Over on Flame we talked about the Luminosity Function.

The Absolute Luminosity Function by ER December 12, 2015 10:32 am

Now we will discuss a related astronomical topic, the Mass Function. The Mass Function, or MF,
is a graph of the masses of stars plotted against their numbers. The mass is plotted on the X-axis,
the number of stars (or other type objects) is plotted on the Y-axis. The MF is not a directly observable characteristic of astronomical populations, only those stars that are members of binary systems can have their masses directly determined (through Newton’s Laws). Fortunately, binary stellar systems are quite common, so we have been able to accumulate a mass of data correllating the masses of stars with their brightness, shape, size, color, temperature, chemical composition, luminosity, age and so on, so that we can now infer the masses of stars even when they are not a member of a gravitationally bound system. All of these characteristics are related, but in no simple, straightforward way. The properties of a star also change over time, during the course of its evolution, so untangling how all these parameters influence each other has been one of the major programs (and triumphs) of astrophysics in the twentieth century.

Mass is the single characteristic of a star that determines most its subsequent evolution and appearance over time. All those other properties I mentioned above can change, but for the most part, a star is born with a given mass and it maintains that mass throughout its history. There are exceptions, but in general, the mass of a star at birth determines its life and its destiny.

A good working definition of “star” is a compact sphere of ionized gas illuminated by an interior thermonuclear energy source. Less massive objects that cannot generate the core temperatures and pressures to support fusion processes are called “brown dwarfs”, although they are not really brown at all. They look like faint red dwarf stars, but their luminosity is the result of the heat generated by their gravitational collapse. They shine, essentially, from friction. Relict gravitational energy causes them to glow. Eventually, they will cool off and radiate their energy into space, becoming lumps of cold slag floating through the darkness. We suspect there are lots of brown dwarfs, but its hard to tell. They are faint and hard to see, even when they are lit up; only a handful are known, but they may be the most common objects in the universe. Even less massive objects can form, but they may not be heavy enough to generate the gravitational energy to fully ionize their gases. We call these “gas giant planets” like Jupiter or Saturn. Both of them also glow faintly (in the infrared) with gravitational energy but we see them primarily from reflected sunlight. The boundary line between gas giants and brown dwarfs is hazy and indistinct, in fact, it is purely arbitrary. Gas giant planets are simply the least massive, leftmost members of the mass function. brown dwarfs are the next bin of the histogram. It is said that Jupiter is a “failed star”, that is, if it had been a bit more massive, it might have generated enough heat and pressure in its core to start fusion reactions and we would now live in a binary star system. Jupiter was probably a brown dwarf soon after the solar system was formed, but it has since cooled off enough to stop glowing, and its gases un-ionized ( became electrically neutral). Jupiter’s mass is about a thousandth that of the Sun.

So at the bottom end of the mass function we have gas giant planets, and then brown dwarfs. The next step up is a star with just enough mass to generate the core conditions that will support fusion processes. The conversion of hydrogen to helium generates heat and light which stops the gravitational collapse of the primordial gas cloud. We call these stars red dwarfs, faint red stars which can be seen only with large telescopes, and which are probably the most common of the true stars, even though we can only see some of the nearby ones. What we know now about the mass function is that the lighter stars are the most common, and they become much scarcer at higher and higher masses. The faintest red dwarfs are about 7.5 percent the mass of our Sun. The heaviest are about half the mass of our Sun.

Protostars are born when they condense from the gas clouds of some nebula. As they collapse gravitationally, they glow from released gravitational energy, until the back-pressure of the gas laws stop the collapse, and the heat of friction is slowly emitted as light into space. If the protostar is massive enough to begin with, fusion reactions in the core start up pushing back the weight of the outer layers and the star stabilizes at a certain level of luminosity and temperature. This is a condition known as hydrostatic equilibrium, and this conflict between the weight of the star’s outer layers and the push back from thermonuclear fusion in its core stabilizes it at a certain size, luminosity, temperature and color at which it will continue throughout its life. When a star reaches this condition of equilibrium it is said to be on the Main Sequence,. You may recall the term “main sequence” from one of my earlier essays, it is that line on the color-magnitude (or Hertzsprung-Russel) diagram, a plot of the stellar luminosity vs their color (color is highly correlated with surface temperature).

IOW, when a star is formed, it quickly stabilizes into a condition of constant luminosity, surface temperature, and size, which corresponds to a specific point on the main sequence. And it stays there, for most of its life, at a predetermined color, temperature, size and brightness, until something changes (like it runs out of nuclear hydrogen fuel in its core). Our sun is a main sequence star, a G-type subdwarf, and it has been on the main sequence for about 5 billion years now, and will remain there until its core runs out of hydrogen about 5 billion years from now. The G refers to its color, yellow, which in turn is the result of its surface temperature. Evolving off the main sequence determines the final destiny of a star, it will quickly go through some pretty significant changes, and either blow itself up in a colossal explosion, or eventually settle down and become a red or white dwarf

So for most stars, they form out of the gas cloud, light up when fusion starts up in their cores and they settle down for a long dull life on the main sequence until they start running out of hydrogen.
Where on the main sequence they wind up, (hence, at what luminosity and temperature) depends on only one thing: how massive they are. The least massive ones will remain red dwarfs for a long, long, time. In fact, the universe isn’t old enough yet for any of them to have evolved off the main sequence. The most massive will start off on the main sequence as bright hot blue supergiants and they will evolve off the main sequence quite quickly, perhaps just a few million years. They burn through their fuel in a relatively short time. The mass range between stars goes from roughly 1/10 solar masses, to about 100 solar masses. And the mass function we have derived tells us faint, old, cool, stars far outnumber the bright hot young ones. Our own Sun is of intermediate brightness and temperature, and is about half-way through its main sequence lifetime.

Although the evolution of a star after it leaves the main sequence only occupies a very small percentage of its total life span, it will be the most spectacular and interesting period of its life (unless you happen to live on a planet orbiting it!). As the star runs out of fuel, its core will shrink until it gets hot enough to fuse other elements, in this way, all the elements up to iron are formed in stellar cores. However, as the core shrinks, the outer layers of the star may balloon enormously, forming a red giant or supergiant, with a relatively cool surface but a very hot, high density core. Hydrostatic equilibrium keeps the star together, internal energy holding up the weight of the outer layers, as elements of higher atomic number fuse in the core. Eventually, the star runs out of every source of nuclear fuel, and it will eventually shrink back down to red dwarf stage. However the most massive stars (about 1.4 solar masses or more, a very tiny fraction of all stars) are too massive, their cores become unstable due to quantum effects, and they eventually blow themselves up in spectacular supernova explosions. The elements heavier than iron are formed by neutron absorption as the star blows itself apart, and the matter of the periodic table is rendered up to the interstellar medium so that new stars can be born with rocky planets suitable for folks like us.
What’s left after the supernova also depends on the initial stellar mass. Going from light to heavy, a white dwarf, a neutron star, or a black hole.

Stars below the Chandrasekhar Limit of 1.4 solar masses either achieve stability by going through a long series of ordinary nova outbursts, or shed excess mass by simple outgassing (the planetary nebula phase) and eventually settle down as a white dwarf.

So although stars may vary enormously in size, luminosity, temperature, and color, and they may change all those properties over time in the course of their evolution, their destiny is decided at birth solely by their mass. The mass function is the key to understanding stellar structure and evolution.

Its unlikely we understand all there is to know about this, but over the last half of the twentieth century most of the blank spaces in this incredible story have been filled in. It is one of the most remarkable achievements of the human intellect, and it was mostly accomplished during my lifetime, by at most a few hundred men.