The calculated values agree better with Weinstein's values than with Airy's, which may be due either to better or more modern methods of observation or to the proximity of Greenwich to the sea. The agreement with Weinstein's data seems fairly satisfactory.

As regards the easterly E.M.F. of the diurnal earth currents, the agreement is less good. This is not unnatural, on account of the failure of the magnetic potential function to represent properly the northerly component of the magnetic force*, on which the easterly earth currents largely depend. The amplitude and character of the observed and calculated variation show very fair agreement, but there is a phase-difference of about 24 hours.

The local and irregular earth currents will next be considered. These are often large, e.g. a potential difference of between seven and eight hundred volts was found between earth plates 500 kilometres apart in 1859, and potential gradients of half a volt per mile have often been recorded since. The areas specially affected may vary within wide limits. The more local the area affected by a disturbance and its associated earth currents, the higher will be the degree and order of the leading terms when the field is analysed into its component spherical harmonics. The disturbance will be supposed oscillatory, so that in the time factor eat of Part I the appropriate value of a will be up, where the frequency p is supposed to be quite independent of the degree n and order m of the tesseral harmonics; m and n will be supposed fairly large, say from 5 to 10, in order to represent a disturbance completing its range over a relatively small fraction of the earth's surface. The period 2π/p may be anything from a few hours down to a few seconds.

The question will be illustrated chiefly in connection with the variations in the vertical magnetic force. Table X shows for latitude 60° N., and for a field varying in one harmonic component only, the amplitude of variation in the vertical force H, corresponding to an earth voltage of 5 volt per mile; the variations are of course proportional to one another. The values of n and m refer to the degree and order of the few typical harmonic components considered, while the periods dealt with vary from 2 to 30 minutes.

The corresponding variation of the horizontal force is less readily calculated, since it depends on the conductivity of the crust and core of the earth. With the data derived in § 11 from land

* Cf. Phil. Trans. A 218, p. 23, 1919.

stations only (immediately after Table V) the amplitudes of the resultant horizontal force variation corresponding to the first line of the last table above would be

Oscillations with so large an amplitude as 892 y in the horizontal force are rarely if ever observed, indicating that the surface potential gradient in the earth approaches half a volt per mile only when the period is shorter than 30 minutes.

OF THE

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PHILOSOPHICAL SOCIETY

VOLUME XXII. No. XXVI. pp. 483-517.

XXVI. THE ESCAPE OF MOLECULES FROM AN ATMOSPHERE, WITH SPECIAL REFERENCE TO THE BOUNDARY

OF A GASEOUS STAR

BY

E. A. MILNE, M.A.,

TRINITY COLLEGE, CAMBRIDGE

CAMBRIDGE

AT THE UNIVERSITY PRESS

M.DCCCC.XXIII

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XXVI. The Escape of Molecules from an Atmosphere, with special

reference to the Boundary of a Gaseous Star.

By E. A. MILNE.

[Communicated by Prof. H. F. Newall, F.R.S., Director of the Solar Physics Observatory, Cambridge. Received 15 January, 1923.]

§ 1. Scope of the paper. It has recently been suggested* that since for a giant star of the observed size of a Orionis the value of gravity at the surface can be at most a fraction of that at the surface of the moon and since the moon is unable to retain an atmosphere, therefore the atmospheres of such stars would be rapidly dissipated, and consequently the stars themselves if assumed to be gaseous would be dissipated also. The fallacy in this argument arises from the fact that the rate of dissipation of an atmosphere depends not on the gravitational acceleration but on the gravitational potential, since it is the latter which determines the critical velocity of escape of the molecules. The gravitational potential falls off only as the inverse first power of the distance, and it is easily calculated that for a Orionis, in spite of its large radius and consequently low surface value of gravity, the potential at the surface is relatively large, large enough moreover for the loss by diffusion to be inappreciable. The point nevertheless suggests that it would be of interest to apply the detailed theory of the escape of molecules from an atmosphere to stars of various masses and temperatures.

In the case of the atmospheres of the earth, moon and planets the question has received considerable attention. Johnstone Stoney in 1868 pointed out that on the kinetic theory of gases a proportion of the molecules would from time to time attain speeds greater than the critical speed of escape from the gravitational pull of the planet, and that such molecules if moving outwards in the regions of low density where collisions are rare would be permanently lost to the atmosphere; and he afterwards elaborated this in a series of papers†. The subject has also been considered by Sir George Darwin, Cook§, Bryan|| (chiefly from the point of view of the effects of rotation), Emden¶ and others, and a very clear summary of the method and the results of its application has been given by Jeans**. But none of these writers evaluates with any precision the height at which loss becomes appreciable, nor do they determine the critical density corresponding to this height, i.e. the density of the layer from which escape is mainly proceeding; again, one finds no definite estimate of the size of mean free path necessary in order that the chance of a collision may be sufficiently small. The reason appears to be that in certain cases precise knowledge of these quantities is immaterial. Jeans discusses in a general way the height of the critical level, but obtains his final result in a form independent of an evaluation of this height. For the rate of loss from an isothermal atmosphere he derives a formula expressed by the product of the critical density into a function of the critical height (quoted as formula (30)