Implications from Stellar Masses:
A Point-Form Summary.
This section of the course notes, and the associated PowerPoint presentation, makes the following critical points:
from the binary stars, we now know the masses of many thousands of stars of various kinds. We can go back to the HR diagram to see how the position of the stars in that figure depends (if at all) on stellar mass
we discover that there is a strong correlation along the main sequence (MS). The bright blue MS stars are massive (up to 100x the mass of the sun); the faint red MS stars are less massive (at 10% of the mass of the sun or less)
there is no such simple relationship off the main sequence: among the giants, there are stars of a variety of masses. The white dwarfs, however, all seem to be about one solar mass or less
the dependence of mass along the main sequence means that stars cannot evolve along it. There is no way that so much mass can be lost or gained during the course of a star's life. So the sun did not start out as a bright blue star and gradually get dimmer; neither did it start as a dim red star and gradually get hotter and brighter
it is true that stars lose some mass as they age, because mass is being converted to energy. But this is just a tiny fraction of the total original mass
it is easy to understand why the more massive stars are hotter and brighter. The inward pull of gravity is stronger, so the outward pressure that sustains the star must be greater. This requires a higher temperature, which makes the star hotter overall. Moreover, the higher temperature leads to higher rates of nuclear reactions, which eats up the fuel very quickly
the luminosities of the MS stars are directly correlated with their masses. In fact the dependence is a strong one: a star ten times more massive than the sun is a thousand times more luminous. The relationship can be encapsulated in the mass-lummiosity law
this law means that the more massive stars will have dramatically shorter lifetimes than the low-mass stars. (They have more fuel, but chew it up at a prodigious rate.) In this sense, they are unlike animals, where the bigger living creatures tend to have longer lifetimes
Associated Readings from the Text.Please look at: Chapter 16, especially the discussion on page 535.Stellar Masses.Not every star is in a binary pair, but there are enough of them that we have can determine the masses of many thousands of stars. Moreover, the binaries can be of different sorts. Sometimes we have two G stars orbiting each other; elsewhere, we see a blue B star and a red giant of type K in mutual orbit; and so on. We look through all the data to try and find masses for stars of as many different spectral types and luminosities as we can, and then insert the numbers into the HR diagram to see if we can make any sense of it. The principal result is shown in the figure on page 533 of your text. along the main sequence, there is a steady progression of stellar masses. The bright, hot O and B stars are more massive than the sun, by factors of 12, 20, or 40, for instance. Stars near the middle of the main sequence are a few times the mass of the sun (Sirius is an example of such a star). Stars fainter and cooler than the sun are less massive, with the smallest ones shown being about 10 percent of the mass of the sun. by contrast, off the main sequence (i.e. when you look at the red giants and the white dwarfs) there is no such pleasing simplicity. (Unfortunately, the figure in the textbook does not show this explicitly.) Some of the supergiants and giants are massive (up to 10 or 20 times the mass of the sun), but others are just about the same mass as the sun. The white dwarfs are typically about one solar mass or less. Somehow we have to make sense of this! Let us consider a couple of obvious questions -- but first, permit me an important digression.An Important Semantic Point.Astronomers use the expression stellar evolution in a way which may seem strange to you. Let us first remind ourselves what biological evolution means in the Darwinian sense. In the theory of Darwinian evolution, there are random small changes from one generation to the next, changes caused by such things as tiny imperfections in the reproduction of the DNA molecules which carry the genetic code. Since the DNA carries the `blueprint' of the lifeform, small errors will lead to small -- or perhaps not so small! -- changes in the living creature or plant. Any new characteristic may be beneficial, benign, or detrimental. For instance, if an Arctic Hare were to grow a thicker than usual coat of hair, it might better survive the winter storms; but if it sprouted red hair, like that of a red squirrel, it would lose its protective coloration and be easier prey for predators. Since beneficial characteristics make the owners more likely to survive and thus able to reproduce in large numbers, that genetic benefit is more likely to proliferate in the species (the `gene pool') than the detrimental characteristic. Over time, then, the species adapts, without conscious volition -- that is, without `trying to' -- to a form which is better suited to the environment. The biological use of the term, then, implies a change from generation to generation, from parent to offspring. Confusingly, astronomers use the term quite differently. When we speak of stellar evolution, we really mean nothing more than the way in which an individual star changes as it uses up the fuel available to it. This is a completely different use of the expression, one which would be better captured by a term like stellar change or even stellar aging, but unfortunately the term stellar evolution is already firmly in place. You will simply have to get used to what we mean by it!Do Stars Evolve Along the Main Sequence as They Age?There is an immediate conclusion which we can draw from the masses of the various stars in the HR diagram. Stars do not evolve along the main sequence. That is, billions of years from now, the sun will not look like a main sequence O star; nor will it look like a faint main sequence M star. This conclusion should not surprise you, because we already know that the sun has been fairly steady in its brightness for quite a long time. The fossil record tells us that there were primitive life forms in seas of liquid water on the Earth a few billion years ago, a situation which could not have existed if the sun were much hotter or much cooler than at present. So perhaps you were already expecting the conclusion that the sun has been and will continue to be pretty much unchanging for a long time. But now we can prove it in a different way! The point is that all main sequence O stars are much more massive than the sun, and if the sun were ever to turn into such a star it would have to gain an enormous amount of matter from the depths of interstellar space. We don't see this happening for any stars, and moreover we cannot find a plausible mechanism by which this would ever take place. In short, there seems to be no way the sun can gain mass to build itself up to this great extent. (By contrast, of course, people eat lots of food and do grow in height and weight as they mature, thanks to the external sources of nourishment.) Please remember, of course, that there is a brief `building-up' period when a star is first formed. At that time, clouds of gas and dust condense to form a new star. (We described such an episode in Physics 015 when we considered the formation of the Solar System.) But once the star turns on its nuclear furnace, that large-scale accumulation of matter ceases, and thereafter there is little, if any, inflow of material onto a star. The stars, once formed, do not `put on weight'. At the other end of the main sequence, the M stars are all a lot less massive than the sun. Can the sun be losing material as it burns, and will it gradually dwindle in size to that of a small, cool M star? Indeed, there is a germ of truth to this idea: as you know, the sun is generating energy by the slow conversion of some of its mass into radiant energy. But the total effect is actually quite small. The sun will lose only about one-tenth of a percent of its mass, since the hydrogen is not completely converted to pure energy by the nuclear reactions: most of the mass is 'left over' in the form of the helium "ashes". In short, there is no way that the sun can shed enough mass to turn into one of the M stars on the lower main sequence. Inevitably, then, we conclude that the main sequence stars are simply placidly enjoying a long-lasting part of their lives, a period we call their main-sequence lifetimes. During this phase, the observable properties are quite stable: each star displays a particular unchanging combination of temperature and brightness, depending on its mass. The sun has scarcely changed for several billion years at least. In summary: Here and there in interstellar space, a cloud of gas may contract because of its self-gravity and eventually form a star (or perhaps more than one: we will see later that many or most stars form within big gas clouds which fragment to form a whole cluster of stars) Each star that is formed has some fixed mass, greater or less than that of the sun, depending on the size of the cloud from which it formed and the details of how the collapse proceeded The mass determines what kind of a main sequence star it is , and the star remains pretty much unchanging until it eventually runs out of fuel (a topic we will explore a bit later) But what are the red giants and white dwarfs? As we will learn, the red giants are stars which have run out of fuel in their central nuclear furnaces: they are starting their `death throes.' The white dwarfs are the final state, the dying embers if you like, of low-mass stars -- those which are like the sun or smaller.Why Should the Mass Matter So Much?On the main sequence, why is it that the mass is so critical to the nature of a star? The answer is by now a familiar one: it is determined by gravity. To understand the effect, consider a particularly massive star. In such a star, every atom feels an especially strong downward pull from the cumulative effect of all the other atoms: that is, a massive star has lots of self-gravity trying to pull all the material towards the center. The star originally formed because the atoms responded to that gravitational pull by falling inwards. The inrushing particles collided here and there, with the result that all the particles bounce and rush about more vigorously: that is, gravitational potential energy was converted to internal heat. The more massive the star, the more vigourous the fall, and the hotter the star in its inner parts. This heat, once generated, leads to the start-up of nuclear reactions. These reactions in turn maintain the heat for a long time, and the heat provides a sustaining outward pressure which supports the star against further collapse. But to provide a sufficient outward pressure to support such a massive star against its extremely strong self-gravity, the temperature must be very high indeed. Consequently, the massive main sequence stars are hotter than the sun is - both in the core, and at the surface as well, which is why they look blue.The Mass-Luminosity Law.One important piece of information is contained in what is known as the mass-luminosity relation, shown in the following figure. In this figure, we plot the luminosity of each star against its mass. (Remember that, except for the sun itself, the mass determinations come from binary stars of various kinds, as the legend indicates.) A star with a mass ten times that of the sun is more than a thousand times as bright, as you can see. A star which is ten percent the mass of the sun is about one one-thousandth as bright. [There is one important qualification, by the way. This relation only applies to main sequence stars. For stars not on the main sequence, there is no such simple law. Among the giants, for instance, there are several stars of roughly equal luminosity but very different masses, so there is no simple law which describes this. The main sequence stars, however, are particularly straightforward in this important respect.] The mass-luminosity law means that once you know the total mass of a star, you can state with complete confidence what properties it will have while on the main sequence. (Its subsequent evolution is a bit more complex, as we will learn.) But this simple mass-luminosity law has a very profound consequence.The Main-Sequence Lifetimes of the Stars.A remarkable conclusion follows from the mass-luminosity law. Even if you do not know in detail what power supply makes the stars run, you can predict fairly confidently that the massive stars will not last as long as the lower-mass stars! How can you conclude this? It may help to consider a simple analogy. If you have a fireplace at home in which an average log burns for one hour, and you have a supply of ten logs, then it takes little imagination to realise that you can keep the fire going for ten hours. But if you were to throw all the logs into the fire at once, it probably would not last nearly so long because the fire itself would increase in intensity and consume the fuel faster. (If you have enough wood piled in the back yard to keep the house warm all winter, but someone sets fire to the woodpile, it will not burn all winter!) Similar considerations hold for the main sequence stars. Whatever is happening in a star - whether it is chemical combustion as in a wood fire, or nuclear reactions, or some other source of energy - there is only a finite supply of fuel, which is clearly limited by the total amount of matter in the star. Because the stars are all pretty much the same composition, a star ten times the mass of the sun has about ten times as much fuel in total. But the mass-luminosity law tells us that this massive star is using up its fuel at thousands of times the rate that the sun is! This means that its great fuel reserve will be depleted very quickly, and such a star cannot last as long as the sun. Conversely, of course, the smaller main sequence stars have less fuel than the sun, but are using it in a very sparing fashion, and they can last a very long time indeed. Here is an even more straightforward analogy to make this point clear. Suppose you have $1000 in the bank, and are spending $10 a day on simple nourishment. You can keep going for 100 days - a bit more than three months. By comparison, I might be quite wealthy, with $10,000 in the bank. But being ten times as wealthy will not help me if I spend the money at a thousand times the pace that you do! I will blow my entire reserve in a single day of gluttony, and starve thereafter. Back to the stars: we will quantify the effects of the mass-luminosity law to discover that a very massive O star might last only a million years, whereas the sun will have a total main-sequence lifetime of about ten billion years, of which it has so far used up about half. (These numbers are shown in the figure on page 533 of the text.) In other words, some of the very bright blue stars in the sky were not even in existence when the first homo sapiens appeared on the Earth. Meanwhile, the lowest-mass stars can look forward to enormously long main sequence lifetimes. In my usual style of drawing analogies, I pointed out that in this respect the stars are opposite to living creatures on Earth. In the animal kingdom, larger animals -- those with relatively large masses, including whales, elephants, and people -- generally have longer lives than small ones, like ants, shrews, and mayflies. In the astronomical world, by contrast, small stars just barely keep the nuclear fires burning, and last much longer than their more massive brethren. [Shrews are particularly interesting, and remind us of another piece of basic physics. They have fantastically high metabolic rates, and eat and digest nourishment equivalent to several times their own body weight per day. (Imagine a human eating several hundred kilos of high energy food per day!) This situation is forced by the need to keep their body temperatures high. Their difficulty in doing so stems from the simple scaling argument we encountered in the first lectures of Physics 015: small bodies are less able to retain internal heat than are big ones, thanks to the relatively larger surface areas which they present to the outside world.] Previous chapter:Next chapter0: Physics 016: The Course Notes, spring 2005. 1: The Properties of the Sun: 2: What Is The Sun Doing? 3: An Introduction to Thermonuclear Fusion. 4: Probing the Deep Interior of the Sun. 5: The Sun in More Detail. 6: An Introduction to the Stars. 7: Stars and Their Distances: 8: The HR Diagram: 9: Questions Arising from the HR Diagram: 10: The Importance of Binary Stars: 11: Implications from Stellar Masses: 12: Late in the Life of the Sun: 13: The Importance of Star Clusters in Understanding Stellar Evolution: 14: The Chandrasekhar Limit: 15: Supernovae: The Deaths of Massive Stars, 16: Pulsars: 17: Novae: 18: An Introduction to Black Holes: 19: Gravity as Geometry: 20: Finishing Off Black Holes: 21: Star Formation: 22: Dust in the Interstellar Medium: 23: Gas in the ISM: 24: The Size and Shape of Our Galaxy: 25: The Discovery of External Galaxies: 26: Galaxies of All Kinds: 27: The Expanding Universe: 28: Quasars and Active Galaxies: 29: The Hot Big Bang: 30: The Geometry of the Universe: 31: Closing Thoughts: Part 1:Part 2:Part 3: |
(Wednesday, 22 April, 2026.)
