126 



NA TURE 



[June 7, 1900 



LETTERS TO THE EDITOR. 

 [The Editor does not hold himself responsible for opinions ex- 

 pressed by his correspondents. Neither can he undertake 

 to return, or to correspond with the writers of, rejected 

 manuscripts intended for this or any other part of Nature 

 No notice is taken of anonymous communications. '\ 



The Kinetic Theory of Planetary Atmospheres. 



In the paper which I communicated to the Royal Society on 

 April 5, I examined the logical conclusions obtained on the 

 hypothesis that the atmosphere of a planet is distributed ac- 

 cording to the generalised form of the Boltzmann-Maxwell 

 distribution applicable to a gas in a field of external force, with 

 the further generalisation required to take account of the effects 

 of axial rotation. As regards the effects of the planet's attraction 

 on the distribution of density, the expressions assumed to repre- 

 sent these were of the form now generally accepted by writers 

 on the kinetic theory {e.g. Watson and Burbury), and the 

 modifications required in taking account of centrifugal force 

 were investigated by me in 1894, and are in harmony with the 

 conclusions to which Maxwell's investigations tend. In the 

 aforementioned paper I .showed how to calculate a superior limit 

 to the rate at which a planet is losing its atmosphere, and ob- 

 tained the results that helium would be permanently retained at 

 all ordinary temperatures by terrestrial gravitation and vapour of 

 water by the gravitation on Mars ; conclusions with which those 

 deduced by Mr. Cook would appear to be identical, so far as I 

 judge from his letter. 



The objections which naturally suggest themselves to the 

 mode of treatment in this paper are that the distribution in 

 question is that which would be brought about exclusively as 

 the result of molecular encounters, and of the free paths of the 

 molecules between these encounters ; and that it therefore 

 represents the distribution in an atmosphere of uniform tempera- 

 ture. In an actual atmosphere the equilibrium of the lower 

 strata is largely modified by convection currents, so that the 

 adiabatic law, rather than the isothermal law, is applicable. 

 This point I hope to discuss at full length in the second part of 

 the paper ; in the meanwhile, it is hardly likely that any one 

 will suggest that helium escapes from our atmosphere because 

 the upper strata are at a low temperature, but that it would 

 cease to escape if the upper strata were heated up to the same 

 temperature as the lower ones. The point at issue between Dr. 

 Johnstone Stoney and Mr. Cook and myself appears to be how 

 far the Boltzmann-Maxwell distribution represents what happens 

 in the upper strata of the atmosphere. To assert " that in the 

 present state of our knowledge it " (the a priori method as Dr. 

 Stoney calls it) "cannot be made to furnish a valid investig- 

 ation," seems to me tantamount to striking at the very found- 

 ations of our kinetic theories of matter. It may be that these 

 theories will not resist such an attack, but the consequences of 

 the onslaught cannot be properly traced, except by making 

 mathematical determinations in the way that I have done. It 

 appears to me to be just in this very problem of planetary 

 atmospheres that the fundamental assumptions of the 

 kinetic theory are least open to objections. Experiments 

 on the relation of diffusion to temperature led Maxwell 

 to abandon the notion that the molecules of a -gas behave as 

 elastic spheres and to consider the effects of finite intermolecular 

 forces. So far as I am aware, (i) every attempt at a kinetic 

 explanation of the thermcdynamical properties of gases on the 

 latter view involves some assumption which restricts its validity 

 to the limiting case of attentuated gases, where the number of 

 molecules within each other's sphere of influence is a negligible 

 proportion of the whole number, and the duration of an en- 

 counter is negligible in comparison with the time of free motion 

 between encounters. On the other hand, (2) it is amply proved 

 by Watson and Burbury that the Boltzmann-Maxwell distribu- 

 tion, if it hold at any instant, will hold at all future instants in 

 the absence of molecular encounters. (3) Boltzmann's minimum 

 theorem tells us that if encounters take place at random, the 

 molecules tend towards the distribution in question. (4) We are 

 told on good authority that we must regard the Boltzmann- 

 Maxwell law as a theorem in probability. Now the divergence 

 between actual conditions and the assumptions required under 

 heading (i) gets less and less as we ascend in the atmosphere ; 

 (3) gives us reason for believing that the Boltzmann-Maxwell 

 distribution holds at the highest altitudes where encounters not 

 unfrequently take place ; (2) shows that the molecules which are 

 projected from these strata and ascend to still greater altitudes 



NO. 1597, VOL. 62] 



without encountering other molecules remain distributed accord- 

 ing to the same law ; and (4V removes the necessity of taking the 

 size of the element of volume dxdydz into account by telling us 

 that the law represents not merely the number of nvilecules 

 having given limits of velocity occurring in the element, but also 

 the probability of a molecule coming within these limits, and 

 this probability may be as small as we please. 



If helium really does escape from our atmosphere, either 

 there must be a fallacy in the assumptions underlying (i), 

 (2), (3), or (4), and this fallacy must affect numerous pre- 

 vious writings on the kinetic theory, or else our preconceived 

 notions as to the relation between temperature and kinetic 

 energy are at fault. With regard to (4), it may be objected that 

 the error-law fails to apply to events of exceptional occurrence, 

 and therefore that we cannot apply it to calculate the probability 

 of a molecule escaping from the atmosphere when the velocity 

 required would represent an abnormal divergence from the 

 mean. This point was carefully considered by me. It appears, 

 however, to be the accepted view that abnormal divergences are 

 excluded because in practice they never occur, not because their 

 occurrence is far more frequent than the error-law would lead 

 us to suppose. If the methods of the kinetic theory should 

 prove to be inapplicable to rarefied gases as well as to dense 

 assemblages of molecules, and they do not altogether agree with 

 experiment for distributions of intermediate density, the position 

 is indeed a serious one. In face of such a possibility, instead of 

 abandoning our mathematical calculations we ought to push 

 them to their ultimate consequences, in order to arrive at a 

 better understanding of the true state of the case. The escape 

 of gases from the atmospheres of planets is a phenomenon 

 probably more directly dependent on the translational kinetic 

 energy of the molecules than any other property ot gases. The 

 prevailing doctrine that not only is the mean value of this 

 translational kinetic energy proportional to the absolute temper- 

 ature, but the conceptions of temperature and kinetic energy 

 are physically identical, has always seemed to me to require 

 closer investigation than it has as yet received, and it may well 

 be that the kinetic theory of planetary atmospheres furnishes 

 one means of putting this doctrine to a test. 



Plas Gwyn, Bangor, May 26. G. H. Bryax. 



The Severn Bore. 



No one who suffers from scientific curiosity should miss 

 seeing a tidal bore at least once in his life. The locality and 

 conditions under which the Severn Bore can be seen make it an 

 ideal object for a pleasurable excursion. The time to be 

 selected is about twenty-four hours after new or full moon ; the 

 largest spring tides should be chosen, if possible, and an 

 occasion when the light permits both evening and morning bore 

 to be seen. They occur at about 7.30 to 9 o'clock, a.m. and 

 p.m. The visits should therefore be either when the d.ays are 

 long or at full moon. During a recent excursion, I stayed at 

 Newnham-on-Severn, below Gloucester. This is about 

 3 hours 20 minutes from Paddington station, and it is possible 

 to leave this station at 3.15 p.m. and be in time for the evening 

 bore, see the morning bore next day, and be back at Padding- 

 ton by 2.20 p.m. 



On April 29, twelve hours after full moon, I awaited the bore 

 at the south-east corner of Newnham Churchyard. The 

 position is the summit of a cliff situated on the outer bank, and 

 near the centre of the base of a U-shaped bend of the Severn, 

 the limbs of the U being four miles long, and the width bet\Veen 

 the limbs two miles. The prospect is one of the most pleasing 

 in the South of England ; the broad, winding river, emerald 

 pastures abandoned by the wandering channel, miles of rich 

 champagne country, with apple and plum orchards, and the 

 distant range of the Cots wolds. At 6.45 p.m. the bore was 

 sighted as a line of white foam between Aure and Fretherne, 

 rather more than three miles down the river. For a quarter of 

 an hour I watched its march up stream, first wheeling by the 

 left, then advancing up the straight reach, and finally wheeling 

 by the right round the last bend. The wheeling movement is 

 most fascinating to watch. I now hurried down to the ferry, 

 and shoved off the boat into deep water to meet the bore, which 

 was now roaring like a railway train. The water channel was 

 about 200 yards wide ; at high water it is double that width. 

 On the sands of the opposite convex, shallow shore the bore 

 discharged itself obliquely as a curling breaker. Against our 

 rocky shore it was a bursting surge. A rise of level was per- 

 ceptible about ten yards in front of this. In the deep channel 



