Notes compiled by Paul Woodward Department of Astronomy University of Minnesota This week, we will be discussing the way astronomers and astrophysicists have been able to develop a detailed understanding of the structure and evolution of stars. A key point to appreciate is the inability of astronomers to observe this evolution as it proceeds, since the time for a to evolve from birth to death can be billions of years. We can only see a snapshot of the heavens during our lifetimes. We must therefore assume that much of the variety that we observe comes from catching the stars in different stages of their evolution. How we have sorted out which differences are due to evolution and which are due to fundamental differences in the objects involved is a fascinating story. The process of understanding the stars began very simply by trying to classify them in a number of different ways. The first classification was by apparent brightness. However, the brightness of a star is largely an accident of its distance from us. If our brightness categories are to have any close relationship with properties of the stars themselves, and not just of where they happen to be located, then we need to remove the distance factor. We need to classify the stars by luminosity, the amount of energy they radiate per unit time in all directions, not by apparent brightness. The total luminosity is the energy radiated per unit time in all wavelengths of the electromagnetic spectrum.

The apparent brightness of a star decreases as the inverse square of the distance to the star. The luminosity of the star is independent of the distance to the star. It is an intrinsic property of the star itself, not of its position. Since we can only measure the apparent brightness directly, we must somehow also measure the distance to the star in order to calculate the star’s luminosity. If a star is near us, that is, within about 30 light years, then the most reliable method of measuring its distance from us is to observe its parallax. Recently, the Hipparcos satellite measured the positions and parallaxes of 120,000 stars with an accuracy of 0.002 arcsec.

The apparent brightness of a star can be measured in Watts per square meter, or in units of the Sun’s luminosity, L . Some astronomers still use the ancient magnitude system, devised by Hipparchus (c. 190 - 120 B.C.). In this system, stars of first magnitude were brightest, stars of second magnitude somewhat less bright, and so on. The magnitude system has been modernized, and now the brightest star, Sirius, has a magnitude of -1 (two magnitudes brighter than 1). Absolute magnitudes are like luminosities. They are the apparent magnitudes that stars would have if they were placed at a distance of 10 from the Sun. A is a distance at which a star has a parallax of one arc second. Thus a parsec is 3.26 light years.

Once we get the distances to stars from their parallaxes, we can compute their intrinsic luminosities from their apparent brightness, and this allows us to begin to really learn something about the stars. In particular, we can learn how many stars there are in various luminosity bins. This is shown on the previous slide. We will see that when we put this together with other information, we can learn how many stars have what masses and how many stars are likely to be in which phases of their evolution from formation to “death.” We do not live long enough to watch stars evolve. Therefore we must observe a great many stars to put the story of what stars are and what they do during their “lifetimes” into a clear picture. The work of classifying the stars in order to get the essential information to do this was extremely tedious and was done largely by women, it turns out. But there is no other way to discover what the stars are. By engaging intelligent and dilligent women to do this work, male astronomers of the day gave up the opportunity to make many very important discoveries and ceded this fundamental role to women astronomers. They may not have seen it that way at the time, but we do so now. These women astronomers showed themselves perfectly able to make their own discoveries from their work, and they brought about great advances in our understanding of the stars. In addition to classifying stars by their luminosities, we can also classify them by their colors. Because of the properties of the black body spectrum discussed near the beginning of the course, the color of a star is fairly directly related to its surface temperature. The Color of Stars Figure 18-4 Universe – Kaufmann Blackbody Curves Figure 17-4 Universe - Kaufmann Wien’s Displacement Law – temperature vs. maximum intensity wavelength Colors of stars can be measured quantitatively and repeatably by measuring their luminosities in separate regions of the spectrum that are selected by colored filters. Still more information can be obtained by examining the spectrum of the star. Lines in the spectrum from highly ionized elements indicate high surface temperatures, while lines of molecules (which dissociate at high temperatures) indicate low surface temperatures. Astronomers have thus devised a classification system of spectral types, according to what lines of what strength are observed in the spectrum of a star. These spectral types are related directly to stellar surface temperature. In order of decreasing temperature, they are O, B, A, F, G, K, M.

Measuring temperatures of Stars from spectra - composition of M stars Like many classification schemes in astronomy, this one for stellar spectra turned out to be weird because it was originally devised in a fairly complete absence of understanding of what the spectra meant. Only much later was it understood, and rationalized, but, characteristically, the classification scheme retained its original nomenclature. This now seems to us obscure. In the 1880s, Edward Pickering, of Harvard College Observatory, suggested that the stellar spectra be classified according to the strength of their hydrogen lines. Type A had the strongest hydrogen lines, then came type B, and so on to type O, with the weakest hydrogen lines. Pickering hired several women, among them Williamina Fleming (1857-1911) to classify the stellar spectra in this way, and Ms. Fleming classified 10,000 stars by 1890. In 1896, Annie Jump Cannon (1863-1941) joined Pickering’s team at Harvard. Building on the work of Fleming and of Antonia Maury, Cannon realized that the spectral classes fell into a natural order when more than just the hydrogen lines were considered, but this order was not the A,B,C order originally proposed. Some of the classes overlapped and some could therefore be eliminated. Cannon determined that the natural order of spectral classes was OBAFGKM. Cannon also added the numerical subclasses that we use today. During her lifetime she personally classified over 400,000 stars. She was the first woman ever awarded an honorary degree by Oxford University.

It fell to Cecilia Payne-Gaposchkin to figure out why Cannon’s spectral sequence was the natural one. She was educated at Cambridge, and worked with Rutherford, but chose to come to the U.S. because Harlow Shapley at Harvard gave women the opportunity to play important roles in the science of astronomy (or at least more important roles than seemed available to women in science in England at the time). Payne-Gaposchkin showed that the differences between the spectra of the different classifications did not reflect different chemical compositions but instead reflected different ionization levels of the emitting atoms. O stars, for example, have weak hydrogen emission lines because at their high surface temperatures nearly all the hydrogen is ionized. Payne-Gaposchkin published her dissertation in 1925.

Williamina Paton Stevens Fleming (1857-1911)

Mina Stevens Fleming, the first to discover stars called "white dwarfs", was born May 15, 1857 in Dundee, Scotland. She attended public schools in Dundee and then taught in Dundee from age fourteen until her marriage to James Fleming in 1877. The couple emigrated to Boston when she was twenty-one. A year later she was abandoned by her husband while pregnant with their child. To support herself and the baby, Mina Fleming obtained work as a maid in the home of Prof. Edward Pickering, the director of the Harvard Observatory. Pickering was unhappy with the work performed by his male employees and declared that his maid could do a better job than they did. In fact, he hired her in 1881 to do clerical work and some mathematical calculations at the Observatory. Fleming soon proved that she was also capable of doing science. She devised a system of classifying stars according to their spectra, a distinctive pattern produced by each star when its light is passed through a prism. She used this system, which was later named after her, to catalog successfully over 10,000 stars within the next nine years. This work was published in 1890 in a book titled Draper Catalogue of Stellar Spectra. Her duties were expanded and she was put in charge of dozens of young women hired to do mathematical computations, the work nowadays done by computers. She also edited all publications issued by the observatory. The quality of her work was so superior that in 1898 Harvard Corporation appointed her curator of astronomical photographs. This was the first such appointment given to a woman. In 1906 she was the first American woman elected to the Royal Astronomical Society. In 1907 she published a study of 222 variable stars she had discovered. A British astronomer made the following observation: "Many astronomers are deservedly proud to have discovered one...the discovery of 222...is an achievement bordering on the marvellous." Her achievement is especially noteworthy when one takes into account that she had no formal higher education. In 1910 she published her discovery of "white dwarfs," stars that are very hot and dense and appear bluish or white in color. "White dwarfs" are believed to be stars in a final stage of their existence. Williamina Fleming died May 21, 1911 in Boston, Massachusetts. Henrietta Leavitt, another of Pickering’s female hires at the Harvard College Observatory, discovered, in 1912, a correlation between the period and luminosity of Cepheid variable stars Henrietta Leavitt joined the Harvard College Observatory as a volunteer in 1895. She was appointed to the permanent staff in 1902, and eventually became chief of the photometry department. She worked there for the rest of her life. Leavitt discovered 2,400 variable stars, about half of the known total in her day. Through these discoveries came her most important contribution to the field: the study of cepheid variable stars in the Magellenic Clouds -- the Milky Way's two companion . By intense observation and mathematical calculation, Leavitt realized that with cepheid variable stars (which change brightness with great regularity), there is a direct correlation between a star's magnitude (degree of brightness) and the length of time it is most luminous. The brighter the star is overall, the longer the period of luminosity. Since the cepheids in the Magellanic Clouds were all about the same distance from Earth, Leavitt concluded that the period, or time it took to complete one cycle of dimming and brightening, was related to the star's magnitude, not distance. Yet magnitude itself allowed you to calculate distance. Leavitt published her findings in 1912 -- in a chart of 25 cepheid periods and their apparent brightness. Using this, astronomers only needed to know the period of a cepheid variable to figure out how bright, and therefore how far away it was. Until then, methods for measuring distances in space only worked within about 100 light years. With Leavitt's findings, distances of cepheids could be determined up to 10 million light years. This became the "yardstick to the universe" used by Edwin Hubble and others to make discoveries that changed our view of our and the universe. Henrietta Leavitt, 1868-1921 Stars continued Notes compiled by Paul Woodward Department of Astronomy University of Minnesota Measuring stellar distances and surface temperatures is, from our discussion last time, straightforward, at least for nearby stars. But measuring the masses of stars is much more difficult. The most direct method is to apply Newton’s theory of gravity, which allows the mass of a star to be deduced if the average orbital separation and orbital period of a low mass satellite can be determined. At the great distances of the stars, it is extremely difficult to observe the orbital motions of planets. But the orbital motions of two stars about each other can in fact be observed, and can be used to yield an estimate of the masses of the stars (under favorable circumstances). A young binary star is indicated by the circle. Binary Star Alpha Centauri is a 3-star system, consisting of Alpha Centauri A, about the same size as the Sun, Alpha Centauri B, second largest of the 3 stars, at about the same distance from Alpha Centauri A as Pluto is from the Sun, and finally Proxima Centauri, orbiting Alpha Centauri at a distance of 1/10 of a light- year, taking thousands of years to complete one orbit. α (left) and β (right) Centauri

Sometimes we can observe a star slowly moving back and forth in the sky, as if it were orbiting about a companion too faint to be seen. This is how astronomers discovered that the bright star Sirius must have a dim companion. Modern telescopes make both Sirius A and Sirius B (a ) visible.

In most cases, we cannot observe both stars directly as a visual binary. Then the binary nature of the star may be evident through periodic light variations due to eclipses (eclipsing binaries) or to Doppler shifts in the spectrum do to the orbital motion (spectroscopic binaries). Eclipsing Binary

A variety of different possible configurations for eclipsing binary stars, with the associated light curves. A variety of different possible configuratio ns for eclipsing binary stars, with the associated light curves. In order to determine the masses of the stars in an eclipsing binary, we need to measure the average orbital separation as well as the period. If the binary is too distant for us to measure the orbital separation directly, we can use the Doppler shifts in the spectra of the two stars as they orbit in order to deduce the orbital separation from the orbital velocities. Binaries that are eclipsing are most useful, because the Doppler shifts give us the orbital velocities unambiguously. From the fact that the stars are eclipsing, we know that their orbital plane is seen edge on. By measuring the lengths (in time) of the eclipses, and knowing the orbital periods and mean separations, we can directly deduce the radii of the two stars. Binaries detected only by the periodic Doppler shifts in their spectra are called spectroscopic binaries.

Working out the orbits of binary stars using Newton’s theory of gravitation produces the following results: 1. Both stars have elliptical orbits about their mutual center of mass. 2. These elliptical orbits both have the same shape, but different sizes. 3. Each star, as viewed from the other, executes an elliptical orbit, and this orbit again has the same shape, but a different size. 4. The two stars are always located on opposite sides of their center of mass (of course), with the more massive star closer and the less massive star further away. 5. Their orbital motion is most rapid when they are closest to each other, and slowest when they are most widely separated.

We have now seen how to determine observationally the absolute luminosity, the surface temperature, and, in a very limited number of cases, the mass of a star. It is a natural idea to plot these intrinsic properties of stars against each other in order to see if there are any correlations between them. Since stars with precisely known masses are so rare, it is natural to first plot the luminosities of stars against their surface temperatures (or colors). Such a plot is called a Hertzsprung-Russell (or H-R) diagram, because these people thought of doing this in the early 1900s. If we make such a plot for the nearby stars, we quickly discover that, in the main, the colors and luminosities of stars are well correlated and follow the line indicated in the next slide as the “main sequence.” Color-Luminosity Relation for 16,600 Nearby Stars as measured by the Hipparcos satellite. At the right, red dots are the 46 nearest stars, while blue dots are the 97 brightest stars. Of course, we would not use the name “main sequence” to describe this color-luminosity relation if there were not a few stars that do not follow this rule. A more complete graph appears on the next slide. Most stars fall along the main sequence, as indicated by the number of dots shown along this diagonal line in the plot. The few stars located in the extreme upper right of the plot are called supergiants. At their low surface temperatures, in order to be so luminous they must be extremely large. (Remember that the amount of energy radiated by a black body is proportional to its temperature raised to the fourth power.) Just below the supergiants in the plot is a group of stars we call giants, since they too must be awfully large. In the lower left portion of the plot are stars we call white dwarfs. Since they are hot and also dim, they must be small. Fig. 15.10 Basic H-R Diagram On the next slide, we see more or less the same diagram from the second edition, rather than the third, of your text book. Note that the shape of the main sequence is different. This is because it changed when the results for the 16,600 nearest stars came in from the Hipparcos satellite. This satellite allowed very accurate determinations of stellar parallax, and hence of stellar distance, so that we could at last accurately convert observed stellar brightness into stellar luminosity. This older, inaccurate slide is included as example of the dynamic nature of our knowledge in astronomy. Newer and better observations continue to improve our knowledge of even such well studied and well understood objects as normal stars. Fig. 15.10 Basic H-R Diagram We have been using the properties of the blackbody radiation spectrum to determine the sizes of stars on the H-R diagram – dwarfs, normal stars, giants, and supergiants. Because we know how the power radiated by a unit of surface area of a star varies with its temperature (through the blackbody law, the Stefan-Boltzmann law), for each point in the H-R diagram we can compute the radius of the spherical star that would emit that luminosity with that surface temperature. The result is that we can put stellar radius lines on the H-R diagram, so that we can read off the radius of any star plotted there. Such an annotated H-R diagram follows after 2 “refresher” slides on the blackbody spectrum, taken from the notes from the second week of class. Blackbody Spectra: 1. Common experience, especially for those who like to play with fire, shows that as an object is heated, it glows more and more brightly, first with a reddish color, then yellow, then ultimately white. 2. Stefan and Boltzmann quantified this knowledge through the Stefan-Boltzmann law, which states that the energy emitted per second per unit area from the surface of a perfectly light- absorbing object (a “blackbody”) is proportional to the temperature of the object raised to the fourth power: E = σT4 3. The dominant wavelength at which this light energy is emitted decreases as (1/T), or equivalently, the frequency of the dominant emitted light is proportional to the temperature of the object. This is call Wien’s law.

Fig. 15.11 Stellar radii on the H-R diagram. One thing to note about the H-R diagram is that its axes are set up so that one major tick mark represents a factor of 10. This “log-log” type of plot allows us to include in one picture the vast range of stellar types that we find in the sky. You should also note that the x-axis is set up backwards. This is annoying to physicists, but astronomers are quite comfortable with it. They do it this way because they have always done it this way. We had a long discussion of how we can measure the masses of stars. Although not too many stellar masses may be directly determined in this way, enough have been to lead to the following conclusion: as we go along the main sequence from the dim, red stars at the bottom right to the luminous, blue stars at the upper left, the masses of the stars increase. Theoretical calculations which we will discuss next week have given us the understanding that the stars along the main sequence are all burning hydrogen into helium in their cores. These calculations of the evolution of stars over their lifetimes also indicate that stars spend most of their lives, or at least most of their lives while they are bright enough for us to see them, in this core hydrogen burning stage. This theoretical conclusion thus “explains” why most of the stars fall on the main sequence in the H-R diagram. The theoretical calculations, together with the few observationally determined stellar masses, then allow us to mark mass values along the main sequence. The theory also allows us to mark main sequence lifetimes for the stars as well on the diagram. Fig. 15.12 Main sequence masses and lifetimes. Fig. 15.12 Main sequence masses and lifetimes. We can understand the correlation of stellar mass with stellar luminosity as follows. A star of greater mass must have a greater pressure at the center of its core in order to hold up the greater weight of its outer layers. This greater pressure is achieved by a combination of higher density and higher temperature. Both of these tend to increase the rates of the nuclear reactions that burn hydrogen into helium. The higher nuclear reaction rates in the cores of these stars cause more energy to be released per unit time, so naturally the stellar luminosity (which is the energy radiated into space per unit time) is larger. Because a star only 10 times more massive than the sun is over 1000 times more luminous, it is easy to see that this star cannot last as long. Thus main sequence lifetime decreases with mass. The fact that we see massive stars at all therefore implies that they must have formed recently, and therefore that the star formation process operates continually. We will see that it is the massive stars that generate the elements heavier than hydrogen and helium. Their short lifetimes therefore allowed these products to be mixed into the interstellar medium rapidly, so that most lower mass stars (like the Sun) contain heavy elements, even though many of them were formed billions of years ago. The theoretical calculations of stellar evolution tell us that when stars consume all the hydrogen fuel in their cores, they move in the H-R diagram off of the main sequence. They travel to the right in the diagram, developing larger radii and cooler surface temperatures. The observational confirmation of this phenomenon comes from the study of star clusters.

Stars form from large clouds of interstellar gas. Many theoretical ideas that we will encounter later on in this course lead us to believe that these gas clouds must be quite massive before they can have enough gravitational force to overcome their pressure support and collapse. As a result, stars usually tend to form in groups. These groups, called star clusters, can be small or large. The smaller ones are called “open clusters,” because the stars tend not to be very densely packed within them. One reason for this “open” nature of small star clusters is that after their formation, the new stars tend to expel any leftover gas from the region. The gas thus expelled can actually contain more mass than the new stars. In this case, the new star cluster expands, since the gravitational attraction of the gas no longer helps to hold the cluster together. The Pleiades, shown on the next slide is a typical open cluster. Only a few of its stars are easily visible to the naked eye, but this cluster contains thousands of stars. A typical size for an open cluster is about 30 light-years, in which several thousand stars are found. The largest star clusters are the globular clusters. These are very densely packed and can contain as many as a million stars. A typical size for a globular cluster is about 60 to 150 light-years, with the innermost region packing up to 10,000 stars within just a few light-years. Open star clusters are found only in the disk of our galaxy, while globular clusters are found both in the disk and in the galaxy’s halo, a roughly spherical region enveloping the galaxy. There are good theoretical reasons to believe that globular clusters may be extremely old, old enough to have formed from gas with very low concentrations of elements heavier than hydrogen and helium.

Star clusters are useful to us for two major reasons: 1. All the stars in a cluster presumably formed at nearly the same time. 2. All the stars in a cluster are at nearly the same distance from us. These two properties of star clusters allow us to remove two major uncertainties in studying stars, namely uncertainties in their ages and in their distances. We may not know the precise age or distance of a star cluster, but we know the relative ages and distances of the stars within it quite precisely. Because all the stars in a cluster are at about the same distance, there is only a single common factor between all their apparent brightnesses and their absolute luminosities. Therefore we can plot the cluster stars on an H-R diagram using their apparent brightnesses rather than their luminosities. The resulting plot should show the main sequence clearly. Since we know the absolute position of the standard main sequence, from studying nearby stars whose distances are known, we may then slide the cluster H-R diagram up or down until the main sequence fits our standard. The amount that we had to slide up or down will tell us the conversion factor from apparent brightness to luminosity, and from this factor we can calculate the distance to the cluster. This procedure is called main sequence fitting.

The short main sequence lifetimes of massive stars, together with the nearly simultaneous formation of all the cluster stars, tells us that for an old cluster, all the massive stars will have left the main sequence. A younger cluster will have more massive stars still located on the main sequence, burning hydrogen in their cores. If our theory of stellar structure and evolution is good enough, we may then calculate the main sequence lifetime of the most massive cluster star still on the main sequence, and we will then have the age of all the stars in the cluster. We may use this technique to date each cluster. We can also look for consistency of this theoretical calculation with other knowledge in order to reinforce our confidence in the theory. For example, globular clusters in the galaxy’s halo, which we believe to be very old for other reasons, turn out indeed to be very old when dated by this method. Also, the Pleiades, which still has some of its original nebulosity, is indeed young. The Pleiades Star Cluster Star MEROPE in the Pleiades Cluster in Taurus, showing nebulosity. 60-inch. Fig. 15.19 An H-R diagram for the stars of the Pleiades. Note that the upper main sequence stars are missing. Fig. 15.19 An H-R diagram for the stars of the Pleiades. Note that the upper main sequence stars are missing. This diagram uses recent data from the Hipparcos satellite. Fig. 15.12 Main sequence masses and lifetimes. The main sequence turnoff point for the Pleiades is at spectral type B6, which has a main sequence lifetime of about 100 million years.

A fine point in determining cluster ages from main sequence turnoff points is that for very young clusters, the low-mass stars may not actually have had time to reach the main sequence. We will come back to this point later, when we discuss stellar evolution, but it is illustrated on the next slide. This effect can also be detected in the H-R diagram for the Pleiades. There the very reddest stars have not yet had time to reach the main sequence. Young Cluster and Evolutionary Models Stars from 4 clusters with very different ages. h + χ Persei is only about 14 million years old. Stars at the main sequence turn-off point for NGC 188 are slightly more massive and luminous than the Sun, indicating an age of 7 billion years.

As the main sequence turnoff points for globular clusters indicate, these clusters are indeed ancient. They are indeed so old that their stars formed from gas that was not yet enriched in heavy elements by the explosions and winds of earlier generations of stars. The different chemical compositions of globular cluster stars means that we must take particular care in interpreting their main sequence turnoff points. This is illustrated on the next slide. In fact, globular clusters are some of the oldest things we have been able to find to date. They are so old that they have been used to constrain estimates of the age of the universe (the universe must be at least as old as its globular clusters). Old Clusters of Different Heavy Elements Abundances If we put all these H-R diagrams together, we see that indeed our picture of stellar evolution and our interpretation of the main sequence as marking the core hydrogen burning phase of a star’s life is consistent.

Stars continued Notes compiled by Paul Woodward Department of Astronomy University of Minnesota We now begin looking at stars by tracing their lives, beginning with their formation out of huge interstellar gas clouds and ending with their expulsion of their outer envelopes or simply with an explosion. This subject is covered in your textbook in Chapter 17. You should review that chapter thoroughly. When discussing the formation of the , we already talked quite a lot about the early stages of the star formation process, going from a large cloud of interstellar gas to a protostar surrounded by a protoplanetary disk. Therefore we will skip that part of the story here. Fig. 16.3 Artist’s concept of a collapsing stellar disk. The bipolar jets emanating from compact objects at the centers of rotating disks of gas, as shown on the previous slide, seem to be a common feature of all such systems, big and small. We will encounter them again when we discuss double star systems, and yet again when we discuss the nuclei of galaxies. This is a feature of astronomy, we find again and again close similarities in the behavior of systems that differ in scale by factors as large as a thousand or even a million. The key point is that these systems do not differ in any fundamental aspects other than sheer scale. In the early 2000s these bipolar jets were identified as a mechanism for removing angular momentum from a collapsing disk, allowing the central stellar object to form.

If we think about the glob of a larger gas cloud that collapses to form a single star, we can distinguish 4 stages of this contraction process under gravity, once this glob has contracted sufficiently to establish its distinct identity. We consider its collapse beginning after it has become opaque to radiation, so that it makes sense to talk about its “surface,” the surface region from which light can escape into space. 1. First, the surface temperature of the gas cloud is very low, but its surface area is very large. As it contracts, liberating gravitational potential energy as heat, this heat is radiated away into space very efficiently. The luminosity of the gas cloud is therefore very high, perhaps as much as 100 times as high as it will be once the star reaches the main sequence and begins burning hydrogen in its core. However, this radiation, because of the low surface temperature of the cloud, will appear mostly in the infrared. 2. Second, as the gas cloud continues to collapse, radiating into space essentially all of the heat generated from gravitational potential energy release, its surface temperature warms only slightly. However, as the collapse proceeds, the surface area of the cloud is diminished, so that its luminosity is diminished accordingly. This stage lasts only a few million years (for a protostar of about one solar mass). 3. Third, release of gravitational potential energy, together with the reduced radiating surface area, cause the interior of the protostar to heat steadily. In this third stage the core temperature exceeds a few million degrees K, so that hydrogen fusion to form helium begins in the center and significantly slows the continuing collapse of the protostar. Continued shrinkage and continued heating give slight increases in the luminosity over about 10 million years (for a one solar mass protostar). 4. Fourth, shrinkage of the protostar and increase of the rate of fusion in its core continue for a few tens of millions of years until the fusion rate finally establishes gravitational equilibrium, and the star is said to be on the main sequence. For whatever it is worth, we can plot such a protostellar evolutionary track on the H-R diagram. Stars of different masses of course follow different tracks on the diagram, but they all follow the same 4 stages of protostellar evolution outlined above.

Fig. 16.6 Life tracks from protostar to main sequence star for stars of different masses. You may remember that we said previously that the H-R diagram for the Pleiades cluster indicates that the cluster is about 60 million years old. You may also remember that this determination was made from the main sequence turn-off point. There were some stars in the cluster on the main sequence, and some more massive stars had already begun to move off to the right of the main sequence. However, the least luminous stars in the cluster, the lowest mass stars, were mostly located a bit above the main sequence. Now you can understand that these low-mass stars have not yet reached the main sequence for the Pleiades cluster. The previous slide indicates that only stars of about one solar mass or more have had time to reach the main sequence by now. From observing star clusters, we can roughly determine the relative frequencies of formation of main sequence stars of different masses. This initial mass function, as it is called, strongly favors low- mass stars. For every star formed between 10 and 100 solar masses, we find roughly 10 stars between 2 and 10 solar masses, and a few hundred stars below half a solar mass. We have never conclusively seen any star of greater than about 100 solar masses. It is believed, as the British astronomer Eddington pointed out, that a star of such great mass would generate so much energy in its core that the outward streaming radiation would tear it apart. (People still argue about this.) We are not sure where the star formation story ends on the low mass end of the scale. As very low mass protostars (say, below 0.08 solar masses) contract under gravity, we believe that a bizarre physical phenomenon called degeneracy pressure would halt the collapse before efficient, self-sustaining hydrogen fusion could begin in the core. Such low mass stars fill the gap between stars and planets. It is not clear whether Jupiter-sized objects form individually from collapsing gas clouds, without being planets forming within the protostellar disks of larger objects. Such a “star,” or “brown dwarf,” with 0.05 solar masses, Gliese 229B, was discovered orbiting a “real” star, Gliese 229A in 1995. The brown dwarf Gliese 229B. This object was detected with a 1.5 m telescope on the ground (left) but the provided a much sharper image (right). The small companion, Gliese 229B, has a mass of only 20 to 50 times that of Jupiter. Gliese 229B is the companion of an ordinary star, but it has a luminosity of only 2 to 4 millionths that of the sun. Its spectrum resembles that of Jupiter. It has a lot of methane and a surface temperature of 600ºC to 700ºC. Early models of the contraction of protostars to form main sequence stars were based upon rough scenarios like the one we have just outlined. The Japanese astronomer Hyashi built models assuming this sort of “quasi-static” evolution, that is, a slow progress through a series of physical states each of which was very nearly in equilibrium. Unfortunately, life is not this simple. First Karl-Heinz Winkler, working in Germany, and later Frank Shu, at Berkeley, worked out detailed models for the formation of single solar mass stars that we now think are basically correct. These models reveal that the protostar is not in equilibrium, but instead its surface is marked by an extremely strong shock front, in which infalling gas from the surrounding cloud is suddenly decelerated upon striking the surface of the protostar. It is from this shock front that a huge amount of kinetic energy from the infalling gas, just converted into heat in the shock, is radiated into space. The dynamics of this situation determines much of the internal structure of the protostar. The loss of huge amounts of heat of condensation from the shock, essentially before that energy can become incorporated into the protostar, greatly affects the course of such a protostar’s development. These models, of course, did not include rotation or magnetic field affects, which still remain to be treated properly in such work. These theoretical studies have led us to the conclusion that the internal structures of main sequence stars of different masses are dramatically different. High mass stars generate nuclear energy prodigiously in their cores. They produce so much energy so rapidly that “conduction” by radiation diffusion cannot transport it outward fast enough. Instead, we believe that their cores are fully convective. Nevertheless, outside their cores, where the gas is still very hot and ionized, radiation diffusion works well in carrying the energy all the way out to the surface. Medium mass stars like our Sun transport liberated nuclear energy outward from their cores by radiation diffusion. In a layer near the surface, the gas becomes too opaque for efficient radiation diffusion, and convection takes over. Low mass stars are believed to be convective from their surfaces right down to their cores. Such stars with fast rotation rates can produce very powerful flares as a result of the winding up of their magnetic fields by their deep convection zones. Proxima Centauri is such a “flare star.” Low mass stars gradually burn their core hydrogen, reducing the number of independent particles (four protons are replaced by a single helium nucleus) in their cores. In this process, their cores shrink, and they grow gradually a bit more luminous (the sun is thought to have increased in luminosity by perhaps 30% over the 4.7 billion years of the Earth’s history to date).

In the Sun, hydrogen nuclei are combined to produce helium nuclei (and extra energy in various forms). Because helium nuclei usually have two protons and two neutrons, we have to combine four hydrogen nuclei (protons) to get one helium nucleus. This is incredibly unlikely to happen in a single event. Instead, it proceeds in stages, each involving only the collision of two particles.

The previous slide illustrates the primary nuclear reaction chain in the Sun. It provides about 86% of the Sun’s energy. Together with the alternate (second) reaction chain on the next slide, which involves berylium and lithium as intermediate products, these two reaction sequences produce 98.5% of the Sun’s energy. The remainder of the Sun’s energy is produced by the CNO- cycle, the third reaction chain (on the next slide) involving carbon, nitrogen, and oxygen. These alternate reaction chains are included here for correctness, but all you need concern yourself with is understanding the primary chain shown on the previous slide. Do not memorize these nuclear reactions. They are given here for completeness and to satisfy the curious. Now let’s consider what happens to a star like the Sun when it runs out of hydrogen fuel in its core. It’s subsequent evolution is called the “red giant phenomenon,” and it is one of the early triumphs of computational science. It was stellar evolution models that made us understand that red giant stars are not separate sorts of objects that have always been that way since their formation. These computer models, which take stars from one equilibrium state to the next, in which the nuclear makeup is slightly altered by the star’s burning, were the only way that we were able to figure out that red giant stars are just the evolved states of main sequence stars. When hydrogen is exhausted in the core of a star like the Sun, the inert helium there does not generate further nuclear energy, so it contracts under the crushing weight of the gas above it. The contraction of the helium core releases gravitational potential energy, so it actually heats up. The layers above the core, which still contain unburnt hydrogen, contract and heat up as well. The layer of hydrogen just above the helium core becomes so hot that it begins to burn, and this process actually generates more nuclear energy than the core hydrogen burning did when the star was on the main sequence. The star can eventually increase in luminosity by up to 4 orders of magnitude. This process takes about a billion years, for a star like the Sun, and longer for less massive stars. In order to radiate all this luminosity into space, the layers above the hydrogen burning shell expand enormously (by about a factor of 100 in radius) and become fully convective. The now far greater surface area produces the much greater luminosity, ironically, at a somewhat reduced surface temperature. The star has thus become a red giant. As newly produced helium adds to the mass of the inert core, its greater gravity causes it to shrink still further. The hydrogen burning shell shrinks along with the core, growing hotter and denser. This makes the hydrogen burning rate increase in the shell, which increases the star’s luminosity still further. This vicious cycle feeds upon itself until the temperature in the helium core reaches about 100 million degrees K, at which point helium nuclei can fuse in the core to produce carbon. The process by which the core of a star can get hotter, rather than cooler, once its source of nuclear heat generation ceases is a bit odd, but we can understand it as follows. We begin when nuclear reactions are continually generating heat in the core. This heat produces pressure, which supports the core against gravity at its present radius. The heat generation is balanced by escape of heat due to radiation, conduction, or convection from the surface region of the core. So the pressure, the gravity, and the radius of the core remain constant. Now suppose that the nuclear reactions shut off from lack of further fuel. The escape of heat does not shut off. The escaping heat would cause the core to cool off, if only its gravity and its radius were to remain constant, as before. But the escape of heat causes a reduction of the pressure supporting the core, so that it contracts under its gravity. We can think of this contraction as generating ordered inward motion, which has an associated kinetic energy derived from the liberation of gravitational potential energy. We can also think of the inward moving material as colliding with other material moving inward and toward it, so that the kinetic energy of the ordered motion is transformed into kinetic energy of disordered motion, which we call heat. This additional heat will raise the pressure of the gas. If the pressure rises enough, the collapse will stop, or at least pause or slow, since heat continues to escape from the surface. How much added pressure does it take to arrest the collapse? You might think that all we have to do is to replace the heat energy that escaped from the surface of the core with heat energy generated in the collapse from release of gravitational potential energy. But this cannot be true, since the collapse makes the core smaller. The core’s mass is still the same, but now its smaller size means that its gravity is stronger. This in turn means that we need more pressure than before in order to support it at this smaller size. Therefore, if the collapse is slowed and nearly stopped, we know that the pressure must be higher than before. In fact, the temperature, which is proportional to the ratio of the pressure to the density, must be higher to counter the stronger gravity at this smaller core radius. Once hydrogen fusion ignites in a shell around the core, it eventually burns much more brightly than the hydrogen burning in the star’s core. We can understand this also. The energy generation rate we had with core hydrogen burning had to be sufficient to provide enough pressure to hold up all the material overlying the core against the crush of gravity. It had to do this while matching the rate at which energy was escaping from the star through its surface. Now the energy generated in the hydrogen burning shell must still support all this overlying material as well as the dead weight of helium accumulating in the core. The helium must be heated sufficiently by the hydrogen burning shell to hold itself and the overlying material up. It’s like trying to carry on your shoulders a child who keeps growing older and heavier – it is quite a burden. One might think this burden of the helium core might be light. No way. All this hellium is really, really close together. It finds itself extremely gravitationally attractive. Keeping its nuclei and electrons whipping around fast enough so that it does not come completely together is a difficult job requiring lots and lots of energy.

Because helium nuclei have twice the charge of hydrogen nuclei, they repel each other more strongly. They must therefore be moving faster in order to strike each other hard enough to overcome this repulsion and form beryllium. (The beryllium would split back into two helium nuclei, but under the conditions in a helium burning stellar core another helium nucleus can come along before this happens, so that a stable carbon nucleus can be created.) Helium fusion in a red giant star ignites suddenly, in what is called a . When the helium first ignites, the core is supported by degeneracy pressure, and its total pressure does not increase much as its temperature, and the helium fusion rate, shoot up. After the helium flash, the core does expand against gravity, expanding and cooling the hydrogen burning shell. Degeneracy pressure is a concept that will pop up again and again in our discussions of stellar evolution. It can be understood, at least qualitatively, by the following analogy. Think about a room. Now put a whole lot of billiard balls into the room. Just pour them in until they fill up, say, a quarter of the room’s volume. They will be in the lower quarter of the room. If you try to squeeze them so that they take up less volume, you will probably be unable to do so (let’s say that you can’t squeeze hard enough to pulverize them, so that all the little spaces between them can be filled with the powder produced). The billiard balls now act like helium gas exerting degeneracy pressure. The billiard balls are touching. You can’t get them any closer to each other no matter how hard you push. Now let’s put some energy into this system of billiard balls by picking them up and throwing them every which way. Let’s imagine that we can do this somehow without getting hit, although the billiard balls will all hit each other. The billiard balls have a lot more kinetic energy of disordered motion than before. They are a lot hotter. And they must now take up a lot more room. If you were to turn a winch and pull the ceiling of the room downward, you could make the billiard balls take up less room But this would take a lot of work, because the billiard balls would be hitting the ceiling pretty hard, especially as the volume available to them got closer and closer to a quarter of the original room volume (when they would all be touching once again). Without crushing the billiard balls, you could not make them take up less volume than a quarter of the original room. Degeneracy pressure is like the force of resistance that the billiard balls exert when they are all touching and you try to squeeze them into a smaller volume. Normal gas pressure is like the force that the billiard balls exert against the ceiling of the room as you lower it with your winch against the force of their bouncing off of it. Normal gas pressure is strong, but you can overcome it if you press hard enough, and you can make the gas squeeze into a smaller volume. Degeneracy pressure is stronger, in the sense that you cannot overcome it and squeeze the material into a smaller volume unless you squeeze so hard that the material changes its fundamental nature. An example of such a change is when the particles of the material fuse into different particles that take up less space. This is like pulverizing the billiard balls, or like combining protons with electrons in a star to form neutrons, which take up dramatically less space, believe it or not. Fig. 16.10a Core structure of a helium-burning star. Fig. 16.10b Relative sizes of a low-mass star as a main-sequence star, a red giant, and as a helium-burning star. It turns out that the helium burning cores of all low-mass stars fuse helium into carbon at about the same rate. Therefore these stars all have about the same luminosity. However, these stars can have different masses, based upon how much mass they started out with and how much mass they lost in stellar winds during their red giant phases. Stars that lost more mass end up with smaller radii and higher surface temperatures. These helium burning stars therefore occupy a horizontal branch in the H-R diagram of a star cluster.

The core helium of a low-mass helium-burning star runs out in about a hundred million years. Once helium is exhausted in the core, the core again shrinks and heats up, helium begins burning in a shell around this core, and hydrogen continues burning in a shell around the helium region. Once again the luminosity increases to new heights as the core size shrinks, and the outer layers of the star puff up again to a greater extent than ever. Computer models show that the helium burning in the shell spikes upward every few thousand years in a series of thermal pulses. For a one solar mass star, this stage can last less than a million years. In order to ignite carbon in the core of such a star, the temperature must rise to about 600 million degrees K. For low-mass stars, degeneracy pressure halts the shrinkage of the core before this very high temperature can be reached. Such stars have huge stellar winds, and during the thermal pulses, carbon can be dredged up from the core and brought to the surface and into the wind by convection. Red giants with high carbon concentrations in their atmospheres are called carbon stars. Such carbon in the stellar wind can form dust grains, because of the very low surface temperatures of these stars. During the final stages of the evolution of a low-mass star, the wind from the star becomes very great. Ultimately, all the mass of the envelope surrounding the inert, degenerate carbon core is ejected to form a , which is set aglow by the ultraviolet radiation of the cooling, but still very hot core. This luminous nebula is called a . The degenerate carbon core is called a white dwarf. The textbook discusses what humans might do 5 billion years hence, when the Sun begins to become a subgiant, and the rest. This discussion is interesting enough to warrant reading, once. But 5 billion years is a long time. The next slide summarizes the life story of a low-mass star, before we go on to look at some planetary nebulae.

In between the helium burning shell and the hydrogen burning shell above it is a region containing a mixture of helium from hydrogen burning and also carbon from the burning of helium. In the last stages of the star’s life as a giant (as an AGB star), before it expels its outer hydrogen-rich envelope, the helium shell burns intermittently in a series of “helium shell flashes” at intervals of about a thousand years. These brief (2-year) flashes mix carbon from helium burning into the layers above the helium-free core of carbon and oxygen. When the helium shell flashes into action, the hydrogen shell is lifted upward by the greatly increased pressure generated by helium burning, and hydrogen burning ceases. After the flash, as hydrogen begins burning again, the products of helium burning (mainly carbon) are “dredged up” by the convection of the outer envelope and carried to the surface of the star. Once the carbon gets to the surface of the star, where the temperature is relatively low, it can condense to form dust grains. These dark dust grains absorb the light coming out from the interior of the star and are pushed outward by this radiation pressure to create a strong stellar wind that carries off a portion of the outer layer of gases. This process repeats with each successive helium shell flash, until eventually all the outer material of the star is expelled, and the inner core of carbon and oxygen remains. We call this exposed carbon/oxygen core a white dwarf, because its surface is “white hot” and its radius is tiny (comparable to the radius of the earth). Although carbon is the main product of helium burning in the star, other reactions also occur. When hydrogen gets mixed in with the carbon above the helium burning shell, it can fuse with the carbon to form radioactive nitrogen. This nitrogen-13 decays to form carbon-13, and this becomes a source of neutrons that get added to heavy elements that exist in trace quantities in this region of the star. Over time, exposure to these neutrons builds ever heavier elements, including such very heavy elements as gold and lead. All these trace constituents of the gas in the region get dredged up by the envelope convection, carried to the stellar surface, and expelled into the surrounding interstellar environment. From there, they can become incorporated in a later generation of stars and/or planets. The following slides show a hemisphere of the AGB star’s interior. We are looking out from the center of the star, and see the helium shell flash convection zone like a dome over us. We have made all the helium and carbon (and some oxygen) in the convection zone above the helium burning shell transparent. What we see in these images is the unburned hydrogen/helium mixture that is being drawn down with the convection flow, despite its greater buoyancy, toward the helium burning shell. At sufficient depth, the temperature will be great enough for this hydrogen to fuse with the carbon-12 to form nitrogen, which will decay into carbon-13, releasing enormous amounts of energy (the “hydrogen ingestion flash”). The carbon-13 becomes a neutron source to build heavier elements via the process of “slow neutron capture” (the “s-process”). These heavy elements are later swept into the outer envelope.

The Owl Nebula, on the following slide, gives an idea why these nebulae, formed by mass ejections from dying stars, are called planetary nebulae – with poor telescopes they look like planets. Planetary Nebula, the “Owl” in Ursa Major (NGC 3587) Planetary Nebula, Abell 39, a textbook example; 6 light years across, 7000 light years distant A small collection of Hubble Space Telescope images of planetary nebulae is shown on the next few slides. The mass of gas ejected from a star in this manner can be up to 75% of the total. The first of these planetary nebulae, the Stingray Nebula, is so young that only 20 years ago its gas was not hot enough to emit light. However, the temperature of the central star has increased rapidly, so we will be able to witness the formation of this planetary nebula, a process that may take only 100 years or so. The ultraviolet light from the central white dwarf star causes the surrounding nebula to glow. Because Henize 1357 is 18,000 light-years away, only the Hubble Space Telescope can resolve its structure. The central star is in a binary system, and its companion could be responsible for the dense ring of gas surrounding the star that has shaped the nebula, making the ejected gas form two bubbles. The Stingray Nebula, or Henize 1357, HST image. The Egg Nebula, CRL 2688, is likely to have first appeared a few hundred years ago. The arc structures in this nebula, visible on the next slide, have been interpreted as shells of gas ejected previously by the central star at intervals of perhaps 100 to 500 years (could these have been the periodic helium flashes predicted by stellar evolution models?). In the infrared (shown on the second slide), there is a dumbbell structure of molecular hydrogen. Matter streams out along the polar axis at 100 km/sec and collides with previously ejected gas moving at only 20 km/sec. The matter in the two polar streams darkens the centers of the two cones of visible light from the central star. The Egg Nebula, CRL 2688 The Hourglass Nebula, MyCn18, discovered by Margaret Mayall and Annie Cannon, at first fits the standard model of planetary nebula formation. The idea is that matter is ejected from the central star in wind-like flows that are episodic, and that grow faster and faster. We believe that the earlier ejected matter is denser at the equator than at the poles, and that it therefore channels the new, more rapidly ejected gas into an hourglass shape. The Hourglass Nebula shows this structure beautifully, with the walls of the hourglass showing detailed structure that may either be related to the episodes of the earlier, slower winds or to the interaction of an energetic stream of gas with the walls of this channel. The Hourglass Nebula is at a distance of 8,000 light-years. Hubble Space Telescope image of the Hourglass Nebula, MyCn 18. “Snowplow” model of planetary Nebula Internal structure of main sequence star and helium burning star Internal structure of one solar mass star in second red giant stage Internal structure of star – 15 solar mass in late second red giant stage

The ultimate fate of a star depends mainly upon the mass that it had when it was on the main sequence, burning hydrogen to form helium in its core.