Monday, January 31, 2011

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.