The Life Expectancy of a Star

A star dies when it consumes its nuclear fuel, its mass. We might be tempted to conclude that the greater the supply of fuel (the more massive the star), the longer it will live; however, a star’s life span is also determined by how rapidly it burns its fuel. The more luminous a star, the more rapid the rate of consumption. Thus stellar lifetime is directly proportional to stellar mass and inversely proportional to stellar luminosity (how fast it burns). An analogy: A car with a large fuel tank (say a new Ford Excursion that gets 4–8 mpg) may have a much smaller range than a car with a small fuel tank (a Saturn which might get 30–40 mpg). The key? The Saturn gets much better mileage, and thus can go farther with the limited fuel it has.
Thus, while O- and B-type giants are 10 to 20 times more massive than the our G-type sun, their luminosity is thousands of times greater. Therefore, these most massive stars live much briefer lives (a few million years) than those with less fuel but more modest appetites for it.
A B-type star such as Rigel, 10 times more massive than the sun and 44,000 times more luminous, will live 20 106 years, or 20 million years. For comparison, 65 million years ago, dinosaurs roamed the earth! The G-type sun may be expected to burn for 10,000 106 years (ten billion years). Our red dwarf neighbor, Proxima Centauri, an M-type star that is 1⁄10 the mass of the sun (and 1⁄100 that of Rigel), is only 0.00006 times as luminous as the sun, so will consume its modest mass at a much slower rate and may be expected to live more than the current age of the universe. In the next two chapters we will see how stars go through their lives, and how they grow old and die.

Understanding Stellar Mass

The overall orderliness of the main sequence suggests that the properties of stars are not random. In fact, a star’s exact position on the main sequence and its evolution are functions of only two properties: composition and mass.
Composition can be evaluated if we have a spectrum of the star, its fingerprint. But how can we determine the mass of a star?
Fortunately, most stars don’t travel solo, but in pairs known as binaries. (Our sun is an exception to this rule.) Binary stars orbit one another.
Some binaries are clearly visible from the earth and are called visual binaries, while others are so distant that, even with powerful telescopes, they cannot be resolved into two distinct visual objects. Nevertheless, these can be observed by noting the Doppler shifts in their spectral lines as they orbit one another. These binary systems are called spectroscopic binaries. Rarely, we are positioned so that the orbit of one star in the binary system periodically brings it in front of its partner. From these eclipsing binaries we can monitor the variations of light emitted from the system, thereby gathering information about orbital motion, mass, and even stellar radii.
However we observe the orbital behavior of binaries, the key pieces of information sought are orbital period (how long it takes one star to orbit the other) and the size of the orbit. Once these are known, Kepler’s third law can be used to calculate the combined mass of the binary system.
Why is mass so important? Mass determines the fate of the star. It sets the star’s place along the main sequence and it also dictates its life span.

Making the Main Sequence

Working independently, two astronomers, Ejnar Hertsprung (1873–1967) of Denmark and Henry Norris Russell (1877–1957) of the United States studied the relationship between the luminosity of stars and their surface temperatures. Their work (Hertsprung began about 1911) was built on the classification scheme of another woman from the Harvard College Observatory, Antonia Maury. She first classified stars both by the lines observed and the width or shape of the lines. Her scheme was an important step toward realizing that stars of the same temperature could have different luminosity. Plotting the relationship between temperature and luminosity graphically (in what is now known as a Hertzsprung-Russell diagram or H-R diagram), these two men discovered that most stars fall into a well-defined region of the graph. That is, the hotter stars tend to be the most luminous, while the cooler stars are the least luminous.
The region of the temperature luminosity plot where most stars reside is called the main sequence. Most stars are there, because as we will discover, that is where they spend the majority of their lives. Stars that are not on the main sequence are called giants or dwarfs, and we will see how stars leave the main sequence and end up in the far corners of the temperature-luminosity plot.

Sorting the Stars by Size

The radius of a star can be determined from the luminosity of the star (which can be determined if the distance is known) and its surface temperature (from its spectral type). It turns out that stars fall into several distinct classes. In sorting the stars by size, astronomers use a vocabulary that sounds as if it came from a fairytale:

• A giant is a star whose radius is between 10 and 100 times that of the sun.
• A supergiant is a star whose radius is more than 100 times that of the sun. Stars of up to 1,000 solar radii are known.
• A dwarf star has a radius similar to or smaller than the sun.

How Hot Is Hot?

Stars are too distant to stick a thermometer under their tongue. We can’t even do that with our own star, the sun. But you can get a pretty good feel for a star’s temperature simply by looking at its color.
The temperature of a distant object is generally measured by evaluating its apparent brightness at several frequencies in terms of a blackbody curve. The wavelength of the peak intensity of the radiation emitted by the object can be used to measure the object’s temperature. For example, a hot star (with a surface temperature of about 20,000 K) will peak near the ultraviolet end of the spectrum and will produce a blue visible light. At about 7,000 K, a star will look yellowish-white. A star with a surface temperature of about 6,000 K—such as our sun—appears yellow. At temperatures as low as 4,000 K, orange predominates, and at 3,000 K, red.
So simply looking at a star’s color can tell you about its relative temperature. A star that looks blue or white has a much higher surface temperature than a star that looks red or yellow.

Creating a Star Scale of Magnitude

So astronomers have learned to be very careful when classifying stars according to apparent brightness. Classifying stars according to their magnitude seemed a good idea to Hipparchus (in the second century B.C.E.) when he came up with a 6-degree scale, ranging from 1, the brightest stars, to 6, those just barely visible. Unfortunately, this somewhat cumbersome and awkward system (higher magnitudes are fainter, and the brightest objects have negative magnitudes) has persisted to this day.
Hipparchus’ scale has been expanded and refined over the years. The intervals between magnitudes have been regularized, so that a difference of 1 in magnitude corresponds to a difference of about 2.5 in brightness. Thus, a magnitude 1 star is 2.5 ×2.5 ×2.5 ×2.5 ×2.5=100 times brighter than a magnitude 6 star. Because we are no longer limited to viewing the sky with our eyes, and larger apertures collect more light, magnitudes greater than (that is, fainter than) 6 appear on the scale. Objects brighter than the brightest stars may also be included, their magnitudes expressed as negative numbers. Thus the full moon has a magnitude of –12.5 and the sun, –26.8. In order to make more useful comparisons between stars at varying distances, astronomers differentiate between apparent magnitude and absolute magnitude, defining the latter, by convention, as an object’s apparent magnitude when it is at a distance of 10 parsecs from the observer. This convention cancels out distance as a factor in brightness and is therefore an intrinsic property of the star.

Luminosity Versus Apparent Brightness

Ask an astronomer this question, and she will respond that the flashlight, a few feet from your eyes, is apparently brighter than the distant headlights, but that the headlights are more luminous. Luminosity is the total energy radiated by a star each second. Luminosity is a quality intrinsic to the star; brightness may or may not be intrinsic. Absolute brightness is another name for luminosity, but apparent brightness is the fraction of energy emitted by a star that eventually strikes some surface or detection device (including our eyes). Apparent brightness varies with distance. The farther away an object is, the lower its apparent brightness.
Simply put, a very luminous star that is very far away from the earth can appear much fainter than a less luminous star that is much closer to the earth. Thus, although the Sun is the brightest star in the sky, it is not by any means the most luminous.

Do Stars Move?

The ancients believed that the stars were embedded in a distant spherical bowl and moved in unison, never changing their relative positions. We know now, of course, that the daily motion of the stars is due to the earth’s rotation. Yet the stars move, too; however, their great distance from us makes that movement difficult to perceive, except over long periods of time. A jet high in the sky, for example, can appear to be moving rather slowly, yet we know that it has to be moving fast just to stay aloft and its apparent slowness is a result of its distance. Astronomers think of stellar movement in three dimensions:

• The transverse component of motion is perpendicular to our line of sight—that is, movement
• across the sky. This motion can be measured directly.
• The radial component is stellar movement toward or away from us. This motion must be measured from a star’s spectrum.
• The actual motion of a star is calculated by combining the transverse and radial components.

The transverse component can be measured by carefully comparing photographs of a given piece of the sky taken at different times and measuring the angle of displacement of one star relative to background stars (in arcseconds).
This stellar movement is called proper motion. A star’s distance can be used to translate the angular proper motion thus measured into a transverse velocity in km/s. In our analogy: If you knew how far away that airplane in the sky was, you could turn its apparently slow movement into a true velocity.
Determining the radial component of a star’s motion involves an entirely different process. By studying the spectrum of the target star (which shows the light emitted and absorbed by a star at particular frequencies), astronomers can calculate the star’s approaching or receding velocity. Certain elements and molecules show up in a star’s spectrum as absorption lines (see Chapter 7). The frequencies of particular absorption lines are known if the source is at rest, but if the star is moving toward or away from us, the lines will get shifted. A fast-moving star will have its lines shifted more than a slow-moving one. This phenomenon, more familiar with sound waves, is known as the Doppler effect.
How fast do stars move? And what is the fixed background against which the movement can be measured? For a car, it’s easy enough to say that it’s moving at 45 miles per hour relative to the road. But there are no freeways in space. Stellar speeds can be given relative to the earth, relative to the sun, or relative to the center of the Milky Way. Astronomers always have to specify which reference frame they are using when they give a velocity. Stars in our neighborhood typically move at tens of kilometers per second relative to the sun.

Nearest and Farthest

Other than the sun, the star closest to us is Alpha Centauri, which has the largest known stellar parallax of 0.76 arc seconds. In general, the distance to a star in parsecs (abbreviated pc) is equal to 1 divided by the stellar parallax in arcseconds—or conversely, its parallax will be equal to 1 divided by the distance in parsecs. The measured parallax, in any case, will be a very small angle (less than an arcsecond). Recall that the moon takes up about 1,800” on the sky when full, so the parallax measured for Alpha Centauri is about 1⁄2000 the diameter of the full moon! Using the rule above to convert parallax into distance, we find that Alpha Centauri is about 1.3 pc or 4.2 light-years away. On average, stars in our Galaxy are separated by 7 light-years. So Alpha Centauri is even closer than “normal.” If a star were 10 pc away, it would have a parallax of 1⁄10 or 0.1”.
The farthest stellar distances that can be measured using parallax are about 100 parsecs (333 light-years). Stars at this distance have a parallax of 1⁄100” or 0.01”. That apparent motion is the smallest that we can measure with our best telescopes. Within our own Galaxy, most stars are even farther away than this. As telescope resolutions improve with the addition of adaptive optics, this outer limit will be pushed farther out.

How Far Away Are the Stars?

Like the campsite separated from you by the Grand Canyon, the stars are not directly accessible to measurement. However, if you can establish two view points along a baseline, you can use triangulation to measure the distance to a given star.
There is just one problem.
Take a piece of paper. Draw a line one inch long. This line is the baseline of your triangle. Measure up from that line, say, one inch, and make a point. Now connect the ends of your baseline to that point. You have a nice, normal looking triangle. But if you place your point several feet from the baseline, then connect the ends of the baseline to it, you will have an extremely long and skinny triangle, with angles that are very difficult to measure accurately, because they will both be close to 90 degrees. If you move your point several miles away, and keep a 1-inch baseline, the difference in the angles at Points A and B of your baseline will be just about impossible to measure. They will both seem like right angles. For practical purposes, a 1-inch baseline is just not long enough to measure distances of a few miles away. Now recall that if our Earth is a golf ball (about 1 inch in diameter), that the nearest star, to scale, would be 50,000 miles away. So the baseline created by, say, the rotation of the earth on its axis—which would give 2 points 1 inch away in our model—is not nearly large enough to use triangulation to measure the distance to the nearest stars. The diameter of the earth is only so wide. How can we extend the baseline to a useful distance?
The solution is to use the fact that our planet not only rotates on its axis, but also orbits the sun. Observation of the target star is made, say, on February 1, then is made again on August 1, when the earth has orbited 180 degrees from its position six months earlier. In effect, this motion creates a baseline that is 2 A.U. long—that is, twice the distance from the earth to the sun. Observations made at these two times (and these two places) will show the target star apparently shifted relative to the even more distant stars in the background. This shift is called stellar parallax, and by measuring it, we can determine the angle relative to the baseline and thereby use triangulation to calculate the star’s distance.
To get a handle on parallax, hold your index finger in front of you, with your arm extended. Using one eye, line up your finger with some vertical feature, say the edge of the window. Now, keeping you finger where it is, look through the other eye. The change in viewpoint makes your finger appear to move with respect to a background object. In astronomy, your eyes are the position of the earth separated by 6 months, your finger is a nearby star, and the window edge is a distant background star. This method works as long as the star (your finger) is relatively close. If the star is too far away, parallax is no longer effective.

The Parallax Principle

First, how do we know that the nearest stars are so far away? For that matter, how do we know how far away any stars are? We’ve come a long way in this blog, and, on our journey, we have spoken a good deal about distances—by earthly standards, often extraordinary distances. Indeed, the distances astronomers measure are so vast that they use a set of units unique to astronomy. When measuring distances on the earth, meters and kilometers are convenient units. But in the vast spaces between stars and galaxies, such units are inadequate. As we’ll see in this chapter and those that follow, the way astronomers measure distances, and the units they use depend upon how far away the objects are. Distances between a given point on the earth and many objects in the solar system can be measured by radar ranging. Radar, a technology developed shortly before and during World War II, is now quite familiar. Radar can be used to detect and track distant objects by transmitting radio waves, then receiving the echo of the waves the object bounces back (sonar is a similar technique using sound waves). If we multiply the round-trip travel time of the outgoing signal and its incoming echo by the speed of light (which, you’ll recall, is the speed of all electromagnetic radiation, including radio waves), we obtain a figure that is twice the distance to the target object.
Radar ranging works well with objects that return (bounce back) radio signals. But stars, including our sun, tend to absorb rather than return electromagnetic radiation transmitted to them. Moreover, even if we could bounce a signal off a star, most are so distant that we would have to wait thousands of years for the signal to make its round trip—even at the speed of light! Even the nearby Alpha Centauri system would take about eight years to detect with radar ranging, were it even possible.
Another method is used to determine the distance of the stars, a method that was available long before World War II. In fact, it is at least as old as the Greek geometer Euclid, who lived in the third century B.C.E. The technique is called triangulation—an indirect method of measuring distance derived by geometry using a known baseline and two angles from the baseline to the object. Triangulation does not require a right triangle, but the establishment of one 90-degree angle does make the calculation of distance a bit easier. It works like this. Suppose you are on one rim of the Grand Canyon and want to measure the distance from where you are standing to a campsite located on the other rim. You can’t throw a tape measure across the yawning chasm, so you must measure the distance indirectly. You position yourself directly across from the campsite, mark your position, then turn 90 degrees from the canyon and carefully pace off another point a certain distance from your original position. This distance is called your baseline. From this second position, you sight on the campsite. Whereas the angle formed by the baseline and the line of sight at your original position is 90 degrees (you arranged it to be so), the angle formed by the baseline and the line of sight at the second position will be somewhat less than 90 degrees. If you connect the campsite with Point A (your original viewpoint) and the campsite with Point B (the second viewpoint), both of which are joined by the baseline, you will have a right triangle. Now, you can take this right triangle and, with a little work, calculate the distance across the canyon. If you simply make a drawing of your setup, making sure to draw the angles and lengths that you know to scale, you can measure the distance across the canyon from your drawing. Or if you are good at trigonometry, you can readily use the difference between the angles at Points A and B and the length of the baseline to arrive at the distance to the remote campsite.