L E T T E R E A L L A t { E D A Z I O N E

4La respons~bilit~ scientifica degli scritti inseriti in questa rubrica d com~letamente lasciata dalla Direzlone dcl perfodico ai singoli aulorl)

The External Magnetic Field of the Earth.

A . B E I S E R

Department o/ Physics, New York U~iversity, University Heights. New York, N .Y .

(ricevuto il 7 Gennaio 1958)

Recently i t has been discovered (1) ~hat the equivalent geomagnetic dipole required to account for the lati tude variation in cosmic-ray neutron intensi ty is different from the equivalent dipole derived from surface observations. The cosmic-ray data indicate that the dipole ,component in the equatorial plane, as seen by a charged particle incident upon the earth from a distant source, is ap- proximately 45 ~ west of the equatorial component obtained from the harmonic analysis of the surface magnetic field. While not too much is known about the properties of interplanetary matter, there is little doubt that it is a fairly good conductor of electricity, and therefore it seems reasonable to suppose that the origin for the above discrepancy is to be found in magnetic fields set up in this matter by currents there that are induced by the rotation of tile geo- magnetic equatorial dipole.

In the system of co-ordinates we shall use in treating this problem a magnetic �9 dipole of moment M is at the origin, and rotates about the z-axis with the

angular velocity to corresponding to the rotation of the earth. M has the com- ponents

M~ = M cos q~

( 1 ) M v = M sin q~

M~ = 0,

where ~ is the angle between M and the x-axis, and to=~),. The vector

p o t e n t i a l A at the point (x,y,z) due to M is

m x r (2) A - r g

and since ~--~ot, it is a function of time. This varying potential gives rise to an electric field

d E -

~t

in the interplanetary matter, and con- sequen~ly to a current density J given by

(1) j . A, SIMPSON, K . B. FENTON, J , I~&TZMAN (3) -und D. C. ROSE: Phys. Rev., 102, 1648 (1956).

aA J - z E = - - a - - ,

~t

T H E E X T E R N A L IV[AGNETIC F I E L D O F T~IE E A R T H 16i

where a is the conductivity. J has the components

(4)

~wMz cos 9 r 3

a w M z sin ~v

T3

a w M ( y sin ~ + x cos 9) J z

r 3

and evidently corresponds to current loops lying in planes parallel to the moving plane containing both M and %he z-axis.

O~dng to the physical state of the interplanetacry matter spontaneous cur- rents and associated magnetic ~ fields may be expected to exist in it which, beyond some mean distance R from the earth, dominate the field of M. We must *herefore consider J to be contained within a spherical shell whose inner radius R~ is tha t of the earth and whose outer radius R we shall leave unspecified for the moment.

The dipole moment M i of the cur- rents J is

{5) M~=} x,?dv,

where the volume integral is to be %aken over the spherical shell of the prece4ing paragraph. The induced di- pole M ~ turns out to have the com- ponents

( 6 )

3

2 A/I;, = - ~cr~o �9

3

M~ --- 0 �9

Comparing this result with Eq. (1) we see tha t the magnetic field induced in interplanetary mat ter by t h e rotating equatorial dipol~ M is equivalent to that of a dipole of moment

2 2 (7) M i = - z a ~ l ( R 2 - -R . ) 3

rotating i n the same plane and about the same axis a s / ~ / b u t displaced by 90 ~ behind it. Hence this analysis provides an explanation for the difference between the cosmic-ray a n d surface-field equa- torial dipoles, since the net equatorial field influencing the motion of cosmic-ray primaries in the -deinity of the earth is tha~ of M + M ~ beyond R. The field of M + M ~ is a n angle

m i

( 8 ) 0 = g g - l _ M =

R 2 ~ _-- tg-1 z~o)(B 2 - ~)

behind (i.e. to westward) of M. The experimental observation (1) that 0 ~ 45 ~ means that M / ~ M.

The resultant equivalent dipole re- presenting the external magnetic field of the earth, consisting of Md-Mi-d-M~x~, , should be tilted by several degrees farther from the rotational axis t h a n / ) / d - M ~ , as well as being displaced westward. I t seems likely that the small but definite discrepancies between the locations of t h e auroral maximum zones and those predicted on the basis of the surface-field dipole and between the actual positions of auroral corona radiation points and the surface-field magnetic zenith are due to M ~. Certain anomalies in the pro- pagation of radio whistlers may also result ~rom the presence of M i. (A de- tailed consideration of these effects is in preparation.)

From the point of view of hydro- magnetic theory, we can regard the inter- planetary matter as exerting a kind of

~o 11 - I I N u o v o Cimento.

162 n. BEISER

viscous drag on the lines of .force o f ~ / wh ich causes them to lag behind their .normal position with respect to ]//. The work that must be done in overcoming this drag is at the expense of the rota- t ional kinetic energy of the earth, and appears as Joule heat. In order to evaluate the power needed to mainta in the rotat ion of M, we begin by com- put ing the energy W in the magnetic fiel(1 H ~ outside of the earth due to ~//r This is

(9) 2]I i ~

3 R ~ ,

As we have found, M ~ ~ M, while M, the equatorial component of the equivalenf geomagnetic dipole, is 1.59.10 .5 G c m 3; hence W = 3.26. ]023 ergs. The time T required for the free decay of a mag- netic f~eld in a region whose linear di- mensions are of order L and whose con- ductivi ty is a is

(10) z ~ 4 ~ a L 2 �9

Upon setting 0=45 ~ in Eq. (8) we have

1 2

3 "

so that the magnetic field of M / decays with a t ime. scale approximately that involved in th6 rotat ion of the earth.

If now we suppose that the magnetic energy W must be replenished per radian of the earth's rotation, the power re- quired amounts to 2.38.1019 erg/s. This is slightly over twice the mean rate of energy dissipat ion resulting from tidal friction in ~he ocean basins. (Of course,

the above figure is subject to a good deal of uncertainty, perhaps as much as an order of magnitude either way.) I t is therefore possible t h a t the maintenance of M / may account for at least part of the discrepancy between the observe4 rate of lengthening of the day and the~ contribution of tidal friction to it.

Thus far it has not been necessary to evaluate either a or R explicitly. The~ relation between them is given b y Eq. (11). If R ~ 10/~,, which means. tha t a field strength of 10 -4 G due to M is the least that is effective in in- ducing J i n interplanetary space, a N

10 -16emu. If R > lOEb, a is even lower. But s smallest plausible con- duc t iv i ty for cosmic mat ter tha t we can arrive at by orthodox means is

10 -13 emu. However, it, should be pointed out tha t Eq. (~) is valid in a p l a sma such. as interplanetary mat te r only when the plasma is in a steady-state condition with its consti tuent ions havin~ a vanishing mean velocity and in the~ absence of a gravitat ional po t e n t i a l These conditions obvious!y do not obtaia here, With the consequence that Eq. (3} may be employed as it is only by ab- sorbing into the quant i ty a a variety of effects not included in the usual ex- pression for conductivity. ~ An e ] ] e c t i v e

conductivi ty of 10-1~emu or less there- fore does not seem unreasonable, and in fact estimates of its magnitude obtained_ in in'direct ways such as this may con- t r ibute to an undersiandifig of the pro- perties of interplanetary matter.

I am indebted t+ Professor S. A, KORFF for helpful discussions. This work was supported by the joint program of the U.S. Atomic Energy Commission and the U.~. Off~c,e o,f NavM Research,