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# 12.23 Understanding the Main Sequence

Why are distinctive types of stars, such as the main sequence stars, the giants, and the white dwarfs found at different positions on the H-R diagram? The simple answer is: stars have different masses. High-mass main sequence stars are hotter and brighter and bigger than low-mass main sequence stars. Is there any simple relationship between mass and luminosity? How do stars in different parts of the H-R diagram evolve? Can one type of star turn into another type of star? Astronomers use the H-R diagram as a diagnostic tool, with the main sequence as its focal point.

Hertzsprung-Russell diagram showing color and size of stars. Click here for original source URL.

Arthur Eddington. Click here for original source URL.

Much of the systematic work on stellar properties was begun by the English astrophysicist Arthur Eddington in the 1920s and was carried on by other scientists through the 1950s. Eddington showed that the same two opposing influences at work in the Sun are competing in any star: gravity pulls inward on stellar gas while gas pressure pushes outward. This is the principle of hydrostatic equilibrium. In any stable star, these forces are balanced at every point within the star. The internal gravity of a star is determined by its mass, so mass is a fundamental property of a star even though it may be hard to measure directly.

Heat is produced during gravitational contraction of a star. A stable main sequence star is one that has contracted until the inside is hot enough to start nuclear reactions among hydrogen atoms. At this point the interior becomes a stable heat source, radiating light and creating enough outward pressure to counterbalance the inward force of gravity. What had been a contracting ball of gas now becomes a star with a constant size, governed by the release of energy in the interior. All main sequence stars convert hydrogen into helium by the fusion process. The main source of energy in a main-sequence star's interior is nuclear reactions in which hydrogen is consumed. In this sense, all main sequence stars are like the Sun. However, main sequence stars can be a million times more luminous than the Sun or ten thousand times less luminous than the Sun. Astronomers need to explain the amazing range of energy output on the main sequence.

Because stars form with a huge supply of hydrogen, stars remain stable on the main sequence for a relatively long time at a fixed size. If a stable star were magically expanded, its gas would cool and the nuclear reactions would decline, reducing the outward pressure. The outer layers would fall back to their original state. If it were magically compressed, the inside would get denser and the reactions would increase, raising the outward pressure and expanding the star. A main sequence star tends to stay in a stable state due to the hydrostatic equilibrium that exists at each point within the star. Hydrostatic equilibrium is like a "thermostat" for the star, a form of negative feedback that damps out variations and keeps the star stable. The star will remain stable as long as its internal composition and energy production rate stay the same.

Calculations based on these ideas revealed a mass-luminosity relation among main sequence stars. A hydrogen-fusing star more massive than the Sun has higher luminosity and surface temperature than the Sun. Hence, on the H-R diagram it would lie to the upper left of the Sun. Similarly, a star of lower mass would lie to the lower right of the Sun. These stars therefore fall along a line in the H-R diagram — the main sequence. The main sequence is thus explained as the group of stars of different masses that have reached stable configurations and are generating energy by consuming hydrogen in nuclear reactions.

Mathematically, a star’s luminosity is proportional to the mass to the 3.5 power. In solar units, L = M3.5. This is a very steep relationship, as a few examples will show. A star with five times the mass of the Sun will have a luminosity of 53.5 = 5 × 5 × 5 × √ (5) = 280 times that of the Sun. Conversely, a star with only one tenth the mass of the Sun will have a luminosity of 0.13.5 = 0.1 × 0.1 × 0.1 × ?(0.1) = 1/3160 times that of the Sun. Massive stars are like powerful beacons. Low-mass stars are like puny flashlights. Why does luminosity depend so strongly on mass? An increase in mass corresponds to a star with a substantially higher temperature in the core. The luminosity of the star depends on the way radiation diffuses out from the hot central regions, subject to hydrostatic equilibrium at every point. The result of these calculations is a luminosity that depends strongly on temperature, and so on mass. Intriguingly, the energy generation mechanism does not feature prominently in this calculation. That's why astrophysicists like Arthur Eddington were able to understand the main sequence even before the theory of the fusion process was developed!

The main attributes of main sequence stars can be summarized as follows:

• Main sequence stars are converting hydrogen into helium by nuclear fusion.

• Main sequence stars are stable, with an internal structure governed by the principle of hydrostatic equilibrium.

• High mass main sequence stars are larger, more luminous, and have hotter photospheres than low mass main sequence stars.

• The main sequence on the H-R diagram defines a sequence of mass, with luminosity and mass related by L = M3.5.

• Main sequence stars are all in the first stage of stellar evolution, the stage between formation and the end of hydrogen fusion in the core.

The H-R diagram represents smoothly increasing mass and luminosity as we ascend the main sequence. We can now follow the chain of reasoning that explains the main sequence of stars. More massive stars have greater gravity that creates higher temperature in the stellar interior. The higher temperature and the way the energy is transported outwards give both higher luminosity and higher surface temperature. (The same reasoning doesn’t apply to stars that lie off the main sequence; a star's position on other parts of the H-R diagram is not a simple indicator of mass. In other words, giants, supergiants, and white dwarfs are stars that do not have exactly the same energy generation, composition, or energy transport as main sequence stars.)

A census of main sequence stars shows most of them to be less massive than the Sun. One reason for this is that the star-formation process tends to make many more low-mass stars than high-mass stars. Another reason is that lifetime on the main sequence increases dramatically as mass decreases — low mass stars live much longer than high mass stars.