Encyclopædia Britannica, Ninth Edition/Tides/Chapter 9
IX. Tidal Friction.
§ 46. General Explanation.
General explanation of tidal friction.The investigation of the tides of a viscous sphere has led us to the consideration of a frictionally retarded tide. The effects of tidal friction are of such general interest that we give a sketch of the principal results without the aid of mathematical symbols. In fig. 8 the paper is supposed to be the plane of the orbit of a satellite M revolving in the direction of the arrow about the planet C, which rotates in the direction of the arrow about an axis perpendicular to the paper. The rotation of the planet is supposed to be more rapid than that of the satellite, so that the day is shorter than the month. Let us suppose that the planet is either entirely fluid, or has an ocean of such depth that it is high water under or nearly under the satellite. When there is no friction, with the satellite at TO, the planet is elongated into the ellipsoidal shape shown, cutting the mean sphere, which is dotted. But, when there is friction in the fluid motion, the tide is retarded, and high tide occurs after the satellite has passed the meridian. Then, if we keep the same figure to represent the tidal elongation, the satellite must be at m, instead of at TO. If we number the four quadrants as shown, the satellite must be in quadrant 1. The protuberance P is nearer to the satellite than P, and the deficiency Q is further away than the deficiency Q’. Hence the resultant action of the planet on the satellite must be in some such direction as MN. The action of the satellite on the planet is equal and opposite, and the force in NM, not being through the planet's centre, must produce a retarding couple on the planet's rotation, the magnitude of which depends on the length of the arm CN. This tidal frictional couple varies as the height of the tide, and also depends on the sa tellite's distance; its in tensity in fact varies as Planet's rotation retarded, the square of the tidegenerating force, and therefore as the inverse sixth power of the satellite's distance. Thus tidal friction must retard the planetary rotation. Let us now con sider its effect on the satellite. If the force acting on M be resolved along and perpendicular to the direction CM, the perpendicular component tends to accelerate the satellite's velocity. It alone would carry the satellite further from C than it would be dragged back by the central force towards C. The satellite would describe a spiral, the coils of which would be very nearly circular and very nearly coincident. If now we resolve the central component force along CM tangentially and perpendicular to the spiral, the tangential component tends to retard the velocity of the satellite, whereas the disturbing force, already considered, tends to accelerate it. With Satellite's the gravitational law of force between the two bodies the retardavelocity tion must prevail over the acceleration.<ref>This way of presenting the action of tidal friction is due to Professor Stokes.<ref> The moment of momentum of the whole system remains unchanged, and that of the planetary rotation diminishes, so that the orbital moment of momentum must increase; now orbital moment of momentum in creases with increasing distance and diminishing linear and angular velocity of the satellite. The action of tidal friction may appear somewhat paradoxical, but it is the exact converse of the accelera tion of the linear and angular velocity and the diminution of dis tance of a satellite moving through a resisting medium. The latter result is generally more familiar than the action of tidal friction, and it may help the reader to realize the result in the present case. Tidal friction then diminishes planetary rotation, increases the satellite's distance, and diminishes the orbital angular velocity. The comparative rate of diminution of the two angular velocities is generally very different. If the satellite be close to the planet the rate of increase of the satellite's periodic time or month is large compared with the rate of increase of the period of planetary rota tion or day; but if the satellite is far off the converse is true. Hence, if the satellite starts very near the planet, with the month a little longer than the day, as the satellite recedes the month soon increases, so that it contains many days. The number of days in the month attains a maximum and then diminishes. Finally the two angular velocities subside to a second identity, the day and month being identical and both very long.
We have supposed that the ocean is of such depth that the tides are direct; if, however, they are inverted, with low water under or nearly under the satellite, friction, instead of retarding, accelerates the tide; and it would be easy by drawing another figure to see that the whole of the above conclusions hold equally true with inverted tides.
§ 47. Exact Investigation of the Secular Effects of Tidal Friction.
Tidal friction.The general conclusions of the last section are of such wide interest that we proceed to a rigorous discussion of the principal effects of tidal friction in the elementary case of the circular orbit. In order, however, to abridge the investigation we shall only consider the case when the planetary rotation is more rapid than the satellite's orbital motion.
Suppose an attractive particle or satellite of mass m to be moving in a circular orbit, with an angular velocity O, round a planet of mass M, and suppose the planet to be rotating about an axis perpendicular to the plane of the orbit, with an angular velocity n; suppose, also, the mass of the planet to be partially or wholly imperfectly elastic or viscous, or that there are oceans on the sur face of the planet; then the attraction of the satellite must produce a relative motion in the parts of the planet, and that motion must be subject to friction, or, in other words, there must be frictional Energy tides of some sort or other. The system must accordingly be losing dimin- energy by friction, and its configuration must change in such a way ished by that its whole energy diminishes. Such a system does not differ lotion, much from those of actual planets and satellites, and, therefore, the results deduced in this hypothetical case must agree pretty closely with the actual course of evolution, provided that time enough has been and will be given for such changes. Let C be the moment of inertia of the planet about its axis of rotation, r the distance of the satellite from the centre of the planet, k the resultant moment of momentum of the whole system, e the whole energy, both kinetic and potential, of the system. It is assumed that the figure of the planet and the distribution of its internal density are such that the attraction of the sajllite causes no couple about any axis perpen dicular to that of rotation. A special system of units of mass, length, and time will now be adopted such that the analytical re sults are reduced to their simplest forms. Let the unit of mass be Mm/(M+m). Let the unit of length y be such a distance that the moment of inertia of the planet about its axis of rotation may be equal to the moment of inertia of the planet and satellite, treated as particles, about their centre of inertia, when distant y apart from one another. This condition gives
Let the unit of time T be the time in which the satellite revolves through 57 3 about the planet, when the satellite's radius vector is equal to y. In this case I/T is the satellite's orbital angular velocity, and by the law of periodic times we have where /* is the attraction between unit masses at unit distance. Then by substitution for y p?(Mm?
This system of units will be found to make the three following Special functions each equal to unity, viz., i^Mm (M+m)~l, fj.Mm, and C. units. The units are in fact derived from the consideration that these functions are each to be unity. In the case of the earth and moon, if we take the moon's mass as ^d of the earth's and the earth's moment of inertia as ^J/a 2 (as is very nearly the case), it may easily be shown that the unit of mass is -fa of the earth's mass, the unit of length 5 26 earth's radii or 33,506 kilometres (20,807 miles), and the unit of time 2 hrs. 41 minutes. In these units the present angular velocity of the earth's diurnal rotation is expressed by 7044, and the moon's present radius vector by 11-454. The two Moment bodies being supposed to revolve in circles about their common of mocentre of inertia with an angular velocity fi, the moment of momen- mentum. turn of orbital motion is Af+m Then, by the law of periodic times in a circular orbit, whence fir 2 =( The moment of momentum of orbital motion
and in the special units this is equal to ri. The moment of momentum of the planet's rotation is Cn, and C= 1 in the special units. Therefore ^_ n + r i,...(102). Since the moon's present radius vector is 1T454, it follows that the orbital momentum of the moon is 3 384. Adding to this the rotational momentum of the earth, which is 704, we obtain 4 088 for the total moment of momentum of the moon and earth. The ratio of the orbital to the rotational momentum is 4 - 80, so that the total moment of momentum of the system would, but for the obliquity of the ecliptic, be 5 80 times that of the earth's rotation. Again, the kinetic energy of orbital motion is The kinetic energy of the planet's rotation is %Cn z . The potential energy of the system is - p-Mmfr. Adding the three energies to gether, and transforming into the special units, we have 2e=?i 2 -l/r ........................ (103). Now let x=r$, y = n, Y=2e. It will be noticed that x, the moment of momentum of orbital motion, is equal to the square root of the satellite's distance from the planet. Then equations (102) and (103) become h = y + x .............................. (104). Y=y 2 -l/y?=(h-x?-llx i ............... (105). (104) is the equation of conservation of moment of momentum, or, shortly, the equation of momentum; (105) is the equation of energy. Now consider a system started with given positive moment of momentum h; and we have all sorts of ways in which it may be started. If the two rotations be of opposite kinds, it is clear that Maxiwe may start the system with any amount of energy however great, mumand but the true maxima and minima of energy compatible with the minimum given moment of momentum are supplied by dY/dx=Q, energy. or a;-A + l/x 3 =0, that is to say, se 4 - far 5 + 1 = ........................ (106). We shall presently see that this quartic has either two real roots No rela tive mo tion be tween satellite and planet when energy maxi mum or minimum Equa tions of mo mentum, energy, and no relative motion. Graphical illustra tion. and two imaginary, or all imaginary roots. The quartic may be derived from quite a different consideration, viz., by finding the con dition under which the satellite may move round the planet so that the planet shall always show the same face to the satellite, in fact, so that they move as parts of one rigid body. The condition is simply that the satellite's orbital angular velocity l=n, the planet's angular velocity of rotation, or y=l/x 3, since n=y and ri=O~J=a;. By substituting this value of y in the equation of momentum (104), we get as before x*-hx 3 + l = Q At present we have only obtained one result, viz., that, if with given moment of momentum it is possible to set the satellite and planet moving as a rigid body, it is possible to do so in two ways, and one of these ways requires a maximum amount of energy and the other a minimum; from this it is clear that one must be a rapid rotation with the satellite near the planet and the other a slow one with the satellite remote from the planet. In the three equations h = y + x (107), F=(A-a;) 2 -l/x 2 (108), aty=l (109), (107) is the equation of momentum, (108) that of energy, and (109) may be called the equation of rigidity, since it indicates that the two bodies move as though parts of one rigid body. To illustrate these equations geometrically, we may take as abscissa x, which is the moment of momentum of orbital motion, so that the axis of x may be called the axis of orbital momentum. Also, for equations (107) and (109) we may take as ordinate y, which is the moment of momentum of the planet's rotation, so that the axis of y may be called the axis of rotational momentum. For (108) we may take as ordinate Y, which is twice the energy of the system, so that the axis of Y may be called the axis of energy. Then, as it will be convenient to exhibit all three curves in the same figure, with a parallel axis of x, we must have the axis of energy identical with that of rotational momentum. It will not be necessary to consider the case where the resultant moment of momentum h is negative, because this would only be equivalent to i-eversing all the rotations; h is therefore to be taken as essentially positive. Then the line of momentum whose equation is (107) is a straight line inclined at 45 to either axis, having positive intercepts on both axes. The curve of rigidity whose equation is (109) is clearly of the same nature as a rectangular hyperbola, but it has a much more rapid rate of approach to the axis of orbital momentum than to that of rotational momentum. The intersections (if any) of the curve of rigidity with the line of momentum have abscissae which are the two roots of the quartic x 4 - hx 3 +l = 0. The quartic has, therefore, two real roots or all imaginary roots. Then, since x=/r, the intersection which is more remote from the origin indicates a configuration where the satellite is remote from the planet; the other gives the configuration where the satellite is closer to the planet. We have already learnt that these two cor respond respectively to minimum and maximum energy. When x is very large, the equation to the curve of energy is Y= (h - a;) 2, which is the equation to a parabola with a vertical axis parallel to Y and distant h from the origin, so that the axis of the para bola passes through the intersection of the line of momentum with the axis of orbital momentum. When x is very small, the equation becomes Y= - 1/z 2 . Hence the axis of Y is asymptotic on both sides to the curve of energy. Then, if the line of mo mentum intersects the curve of rigidity, the curve of energy has a maximum vertically underneath the point of intersection nearer the origin and a minimum underneath the point more remote. But, if there are no intersections, it has no maximum or minimum. Fig. 9 shows these curves when drawn to scale for the case of the earth and moon, that is to say, with h =4. The points a and b, which are the maximum and minimum of the curve of energy, are supposed to be on the same ordinates as A and B, the intersections of the curve of rigidity with the line of momentum. The in tersection of the line of momentum with the axis of orbital momentum is denoted by D, but in a figure of this size it necesFig. 9. sarily remains indistinguishable from B. As the zero of energy is quite arbitrary, the origin for the energy curve is displaced down wards, and this prevents the two curves from crossing one another in a confusing manner. On account of the limitation imposed we neglect the case where the quartic has no real roots. Every point of the line of momentum gives by its abscissa and ordinate the square root of the satellite's distance and the rotation of the planet, and the ordinate of the energy curve gives the energy corresponding to each distance of the satellite. Part of the figure has no physical meaning, for it is impossible for the satellite to move round tho planet at a distance less than the sum of the radii of the planet and satellite. For example, the moon's diameter being about 2200 rniles, and the earth's about 8000, the moon's distance cannot be less than 5100 miles. Accordingly a strip is marked off and shaded on each side of the vertical axis within which the figure has no physical meaning. The point P indicates the present configura tion of the earth and moon. The curve of rigidity x 3 y="i is the same for all values of h, and by moving the line of momentum parallel to itself nearer to or further from the origin, we may represent all possible moments of momentum of the whole system. The smallest amount of moment of momentum with which it is Least mopossible to set the system moving as a rigid body, with centrifugal mentum force enough to balance the mutual attraction, is when the line of for which momentum touches the curve of rigidity. The condition for this no relais clearly that the equation x 4 - hx 3 + 1 = should have equal roots, tive moIf it has equal roots, each root must be f A, and therefore tion pos8ible whence & 4 =4 4 /3 3 or 7i=4/3 J =175. The actual value of h for the moon and earth is about 4; hence, if the moon-earth system were started with less than % of its actual moment of momentum, it would not be possible for the two bodies to move so that the earth Maxishould always show the same face to the moon. Again, if we travel mum along the line of momentum, there must be some point for which number yx 3 is a maximum, and since yx 3 =n/ft there must be some point of days in for which the number of planetary rotations is greatest during one month. revolution of the satellite; or, shortly, there must be some con figuration for which there is a maximum number of days in the month. Now yx 3 is equal to x 3 (h-x), and this is a maximum when a; =|A and the maximum number of days in the month is ($h) 3 (h- |/i) or 3 3 /i. 4 /4 4; if A is equal to 4, as is nearly the case for the earth and moon, this becomes 27. Hence it follows that we now have very nearly the maximum number of days in the month. A more accurate investigation in a paper on the "Precession of a Viscous Spheroid" in Phil. Trans., part i., 1879, showed that, taking account of solar tidal friction and of the obliquity to the ecliptic, the maximum number of days is about 29, and that we have already passed through the phase of maximum. We will now consider the Discusphysical meaning of the figure. It is assumed that the resultant sion of moment of momentum of the whole system corresponds to a positive figure. rotation. Now imagine two points with the same abscissa, one on the momentum line and the other on the energy curve, and suppose the one on the energy curve to guide that on the momentum line. Then, since we are supposing frictional tides to be raised on the planet, the energy must degrade, and however the two points are set initially the point on the energy curve must always slide down a slope, carrying with it the other point. Looking at the figure, we see that there are four slopes in the energy curve, two running down to the planet and two down to the minimum. There are therefore four ways in which the system may degrade, according to the way it was started; but we shall only consider one, that corresponding to the portion AB&a of the figure. For the part of the line of momentum AB the month is longer than the day, and this is the case with all known satellites except the nearer one of History of Mars. Now, if a satellite be placed in the condition A that is to satellite say, moving rapidly round a planet which always shows the same as energy face to the satellite the condition is clearly dynamically unstable, degrades. for the least disturbance will determine whether the system shall degrade down the slopes ac or ab that is to say, whether it falls into or recedes from the planet. If the equilibrium breaks down by the satellite receding, the recession will go on until the system has reached the state corresponding to B. It is clear that, if the intersection of the edge of the shaded strip with the line of mo mentum be identical with the point A, which indicates that the satellite is just touching the planet, then the two bodies are in effect parts of a single body in an unstable configuration. If, therefore, the moon was originally part of the earth, we should expect to find this identity. Now in fig. 9, drawn to scale to re present the earth and moon, there is so close an approach between the edge of the shaded band and the intersection of the line of momentum and curve of rigidity that it would be scarcely possible to distinguish them. Hence, there seems a probability that the two bodies once formed parts of a single one, which broke up in consequence of some kind of instability. This view is confirmed by the more detailed consideration of the case in the paper on the "Precession of a Viscous Spheroid," already referred to, and sub sequent papers, in the Philosophical Transactions of the Royal Society. 1 1 For further consideration of this subject see a series of papers by Mr G. H. Darwin, in Proceed, and Trans, of the Royal Society from 1878 to 1881, and Appendix Q (b) to part ii. vol. i. of Thomson and Tait's Nat. Phil., 1883. § 48. Amount of Tidal Retardation of Earth's Rotation.
Acceleration of moon's motion due to tidal friction.With respect to the actual amount of retardation of the earth's rotation, we quote the following from Thomson and Tait's Nat. Phil. (1883), § 830.[1]
"In observational astronomy the earth's rotation serves as a timekeeper, and thus a retardation of terrestrial rotation will appear astronomically as an acceleration of the motion of the heavenly bodies. It is only in the case of the moon's motion that such an apparent acceleration can be possibly detected. Now, as Laplace first pointed out, there must be a slow variation in the moon's mean motion arising from the secular changes in the eccentricity of the earth's orbit around the sun. At the present time, and for several thousand years in the future, the variation in the moon's motion is and will be an acceleration. Laplace's theoretical calculation of the amount of that acceleration appeared to agree well with the results which were in his day accepted as represent ing the facts of observation. But in 1853 Adams showed that Laplace's reasoning was at fault, and that the numerical results of Damoiseau's and Plana's theories with reference to it consequently require to be sensibly altered. Hansen's theory of the secular acceleration is vitiated by an error of principle similar to that which affects the theories of Damoiseau and Plana; but, the mathe matical process which he followed being different from theirs, he arrived at somewhat different results. From the erroneous theory Hansen found the value of 12″·18 for the coefficient of the term in the moon's mean longitude depending on the square of the time, the unit of time being a century; in a later computation given in his Darlegung he found the coefficient to be 12″·56.[2]
"In 1859 Adams communicated to Delaunay his final result, namely, that the coefficient of this term appears from a correctly con ducted investigation to be 5" 7, so that at the end of a century the moon is 5"7 before the position it would have had at the same time if its mean angular velocity had remained the same as at the begin ning of the century. Delaunay verified this result, and added some further small terms which increased the coefficient from 5″·7 to 6″·l.
Various estimates of amount."Now, according to Airy, Hansen's value of the advance represents very well the circumstances of the eclipses of Agathocles, Larissa, and Thales, but is if anything too small. Newcomb, on the other hand, is inclined from an elaborate discussion of the ancient eclipses to believe Hansen's value to be too large, and gives two competing values, viz., 8″·4 and 10″·9.[3]
"In any case it follows that the value of the advance as theoreti cally deduced from all the causes, known up to the present time to be operative, is smaller than that which agrees with observation. In what follows 12" is taken as the observational value of the advance, and 6" as the explained part of this phenomenon. About the beginning of 1866 Delaunay suggested that the true explana tion of the discrepancy might be a retardation of the earth's rota tion by tidal friction. Using this hypothesis, and allowing for the consequent retardation of the moon's mean motion by tidal reaction, Numeri- Adams, in an estimate which he has communicated to us, founded cal result on the rough assumption that the parts of the earth's retardation as to due to solar and lunar tides are as the squares of the respective earth's tide-generating forces, finds 22 sec. as the error by which the earth, rotation, regarded as a time-keeper, would in a century get behind a perfect clock rated at the beginning of the century. Thus at the end of a century a meridian of the earth is 330" behind the position in which it would have been if the earth had continued to rotate with the same angular velocity which it had at the beginning of the century. . . . " Whatever be the value of the retardation of the earth's rotation it is necessarily the result of several causes, of which tidal friction is almost certainly preponderant. If we accept Adams's estimate as applicable to the outcome of the various concurring causes, then, if the rate of retardation giving the integral effect were uniform, the earth as a time-keeper would be going slower by 22 of a second per y ear in the middle, and by 44 of a second per year at the end, than at the beginning of the century. The latter is ^ - - s of the present angular velocity; and, if the rate of retardation had been uniform during ten million centuries past, the earth must have been rotat ing faster by about one-seventh than at present, and the centrifugal force must have been greater in the proportion of 81 7- toThom717 2 or of 67 to 51. If the consolidation took place then or earlier, son's arthe ellipticity of the upper layers must have been -5^ instead of gurnent about -3^1 as it is at present. It must necessarily remain uncertain a's to age whether the earth would from time to time adjust itself completely O f con . to a figure of equilibrium adapted to the rotation. But it is clear solidathat a want of complete adjustment would leave traces in a pre- tion of ponderance of land in equatorial regions. The existence of large earth, continents and the great effective rigidity of the earth's mass render it improbable that the adjustments, if any, to the appropri ate figure of equilibrium would be complete. The fact then that the continents are arranged along meridians rather than in an equatorial belt affords some degree of proof that the consolidation of the earth took place at a time when the diurnal rotation differed but little from its present value. It is probable, therefore, that the date of consolidation is considerably more recent than a thou sand million years ago. It is proper, however, to add that Adams Great lays but little stress on the actual numerical values which have uncerbeen used in this computation, and is of opinion that the amount taiuty of tidal retardation of the earth's rotation is quite uncertain."
§ 49. Effects of Tidal Friction on the Elements of the Moon's Orbit and on the Earth's Rotation.
It would be impossible within the limits of the present article to Effects of discuss completely the effects of tidal friction; we therefore confine tidal ourselves to certain general considerations which throw light on friction, the nature of those effects. We have in the preceding sections supposed that the planet's axis is perpendicular to the orbit of the satellite, and that the latter is circular; we shall now suppose the orbit to be oblique to the equator and eccentric, and shall also consider some of the effects of the solar perturbation of the moonearth system. For the sake of brevity the planet will be called the earth, and the satellite the moon. The complete investigation was carried out on the hypothesis that the planet was a viscous spheroid, because this was the only theory of frictionally resisted tides which had been worked out. Although the results would be practically the same for any system of frictionally resisted tides, we shall speak below of the planet or earth as a viscous body.[4]
We shall show that if the tidal retardation be small the obliquity Qbliof the ecliptic increases, the earth's rotation is retarded, and the quity of moon's distance and periodic time are increased. Fig. 10 represents the the earth as seen from above the south pole, so that S is the pole and the outer circle the equator. The earth's rotation is in the direction of the curved arrow at S. The half of the inner circle which is drawn with a full line is a semi-small-circle of south lati tude, and the dotted semicircle is a semi -small -circle in the same north latitude. Generally dotted lines indicate parts of the figure which are below the plane of the Fj 10 paper. It will make the explanation somewhat simpler if we suppose the tides to be raised by a moon and anti-moon diametrically opposite to one another. Let M and M be the projections of the moon and anti-moon on to the terrestrial sphere. If the fluid in which the tides are raised were perfectly frictionless, [5] or if the earth were a perfect fluid or per fectly elastic, the apices of the tidal spheroid would be at M and M . If, however, there is internal friction, due to any sort of viscosity, the tides will lag, and we may suppose the tidal apices to be at T and T . Now suppose the tidal protuberances to be replaced by two equal heavy particles at T and T, which are in stantaneously rigidly connected with the earth. Then the attrac tion of the moon on T is greater than on T, and that of the antimoon on T is greater than on T. The resultant of these forces is clearly a pair of forces acting on the earth in the direction TM, T M . These forces clearly cause a couple about the axis in the equator, which lies in the same meridian as the moon and anti-moon. The direction of the couple is shown by the curved arrows at L,L . If the effects of this couple be compounded with the existing rotation of the earth according to the principle of the gyroscope, the south pole S tends to approach M and the north pole to approach M . Hence, supposing the moon to move in the ecliptic, the inclination of the earth's axis to the ecliptic diminishes, or the obliquity increases. Next the forces TM, T M clearly produce, as in the simpler case considered above, a couple about the earth's polar axis, which tends to retard the diurnal rotation.
This general explanation remains a fair representation of the state of the case so long as the different harmonic constituents of the aggregate tide-wave do not suffer very different amounts of retardation; and this is the case so long as the viscosity is not great, The rigorous resuLt for a viscous planet shows that in general the obliquity will increase, and it appears that, with small viscosity o; the planet, if the period of the satellite be longer than two periods of rotation of the planet, the obliquity Increases, and vice versa. Hence zero obliquity is only dynamically stable when the perioc of the satellite is less than two periods of the planet's rotation.
Inclina- Suppose the motions of the planet and of its solitary satellite to tion of be referred to the invariable plane of the system. The axis oi plane of resultant moment of momentum is normal to this plane, and the orbit component rotations are that of the planet about its axis of figure generally and the orbital motion of the planet and satellite round their corndecreases, mon centre of inertia; the axis of this latter rotation is clearly the normal to the satellite's orbit. Hence the normal to the orbit, the axis of resultant moment of momentum, and the planet's axis of rotation must always lie in one plane. From this it follows that the orbit and the planet's equator must necessarily have a common node on the invariable plane. If either of the component rotations alters in amount or direction, a corresponding change must take place in the other, such as will keep the resultant moment of momentum constant in direction and magnitude. It has been shown that the effect of tidal friction is to increase the distance of the satellite from the planet, and to transfer moment of momentum from that of planetary rotation to that of orbital motion. If, then, the direction of the planet's axis of rotation does not change, it follows that the normal to the lunar orbit must approach the axis of resultant moment of momentum. By drawing a series of parallelograms on the same diameter and keeping one side constant in direction, this may be easily seen to be true. This is equivalent to saying that the inclination of the satellite's orbit will decrease. But this decrease of inclination does not always necessarily take place, for the previous investigations show that another effect of tidal friction may be to increase the obliquity of the planet's equator to the invariable plane, or, in other words, to increase the inclination of the planet's axis to the axis of resultant moment of momentum. Now, if a parallelogram be drawn with a constant diameter, it is seen that by increasing the inclination of one of the sides to the diameter (and even decreasing its length) the inclination of the other side to the diameter may also be in creased. The most favourable case for such a change is when the side whose inclination is increased is nearly as long as the diameter. From this it follows that the inclination of the satellite's orbit to the invariable plane may increase, and that it is most likely to increase, when the moment of momentum of planetary rotation is large com pared with that of the orbital motion. The analytical solution of the problem agrees with these results, for it shows that if the vis cosity of the planet be small the inclination of the orbit always diminishes, but if the viscosity be large, and if the satellite moves with a short periodic time (as estimated in rotations of the planet), the inclination of the orbit will increase. These results convey some idea of the physical causes which may have given rise to the present inclination of the lunar orbit to the ecliptic. For the analytical investigation shows that the inclination of the lunar orbit to a certain plane, which replaces the invariable plane when the solar attraction is introduced, was initially small, that it then increased to a maximum, and that it finally diminished and is still diminishing.
But the laws above referred to would, by themselves, afford a very unsatisfactory explanation of the inclination of the lunar orbit, be cause the sun's attraction is a matter of much importance. It has been found that, if the viscosity of the planet be small, the in clination of the orbit of the solitary satellite to the invariable plane will always diminish; but, when solar influence is introduced, the corresponding statement is not true with regard to the inclination of the lunar orbit to the proper plane, for during one part of the moon's histoiy the inclination to the proper plane would have increased even if the viscosity of the earth had been small.
Eccen- Consider a satellite revolving about a planet in an elliptic orbit, tricity of with a periodic time which is long compared with the period of rota tion of the planet; and suppose that frictional tides are raised on orbit generally the planet. The major axis of the tidal spheroid always points in increases, advance of the satellite, and exercises on it a force which tends to accelerate its linear velocity. When the satellite is in perigee the tides are higher, and this disturbing force is greater than when the satellite is in apogee. The disturbing force may therefore be repre sented as a constant force, always tending to accelerate the motion of the satellite, and as a periodic force which accelerates in perigee and retards in apogee. The constant force causes a secular increase of the satellite's mean distance and a retardation of its mean motion. The accelerating force in perigee causes the satellite to swing out further than it would otherwise have done, so that when it comes round to apogee it is more remote from the planet. The retarding force in apogee acts exactly inversely, and diminishes the perigean distance. Thus, the apogean distance increases and the perigean But it distance diminishes, or, in other words, the eccentricity of the orbit may de- increases. Now consider another case, and suppose the satellite's crease. periodic time to be identical with that of the planet's rotation. Then, when the satellite is in perigee, it is moving faster than the planet rotates, and when in apogee it is moving slower; hence at apogee the tides lag, and at perigee they are accelerated. Now the lagging apogean tides give rise to an accelerating force on the satellite, and increase the perigean distance, whilst the accelerated perigean tides give rise to a retarding force, and decrease the apogean distance. Hence in this case the eccentricity of the orbit will diminish. It follows from these two results that there must be some intermediate periodic time of the satellite for which the eccentricity does not tend to vary.
But the preceding general explanation is in reality somewhat less satisfactory than it seems, because it does not make clear the existence of certain antagonistic influences, to which, however, we shall not refer. The rigorous result, for a viscous planet, shows that in general the eccentricity of the orbit will increase; but, if the obliquity of the planet's equator be nearly 90, or if the viscosity be so great as to approach perfect rigidity, or if the periodic time of the satellite (measured in rotations of" the planet) be short, the eccentricity will slowly diminish. When the viscosity is small the law of variation of eccentricity is very simple: if eleven periods of the satellite occupy a longer time than eighteen rotations of the planet, the eccentricity increases, and vice versa. Hence in the case of small viscosity a circular orbit is only dynamically stable if the eleven periods are shorter than the eighteen rotations.
- ↑ See also G. H. Darwin's Address to Sect. A, Brit. Assoc. meeting, 1886
- ↑ "It appears not unusual for physical astronomers to use an abbreviated phraseology, for specifying accelerations, which needs explanation. Thus, when they speak of the secular acceleration being, e.g., '12″·56 in a century, they mean by acceleration what is more properly the effect of the acceleration on the moon's mean longitude. The correct unabbreviated statement is the acceleration is 25″·12 per century per century. Thus Hansen's result is that in each century the mean motion of the moon is augmented by an angular velocity of 25″·12 per century, so that at the end of a century the mean longitude is greater by 12 of 25″·12 than it would have been had the moon's mean motion remained the same as it was at the beginning of the century. Considering how absurd it would be to speak of a falling body as experiencing an acceleration of 16 feet in a second, or of 64 feet in two seconds, and how false and inconvenient it is to speak of a watch being 20 seconds fast when it is 20 seconds in advance of where it ought to be, we venture to suggest that, to attain clearness and correctness without sacrifice of brevity, 'advance' be substituted for 'acceleration' in the ordinary astronomical phraseology."
- ↑ Researches on the Motion of the Moon, Washington, 1878.
- ↑ These explanations, together with other remarks, are to be found in the abstracts of Mr G. H. Darwin's memoirs in Proc. Roy. Soc., 1878 to 1881.
- ↑ We here suppose the tides not to be inverted. If they are inverted the conclusion is precisely the same.