The coefficient of diffusion of ions in the F2 regions

Research notes

However, if the polarization of the sky is taken into account, then observations made at 90” to the direction of the sun would benefit by a reduction in sky brightness of perhaps ten times (VAN DE HULST, 1952). Unfortunately the aurora1 light would suffer a depletion of 50 per cent, giving a net gain of five. A very serious obstacle is the scattering from the ozone layer. At 25 km this amounts to roughly half the ozone scattering at sea level. This implies an increase in scattering from the sky by a factor of 6 at 25 km. A particularly intense aurora might be ten times as bright as the one we have considered. Even so, there seems to be little chance of detecting it unless observations can be made from an even greater altitude. A. H. JARRETT P. L. BYARD

Uwiversity Observatory St. Andrews, Scotland REFERENCES ALLEN VAN

c. w. DE

HUNTEN

HULST

D. M.

H. C.

1955 1952

Astrophysical &zu&ities. University

of London. The Atmospheres of the Earth and Planets p. 79. University of Chicago Press. J. Atmosph. Terr. Phys. 7, 141.

1955

The coefficient of diffusion of ions in the F2 regions (Received

2 September

1957)

THE writer (FERRARO, 1945) derived an expression for the coefficient of diffusion, D,,, of ions in the ionosphere based on Sutherland’s molecular model. In this the molecules of a gas are treated as smooth rigid elastic spheres surrounded by a weak attractive field of force which for diffusion of ions in an otherwise neutral gas, is due to the attraction between the ion and the charge induced by it on a neutral molecule. Treating this as a conducting sphere the law of force is that of the inverse fifth power of the distance. The value of D,, derived by using Sutherland’s model was found to be b/n, where n is the molecular density of the neutral gas and b is a function of the temperature T and of the molecular weight of the gas which is equal to b =

3.9

x

101’T3”/(T

+ 187)

(1)

for a mean molecular weight of 25. At a temperature of about lOOO”, such as obtains in the F2 region, b is of the order of 10lg, and for this value of the coefficient of diffusion it appears difficult for a stable bank of ionization to persist for long at the level of the F2 region unless the molecular density there is of the order of lOlo mol/ cm3 at least. This conclusion is difficult to reconcile with recent rocket estimates of the molecular density in the F2 layer, which range between 5 x log and 5 x IO* mol/cm3. Because of this discrepancy the writer has reconsidered the assumptions made in deriving equation (1) and finds that the Sutherla,nd model, though not suitable in this case 296

Research

notes

since the attraction between an ion and neutral molecule cannot be regarded as small, nevertheless is not greatly in error at the temperatures obtaining in the F2 layer. The case where the attraction between the molecules is not small was first considered by LANGEVIN (1905), who gave a formula for the mobility of ions in a neutral gas. This formula appears to be in fair agreement with the experimental results, even though no account is taken of the effect of electron transfer. LANGEVIN’S formula for D,, is given by

D,,

f

= ~-~(kT/e)

where p is the density of the gas, m, and ms the masses of the neutral molecules and ion respectively, k is Boltzmann’s constant, e the electronic charge and E the dielectric constant, The non-dimensional parameter A occurring in equation (2) is a function of 1, where

p being the pressure of the gas and cr the sum of the radii function A has been computed by LANGEVIN a,nd a graph of p. 39). It has a maximum value of about 0.6 at I = O-6 and d is 0.505. When il is large, 1A + 2 and equation (2) then D,, for rigid elastic spheres, namely, D,,

=

-

of an ion and molecule. The it is given by TYNDALL (1935, as i --f 0 the limiting value of reduces to the expression for

3

(4)

Hno2

This is because when il is large, the attractive forces become negligible. In the F2 regions both oxygen and nitrogen are largely dissociated into atoms and it becomes difficult to estimate the value of (E - 1) occurring in equations (2) and (3) appropriate to these a t oms. However, E - 1 can sca.rcely exceed its value for the molecules and accordingly we take E - 1 = 5.8 x 10-14, as was done by the author (1945), which corresponds to the value of c for 0,. Taking also c N 3 x lo-* cm, and inserting numerical values for the constants, we find A2m5*64 x 10-4T (5) Thus unless T greatly exceeds lOOO”, A2 is of order unity. For such va,lues of L2, the value of A is about 4 and so may be treated as a constant. As was mentioned by the author (1945) (E - 1)/n is a constant, whence it follows from equation (2) that, D,, cc n-l. Adopting the value of (6 - 1) to be that corresponding to 0, as was done (1945), namely, 5.8 x 10-4, taking A = ij and assuming that the mass of the gas particles is that of an oxygen atom, we find from equation (2) that D,, = 8.69 x 10i5T/n

(6)

The values of nD1, for various values of T are illustrated in the following table; corresponding earlier values derived by the author (1945) are shown in parentheses.

n&s

T x 10-r*

300” 2.61 (4.2)

400” 3.48 (5.3)

500” 4.34 (6.4)

700” 6.1 (8.1)

the

1000°K S.69 (10.1)

It will be seen therefore that the values of nD,, calculated by the author are about double those calculated from LANGEVIN’S formula at ordinary temperatures and not very different at temperatures of the order obtaining in the F2 region. If the value of E - 1 is much smaller than that corresponding to molecular nitrogen or oxygen, the value of D,, to be 297

Research notes considered is that corresponding to the case when the molecules are treated as rigid elastic spheres. Taking the masses of the ions and atoms to be the mass of an oxygen atom, and inserting numerical values for the aarious constants, we find that

For 7’ = 400” and 900” the values of nD,, are 1.47 x 10lg and 2.15 x 10lg respectively that is, a little over three times as large as the values computed from LANQEVIN'S formula,., If the number density of the atmosphere in the F2 region is as low as rocket measurements indicate, and if, as observations a’ppear to suggest, diffusion in the F2 region is inappreciable, it follows that the calculated values of D,, may be too large by a factor of 10, unless diffusion is offset by other factors, such as movements of ions vertically upwards. Alternatively, the level of maximum electron density is below that of maximum ionproduction. One way of reducing the value of D,, to the required value would be to suppose that a stable neutral region of positive and negative ions is present below the F2 layer. When additional ionization is produced in this layer the electrostatic forces between the ions will be according to the inverse square law of the distance, provided there are at least lo6 ions/cm3. Taking the temperature of the F2 layer as 900”, the coefficient of diffusion would then be reduced to one-tenth of the value calculated above. However, the abundance of negative ions in the ionosphere has been investigated by several authors and it would seem that the ratio of the number of ions to electrons per unit) volume is not greatly different from unity in t’he E layer and about 1O-2 or less in the F layer. Thus the suggestion that the coefficient of diffusion may be reduced in this way is not attractive. If the density of the neutral gas is as low as the rocket measurements indicate, there can be little diffusion across the lines of force of t,he earth’s magnetic field. The effect is to reduce the coefficient of diffusion in the ratio of 1 : sin2 1, where 1 is the magnetic inclination of the locality. However, this reduct,ion is inappreciable except’ in the magnetic equatorial belt. V. C. A. FERRARO Queen Mary College University of London REFERENCES FERRARO 1'.C. A. LANGEVIN P. TYNDALL A. M.

1945 1905 1938

Terr.IWag.50, 215. Ann. Chim. Phys. 5, No. 8, 246. The Mobility of Positive Ions in Gases. Cambridge.

298