Encyclopædia Britannica, Ninth Edition/Gravitation/III. Universal Gravitation.
III. Universal Gravitation.
§ 9. Law of Gravitation between Two Masses.—Investigations of the motions of the planets have conducted us to the conclusion that each planet is attracted towards the sun by a force which varies according to the inverse square of the distance. We likewise see that each of the satellites appears to be attracted towards the correspond ing primary planet by a force which obeys the same law. We are therefore tempted to generalize these results into the proposition that any two masses in the universe attract each other with a force u-hich varies according to the inverse square of the distance. Obser vations of the most widely different character have combined to show us that this law, which was discovered by Sir Isaac Newton, is true. It is called the law of Gravitation. We suppose that the distance between the two bodies is so exceed ingly great compared with their dimensions that we may practically consider each of the bodies as a particle. Let m, m be the masses of the two bodies, and let r be the dis tance. The force with which m attracts m is equal in magnitude though opposite in direction to the force with which m attracts m. The reader may perhaps feel some difficulty at first in admitting the truth of this statement. We speak so often of the effects which the attraction of the sun produces on the planets that it may seem strange to hear that each planet reacts on the sun with a force pre cisely equal and opposite to the force with which the sun acts upon the planets. We can, however, by the aid of a simple illustration, show that such is really the case. Siippose the sun and the earth to be placed at rest in space, and abandoned to the influence of their mutual gravitation. It is evident that the two bodies would begin moving towards each other, and would after a certain time come into collision. If, however, the earth and the sun had been separated by a rigid rod, it would then be impossible for the two bodies to move closer to each other, so we must consider whether any other motion could be produced by their mutual attraction. As the two bodies were initially at rest, it is clear that there will be no tendency of the rod to move out of the line in which it was originally placed", and consequently if the rod begin to move at all it must move in that line, and carry with it the earth at one end, and the sun at the other. As, however, the earth and the sun would remain separated at a constant distance, the statical energy due to their separation would remain constant, and therefore if they began to move we should have the kinetic energy of their motion created out of no thing, which is now well known to be impossible. It therefore follows that, under the circumstances we have assumed, the rod would remain for ever at rest. But what are the forces which act iipon the rod ? At one end of the rod the earth presses upon it with a force which is equal to the attraction of the sun upon the earth. At the other end the sun presses upon the rod with a force which is equal to the attraction of the earth upon the sun. But since the rod remains at rest, these two forces must be equal and opposite, and hence the force with which the sun attracts the earth must be equal and opposite to the force with which the earth attracts the sun. If we express the gravitation of two masses in the form f(m, m )-=-r 2 , then m and m must enter symmetrically into the expression, for if the two masses were interchanged the gravitation must not be altered. We should here also advert to a circumstance connected with gravitation which is of the very highest importance. Suppose we take two definite masses (for simplicity, two pounds) separated at a definite distance (for simplicity, one foot), then the gravitation of these two masses to each other is a certain definite force (which we shall subsequently calculate). What we want here to lay special stress upon is that, so far as we know at present, this force appears to be the same whatever be the material of which the two masses are composed. Thus two pounds of iron at a distance of one foot attract each other with the same force as a pound of iron would attract a pound of lead at the same distance. The signifi cance, or perhaps it should be said the vast importance, of this statement is apt to be lost sight of from a somewhat peculiar cause. It must never be forgotten that, when it is asserted that two masses are equal to each other, what is really meant is that, if equal forces were to act upon each of the masses for the same time, the masses would receive the same velocity. As this test of the equality of masses is not practically convenient, the weighing scales have come into use for the purpose ; and though it appears to be true that when two masses have equal " sveights," as tested by the scales, then the masses are themselves equal, yet this is so far from being an obvious or necessary truth that it really is the most remarkable phenomenon connected with gravitation. The expression for the gravitation of two attracting masses must therefore depend solely upon their masses, upon their distance, and upon some specific con stant which is characteristic of the intensity of gravitation. Experiment shows that the gravitation" of a body towards the earth is directly proportional to its mass, and hence we see that the expression must be proportional to m, and as/ (m, m ) must be unaltered by the interchange of m and m , it appears finally that the gravitation of the two masses is emm -=-r 2 , whence is a numerical constant which is equal to the gravitation of two units of mass placed at the unit of distance. To form a de finite conception of the intensity of this force, we take some specific instances. It can be shown that two masses A and B, each contain ing 415,000 tons of matter, and situated at a distance of one statute mile apart, will attract each other with a force of one pound. If the masses of A and B remain the same, and if the distance between them be increased to two miles, then the intensity of the force with which the two masses gravitate together is reduced to one quarter of a pound. If either of the masses were doubled, the distance being unaltered, then the force would be doubled. If both the masses were doubled, then the force would be quadrupled.
§ 10. Motion of a Planet round the Sun.—The effect of gravitation when the bodies are in actual motion must next receive our attention. It may so happen that in consequence of the attrac tion of gravitation one of the bodies will actually describe a circle around the other, so that, notwithstanding the etfect of the attrac tion, the distance between the two bodies remains constant. We shall first explain, by elementary considerations, how it is possible for a planet to continue to revolve in a circular or nearly circular orbit about the sun in its centre ; and then we shall proceed to the more exact consideration of the form of the orbit, and the laws according to which that orbit is described.
Let S represent the sun (fig. 6), and let T be the initial position of the planet. If the planet be simply released, it will immediately begin to fall along the line TS into the sun. If, on the other Land, the planet were initially projected along the line TZ perpendicular to TS, the attraction of the sun at S will deflect the planet from the line TZ which it would otherwise have followed, and compel the planet to move in a curved line. The particular form of curve which the planet will describe depends upon the initial velocity. With a small initial velocity the deflecting power of the sun will have a more speedy effect than is possible when the initial velocity is considerable. The rapidly curving path TX will therefore correspond to a small initial velocity, while the flatter curve TV may be the orbit when the initial velocity is considerable. As the movement proceeds, the velocity of the planet will generally alter. If the planet were moving along the curve TY, it is at every instant after leaving T going farther away from the sun. It is manifest that the planet is thus going against the sun s attraction, and therefore its velocity must be diminishing. On the other hand, when the planet is moving along the curve TX, it is constantly getting nearer the sun, and the effect of the sun s attraction is to increase the velocity. It is therefore plain that for a path somewhere between TX and TY the velocity of the planet must be unaltered by the sun s attraction. With centre S and radius ST describe a circle, and take a point P on that circle exceedingly near to T. With a certain initial velocity it is possible to project the planet so that it shidl describe the arc TP. The attraction of the sun always acts along the radius, and hence in describing the arc TP the planet has at every instant been moving perpendicularly to the sun s attraction. It is manifest that under such circumstances the sun s attraction cannot have altered the velocity, for it would be impossible to give a reason for the velocity having been accelerated which could not be rebutted by an equally valid reason for the velocity having been retarded. We thus see that the planet reaches P with an unchanged velocity, and at that point the direction of the motion is perpendicular to the radius. It is therefore clear that after passing P the planet will again desciibe a small portion of the circle, which will again be followed by another, and so on, i.e. , the planet will continue to move in a circular orbit. We have therefore shown that, if a planet were originally projected with a certain specific velocity in a direction at right angles to the radius connecting the planet and the sun, then the planet would continue for ever to describe a circle around the sun.
§ 11. On the Elliptic Motion of the Planets.—The laws of the motions of the planets were discovered by Kepler by means of cal culations founded upon observations. They may be thus stated : 1. Each planet moves in an ellipse in one focus of which the sun is situated. 2. The radius vector drawn from the sun to the planet sweeps over equal areas in equal times. 3. The squares of the periodic times of the motions of the planets round the sun are in the same ratio as the cubes of their mean distances. These three laws form the foundations of that branch of astronomy which is called Physical Astronomy, and they are generally known as Kepler s Laws. The sun and the planets are all very nearly spherical, and for the present, whenever we speak of the motions of the sun or the planets, we are to understand the motions of the centres of the corresponding spheres. Indeed the diameters of these spheres are so small in comparison with the distances at which they are separated from each other, that we may generally regard them as mere physical points. Thus though the diameter of the sun is ten times greater than the diameter of the greatest planet, Jupiter, yet the sun s dia meter is only the forty-second part of its distance from the nearest planet, Mercury, and it is less than the three-thousandth part of its distance from the outermost planet, Neptune. The orbits of the planets are so little eccentric that at a first glance the majority of them appear to be circular. Among the larger planets the orbits of Mercury and of Mars have the greatest eccen tricity, being about and ^- respectively. Next to these comes the orbit of Saturn, which has an eccentricity of T V Next in order come Jupiter, Uranus, the Earth, Neptune, and Venus, with eccentricities f T?T) TT> T?T> TTS> TIB" respectively. The orbits of some of the minor planets are, however, much more eccentric.
Fig. 7.
§ 12. The Hodograph.—In discussing the actual motion of a planet around the sun, it is very convenient to introduce the elegant con ception of the hodograph. The use of this curve is originally due to Bradley, but for its practical development we are principally indebted to Sir "W. 11. Hamilton. Let AB (fig. 7) be a portion of the path of a particle P acted upon by any forces. From any point draw radii veetores OR 1; OR 2 , OR 3 , OR 4 (fig. 8), parallel to the tan gents to the curve AB drawn at the points, P u P 2 , P 3 , P 4 , and equal to the velocities at those points. Then the curve R 1 R 2 R 3 R 4 Is the hodograph of the orbit of P. To each position P of the Q^!____ particle in the path there is a corresponding point R in the hodograph ; and simultaneously with the motion of P in its orbit we have the motion of R in the hodograph.
The utility of the hodograph depends upon the theorem that the force which acts upon the particle is at any time equal and parallel to the velocity of the corresponding point in the hodograph. This theorem is thus proved. Let P a and P 3 , and therefore R 2 and R 3 , be very close together. The velocity imparted to the particle in passing from P 3 to P 3 in the small time St must, when compounded with the velocity OR. 2 , produce the velocity OR 3 . This increment of velocity must therefore bo equal and parallel to R 2 R 3 . If there fore v denote the velocity of R, we have v$t for the velocity ac quired in the time St. The force acting on P must therefore, by the second law of motion, be equal to v and parallel to the line R a R 3 , which ultimately coincides with the tangent. The theorem is there fore proved.
Fig. 9.
Kepler s second law states that in the movement of a planet around the sun the radius vector from the sun to the planet describes equal areas in equal times. It can be shown that when this is the casr the force which acts upon the planet must necessarily be directed towards the sun. Let the planet be supposed to move from P to P (fig. 9) in the time t with the velocity v. Then, since equal areas are described in equal times, we must have the area OPP equal to hSt, where h is a constant which represents the area described in the unit of time. Let fall OT perpendicular upon the tangent at P, then. irOT = 2A. "We therefore infer that the velocity of a planet is inversely proportional to the perpendicular 4 et fall from the centre of the sun upon the tangent to the orbit. Produce OT to Q, so that OT x OQ is constant ; the radius vector OQ will therefore be pro- portional to the velocity ; and as OQ is perpendicular to the tangent at P, it follows that the locus of Q must be simply the hodograph turned round through 90. Draw two consecutive tangents to tha orbit, and let Q x and Q 2 be the corresponding points (fig. 10). Then since OTj x OQ 1 = OT !1 x OQ 2 , the quadrilateral T^O^T., is inscrib- able in a circle, and therefore the angles OQjQa and OTjT x are equal ; whence it is easily seen that QjQ., is perpendicular to OP. It follows from the principle of the hodograph that the force on P must be directed along OP.
Fig. 10.
Let F (fig. 11) be the focus of the ellipse in which, according to Kepler s first law, the planet is moving, then, from a well-known property of the ellipse, the foot of the perpendicular FT let fall from F on the tangent drawn to the ellipse at the point P lies on the circle of which the diameter is the azis major of the ellipse. The line FT cuts the circle again at Q, and as FT x FQ is constant, the radius vector FQ must be con stantly proportional to the velocity, and there fore the hodograph will be the circle TAQ turned round through 90. "We have already shown that the tangent to the hodograph is parallel to the force. The line CQ must therefore be parallel to FP. If 6 be the angle PFC, then, in consequence of the law of equal description of areas in equal times, Wde + dt must be con stant, and therefore dQ + dt must vary inversely as FP 2 ; but the velocity of Q will be proportional to dfj + dt ; hence the velocity of Q, and therefore the force, will be inversely proportional to FP S . In this way it is shown that the planets are attracted by the sun with a force which varies inversely as the square of the distance.
§ 13. The Problem of Three Bodies.—We have pointed out that the motion of a planet is determined by the mutual attraction which ex ists between the planet and the sun. It follows, however, as a necessary consequence from the law of universal gravitation, that each planet is attracted not only by the sun but by every other planet, and indeed, so far as we know, by every other body in the universe. The effect of the attraction of the sun preponder ates so enormously over the other attractions that the obedience of the planets to Kepler s laws, which would be perfect were the sources of disturbance absent, is still so nearly perfect that the departure therefrom can only be perceived either by very accurate or by very long-continued observations. The refinements of modern observations have, however, brought to light a very large number of perturbations, as they are called, in the motions of the planets and their satellites, which are due to the interference of other bodies. The explanation of these different perturbations by the law of universal gravitation has proved in the great majority of cases triumphantly successful. In any problem where perturbations are involved, we have at least three bodies, viz., the principal body S, a body P which circu lates around S, and a disturbing body P . If we had only the two bodies S and P to consider, then P would describe around S a conic section, of which S was the focus, and the radius vector SP would sweep over equal areas in equal times. By the introduction of the third body P , which attracts both the former bodies S and P, the motion is deranged, the orbit which P describes is no longer a conic section, and its radius vector has ceased to describe equal areas in equal times. The general case of this problem, as we have here described it, is one of excessive complexity. It is, however, a fortu nate circumstance that, up to the present time, astronomers have had but little occasion to attack the problem in its general form. 1 In the solar system we have a great number cf different problems of perturbations to discuss, but there is one feature common to all these problems. This feature is that the perturbing force is very small in comparison with the primary force, bij which the motion of the disturbed body is controlled. In consequence of this peculiarity, the problems of perturbations in the solar system become greatly simplified. The circumstances of the orbits of the planets and their satellites are also happily such as to furnish additional facilities in the solution of the problems of perturba tions. The eccentricities of the orbits are small, and in nearly every case the inclinations are small also. The problems of perturba tions in the solar system are consequently adapted for the methods of successive approximation ; but we should perhaps warn the reader that, even with the assistance so fortunately rendered by the circumstances of the case, the problems are still among the most difficult to which analysis has ever been applied.
Let S, P, P (fig. 12) denote the three bodies of which the masses are M, m, TO . Let the dis tances SP, SP , PP be denoted by r, r , p respectively; then if e denotes, as before, the gravitation between two units of mass separated by the unit of distance, we have on S the forces e ^ and e ~ along SP and SP respectively ; on P the forces e M and e~ along PS and PP respectively ; on P the forces Mm and e- m along P S and P P respectively. r* p 2 In the problem of the perturbation of a planet which is moving around the sun, the absolute motion of the planet in space is not what we are concerned with ; what we do require is the relative motion of the planet with respect to the sun. Suppose that we apply to each unit of mass of the three bodies, forces equal to tmr- 3 parallel to PS and tm r -" 2 parallel to P S, then it is clear that, from the second law of motion, each unit of mass of the system will receive in a small time, so far as these forces are concerned, equal and parallel velocities, and therefore the rela tive motions of the three bodies will not be altered by the introduc tion of these, forces.
1 There can be little doubt that in the case of multiple stars the problem of three or more bodies would often have to be faced in all its ruggedness, but the observations which would be required are certainly wanting, and probably unattainable.
Rut these forces will, so far as the body S is concerned, amount to eMm -f- r" 2 and eMm -f-r 2 in the directions PS and P S respectively, and will therefore neutralize the attractions of P and P upon S. It follows that, so far as the relative motions only are concerned, we may consider the body S as fixed, while the body P is acted upon by the forces em(M + ?H) r 2 on PS tmm p^ on PI" finm r" 2 parallel to P S, with of course similar expressions for the forces on P . The problem of perturbation is therefore reduced to the determina tion of the motion of P in obedience to these forces, while S regarded as fixed. It is plain that, in order to find the motion of P, we should know the motion of P , which is itself disturbed by P ; but as the effect of the perturbation is small, we may, without appreciable error, use the place of P , derived on the supposition that the motion of P is undisturbed, for the purpose of calculating disturbances caused by P in the motion of P. We shall therefore suppose that the orbit of P is an ellipse which is described around the focus S in consequence of an attractive force e/M (M + m )-^-r 2 directed towards S. In computing the disturbed motion of P we transform the three given forces into three equivalent forces, which will be more con venient for our purpose. We take first a force T directed along the radius vector ; secondly, a force V perpendicular to the radius vector, and lying in the plane of the orbit ; and, thirdly, a force W, which is normal to the plane of the orbit. T is to be regarded as positive when it endeavours to increase the distance between P and S ; V is to be regarded as positive when it tends to move P to the same side of the line PS as that in which the direction of its motion tends to carry it ; and W is to be regarded as positive when it tends to raise P to the north of the plane of its orbit.
Fig. 13.
From P (fig. 13) let fall a perpendicular P Q upon the plane of the undisturbed orbit cf P, and let fall from Q a perpendicular QR upon the line SP. The force along PI" can be resolved into a com ponent parallel to QP and another parallel to QP ; and the latter can again be decomposed into components parallel to RQ and RP. Hence we have, as the equivalents to the force on PP , the following three forces which form parts of T, V, W respectively : o :1 P In a similar manner we find for the components of the force parallel to SP, the following three forces which form parts of T, Y, W respectively: SR emwi + -VT Hence we deduce the following expressions : T em(M + m) _ ^^ pT> _. f^ 1 r* p 3 r 3 W tmm Let the line SO, from which the longitudes are reckoned, be drawn in the plane of the undisturbed orbit of P. Let the longi tudes of P and P be I, I respectively, and let the angle P SQ or the latitude of P be denoted by b , then we have QP = r sin b ; SQ = ?- cos?/. KQ =SQ sin (l -l) = r cos b sin (V-l). SR = SQ cos (I -I)- r cos b cos (I 1 - 1). PR = r - r cos b cos (I 1 - 1). With these substitutions, we have for the values of T, Y, W, the following expressions : m e ?/i(M + m) , / 1 1 . T, if _i-^ m ^ r 1 = i -}- )ft)/l I p ^7 I / COb LUb o I 5 / - r 2 p 3 r*J p A // --IV cos& sin^-O- Vp r 3 } - ( ----^V sin V. It will be noticed that the first term of the expression for T con sists of the force which corresponds to the purely elliptic motion. The two remaining terms in T, as well as the whole of V and AV, depend upon the disturbing force, as is evident from the circum stance that they contain in as a factor, and would vanish if m were equal to zero. As b is small, we may neglect its squares and higher powers, so that cos b = l and sin b = b . When these substitutions are made, we see that the expressions for T and V are both independent of b ; and consequently we may, so far as these forces are concerned, consider the motions of the dis turbing and the disturbed body to take place in the same plane. The expression for W contains, however, the first power of the latitude of the disturbing body. This is of course connected with the circumstance that it is only in consequence of the disturbing force W that the disturbed body is induced to leave the plane of its undisturbed motion at all.
§ 14. Calculation of Disturbed Motion.—Observation has shown that, notwithstanding the perturbations, the orbit of each planet differs but little from a circle, of which the sun is the centre. It is further shown by observation that, though the rate at which a planet moves in its orbit is not quite constant, it is still very nearly so. We may make a similar statement with reference to the motion of the moon around the earth. The orbit of the moon is nearly a circle, of which the earth is the centre, and the velocity of the moon in its orbit is nearly constant. These features of the motions of the planets and the moon enable us to replace the more exact formulae by approximate expressions which are much more convenient, while still sufficiently correct. Let p, be the polar co-ordinates of a celestial body which moves nearly iniformly in a nearly circular orbit. The form of the orbit may be expressed by an equation of the type /( P ,0) = 0. It will, however, be more convenient to employ two equations, by means of which the coordinates are each expressed directly in terms of the time. We thus write two equations of the form and by elimination of t from these equations the ordinary equation in polar coordinates is ascertained. With reference to the forms of the functions / x and / 2 , we shall make an assumption. Letxi, Xa> &-> be arbitrary angles ; 1( ca, 2 , &c., arbitrary angu lar velocities ; a,, a 2 , &c., small arbitrary linear magnitudes ; and ?; 2 , r. 2 , &c., small numerical factors, a is an arbitrary linear magni tude, and w is an arbitrary angular velocity. We shall assume that p=^a + a L cos (<V + Xi)+ a 2 cos (^ + Xa) + > & c - = cat + TJ ! sin (u^ + Xj) + 1? 2 sin (ta.J, + Xz) +> & c - To justify our employment of these equations it would really be sufficient for us to state that, as a matter of fact, the motions of all the heavenly bodies which are at present under consideration are capable of being expressed in the forms we have written. It may, however, facilitate the reader in admitting the legitimacy of this assumption, if we point out how exceedingly plausible are the a priori arguments which can be adduced in its favour. The orbits of the celestial bodies are approximately circular, and consequently p must remain approximately constant. The value of p is, in fact, incessantly fluctuating between certain narrow limits which it does not transcend. Thus p is what is called a periodic function of the time. It is necessary that the mode in which the time t enters into the expression for p must fulfil the condition of confining p within narrow limits, notwithstanding the indefinitely great increase of which t is susceptible. It is obvious that this con dition is fulfilled in the form we have assumed for p. Under all circumstances / = !!>TiToT, &c. ; p = <jia l a. 2 , &c. ; and as a v a. 2 , &c., are small quantities it appears that p is neces sarily restricted to narrow limits. By similar reasoning we can justify the equation for 6, for we can show that the angular velocity of the celestial body to which it refers must be approximately uniform. By differentiation, de = o> + T^W! cos (oi-ft + Xi) + i?j*-2 COS Under all circumstances we must have ^ = > T T dt ^o+W^W: = <fco w e& " 1&>1 2 : and as rj^, ri. 2 , &c., are all small quantities, it is obvious that - i s ctt confined within narrow limits, notwithstanding the indefinite augmentation of the time. In addition to the reasons already adduced in justification of the expressions of p and 0, it is to be remarked that the number of dis posable constants a 1} a. 2 , &c., xi> Xa> & c -> ^u ^a; ^ c -> w i> u v ^- c> a, to is practically indefinite, and that consequently the equations can be compelled to exhibit faithfully the peculiarities of any approximately circular orbit described by a particle moving with approximate uniformity.
§ 15. Determination of the Forces.—The orbit which is described by a planet or other celestial body being given by the equations p a + 2a 1 cos ( = o>t + 2,ri 1 sin it is a determinate problem to ascertain the forces by which the motion of the planet is controlled. In making this calculation we shall assume that the squares and higher powers, and also the pro ducts of the small quantities a x , a. 2> &c., TJJ, 7j 2 , &c., may be dis carded. T is to be computed from the well-known formula where we have dt de__ dt dff 2 df 1 de* whence by substitution T - maa? - 2i( 1 c> 1 2 + 2Tj I rtcoo> 1 + a^) cos (co 1 < + Xi)- To compute V, the force perpendicular to the radius vector, we proceed as follows : p * = a? + 22<mj cos dO = ~dt whence we find %dO = 2 P dt " Differentiating we have Substituting this in the ordinary formula v _? d ^ ~ we have finally V= - 2w(2a j coco 1 -l-; 1 ffltf 1 2 ) sin Let us assume, for the sake of brevity, A! = aj + xi ; A 2 = o>. 2 t + Xa 5 &c - &c. &c. then the result to which we have been conducted maybe thus stated: If a body be moving in a nearly circular orbit under the infiv.sace of a radial force equal to - mrtoi 2 - i2F 1 cos A] and a force perpendicular to the radius vector equal to sin then the path which the body describes is defined by the equations r = a + 2 L cos X : ;
§ 16. Deductions from these Expressions.—We proceed to point out a few of the more remarkable deductions from this theorem. If as a first approximation we neglect entirely the small quantities 1} a. 2 , &c., 77 1( rt. 2 , &c., we have r = a; = wt; T=-maca"; V = 0.
In this case the orbit described by the body is a circle of which the radius is a. The angle made by the radius vector to the particle with a fixed axis is proportional to the time. It follows that the velocity with which the particle moves in its orbit is uni- form. Since the angle swept out by the particle in one unit of time is equal to <o, it follows that the angular velocity of the particle is equal to o> ; since the radius is a, the actual velocity with which the particle is moving in its orbit is u>a. In the case now under consideration the force V perpendicular to the radius vector is con stantly equal to zero. Hence the total force which acts upon the particle is always directed along the radius vector, and is equal to - maa>*. The sign - merely expresses that the force which acts upon the particle must be constantly directed towards the centre. Let p denote the periodic time of the motion of the particle in its orbit. Then we must find the angular velocity by dividing the angle 2ir described in the time p by the time p, whence 2ir - = 60. P By substituting this value for o> in the expression for T, we deduce This proves that the force must vary directly as the radius of the orbit of the particle and inversely as the square of the periodic time in which the orbit is described. Let us now consider the case in which a l and 7^ are retained, while the remaining quantities a z , 3 , &c., rj.,, 7j 3 , &c., are all equal to zero. The formula; then are r a + a 1 cos A x ; T = - wia&> 2 - ?naF x cos AJ ; 6 =ut + ri 1 sin AJ ; V= - maG 1 sin A : ; where aF 1 = re 1 (co 2 + a> 1 2 ) + 2r; 1 rcw 1 ; Gj = 2 1 a>co 1 + aijjo^ 2 . It is clear in the first place that the orbit is not a circle, for as AJ depends upon t, it will follow that cos A. x may vary between the limits + 1 and - 1 , so that the radius vector r may also vary between the extreme values a + a l and a - a^ As a 1 is extremely small, it appears that the orbit is still nearly circular with a radius a, but that the particle may sometimes be found at a distance a^ on the inside or outside of the circle. We shall similarly find that the angular velocity with which the radius vector sweeps round is not quite uniform. The average angular velocity is no doubt u, but the actual position of the radius vector differs by the quantity r} 1 sin Aj from what it would have been had its motion been uniform. As r)! sin AJ must vary between the limits + n l ftnd - rjj, it follows that the radius vector can never be at an angle greater than j] l from its mean place, i. e. , the place which it would have occupied had it continued to move uniformly. The orbit is completely defined by the two equations l sin (ea 1 t cos If from these two equations the time t could be eliminated, the result would be the equation in polar coordinates of the path which the particle described. Owing, however, to the fact that the quantity t occurs separately, and also under the form of a sine and cosine, this elimination would be transcendental. If, however, we take advantage of the smallness of / t and g v we can eliminate t with sufficient accuracy for all practical purposes. As is nearly equal to tat, we may assume for t as the first approximation the value 0-j-. If we consider the squares or higher powers of a v and rjj negligible, this value of t may be substi tuted for t under the sine and cosine, and we have This equation in general denotes a curve undulating about the circumference of the circle of which the radius is a. We must now briefly consider the case where the motion of the particle is not confined to a plane. Suppose a third axis OC be drawn through the point perpendicular to the plane which con tains the undisturbed orbit. Let z be the coordinate of the particle parallel to the line OC which is called the axis of z, while x, y denote as usual the coordinates referred to two other rectangular axes. We shall suppose that the motion of the particle is such that the coordinate z can be expressed by the equation 2 = Aj sin 0J + &2 sin 2 + , &c., where h lt Ti z , &c., are lines of constant length ; j8 I; &. 2 , &c. , are angles of the form p ] t + ^ l , ^ + 2 > &c - 5 Pi, Pz, &c -> L tu &c., are constants ; and t denotes the time. Let W denote the force which acts upon the particle P in a direction parallel to the axis of z. Then we have Differentiating the equation 2 = Aj sin #1 - we have _ = Ji, cos B, a but whence dz -,- ~~TJ ^ Pz ^ * 5 COS /Sj + ^g&jj COS Differentiating again d 2 ~ 4P~ ~ whence, finally, Pi ~ Sil1 02 - : &C , &c.
§ 17. The Motion of the Moon.—One of the most important problems to which we may apply the expressions to which we have been conducted is to an examination of the disturbances which the moon experiences in its motion round the earth. The moon would describe a purely elliptic motion around the earth in one of the foci were it not that the presence of the sun disturbs the motion and pro duces certain irregularities. Notwithstanding the vast mass of the sun, these disturbing causes still only slightly derange the moon s motion from what it would be were these disturbing causes absent. The reason of this is that the sun is about 400 times as far from the earth as the moon, and consequently the difference of the effects of the sun upon the earth and the moon is comparatively small. In applying the formula to the case of the moon we denote by S, P, P the earth, the moon, and the sun respectively ; M, m, m signify the masses of the earth, moon, and sun; r, r are the distances of the moon and the sun from the earth ; p is the distance from the moon to the sun ; Z, Z are the longitudes of the moon and the sun measured in the moon s orbit. T, V, W are the forces acting upon the moon, whereof T is along the radius vector, V is perpendicular to the radius vector, and W is perpendicular to the plane of the moon s orbit. Since r-^r 1 is very nearly equal to l-f-400, we may regard this frac tion as so small that its squares and higher powers are negligible. Hence, since cos & = whence or JL- * =3 ,cos(Z-Z )- p.i ? . d r i With these substitutions we have, after a few simple transformations, T= - - - + - - ,. ( 1 + 3 cos 2(Z- Z ) r 2 2 r 3 V w = 3 ??i?,V cog ^_ ^ gin ^ r 3
§ 18. The Variation.—The inequality in the motion of the moon which is known as the variation is independent both of the eccen tricity of the orbit of the moon and of that of the earth. As we shall at present only discuss irregularities in longitude and radius vector, and as we shall neglect small quantities of an order higher than the second, we may assume that the plane of the orbit of the sun coincides with the plane of the orbit of the moon. The radius vector and the longitude may be expressed by the formula? r = a + a/j cos 1 ; l = (at + g^ sin A r We shall assume that 2(Z -Z) = A l5 and since we have + 2 but from 15, n being the mean motion of the moon, T = - mifia - maF x cos A x ; whence, by identifying the two expressions we have from the portions independent of the time, m) _ 1 em a ~" ~~ This is a very important formula, inasmuch as it gives the re lation between the mean motion n and the mean distance a in the disturbed orbit. Suppose that there were no disturbing influence, and that a satellite moved uniformly around the earth in a circular orbit of radius a with a mean motion n , then we have
[ equation.]
Assuming also that the earth moves uniformly round the sun in a circular orbit of radius a with a mean motion n , then or, since M is negligible in comparison with m,
em = n 2 a 3 . With these substitutions, and making? =?i 2 -j- 2, we have finally
o- *a A —w V 2 J
If we identify the coefficients of cos in the two expressions for T, we have
), 3 emm a
whence, by substitution and by neglecting small quantities,
We have also for V the two expressions
3 emm r . -2-73- sm and - maG l sin A:;
whence by identifying the coefficients
where n^ denotes the rate at which A: alters.
Solving the two equations, we deduce the following expressions:
= _ 3w 7i%! + 2?t 3 . ff = 3w Ti 2 ^^ 2 4 2w_ 1 _+_3 2 ) 2 7i"i 3 -77 2 7l 1 2 ~ "7l 1 2 (?l 1 ^ n 2 )~
We can now calculate the value of /j and q^ numerically. For
i _ 9 - 9 365-256 - 27 "322 - . .. . - 2 ~ - 2 =1 8504 365-256 1 178-72 /,=
-0-007204 71
and also
from these we deduce
The results at which we have arrived may be thus summarily stated, using, however, a more accurate value of g 1 than that which is formed by this method:
If we suppose the orbit of the moon to coincide with the plane of the ecliptic, and if we neglect the ellipticity of the orbit of the earth around the sun and that of the moon around the earth; if n denotes the mean motion of the earth around the sun, and n the mean motion of the moon around the earth; and, finally, if n and a be connected by the equation
7i 2 3 = 0-9972
then we have for the motion of the moon the equations
r = af 1-0-007204 cos (2nt-2n f) " sin (2nt-2n't).
We thus see that the motion of the moon, on the hypothesis which we have assumed, is different from a uniform circular motion. The distance from the moon to the earth is sometimes 1-1 39th part greater or less than its mean value. And the longitude of the moon is sometimes 39 30" in advance of or behind what it would be on the supposition that the moon was moving uniformly.
When the distance is the greatest, we have cos (2nt 2717)= -1; whence n, ~,, 2nt-2nt = ir or 3?r;
and it follows that the distance of the moon from the earth is great est at quadratures. [1]When the distance is least, then
cos (2nt-2n't)= +1, whence 2nt-2n't = or 2-ir;
consequently the moon is nearest to the earth at syzygy. It thus appears that the orbit of the moon as modified by the disturbing influence of the variation resembles an oval of which the earth is the centre, and of which the minor axis is constantly in the line of syzygies.
The mean longitude of the moon and the true longitude coincide when
This condition is fulfilled both at syzygy and quadrature; conse quently the mean place of the moon and its true place coincidewhen the moon is either in syzygy or in quadrature. The true place of the moon is at its greatest distance in advance of the mean place when
sin (2nt- 2n't) = l.
This condition is fulfilled at the middle points of the first and third quadrants, while at the middle points of the second and fourth quadrants
sin (2nt - 2n't) = - 1;
and therefore the moon is behind its mean place in the second and fourth quadrants.
After new moon, the distance between the moon and the earth gradually increases, and the apparent velocity of the moon also in creases until, when the moon is three or four days old, it has advanced 39 beyond its mean place; the velocity then begins to diminish, though the distance goes on increasing, until at first quarter the distance has attained a maximum. After first quarter the distance diminishes, and the moon falls behind its mean place, the maximum distance of 39 behind the mean place being reached about 11 days after new moon. At full moon the distance has become a minimum, and the mean place and true place coincide. At 18 days the true place has again gained 39 on the mean place, but the distance increases, and at third quarter the distance is again a maximum, and the true and the mean place coincide. After passing third quarter the distance diminishes, and the true place falls behind the mean place, the difference attaining a maximum on the 26th day, after which the true place gains on the mean place,. with which it coincides at new moon, when also the distance is again a minimum.
Since the amount of this irregularity in longitude is so consider able, being in fact larger than the diameter of the moon itself, it is very appreciable even in comparatively coarse observations. It was discovered by observation by Tycho Brahe, by whom it was named the variation.
It would lead us too far to endeavour to trace out any of the other irregularities by which the motion of the moon is deranged. We have taken the variation merely as an illustration of one of the numerous corrections which the law of gravitation has explained. The accordance which subsists between the values of these corrections as computed by theory and as determined by observation affords the most conclusive evidence of the truth of the law of universal gravitation.(r. s. b.)
- ↑ 1 When the approximation is carried sufficiently far it is found that the coefficient of the variation is 39′ 30″.