be accommodated by making the star smaller. But as the temperature reaches ten million Kelvins, nuclear fusion begins, in the form of the conversion of hydrogen to helium via the proton-proton cycle (the same energy prodcution method as in the core of the Sun).
The energy production of the proton-proton cycle is dependent on the density and temperature of the gas, approximately (for the temperature range of interest) as the square of the density, and the cube of the temperature:
The density is involved because the more particles there are per cubic inch, the more particles are available for collisions (and hence fusion reactions), and the less distance they have to go to collide with other particles, which also increases the rate of collisions. The temperature is involved because the particles which fuse are positively chargedprotons, which repel each other throughout their approach to each other. In fact, at the low temperatures in stars (low, that is, compared to the temperatures simulated in nuclear accelerators), the particles almost never get close enough to react with each other, and a typical proton bounces off of other protons hundreds of millions of times per second, for over thirty million seconds per year, and in the entire ten billion year Main Sequence lifetime of the Sun, has only one collision that results in a nuclear fusion. That is, only one collision in tens of thousands of millions of millions of billions of collisions accomplishes anything, and if it weren't for the tremendous number of partciles running around, hardly anything would ever happen. But since each cubic inch in the central hundred thousand miles and thirty thousand Earth masses of the Sun's core contains hundreds of billions of trillions of protons, although the chance of anything happening to any one proton is very close to zero, the chance that out of all the protons something will happen is so large that every cubic inch has many reactions per second, and half a billion tons of protons undergo fusion every single second, creating the four hundred trillion trillion watts of photons that leaks from layer to layer, and is lost at the surface.
Now, let's consider a star that has just barely become hot enough for nuclear "burning" to begin, deep in its core. As it radiates heat, very little is replaced at first by the fusion reactions in the core, and so the star continues to contract at very nearly the same rate as before. In the case of the Sun, this is estimated to have been about twenty feet per year just before it reached the Main Sequence, while for a much bigger, brighter star it might be several miles per year, and for a much smaller, fainter star, a few inches per year.
If the nuclear fusion made no difference in the rate of contraction of the star, then as it contracted, year after year, the density and temperature would go up at a gradually increasing rate, and the rate of nuclear burning would go up, at a much more substantial rate. In fact, if there were no change in the rate of contraction as a result of the nuclear burning, the increase in density and temperature that resulted from that contraction would cause the energy production to increase at an exponential rate, going faster and faster, until it became so tremendous that it completely stopped and reversed the contraction of the star, ejected its exterior layers, and caused the fury of the star's nuclear reactions to become visible at the surface. And as discussed in the box below, this is a very common misconception among introductory students.
How fusion would begin in a star with constant contraction
At first, energy production would be very small; but as the star continued to contract, density and temperature would rapidly increase, energy production would rise as the fifth power of that rapid increase, and become catastrophically high in a relatively short periods of time.
Nuclear Fusion Is NOT Why Stars Are Hot And Bright|
Before going further, I want to address a very common error in thinking about the heat source of stars. Suppose you were asked, why is the Sun so hot and bright? The obvious answer is the fusion of hydrogen to helium in the core of the Sun. But that is wrong. The fusion of hydrogen to helium is what keeps the Sun hot and bright, but is not what made it that way. What made it that way was the heating of its gases by extreme compression -- its collapse and contraction from a cloud of interstellar gas hundreds of thousands of AUs in size, to a ball of plasma less than one two-hundredth of an AU in size. During that reduction of tens of millions of times in diameter, and thousands of millions of trillions (approximately 21 zeros) in volume, the gas inside the Sun became very, very hot -- more than ten million Kelvins hot -- and as a result, thermonuclear fusion began, which maintains the Sun's heat and brightness to this day, and will continue to do so for billions of years to come. But -- and this is a very important "but" -- even before nuclear reactions began in the core of the Sun, it was nearly as hot and bright as it is, and if those reactions had never begun, it would have continued to contract, and would have become even hotter and brighter than it is. It just wouldn't have lasted as long.
The Onset of Nuclear Fusion, and the End of Protostellar Contraction
The way in which things happen as stars approach the Main Sequence is shown in the diagram below. Prior to the beginning of nuclear fusion, the protostar is radiating a certain amount of heat (the difference between the top line, "heat radiated at the surface", and the bottom line, "no fusion energy production"). This causes the protostar to contract at a certain rate (in the case of the proto-Sun, about twenty feet per year).
When nuclear reactions first begin (near the left side of the diagram, at line "1"), the amount of energy they produce is very small, because the density and temperature are just barely big enough to produce any fusion. This insignificant energy production doesn't do anything to reduce the heat loss, so the star continues to contract at approximately its original rate, and energy production rapidly increases, exactly as in the case of constant contraction, discussed above. However, as time goes on, and the energy production becomes more and more significant, the net loss of heat, indicated by the difference between the heat loss by radiation and the current production of energy through fusion (and the vertical red line connecting them in the diagram) becomes smaller, and the contraction of the protostar slows a little (at line "2", by about 20%, or in the case of the proto-Sun, about four feet per year).
As time goes on, the protostar continues to contract, and the density, temperature, and nuclear fusion rate continue to increase; but since each increase in the fusion rate decreases the net loss of heat, the protostar contracts more and more slowly. By the time the central position (line "3") is reached, the protostar is contracting only a third as fast as originally, and the last position (line "4") involves a heat loss and shrinkage rate less than a tenth of the original contraction (in the case of the proto-Sun, the rate of contraction had gone down, from around twenty feet per year, to less than two feet per year, by this point). And as the energy production by fusion approaches the heat loss at the surface, the net heat loss approaches zero, and the contraction of the protostar slows still further, and completely stops at the moment that the heat loss at the surface and the heat production in the core become exactly the same.
The onset of energy production, as stars approach the Main Sequence (shown in blue)
The exponentially rising (brown) curve represents the onset of energy production if there were no change in the rate of contraction of the protostar. But since that rate is determined by the net heat loss = heat radiated minus energy produced by fusion, as shown in red, the increase in energy production is stretched out, more and more. At first (at 1), the energy production through fusion is small, and the protostar contracts at almost the same rate as before the start of nuclear fusion; but as the fusion reaction goes faster (2) and faster (3) the net heat loss gets smaller and smaller, and the contraction slows, slowing the rate at which energy production increases. As the energy production nears the heat loss (4), the contraction practically stops, and when the energy production is exactly equal to the heat loss (at the far right), the contraction completely stops, and there is no further increase in energy production, for however long the nuclear fusion can replace the loss of heat, and prevent any further contraction of the star.
(STILL WORKING FROM THIS POINT ON)
(Main Sequence = heat loss in core replaces heat loss at surface, so no need to contract, so unlike protostellar contraction stage, in which the size of the star is continually decreasing, and its density, temperature and brightness are continually changing, there is no reason for the star to change its properties, for however long the nuclear fuel can last)
What Is The Main Sequence?
The Main Sequence represents the longest, most stable state of stellar life for all stars. Prior to the start of Main Sequence life, stars are throwing away heat, but have no way to replace it, save by contracting to smaller size. Gravity can provide tremendous amounts of energy to the gases it compresses (as will be detailed below), but if a star like the Sun had to rely on gravity alone, to supply its energy needs, it couldn't shine as it does for more than a few tens of millions of years, without becoming drastically different from its present state. And since the Earth -- and, presumably, the Sun as well -- is 4.5 billion years old, or many hundreds of times older than the tens of millions of years just mentioned, the Sun must have some other energy source, or it would have stopped shining a long, long time ago.
That energy source is, of course, the thermonuclear fusion of hydrogen into helium, via the proton-proton cycle, at a rate of more than half a billion tons per second. As staggering as this rate of nuclear "burning" is in comparison to our puny abilities, the thirty-some thousand Earth masses of hydrogen in the core of the Sun can burn hydrogen at this rate for more than ten billion years, at the present rate, and still not run out of fuel. So there is certainly no need to worry about the fuel running out anytime soon, although we will be discussing the consequences of its running out at some time in the future, in a later page.
All Main Sequence stars, whether like the Sun, much brighter than the Sun, or much fainter than the Sun, work in exactly this same way. There are differences in the details, of course. The larger, hotter, brighter stars at the top of the Main Sequence burn their fuel as much as a million times faster than the Sun, and cannot possibly last more than a few million years, before running out of fuel; and as it happens, the details of how they burn the hydrogen are different than in the Sun, carbon atoms being used to catalyze the hydrogen fusion and make it go much faster, so that the specific process involved is called the CNO or Carbon Cycle. However, the net result of the process is absolutely identical to that in the Sun -- four hydrogen atoms are turned into one helium atom, 0.7% of the mass of the hydrogen is transformed into energy, and that energy is what keeps the star shining so brightly, from day to day, year after year after year.
Similarly, the low-mass stars near the bottom of the Main Sequence also turn hydrogen into helium, and in fact do so in exactly the same way that the Sun does, but much, much more slowly, since they are so faint, and as a result can last for tens of trillions of years, before running out of fuel.