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Perhaps the most interesting elements are $a$ and $P$. The periods range from three months in the case of Mercury to 165 years for Neptune; the age of an earthly octogenarian is less than half of a Neptunian "year." The mean distance of Mercury is some 38,000,000 miles, while that of Neptune is roughly 3,000,000,000. Light, which reaches the Earth from the Sun in 499 seconds, requires more than four hours to make the journey to Neptune. The range of distances is so great that it is impracticable to represent all the planet orbits on a single diagram. The four inner orbits are pictured in Fig. 114, and those of Mars and the four outer planets in Fig. 115.
**Bode's Law.**—The approximate distances of the planets from the Sun may be conveniently remembered by a relation first pointed out by Titius, but commonly known as Bode's law. If we write a series of $4s$ and add to them the numbers o, 3, 3 × 2, 3 × 2 × 2, 3 × 2 × 2 × 2, etc., thus:
![]() A diagram showing the orbits of the four inner planets.
**FIG. 114. ORBITS OF THE FOUR INNER PLANETS**
occupied by most of the asteroids, the relation holds pretty well for all the planets except Neptune, but there it fails completely. There is no known theoretical reason for the "law," and it may be merely a coincidence.
**Kepler's Method of Determining the Form and Size of an Orbit.**—Since the discovery of the law of gravitation and the development of modern mathematical methods, all the elements of the orbit of a planet can be very approximately determined, by a computation that in skilled hands requires less than a day, from three observations of the planet's right ascension and
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4 | THE PATHS OF THE PLANETS 197
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![]() A diagram showing the orbits of the five outer planets.
**FIG. 115. ORBITS OF THE FIVE OUTER PLANETS**
Since the discovery of the law of gravitation and the development of modern mathematical methods, all the elements of the orbit of a planet can be very approximately determined, by a computation that in skilled hands requires less than a day, from three observations of the planet's right ascension and
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declination made on different dates. This method was not available to Kepler, who investigated the planetary orbits and discovered his three great laws by a much more laborious process.
As we have seen, there were available to Kepler the numerous observations of the apparent positions of the planets recorded during many years by Tycho Brahe and his assistants at the great observatory on the island of Hven. The first step was to determine accurately the planet's sidereal period, which could be done in two ways: (1) by observations of the time of conjunction or opposition, and (2) by observations of the time of passage.
The **sidereal period** is the time of one complete orbital revolution, from a given direction among the stars to the same direction again, as seen from the Sun; the **synodic period** is the time from opposition to opposition or from conjunction to conjunction as seen from the Earth. The synodic periods are variable because of the varying speed of the planets in different parts of their orbits; but if we represent by $S$ the mean synodic period of a planet in days, by $P$ its sidereal period, and by $E$ the sidereal period of the Earth ($365.25$), then
$$\frac{1}{S} = \frac{1}{P} - \frac{1}{E} \quad \text{if} \quad E > P,$$
$$\frac{1}{S} = \frac{1}{E} - \frac{1}{P} \quad \text{if} \quad E < P;$$
for each side of the equation represents the average daily gain of the Sun upon the planet in their apparent eastward motion, expressed in fractions of a circumference. Oppositions, quadratures, and conjunctions of the planets played an important part in astrology, and hence records of such observations made long before the time of Tycho were available; from these $S$ could be found, and $P$ could then be determined from one of the above equations.
When a planet is at its node, its latitude is zero as seen from either the Earth or the Sun, since it is then exactly in the plane of the ecliptic. One sidereal period later, it will again be at the node and, although its longitude as seen from the Earth will be different because the Earth occupies a different point of its orbit, the planet's latitude will again be zero. Hence, the interval between two successive times when the planet is seen crossing the ecliptic in the same direction is its sidereal period.
Having the sidereal period, Kepler determined the planet's distance from the Sun at different points of its orbit by triangulation. His first thorough investigation was that of the orbit of Mars, the period of which is 687 days. Suppose that Mars is at the point $C$ of its orbit (Fig. 116) when the Earth is at $A$. After 687 days Mars will be back at $C$, while the Earth will have made one complete revolution and a large part of another, and be at $B$. In the triangle $ASB$ the sides $SA$ and $SB$ are radii of the Earth's orbit, and the angle $ASB$ can be found from the
![]() A diagram showing a triangle with vertices labeled A, B, C. Side AB is labeled with a small circle around it. Side AC is labeled with a small circle around it. Side BC is labeled with a small circle around it.
FIG. 116. KEPLER'S METHOD OF DETERMINING THE ORBIT OF A PLANET
interval of time; hence it is possible to compute trigonometrically the side $AB$ and the angles $SAB$ and $SBA$. Tycho's observations furnished the angles $SAC$ and $SBC$, the planet's elongations when the Earth was at $A$ and $B$, respectively; from these, by subtracting the computed angles $SAB$ and $SBA$, he obtained the angles $CAB$ and $ABC$; and from these and the side $AB$ he computed the sides $AC$ and $BC$ of the triangle $ABC$, giving the distances of Mars from the Earth in terms of the Earth's distance from the Sun. Finally, he obtained $SC$, the radius vector of Mars, from the triangle $SAC$ or $CBS$, having given two sides and included angle. From many pairs of observations separated by the interval of 687 days, he was thus enabled to find the distance of Mars from the Sun at
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many points, and thus to deduce the size and form of the orbit and the speed with which the planet moved, and so to discover his first two laws. The third law resulted later from a comparison of the distances and periods of different planets.
**The Sun as Seen from Different Planets.**—As an observer recedes from the Sun, both its apparent diameter and the intensity of its light and heat become less, the apparent diameter being inversely proportional to the first power of the distance and the intensity to the square. As seen from Mercury the Sun's diameter is about two and one-half times as great at its distance from the Earth; as seen from Neptune its diameter is only a little more than a minute of arc, and to the naked eye it would appear as an intensely bright star without perceptible disk.
Taking the intensity of the Sun's light and heat at the Earth's distance as unity, we have the following values of the intensity at the distances of the various planets:
| U | Q | @ | φ | Q' | b | δ | Ψ |
| 6.7 | 1.0 | 1.00 | 0.43 | 0.04 | 0.01 | 0.003 | 0.001 |
It may well be imagined that there exists a great diversity of climates.
**The Earth as Seen from Other Planets.**—The Earth, being an opaque body, must shine by reflected sunlight and, to an observer on one of its nearest neighbors, must present much the same appearance as the other planets do to us. The most favorable view would be obtained from Venus. When the Earth and Venus are nearest together, at a distance of some 26,000,000 miles, the latter planet is at inferior conjunction, nearly or (on rare occasions) quite directly between us and the Sun, so that her dark side is turned toward us and at the same time the eye is dazzled by the light of the Sun. For an observer on Venus at the same time, the conditions for viewing the Earth would be the reverse of these: the Earth would be at opposition with its illuminated side turned full upon the observer and would appear as a star about six times as bright as Venus appears to us. The Moon would also be plainly seen as a fainter star passing from side to side of the Earth in the course of a month to a distance of about half a degree.
From Mars the view would be less favorable, since for the Martians the Earth, like Venus and Mercury, would seem to oscillate from side to side of the Sun and would reach a "greatest elongation" of 48°, or about the same as that of Venus as seen from the Earth. When Mars and the Earth are nearest together, the Earth would be at inferior conjunction, and near that time a Martian with a good telescope might see both the Earth and the Moon in their crescent phases.
As seen from Jupiter, the Earth is never more than 12° distant from its Sun, and it is likely that a race of Jovians, if equipped with eyes and telescopes similar to ours, would be unaware of the existence of this little globe. This deplorable state of ignorance is of course still more likely to be true of any possible inhabitants of Saturn, Uranus, Neptune, or their satellites.
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THE LAW OF GRAVITATION
CHAPTER X
THE LAW OF GRAVITATION
Isaac Newton (1643-1727).—Kepler's laws of planetary motion were simply descriptive statements of the behavior of the planets. Neither Kepler nor any one else had given an explanation of the force that causes the planets to move in just this way and not another. It was generally supposed that, to keep a planet moving, some "projectile force," acting along a tangent to the orbit, was required, and Kepler seems to have been somewhat inclined to attribute this force to the will of a supernatural being—perhaps an angel that had the charge of each planet—or to invisible spokes that radiated from the Sun and pushed the planets along. It remained for the great English mathematical philosopher Sir Isaac Newton to show that the planetary motions were but manifestations of a universal principle and to derive the mighty generalization known as the Law of Gravitation.
The contributions of Newton to astronomy, mathematics, and physics are numerous and exceedingly important. Before reaching the age of twenty-four he had discovered the binomial theorem, formed his theory of colors, founded the branch of mathematics which he called "fluxions" and which has grown into the modern Calculus, and laid the basis of the law of gravitation. It was not, however, until 1687 that he published his immortal work *Philosophiae Naturalis Principia Mathematica*, commonly known as Newton's Principia, the appearance of which probably marks the greatest forward step ever made in physical science.
**Newton's Laws of Motion.**—The science of mechanics consists of theorems built upon certain laws or principles just as geometry is built upon its familiar axioms. The motions of bodies as ordinarily observed, whether they be the planets or familiar bodies at the surface of the Earth, may be calculated precisely by the system of mechanics built up by Newton. He used as a basis the axioms of Euclidian geometry together with three principles that are known as Newton's laws of motion because they were first definitely stated by Newton, although they were at least partly understood by Galileo and before him, by the great artist-inventor Leonardo da Vinci. These laws of motion are:
I. Every body continues in a state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by a force impressed upon it.
II. The rate of change of motion is proportional to the impressed force, and the change takes place in the direction of the straight line in which the force acts.
III. To every action there is an equal and opposite reaction.
Comment on the First Law.—If a block of wood resting on a level floor be given a sufficient impulse, it will slide along the floor, but if then left to itself it will come quickly to a stop because the frictional force and therefore its effect is greater than any impulse given. A block will slide further before coming to a stop because the friction is less. If the block be placed on smooth ice, a given impulse will carry it further still. We cannot remove from a body all forces such as friction, gravity, etc., without completely removing them; consequently, but experiment shows that, the more nearly we eliminate the forces acting on a body, the more nearly does its motion conform to the law; and Newton assumed that no body can in any circumstances travel forever uniformly in a straight line. Simply to keep moving, then, a planet or any other body needs no force such as the "projectile force" of Aristotle. That which requires an explanation is the curvature of the planet's path and the time taken.
Comment on the Second Law.—By "the rate of change of motion" Newton meant what many later writers call the rate of change of momentum, moment-mass times velocity. In other words, it is what we now call "force." The change of motion of a body whose mass remains constant is thus the mass times the rate of change of velocity, or mass times acceleration.
Up to the time of Galileo it was believed that a one-pound weight would fall ten times as fast as a one-pound weight. Galileo, by a famous experiment at the leaning tower of Pisa, showed that spheres of different masses fell from the tower at different rates. This fact led him to conclude that it is not "the force" gravity, acting on a ten-pound ball is ten times as great as that acting on a one-pound ball, and it therefore produces in the former ten times the "quantity of motion," i.e., the mass times velocity ten times as much as in the latter. Since also their rate of change of velocity, is same. The fact that all bodies, such for example, as a bullet and a feather, do not fall a given distance in air in the same time is due to the difference
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the resistance of the air caused by their different shape or density. If dropped in a tube from which the air has been exhausted, the feather and the bullet fall in the same time.
In the statement of the second law, nothing is said about the condition of rest or motion of the body at the time the force is applied, nor of other forces that may act on it at the same time. It is therefore implied that, whether the body is at rest or in motion, when a force is applied to it, either or not, a given force produces the same change of momentum that it would produce if it alone acted on the body at rest.
Thus, if a body at $A$ (Fig. 117) is acted upon by two equal and opposite forces, one will cause it to move in one second to $B$, and another which would send it in one second to $C$, it must, at the end of one second, arrive at a point $D$ which is at the distance $AB$ from $A$. The direction of the first force and at the distance $AC$ from $A$ as measured in the direction of the second force; that is, perpendicular to line $AB$, and parallel to lines $AB$ and $AC$ are adjacent sides. The important principle of the parallelogram of forces is thus a corollary to the second law of motion.
Comment on the Third Law.--Suppose a man on a raft that floats freely in still water pulls on a rope attached to a similar raft that is heavily loaded, say with scrap iron. Both rafts will move, and the more lightly loaded raft will move faster than the heavier raft. Suppose now that we measure the pull on the rope, the product of the mass of each raft (plus its load) by its rate of change of velocity would be the same—the "action" of the man is met by an equal "reaction" on the part of the inanimate scrap iron. A little reflection will disclose that this applies wherever forces are applied.
The first two laws are sufficient for a discussion of the results of applying various forces to one body; the third is needed for studying the motions of a system of bodies.
The Theorem of Areas.--As an example of Newton's methods and the important results to which they led we may prove the following theorem:
To Prove.--If a body move subject to no forces except one that is directed always toward the same fixed point $O$, the line joining the body to $O$ must pass over equal areas in equal intervals of time.
Proof.--Suppose, first, that the body is subject to no forces whatever; so that, by the first law of motion, it will move uniformly in a straight line. Let the line of its motion be $AC$ (Fig. 118) and let it move the distance $AB$ in one second. In the next second it will travel an equal distance $BC$. Let $O$ be any point not in the line $AC$ and draw $OA$, $OB$ and $OC$. The triangles $OAB$ and $OBC$, having equal bases $AB$ and $BC$ and the common altitude $OK$, are equal in area.
Suppose, however, that upon arriving at $B$ at the end of the first second, the body is acted on by an instantaneous force like a blow from a hammer, say--acting along the line $BO$, and that this force is of such magnitude that, had the body been at rest, it would have been sent to $D$ (Fig. 119) in one second. According to Newton's second law, this instantaneous force will produce a change in motion parallel to the line $BO$, causing the body to be displaced one second a distance $BD$ in the direction of that line; but it will not cause it to lose its original motion which, acting alone, would carry the body in
![]() Fig. 117.
![]() Fig. 118.
![]() Fig. 119.
one second from $B$ to $C$. The actual path is therefore $BE$, the diagonal of the parallelogram of which $BD$ and $BC$ are adjacent sides. Draw $OC$ and $OE$. The triangles $OBC$ and $OBE$ have the common base $OB$ and equal altitudes $CL$ and $EM$, the distance between the parallel lines $BD$ and $CE$. Therefore triangle $OBE$ is equal in area to triangle $OBC$, which we have already shown to be equal to $OAB$.
In the third second, if the body were left to itself, it would according to the first law of motion, move from $E$ to $G$ where $\overline{EG} = BE$ (Fig. 120); but let it again be acted upon, when at $E$, by an instantaneous force directed toward $O$ and of such magnitude as to carry it to $H$, it will be at rest at $E$. It will travel to $H$, the opposite corner of the parallelogram whose sides are $\overline{EG}$ and $\overline{EF}$. The triangles $\overline{OEH}$ and $\overline{OEG}$ are equal
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in area since they have the common base $OE$ and equal altitudes $GP$ and $HQ$.
We now have (Fig. 121)
$$\Delta OAB = \Delta OBE = \Delta OEH;$$
that is, if the body be subject to no forces except one which acts instantaneously toward $O$ at the end of each second, the lines joining $O$ to the positions occupied by the body at the instants of action of the force form triangles of equal area. Our proof will hold as well if we suppose the force to act at intervals of half a second or a millionth of a second, or at intervals that are smaller than any imaginable quantity, however small. In the limit, the intermittent, instantaneous forces are replaced by
![]() Fig. 120
a continuously acting force (still supposed to be always acting along the line joining the body to $O$), and the path of the body becomes a smooth curve of a form depending on the way in which the force varies. Yet, however small the interval at which the force be supposed to act, the area described in any unit of time, as one second, will be the same as that described in any equal unit since it is the sum of the same number of equal areas; and hence, however the force may vary in intensity, so long as it acts always along the line joining the body to $O$ (it may be directed away from $O$ as well as toward $O$), that line will pass over equal areas in equal intervals of time. Q. E. D.
We have placed no limitations upon the force except that it shall always act along the line joining the body to $O$. It may be of constant or variable intensity, or it may be intermittent, or alternately attractive and repulsive; it may vary in intensity directly as the distance, or inversely as the square of the distance, or according to any other law, or even capriciously; and the law of areas proved above must still hold.
This theorem illustrates one application—the only one that we shall attempt to give in detail—that Newton made of his laws of motion to the study of Mechanics. He proved also that the converse is true: If a body be found moving in such a way that the line joining it to any point passes over equal areas in equal intervals of time then the only force (or the resultant) acting on the body must be directed toward (or away from) the point. Kepler had, as we have seen, already shown that each planet moves in that way in regard to the Sun—i. e., that its radius vector describes equal areas in equal times. The conclusion was therefore obvious that *the only force required to explain the motion of any one of the planets is a force directed toward the Sun*.
**Newton's Inferences from Kepler's Laws.—By mathematical processes based upon the laws of motion as in the above example, Newton made an important inference from each of the laws of Kepler. These inferences, including the one just stated, are:**
1. From Kepler's law of areas: *The force that controls the motion of each planet is directed toward the Sun.*
2. From Kepler's first law: *The Sun's attraction for a given planet varies inversely as the square of the planet's distance from the Sun.*
3. From the harmonic law: *The Sun's attraction varies from one planet to another inversely as the square of the distance.*
The second statement means that, as a planet approaches perihelion, the intensity of the attraction rapidly increases; if the distance be halved, the attraction is increased fourfold. It depends both on the fact that the orbit is an ellipse and on the fact that the Sun occupies its focus; if, for instance, Kepler had found that the Sun was in the center of all planetary ellipses, Newton must have concluded that the attraction varied directly as first power of the distance instead of inversely as the square.
The third statement means that the attraction for a planet
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![]() Fig. 121
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of given mass depends only on the distance and not on any other properties of the planets such as temperature, chemical constitution, etc. If Mars were placed at the Earth's distance from the Sun, the Sun's attraction for it, except for the differences of mass, would be the same that it is in the case of the Earth. Newton showed, however, that for planets of appreciable mass, the harmonic law is only approximately true; the precise relation is
$$a_1^3 : a_2^3 : P_1^3 (M + m_1) : P_2^3 (M + m_2)$$
where $M$ is the mass of the Sun and the subscripts refer to the different planets. In every case in the Solar System, $M$ is so much greater than $m$ (the mass of the Sun is about a thousand times that of even Jupiter, the greatest planet) that Kepler's statement of the law, in which the masses are omitted, is very nearly correct.
**The Law of Gravitation.**—It is related that Newton was sitting one day in his garden, reflecting upon the force that bends the planets from their paths and upon the force that holds the Moon in its orbit around the Earth, when his attention was diverted by an apple falling from a tree. It occurred to him that the fall of the apple and the divergence of the Moon and planets from straight lines might be manifestations of the same force. Becoming convinced that this was true, he was eventually led to the Law of Gravitation, which is:
Every particle of matter in the universe attracts every other particle with a force that varies inversely as the square of the distance between them and directly as the product of their masses.
This great law may be expressed very simply in symbols if we let $m_1$ and $m_2$ be the masses of any two particles, $d$ the distance between them, $F$ the force of their attraction, and $G$ a constant; thus:
$$F = G \frac{m_1 m_2}{d^2}$$
Newton did not state the law of gravitation in just these words in the Principia, nor does it appear that he explicitly extended it beyond the Solar System; but the whole Celestial Mechanics, including Newton's own conclusion, is in harmony with the law as above stated, and the elliptical motion of double stars shows that the law holds between them as between the Sun and planets.
**Test of the Law of Gravitation by the Motion of the Moon.**—By means of his principles of "fluxions" Newton was able to prove the difficult proposition that, if the law of gravitation be true, a homogeneous sphere of any size attracts an exterior particle (here) inversely as the square of the distance from its center; that is, as if the whole mass of the sphere were concentrated at its center. He was then in a position to test the law of gravitation by comparing the distance which an apple falls in the first second with the distance that the Moon "falls" toward the Earth, or in other words deviates from a straight line, in the same time.
At a given distance from the Earth and the apple would fall at the same speed just as did Galileo's unequal weights at the leaning tower of Pisa; but here, as Newton knew, the Moon is about sixty times farther from the center of Earth as is the apple, and hence, if they are both controlled by an attraction that varies inversely as the square of their distances from the center of Earth, the fall of the apple in the first second should be 60², or 3,600, times the Moon's departure from a straight path in an equal time. The apple falls in the first second 16.1 feet, or 193 inches; hence the Moon should depart in one second from a straight line 1/3600th of this distance, or 0.0535 inch.
The actual deviation may be easily determined from a knowledge of the Moon's distance from the Earth and the time of its revolution. In Fig. 122 let circle $A$ represent Earth and circle $B$ represent Moon, and let $AP$ represent radius $AB$. Now draw diameter $AP$ and chord $BP$, and draw arc $BP$ at right angles to $AP$. The fall of the Moon toward Earth in one second is therefore $BM$ or its equivalent $AM$. The angle $ABP$ is in reality so short that it may be taken as a straight line, being
![]() A diagram showing a circle representing Earth (circle A), another circle representing Moon (circle B), and a line segment AB connecting their centers. A perpendicular line BP is drawn from point B to line AP. The text describes how to calculate deviations based on these measurements.
Fig. 122. VERIFICATION OF THE LAW OF GRAVITATION
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one side of a right triangle whose hypotenuse is $AP$, and therefore a mean proportional between $AN$ and $AP$; that is,
$$AN = \frac{AB}{AP}$$
But $AB$, the distance traveled along the orbit in one second, is $\pi R/T$, where $T$ is the number of seconds in a month; and, as $AF = 2R$,
$$AN = \frac{\pi^2 R}{T^2}$$
To compare the fall of the Moon with that of the apple, we must have $T$ in seconds and $R$ in inches. Although Newton knew that $R$ was sixty times the Earth's radius, he did not know its exact value. The Earth's radius in inches, and so the computation failed to verify the law. This fact substan-
tially laid aside that part of his work until six years later, when a new determination of the size of the Earth was made by Picard (page 83). Newton then verified this value and found the Moon's deviation from the tangent to coincide with the fall of the apple within one ten-thousandth of an inch.
In Newton's day modern methods of mathematical analysis were unknown and he developed his theorems by the compara-
tively cumbersome processes of elementary Geometry, aided by his principles of "fluxions". Celestial Mechanics was brought to a high state of perfection in the latter half of the eighteenth century by the French mathematicians Laplace and Lagrange by applying to Newton's laws the methods of the calculus.
**Detection of Gravitation between Small Bodies** - A number of experiments have been performed whereby the attraction between two bodies of much smaller mass than that of the Moon or planets has been detected and measured. In 1740 Bouguer detected the deviation of the plumb-line caused by the attraction of Mount Chimborazo in South America, and in 1774 Maskelyne made a similar observation of greater accuracy at Mount Schehallien in Scotland, from which he deduced the mass of the Earth by comparing its attraction with that of the mountain. A more accurate experiment fulfilling the same purpose is that of the **torsion balance**: it was first performed by Lord Cavendish in 1798 and is known as the Cavendish experiment.
Cavendish's torsion balance is shown in Fig. 123. The balance consisted of a light rod about six feet long which was supported in a horizontal position by a slender wire, and which carried at each end a lead ball about two inches in diameter. Suspended at the same level with these balls and on opposite sides of the rod were two 12-inch balls, also of lead. Any rotation of the rod around its axis could therefore be observed by small telescopes directed to small mirrors attached to the ends of the rod. The whole apparatus was enclosed in a case to prevent disturbance by air currents.
![]() Cavendish's Torsion Balance.
Fig. 123. Cavendish's Torsion Balance. h, torsion rod hung by wire q; n, attracted balls hung from its ends; W, W', attracting masses. (From Jones' General Astronomy.)
Suppose the normal position of the rod was x, Fig. 124. With the large balls in the positions W1W1, their attraction, which was resisted only by the twist of the slender wire, brought the rod into position x1x1. The large balls were then turned to W2W2, and their attraction brought the rod into the line x2x2. The total angle moved through by the rod from position r to position s is observed through the telescope, is four times the angle through which it would be moved by the attraction alone, one small ball and one large one. This angle is proportional to the force of attraction and therefore to the resistance due to the torsion of the wire, and the factor of proportionality can be determined by observing the time of
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swing when the balance is disturbed slightly from its position of equilibrium and then released, the large balls $W$ being for this purpose removed. The stiffer the wire the more rapid are the vibrations of the balance.
Cavendish was able with this apparatus to detect and measure the attraction between the balls, but the accuracy of the measurements is impaired by the effect of air currents which could not be entirely eliminated by his enclosing case. The experiment has been repeated a number of times. In 1805 Boys used gold balls 5 mm. in diameter, attracted by lead balls 10 cm. in diameter. The rod of his torsion balance was a mirror 24 mm. long, which was suspended by an exceedingly fine fiber of spun quartz. The gold balls were suspended from the mirror at different levels so that the lead balls might not interfere with one another, and the balance was enclosed in an air-tight case in which the air was partially exhausted. Boys' results are probably the most accurate yet obtained.
**Determination of the Constant of Gravitation.—If we let $f$ represent the attraction between the large and the small ball, $d$ the distance between their centers, and $B$ and $b$ their respective masses, we have by the law of gravitation**
$$f = G \frac{Bb}{d^2}$$
or, solving for $G$,
$$G = \frac{fd^2}{Bb}$$
The masses $B$ and $b$ can be determined by weighing the balls, $d$ can be measured, and $f$ is determined by the experiment; hence the equation gives the numerical value of $G$, the constant of gravitation, which is believed to have the same value throughout-out the universe. The value determined by Boys is $6638 \cdot 10^{-11}$ C.G.S. units, which means that two spheres, each having a mass of one gram, if placed with their centers one centimeter apart, attract each other with a force of $6638 / 100,000,000,000$ dynes.
The dyne is the C.G.S. unit of force, and is defined as the force required to give a mass of one gram an acceleration of one centimeter per second per second. The weight of a body at the Earth's surface, which is the force of attraction between the Earth and that body, is approximately denoted by one dyne. It is about 0.6 cm., per second per second; hence, the weight of a body whose mass is one gram is 0.64 dynes.
The force of gravitation acting between bodies of ordinary size is exceedingly small when compared, for instance, with the attraction between two magnets. An illustration involving larger masses perhaps makes this clearer. Suppose two spheres of mass $10^8$ grams each (iron balls 100 feet in diameter would have about this mass), placed with their centers one kilometer ($10^8$ cm., 0.62 mile) apart. No matter what buildings or other obstacles intervene, the two spheres will attract each other, but the attraction will be only 66,580 dynes, a force equal to the weight of a body whose mass is 68 grams, or less than three ounces. Even if all friction and other forces did not affect their motion were eliminated, either ball could be kept from moving by a slight pressure of the finger. If perfectly free from other forces, the balls will begin to approach each other, but so slowly that a man must be required to detect the motion. At the end of an hour, each ball will have moved only about 4 cm., centimeters, and will then be moving at the rate of a fourth of a millimeter per second. This speed will be accelerated at a slowly increasing rate, and at the end of eighty-four hours the two balls will come together at a relative velocity of 2.1 cm/sec.
**Determination of the Mass of the Earth.—Let $E$ be the mass of the Earth, and $w$ the weight of the small ball used in the Cavendish experiment. The centers of the Earth and the ball are a distance $r$ apart, where $r$ is the radius of the Earth. The weight $w$ is simply the force of attraction between the ball and the Earth, and is found by multiplying the mass $b$ of $b$
![]() A diagram showing two spheres attracting each other through space.
Fig. 124.-Method of Using Torsion Balance (From Jones' General Astronomy)
out the universe. The value determined by Boys is $6638 \cdot 10^{-11}$ C.G.S. units, which means that two spheres, each having a mass of one gram, if placed with their centers one centimeter apart, attract each other with a force of $6638 / 100,000,000,000$ dynes.
The dyne is the C.G.S. unit of force, and is defined as the force required to give a mass of one gram an acceleration of one centimeter per second per second. The weight of a body at the Earth's surface, which is the force of attraction between the Earth and that body, is approximately denoted by one dyne. It is about 0.6 cm., per second per second; hence, the weight of a body whose mass is one gram is 0.64 dynes.
The force of gravitation acting between bodies of ordinary size is exceedingly small when compared, for instance, with the attraction between two magnets. An illustration involving larger masses perhaps makes this clearer. Suppose two spheres of mass $10^8$ grams each (iron balls 100 feet in diameter would have about this mass), placed with their centers one kilometer ($10^8$ cm., 0.62 mile) apart. No matter what buildings or other obstacles intervene, the two spheres will attract each other, but the attraction will be only 66,580 dynes, a force equal to the weight of a body whose mass is 68 grams, or less than three ounces. Even if all friction and other forces did not affect their motion were eliminated, either ball could be kept from moving by a slight pressure of the finger. If perfectly free from other forces, the balls will begin to approach each other, but so slowly that a man must be required to detect the motion. At the end of an hour, each ball will have moved only about 4 cm., centimeters, and will then be moving at the rate of a fourth of a millimeter per second. This speed will be accelerated at a slowly increasing rate, and at the end of eighty-four hours the two balls will come together at a relative velocity of 2.1 cm/sec.
**Determination of the Mass of the Earth.—Let $E$ be the mass of the Earth**, and **$w$** **the weight** **of** **the small ball used in** **the Cavendish experiment** **:** **The centers** **of** **the Earth** **and** **the ball are** **a distance** **\( r \)** **apart,** **where \( r \)** **is** **the radius** **of** **the Earth** **:** **The weight \( w \)** **is simply** **the force** **of attraction** **between** **the ball** **and** **the Earth,** **and** **is found by multiplying** **the mass \( b \)** **of \( b \)**
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ball by the quantity $g$; also, it is given by the law of gravita-
tion,
$$w = G \frac{E b}{r^2}$$
Solving for $E$,
$$E = w r^2 / G b,$$
or, substituting the value of $G$ found in the preceding section,
$$E = w r^2 B / j d^2.$$
From this equation, expressing the forces $w$ and $f$ in dynes,
the mass $B$ in grams, and the distances $r$ and $d$ in centimeters,
it is possible to compute $E$ in grams. The result is $7 \cdot 10^{19}$ grams, or $6 \cdot 10^{18}$ tons.
The Earth's mass and that of the vastly larger Sun are so great that, despite the smallness of $G$, the force of their mutual attraction is sufficiently strong to pull asunder a solid steel rod nearly 3,000 miles in diameter; and yet, because of the Earth's great inertia, this force causes the Earth to deviate from a straight line only about one-ninth of an inch while traveling eighteen miles along its orbit.
Calculation of the Gravitational Gravity of a Heavenly Body.
Since a sphere attracts an external body as if its mass were all situated at its center, if $m$ represent the mass of the Earth, $r$ its radius, and $g$ its attraction for unit mass at its surface, the law of gravitation gives
$$g = K m/r^2$$
where $K$ is a factor of proportionality. Similarly, if $m'$, $r'$, and $g'$ represent the corresponding quantities for any other sphere,
$$g' = K m'/r'^2.$$
Hence, dividing the second equation by the first,
$$g'/g = m'r'/mr^2.$$
The quantity on the left has already been defined as the
superficial gravity of the sphere (page 118). It may readily be computed by the last equation when the body's mass and radius are known in terms of the Earth's.
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215
The Problem of Two Bodies.—The following problem has been completely solved, first by Newton and later by other mathematicians who used modern and more powerful methods:
Given the masses of two particles or spheres which are subject to their mutual gravitational attraction and to no other force; given also their positions and velocities at any moment; to determine the orbits which they will follow and their positions at any other time.
Strictly speaking, the conditions of this problem are never fulfilled in nature, since every particle is attracted by all others instead of by a single one; but they are fulfilled approximately in the case of each planet and the Sun, since the attraction of the latter vastly preponderates over that of the other planets and the stars; and probably still more nearly in the cases of some double stars. The details of the solution of the problem cannot be given here, but some of the important results may be mentioned as follows:
1. The position of the center of mass (page 118) of the two bodies is unaffected by their attraction, and it remains at rest or in uniform rectilinear motion.
2. The two bodies will follow orbits around their common center of mass which are similar in form but of size inversely proportional to their masses, the more massive body moving
![]() Fig. 125. ORBITS OF SERIES AND ITS COMPANION
Orbits of Series and Its Companion
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ASTRONOMY
in the smaller orbit. The orbit of each body relative to the other will be a curve similar to its orbit relative to the center of mass, but of a size proportional to the sum of the masses of the two bodies.
Observations of the right ascension and declination of Sirius made with meridian circles before 1844 showed that this brightest of stars was moving slowly in a little orbit upon the celestial sphere, attracted by Bessel to the attraction of an unseen companion. In 1862, during a stay at the north-ten-thousandth objective of the Deutsches Observatorium, the companion was seen by Clark, and it has since been observed in different parts of the sky. It was discovered fifty-two years to circumvolve. Fig. 125 shows the orbits of the two stars, the fainter of which is excelled about four times by the brighter. \(A\) is the center of mass, and the continuous curves are the absolute orbits of the two stars. The small ellipse belonging to the larger star, \(A\), is a part of the small star relative to the bright one, as determined by micrometer observations. The inclination angle, is shown by the dotted ellipse.
3. The orbits of the two bodies will not necessarily be ellipses, but must in every case be one of the species of curves known as conic sections, of which the ellipse is a member; and the center of mass will be at the focus of the conic.
**THE LAW OF GRAVITATION**
passing at right angles to the axis, as at \(N M\), the curve of intersection is a circle of a size depending on the angle of the cone and the distance of the cutting plane from the vertex. If the plane cuts through all three foci, then this curve becomes an ellipse, as at \(F E\); and if the eccentricity of the ellipse increases until the plane makes an angle with the axis equal to half the vertical angle of the cone as \(D F\), while still passing through all three foci, then it becomes a parabola.
If we pass through only two foci inclined, as at \(F D\), then we get a curve above the vertex as well as below, and the curve becomes a hyperbola. The circle and parabola are thus seen to be limiting cases of the ellipse. Parabolas as well as ellipses and hyperbolas have various shapes as well as sizes. If the cutting plane passes through the vertex of the cone and ellipse each degenerate to a point, while the parabola narrows to a line and the hyperbola opens out into two infinite lines.
The conic sections were worked out from the above point of view by Apollonius of Perga about 230 b.c. In modern analytic geometry the curves are treated by means of their equations in rectangular coordinates. Their properties are then deduced.
The ellipse is a closed curve, returning into itself as does the circle. The parabola does not return into itself, but extends to infinity. A body moving around the Sun on a path such as \(O C\) would always approach nearer and nearer to it without ever reaching it. The parabola may be regarded as an ellipse with its second focus infinitely removed from the first; the distance between its foci is therefore infinite and so is its semi-axis \(x\), and so is every triangle formed by any point on its surface and its two infinite foci. The second focus is behind the first, and the eccentricity is greater than 1. In the case of an attractive force varying inversely as the square of the distance, the body will move on the branch of the hyperbola that is concave to the center of attraction; if the force were repulsive, it would move on the convex branch. The ellipse is the locus of points whose distances from two foci differ by a constant amount; this constant is called \(S\). The difference of whose distances from the foci is constant. In Fig. 127 are shown an ellipse, a parabola, and a hyperbola which have a common focus at \(S\).
**Speed in the Relative Orbit.**—In solution of the two-body problem it is shown that speed of one of the bodies
![]() Fig. 126. The Conic Sections (From Young's Manual of Astronomy)
The conic sections are so called because each may be defined as the intersection of a plane with a right circular cone, that is, with surfaces resulting from different positions of cutting planes. A right circular cone (Fig. 126) is defined as the surface generated by a straight line which passes always through a fixed point \(C\) called **the vertex**, while one point of this line travels around a circle whose center is at \(C\). This section is perpendicular drawn from \(V\) to plane of circle. This perpendiculor, \(V C\), is called **the axis** or **the cone**. If some surface is cut by plane
![]() Fig. 127. Conical Cones.
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in its orbit around the other is always the same at a given distance $r$, irrespective of the direction of one body from the other, and is given by the formula
$$V^2 = k^2(m_1 + m_2)\left(\frac{2}{r} - \frac{1}{a}\right)$$
where $a$ is the semi-major axis of the orbit and $k$ a constant. For a given pair of bodies, $m_1$ and $m_2$ are also constant, and we may write
$$V^2 = \mu(2/r - 1/a).$$
If the orbit be circular, $a = r$ and $V^2 = \mu/r$; if it be parabolic, $a = \infty$ and $V^2 = 2\mu/r$. The velocity of a body in a parabola at a given distance from the Sun is therefore $\sqrt{2}$ times that of a body moving in a circle at the same distance. Meteors are observed to enter the Earth's atmosphere at an average speed of about twenty-six miles a second, which is $\sqrt{5}$ times eighteen miles per second, the orbital speed of the Earth. It is therefore inferred that these meteors travel on orbits that are either parabolas or very long ellipses.
Representing the velocity $\sqrt{2u/r}$, which is called the **parabolic velocity**, by $U$, the equation for the velocity in an orbit of semi-major axis $a$ becomes
$$V^2 = U^2 - \mu/a,$$
from which
$$a = \frac{\mu}{U^2 - V^2}.$$
Therefore, if a planet were launched in an orbit at a given distance $r$ from the Sun, the length of the major axis of the orbit would be the greater the greater the speed of projection, for an increase in $V$ decreases the denominator and hence increases $a$. If $V = U$, the orbit is of course parabolic; if $V < U$, it is elliptic, and if $V > U$ it is hyperbolic. Fig. 128 shows a number of orbits described by planets projected from the same point in the same direction, but with different speeds. If projected in different directions with the same speed, the bodies would move in orbits of the same length but of different eccentricities and lines of apsides, as shown in Fig. 129.
**Projectiles Near the Earth; Escape of Atmospheres.**
When a ball is tossed into the air, it forms with the Earth a two-body system, and, if there were no friction, the ball would move along an ellipse having the center of the Earth as its distant focus and the highest point to which it is thrown as its apogee. Since the ball is stopped by the rarefied atmosphere after describing only a short arc, the orbit is sensibly a parabola. If the speed of projection is increased, the major axis of the orbit is also increased, and the ball rises higher. The greatest speed yet attained by a projectile from a gun (the German "Big Bertha") is less than a mile a second, and the greatest height of projection about twenty-four miles—very small in comparison with the dimensions of the Earth. If the speed were increased until it equaled the parabolic velocity due to the Earth's attraction, the projectile would move on a parabola and depart an infinite distance from the Earth, never to return. The parabolic velocity at the surface of a planet is therefore sometimes called the **velocity of escape**. It is equal to $\sqrt{2g\rho}$ where $\rho$ is the radius of the planet and $g$ is to be computed
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![]() A diagram showing three orbits projected in different directions with different speeds.
Fig. 128. Orbits of Bodies Projected in the Same Direction with Different Speeds
![]() A diagram showing three orbits projected in different directions with different speeds.
Fig. 129. Orbits of Bodies Projected with Same Speed in Different Directions
| Body |
Velocity of Escape |
| Earth. |
11,180 miles, or
6.95 miles, per sec. |
| Moon. |
3,506 *
61,700 *
643.30 * |
| Sun. |
61,700 *
643.30 * |
| Mercury. |
3,106 *
6.51 * |
| Venus. |
10,475 *
6.32 * |
| Mars. |
67,100 *
8.58 * |
| Jupiter. |
67,100 *
8.58 * |
| Saturn. |
37,850 *
73.53 * |
| Uranus. |
25,160 *
14.40 * |
| Neptune. |
20,820 *
12.95 * |
We are assured by physicists that any body of gas, such as the atmosphere of a planet, consists of molecules which are in rapid motion, each molecule moving in a straight line until it collides with another molecule or the wall of the containing vessel. The mean velocity of the molecules depends on the temperature and pressure of the gas, but that of any individual may be indefinitely increased or lessened by collision with others. A molecule near the upper limit of an atmosphere may thus attain a velocity exceeding the velocity of escape, and it is possible that the absence of an atmosphere on the Moon, and on other bodies for which the velocity of escape is small, may be explained on this basis.
**Determination of the Mass of a Heavenly Body.—** In the mathematical treatment of the problem of two bodies, it is shown that
$$m_1 + m_2 = \frac{a^3}{P^2},$$
where the masses $m_1$ and $m_2$ are expressed in terms of the Sun's mass, the mean distance $a$ in astronomic units, and the period $P$ in years. This formula gives at once the combined masses of a planet and its satellite, or of the components of a double star, when the period and mean distance have been determined by observation. As the planets Mercury and Venus have no known satellites, their masses can be determined only by their attractions for other planets or for occasional comets that pass near them, and are not very accurately known.
The mass of the Sun is found in terms of that of the Earth-Moon system by a simple application of Kepler's third law as modified by Newton; for if $M$ be the sum of the masses of the Earth and Sun, $m$ one of those of the Earth and Moon, $A$ and $a$ the semi-major axes of the orbits of the Earth and Moon, and $P$ and $\rho$ the lengths of the sidereal year and the sidereal month, then
$$M : m :: \frac{A^3}{P^2} : \frac{a^3}{\rho^2}.$$
**The Problem of Three Bodies.—** The problem of the motion of three mutually attracting bodies is just as determinate as that of two—that is, a given set of initial conditions leads as certainly to a definite result; but the problem is of such difficulty and complexity that only in special cases is it solved by most powerful of modern analysis capable of producing formulae by which this result can be computed. Of these special cases two are Lagrange; in one, the three bodies remain always in a rotating straight line; and in the other they remain at the vertices of an equilateral triangle.
If an infinitesimal body were launched with the proper velocity at a point about 61,700 miles from the Earth on the side directly opposite the Sun, and then not disturbed by any outside force, it would remain in that relative position and lie like the Earth revolve around the Sun in a period of about eight hours.
This body would be attracted equally by all three planets disturbed ever so slightly—and it certainly would be disturbed by the attraction of both other planets and the Moon—it would depart from the position indefinitely. The dim lights known as asteroids are probably composed generally by minute particles to sunlight reflected from tiny bodies which, moving through interplanetary space, are caught temporarily at the anti-solar point by a sort of "dynamic whirlpool."
There are many examples to illustrate this fact, and examples are found in the motion of certain of the asteroids known as the Trojan group (page 347). Each of these asteroids revolves at the mean distance of Jupiter from the Sun and at the same time oscillates slowly about a point equally distant from the Sun and Jupiter.
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