(8d) Parallax

The greater the distance, the smaller the parallax. Move your head from side to side and the view of the computer in front of you will change, but not the view of distant buildings or mountains.

The previous section told how Hipparchus estimated the Moon's distance, by using an eclipse of the Sun to establish that the Moon had a parallax of about 0.1 degrees when viewed from sites 1000 km apart. Here two other ways of estimating distance will be described, based on the parallax.

Estimating distance outdoors

The drawing shows a schematic view of the situation from above (not to scale). To estimate the distance to the landmark A, you do the following:

1.   Stretch your arm forward and extend your thumb, so that your thumbnail faces your eyes. Close one eye (A') and move your thumb so that your thumbnail covers the landmark A.

2.   Then open the other eye (A') and close the first (B'), without moving your thumb. You may find that your thumbnail is now in front of some other feature in the landscape, at about the same distance as A, marked B.

3.   Estimate the distance AB, by comparing it to the estimated heights of trees, widths of buildings, distances between power-line poles, lengths of cars etc. The distance to the landmark is 10 times the distance AB.   Why does this work? Because even though people vary in size, the proportions of the average human body are fairly constant, and for most people, the distance of the outstretched thumb is about 10 times the distance between the eyes.

     The angle a at the tip of the triangle A'B'C formed by the eyes and the thumb is the parallax of your thumb. That triangle has the same proportions as the much larger triangle ABC, and therefore, if the distance B'C to the thumb is 10 times the distance A'B' between the eyes, the distance AC to the far landmark is also 10 times the distance AB.

   How far to a Star?

     When estimating the distance to a very distant object, our "baseline" between the two points of observation better be large, too. The most distant objects our eyes can see are the stars, and they are very far indeed: light which moves at 300,000 kilometers (186,000 miles) per second, would take years, often many years, to reach them. The Sun's light needs 500 seconds to reach Earth, a bit over 8 minutes, and about 5.5 hours to reach the average distance of Pluto, the most distant planet. A "light year" is about 1600 times further, an enormous distance.

     The biggest baseline available for measuring such distances is the diameter of the Earth's orbit, 300,000,000 kilometers. The Earth's motion around the Sun makes it move back and forth in space, so that on dates separated by half a year, its positions are 300,000,000 kilometers apart. In addition, the entire solar system also moves through space, but that motion is not periodic and therefore its effects can be separated.

     And how much do the stars shift when viewed from two points 300,000,000 km apart? Actually, very, very little. For many years astronomers struggled in vain to observe the difference. Only in 1838 were definite parallaxes measured for some of the nearest stars--for Alpha Centauri by Henderson from South Africa, for Vega by Friedrich von Struve and for 61 Cygni by Friedrich Bessel.

     Such observations demand enormous precision. Where a circle is divided into 360 degrees (360°), each degree is divided into 60 minutes (60')--also called "minutes of arc" to distinguish them from minutes of time--and each minute contains 60 seconds of arc (60"). All observed parallaxes are less than 1", at the limit of the resolving power of even large ground-based telescopes.

     In measuring star distances, astronomers frequently use the parsec, the distance to a star whose yearly parallax is 1". One parsec equals 3.26 light years, but as already noted, no star is that close to us. Alpha Centauri, the sun-like star nearest to our solar system, has a distance of 4.3 years and a parallax of 0.75". It is located high in the southern skies and you need to be south of the equator to see it well.