## On MethodsHow 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 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 " 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 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, 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 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. 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 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 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 will frequently recur, and always in connection with the measurement of celestial distances. "Do not let us fear," wrote Lalande in his "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 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 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 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, Fig. 83.—Measurement of the distance of the Sun. Thus, it has been found that the half-diameter of the Earth viewed from the Sun measures 8.82″. Now, we know that an object presenting an angle of one degree is at a distance of 57 times its length. The same object, if it subtends an angle of a minute, or the sixtieth part of a degree, indicates by the measurement of its angle that it is 60 times more distant, Finally, an object that measures one second, or the sixtieth part of a minute, is at a distance of 206,265 times its length. Hence we find that the Earth is at a distance from the Sun of 206265⁄8.82—that is, 23,386 times its half-diameter, that is, 149,000,000 kilometers (93,000,000 miles). This measurement again is as precise and certain as that of the Moon. I hope my readers will easily grasp this simple method of triangulation, the result of which indicates to us with absolute certainty the distance of the two great celestial torches to which we owe the radiant light of day and the gentle illumination of our nights. The distance of the Sun has, moreover, been confirmed by other means, whose results agree perfectly with the preceding. The two principal are based on the velocity of light. The propagation of light is not instantaneous, and notwithstanding the extreme rapidity of its movements, a certain time is required for its transmission from one point to another. On the Earth, this velocity has been measured as 300,000 kilometers (186,000 miles) per second. To come from Jupiter to the Earth, it requires thirty to forty minutes, according to the distance of the planet. Now, in examining the eclipses of Jupiter's satellites, it has been discovered that there is a difference of 16 minutes, 34 seconds in the moment of their occurrence, according as Jupiter is on one side or on the other of the Sun, relatively to the Earth, at the minimum and maximum distance. If the light takes 16 minutes, 34 seconds to traverse the terrestrial orbit, it must take less than that time, or 8 minutes, 17 seconds, to come to us from the Sun, which is situated at the center. Knowing the velocity of light, the distance of the Sun is easily found by multiplying 300,000 by 8 minutes, 17 seconds, or 497 seconds, which gives about 149,000,000 kilometers (93,000,000 miles). Another method founded upon the velocity of light again gives a confirmatory result. A familiar example will explain it: Let us imagine ourselves exposed to a vertical rain; the degree of inclination of our umbrella will depend on the relation between our speed and that of the drops of rain. The more quickly we run, the more we need to dip our umbrella in order not to meet the drops of water. Now the same thing occurs for light. The stars, disseminated in space, shed floods of light upon the Heavens. If the Earth were motionless, the luminous rays would reach us directly. But our planet is spinning, racing, with the utmost speed, and in our astronomical observations we are forced to follow its movements, and to incline our telescopes in the direction of its advance. This phenomenon, known under the name of The length of orbit being thus discovered, one can calculate its diameter, the half of which is exactly the distance of the Sun. We may cite one last method, whose data, based upon attraction, are provided by the motions of our satellite. The Moon is a little disturbed in the regularity of her course round the Earth by the influence of the powerful Sun. As the attraction varies inversely with the square of the distance, the distance may be determined by analyzing the effect it has upon the Moon. Other means, on which we will not enlarge in this summary of the methods employed for determinations, confirm the precisions of these measurements with certainty. Our readers must forgive us for dwelling at some length upon the distance of the orb of day, since this measurement is of the highest importance; it serves as the base for the valuation of all stellar distances, and may be considered as the meter of the universe. This radiant Sun to which we owe so much is therefore enthroned in space at a distance of 149,000,000 kilometers (93,000,000 miles) from here. Its vast brazier must indeed be powerful for its influence to be exerted upon us to such a manifest extent, it being the very condition of our existence, and reaching out as far as Neptune, thirty times more remote than ourselves from the solar focus. It is on account of its great distance that the Sun appears to us no larger than the Moon, which is only 384,000 kilometers (238,000 miles) from here, and is itself illuminated by the brilliancy of this splendid orb. No terrestrial distance admits of our conceiving of this distance. Yet, if we associate the idea of space with the idea of time, as we have already done for the Moon, we may attempt to picture this abyss. The train cited just now would, if started at a speed of a kilometer a minute, arrive at the Sun after an uninterrupted course of 283 years, and taking as long to return to the Earth the total would be 566 years. Fourteen generations of stokers would be employed on this celestial excursion before the bold travelers could bring back news of the expedition to us. Sound is transmitted through the air at a velocity of 340 meters (1,115 feet) per second. If our atmosphere reached to the Sun, the noise of an explosion sufficiently formidable to be heard here would only reach us at the end of 13 years, 9 months. But the more rapid carriers, such as the telegraph, would leap across to the orb of day in 8 minutes, 17 seconds. Our imagination is confounded before this gulf of 93,000,000 miles, across which we see our dazzling Sun, whose burning rays fly rapidly through space in order to reach us. And now let us see how the distances of the planets were determined. We will leave aside the method of which we have been speaking; that now to be employed is quite different, but equally precise in its results. It is obvious that the revolution of a planet round the Sun will be longer in proportion as the distance is greater, and the orbit that has to be traveled vaster. This is simple. But the most curious thing is that there is a geometric proportion in the relations between the duration of the revolutions of the planets and their distances. This proportion was discovered by Kepler, after thirty years of research, and embodied in the following formula: "The squares of the times of revolution of the planets round the Sun (the periodic times) are proportional to the cubes of their mean distances from the Sun." This is enough to alarm the boldest reader. And yet, if we unravel this somewhat incomprehensible phrase, we are struck with its simplicity. What is a square? We all know this much; it is taught to children of ten years old. But lest it has slipped your memory: a square is simply a number multiplied by itself. Thus: 2 × 2 = 4; 4 is the square of 2. Four times 4 is 16; 16 is the square of 4. And so on, indefinitely. Now, what is a cube? It is no more difficult. It is a number multiplied twice by itself. For instance: 2 multiplied by 2 and again by 2 equals 8. So 8 is the cube of 2. 3 × 3 × 3 = 27; 27 is the cube of 3, and so on. Now let us take an example that will show the simplicity and precision of the formula enunciated above. Let us choose a planet, no matter which. Say, Jupiter, the giant of the worlds. He is the Lord of our planetary group. This colossal star is five times (precisely, 5.2) as far from us as the Sun. Multiply this number twice by itself 5.2 × 5.2 × 5.2 = 140. On the other hand, the revolution of Jupiter takes almost twelve years (11.85). This number multiplied by itself also equals 140. The square of the number 11.85 is equal to the cube of the number 5.2. This very simple law regulates all the heavenly bodies. Thus, to find the distance of a planet, it is sufficient to observe the time of its revolution, then to discover the square of the given number by multiplying it into itself. The result of the operation gives simultaneously the cube of the number that represents the distance. To express this distance in kilometers (or miles), it is sufficient to multiply it by 149,000,000 (in miles 93,000,000), the key to the system of the world. Nothing, then, could be less complicated than the definition of these methods. A few moments of attention reveal to us in their majestic simplicity the immutable laws that preside over the immense harmony of the Heavens. But we must not confine ourselves to our own solar province. We have yet to speak of the stars that reign in infinite space far beyond our radiant Sun. Strange and audacious as it may appear, the human mind is able to cross these heights, to rise on the wings of genius to these distant suns, and to plumb the depths of the abyss that separates us from these celestial kingdoms. Here, we return to our first method, that of triangulation. And the distance that separates us from the Sun must serve in calculating the distances of the stars. The Earth, spinning round the Sun at a distance of 149,000,000 kilometers (93,000,000 miles), describes a circumference, or rather an ellipse, of 936,000,000 kilometers (580,320,000 miles), which it travels over in a year. The distance of any point of the terrestrial orbit from the diametrically opposite point which it passes six months later is 298,000,000 kilometers (184,760,000 miles), The difficulty in exact measurements of the distance of a star consists in observing the little luminous point persistently for a whole year, to see if this star is stationary, or if it describes a minute ellipse reproducing in perspective the annual revolution of the Earth. If it remains fixed, it is lost in such depths of space that it is impossible to gage the distance, and our 298,000,000 kilometers have no meaning in view of such an abyss. If, on the contrary, it is displaced, it will in the year describe a minute ellipse, which is only the reflection, the perspective in miniature, of the revolution of our planet round the Sun. The annual parallax of a star is the angle under which one would see the radius, or half-diameter, of the terrestrial orbit from it. This radius of 149,000,000 kilometers (93,000,000 miles) is indeed, as previously observed, the unit, the meter of celestial measures. The angle is of course smaller in proportion as the star is more distant, and the apparent motion of the star diminishes in the same proportion. But the stars are all so distant that their annual displacement of perspective is almost imperceptible, and very exact instruments are required for its detection. Fig. 84.—Small apparent ellipses described by the stars as a result of the annual displacement of the Earth. The researches of the astronomers have proved that there is not one star for which the parallax is equal to that of another. The minuteness of this angle, and the extraordinary difficulties experienced in measuring the distance of the stars, will be appreciated from the fact that the value of a second is so small that the displacement of any star corresponding with it could be covered by a spider's thread. A second of arc corresponds to the size of an object at a distance of 206,265 times its diameter; to a millimeter seen at 206 meters' distance; to a hair, 1⁄10 of a millimeter in thickness, at 20 meters' distance (more invisible to the naked eye). And yet this value is in excess of those actually obtained. In fact:—the apparent displacement of the nearest star is calculated at 75⁄100 of a second (0.75″), All the rest are much more remote, and the procession is extended to infinity. We can not conceive directly of such distances, and in order to imagine them we must again measure space by time. In order to cover the distance that separates us from our neighbor, α of Centaur, If some appalling explosion occurred in this star, and the sound in its flight of 340 meters (1,115 feet) per second were able to cross the void that separates us from it, the noise of this explosion would only reach us in 3,000,000 years. A train started at a speed of 106 kilometers (65 miles) per hour would have to run for 46,000,000 years, in order to reach this star, our neighbor in the celestial kingdom. The distance of some thirty of the stars has been determined, but the results are dubious. The dazzling Sirius reigns 92,000,000,000,000 kilometers (57,000,000,000,000 miles), the pale Vega at 204,000,000,000,000. Each of these magnificent stars must be a huge sun to burn at such a distance with such luminosity. Some are millions of times larger than the Earth. Most of them are more voluminous than our Sun. On all sides they scintillate at inaccessible distances, and their light strays a long while in space before it encounters the Earth. The luminous ray that we receive to-day from some pale star hardly perceptible to our eyes—so enormous is its distance—may perhaps bring us the last emanation of a sun that expired thousands of years ago. If these methods have been clear to my readers, they may also be interested perhaps in knowing the means employed in weighing the worlds. The process is as simple and as clear as those of which we have been speaking.
Here, figures replace weights. Ladies proverbially dislike figures: yet it would be easier for some society dame to weigh the Sun at the point of her pen, by writing down a few columns of figures with a little care, than to weigh a 12 kilogram case of fruit, or a dress-basket of 35 kilos, by direct methods. Weighing the Sun is an amusement like any other, and a change of occupation. If the Moon were not attracted by the Earth, she would glide through the Heavens along an indefinite straight line, escaping at the tangent. But in virtue of the attraction that governs the movements of all the Heavenly bodies, our satellite at a distance of 60 times the terrestrial half-diameter revolves round us in 27 days, 7 hours, 43 minutes, 111⁄2 seconds, continually leaving the straight line to approach the Earth, and describing an almost circular orbit in space. If at any moment we trace an arc of the lunar orbit, and if a tangent is taken to this arc, the deviation from the straight line caused by the attraction of our planet is found to be 11⁄3 millimeter per second. This is the quantity by which the Moon drops toward us in each second, during which she accomplishes 1,017 meters of her orbit. On the other hand, no body can fall unless it be attracted, drawn by another body of a more powerful mass. Beings, animals, objects, adhere to the soil, and weigh upon the Earth, because they are constantly attracted to it by an irresistible force. Weight and universal attraction are one and the same force. On the other hand, it can be determined that if an object is left to itself upon the surface of the Earth, it drops 4.90 meters during the first second of its fall. We also know that attraction diminishes with the square of the distance, and that if we could raise a stone to the height of the Moon, and then abandon it to the attraction of our planet, it would in the first second fall 4.90 meters divided by the square of 60, or 3,600—that is, of 11⁄3 millimeters, exactly the quantity by which the Moon deviates from the straight line she would pursue if the Earth were not influencing her. The reasoning just stated for the Moon is equally applicable to the Sun. The distance of the Sun is 23,386 times the radius of the Earth. In order to know how much the intensity of terrestrial weight would be diminished at such a distance, we should look, in the first place, for the square of the number representing the distance—that is, 23,386 multiplied by itself, = 546,905,000. If we divide 4.90 meters, which represents the attractive force of our planet, by this number, we get 9⁄1,000,000 of a millimeter, and we see that at the distance of the Sun, the Earth's attraction would really be almost Now let us do for our planet what we did for its satellite. Let us trace the annual orbit of the terrestrial globe round the central orb, and we shall find that the Earth falls in each second 2.9 millimeters toward the Sun. This proportion gives the attractive force of the Sun in relation to that of the Earth, and proves that the Sun is 324,000 times more powerful than our world, for 2.9 millimeters divided by 0.000,009 equals 324,000, if worked out into the ultimate fractions neglected here for the sake of simplicity. A great number of stars have been weighed by the same method. Their mass is estimated by the movement of a satellite round them, and it is by this method that we are able to affirm that Jupiter is 310 times heavier than the Earth, Saturn 92 times, Neptune 16 times, Uranus 14 times, while Mars is much less heavy, its weight being only two-thirds that of our own. The planets which have no satellites have been weighed by the perturbations which they cause in other stars, or in the imprudent comets that sometimes tarry in their vicinity. Mercury weighs very much less than the Earth (only 6⁄100) and Venus about 8⁄10. So the beautiful star of the evening and morning is not so light as her name might imply, and there is no great difference between her weight and our own. As the Moon has no secondary body submitted to her influence, her weight has been calculated by reckoning the amount of water she attracts at each tide in the ocean, or by observing the effects of her attraction on the terrestrial globe. When the Moon is before us, in the last quarter, she makes us travel faster, whereas in the first quarter, when she is behind, she delays us. All the calculations agree in showing us that the orb of night is 81 times less heavy than our planet. There is nearly as much difference in weight between the Earth and the Moon as between an orange and a grape. Not content with weighing the planets of our system, astronomers have investigated the weight of the stars. How have they been enabled to ascertain the quantity of matter which constitutes these distant Suns—incandescent globes of fire scattered in the depths of space? They have resorted to the same method, and it is by the study of the attractive influence of a sun upon some other contiguous neighboring star, that the weight of a few of these has been calculated. Of course this method can only be applied to those double stars of which the distance is known. It has been discovered that some of the tiny stars that we can hardly see twinkling in the depths of the azure sky are enormous suns, larger and heavier than our own, and millions of times more voluminous than the Earth. Our planet is only a grain of dust floating in the immensity of Heaven. Yet this atom of infinity is the cradle of an immense creation incessantly renewed, and perpetually transformed by the accumulated centuries. And what diversity exists in this army of worlds and suns, whose regular harmonious march obeys a mute order.... But we have as yet said nothing about weight on the surface of the worlds, and I see signs of impatience in my readers, for after so much simple if unpoetical demonstration, they will certainly ask me for the explanation that will prove to them that a kilogram transported to Jupiter or Mars would weigh more or less than here. Give me your attention five minutes longer, and I will restore your faith in the astronomers. It must not be supposed that objects at the surface of a world like Jupiter, 310 times heavier than our own, weigh 310 times more. That would be a serious error. In that case we should have to assume that a kilogram transported to the surface of the Sun would there weigh 324,000 times more, or 324,000 kilograms. That would be correct if these orbs were of the same dimensions as the Earth. But to speak, for instance, only of the divine Sun, we know that he is 108 times larger than our little planet. Now, weight at the surface of a celestial body depends not only on its mass, but also on its diameter. In order to know the weight of any body upon the surface of the Sun, we must argue as follows: Since a body placed upon the surface of the Sun is 108 times farther from its center than it is upon a globe of the dimensions of the Earth, and since, on the other hand, attraction diminishes with the square of the distance, the intensity of the weight would there be 108 multiplied by 108, or 11,700 times weaker. Now divide the number representing the mass, This reasoning as just stated for the Sun may be applied to the other stars. We know that on the surface of Jupiter the intensity of weight is twice and a third times as great as here, while on Mars it only equals 37⁄100. On the surface of Mercury, weight is nearly twice as small again as here. On Neptune it is approximately equal to our own. With deference to the Selenites, everything is at its lightest on the Moon: a man weighing 70 kilograms on the Earth would not weigh more than 12 kilos there. So all tastes can be provided for: the only thing to be regretted is that one can not choose one's planet with the same facility as one's residence upon the Earth. Life Universal And Eternal Our Star The Sun Feedback |