SCIENCE. 



23 



charge of the Observatory, that the instruments in use are 

 a Clark equitorial telescope, focal length io}4 feet, aperture 

 8% inches ; a portable equatorial made by John Byrne, of 

 New York, aperture 4.3 inches ; a Howard sideral clock ; 

 a Howard mean-time clock, a Bond sideral chronometer, 

 a Fauth transit instrument with telescope of 3 inches 

 aperture, a Clark chronogragh ; meteorological apparatus, 

 and a complete set of Johnson's large astronomical maps, 

 recently imported. By courtesy of Lieut. Edw. Maguire, 

 Chief Engineer of the Department of Dakota, the 

 Observatory has also the use of an excellent zenith tele- 

 scope for special work'. 



The time of the Observatory is the standard for the 

 State of Minnesota and parts of those States adjoining, 

 and given to the railroad companies daily by telegraph. 

 The distribution of the time of the Northfield meridian 

 by the aid of excellent instruments, is said to be easy, 

 exact and reliable. 



The object of erecting this Astronomical Observatory 

 appears to have been three-fold. 1. To give instruction 

 to undergraduate students. 2. To offer opportunities for 

 a complete course of study in Theoretical and Practical 

 Astronomy. 3. To aid in useful investigations. 



ON THE LIMIT OF PLANETARY STABILITY. 



By Professor Daniel Kirkwood. 



Laplace, in his Systeme du Monde, pointed out the 

 limit at which, according to his estimate, the moon's at- 

 traction could have retained an elastic atmosphere.* 

 The question of a satellite's stability was also considered 

 by the late Professor Vaughan, of Cincinnati. t I have 

 seen no attempt, however, to obtain for the different 

 members of our system any definite numerical results. 

 In the present paper it is proposed to find the approxi- 

 mate limits of stability in the cases of the eight major 

 planets and certain of the satellites, on the hypothesis 

 that their primitive condition was either liquid or 

 gaseous. 



Let M = the mass of the larger or central body, 

 m = that of the dependent planet or satellite, 

 x = the distance from the centre of the former to 



the limit of stability of the latter, 

 a = the distance between their centres ; then, j 

 since the disturbing or separating force of the larger 

 upon the smaller mass is the difference between the at- 

 traction of the former on the nearest point of the surface 

 of the latter and that on its centre of gravity, we have 

 MM m 

 ~x> ~ ~aS ~~ (a—xf W$ 

 or putting a = 1 and reducing, 

 m 



X i — 2X i + JJX i + 2X = I. (2) 



If we adopt the masses and distances given in New- 

 comb's Popular Astronomy and solve equation (2) for 

 each of the eight principal planets we shall obtain the 

 distance from the centre of each to its limit of stability, 

 as given in the second column of the following table. ' 

 If, moreover, the planets, with their present masses, be 

 reduced to the sun's mean density their radii as stated i 

 in the third column are found by the formula 



r» = 430,000 (^f) ■ 



and the respective ratios of the limits of stability to these 

 radii are seen in column fourth. 



* Syst. du Monde, B. IV., Ch. X. 



t Pop. Sci. Monthly for Sept. 1878. See also the Proc. of the A. A. A. S. 

 for 1856. 



t We neglect the centrifugal force due to the planet's rotation, as the 

 modification would be slight and we propose to obtain merely approxi - 

 mate results. 



Table. 



Planet. 





r„ 



r> 

 K n 



r" 



Mercurv 



165,165 ms 



2,514 6 ms 



65-7 





701,746 



5,7i9-2 



122.7 



Earth 



1,059,386 



6,242.7 



169.7 



Mars 



764,900 



2,951.1 



259.2 



Jupiter 



37.354.287 



42,335 



882.35 





4S.859.38i 



28,317 



1619.48 





49,512,900 



15.209 



3255-5I 



Neptune 



81,663,510 



16,009 



5101.10 



On the assumption that in each case the mean density 

 of the separated mass was equal to that of the central 

 body, the sun's present radius multiplied by the respec- 

 tive numbers in column fourth will give the radii of the 

 solar nebula when the planets extended to their respec- 

 tive limits of stability. These radii are less than the 

 mean distances of the planets in the ratio of i to 1.265. 

 This fact may have some significance in regard to the 

 former oblateness of the solar nebula or the law of its 

 density. 



The Earth and the Moon. — For the moon, which in 

 perigee approaches within 221,500 miles of the earth, the 

 limit of stability is about 38,000 miles. Were the 

 moon's density reduced to that of the earth its radius 

 would be 916 miles, the ratio of which to the limit of 

 stability is 1 : 41.6. The moon's least distance dimin- 

 ished by 38,000 miles is 183,500 miles. If our satellite 

 originally extended to the limi', and if the moon and the 

 earth had the same form and density, the radius of the 

 latter was [65,000 miles. 



The Martian System. — The diameter of Phobos, ac- 

 cording to Prof. Pickering, is 5.57 miles. If its density, 

 therefore, be equal to that of Mars the limit of stability 

 is about two miles exterior to the surface ; or, if the 

 density be to that of the primary in the same ratio as the 

 density of the moon to that of the earth, the limit is less 

 than a mile from the surface of the satellite ; and finally 

 if the density were no greater than that of water the 

 satellite, if fluid, would be unstable, the limit being ac- 

 tually within the surface. Since, therefore, the satellite 

 could never have existed at its present distance in a neb- 

 ular state, it must follow, if any form of the nebular 

 hypothesis is to be accepted, that its original distance 

 was much greater than the present. Can we r find a 

 probable cause for this ancient disturbance.'' 



If we suppose the former period of Mars to have been 

 very nearly one-sixth that of Jupiter the close commen- 

 surability would render the orbit of Mars more and 

 more eccentric. The planet in perihelion would thus 

 pass through the sun's atmosphere, or rather through 

 the outermost equatorial zone of the solar nebula. This 

 resisting medium would not only accelerate the motion 

 of Mars but also in a much greater degree that of his 

 extremely small satellite. The solar mass contracting 

 more rapidly than the orbit of Mars would finally leave 

 the latter moving in an eccentric path without sensible 

 resistance. 



Other Secondary Systems.— For the first satellite of 

 Jupiter the limit is 5250 miles, or 4^ times the radius of 

 the satellite. For Mimas, the innermost satellite of 

 Saturn, it is less than twice the radius. The rings of 

 Saturn, in all probability, could not exist as three satel- 

 lites, the limits of stability being interior to the surface.* 



The effect of preturbation in the dismemberment of 

 comets is known to all astronomers. The nucleus of the 

 great comet of 1880, which approached within less than 

 100,000 miles of the sun's surface, must have had a den- 



* It has been recently shown that liessel's mass of the ring is much 

 greater than the true value. 



