AN ELECTRON ORBIT IN THE MAGNETIC EQUATORIAL PLANE OF THE EARTH.

BY

W. F. G. SWANN,

Director of the Bartol Research Foundation of The Franklin Institute.

BARTOL RESEARCH The problem of the orbits of electrons in FOUNVATION. the earth's magnetic field has been treated

co,mmcation No.o. extensively by St6rmer i from the theoretical side, and by Birkeland 2 from the experimental side, chiefly in relation to the theory of the aurora. In connection with that problem, the orbit of an electron in the equatorial plane did not assume a role of primary importance, as the main point of interest concerned the possibility of explaining how electron orbits could penetrate to sufficiently low latitudes to coincide with the auroral latitudes. In view, however, of the recent recognition of the possibilities of electron energies much higher than would have seemed reasonable a few years ago, and particularly of the recent experiments by Prof. A. H. Compton on the variation of cosmic-ray intensity with latitude, the problem of electronic motion in the magnetic equatorial plane at zero latitude becomes of importance. The formula for this case is, of course, deducible immediately from the more general case. Since, however, it can be deduced from particularly elementary considerations for motion in the magnetic equatorial plane, it may be of interest to record here a derivation, particularly as the derivation serves to emphasize certain important features.

If r is the radius vector to a point P on a curve, o the radius of curvature at P, p the perpendicular distance from the origin to the tangent at P, and r0 the length of the tangent

1 St6rmer, Carl, Naturwissenschaften, 17, 643-651. The problem has also been attacked by P. Epstein: " Note on the Nature of Cosmic Rays," Proc. Nat ' l Acad. Sci., I69 658, I93 o.

2 Birkeland, K.: " The Norwegian Aurora Polaris Expedition . . . . "

465

466 W . F . G . SWANN. [J. F. I.

f rom the origin to the curve, then, as is well known,

d p r - i f r > r0, (I)

dr p

d p r . . . . if r < r0. (2) dr p

p is to be inser ted as posit ive if the radius vector str ikes the curve on its concave side a t P and negat ive if it s tr ikes it on its convex side.

In the case where there is no t angen t to the curve f rom the origin, the formula

d p r - ( 3 ) dr p

holds for all exist ing values of r.

If M is the ea r th ' s magne t ic momen t , m the relat ivist ic electronic mass, v the veloci ty of the electron, e the electronic charge, and H the magnetic-f ield intensi ty , since v~/p is the accelerat ion

mC M = H e y = --~ ev.

p

T h e veloci ty of the part icle is cons tan t along the path , since the force is perpendicular to the path . Since we are deal ing wi th a part icle whose veloci ty is near ly equal to c, the veloci ty of light, v = c. Moreover , the energy of the part icle is ( m - rno)c 2 which is ve ry approx ima te ly equal to rnc 2.

If V is the so-called energy in volts,

Ve X IO 8 = m c 2,

where e is in e lec t romagnet ic uni ts in both cases. Thus ,

Vr 3 × lO 8 P = M c

Hence,

For convenience, wri te

d p M c d r = 4- ~o 8 Vr----- ~ • (4)

g c Io 8 V - p g ' (5)

Oct., 1932.] A N ELECTRON ORBIT. 467

dp= ~ PS. d r r ~

For r > r0, we use the plus sign

p = A P°~, f

where A is a cons tan t which represents the value of p at r = oo, when the orbi t goes to infinity.

For r = to, p = o, so t h a t

A = P°-~. (6) r0

T h u s

p = po2(r~ - I ) for r > ro. (7)

For r < r0, we have to use the negat ive sign in (4), and

p = K + p o j . r

P u t t i n g r = ro at p = o, we find

K = - po2/ro,

so t h a t

p = p o 2 ( I - r ~ ) for r < r0. (8)

T h e m i n i m u m dis tance of approach is Pm obta ined by pu t t i ng r = p i n (8). T h u s

P u t Pm/po = x . T h e n (9) becomes

i Po X . . . . . ( IO)

X rO

F r o m (IO), wri t ing p o / r o = 7, we find

d x d x d x - x ~ + ~-G, + ~ = o, d-~ = ~ +---G"

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468 W . F . G . SWANN. [J. F. I.

Since x and 7 are posit ive, x diminishes cont inua l ly wi th increase of 7, and has its least va lue for the largest possible va lue of 7, i.e., for the smallest va lue of r0. We proceed to find the smallest possible va lue of r0.

P u t r = r0(I + ¢). Then , f rom (7),

_ =Po r r0 2 (I + e) ~

Now, for a given r0, ~/(I + ~)~ has a m a x i m u m for ~ = I. T h e value of this m a x i m u m is

4ro 2

By increasing p,~ we can d iminish r0 by (6). We can cont inue to d iminish ro in this m a n n e r so long as (p/r)max ~ I . T h e smallest possible va lue of r0 is, therefore, g iven by

/'02 r-~ = 4,

i.e., P___0 = 2. ( I I ) r0

It is of in teres t to observe t h a t the orbi t for which r0 has this value is perpendicular to the radius vec tor f rom the origin a t the plane where p / r has its m a x i m u m value. Th is orbi t does no t go to infini ty bu t takes the course qual i ta t ive ly shown in Fig. I, A. However , for a value of r0 infinitely near to the value for this orbit , we find an orbi t which does go to infinity. For, pu t

P0 (I + ~)

where ~ is a small quan t i ty . Then , on subs t i tu t ing in (7),

P = P°2 p:0(I

If we a t t e m p t to find a value of r for which p = r, we find

2po r = - - r + p o 2 = o,

I + a

r --

D

I 5 4- o~ I-~c~

This is never real for positive c~ unless a = o. Hence, for to, infinitesimally greater than po/2, the orbit goes to infinity, as in Fig. I, B, and this orbit gives the minimum distance of

A

Oct., I932.] AN ELECTRON ORBIT. 469

approach for practical purposes. Inserting this minimum value of r0 in (IO) to give the minimum value of x, we find

the solution of which is

Therefore,

Hence, from (5)

I X = - - - 2 ,

X

x = 4 ~ - I.

p~ = ( ~ / 2 - I )p0 = o.41 Po.

Mc V =

5.9 × IOs Pm ~

470 W . F . G . SWANN. [J. F. I .

Put t ing Pm equal to R, the radius of the earth, and remember- ing tha t the field H0 at the surface is M / R 3, we have

HocR V =

5.9 X IO s"

Pu t t ing H0 = o.31, R = 6.4 >( lO 8 , V = lO l° volts. I t is of interest to calculate, for this case p®, the dis tance

from the origin to the a s y m p t o t e to the curve at r = oo. Since A = p®, for this case, we have

f l o 2 2 2

P® = r0 2p0 Pm O"41R" (12)

p~ = 5R.

In other words, the electron which jus t grazes the ear th 's surface is one which a t infinity t ravels along a line whose perpendicular dis tance from the ear th 's center is five t imes the ear th ' s radius. Moreover , as a l ready shown, this orbi t is pract ical ly perpendicular to the radius vec tor from the origin a t the value of r obta ined b y put t ing p = r in (7), i.e., in view of ( I I ) , at r = P0, i.e., a t r = 2.5 R.

For the case such as is represented b y (3) for all values of r, and where the origin lies inside the curve, i.e., for the case where the orbi t passes around the earth, bu t does no t make a loop as in Fig. 1, B.

p0 2 p = p . - - - - . (I3)

r

No w p is a lways less than or a t most equal to r. Thus,

P __ P® p02 <~ I .

r r r 2

Pu t p-2~ = X, and r/po = y. Then P.

X 1 y y2

ow() ,) f has a max imum for y = 2/X; and, for this

Oct., 1932.] A N ELECTRON ORBIT. 471

value it is equal to X~/4. Hence,

~ -<4 and

p~ --< 2p0.

Now from (I3), the nearest distance of approach is given by Pro, where

p r o = p _ p°~ . pm

P u t p m / p 0 - - x. T h e n x + t - - ) , = o. X

x±~/~ - 4 X =

2

This gives a real value only if X2 >_ 4. But we must have X~ ---< 4. Hence, the only possibility is X = 2, and x = I. In other words, p~ = P0. The orbit in question is none other than the orbit of Fig. I, B, with the loop missed out. The parts P and Q of the orbit join continuously at the point D for this limiting case, so tha t both orbits are possible, one with the loop and one wi thout it.

In connection with the foregoing consideration, it is worth while to point out tha t in considering celestial bodies as the origin of electron rays, one must pay a t tent ion to the radiation coming from bodies on the side of the ear th remote from tha t on which we are stationed at the instant.

BARTOL RESEARCH FOUNDATION OF THE FRANKLIN INSTITUTE,

August 6, 1932.