Online Astronomy eText: Stellar Evolution
The Late Main-Sequence Life of the Sun
(also see The Main Sequence and Outline of Stellar Death)
Summary of Main Sequence Life
     The Main Sequence represents a long, stable stage of stellar life in which the thermonuclear fusion of hydrogen to helium in the core replaces the heat escaping at the surface. Prior to reaching the Main Sequence the star's energy source is a gradual contraction. The compression of its gases raises the density, temperature, brightness, pressure and gravitational force in its interior and changes its external appearance, so that the dot representing its properties (in a Hertzsprung-Russell Diagram) moves downward or to the left over a period of time.
     Once nuclear fusion begins, however, the net loss of heat which has to be replaced by gravitational contraction is reduced, so the contraction slows as the rate of nuclear fusion increases. When the rate of nuclear fusion is equal to the heat loss at the surface the star undergoes an adjustment of its structure, from one in which the heat loss is replaced throughout the star (by its overall contraction) to one in which the energy is produced only in the core and the rest of the star just passes the heat to the surface, and the contraction of the star comes to an end. For however long the fuel in the core lasts (as discussed in The Mass-Luminosity Diagram and the Lifetime of Main Sequence Stars), there is no obvious reason for the star to change, so year after year the dot representing the star's properties remains fixed in its Main Sequence position in the HR Diagram.

How The Sun Changes As It Ages
     To a good first approximation a star like the Sun remains the same throughout its Main Sequence lifetime, but it does change as it ages, just a little bit at first, but more and more the older it gets. In fact, as shown below those changes have probably made the Sun about 25% larger and 50% brighter than when it first formed, 4 1/2 billion years ago.
     When the Sun was first visible (as a protostar) it had just finished its ionization collapse, and was undoubtedly undergoing violent vertical mixing throughout its interior. This mixing continued as it moved downward in the HR Diagram, but as its core became hotter the contribution of absorption to opacity gradually diminished, and as the bottom of the convective zone moved toward the surface, the dot representing the Sun's characteristics began to move to the left in HR Diagram. Its present position corresponds to its present structure, in which the convective zone only reaches about a third of the way down into the solar interior. But even though the central regions were no longer being mixed, there was no reason for them to become unmixed either, so at the time the Sun reached the Main Sequence (that is, achieved energy production from proton-proton cycle hydrogen fusion at a rate equal to its luminosity), it should still have been thoroughly mixed, with about 75% of its mass in the form of hydrogen atoms (or more accurately, bare protons and free electrons), 25% in the form of helium atoms (or more accurately, alpha-particles and free electrons), and about 2% in the form of heavier atomic nuclei (primarily carbon, nitrogen, oxygen and neon). Since this composition would have been the same throughout the Sun, a graph showing the relative proportions of different materials in different regions consists of straight horizontal lines, as shown below.
Diagram showing the composition of the Sun at the start of its Main Sequence life
The composition of the Sun at the start of its Main Sequence life.
The proportion of different elements was the same everywhere.

     Although the composition of the young Sun was uniform, its energy production was not. The energy production per cubic foot was higher near the center, where the density and temperature are higher, and lower further out, where the density and temperature are lower, and becomes negligible within 10% or so of the Solar radius from the center. In a thoroughly mixed lower-Main Sequence star (a third or less the Sun's radius), the helium produced by hydrogen fusion would be mixed with the rest of the star, and fresh hydrogen from the outer regions mixed with the interior, so the composition of the star would remain uniform. As such a star ages, the line showing the proportion of hydrogen and helium would gradually go down, but would remain a straight horizontal line. As a result, in very old thoroughly mixed stars, the proportion of hydrogen to helium would be noticeably different than in very young stars; but since such stars last for trillions of years, none of them has used more than 1% of their hydrogen, and any changes in their ratio of hydrogen to helium are undetectable.
     But the Sun is not thoroughly mixed, since the convective zone only reaches a third of the way into the interior; and as hydrogen is turned to helium in the core, the line showing the proportion of hydrogen to helium would go down in the core while remaining the same outside the core, as shown below. (Note: The rate of hydrogen burning at different locations and the corresponding change in the composition of the core are shown by straight lines to simplify drawing the illustration. A truly accurate graph would use curves instead of straight lines.)
Diagram showing the composition of the Sun as it begins to age
The composition of the Sun as it begins to age.
Hydrogen is turned to helium at a rate proportional to the rate of fusion.

     As (roughly) shown above, the change in the composition of the core is more or less a mirror image of the rate of hydrogen burning. Toward the outside of the core, where hydrogen is hardly being burnt, there is very little change in the composition; but as you move toward the center, where hydrogen is being burnt at greater and greater rates, the amount of hydrogen used over the eons (for this discussion, an eon is one billion years) is greater and greater as well, and the remaining hydrogen is less and less. In other words, where the hydrogen is being rapidly used, there is less of it left. The three diagonal lines (roughly) represent the remaining hydrogen at 1, 3 and 5 billion years after the start of Main Sequence life.
     Now if the rate of fusion were exactly as shown in the diagram -- proportional to the square of the density and the cube of the temperature -- then the declining proportion of hydrogen would be of little importance. But since hydrogen is the fuel that is involved in the reaction, the appropriate density is not the total density, which barely changes (since the helium produced by fusion has nearly the same mass as the hydrogen used up), but the density of hydrogen, and as that goes down the rate of hydrogen burning should be reduced as well, as shown below.
Diagram showing how the energy output of the Sun's core should decrease as it runs out of hydrogen
As the Sun runs out of hydrogen, its energy output should decrease.

     As shown above, the conversion of hydrogen to helium, by reducing the hydrogen density dH, should reduce the energy production in the core. But this cannot happen, because the star will only remain stable if the energy production in the core is the same as the rate of heat loss at its surface. So something has to change to make up for the poorer quality of the fuel. What happens is the core contracts, increasing its density and temperature until the energy production is the same as the rate of energy loss. The contraction involved is not very fast -- only about 1/1000th as fast as the contraction that the Sun had before it became a Main Sequence star -- because it doesn't have to replace the "lost" energy production. It just has to give the density and temperature a little boost, so that the now poorer-quality fuel can be used at the same rate as before.
     Suppose, for example, that the Sun had used up about half the hydrogen in its central core (which isn't far from the truth). If no adjustments were made that would reduce the energy production in that region by 75%. But the energy production is proportional to the square of the density, which is inversely proportional to the cube of the size of the region, and the cube of the temperature, which is inversely proportional to the size of the region. In other words, a 1% reduction in the size of a given region would increase the temperature by 1%, the density by 3%, and the energy production rate by nearly 10% (a little over 3% because of the temperature increase, and a little over 6% because of the density increase), and (taking compounding into effect) a reduction of just over 5% of the central core's radius would nearly double its energy production.
    The problem with this is, the outer portions of the core have nearly the same amount of hydrogen as ever, so they hardly need to contract at all, while the inner portions have considerably less hydrogen, so they need to contract significantly more. So to exactly reproduce the original energy production rate, the central portions would contract 10% or more (in the 5 billion years under discussion), while the outer portions would hardly contract at all. And if the inner regions contract a lot, the outer regions will be forced by the weight of the gases above them (which is most of the weight of the star) to follow the inner regions as they contract, causing them to become hotter and denser and brighter than they need to be to maintain their original energy production. So if the inner core is allowed to contract enough to maintain its original brightness the core as a whole will become much brighter, as shown below.
Diagram showing how a uniform contraction of the Sun's core would cause it to produce too much energy
If the central core contracts enough to offset its lower hydrogen content,
the outer core contracts too much, and the core becomes brighter than ever.

     What actually happens is a sort of compromise. Since the outer core is producing more energy than needed, the central core doesn't have to produce as much as it originally did to maintain the overall energy of the star. It still has to be brighter and hotter than the surrounding regions, so it still has to produce some energy; but it can get by with a somewhat reduced energy production and a somewhat reduced contraction, so that the outer core doesn't have to contract too much or become too bright. And if the reduced energy production in the central core balanced the excess energy production in the outer core, the core as a whole might remain constant in brightness. But as in all circumstances in which we try to change the conditions inside a gaseous object, the brightness does not remain constant. Just as, if we tried to cool off a protostar it would contract, compress its gases until they were hotter and denser than ever, and (for stars like the Sun) become brighter than ever, the contraction of the core caused by the poorer quality fuel not only overcomes the lower energy production that would otherwise have occurred, but increases the brightness of the star, as shown below.
Diagram showing the compromise in energy production in different parts of the Sun's core as it begins to run out of fuel
The central core contracts less than it needs to and becomes fainter,
while the outer core contracts more than it needs to and becomes brighter.
(previous energy production labels removed to reduce clutter and confusion)

Summary, and Consequences for the Sun
     As the core of the Sun runs out of hydrogen it must very slowly contract to make up for the poorer fuel quality. The central core needs to contract more than the outer core to maintain constant brightness; but the weight of the rest of the star forces them to contract together, with the result that although the central core contracts less than needed to maintain its brightness, the outer core more than makes up for it, and the overall brightness of the core increases, instead of decreasing, as it runs out of fuel. As time goes on the core uses more and more of its hydrogen and contracts more and more, which makes the star brighter and brighter. At first this effect is negligible, and the one-billion year old Sun hardly differed from the newly formed Sun. But as the eons go by and hydrogen becomes less and less abundant, the rate of contraction of the core increases, it becomes brighter and brighter, and as the core increases in brightness the excess radiation flooding outward causes the rest of the star to swell to larger and larger size and greater and greater brightness itself (since it must have the same brightness as the core). As a result, the Sun is now about 25% larger and 50% brighter than when first formed, and is growing about 5% larger and 10% brighter every billion years.

Consequences for the Main Sequence
     But if the Sun is now larger and brighter than it was when it reached the Main Sequence, that means that it was once only 70% as bright as now; and if it continues to get larger and brighter, it will eventually be much brighter than now. In other words, stars like the Sun do not maintain constant brightness while on the Main Sequence, but gradually increase in brightness. And by the time the Sun is nearly finished with its Main Sequence life, it will be about half again as large and twice as bright as when it became a Main Sequence star. But this means that Main Sequence stars of the same mass do not have the same properties, as was implied in earlier pages, but some variation in those properties, according to -- among other things -- how old they are. In other words, the Main Sequence has some "fuzziness" or "thickness".
     Of course in observing the properties of stars, we would expect the Main Sequence to be fuzzy, because we have to know exactly how bright the stars are and exactly how hot to know where to plot the dots that represent their properties, and there are always some uncertainties in those measurements and the resulting values. In addition, not all stars have the same composition. Stars in very old clusters of stars formed more than ten billion years ago have very few atoms other than hydrogen and helium (often less than one percent by mass), while stars formed four to eight billion years ago such as the Sun usually have a greater proportion of heavy atoms (about two percent by mass), and stars in some young clusters have as much as four or five percent of heavy atoms by mass. This difference in composition is caused by the gradual transformation of interstellar clouds into stars (so that there is less hydrogen and helium than originally), and the mixing of the ashes of dead stars with the remaining gases (usually in iron-core-collapse supernovae), with the result being a gradual increase in the percentage of heavy atoms inside the clouds of gas that form new stars.

Consequences for the Sun (and the Earth)
(future additions to this discussion, probably in new pages)
Not all the core's fuel is used while on Main Sequence, leading to Red Giant phase
Past and future effects on the Earth's climate (will life on Earth survive?)
The Fate of the Earth (will the Earth itself survive?)