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S 102 10 1

10-1

3 x 104

3 x 103

Evolutionary tracks away from the main sequence on an HR diagram. Each track is marked by the mass for the model.The dashed line is the zero-age main sequence (ZAMS).

3 x 104

3 x 103

therefore increases, and the evolutionary track moves vertically. The star is then a red giant.

By the time the star becomes a red giant, the energy transport in the envelope is convective. This is because of the large value of — dT/dr. The analogous situation on Earth involves the heating of the atmosphere. Sunlight heats the ground, and then infrared radiation from the ground heats the air. (This explains why the air is cooler at high altitudes; it is farther from the direct heat source, the ground.) In this situation, we say that the energy transport is radiative. However, if —dT/dr becomes larger, then —dP/dr, the rate at which the pressure falls, also becomes large. The air that is heated near the ground expands slightly, and becomes very buoyant, being driven upward by the pressure difference between the bottom and top of any parcel of air. The hot air rising, being replaced by cool air falling, known as convection, becomes the dominant mode of energy transport.

We now look at the evolution of the core while the star is becoming a red giant. The temperature of the core climbs to 108 K. This is hot enough for the triple-alpha process to take place (equations 9.11 and 9.12), fusing the helium into carbon. The density is so high that the material no longer behaves like an ideal gas. This is called a degenerate

Evolutionary tracks away from the main sequence on an HR diagram. Each track is marked by the mass for the model.The dashed line is the zero-age main sequence (ZAMS).

gas. We will discuss degenerate gases in Section 10.4, but for now we note that the equation of state is very different for a degenerate gas. In an ideal gas, when the triple-alpha process starts, the extra energy generated causes an increase in pressure, which causes the gas to expand, slowing the reaction rate. This keeps the reactions going slowly. In a degenerate gas the pressure doesn't depend on temperature and no such safety valve exists. The conversion of helium to carbon takes place very quickly. We call this sudden release of energy the helium flash. The energy released causes a brief increase in stellar luminosity.

Following the helium flash the energy production decreases. The core is no longer degenerate, and steady fusion of helium to carbon takes place. This region is surrounded by a shell in which hydrogen is still being converted into helium. At this point the star reaches the horizontal branch on the HR diagram. The outer layers of the star are weakly held to the star, since they are so far from the center. The star begins to undergo mass loss. The subsequent evolution depends on the amount of mass that is lost.

Eventually all the helium in the core is converted into carbon and oxygen. The temperature is not high enough for further fusion, and the core again begins to contract. A helium-burning shell develops, and the rate of energy production again increases. The envelope of the star again expands. On the HR diagram the evolutionary track ascends the giant branch again, reaching what is called the asymptotic giant branch. Stars on the asymptotic giant branch are more luminous than red giants. The star can briefly become large enough to become a red supergiant at this stage. The star can also undergo oscillations in the rate of nuclear energy generation.

10.1.2 High mass stars

More massive stars live a shorter lifetime on the main sequence than do lower mass stars. As with the lower mass stars, the main sequence lifetime for higher mass stars ends when the hydrogen in the core is used up. The core then begins to contract, and the temperature for helium fusion to heavier elements is quickly reached. The helium fusion takes place before the core can become degenerate. Therefore, in contrast with the helium flash in lower mass stars, the helium burning in more massive stars takes place steadily. At this point, the star has a helium-burning core with a hydrogen-burning shell around it (Fig. 10.3).

When the helium in the core is exhausted, the temperature is high enough for the carbon and oxygen to fuse into even heavier elements. At this time, we have a carbon- and oxygen-burning core, surrounded by a helium-burning shell, which in turn is surrounded by a hydrogen-burning shell. As heavier elements are built up, the core develops more layers.

As the luminosity of the core increases, the outer layers of the star expand. The atmosphere cools with the expansion, but the size increases sufficiently for the luminosity to increase. At this point the envelope is convective, and the temperature gradient is limited by the adia-batic lapse rate. So the envelope must grow to a large size to accommodate the large temperature difference between the core and the surface. Eventually, the radius of the star reaches about 103 R©. At this point the star is called a red supergiant.

(Not to Scale)

Shells in the core of a high mass star as it evolves away from the main sequence. (a) The core is only a small fraction of the total radius. (b) In the core, there is a succession of shells of different composition. Each shell has exhausted the fuels that are still burning in shells farther out.

(Not to Scale)

H Burning He Burning C Burning Ne Burning O Burning

H Burning He Burning C Burning Ne Burning O Burning

Shells in the core of a high mass star as it evolves away from the main sequence. (a) The core is only a small fraction of the total radius. (b) In the core, there is a succession of shells of different composition. Each shell has exhausted the fuels that are still burning in shells farther out.

10.2 I Cepheid variables 10.2.1 Variable stars

If we monitor the brightnesses of certain stars, we find that many oscillate with time. These are known as variable stars. The periods of variability range from seconds to years. We have already seen that eclipsing binaries appear as variables. However, many stars have luminosity variations associated with physical changes in the stars themselves (rather than simply by eclipsing one another).

Since we will be using specific stars as examples, we will briefly mention systems for naming normal and variable stars. The bright stars are named, in order of brightness within their constellation, by a Greek letter, followed by the Latin genitive form of the constellation name. An example is a Orionis (abbreviated as a Ori). Some of the brightest stars are also known by their ancient names. For example, a Ori is Betelgeuse. Variable stars are listed in order of discovery within a given constellation. The first is designated R (e.g. R Ori), the next S, and so on to Z. After that, two letters are used, starting with RR, RS to RZ, then SS to SZ, and so on, until ZZ is reached. Then comes AA through AZ, BB through BZ, and so on to QZ. (The letter J is never used because of possible confusion with I.) This gives a total of 334 variable stars per constellation. Beyond that, numbers starting with 335, preceded by a V (for variable), are used (e.g. V335 Ori, V336 Ori, etc.)

For any particular star, we are interested in producing a light curve, a graph of its magnitude as a function of time. Studies of variable stars often require very long term monitoring. In some cases, it is possible to recover information on a star's variability from plate archives. When photographic plates are taken at an observatory, the astronomer who took them is often required to return the plates when that astronomer's work has been completed. The astronomer may be interested in only one star on the plate, but it contains a record of many stars. With the advent of CCD observations, archives are no longer being kept in the same manner. Observations of many variable stars can be so time consuming that it has become an area of astronomy where amateur observers have been able to make major contributions, generally coordinated by the American Association of Variable Star Observers (AAVSO). (In measuring light curves, we often measure time in Julian days, the number of days since noon on January 1 4713 bc, or modified Julian days, the number of days since the beginning of the Besselian year 1950 (see Appendix F for a further discussion of timekeeping).

We distinguish different types of variable stars by such things as their period and the magnitude range. A particular class of variable is generally named after the prototype of the class, either the first or most prominent star with the distinguishing properties of the class. In this section, we look at a few examples of the most important types of variables. Different types of variables appear in different parts of the HR diagram, as shown in Fig. 10.4. These bright stars were named before their variable nature was known, so they do not follow the naming convention discussed above.

Mira variables are named after the prototype (a star also known as O Ceti). These stars have periods of about three months to two years, or even

Temperature (K) 50,000 10,000 5,000 3,500

Temperature (K) 50,000 10,000 5,000 3,500

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