Chapter IV

THE FORM, MAGNITUDE, WEIGHT, AND DENSITY OF THE LUNAR GLOBE.

We have not hitherto had occasion to refer to what we may term the physical elements of the moon: by which we mean the various data concerning form, size, weight, density, &c. of that body, derived from observation and calculation. To this purpose, therefore, we will now devote a few pages, confining ourselves to such matters as specially bear upon the requirements of our subject, omitting such as are irrelevant to our purpose, and touching but lightly upon such as are commonly known, or are explained in ordinary elementary treatises on astronomy.

First, then, as regards the form of the moon. The form of the lunar disc, when fully illuminated, we perceive to be a perfect circle; that is to say, the measured diameters in all directions are equal; and we are therefore led to infer that the real form of the moon is that of a perfect sphere. We know that the earth and the rest of the planets of our system are spheroidal, or more or less flattened at the poles, and we also know that this flattening is a consequence of axial rotation; the extent of the flattening, or the oblateness of the spheroid, depending upon the speed of that rotation. But in the case of the moon the axial rotation is so slow that the flattening produced thereby, although it must exist, is so slight as to be imperceptible to our observation. We might therefore conclude that the moon is a perfectly spherical body, did not theory step in to show us that there is another cause by which its form is disturbed. Assuming the moon to have been once in a fluid state, it is demonstrable that the attraction of the earth would accumulate a mass of matter, like a tidal elevation, in the direction of a line joining the centres of the two bodies: and as a consequence, the real shape of the moon must be an ellipsoid, or somewhat egg-shaped body, the major axis of which is directed towards the earth. That some such phenomenon has obtained is evident from the coincidence of the times of orbital revolution and axial rotation of the lunar sphere. “It would be against all probability,” says Laplace, “to suppose that these two motions had been at their origin perfectly equal;” but it is sufficient that their primitive difference was but small, in which case the constant attraction by the earth of the protuberant part of the moon would establish the equality which at present exists.

It is, however, sufficient for all purposes with which we are concerned to regard the moon as a sphere, and the next point to be considered is its size. To determine this, two data are necessary—its apparent or angular diameter, and its distance from the earth. The first of these is obtained by measuring the angle comprised between two lines directed from the eye to two opposite “limbs” or edges of the moon. If, for instance, we were to take a pair of compasses and, placing the joint at the eye, open out the legs till the two points appear to touch two opposite edges of the moon, the two legs would be inclined at an angle which would represent the diameter of the moon, and this angle we could measure by applying a divided arc or protractor to the compasses. In practice this measurement is made by means of telescopes attached to accurately divided circles; the difference between the readings of the circle when the telescope is directed to opposite limbs of the moon giving its angular diameter at the time of the observation. But from the fact that the orbit of the moon is an ellipse, it is evident that she is at some times much nearer to us than at others, and, as a consequence, her apparent magnitude is variable: there is also a slight variation depending upon the altitude of the moon at the time of the measure; the mean diameter, however, or the diameter at mean distance from the centre of the earth has, from long course of observation, been found to be 31′ 9″.

To convert this apparent angular diameter into real linear measurement, it is necessary to know either the distance of the moon from the earth, or in astronomical language as leading to a knowledge of that distance, what is the amount of the moon’s parallax. Parallax, generally, is an apparent change of position of an object arising from change of the point of view. The parallax of a heavenly body is the angle which the earth would subtend if it were seen from that body. Supposing an observer on the moon could measure the earth’s angular diameter, just as we measure that of the moon, his measurement would represent what is called the parallax of the moon. But we cannot go to the moon to make such a measurement; nevertheless there is a simple method, explained in most treatises on astronomy, which consists in observing the moon from stations on the earth widely separated, and by which we can obtain a precisely similar result. Without detailing the process, it is sufficient for us to know that the angle which would be subtended by the earth if seen from the moon, or the moon’s parallax, is according to the latest determination, equal to 1° 54′ 5″. This value, however, varies considerably with the variations of distance due to the elliptic orbit of the moon: the number we have given represents the mean parallax, or the parallax at mean distance.

But we have to turn these angular measurements into miles. To effect this we have only to work a simple rule of three sum. It will easily be understood that, as the angular diameter of the earth seen from the moon is to the angular diameter of the moon seen from the earth, so is the diameter of the earth in miles to the diameter of the moon in miles. The diameter of the earth we know to be 7912 miles: putting this therefore in its proper place in the proportion sum, and duly working it out by the schoolboy’s rule, we get:—

1°. 54′. 5″ : 31′. 9″ :: 7912 miles. : 2160 miles.

And 2160 miles is therefore the diameter of the lunar globe.

Knowing the diameter, we can easily obtain the other elements of magnitude. According to the well-known relation of the diameter of a sphere to its area, we find the area of the moon to be 14,657,000 square miles: or half that number, 7,328,500 miles, as the area of the hemisphere at any one time presented to our view. And similarly, from the relation of the solidity of a sphere to its diameter, we find the solid contents of the moon to be 5276 millions of cubic miles of matter.

Comparing these data with corresponding dimensions of the earth, we find that the diameter of the moon is 1/3·665; the area 1/13·4245; and the volume 1/49·1865, of the respective elements of the earth. Those who prefer a graphical to a numerical comparison, may judge of the sizes of the two bodies by the accompanying illustration (Fig. 10). To gain an idea of their distance from each other it is necessary to suppose the two discs in the diagram to be 30 inches apart; the real distance of the moon from the earth being about 238,790 miles at its mean position.

Next, we come to what is technically termed the _mass_, but what in common language we may call the _weight_ of the moon. It is important to know this, because the weight of a body taken in connection with its size furnishes us with a knowledge of its density, or the specific gravity of the material of which it is composed. But it is not quite so easy to determine the mass as the dimensions of the moon: to _measure_ it, we have seen is easy enough; to _weigh_ it is a comparatively difficult matter. To solve the problem we have to appeal to Newton’s law of universal gravitation. This law teaches us that every particle of matter in the universe attracts every other particle with a force which is _directly proportional to the mass_, and inversely proportional to the distance of the attracting particles. There are several methods by which this law is applied to the measurement of the mass of the moon. One of the simplest is by the agency of the Tides. We know that the moon, attracting the waters, produces a certain amount of elevation of the aqueous covering of the earth; and we know that the sun produces also a like elevation, but to a much smaller extent, by reason of its much greater distance. Now measuring accurately the heights of the solar and lunar tides, and making allowance for the difference of distance of the sun and moon from the earth, we can compare directly the effect that is due to the sun with the effect that is due to the moon: and since the masses of the two bodies are just in proportion to the effects they produce, it is evident that we have a comparison between the mass of the sun and that of the moon; and knowing what is the sun’s mass we can, by simple proportion, find that of the moon. Another method is as follows:—The moon is retained in her orbital path by the attraction of the earth; if it were not for this attraction she would fly off from her course in a tangential line. She has thus a constant tendency to quit her orbit, which the earth’s attraction as constantly overcomes. It is evident from this that the earth pulls the moon towards itself by a definite amount in every second of time. But while the earth is pulling the moon, the moon is also pulling the earth: they are pulling each other together; and moreover each is exerting a pull which is _proportional to its mass_. Knowing, then, the mass of the earth, which we do with considerable accuracy, we can find what share of the whole pulling force is due to it, the residue being the moon’s share: the proportion which this residue bears to the earth’s share gives us the proportion of the moon’s mass to that of the earth, and hence the mass of the moon.

There are yet two other methods: one depending upon the phenomena of nutation, or the attraction of the sun and moon upon the protruberant matter of the terrestrial spheroid; and the other upon a displacement of the centre of gravity of the earth and moon, which shows itself in observations of the sun. By each and all of these methods has the lunar mass been at various times determined, and it has been found, as the latest and best accepted value, that the mass of the moon is _one-eightieth_ that of the earth.

From the known diameter of the earth we ascertain that its volume is 259,360 millions of cubic miles: and from the various experiments that have been made to determine the mean density of the earth, it has been found that that mean density is about 5½ times that of water; that is to say, the earth weighs 5½ times heavier than would a sphere of water of equal size. Now a cubic foot of water weighs 62·3211 pounds, and from this we can find by simple multiplication what is the weight of a cubic mile of water, and, similarly, what would be the weight of 259,360 cubic miles of water, and the last result multiplied by 5½ will give the weight of the earth in tons: The calculation, although extremely simple, involves a confusing heap of figures; but the result, which is all that concerns us, is, that the weight of the earth is 5842 trillions of tons: and since, as we have above stated, the mass of the earth is 80 times that of the moon, it follows that the weight of the moon is 73 trillions of tons.

The cubical contents of a body compared with its weight gives us its density. In the moon we have 5276 millions of cubic miles of matter, the total weight of which is 73 trillions of tons. Now, 5276 millions of cubic miles of water would weigh about 21½ trillions of tons; and as this number is to 73 as 1 is to 3·4, it is clear that the density of the lunar matter is 3·4 greater than water: and inasmuch as the earth is 5½ times denser than water, we see that the moon is about 0·62 as dense as the earth, or that the material of the moon is lighter, bulk for bulk, than the mean material of the terraqueous globe in the proportion of 62 to 100, or, nearly, 6 to 10. This specific gravity of the lunar material (3·4) we may remark is about the same as that of flint glass or the diamond: and curiously enough it nearly coincides with that of some of the aërolites that have from time to time fallen to the earth; hence support has been claimed for the theory that these bodies were originally fragments of lunar matter, probably ejected at some time from the lunar volcanoes with such force as to propel them so far within the sphere of the earth’s attraction that they have ultimately been drawn to its surface.

Reverting, now, to the mass of the moon: we must bear in mind that the mass or weight of a planetary body determines the weight of all objects on its surface. What we call a pound on the earth, would not be a pound on the moon; for the following reason:—When we say that such and such an object weighs so much, we really mean that it is attracted towards the earth with a certain force depending upon its own weight. This attraction we call gravity; and the falling of a weight to the earth is an example of the action of the law of universal gravitation. The earth and the weight fall together—or are held together if the weight is in contact with the earth—with a force which depends directly upon the mass of the two, and upon the distance between them. Newton proved that the attraction of a sphere upon external objects is precisely as if the whole of its matter were contained at its centre. So that the attractive force of the earth upon a ton weight at its surface, is the attraction which 5842 trillions of tons exert upon one ton situated 3956 miles (the radius of the earth) distant. If the weight of the earth were only half the above quantity, it is clear that the attraction would be only half what it is; and hence the ton weight, being pulled by only half the force, would only be equal to half a ton; that is to say, only half as much muscular force (or any other force but gravity) would be required to lift it. It is plain, therefore, that what weighs a pound on the earth could not weigh a pound on the moon, which is only 1/80 of the weight of the earth. What, then, is the relation between a pound on the earth and the same mass of matter on the moon? It would seem, since the moon’s mass is 1/80 of the earth, that the pound transported to the moon ought to weigh the eightieth part of a pound there; and so it would if the distance from the centre of the moon to its surface were the same as the distance of the centre of the earth from its surface. But the radius of the moon is only 1/3·665 that of the earth; and the force of gravity varies _inversely as the square of the distance_ between the centres of the gravitating masses. So that the attraction by the moon of a body at its surface, as compared with that of the earth, is 1/80 multiplied by the square of 1/3·665; and this, worked out, is equal to 1/6. The force of gravity upon the moon is, therefore, 1/6 of that on the earth; and hence a pound upon the earth would be little more than 2½ ounces on the moon; and it follows as a consequence that any force, such as muscular exertion, or the energy of chemical, plutonic or explosive forces, would be six times more effective upon the moon than upon the earth. A man who could jump six feet from the earth, could with the same muscular effort jump thirty-six feet from the moon; the explosive energy that would project a body a mile above the earth would project a like body six miles above the surface of the moon.

It is the practice, in elementary and popular treatises on astronomy, to state merely the numerical results in giving data such as those embodied in the foregoing pages; and uninitiated readers, not knowing the means by which the figures are arrived at, are sometimes disposed to regard them with a certain amount of doubt or uncertainty. On this account we have thought it advisable to give, in as brief and concise a form as possible, the various steps by which these seemingly unattainable results are obtained.

The data explained in the foregoing text are here collected to facilitate reference.

Diameter of Moon 2160 miles 1/3·665 that of earth.
Area 14,657,000 square miles 1/13·424 ” ”
Area of the 7,328,500 square miles
visible
hemisphere
Solid contents 5276 millions of cubic miles 1/49·186 ” ”
Mass 73 trillions of tons 1/80 ” ”
Density 3·39 (water = 1) 0·62 ” ”
Force of 1/6 ” ”
gravity at
surface
Mean distance 238,790 miles.
from earth