1920AJ. G R? Q 35P ASTEOSOMICAL JOÜENAI FOUNDED by B. A

g r? Q 35p H A F0RD c H Zl *m HAVE^FôRD, PA ASTEOSOMICAL JOÜENAI FOUNDED BY B. A. GOULD 1920AJ. No. 775 VOli. XXXIII ALBANY, N. Y., 1920, SEPTEMBER 15 NO. 7 ON THE ORIGIN OF PERIODIC COMETS, By HENRY NORRIS RUSSELL. It is well known that certain short period comets with a period less than half that of Jupiter should be have been diverted into their present orbits by close 126. Those with periods less than Jupiter's should encounters with Jupiter, previous to which their number 839: with periods less than twice Jupiter's periods were much longer than at present; and for 2670: and so on. this reason the numerous comets with periods between (c) Of the captured comets the majority will be five and nine years are often described as belonging to moving in direct orbits. Newton calculates that out Jupiter's “family.” It is also generally supposed that of the 839 comets with period less than Jupiter's 257 the comets of somewhat longer period belong, in the should have inclinations less than 30°, and only 51 same sense, to the “families” of Saturn, Uranus and inclinations exceeding 150°. Moreover, those with Neptune. Strong evidence against the real existence of direct motions are much more likely to have their Neptune's family has however been presented by H. C. periods still further shortened at a subsequent en- Wilson (1). counter than to have them lengthened, while the It is the purpose of the present discussion to examine reverse is the case for the comets with retrograde the validity of the common supposition as regards motion. captures by the outer planets, and to extend the (d) Finally, and obviously, the orbits of the cap- results to comets of still longer period. tured comets will continue to pass near to those of the capturing planet, until they are gradually shifted to a greater distance by the cumulative effect of planetary I — Theory perturbations of the ordinary type. Further close encounters with the original capturing planet, or with The theory of the capture of comets by planets has some other planet, may be excluded from this state- been entensively studied, notably by H. A. Newton ment, if we agree to assign a comet to the family of the (la) from whose discussion the following results may be planet which it last encounters. The probability of quoted: such additional encounters, however, is enormously (a) It is possible that an originally parabolic orbit greater for a captured comet than for one moving at may be transformed by a single encounter with any random, and encounters with a second planet are one of the major planets into an elliptic orbit with possible if the inclination of the orbit is small after period shorter than that of the planet, and even that capture. These factors, as Newton shows, may the comet may be stopped dead, so that it falls into operate to cause a very considerable increase in the the Sun. number of comets with short period, direct motion, (b) Such great perturbations will occur only when and small inclinations. the encounter is very close, so that they will be rare, Certain other results, not expressly stated by New- and for every case in which they happen, there will be ton, but following from his equations, may be men- a much larger number in which the perturbations are tioned. smaller, and the period after the encounter is longer. (e) Newton proved that, if a planet of mass m Newton calculates that, out of 109 comets which moves in a circular orbit of radius r, and a very large enter a sphere described about the Sun, with radius number of comets, moving at random in parabolic equal that of Jupiter's orbit (assumed circular) the orbits, approach the Sun within the distance r, the number which have their orbits changed into ellipses fraction F of the whole number which will have their (49) © American Astronomical Society Provided by the NASA Astrophysics Data System tí O'!1 ^r 50 THE ASTRONOMICAL JOURNAL N°- 775 00 orbits changed, by a single encounter with the planet, captured comets will have large perihelion distances, into ellipses of mean distances less than A will be and escape observation. If we assume that the same fraction of them have perihelion distances less than q F = (1) as in the case of the original parabolic comets, (which 1920AJ. should be nearly true in the majority of cases, where the perturbations are small) we find, for the number of where the range of integration extends over all values captured comets which will have mean distances less of s greater than \/2 - 1, which make the integrand than a and perihelion distances less than q the fraction positive. If in this expression we set 4 m2 a? q 2 of all those wbich enter the unit-sphere. 2 3 r a — r = (X — l)a, This expression holds good when a is large compared with r; but, if q is small compared with r it remains a the limits of integration are 1 — X and 1 + X provided good approximation for smaller values of a, — the that X < 2 — \/2, but are \/2 — 1 and 1 + X if X 4 decrease in the coefficient F, which is when a is exceeds this limit. In the former case 3 2 2 large, being compensated by the concentration of the 4 m a 3 F X (2) perihelion distances toward small values which nec- 3 ~F‘ essarily occurs when a is less than r. and in the latter If we set q = 2, corresponding roughly to the limit beyond which comets are likely to escape observation, 1+V2 4 s 2 \/ 2 - 1 and a = 100, we find that, out of 100,000,000 comets F = X + g X + X - (3) which enter a sphere described about the Sun of radius equal to the Earth’s mean distance, the number which These equations, though not given expressly by are changed by capture into comets visible from the Newton, must have been used by him in deriving the Earth, and of period less than 1000 years, by the results given under (6) above. action of Jupiter will be 90,000; by Saturn 2400; When a is considerably greater than R the factor by Uranus 14; and by Neptune 8. in parenthesis in (3) may be expanded in a series If the limiting period is made shorter, the relative predominance of Jupiter will be still greater. 4 2 + \/2 r2 r3 2 3 (/) The effects of an individual encounter are 3 8 a 96 a given by the equations 2 2 2 2 JL = 1 4- :L f = _s_ A + d + h sin 6 which is so rapidly convergent that for all practical a purposes we may write a" a' a Am A cos 6 -\- h sin- 6 ^ ^ 4 m2 a2 In (4) a' and a" are the comeLs mean distances before 3 r2 and after the encounter. If the original orbit is parabolic, the mean distance after the encounter is a. whenever a > 3r. In equation (5) cl is the least 'distance between the The number of comets available for capture is, how- planet’s orbit and the undisturbed orbit of the comet: ever, greater for the remoter planets. Newton shows s the ratio of the undisturbed velocity of the comet, that, if the comets are moving at random in parabolic relative to the planet, to that of the planet, relative orbits, the number which enter a sphere of radius r, to the Sun; 6 the angle between these two velocity described about the Sun as centre, per unit of time vectors: h the distance which the planet has still to will be proportional to r — from which it follows that go to reach the point on its orbit which is nearest the the fraction of this whole number which have peri- comet’s undisturbed orbit at the instant when the comet, if undisturbed, would reach the point on its helion distances less than q will be ^ . Hence the own orbit which is nearest the planet’s orbit; and A whole number of comets which during a given inter- the real semi-axis of the hyperbolic orbit which the val of time, are diverted by a given planet into elliptic comet describes about the planet during the en- 4 m2 a2 counter. orbits of mean distance less than a is ^ times These equations have been derived by Newton the number which approach the Sun within the unit with the usual approximations, (neglecting the per- of distance during this interval. But many of these turbations of the comet by the planet when it is © American Astronomical Society • Provided by the NASA Astrophysics Data System tí O'!1 ^r ^ N°- 775 TW W A‘B T R O-.N- Q M I C A L; J 0 U R N A L ,51 remote from the latter, and by the ¿Shm when it is If now we set d =■ p sin \p, h sin d = p cos. (5) be- near the planet) but are correct to the first order of ; comes 1 < : v small quantities, and applicable to all encounters, 2 1920AJ. whatever the eccentricity of the planet’s orbit. If the - . _ sA , , . see a , :., . Am cos 6 + tan a sin B cos \f/ . , . latter is circular, and of radius cq, then A = ; r in the more general case, if r is the distance of the planet For the present purpose we may, treat the, planet’s from the Sun at the time of the encounter, and ai its orbit as circular, and set mean' distance m ai r s2 (2ai — r) We then find If the orbit along which the comet approaches the planet is given, all the quantities which appear fin q _ 16m (5) are fixed with exception of K, which defines the a2 m circumstances of the encounter, and may be so de- termined as to make the perturbation a maximum.

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