Appendix: Parallax and Distance

One of the most important breakthroughs in astronomy was the development of accurate distance determination methods for the nearby stars. Distances in astronomy are elusive, and astronomers use many methods to determine them. Our knowledge of the size and scale of the Universe is based upon some fundamental distances, and the distances to the nearby stars are one pillar upon which many theories rest. How certain are we about the distance to a nearby star?

To find the distance to a nearby star, astronomers measure something called the parallax angle of a star. Generally, parallax is the apparent shift in an object's position resulting from the observer's changing perspective. We experience this effect just by closing one eye, then viewing the same scene with only the other eye open. Of course, the scene must be nearby, as our eyes can perceive depth only when objects are relatively close to us.

We know that stars are very far away, so we cannot use our eyes to see the parallax of stars. We need a larger baseline than the separation between our eyes. The longer baseline that allows us to see parallax in stars is the motion of the Earth around the Sun. Six months from now, Earth will have moved through half its orbit and will be located opposite the Sun from where it is today. The baseline between these two points is 2 times the Earth-Sun distance, or 2 astronomical units.

This baseline, which is almost 300 million kilometers (186 million miles), still does not reveal the apparent motion of the stars to our eyes. However, using a telescope, astronomers can measure these tiny apparent motions throughout the course of the year. But what do we observe?

Let's think about the situation. We have the Sun, Earth in orbit around it, and a distant star. The Sun-Earth distance, R, is known to be 1 AU, and the angle between the line connecting the Sun and star and the line connecting Earth and the star, p, is the parallax angle that is measured. Given these two quantities, we can solve the triangle for the distance, d, between the Sun and the star.

If you recall your trigonometric equations, you can express the solution to the triangle in this equation:

tan(p) = $\displaystyle { rac{{R}}{{d}}}$

When the angle p is very small, as it is here (the parallax angle to the nearest star, Proxima Centauri, is only 0.75 arcseconds = 0.^o00021), tan(p) is nearly zero, which means that R must be much, much greater than d. And indeed it is! When the parallax angle is one arcsecond, the distance to the star is defined to be one parallax-second, or parsec , a fundamental unit of distance in astronomy. A star that is 1 parsec from the Sun appears to subtend 1 second of arc in the sky as Earth orbits the Sun. A star of 10 parsecs, like the star Pollux in Gemini, has a parallax angle of about 1/10 arcseconds (in fact, it is 0.0967 arcseconds, close to 0.1).

This very simple relation yields insight toward the scale of the nearby stars. This understanding came in 1838 thanks to the observations of Friedrich Bessel at the Prussian Observatory in Berlin, Germany. He measured the parallax angle of the star 61 Cygni to be about a third of an arcsecond, yielding the first reliable distance to a nearby star (which turns out to be 3.2 parsecs, or about 10 light-years). Of course, at that time the properties of light were yet to be discovered, so these enormous distances were described in parsecs, not light-years.

You might think that this solves all of our problems in understanding the distance to objects, but it does not. Just as the baseline for the human eye is too small to see the effects of parallax, or depth, the baseline of Earth's orbit is also minute compared with the scales of the Milky Way, let alone the Universe.

Photometric Parallax

For stars that are too far away to measure their parallax angle, astronomers derive a photometric parallax for use in determining their distance. The photometric parallax is not an angle and is not even based in geometry. Rather, it is based on an analysis of the light of the star.

For stars with a trigonometric parallax that can be measured with little uncertainty, astronomers look at the light from the stars and classify them by spectral types according to the absorption and emission lines that appear in their spectra. If the star has a measured apparent brightness, m, as seen from Earth, and its distance d is known from its trigonometric parallax, then its intrinsic brightness, or absolute magnitude M_V, can be calculated with this equation:

m - M_V = - 5 + 5 log(d )

From these classifications, astronomers can determine whether it is a hot, intrinsically bright star or a cool, dimmer star. Then the star's intrinsic brightness and its spectral type can be calibrated for stars with known distances.

Astronomers use this calibration to determine the distance to stars that are too far away to measure trigonometric parallax angles. By observing the star's spectrum, they infer intrinsic brightness. This, along with the apparent magnitude of the star, is used in the previous equation, which, rearranged to solve for the distance, is given by

d = 10

The notion of photometric parallax, assuming the intrinsic brightness of an object, is common in astronomy for determining distances to stars, molecular clouds, and supernovae. It is plagued by large uncertainties, but in some cases, it is the best clue we have to determine an object's distance.

AMNH Star Distances

In the AMNH star catalog, a star's distance is derived from a weighted mean value of the trigonometric and photometric parallax. The uncertainties of each are factored in, and then a final distance is computed. If the trigonometric parallax measurement is of high quality, then it will be the primary measurement used in determining the distance. If the measurement has a high uncertainty, then the photometric parallax will be brought in for comparison. For about 45% of the stars, the final distance is a combination of the trigonometric and the photometric parallax.

Digital Universe Atlas

Digital Universe Guide

Appendix: Parallax and Distance

Appendix: Parallax and Distance

Photometric Parallax

AMNH Star Distances