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.
[Illustration: Fig. 10.]
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