
Index
5c. Coordinates 6. The Calendar 6a. Jewish Calendar 7.Precession 8. The Round Earth 8a. The Horizon 8b. Parallax 8c. Moon dist. (1) 8d. Moon dist. (2) 9a. Earth orbits Sun? 9b. The Planets 9c. Copernicus to Galileo 10. Kepler's Laws 10a. Scale of Solar Sys. 11a. Ellipses and First Law 
"PreTrigonometry"Section M7 describes the basic problem of trigonometry (drawing on the left): finding the distance to some faraway point C, given the directions at which C appears from the two ends of a measured baseline AB. This problem becomes somewhat simpler if:

Draw a circle around the point C, with radius r, passing through A and B (drawing above). Since the angle α is so small, the length of the straightline "baseline" b (drawing on the right; distance AB renamed) is not much different from the arc of the circle passing A and B. Let us assume the two are the same (that is the approximation made here). The length of a circular arc is proportional to the angle it covers, and since 
2π r covers an angle 360° we get and dividing by 2π
Therefore, if we know b, we can deduce r. For instance, if we know that α = 5.73°, 2 π α = 36° and we get (approximately) Estimating distance outdoorsThis method can be used in a way useful to hikers and scouts. Suppose you want to estimate the distance to some faraway landmarke.g. a building, tree or water tower.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:


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 hours to reach the the distant planet Pluto. 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 starsfor 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 timeand 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 groundbased telescopes. In measuring star distances, astronomers frequently use the parsec, the distance to a star whose yearly parallax is 1"one second of arc. One parsec equals 3.26 light years, but as already noted, no star is that close to us. Alpha Centauri, the sunlike star nearest to our solar system, has a distance of 4.3 years and a parallax of 0.75". Alpha Centauri is not a name, but a designation. Astronomers designate stars in each constellation by letters of the Greek alphabetalpha, beta, gamma, delta and so forth, and "Alpha Centauri" means the brightest star in the constellation of Centaurus, located high in the southern skies. You need to be south of the equator to see it well.
Questions from Users: *** Are all stars we see suns? Next Stop: #8c. How Distant is the Moon?1 Timeline Glossary Back to the Master List
Author and Curator: Dr. David P. Stern 