The Masses of Stars
Lecture 8 --The Sizes and Masses of Stars

So far, we've talked about stellar luminosity, temperature, and size. Another important property to understand is stellar mass. Stars can be ``weighed'' using the laws of Kepler and Newton. Recall that Newton's modification to Kepler's 3rd law is (M + m) P^2 = a^3, where P is the orbital period (in years), a the semi-major axis of the orbit (in astronomical units), and M and m the masses of the two objects (in solar masses). If we could identify binary stars that orbit around each other, and if we could measure their period and their semi-major axis, we could measure their mass.

The universe is kind to us. Perhaps 1/2 of all stars are binary stars. Astronomers have different names for these stars, which depend on how we identify and perceive their binary nature. Here's a list of terminology; note that the same stars can occasionally fall into more than one catagory.

Optical Double. This is not a binary star at all. An optical double is just two stars that appear next to each other on the sky.

Visual Binary. Like an optical double, a visual binary occurs when you actually see both stars. Unlike the optical double, however, in this case, both stars are really together in space and going around in each. Our nearest star, proxima Centauri, is part of a visual binary (actually trinary). Visual binaries typically have orbital periods of centuries.

Spectrum Binary. In this case, the two stars are so close together that an astronomer only sees one object. However, the spectrum of the object gives the binary nature of the star away. For example, suppose the stellar spectrum shows absorption due to helium, and absorption due to molecules. Molecular absorption only occurs in the coolest stars; helium only appears in the hottest stars. Since the star cannot be both hot and cool at the same time, there must be two stars. Spectrum binaries are interesting, but not usually of much use.

Spectroscopic Binary. This is an extremely important type of binary star. For spectroscopic binaries, the two stars are so close together that an astronomer only sees one object. However, over time, the astronomer will see that the Doppler shift (as measured through its absorption lines) change. First, the star will be moving towards us; then away from us; then towards us again. Through Newton's first law, this can only happen if there's a force at work -- the gravity from another star. Spectroscopic binaries can have periods of months, days, hours, or even minutes!

Spectroscopic binaries are important because of what they can tell us. First, by monitoring the Doppler shift, one can time how long it takes to complete one forward-backward-forward cycle. That's the star's period. Next, again, from the Doppler shift, one can determine the star's velocity. Velocity times time equals distance, so this gives you the size of the star's orbit, and, with a minimal amount of math, the orbit's semi-major axis. The law of gravity then gives you the total mass of the two stars.

If one star is very bright, while the other is very faint, then you will only see the absorption lines of the bright star. In this case, the star is a single-line spectroscopic binary and you can't do any more with it. However, if both stars are about equal brightness, then you may see absorption lines from both stars. While one star is moving towards you, the other will be moving away from you. The relative speeds of the stars tell you the relative mass: the more massive star will be moving slowly, while the less massive star will be moving rapidly. In this double-line spectroscopic binary case, you can not only measure the total mass of both stars together, but the mass ratio of the stars. You thus have individual masses.

There is one limitation to spectroscopic binaries. The Doppler shift only gives you the motion of the star in your direction; you don't see the motion that is in the plane of the sky. Thus, depending on the inclination, you may underestimate the star's velocity and mass. (In the extreme case where both stars are in the plane of the sky, there will be no Doppler shift at all, even though the stars may be moving rapidly.)

A site which allows you to experiment with changing the mass and orbital elements of a spectroscopic binary system is linked here.

Eclipsing Binary. Eclipsing binaries are like a spectroscopic binaries in that the astronomer only sees one object. However, in this case, the stars are oriented in such a way as that one occasionally gets in the way of the other, i.e., one star eclipses the other. When this happens, the light from the system decreases. By following the object's light curve, one can measure the star's period. Eclipsing binaries are important because for these stars, you know the orientation; the plane of the orbit is along your line-of-sight.

If you measure the absorption lines of an eclipsing binary, you'll find it to be a spectroscopic binary. Since the inclination of the stars is known, then there is no ambiguity about the mass determination from the Doppler shift. Moreover, by timing how long it takes for the stars to move into and out of eclipse, and by noting how fast the stars are moving, it is possible to use these systems to measure not only a star's mass but also its size.

A site which allows you to experiment with eclipsing binary stars is linked here.

Using spectroscopic and eclipsing binary systems, it is possible to measure stellar masses. From Newton's second law, the relative speeds of the stars gives you their relative masses: the faster the star is moving, the lighter it is. The size of each star's orbit is easily measured from its velocity (obtained through the Doppler shift) and the period of the orbit. (Distance equals velocity times time.) From the size of the orbit comes the semi-major axis, and, once the semi-major axis and the period are known, the total mass of the system follows from (Newton's modification of) Kepler's third law. So you know everything!

Once you have the stellar masses, you can compare them to the positions of the stars on the HR diagram. When you do this, a pattern emerges.

Along the main sequence, there is a smooth change in mass. Faint, red main sequence stars have low mass, perhaps only 0.1 that of the Sun (ie., 0.1 of a solar mass). Bright blue main sequence stars have large masses, up to around 60 solar masses. (G main sequence stars like the Sun, of course, have around 1 solar mass.) This is the main-sequence mass-luminosity relation. In addition, white dwarf stars are mostly less than 1 solar mass, and are never more than 1.4 solar masses. However, there is no pattern to the red giant stars: some red giants have masses like the Sun, while others are much more massive.

How Stars Don't Work

In order to explain the patterns in the HR diagram, we need to understand how stars work. But, as you shall see, it's even more important to understand how stars don't work. Let me explain.

Let's consider the Sun. The Sun is, essentially, an extremely massive ball of gas. (Like most things in the universe, 9 out of 10 atoms in the Sun is hydrogen, and 9 out of 10 of what's left is helium.) On the surface of the Sun, there is considerable surface gravity (roughly 50 times that of the earth). So what's holds up the Sun? What stops gravity from pulling all the gas down to a point at the center?

The answer is gas pressure. The same thing that holds up a bicycle tire holds up the Sun. The atoms of gas inside the Sun are bouncing left and right off each other, and by doing so, they create a force outward -- a pressure. But appreciate this: through the equation of state, gas pressure is locked hand-in-hand with temperature. If you increase the gas pressure you increase the temperature. (Use a bicycle pump to pressurize a tire, and then feel pump, if you don't believe this.) If you decrease the temperature, you slow down the atoms, and you decrease the gas pressure. You can't move one without affecting the other.

Now consider: with all the mass above it, the gas at the center of the Sun is under tremendous pressure. This means that gas at the center must be very hot, over 10 million degrees. Since the outside of the Sun is much cooler than this, the heat at the center of the Sun must flow out.

There are three ways in which heat can be transported. Heat can move through conduction --- fast moving (hot) free electrons crash into slow moving (cool) free electrons and give up some of their energy. This is (almost) never important in astronomy. Heat can move through convection. This is the equivalent of taking the hot water of a bath tub and stirring it in with the cold water. By and large, this doesn't happen in the Sun either. The third method of heat transport is through radiation. The hot gas at the center of the Sun radiates light (through the blackbody law). The photons travels a very short way before being absorbed, and the absorbing material gains the energy. This material then radiates the energy way, and the new photons again get re-absorbed. In this random walk manner, energy percolates outward. This is how the heat at the center of the Sun gets out. But it does take some time --- about 10 million years.

Now consider what is happening to the center of the Sun during this time. Because the heat is leaking out, the center of the Sun is becoming slightly cooler. A cooler temperature means that the gas pressure is becoming slightly smaller, and the pressure that resists the pull of gravity is slightly less. So the outside of the Sun shrinks a little. This additional shrinking reduces the radius of the star and, since the mass remains the same, the effect of gravity increases. This puts the Sun's center under a bit more pressure. More pressure means more temperature. More temperature means more heat and more radiation, which then leaks out. And the process continues.

William Thompson, better known as Lord Kelvin, considered this scenario in the middle of the 19th century. He realized that the entire energy of the Sun could be produced if the Sun was shrinking about 1 centimeter per year. He also calculated that through gravitational contraction, the Sun could successfully maintain its brightness for about 40 million years. After that, larger structural changes would occur. As a result, Lord Kelvin concluded that the Sun (and the earth) could not be older than about 40 million years, and the slow biological evolution scenario proposed by Charles Darwin had to be wrong.