Astronomy for Everybody/Part 4/Chapter 10

Distances in the heavens may be determined by a method similar to that employed by an engineer in determining the distance of an inaccessible object—say a mountain peak. Two points, A and B, are taken as a base line from which to measure the distance of a third point, C. Setting up his instrument at A, the engineer measures the angle between B and C. Setting it up at B he measures the angle between A and C. Since the sum of the three angles of a triangle is always one hundred and eighty degrees, the angle at C is found by

subtracting the sum of the angles at A and B from that quantity. It will readily be seen that the angle at C is that subtended by the base line as it would appear if viewed by an observer at C. Such an angle is, in a general way, called a parallax. It is the difference of direction of the point C as seen from the points A and B.

It will readily be seen that, with a given base line, ​the greater the distance of the object the less will be its parallax. At a sufficiently great distance the latter will be so small that the observer cannot get any evidence of it. To all appearance the lines B C and A C will then have the same direction. The distance at which the parallax cannot be made out depends, of course, on the accuracy of the measurement, and the length of the base line.

The moon being the nearest of all the heavenly bodies has the largest parallax. Its distance can therefore be determined with the greatest precision by measurement. Even Ptolemy, who lived only one or two centuries after Christ, was able to make an approximate measure of the distance of the moon. But the parallax of a planet is so small that it can be determined only with the most refined instruments.

The ends of the base line used in the determination may be any two points on the earth's surface—say the observatories of Greenwich and the Cape of Good Hope. In the case of the transits of Venus, which we have already described, there were a number of different stations at various points on the earth's surface, from which the direction of Venus at the beginning and end of its transit could be inferred. This method of determining distances is called triangulation.

The idea of a triangulation, as thus set forth, gives an understanding only of the general principle involved in the problem. One can readily see that it would be out of the question for two observers in distant parts of the earth to get the exact direction of a planet at the same ​moment of time. The actual determination of the parallax requires a combination of observations too complex to be set forth in the present book, but the fundamental principle is that just explained.

In order to get the dimensions of the whole solar system, it is only necessary to know the distance of any one planet from us at any given moment. The orbits and motions of all the planets are mapped down with the greatest possible exactness, but with the map before us we are in the position that one would be who had a very exact map of a country, only there was no scale of miles upon it. So he would be unable to measure the distance from one point to another on his map until he knew the scale. It is the scale of our map of the solar system which the astronomer stands in need of and which he has not, even with the most refined instruments, yet been able to determine as accurately as he could wish.

The fundamental unit aimed at is that already described—the mean distance of the earth from the sun. Measures of parallax are by no means the only method of determining this distance. Within the last fifty years other methods have been developed, some of which are fully as accurate as the best measures of parallax, perhaps even more so.

One of the most simple and striking of these methods makes use of the velocity of light. By observations of Jupiter's satellites, made when the earth was at different points of its orbit, it has been found that light passes ​over a distance equal to that of the earth from the sun in about eight minutes twenty seconds, or five hundred seconds. This determination has been more accurately made in another way by the aberration of the stars. This is a slight change in their position due to the combined motion of the earth and the ray of light by which we see the star. By accurate observations on the aberration, it is found that light travels from the earth to the sun in almost exactly 499.6 seconds. It follows that if we can find how far light will travel in one second, we can determine the distance of the sun by multiplying the result by 499.6. The measurement of the velocity of light is one of the most difficult problems in physics, as it requires the measurement of intervals of time only a few millionths of a second in duration. Those who are interested in the subject will see the method of doing this explained in special treatises; at present it is sufficient to say that light is found to travel 299,860 kilometres, or 186,300 miles in a second. Multiply this by 499.6 and we have 93,075,480 miles for the distance of the sun from the earth.

A third method rests on the measures of the sun's gravitation upon the moon. One effect of this is that, as the moon performs its monthly revolution round the earth, it is at its first quarter a little more than two minutes behind its average position, to which it catches up at full moon, and passes; so that at last quarter it is two minutes ahead of the mean position. Toward new ​moon it falls behind again to the average place. Thus a slight swing goes on in unison with the moon's motion around the earth. The amount of this swing is inversely proportional to the distance of the sun. Hence, by measuring this amount, its distance may be determined. As in other astronomical measurements, the difficulty of the determination is very great. A swing like this is very hard to measure without error; moreover, the problem of determining just how much swing the sun would produce at a given distance is one of the difficult problems of celestial mechanics, which has not yet been solved so satisfactorily as to leave no doubt whatever on the result.

The fourth method also rests on gravitation. If we only knew the exact relation between the mass of the earth and that of the sun; that is to say, if we could determine precisely how many times heavier the sun is than the earth, we could compute at what distance the earth must be placed from the sun in order to revolve around it in one year. The only difficulty, therefore, is to weigh the earth against the sun. This is most exactly done by finding the change in the position of the orbit of Venus produced by the earth's attraction. By comparing the positions of the orbit of Venus by its transits in 1761, 1769, 1874, and 1882, it is found that the orbit has a progressive motion, indicating that the mass of the sun is 332,600 times that of the earth and moon combined. Thus we are enabled to compute the distance of the sun by still another method.

We have described four methods of making this fundamental determination in astronomy, and in order that the reader may see just what degree of certainty and precision astronomical theory and measurements have reached, we give the separate results of these methods. The first column shows the parallax of the sun, which is the quantity actually used by astronomers. It is the same thing as the angle under which the equatorial radius of the earth would be seen by an observer at the distance of the sun from us. This is followed by the accompanying distance in miles.

The difference between these results is no greater than the liability of error wherever mathematical demonstrations and instrumental measurements of such extreme minuteness and complexity as these are required. From the close agreement between results reached by methods so widely different in their principles, we have a striking proof of the correctness of the astronomical views of the universe. Yet discrepancies exceeding a hundred thousand miles will not be tolerated by astronomers longer than is absolutely necessary.