352 



NA TURE 



[February 13, 1902 



the heat-generatinp effect of the contractile force exceeds (lie 

 loss by outward radiation, as well as another period when the 

 declining temperature of the star indicates an excess of the 

 heat-dissipating over the heat-producing forces. Which of 

 these conditions, at the present moment, prevails on our sun 

 can so far be only a matter of conjecture. In this respect, 

 therefore, an assumption has to be made. The following in- 

 quiry applies to the case of a star on which the generation of 

 energy by contraction falls short of the loss of energy by radia- 

 tion. Whether the results of this investigation may be applied 

 to the case of our .sun must, then, depend on the further question 

 whether the sun really belongs to those stars the temperature 

 of which is declining. So far as I know, this latter opinion is at 

 present held by the great majority of astrophysicists. 



If on a star the loss of energy exceeds the production, the 

 kinetic energy of its molecules, and consequently its absolute 

 temperature, must decrease. Hence if the temperature of a 

 layer, a„, at a certain distance, p„, from the centre was T, at 

 the epoch /,, it will be T; at a later epoch /„, where T2<T,. 

 Now let (i| be the level of the photosphere— or the level of 

 maximum incandescence, and therefore also of maximum radia- 

 tion — at the epoch t^. In consequence of deficient contraction 

 the temperature of this layer must decrease, and the materials 

 composing it must cool down, so that, at the subse()uent epoch 

 /o, the level of maximum incandescence will have shifted 

 towards a layer, a„ nearer to the star's centre, where the 

 temperature is still sufficiently high to maintain the incan- 

 descent state of all the particles. The space between a, and a, 

 will then be occupied by particles in a less luminous state which 

 act as an absorbing screen on the radiation emanating from ag. 

 Whatever fraction of the total radiation which originally left 

 the photosphere at a^ is thus stopped in its outward progress will 

 be in part absorbed by, and in part reflected from, the inter- 

 vening particles of the layer a, a.,, and there can be no doubt 

 that some at least of this arrested energy will ultimately be 

 thrown back to a, from which it started. The layer a]<!o must 

 therefore act on the photospheric radiation in the same way as 

 do our atmosphere and its clouds on the radiation from the soil. 

 We are quite familiar with the fact that clear nights are, as a 

 rule, cooler than cloudy ones, and we explain this phenomenon 

 by the assumption that on clear nights radiation from the soil 

 into space goes on more freely than when clouds offer an effective 

 impediment to the dissipation of radiant energy. 



We conclude, then, that the progressive coolmgof thestar leads 

 to the formation of an absorbing envelope above its photo- 

 sphere, by which the disproportion between the generation and 

 loss of energy is reduced. But if, under the conditions at the 

 epoch t^, the amount of energy actually radiated into space still 

 exceeds what is produced by contraction, the photosphere will 

 move to (73, still nearer to the centre, and the quantity of ab- 

 sorbing matter in the layer .2,(73 ^'" ^^ further increased. Now 

 although «3 emits the same quantity of energy as did a, and a, 

 at the former epochs, the total amount of radiation emerging 

 into space must, at the epoch Z^, be less than it was at A and /,. 

 Thus the opacity of the cooled atmosphere gradually increases 

 as time goes on, and the total radiation of the star becomes less 

 and less. Since no force is present to interfere with the cooling 

 of the layers a^, a.j . . . , a moment /„ must eventually be 

 reached at which the photosphere at a„, through reflection from 

 all the layers above it, receives back so much of its radiation 

 that its total expenditure of energy is exactly counterbalanced 

 by the energy contributed by the contractile forces. 



This result appears to be of eminent importance. For it 

 .shows that even on a star with deficient coniraclion the exact 

 compensation of the loss of energy may still be possible from a 

 certain layer downwards. This state, so exceedingly imporiant 

 for the conservation of energy within the star, is brought about 

 by the progres.sive cooling of its superficial layers, which thereby 

 increase their power of absorption and thus offer a more and 

 more effective check to the radiation from the incandescent 

 layers below. 



Here, now, we are confronted with a question which leads us 

 at once to the principal object of this inquiry : Can the slate of 

 thermal equilibrium thus eventually attained by the layer a„ be 

 permanent? The answer is clearly negative. For when <i„ has 

 arrived at this stale, none of the layers a,, a„ . . . a„_, outside 

 a„ have reached the same condition. Their cooling is bound to 

 go on, and consequently their ability to absorb and reflect the 

 heat emanating from the layer a„ must still further increase even 

 after the establishment of thermal equilibrium at (i„. But, owing 



to this increasing amount of reflection towards it, the layer a„ 

 will now dissipate even less energy than is required for the 

 maintenance of thermal equilibrium, and therefore must become 

 onerhealed. It thus comes to pass that, while the function of 

 the absorbing envelope is that of reducing as much as possible 

 the waste of energy from the photospheric layers, it is, by the 

 very nature of this process, compelled to overdo its work, and 

 to preserve finally too much energy within the star. 



Now by this gradual overheating of the inner layers the 

 vertical temperature-gradient must increase more and more, 

 until it reaches a degree of steepness at which the permanence 

 of a mechanical equilibrium becomes impossible. In such a 

 case the overheated gaseous matter will force its way outwards 

 and will break through the "cloak" of absorbing elements 

 above it. But the overheated matter will not at once obey its 

 molecular impulse to escape into higher levels. We must 

 remember that there exists a powerful system of convection 

 currents between the interior'and the surface of the sun, and that 

 the overheated particles may for some time be swept along the 

 paths of these currents and may thus be forcibly detained in 

 levels inconsistent with their increased temperature, so that 

 their state of equilibrium is rendered unstable. This will pro- 

 duce a tension which increases in course of time until the 

 upward tendency of the overheated particles becomes strong 

 enough to overcome the resistance of the currents. At such a 

 critical moment even a slight disturbance will be sufficient to 

 induce the upward motion so long restrained, thus giving rise to 

 a solar eruption. The cause of a solar outburst is therefore 

 to be found in the temporary existence of an excessively great 

 vertical temperature gradient caused by progressive cooling of 

 the outer atmospheric layers and the ensuing overheating of 

 the inner photospheric layers. 



From this exceedingly simple principle we are able to deduce 

 an analytical demonstration of the periodicity of solar phenomena 

 which explains all the characteristics of the sunspot curve 

 hitherto observed. Obviously, the problem consists in demon- 

 strating the changes in the amount of outward radiation which 

 are caused, on the one hand, by the increase of absorptive power 

 of the atmosphere in consequence of its progressive cooling, 

 and, on the other, by the reduction of absorptive power of this 

 same atmosphere in consequence of the " clarifying " action of 

 eruptions which, by breaking through the " veil,'' .iiminish the 

 number of cooled absorbing elements at ihc localities of erup- 

 tion. I shall not enter upon this part of my investigation in 

 the present note beyond stating that it is a simple application of 

 Bouguer- Lambert's formula for the extinction of light and heat 

 in an absorbing medium. The energy 3 of the radiation leaving 

 the upper limit of the atmosphere is found by the differential 

 equation 



d^h . dS, 



dl- 



dl 



a0d 



where t denotes the time reckoned from the moment when the 

 photospheric layer a„ has attained its state of thermal 

 equilibrium, and a and 6 represent constants, the former of 

 which depends on the rate ol cooling of the atmosphere, the 

 latter on the action of the eruptions. The integral of this 

 eciuation will thus give us the changes in the r.idiating power of 

 the sun towards a point in the universe. Considering that the 

 intensity of the dynamical phenomena at the solar surface must 

 depend on the excess of energy preserved to the sun beyond 

 what he requires for the maintenance of thermal equilibrium at 

 a,„ we arrive at the following theoretical equation for the 

 frequency of eruptions and spots : — 



{■ 



-)1 



where / is the period and 



A, = - Jo -I- i^'iar + aB 

 Aj= -Jo- ^Jo'^ -1-0(3. 



It is readily seen that r starts from zero at the moment t = o, 

 and that it reverts to zero at the moment /=/>. Between these 

 two moments r attains a maximum, and we find the time when 

 this occurs from the equ.ation 



-i-^i/ 



.(A, - A..)/,„ 



- <'^3/>' 



NO. 1685, VOL. 65] 



