Astronomy for Amateurs

How Celestial Distances are Determined, and how the Sun is Weighed I will not do my readers the injustice to suppose that they will be alarmed at the title of this Lesson, and that they do not employ some "method" in their own lives. I even assume that if they have been good enough to take me on faith when I have spoken of the distances of the Sun and Moon, and Stars, or of the weight of bodies at the surface of Mars, they retain some curiosity as to how the astronomers solve these problems. Hence it will be as interesting as it is useful to complete the preceding statements by a brief summary of the methods employed for acquiring these bold conclusions. The Sun seems to touch the Earth when it disappears in the purple mists of twilight: an immense abyss separates us from it. The stars go hand in hand down the constellated sky; and yet one can not think of their inconceivable distance without a shiver. Our neighbor, Moon, floats in space, a stone's throw[Pg 288] from us: but without calculation we should never know the distance, which remains an impassable desert to us. The best educated persons sometimes find it difficult to admit that these distances of Sun and Moon are better determined and more precise than those of certain points on our minute planet. Hence, it is of particular moment for us to give an exact account of the means employed in determining them. The calculation of these distances is made by "triangulation." This process is the same that surveyors use in the measurement of terrestrial distances. There is nothing very alarming about it. If the word repels us a little at first, it is from its appearance only. When the distance of an object is unknown, the only means of expressing its apparent size is by measurement of the angle which it subtends before our eyes. We all know that an object appears smaller, in proposition with its distance from us. This diminution is not a matter of chance. It is geometric, and proportional to the distance. Every object removed to a distance of 57 times its diameter measures an angle of 1 degree, whatever its real dimensions. Thus a sphere 1 meter in diameter measures exactly 1 degree, if we see it at a distance of 57 meters. A statue measuring 1.80 meters (about 5 ft. 8 in.) will be equal to an angle of 1 degree, if distant 57 times its height, that is to say, at[Pg 289] 102.60 meters. A sheet of paper, size 1 decimeter, seen at 5.70 meters, represents the same magnitude. In length, a degree is the 57th part of the radius of a circle, i.e., from the circumference to the center. The measurement of an angle is expressed in parts of the circumference. Now, what is an angle of a degree? It is the 360th part of any circumference. On a table 3.60 meters round, an angle of one degree is a centimeter, seen from the center of the table. Trace on a sheet of paper a circle 0.360 meters round—an angle of 1 degree is a millimeter. Fig. 80.—Measurement of Angles. If the circumference of a circus measuring 180 meters be divided into 360 places, each measuring 0.50 meters in width, then when the circus is full a person placed at the center will see each spectator occupying an angle of 1 degree. The angle does not alter with the distance, and whether it be measured at 1 meter, 10 meters, 100 kilometers, or in the infinite spaces of Heaven, it is always the same angle. Whether a degree be represented by a meter or a kilometer, it always remains a degree. As angles measuring less than a degree often have to be calculated, this angle has been subdivided into 60 parts, to which the name of[Pg 290] minutes has been given, and each minute into 60 parts or seconds. Written short, the degree is indicated by a little zero (°) placed above the figure; the minute by an apostrophe (′), and the second by two (″). These minutes and seconds of arc have no relation with the same terms as employed for the division of the duration of time. These latter ought never to be written with the signs of abbreviation just indicated, though journalists nowadays set a somewhat pedantic example, by writing, e.g., for an automobile race, 4h. 18′ 30″, instead of 4h. 18m. 30s. This makes clear the distinction between the relative measure of an angle and the absolute measures, such, for instance, as the meter. Thus, a degree may be measured on this page, while a second (the 3,600th part of a degree) measured in the sky may correspond to millions of kilometers. Now the measure of the Moon's diameter gives us an angle of a little more than half a degree. If it were exactly half a degree, we should know by that that it was 114 times the breadth of its disk away from us. But it is a little less, since we have more than half a degree (31′), and the geometric ratio tells us that the distance of our satellite is 110 times its diameter. Hence we have very simply obtained a first idea of the distance of the Moon by the measure of its diameter.[Pg 291] Nothing could be simpler than this method. The first step is made. Let us continue. This approximation tells us nothing as yet of the real distance of the orb of night. In order to know this distance in miles, we need to know the width in miles of the lunar disk. Fig. 81.—Division of the Circumference into 360 degrees. This problem has been solved, as follows: Two observers go as far as possible from each other, and observe the Moon simultaneously, from two stations situated on the same meridian, but having a wide [Pg 292]difference of latitude. The distance that separates the two points of observation forms the base of a triangle, of which the two long sides come together on the Moon. Fig. 82.—Measurement of the distance of the Moon. It is by this proceeding that the distance of our satellite was finally established, in 1751 and 1752, by two French astronomers, Lalande and Lacaille; the former observing at Berlin, the latter at the Cape of Good Hope. The result of their combined observations showed that the angle formed at the center of the lunar disk by the half-diameter of the Earth is 57 minutes of arc (a little less than a degree). This is known as the parallax of the Moon. Here is a more or less alarming word; yet it is one that we can not dispense with in discussing the distance of the stars. This astronomical term will soon become familiar in the course of the present lesson, where it[Pg 293] will frequently recur, and always in connection with the measurement of celestial distances. "Do not let us fear," wrote Lalande in his Astronomie des Dames, "do not let us fear to use the term parallax, despite its scientific aspect; it is convenient, and this term explains a very simple and very familiar effect." "If one is at the play," he continues, "behind a woman whose hat is too large, and prevents one from seeing the stage [written a hundred years ago!], one leans to the left or right, one rises or stoops: all this is a parallax, a diversity of aspect, in virtue of which the hat appears to correspond with another part of the theater from that in which are the actors." "It is thus," he adds, "that there may be an eclipse of the Sun in Africa and none for us, and that we see the Sun perfectly, because we are high enough to prevent the Moon's hiding it from us." See how simple it is. This parallax of 57 minutes proves that the Earth is removed from the Moon at a distance of about 60 times its half-diameter (precisely, 60.27). From this to the distance of the Moon in kilometers is only a step, because it suffices to multiply the half-diameter of the Earth, which is 6,371 kilometers (3,950 miles) by this number. The distance of our satellite, accordingly, is 6,371 kilometers, multiplied by 60.27—that is, 384,000 kilometers (238,000 miles). The[Pg 294] parallax of the Moon not only tells us definitely the distance of our planet, but also permits us to calculate its real volume by the measure of its apparent volume. As the diameter of the Moon seen from the Earth subtends an angle of 31′, while that of the Earth seen from the Moon is 114′, the real diameter of the orb of night must be to that of the terrestrial globe in the relation of 273 to 1,000. That is a little more than a quarter, or 3,480 kilometers (2,157 miles), the diameter of our planet being 12,742 kilometers (7,900 miles). This distance, calculated thus by geometry, is positively determined with greater precision than that employed in the ordinary measurements of terrestrial distances, such as the length of a road, or of a railway. This statement may seem to be a romance to many, but it is undeniable that the distance separating the Earth from the Moon is measured with greater care than, for instance, the length of the road from Paris to Marseilles, or the weight of a pound of sugar at the grocer's. (And we may add without comment, that the astronomers are incomparably more conscientious in their measurements than the most scrupulous shop-keepers.) Had we conveyed ourselves to the Moon in order to determine its distance and its diameter directly, we should have arrived at no greater precision, and we[Pg 295] should, moreover, have had to plan out a journey which in itself is the most insurmountable of all the problems. The Moon is at the frontier of our little terrestrial province: one might say that it traces the limits of our domain in space. And yet, a distance of 384,000 kilometers (238,000 miles) separates the planet from the satellite. This space is insignificant in the immeasurable distances of Heaven: for the Saturnians (if such exist!) the Earth and the Moon are confounded in one tiny star; but for the inhabitants of our globe, the distance is beyond all to which we are accustomed. Let us try, however, to span it in thought. A cannon-ball at constant speed of 500 meters (547 yards) per second would travel 8 days, 5 hours to reach the Moon. A train started at a speed of one kilometer per minute, would arrive at the end of an uninterrupted journey in 384,000 minutes, or 6,400 hours, or 266 days, 16 hours. And in less than the time it takes to write the name of the Queen of Night, a telegraphic message would convey our news to the Moon in one and a quarter seconds. Long-distance travelers who have been round the world some dozen times have journeyed a greater distance. The other stars (beginning with the Sun) are incomparably farther from us. Yet it has been found[Pg 296] possible to determine their distances, and the same method has been employed. But it will at once be seen that different measures are required in calculating the distance of the Sun, 388 times farther from us than the Moon, for from here to the orb of day is 12,000 times the breadth of our planet. Here we must not think of erecting a triangle with the diameter of the Earth for its base: the two ideal lines drawn from the extremities of this diameter would come together between the Earth and the Sun; there would be no triangle, and the measurement would be absurd. In order to measure the distance which separates the Earth from the Sun, we have recourse to the fine planet Venus, whose orbit is situated inside the terrestrial orbit. Owing to the combination of the Earth's motion with that of the Star of the Morning and Evening, the capricious Venus passes in front of the Sun at the curious intervals of 8 years, 1131⁄2 years less 8 years, 8 years, 1131⁄2 years plus 8 years. Thus there was a transit in June, 1761, then another 8 years after, in June, 1769. The next occurred 1131⁄2 years less 8 years, i.e., 1051⁄2 years after the preceding, in December, 1874; the next