Orders of Magnitude

If we reflect on what we need today to characterize an object or an event, we quickly find that we need to measure three quantities: a length, such as the number of miles separating two localities; a time interval, such as the time it takes to perform a certain piece of music: and a mass, such as the mass that an elevator can carry.

To give you an idea of how wide a range these three quantities can be measured over, let us look at this picture showing typical lengths, time intervals, and mass occurring in Nature. The units standardly used in the sciences are: meter for length, Kg for mass, second for time. However, in astronomy some other units are also used, as we will see later.

Although in many cases it is necessary to know, say, a certain length very precisely - and in this case we will write this length with as many digits as we can measure - in other cases we need to specify it only roughly. For example, the diameter of a cylinder in a car engine has to be specified as errors in machining can take us - for example, 2.125 inches, where it is understood that the error- unless specified differently- is in the last digit, i.e., the digit "5". Thus we can expect cylinders made with diameters of 2.126 as well 2.124 inches. But in astronomy, distances cannot be measured so precisely. Thus, we are often satisfied when we know distances within a factor of 10 (or, as is often said, one order of magnitude).

Let us take another example. Suppose we want to measure the diameter of the Earth. You might think we had to wait for the launching of satellites to get a picture of the Earth from above in order to estimate its diameter. It turns out that over 2,200 years before a learned man, Eratosthenes, found out a clever way to measure it. Let us follow his argument. First he convinced himself that the Earth was not flat, but had a curvature. This can be deduced by noticing that distant mountains or tall structures have their lower parts cut-off from view. Second, he noticed that a pole in the city of Syene in the upper Nile region (in Egypt) cast no shadow at noon (during the day when the Sun is highest in the sky - June 21), while a pole in Alexandria, (in the Nile Delta) cast a shadow at the same time of the day. By measuring the length of the shadow (length s in the figure below) he figured out, using simple trigonometric relations, the radius R of the Earth (see the figure below for details). He found a value of 40,000 stadia. The stadium (plural is stadia) was a unit of length; unfortunately, it corresponded to different actual lengths at different times in history. One likely estimate gives that 1 stadium = 0.17 Km (1 Km is 1,000 meters and approximately 0.6 miles), then his value for the radius R is 0.17 x 40,000 = 6,800 Km, remarkably close to the actual value of 6,378 Km. If we used another estimate for the stadium, his result would be 14 % bigger (1.14 x 6,378 =7,270 Km); this would still be a remarkable achievement. Previous measurements, for example by Aristotle 200 years earlier, were much more off the mark.