582 



NATURE 



[October ii, 1900 



THE STABILITY OF A SWARM OF METEOR- 

 ITES AND OF A PLANET AND SATELLITE. 

 THE problem of the stability of a swarm of meteorites which 

 is under the action of its own gravity and the attraction of 

 the sun, that is to say the determination of the condition under 

 which the swarm will remain unbroken up by the tidal action of 

 the sun, has been dealt with by Schiaparelli, M. Luc Picard and 

 M. Charlier {Bulletin de P Academic de St. Petersbourg, t. xxxii. 

 No. 3). The result obtained by the first assigns a much wider 

 limit of stability to such a system than that arrived at by the 

 other two investigators mentioned ; but there cannot, I think, 

 be any doubt of the greater correctness in actual cases of the 

 narrower limit. The problem is intimately related to the still 

 more interesting question of the stability of the earth-moon 

 system, which was treated by Mr. G. W. Hill in a remarkable 

 paper in the American Journal of Mathematics for 1878. This 

 again is, in another form, the problem treated still earlier with a 

 special object in view by Edouard Roche {Mini. Acad, de Mont- 

 pellier, vol. i. 1847-50 ; see also Annates de PObservatoire, t. v.), 

 when he arrived at his result concerning the limiting relation 

 between the distance of a satellite from a primary and the 

 <liameter of the primary, which must hold in order that the satel- 

 lite, held together by its own gravitation only, may just not 

 break up under the tidal forces due to the primary, and his 

 corresponding result for a planet's or satellite's atmosphere. 



These investigations, though of great general interest, are not 

 so well known as might be expected, and one object of this 

 paper is to give some slight account of them. An abstract of 

 the work of Charlier and Hill is given also in Dr. Routh's 

 recently published work on "Dynamics of a Particle." But I 

 wish also to point out how the main conclusions of Charlier, that 

 of Roche with respect to a planet's atmosphere, and more 

 indirectly the result of Hill, can be obtained by means of 

 elementary considerations. 



The problems just referred to have been treated by very 

 different methods. Schiaparelli's discussion is a direct attack 

 of a somewhat long and involved nature ; those of MM. Picard 

 and Charlier {Bulletin de P Acadhnie de St. Petersbourg, t. 

 xxxii. No. 3 ; see also Tisserand, Mecanique Celeste, t. iv.) 

 make use of the method of revolving axes. The radius vector 

 from the centre of the sun to the centre of the meteoric swarm 

 is supposed to revolve with angular velocity, n say, about the 

 centre of the sun as a fixed point ; then the motion of a particle 

 of the swarm is referred to three directions at right angles to 

 one another having their origin at the centre of the swarm, and 

 turning with the radius vector just specified. These axes may 

 be taken as an axis of \ towards the sun, an axis of ij at right 

 angles to this in the plane of motion of the centre, and an 

 axis of ^ at right angles to this plane. Then equations of motion 

 relative to these moving axes are written down for a particle the 

 component distances of which from the centre are |, ?/, ^, it being 

 supposed in the first place that the distance r of the centre of the 

 swarm from the sun, and the angular velocity n of the radius 

 vector are both variable. Approximate values of the forces are 

 obtained by supposing that |, tj, C are small in comparison with 

 r, and that r, and therefore also n, is constant. When account 

 is taken of the condition that must hold for the central particle, 

 the equations assume the very simple form 



^■-2«7J-(3«2_^)| = 0, iJ4-2«i + /UT; = 0, "C+(«'^ + /i)C=0. 



The value of ju is ^irks, where k is the gravitation constant 

 and s is the average density of the portion of the swarm within 

 the spherical surface on which the particle lies, supposed 

 symmetrical about the centre. Considering only particles in the 

 plane of |, 17, the values of these co-ordinates are supposed to 

 oscillate about certain constant values, so that |= a cos (»/-)- e), 

 ■f\ = b%\x\. (a)/-fe). That is, each particle is supposed to revolve 

 in an ellipse, the centre of which is the centre of the swarm, 

 and of which one axis is along the line of centres and the other 

 perpendicular to that line. The ellipse is a circle '\i a = b, and 

 wis then the angular velocity of the relative motion of the particle 

 about the centre. These values substituted in the first two 

 equations of motion lead to the condition 



(«^ - /i)(w^ -f 3«^ - m) "" 4w'^«' = o. 



Now CD is ittf, if f be the frequency of oscillation ; and if the 

 oscillation be stable,/ will have a positive real value. The roots 

 of the quadratic in aP- just written must therefore be real and 

 positive ; and it is not hard to see that the required condition 



NO. 1 61 5, VOL. 62] 



for this is /u>3«*. This gives, when /* is replaced by \-ifks, 

 and m'^ by k'MIr^, where M is the mass of the sun (for we have 

 for the central particle n^r=^M/r^), the inequality 



^wr'> 3M. 



In order, therefore, that the swarm of small particles may keep 

 together, it is necessary that its average density be greater than 

 that of a spherical distribution of matter of radius equal to the 

 sutHs distance and of three times the sun^s mass. The problem 

 for an elliptic orbit of eccentricity e has been considered by 

 M. Callandreau {Bulletin Astronotnique, 1896). The condition 

 ^>3«2isin this case replaced by /u>3«'^-t-5e^«''. The swarm 

 is therefore rendered less stable by the eccentricity. 



It is to be remembered that the effect of the distortion of the 

 swarm by the tidal force of the sun is neglected, and it does not 

 seem of much importance to consider eccentricity of orbit so 

 long as the assumption of sphericity of figure is maintained. 



Since the equation of condition stated above must be satisfied, 

 only values of w consistent therewith, and with the inequality 

 fj.>^n^, are admissible. Thus if « = o, that is if there be no 

 revolution of any particle about the centre of the swarm, the 

 equation gives /x = 3«^, and the inequality is not fulfilled. This 

 is a limiting case between stability and instability. 



Now let the differential equations of motion referred to above 

 (from which the more roughly approximate equations quoted are 

 derived) be modified for the case ir\. which the swarm is replaced 

 by a planet of given mass m, and the particle considered by a 

 satellite of mass m' at the external point |, tj, C. then let them 

 be multiplied by i, rj, f respectively, integrated, and added. 

 Thereby will be obtained the equation of kinetic energy for the 

 relative motion, commonly called Jacobi's equation. This has, 

 if l^ + v^ + C^, the square of the resultant relative velocity, be 

 denoted by v^, the form 



•"{ 



(;'-|)2 



■}J 



-t- C-o 



C^ r= s/^r- 1)-^ + v + c^ 



where 



H = k{m + m'), p= >y|''^ 



C is a constant, and r is, as before, the distance of the centre of 

 the sun from that of the planet. 



Now when v^ has a given value, the satellite must have its 

 centre on the surface of which the equation is obtained by 

 placing that value in the equation just written. Hence, since v'^ 

 is positive, the satellite cannot pass across the surface for which 

 v"^ = o, that is the surface for which 



+ 2 



r\ 



{r-O'+V 



}-C = o; 



by putting C=o in this we obtain the equation of the curve in 

 which the surface intersects the plane of |, rj. An investigation 

 of the surface shows that if C be positive the surface consists of 

 three sheets, of which two are closed and surround the sun and 

 the planet respectively ; and the third is asymptotic to a surface of 

 revolution about an axis passing through the sun's centre perpen- 

 dicular to the ecliptic, and surrounds the two closed surfaces. 

 Within the closed surfaces, or outside the third surface, v^ is 

 positive ; between the closed surfaces and the outer asympto- 

 tically cylindrical surface v'^ is negative, and therefore v is 

 imaginary. The satellite must therefore be within one of the 

 closed surfaces, or beyond the outer surface ; in either case it 

 cannot cross the surface of zero velocity. 



When the proper values of the quantities for the earth-moon 

 system are inserted, it is found that the moon is within the 

 closed sheet surrounding the earth, from which, therefore, it 

 cannot escape. The distance of the moon's centre from the earth, 

 Mr. Hill has calculated, cannot exceed 109*694 equatorial radii of 

 the earth. The result is based, of course, on the assumption 

 that the eccentricity of the earth's orbit may be neglected. 



If, besides neglecting the eccentricity, we suppose the moon 

 to move in the plane of the ecliptic, and to be so distant that 

 we may neglect terms in tj, the equation of the curve of no 

 velocity in the plane of the ecliptic is 



I^ + In^^^c, 

 P 2 

 or if I = p cos 



3«2cos^9. p^-cp + 2fj. — o, 



where c is another constant. 



