Such observations demand enormous precision. 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. For many years astronomers struggled in vain to observe the difference. In addition, the entire solar system also moves through space, but that motion is not periodic and therefore its effects can be separated.Īnd how much do the stars shift when viewed from two points 300,000,000 km apart? Actually, very, very little. 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. The biggest baseline available for measuring such distances is the diameter of the Earth's orbit, 300,000,000 kilometers. A "light year" is about 1600 times further, an enormous distance. 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. 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. When estimating the distance to a very distant object, our "baseline" between the two points of observation better be large, too. The triangle A'B'C 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. That angle is the parallax of your thumb, viewed from your eyes. 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 angle between the lines from the eyes (A',B') to the outstretched thumb is about 6°, close enough to the value 5.73° for which the ratio 1:10 was found in an earlier part of this section. The distance to the landmark is 10 times the distance AB. Estimate the true distance AB, by comparing it to the estimated heights of trees, widths of buildings, distances between power-line poles, lengths of cars etc.It will now appear that your thumbnail has moved: it is no longer in front of landmark A, but in front of some other point at the same distance, marked as B in the drawing. Then open the eye you had closed (A') and close the one (B') with which you looked before, without moving your thumb.Close one eye (A') and move your thumb so that, looking with your open eye (B'), you see your thumbnail covering the landmark A. Stretch your arm forward and extend your thumb, so that your thumbnail faces your eyes.To estimate the distance to the landmark A, you do the following: The drawing shows a schematic view of the situation from above (not to scale). Suppose you want to estimate the distance to some far-away landmark-e.g. This method can be used in a way useful to hikers and scouts. For instance, if we know that α = 5.73°, 2 π α = 36° and we get (approximately) Therefore, if we know b, we can deduce r. The length of a circular arc is proportional to the angle it covers, and since Let us assume the two are the same (that is the approximation made here). Since the angle α is so small, the length of the straight-line "baseline" b (drawing on the right distance AB renamed) is not much different from the arc of the circle passing A and B. (The Greek mathematician Archimedes derived π to about 4-figure accuracy, though he expressed it differently, since decimal fractions only appeared in Europe some 1000 years later.)ĭraw a circle around the point C, with radius r, passing through A and B (drawing above). They knew that the length of a circle of radius r was 2πr, where π (a modern notation, not one of the Greeks, even though π is part of their alphabet) stands for a number a little larger than 3, approximately The method presented here was already used by the ancient Greeks more than 2000 years ago. We do not ask for great accuracy, but are satisfied with an approximate value of the distance-say, within 1%.That means that the angle α between AC and BC is small that angle is known as the parallax of C, as viewed from AB. The length c of the baseline AB is much less than r.The baseline is perpendicular to the line from its middle to the object, so that the triangle ABC is symmetric.This problem becomes somewhat simpler if: "Pre-Trigonometry" Section M-7 describes the basic problem of trigonometry (drawing on the left): finding the distance to some far-away point C, given the directions at which C appears from the two ends of a measured baseline AB.
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