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.

Radius, Luminosity, Temperature: A Key Relationship

We don’t have to give up on measuring the sizes of stars, however. We just have to be more clever. What astronomers do is determine the temperature and mass of a star, which can be done using a star’s color and spectrum. Then, using numerical models of how stars hold together, they derive the quantity that they are interested in (radius, for example). It is akin to looking out over a parking lot and seeing a Cadillac. Now, you may not know its size, but you know (consulting a chart) that this model of Cadillac is 18.5 feet long. You can see clearly that it is indeed this particular model of Cadillac, so you know its length, even though you didn’t actually measure it with a ruler.
Stefan’s law states that a star’s luminosity (its wattage, or the rate at which it emits energy into space) is proportional to the fourth power to the star’s surface temperature. This relationship can be extended further. A star’s luminosity is not only related to its temperature, but to its surface area. Heat the head of a pin to 400 degrees F and a large metal plate to the same temperature. Which will radiate more heat? Obviously, the object with the larger surface area. Given the same surface temperature, a larger body will always radiate more energy than a smaller one.
This relationship can be expressed in this way: A star’s luminosity is proportional to the square of its radius (that’s the surface area term) times its surface temperature to the fourth power (luminosity ×radius2 ×temperature4). Thus, if we know a star’s luminosity and temperature (which can be measured by available astronomical instruments), we can calculate its radius. How do we measure a star’s luminosity and temperature? Let’s see.

Your Standard Solar Model

By combining theoretical modeling of the sun’s (unobservable) interior with observations of the energy that the sun produces, astronomers have come to an agreement on what is called a standard solar model, a mathematically-based picture of the structure of the sun. The model seeks to explain the observable properties of the sun and also describe properties of its unobservable interior. With the standard solar model, we can begin to describe some of the interior regions—regions hidden, beneath the photosphere, from direct observation. Below the photosphere is the convection zone, some 124,000 miles (200,000 km) thick. Below this is the radiation zone, 186,000 miles (300,000 km) thick, which surrounds a core with a radius of 124,000 miles (200,000 km).
The sun’s core is tremendously dense (150,000 kg/m3) and tremendously hot: some 15,000,000 K. We can’t stick a thermometer in the sun’s core, so how do we know it’s that hot? If we look at the energy emerging from the sun’s surface, we can work backward to the conditions that must prevail at the sun’s core. At this density and temperature, nuclear fusion is continuous, with particles always in violent motion. The sun’s core is a giant nuclear fusion reactor.
At the very high temperatures of the core, all matter is completely ionized—stripped of its negatively charged electrons. As a result, photons (packets of electromagnetic energy) move slowly out of the core into the next layer of the sun’s interior, the radiation zone.
Here the temperature is lower, and photons emitted from the core of the sun interact continuously with the charged particles located there, being absorbed and re-emitted. While the photons remain in the radiation zone, heating it and losing energy, some of their energy escapes into the convection zone, which in effect, boils like water on a stove so that hot gases rise to the photosphere and cool gases sink back into the convection zone. Convective cells become smaller and smaller, eventually becoming visible as granules at the solar surface. Thus, by convection, huge amounts of energy reach the surface of the sun. At the sun’s surface, a variety of processes give rise to the electromagnetic radiation that we detect from the earth. Atoms and molecules in the sun’s photosphere absorb some of the photons at particular wavelengths, giving rise to the sun’s absorption-line spectrum. Most of the radiation from a star that has the surface temperature of the sun is emitted in the visible part of the spectrum.

Chain Reactions in the Sun

The sun generates energy by the converting the hydrogen in its core to helium. The details are complex, but we may content ourselves with an overview. When temperatures and pressures are sufficiently high (temperatures of about 10 million K are required) 4 hydrogen nuclei (which are protons, positively charged particles) can combine to create the nucleus of a helium atom (2 protons and 2 neutrons). Now the mass of the helium nucleus created is slightly less than that of the four protons that were needed to create it. That small difference in mass is converted into energy in the fusion process. One of the simplest fusion reactions involves the production of deuterium (a hydrogen isotope) from a proton and a neutron. When these two particles collide with enough velocity, they create a deuterium nucleus (consisting of a proton and a neutron) and the excess energy is given off as a gamma ray photon. In the sun, this process proceeds on a massive scale, liberating the energy that lights up our daytime skies. That’s a 4 ×1026 watt lightbulb up there, remember.

Fission in the Sun

On December 2, 1942, Enrico Fermi, an Italian physicist who had fled his fascistoppressed native land for the United States, withdrew a control rod from an “atomic pile” that had been set up in a squash court beneath the stands of the University of Chicago’s Stagg Field. This action initiated the world’s first self-sustaining atomic chain reaction. Fermi and his team had invented the nuclear reactor, and the world hasn’t been the same since.
Nuclear fission is a nuclear reaction in which an atomic nucleus splits into fragments, thereby releasing energy. In a fission reactor, such as the one Fermi was instrumental in creating, the process of fission is controlled and self-sustaining, so that the splitting of one atom leads to the splitting of others, each fission liberating more energy.
Nuclear fission is capable of liberating a great deal of energy, whether in the form of a controlled sustained chain reaction or in a single great explosion, like an atomic bomb. Yet even the powerful fission process cannot account for the tremendous amount of energy the sun generates so consistently. We must look to another process: nuclear fusion.
Whereas nuclear fission liberates energy by splitting atomic nuclei, nuclear fusion produces energy by joining them, combining light atomic nuclei into heavier ones. In the process, the combined mass of two nuclei in a third nucleus is less than the total mass of the original two nuclei. The mass is not simply lost, but converted into energy. A lot of energy. One of the by-products of nuclear fusion reactions is a tiny neutral particle called the neutrino. The fusion reactions themselves produce high energy gamma ray radiation, but those photons are converted into mostly visible light by the time their energy reaches the surface of the sun. Neutrinos, with no charge to slow them down, come streaming straight out of the sun’s core. The numbers that we detect give us great insight into a region of the sun that is otherwise inaccessible.