A Text-Book of Astronomy Part 5

The word _ma.s.s_, in the statement of the law of gravitation, means the quant.i.ty of matter contained in the body, and if we represent by the letters _m'_ and _m''_ the respective quant.i.ties of matter contained in the two bodies whose distance from each other is _r_, we shall have, in accordance with the law of gravitation, the following mathematical expression for the force, _F_, which acts between them:

F = k {m'm''/r^{2}}.

This equation, which is the general mathematical expression for the law of gravitation, may be made to yield some curious results. Thus, if we select two bullets, each having a ma.s.s of 1 gram, and place them so that their centers are 1 centimeter apart, the above expression for the force exerted between them becomes

F = k {(1 1)/1^{2}} = k,

from which it appears that the coefficient _k_ is the force exerted between these bodies. This is called the gravitation constant, and it evidently furnishes a measure of the specific intensity with which one particle of matter attracts another. Elaborate experiments which have been made to determine the amount of this force show that it is surprisingly small, for in the case of the two bullets whose ma.s.s of 1 gram each is supposed to be concentrated into an indefinitely small s.p.a.ce, gravity would have to operate between them continuously for more than forty minutes in order to pull them together, although they were separated by only 1 centimeter to start with, and nothing save their own inertia opposed their movements. It is only when one or both of the ma.s.ses _m'_, _m''_ are very great that the force of gravity becomes large, and the weight of bodies at the surface of the earth is considerable because of the great quant.i.ty of matter which goes to make up the earth. Many of the heavenly bodies are much more ma.s.sive than the earth, as the mathematical astronomers have found by applying the law of gravitation to determine numerically their ma.s.ses, or, in more popular language, to "weigh" them.

The student should observe that the two terms ma.s.s and weight are not synonymous; ma.s.s is defined above as the quant.i.ty of matter contained in a body, while weight is the force with which the earth attracts that body, and in accordance with the law of gravitation its weight depends upon its distance from the center of the earth, while its ma.s.s is quite independent of its position with respect to the earth.

By the third law of motion the earth is pulled toward a falling body just as strongly as the body is pulled toward the earth--i. e., by a force equal to the weight of the body. How much does the earth rise toward the body?

38. THE MOTION OF A PLANET.--In Fig. 20 _S_ represents the sun and _P_ a planet or other celestial body, which for the moment is moving along the straight line _P 1_. In accordance with the first law of motion it would continue to move along this line with uniform velocity if no external force acted upon it; but such a force, the sun's attraction, is acting, and by virtue of this attraction the body is pulled aside from the line _P 1_.

Knowing the velocity and direction of the body's motion and the force with which the sun attracts it, the mathematician is able to apply Newton's laws of motion so as to determine the path of the body, and a few of the possible orbits are shown in the figure where the short cross stroke marks the point of each orbit which is nearest to the sun. This point is called the _perihelion_.

Without any formal application of mathematics we may readily see that the swifter the motion of the body at _P_ the shorter will be the time during which it is subjected to the sun's attraction at close range, and therefore the force exerted by the sun, and the resulting change of motion, will be small, as in the orbits _P 1_ and _P 2_.

On the other hand, _P 5_ and _P 6_ represent orbits in which the velocity at _P_ was comparatively small, and the resulting change of motion greater than would be possible for a more swiftly moving body.

What would be the orbit if the velocity at _P_ were reduced to nothing at all?

What would be the effect if the body starting at _P_ moved directly away from _1_?

[Ill.u.s.tration: FIG. 20.--Different kinds of orbits.]

The student should not fail to observe that the sun's attraction tends to pull the body at _P_ forward along its path, and therefore increases its velocity, and that this influence continues until the planet reaches perihelion, at which point it attains its greatest velocity, and the force of the sun's attraction is wholly expended in changing the direction of its motion. After the planet has pa.s.sed perihelion the sun begins to pull backward and to r.e.t.a.r.d the motion in just the same measure that before perihelion pa.s.sage it increased it, so that the two halves of the orbit on opposite sides of a line drawn from the perihelion through the sun are exactly alike. We may here note the explanation of Kepler's second law: when the planet is near the sun it moves faster, and the radius vector changes its direction more rapidly than when the planet is remote from the sun on account of the greater force with which it is attracted, and the exact relation between the rates at which the radius vector turns in different parts of the orbit, as given by the second law, depends upon the changes in this force.

When the velocity is not too great, the sun's backward pull, after a planet has pa.s.sed perihelion, finally overcomes it and turns the planet toward the sun again, in such a way that it comes back to the point _P_, moving in the same direction and with the same speed as before--i. e., it has gone around the sun in an orbit like _P 6_ or _P 4_, an ellipse, along which it will continue to move ever after. But we must not fail to note that this return into the same orbit is a consequence of the last line in the statement of the law of gravitation (p. 54), and that, if the magnitude of this force were inversely as the cube of the distance or any other proportion than the square, the orbit would be something very different. If the velocity is too great for the sun's attraction to overcome, the orbit will be a hyperbola, like _P 2_, along which the body will move away never to return, while a velocity just at the limit of what the sun can control gives an orbit like _P 3_, a parabola, along which the body moves with _parabolic velocity_, which is ever diminishing as the body gets farther from the sun, but is always just sufficient to keep it from returning. If the earth's velocity could be increased 41 per cent, from 19 up to 27 miles per second, it would have parabolic velocity, and would quit the sun's company.

The summation of the whole matter is that the orbit in which a body moves around the sun, or past the sun, depends upon its velocity and if this velocity and the direction of the motion at any one point in the orbit are known the whole orbit is determined by them, and the position of the planet in its...o...b..t for past as well as future times can be determined through the application of Newton's laws; and the same is true for any other heavenly body--moon, comet, meteor, etc. It is in this way that astronomers are able to predict, years in advance, in what particular part of the sky a given planet will appear at a given time.

It is sometimes a source of wonder that the planets move in ellipses instead of circles, but it is easily seen from Fig. 20 that the planet, _P_, could not by any possibility move in a circle, since the direction of its motion at _P_ is not at right angles with the line joining it to the sun as it must be in a circular orbit, and even if it were perpendicular to the radius vector the planet must needs have exactly the right velocity given to it at this point, since either more or less speed would change the circle into an ellipse. In order to produce circular motion there must be a balancing of conditions as nice as is required to make a pin stand upon its point, and the really surprising thing is that the orbits of the planets should be so nearly circular as they are. If the orbit of the earth were drawn accurately to scale, the untrained eye would not detect the slightest deviation from a true circle, and even the orbit of Mercury (Fig. 17), which is much more eccentric than that of the earth, might almost pa.s.s for a circle.

[Ill.u.s.tration: FIG. 21. An impossible orbit.]

The orbit _P 2_, which lies between the parabola and the straight line, is called in geometry a hyperbola, and Newton succeeded in proving from the law of gravitation that a body might move under the sun's attraction in a hyperbola as well as in a parabola or ellipse; but it must move in some one of these curves; no other orbit is possible.[1] Thus it would not be possible for a body moving under the law of gravitation to describe about the sun any such orbit as is shown in Fig. 21. If the body pa.s.ses a second time through any point of its...o...b..t, such as _P_ in the figure, then it must retrace, time after time, the whole path that it first traversed in getting from _P_ around to _P_ again--i. e., the orbit must be an ellipse.

[1] The circle and straight line are considered to be special cases of these curves, which, taken collectively, are called the conic sections.

Newton also proved that Kepler's three laws are mere corollaries from the law of gravitation, and that to be strictly correct the third law must be slightly altered so as to take into account the ma.s.ses of the planets. These are, however, so small in comparison with that of the sun, that the correction is of comparatively little moment.

39. PERTURBATIONS.--In what precedes we have considered the motion of a planet under the influence of no other force than the sun's attraction, while in fact, as the law of gravitation a.s.serts, every other body in the universe is in some measure attracting it and changing its motion.

The resulting disturbances in the motion of the attracted body are called _perturbations_, but for the most part these are insignificant, because the bodies by whose disturbing attractions they are caused are either very small or very remote, and it is only when our moving planet, _P_, comes under the influence of some great disturbing power like Jupiter or one of the other planets that the perturbations caused by their influence need to be taken into account.

The problem of the motion of three bodies--sun, Jupiter, planet--which must then be dealt with is vastly more complicated than that which we have considered, and the ablest mathematicians and astronomers have not been able to furnish a complete solution for it, although they have worked upon the problem for two centuries, and have developed an immense amount of detailed information concerning it.

[Ill.u.s.tration: THE LICK OBSERVATORY, MOUNT HAMILTON, CAL.]

In general each planet works ceaselessly upon the orbit of every other, changing its size and shape and position, backward and forward in accordance with the law of gravitation, and it is a question of serious moment how far this process may extend. If the diameter of the earth's...o...b..t were very much increased or diminished by the perturbing action of the other planets, the amount of heat received from the sun would be correspondingly changed, and the earth, perhaps, be rendered unfit for the support of life. The tipping of the plane of the earth's...o...b..t into a new position might also produce serious consequences; but the great French mathematician of a century ago, Laplace, succeeded in proving from the law of gravitation that although both of these changes are actually in progress they can not, at least for millions of years, go far enough to prove of serious consequence, and the same is true for all the other planets, unless here and there an asteroid may prove an exception to the rule.

The precession (Chapter V) is a striking ill.u.s.tration of a perturbation of slightly different character from the above, and another is found in connection with the plane of the moon's...o...b..t. It will be remembered that the moon in its motion among the stars never goes far from the ecliptic, but in a complete circuit of the heavens crosses it twice, once in going from south to north and once in the opposite direction.

The points at which it crosses the ecliptic are called the _nodes_, and under the perturbing influence of the sun these nodes move westward along the ecliptic about twenty degrees per year, an extraordinarily rapid perturbation, and one of great consequence in the theory of eclipses.

[Ill.u.s.tration: FIG. 22.--A planet subject to great perturbations by Jupiter.]

40. WEIGHING THE PLANETS.--Although these perturbations can not be considered dangerous, they are interesting since they furnish a method for weighing the planets which produce them. From the law of gravitation we learn that the ability of a planet to produce perturbations depends directly upon its ma.s.s, since the force _F_ which it exerts contains this ma.s.s, _m'_, as a factor. So, too, the divisor _r^{2}_ in the expression for the force shows that the distance between the disturbing and disturbed bodies is a matter of great consequence, for the smaller the distance the greater the force. When, therefore, the ma.s.s of a planet such as Jupiter is to be determined from the perturbations it produces, it is customary to select some such opportunity as is presented in Fig. 22, where one of the small planets, called asteroids, is represented as moving in a very eccentric orbit, which at one point approaches close to the orbit of Jupiter, and at another place comes near to the orbit of the earth. For the most part Jupiter will not exert any very great disturbing influence upon a planet moving in such an orbit as this, since it is only at rare intervals that the asteroid and Jupiter approach so close to each other, as is shown in the figure. The time during which the asteroid is little affected by the attraction of Jupiter is used to study the motion given to it by the sun's attraction--that is, to determine carefully the undisturbed orbit in which it moves; but there comes a time at which the asteroid pa.s.ses close to Jupiter, as shown in the figure, and the orbital motion which the sun imparts to it will then be greatly disturbed, and when the planet next comes round to the part of its...o...b..t near the earth the effect of these disturbances upon its apparent position in the sky will be exaggerated by its close proximity to the earth. If now the astronomer observes the actual position of the asteroid in the sky, its right ascension and declination, and compares these with the position a.s.signed to the planet by the law of gravitation when the attraction of Jupiter is ignored, the differences between the observed right ascensions and declinations and those computed upon the theory of undisturbed motion will measure the influence that Jupiter has had upon the asteroid, and the amount by which Jupiter has shifted it, compared with the amount by which the sun has moved it--that is, with the motion in its...o...b..t--furnishes the ma.s.s of Jupiter expressed as a fractional part of the ma.s.s of the sun.

There has been determined in this manner the ma.s.s of every planet in the solar system which is large enough to produce any appreciable perturbation, and all these ma.s.ses prove to be exceedingly small fractions of the ma.s.s of the sun, as may be seen from the following table, in which is given opposite the name of each planet the number by which the ma.s.s of the sun must be divided in order to get the ma.s.s of the planet:

Mercury 7,000,000 (?) Venus 408,000 Earth 329,000 Mars 3,093,500 Jupiter 1,047.4 Saturn 3,502 Ura.n.u.s 22,800 Neptune 19,700

It is to be especially noted that the ma.s.s given for each planet includes the ma.s.s of all the satellites which attend it, since their influence was felt in the perturbations from which the ma.s.s was derived.

Thus the ma.s.s a.s.signed to the earth is the combined ma.s.s of earth and moon.

41. DISCOVERY OF NEPTUNE.--The most famous example of perturbations is found in connection with the discovery, in the year 1846, of Neptune, the outermost planet of the solar system. For many years the motion of Ura.n.u.s, his next neighbor, had proved a puzzle to astronomers. In accordance with Kepler's first law this planet should move in an ellipse having the sun at one of its foci, but no ellipse could be found which exactly fitted its observed path among the stars, although, to be sure, the misfit was not very p.r.o.nounced. Astronomers surmised that the small deviations of Ura.n.u.s from the best path which theory combined with observation could a.s.sign, were due to perturbations in its motion caused by an unknown planet more remote from the sun--a thing easy to conjecture but hard to prove, and harder still to find the unknown disturber. But almost simultaneously two young men, Adams in England and Le Verrier in France, attacked the problem quite independently of each other, and carried it to a successful solution, showing that if the irregularities in the motion of Ura.n.u.s were indeed caused by an unknown planet, then that planet must, in September, 1846, be in the direction of the constellation Aquarius; and there it was found on September 23d by the astronomers of the Berlin Observatory whom Le Verrier had invited to search for it, and found within a degree of the exact point which the law of gravitation in his hands had a.s.signed to it.

This working backward from the perturbations experienced by Ura.n.u.s to the cause which produced them is justly regarded as one of the greatest scientific achievements of the human intellect, and it is worthy of note that we are approaching the time at which it may be repeated, for Neptune now behaves much as did Ura.n.u.s three quarters of a century ago, and the most plausible explanation which can be offered for these anomalies in its path is that the bounds of the solar system must be again enlarged to include another disturbing planet.

42. THE SHAPE OF A PLANET.--There is an effect of gravitation not yet touched upon, which is of considerable interest and wide application in astronomy--viz., its influence in determining the shape of the heavenly bodies. The earth is a globe because every part of it is drawn toward the center by the attraction of the other parts, and if this attraction on its surface were everywhere of equal force the material of the earth would be crushed by it into a truly spherical form, no matter what may have been the shape in which it was originally made. But such is not the real condition of the earth, for its diurnal rotation develops in every particle of its body a force which is sometimes called _centrifugal_, but which is really nothing more than the inertia of its particles, which tend at every moment to keep unchanged the direction of their motion and which thus resist the attraction that pulls them into a circular path marked out by the earth's rotation, just as a stone tied at the end of a string and swung swiftly in a circle pulls upon the string and opposes the constraint which keeps it moving in a circle. A few experiments with such a stone will show that the faster it goes the harder does it pull upon the string, and the same is true of each particle of the earth, the swiftly moving ones near the equator having a greater centrifugal force than the slow ones near the poles. At the equator the centrifugal force is directly opposed to the force of gravity, and in effect diminishes it, so that, comparatively, there is an excess of gravity at the poles which compresses the earth along its axis and causes it to bulge out at the equator until a balance is thus restored. As we have learned from the study of geography, in the case of the earth, this compression amounts to about 27 miles, but in the larger planets, Jupiter and Saturn, it is much greater, amounting to several thousand miles.

But rotation is not the only influence that tends to pull a planet out of shape. The attraction which the earth exerts upon the moon is stronger on the near side and weaker on the far side of our satellite than at its center, and this difference of attraction tends to warp the moon, as is ill.u.s.trated in Fig. 23 where _1_, _2_, and _3_ represent pieces of iron of equal ma.s.s placed in line on a table near a horseshoe magnet, _H_. Each piece of iron is attracted by the magnet and is held back by a weight to which it is fastened by means of a cord running over a pulley, _P_, at the edge of the table. These weights are all to be supposed equally heavy and each of them pulls upon its piece of iron with a force just sufficient to balance the attraction of the magnet for the middle piece, No. _2_. It is clear that under this arrangement No.

_2_ will move neither to the right nor to the left, since the forces exerted upon it by the magnet and the weight just balance each other.

Upon No. _1_, however, the magnet pulls harder than upon No. _2_, because it is nearer and its pull therefore more than balances the force exerted by the weight, so that No. _1_ will be pulled away from No. _2_ and will stretch the elastic cords, which are represented by the lines joining _1_ and _2_, until their tension, together with the force exerted by the weight, just balances the attraction of the magnet. For No. _3_, the force exerted by the magnet is less than that of the weight, and it will also be pulled away from No. _2_ until its elastic cords are stretched to the proper tension. The net result is that the three blocks which, without the magnet's influence, would be held close together by the elastic cords, are pulled apart by this outside force as far as the resistance of the cords will permit.

[Ill.u.s.tration: FIG. 23.--Tide-raising forces.]

An entirely a.n.a.logous set of forces produces a similar effect upon the shape of the moon. The elastic cords of Fig. 23 stand for the attraction of gravitation by which all the parts of the moon are bound together.

The magnet represents the earth pulling with unequal force upon different parts of the moon. The weights are the inertia of the moon in its...o...b..tal motion which, as we have seen in a previous section, upon the whole just balances the earth's attraction and keeps the moon from falling into it. The effect of these forces is to stretch out the moon along a line pointing toward the earth, just as the blocks were stretched out along the line of the magnet, and to make this diameter of the moon slightly but permanently longer than the others.

[Ill.u.s.tration: FIG. 24.--The tides.]

THE TIDES.--Similarly the moon and the sun attract opposite sides of the earth with different forces and feebly tend to pull it out of shape. But here a new element comes into play: the earth turns so rapidly upon its axis that its solid parts have no time in which to yield sensibly to the strains, which shift rapidly from one diameter to another as different parts of the earth are turned toward the moon, and it is chiefly the waters of the sea which respond to the distorting effect of the sun's and moon's attraction. These are heaped up on opposite sides of the earth so as to produce a slight elongation of its diameter, and Fig. 24 shows how by the earth's rotation this swelling of the waters is swept out from under the moon and is pulled back by the moon until it finally takes up some such position as that shown in the figure where the effect of the earth's rotation in carrying it one way is just balanced by the moon's attraction urging it back on line with the moon. This heaping up of the waters is called a _tide_. If _I_ in the figure represents a little island in the sea the waters which surround it will of course accompany it in its diurnal rotation about the earth's axis, but whenever the island comes back to the position _I_, the waters will swell up as a part of the tidal wave and will encroach upon the land in what is called high tide or flood tide. So too when they reach _I''_, half a day later, they will again rise in flood tide, and midway between these points, at _I'_, the waters must subside, giving low or ebb tide.

The height of the tide raised by the moon in the open sea is only a very few feet, and the tide raised by the sun is even less, but along the coast of a continent, in bays and angles of the sh.o.r.e, it often happens that a broad but low tidal wave is forced into a narrow corner, and then the rise of the water may be many feet, especially when the solar tide and the lunar tide come in together, as they do twice in every month, at new and full moon. Why do they come together at these times instead of some other?

Small as are these tidal effects, it is worth noting that they may in certain cases be very much greater--e. g., if the moon were as ma.s.sive as is the sun its tidal effect would be some millions of times greater than it now is and would suffice to grind the earth into fragments.

Although the earth escapes this fate, some other bodies are not so fortunate, and we shall see in later chapters some evidence of their disintegration.

43. THE SCOPE OF THE LAW OF GRAVITATION.--In all the domain of physical science there is no other law so famous as the Newtonian law of gravitation; none other that has been so dwelt upon, studied, and elaborated by astronomers and mathematicians, and perhaps none that can be considered so indisputably proved. Over and over again mathematical a.n.a.lysis, based upon this law, has pointed out conclusions which, though hitherto unsuspected, have afterward been found true, as when Newton himself derived as a corollary from this law that the earth ought to be flattened at the poles--a thing not known at that time, and not proved by actual measurement until long afterward. It is, in fact, this capacity for predicting the unknown and for explaining in minutest detail the complicated phenomena of the heavens and the earth that const.i.tutes the real proof of the law of gravitation, and it is therefore worth while to note that at the present time there are a very few points at which the law fails to furnish a satisfactory account of things observed. Chief among these is the case of the planet Mercury, the long diameter of whose orbit is slowly turning around in a way for which the law of gravitation as yet furnishes no explanation. Whether this is because the law itself is inaccurate or incomplete, or whether it only marks a case in which astronomers have not yet properly applied the law and traced out its consequences, we do not know; but whether it be the one or the other, this and other similar cases show that even here, in its most perfect chapter, astronomy still remains an incomplete science.

CHAPTER V

THE EARTH AS A PLANET