CHAPTER XII.
THE MASSES OF CELESTIAL BODIES.
MASS AND WEIGHT.—As a matter of daily experience, we know that a certain effort is required to prevent a body from falling to the ground, and the larger the bulk of any particular kind of matter, the greater is the effort demanded. Again, equal bulks of different kinds of matter require unequal efforts to sustain them in the hand. From facts such as these we get the idea of _weight_, and we say that one body is heavier than another when it has the greater tendency to fall to the ground. For the purposes of everyday life, the weight of a body is used as a measure of the quantity of matter which it contains, and the standard of weight in our own country is that of a certain piece of platinum kept at the Exchequer Office, in London, which is called a _pound_. The weight of the same piece of matter varies at different parts of the earth’s surface, and also at different distances from the ground, and it is evident, therefore, that weight is not a very scientific measure of the quantity of matter which a body contains. The standard of comparison must be one which is invariable not only in all parts of the earth, but, if we wish to investigate the quantity of matter in the celestial bodies, it must be unalterable through all parts of the universe.
One’s first idea is that the bulk, or space which a body occupies, will furnish a means of measuring the quantity of matter which it contains, but here again we find that the volume of a body can be varied without either adding to or subtracting from it, its weight remaining constant. A piece of ice, for example, occupies a greater space than an equal weight of water.
It is evident then that some other property of matter must be used as a measure of quantities. Now, there is every reason to believe that the same piece of matter, in whatever part of space it may be situated, requires the same force to set it moving with the same speed in a given time. By the continued application of a force, a body will first be set in motion, and at the end of a second it will have a certain speed; in the next second the velocity will have increased by an amount equal to that acquired at the end of the first second, and so on for subsequent intervals. For example, if at the end of a second the velocity were 3 feet per second, at the end of the next second it would be 6 feet per second, and after other equal intervals it would be successively 9, 12, 15, and so on. In this way the velocity is increased uniformly, and is said to be uniformly accelerated, while the gain per second is called the _acceleration_. The greater the force applied, the greater will be the acceleration it produces, and the acceleration can be used as a measure of the force at work.
If the same force be applied to different quantities of the same substance, the acceleration produced will be in inverse proportion to the quantities. We thus arrive at the important result that two bodies, whatever their nature, contain equal quantities of matter, or have equal _masses_, when equal forces give them the same acceleration. The mass of a body can thus be ascertained by observing the acceleration due to the action of a known force.
As a matter of observation, it is found that all bodies, whatever their composition or size, fall to the ground from the same height in the same time if the observations be made at one place. This means that the forces corresponding to weights produce equal accelerations in all bodies at the same place, and it follows, therefore, that the weights of bodies at the earth’s surface, are proportional to their masses. Hence, it is that weight can be practically employed in comparing masses, or quantities of matter, for the purposes of everyday life. It must be clearly understood, however, that a _mass_ of a pound is in reality quite distinct from a _weight_ of a pound, the former specifying a certain quantity of matter, and the latter its tendency to fall towards the earth.
THE LAW OF GRAVITATION.—The idea that weight is due to the attraction of the earth for all bodies in its neighbourhood was first suggested by Newton, and an extension of this idea led him to formulate the great law which underlies the whole science of astronomy. All bodies near the earth’s surface are acted upon by forces proportional to their masses, and the same acceleration is produced in all of them if they are allowed to fall to the ground. Falling freely for a second, all bodies whatsoever, when the resistance of the air is eliminated, pass through a little over 16 feet, and acquire a velocity of just over 32 feet per second. The acceleration due to gravity is thus 32⅙ feet per second for bodies near the earth’s surface. If the experiment be made at the top of a high mountain, the distance fallen through and the acceleration acquired in a second is found to be less.
If we could ascend still higher, the acceleration produced in falling bodies would be again reduced, and, in the light of what has gone before, it is evident that the force with which bodies tend to fall to the earth is diminished as the distance from the earth’s surface is increased. It was such considerations as these which led Sir Isaac Newton to formulate the law that _the force with which a body is attracted towards the earth diminishes in inverse proportion to the square of the distance from the earth’s centre_. Terrestrial means of testing the truth of this statement are obviously very limited, and hence it was that Newton looked to the moon for its verification. If the law holds good at the distance of the moon, an object so far removed and not acted upon by other forces, should fall towards the earth, and as its distance is about sixty times that of a body at the surface from the centre of the earth, the acceleration produced should be only ¹⁄₃₆₀₀th part of that imparted to bodies near the surface. In other words, since a body near the surface falls through 16 feet in the first second, one at the moon’s distance should only fall through about ¹⁄₂₀th of an inch. If, then, the moon be subject to the earth’s attraction, this fall towards the earth must be exhibited in some form or other, although the fact that the moon does not fall down upon the earth shows that there is some counteracting tendency.
Observations have shown us that the moon moves in a curved path. It has been put in motion somehow, and since there is no reason why it should turn to one side or the other, or come to rest, unless some forces are acting upon it, it would tend to go on uniformly in a straight line for ever. That its movement is curvilinear is at once an indication of the action of a force besides that which originally set it in motion. This force is directed towards the earth, and the moon is drawn out of its rectilinear path just as far in any specified time as it would fall towards the earth if at rest.
Let E and M in Fig. 44 represent the earth and moon respectively. Then, if the moon were not hindered in any way, it would move in the direction M _b_, and would reach the point _b_, let us say, at the end of a second. It is, however, found to be at the point _a_, and it has therefore fallen towards the earth through the distance _b a_. The size of the moon’s orbit and the angle through which it moves in a second being known, it is easy to calculate the distance _a b_, which is found to be about ¹⁄₂₀th of an inch, as demanded by Newton’s law.
[Illustration:
FIG. 44.—_The Moon’s Curvilinear Path._ ]
In his first attempt to thus verify the law of gravitation, Newton failed for the want of a sufficiently accurate knowledge of the earth’s diameter, but a few years later a new arc of meridian was measured, and he had the untold satisfaction of demonstrating its truth.
The curved path of the moon is, indeed, similar to that of a projectile. A cannon ball thrown out horizontally will reach the ground after describing a curved path; but if it could be projected from a great elevation, with sufficient velocity, its forward movement would prevent its ever reaching the earth’s surface at all, and a new satellite of the earth would have been manufactured.
The same kind of reasoning can be applied to the paths of the earth and planets around the sun, and Newton demonstrated that the laws of Kepler were a necessary consequence of the law of gravitation extended beyond the system of the earth and moon. By mathematical reasoning it was proved that if one body describes an elliptic orbit around another, and the line joining them describes equal areas in equal times, the attractive force must be directed to the central body, and, moreover, must vary inversely as the square of the distance between the two bodies. In this way the movements of the planets round the sun are perfectly explained by supposing that an attractive force, similar to that which causes bodies to fall to the earth’s surface, is exerted between all masses of matter, and hence the origin of the term _Universal Gravitation_. In its complete form, the law of gravitation states that “any particle of matter attracts any other particle with a force which varies directly as the product of the masses, and inversely as the square of the distance between them.”
Confirmation of this grand law, which controls the movements of all the vast array of heavenly bodies, is furnished by many other phenomena. We see one of its effects in the tides, and another in the disturbances of the movements of planets brought about by their mutual attractions. Even in the depths of stellar space the same law holds good for those systems of stars which are sufficiently close together for their attractions to produce effects which we can study at our immense distance from them.
The cause of gravity is still one of the greatest mysteries of physical science, although many ingenious attempts have been made to furnish an explanation of its mode of action.
MASS OF THE SUN.—When we know the distance of the sun, and the time in which the earth travels completely round it, it is easy to calculate the fall of the earth towards the sun in the same way that the moon’s fall towards the earth is determined.
The distance which a body 93,000,000 miles distant falls towards the sun in a second is thus found to be 0·116 of an inch. A body at the earth’s surface is about 4,000 miles from the centre, and it falls 16¹⁄₁₂ feet in a second; if removed to a distance of 93,000,000 miles, its fall towards the earth would be reduced inversely as the squares of 4,000 and 93,000,000, and would amount to ·000,000,349 of an inch. This is only 1/332,000th part the fall due to the sun’s attraction, and hence it is concluded that the mass of the sun is 332,000 times that of the earth.
Strictly speaking, the accelerations produced by the sun and earth should be compared, but the fall during the first second is proportional to the acceleration due to gravity, and the same result is therefore obtained. It may be observed also that the fall of the earth towards the sun would not be appreciably effected if it were twice the size. All bodies fall towards the earth at the same rate, whatever their weights, and so in the case of a planet, the distance fallen towards the central sun is independent of the planet’s mass; the greater the mass the greater the attractive force.
The sun occupies about 1,300,000 times the space occupied by the earth, and as its mass is only 332,000 times that of the earth, it follows that the sun’s density is only about a quarter that of the earth.
MASSES OF PLANETS.—The process employed for the determination of the sun’s mass can be utilised for finding the masses of those planets which are accompanied by satellites. From the known distance of the planet, the size of the orbit of a satellite can be calculated in miles, and knowing the period of revolution of the satellite, its fall towards the planet can be determined. This fall is then compared with that of the planet’s fall towards the sun, and the mass of the planet in terms of the sun’s mass is thus arrived at.
A convenient way of employing this method is to make use of a modification of Kepler’s third law. If _m_ be the mass of a planet in terms of the sun’s mass, M, _a_ and T respectively denote the semi-axis major of the orbit of the planet and its time of revolution round the sun; _a′_ and T similar quantities pertaining to the satellites’ revolution round the planet: The following formula gives the relation of the masses:—
_m_/M = (_a′_/_a_)^3 (T/T′)^2
This formula can be applied in the case of Mars, Jupiter, Saturn, Uranus, and Neptune, but fails in the case of Mercury, Venus, and the asteroids, which, so far as we know, have no satellites.
The mass of Jupiter obtained in this way can be further checked by the influence of this giant planet upon other bodies in its neighbourhood. This planet has such an enormous mass that it produces very notable effects on the motions of Saturn, the asteroids, and of comets which travel in its neighbourhood, and, by measuring the amounts of these _perturbations_, the mass of the planet can be deduced.
This method of perturbations is at present the only one by which we can obtain a knowledge of the masses of those planets which have no satellites. The motion of Mercury is disturbed by its nearest neighbours, Venus and the earth; that of Venus by the earth and Mercury. The differences between the observed positions of the planets and those calculated on the supposition that the others did not affect them, give the necessary data for the computation of the masses. The process, however, is one requiring profound mathematical knowledge, and even yet the mass of Mercury is not very certainly known.
The asteroids, again, present no little difficulty. Their feeble light and small size point to small masses, and their mutual perturbations are almost insensible, except when two of them come into line with the sun. They produce no appreciable effects upon the movements of comets, so that it is almost impossible to determine their individual masses. Each asteroid, however, tends to produce a revolution of the major axis of the orbit of the nearest planet, Mars, and all tend to give it a motion in the same direction. If the total mass of all the asteroids put together were a quarter of the earth’s mass, a measurable displacement of the position of Mars would be produced. Professor Newcomb has recently shown that such a displacement actually occurs, but cannot amount to more than 5″·5 per century. From this it has been recently calculated that the total mass of the asteroids is probably about ¹⁄₁₁₅th that of the earth’s mass.
MASS OF THE MOON.—As the moon has no satellite, we must again have recourse to indirect methods if we wish to know anything as to its mass. Various processes are open to us; but although the moon is so near to us, it is more difficult to determine its mass than that of the most remote planet in our system.
It has already been explained (p. 77) that as the earth is accompanied by the moon, it is really the centre of gravity of the two bodies which obeys the laws of planetary movement. As this point lies between the centres of the two bodies, at distances which are in inverse proportion to the masses, the centre of the earth describes a small monthly orbit, which, as we have already seen, produces a small monthly inequality in the sun’s apparent movement.
By a careful investigation of this monthly oscillation of the sun, it has been found that the centre of gravity of the earth and moon must lie within the earth at a distance of about 2,900 miles from the centre. This is about ¹⁄₈₁th of the moon’s distance, whence it follows that the mass of the moon is about ¹⁄₈₁th that of the earth.
Other methods of ascertaining the moon’s mass are also available. Among these are the investigation of the parts played by the moon in the production of the tides which swell our shores, and in the displacement of the earth’s axis which causes “nutation.”
MASSES OF SATELLITES.—The earth’s satellite is of exceptional magnitude in comparison with its primary, and the method of finding its mass from the situation of the centre of gravity cannot be applied to the satellites attending other planets. In the case of the satellites of Jupiter and Saturn, the masses have been approximately determined by their mutual perturbations, these generally resulting in a revolution of the major axes of the orbits. Even this method fails for the satellites of Mars, Uranus, and Neptune, so that practically nothing is known with regard to their masses.
MASS AND DENSITY OF THE EARTH.—So far we have been concerned entirely with relative masses, referring the masses of the various orders of the heavenly bodies either to the earth or sun. Although this is usually all that is required for astronomical purposes, it is of great interest to determine the absolute mass of the earth, and from this the absolute masses of the heavenly bodies can at once be deduced.
We already know the dimensions of the earth, and therefore the number of cubic miles or feet which it occupies. We know also the weight or mass of a cubic foot of water or lead, and if the earth were of uniform specific gravity throughout its bulk, and composed of water or lead, we could at once calculate its total mass. It is, however, neither water nor lead; but if we can compare the mass of the earth with what it would be if composed of either of these substances, we can deduce either its mass or its specific gravity.
A very simple method of “weighing” the earth has been employed with much success by Professor Poynting. The experiment was carried out at the Mason Science College, Birmingham, with a large bullion balance in which the beam was 123 centimetres long. Two spheres of lead and antimony, each weighing about 21 kilograms, were suspended from the arms of the balance. Another sphere of lead and antimony, weighing 153 kilograms, was successively brought by means of a turn-table under each of the two smaller weights. The alteration in the weights of the attracted balls were measured by observing the deflection of the beam, this being immensely magnified by a simple optical arrangement in which a mirror reflecting a pencil of light was made to turn through 150 times the angle moved through by the beam itself. The weight corresponding to a given deflection of the beam was determined by observing the disturbance produced by the addition of “riders” of known weights. In order to reduce the chances of error, the large weight was balanced on the turn-table by another mass of half the weight and at twice the distance from the centre, this being necessary in order that the attracting weight should rotate horizontally. The effect of this additional mass was calculated and allowed for, and the weighings were also repeated with the weights in various positions. The principle of the subsequent calculation is briefly as follows:—A mass A of lead and antimony of known bulk attracts another mass B with the force measured; if A were of the same size as the earth, the attraction would be increased by as many times as the earth is larger than A. If the average specific gravity of the earth were the same as that of the mass A, this calculated attraction would be equal to the weight of B. The ratio of this calculated weight of B to the actual weight accordingly gives the proportion between the specific gravity of the experimental ball and the average specific gravity of the whole earth. From this experiment it was estimated that the mean density of the earth is 5·4934 times that of water.
The same principle is applied in the case of the famous Cavendish experiment, and its subsequent modifications by Baily, Cornu, and Boys.
Another method of finding the earth’s density, and therefore its mass, is chiefly of historical interest. This is known as the “mountain method,” and was carried out in 1774 by Maskelyne, Hutton and Playfair on the Schiehallion Mountain, in Perthshire. A plumb-line suspended at the north side of the mountain is drawn towards the mountain, and so will not hang quite vertically. If removed to the opposite side of the mountain it will be deflected in the reverse direction. The amount of this deflection can be measured by reference to the stars, the positions of which are in no wise influenced by the attraction of the mountain. A survey of the mountain was next made in order to determine its bulk, and then the average specific gravity of the rocks composing it was determined with the greatest possible accuracy.
The volume of the earth is 9,933 times that of the mountain, and its attraction would be this number of times greater if it were composed of the same materials as the mountain throughout. It was found to be in reality 17,781 times as great as the attraction of the mountain, and as this is 1·79 times 9,933, it follows that the average specific gravity of the matter composing the earth would be 1·79 times that of the rocks which build up Schiehallion. The mean specific gravity of the rocks being 2·8, the mean density of the earth was thus found to be 5·012 times that of water.
As a general result of all the observations which have been made, the value of the earth’s density may with much probability be considered to be not far from 5·576, or a little over 5½ times that of water.
Whatever may be the composition of the earth’s interior, it is clear that the density must increase as the centre is approached.
This knowledge of the earth’s density, in conjunction with the known number of cubic miles occupied by the earth, readily enables us to determine that the total mass of the earth is about 6,000,000,000,000,000,000,000 tons.