Measuring Distances to Stars

Astronomers use several techniques for discovering how far away an object is. The first is called trigonometric parallax and is based on geometry, but it is only good for up to about 500 light-years. The principle behind this method is elegantly simple: Earth orbits the Sun at a known radius and when the Earth is at opposite ends of its orbit it results in a star appearing in a slightly different positions against distant background stars that allow us to use simple trigonometry to calculate how far away it is (see diagram below). The parallax (symbolized by the Greek letter, Θ) is defined as the angular size of an elliptical arc that the star seems to trace against the background of space. Since,

tan Θ = r/d

where tan refers to the tangent of a triangle, r is the radius of the Earthís orbit (equal to 1 A.U.), and d is the distance to the star. Since an astronomer can determine the parallax by comparing photographs taken in, say, June and December and the Earthís radius is well-established value, calculation of the distance follows easily!

Trignometric parallax
You can quickly demonstrate the idea behind trigonometric parallax to yourself by placing one finger in front of you and keeping it in that position. Close your right eye and make a mental note of your fingerís position against the background. Now close your left eye and view your finger again Ė note how the position against the background has changed! This is the same principle behind the trigonometric parallax method used by astronomers. Just like your finger seems to move based on which eye is open, a star appears to move against the background of space due to the Earthís movement around the Sun.

For stars beyond 500 light-years away the techniques for determining distances must get more complicated because of the limits of measuring tiny changes in a starís apparent change in position. The first such technique, called spectroscopic parallax, makes use of a known relationship between a starís color and its magnitude (i.e., its brightness). A starís magnitude can be measured in two ways: by its apparent magnitude (that is, the brightness we measure from Earth, which is dependent not only on its temperature but also on how far away it is from us) and by its absolute magnitude (that is, the brightness as measured from an arbitrary standard distance of 10 parsecs (= 32.6 light-years), which is only dependent the starís temperature). We can determine a starís absolute magnitude by virtue of the fact that back in the early 1900s, two astronomers, Ejnar Hertzsprung and Henry Norris Russell, made a graph relating the absolute magnitude of the ordinary stars in our galaxy (called ďmain sequence starsĒ) to their color/temperature. Since most stars fall on a narrow line, called the ďmain sequenceĒ, astronomers can deduce a starís absolute magnitude to within about one magnitude. Such main sequence stars represent about 90% of the stars (including our Sun), with the other 10% being white dwarf and red giant stars. Using our own Sun as a source of calibration astronomers can determine a stars temperature from its color, and from its temperature we can look up the absolute magnitude on the Hertzsprung- Russell diagram.

A Hertzsprung- Russell diagram that relates star temperature to magnitude (Courtesy: NASA)
Since it is known that a starís absolute magnitude decreases by a square of its distance from Earth, one can simply calculate the distance to Earth by the following equation:

m = M/ d2

where m is the apparent magnitude, M is the absolute magnitude, and d is the distance to Earth. Spectroscopic parallax works for stars as far away as 150,000 light-years away Ė just about beyond the Milky Way Galaxy.

For measuring the distance to stars in other galaxies (the Large Magellanic Cloud is the nearest at 160,000 light-years away) astronomers must measure the magnitude of stars that vary a little in their brightness, called Cepheid Variables. Cephied Variables are main-sequence stars in ďold ageĒ just prior to death. Such pulsating variable stars have a period over which they go from maximum brightness to minimum brightness and then back to maximum brightness. In addition, the starís period is directly related to its absolute magnitude (i.e., the greater its absolute magnitude, the longer its period), as discovered by Henrietta Leavitt (1868 Ė 1921). Since Cephied variable stars are rather abundant in space, astronomers simply measure the starís period, determine its absolute magnitude and then, together with the relative magnitude that can also be measured, use the equation above to determine distance. For the sake of brevity, some of the details about measuring very far away stars and galaxies have been omitted. For instance, at a certain point astronomers must include the expansion of the Universe into their calculations of distances. However, this discussion of the techniques used by astronomers to determine distances should give you a general idea of how such measurements are possible.