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On H. Alfvén's theory of the effect of magnetic storms on cosmic ray intensity
ON H. ALFV~N'S THEORY OF THE EFFECT OF MAGNETIC STORMS ON COSMIC RAY INTENSITY * BY W. F. G. S W A N N 1 ABSTRACT
H. Alfv~n's theory envisages a stream of gas shot out from a place on the sun where there is a large magnetic field. The gas, supposed highly conducting, carries the magnetic field H with it, and the field is supposed to be perpendicular to the velocity v. A stationary observer observes an electric polarization perpendicular to H and v, and the potential change across the beam of width h is calculated as Hvh/C. A cosmic ray crossing the beam is supposed to change its energy by Hohe/C, and by making H or h large this change of energy can be made as great as is desired. Alfvdn calculates a 10 per cent change for the energy of a 3 X 10 ~°e.v. ray on crossing a beam of width 5 X 10TM cm. carrying a magnetic field of 3 X 10-4 gauss and travelling with a velocity of 2 X 10 s cm./sec. The present paper recognizes t h a t an observer moving with the beam observes no change of energy, and by a relativity transformation, it is shown t h a t a fixed observer cannot observe a change of energy greater than 2vW/C, where W is the energy of the cosmic ray. This change amounts to only about 1.4 per cent. The discrepancy between the conclusions from relativity and those drawn by AlfvSn are shown to arise from the fact t h a t the cosmic ray path in the example considered by Alfv~n would have, in the beam, a radius of curvature 15 times smaller than the beam width and so could never cross it. Thus relativity gives for the change of energy of the beam an upper limit of the order 2v/C times the energy of the beam, and this limit can never be exceeded, however large H or h may be chosen. An analysis of the considerations resulting from the assumption of a finite conductivity for the space external to the beam shows t h a t the existence of such a conductivity does not change the conclusions. INTRODUCTION
H. Alfv6n ~ has proposed an ingenious theory to explain the effect of magnetic storms on cosmic ray intensity. It is founded upon the assumption t h a t a mass of highly conducting gas originating in a region where there is a large magnetic field emerges a s a stream, carrying the magnetic field with it as a result of its high conductivity. T h e case is considered in which the magnetization is perpendicular to the velocity of the stream, in which case the motion of the matter, in its own magnetic field, brings about a condition in which a stationary observer sees a charge displacement in a direction perpendicular to the velocity and to the magnetic field: Alfv6n's theory, as presented, is confined to * Assisted in part b y the Joint Program of the U. S. Office of Naval Research and the U. S. Atomic Energy Commission. 1 Director, Bartol Research Foundation of The Franklin Institute, Swarthmore, Pa. 2 H. ALFV~N, Phys. Rev., Vol. 75, p. 1732(1949). a In all t h a t follows, we shall denote by S the so-called laboratory system with respect to which the motion is observed. S ' shall denote a system moving with the gas. The velocity, v, shall be parallel to the axis of x . The magnetic field, H, shall be along the axis of g which shall be perpendicular to the plane of the paper, so t h a t the axis of y shall be in the plane of the paper and perpendicular to the velocity. I9I
I92
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the case where the stream of gas moves in a non-conducting medium, and our discussion will be limited, in the first instance, to this case. T h e a r g u m e n t of Alfv6n is to the effect t h a t the field along the axis of y is - vH/C, and the potential change of a cosmic r a y in crossing the b e a m is vhH/C, where h is the width of the beam and C is the velocity of light. It is thus concluded t h a t if T is the kinetic energy, the change in kinetic energy in crossing the beam is ~T, where ~T = vhHe/C,
(1)
where e is the charge on the particle. T h e change of kinetic energy leads to an expression for the change of cosmic r a y intensity, in a m a n n e r which does not concern us here, 4 since our problem involves the change in kinetic energy, and in particular the relation (I). DISCUSSION OF T ~ m TrmORY
Relativistic Considerations In connection with the derivation and implications of Eq. 1, one m u s t observe t h a t we are here dealing with a problem in which the magnetic field moves with the m a t t e r . It will be convenient to conc e n t r a t e our a t t e n t i o n upon a finite slab, as represented in Fig. 1.
X
J f f f J
J
I
I
J
J
J
f I !
f
J
J J Jr J J Jr
Jfff
J Jr f J J J J ~
i
~
FIG. 1. 4 Incidentally, there appears to be an error in Alfv~n's calculation of the relation between change of intensity and change of energy. This matter is discussed in the present writer's paper "Cosmic Rays," JouR. FRANKLIN INST., Vol. 251, pp. 120-155 (1951). See in particular pages 146-147.
Mar., 1 9 5 4 . 1
ALFVI~N'S THEORY OF MAGNETIC STORMS
193
N o w we m u s t observe t h a t in the s y s t e m S' which moves with the slab, there is no electric field, b u t only a magnetic field. As a consequence, the cosmic r a y particle never gains a n y e n e r g y in t e r m s of m e a s u r e m e n t s m a d e in S'. If W is the e n e r g y of the particle ~ and M the m o m e n t u m , we have t h a t Mx, M~, M,, i W / C constitutes a 4-vector. T h u s e WU~, WU~, WU~, iWC constitutes a 4-vector where U is the velocity of the cosmic r a y particle in S. T h u s if dashed letters refer to the s y s t e m S', and if 3 = (1 - v2/C2)-i, W=
3
vU~'W' )
W'÷
C---7 -
.
(2)
If, in the s y s t e m S', the particle experiences a change in Ux', from Uxl' to Ux~', then since W' does not change in S', we shall have a change in W, given b y W2 - W1, where W2 -- Wl
= ~W'(
Ux2 ! - - U z l t ) V
C2
(3)
Now the greatest possible value of U . , ' - U.I' is ± 2C. Thus, if [ W~ - W1 [ is the absolute value of the change of W, we have IW2-
23W'v
W,I < ~
(4)
F r o m (2) VUxl !
Wl = 3W'
1÷ ~
]
so t h a t
IW - Wll
2 Wlv
< c
VUxlt )"
N o w the smallest possible value of (1 + vU, I'/C 2) is (1 - v/C). Hence
IW
-W,I <
ITs - TI[ < ~
2v
2Wiv (T, + mock),
(s)
and unless v is comparable with C 2v
ITs - T,I < U (T, + m0C2).
(6)
5 It is to be observed that T ffi W -- m06n. e The essentials of the derivation immediately following are given in my paper cited under footnote 4. See page 134.
I94
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[J. F. L
A lfv£n' s Numerical Conclusions N o w Alfv~n ~ considers a cosmic r a y of e n e r g y 3 X 10 l° e.v., and demands, t h r o u g h his formula (1) a ten per cent change, with h = 5 X 10 TM, v = 2 X 108 cm./sec. He finds H = 3 X 10 -4. W i t h the above value of v, the q u a n t i t y / 3 is practically unity, and our formula (6) gives T1
1 + 3 X 101°]
< ___~
< 1.37 X 10 -~. Thus, Alfvfin, from the aforesaid data, calculates a 10 per cent change in T, and we have shown t h a t the m a x i m u m possible change is 1.37 per cent. Moreover, in calculating the value we h a v e not had occasion to utilize a n y q u a n t i t y b u t v. W h a t is the origin of the disc r e p a n c y in this m a t t e r ?
Origin of the Discrepancy It will be shown t h a t the aforesaid discrepancy lies chiefly in the fact t h a t Alfv6n's m a g n e t i c field H = 3 X 10-* would never allow a cosmic r a y particle to pass t h r o u g h the slab. T h e easiest w a y to see this is from consideration of radius of c u r v a t u r e . Let us consider m a t t e r s in S'. If p is the radius of c u r v a t u r e of our cosmic r a y in the field H', we have, if m is the relativistic mass, m U '2
H ' e U'
p C mU'C mC~(U') W' ( ~ ) O - H'e - H'e --C = ~ee
(7)
p < W'/H'e. For 7 W' = 3 X 10 l ° e . v . a n d H =
3 X 10 -4
3 × 101°
101~ P < 300 X 3 X 10-' = --3-" T h u s the radius of c u r v a t u r e is fifteen times less t h a n the thickness (5 X 101~ cm.) assumed for the slab. A particle entering the slab in S' could n o t possibly travel a distance greater t h a n 2p parallel to the axis of y before r e t u r n i n g and emerging from the slab. Thus, no particle can p e n e t r a t e right through the slab u n d e r the conditions cited. N o second or third " t r y " in the e v e n t of the particle's seeking r e - e n t r y can be a n y more successful. Thus, the example cited b y Alfv6n represents a situation impossible of realization. We need not concern ourselves with the difference b e t w e e n IV' and W in the calculation.
Mar., I 9 5 4 . ]
I95
ALFV~N'S THEORY OF MAGNETIC STORMS
I t is of interest to trace the above story by a n o t h e r p a t h s t a r t i n g w i t h Alfv6n's expression (1), viz. vhHe
6T=~
C
which we m u s t assume to certainly fail if h > 20.
Since (8)
6T -
OvhH' e
C N o w we can s u b s t i t u t e for He' in t e r m s of p by using (7).
6T=
Thus
W' U'vh pC2
~T increases with increase of h, and its m a x i m u m value is reached w h e n h = 2p, t h u s 2/3v ~T <-~--( ~ ) W' < ~ W' (9) which is the s a m e as the result (4) o b t a i n e d directly from the relativity transformation.
Connection between Relativistic Formula and Alfv~n's Formula when the Latter is Applicable On the o t h e r hand, we can show t h a t if the conditions of smallness of change of energy are satisfied, t h e n Alfv6n's result follows from the relativity t r a n s f o r m a t i o n . T h u s if 01' is the angle m a d e by the cosmic ray with the plane of the slab a t e n t r a n c e in S' a n d 02' is the angle m a d e a t emergence, w h e n emergence occurs, we have, referring to (3) UJ
-
Uxl' = U' cos 02' -
U' cos 01'.
Now, if ds' is an e l e m e n t of p a t h in the slab
dy = sin 0' ds' = p sin 8' dO' where p is the radius of curvature, which, in this case, is constant. Thus -
P
h = p ( c o s 0~' -
c o s 01') = ~
Now m U '~
H'e U'
p
C
(U~'
-
Uz,').
(10)
196
W.F.G.
SWANN
[J. F. I.
Hence, from (10) Ux2
~
Uxl ! --
H'eh mC
H'ehC W'
so t h a t (3), which is derived from relativity, yields W2-
W1 --
3H' ehv C
or, since, from (8), Hz = BH' IT2-
T1]=IW2-
Wll-
Hehv C
(11)
which is Alfv6n's result limited however, as aforesaid, to an upper limit defined by (5), a limit which a m o u n t s to a fraction 2v/C of the energy itself, and which is in general much less than 10 per cent. No assumption of large magnetic field or large value of h can enable (W2 - W1) to assume a value larger than that determined by (5). It is of some interest to inquire how the relativity transformation symbolized by its results in (5) seems to " k n o w " of the impossibility of penetrating the slab under the conditions stated. T h e fact is t h a t the transformation from the pure magnetic field in S' to the combined electric and magnetic fields E and H in S is one obeying relativity. Also the force equation of electrodynamics which tells us t h a t there is no gain of energy of the particle in S' while there m a y be a loss or gain in S is also an equation obeying relativity; and if these equations had permitted the passage of the particle through the slab under conditions in which (11) had given a greater value of W2 - WI than was permitted by (5), an algebraical inconsistency would have been involved. Case of a Continuous Beam In Alfv6n's paper, he assumes a continuous stream of gas emerging from the sun. If such a gas spreads on its journey, the value of v will vary. This appears at first sight to introduce a complication. However, the complication is obviously trivial as regards its practical effects. Thus, consider any portion of the beam through which the p a t h of a cosmic ray particle is contemplated. If S' applies to this portion of the beam, and if v is its velocity, the motion in S' will be determined by the magnetic field in S' and by an electric field which comes essentially from the charge displacement in other parts of the beam which do not move with the velocity v.8 The field produced in S' by these displacements, a m o u n t i n g as it does to the field produced by the charges on a "condenser" when evaluated outside of the region between the plates of the condenser, will be negligible for our purposes. A s T h e r e is, of course, no charge d i s p l a c e m e n t in S' a t t h e portion m o v i n g with velocity v.
Mar., 1954.]
ALFVEN'S THEORY
OF M A G N E T I C
]97
STORMS
formal, exact, analysis of all t h a t is here involved would result in a lengthy presentation whose end point would, in the nature of things, involve b u t a small correction on the conclusions which would be drawn by ignoring it. CASE WHERE THE SLAB MOVES IN A CONDUCTING MEDIUM
Limiting Case of Infinite Conductivity T h e fact t h a t we have supposed the slab to carry its internal magnetic field with it demands, for logical consistency, t h a t the external medium, if infinitely conducting, shall resist change of magnetic field within itself to the extent t h a t the magnetic field H remains zero everywhere outside of the slab as viewed in the system S. If the conductivity of the region external to the slab is ohmic as viewed in S, then even though in the limit it tends to infinity, no space charge will arise in S, and consequently 9 by the relativity transformation, there will be no space density in S'. T h e only charge of any kind will be on the surface of the slab. F r o m s y m m e t r y the only possible surface charge will be a density ~° ~' at y = h/2 and - ~' at y = - hi2. T h e field c o m p o n e n t Eyo' in S' just outside the slab at its mid-point will be a small negative fraction - , of the field inside the slab which is - 4 ~ ' . T h u s Evo' = 4*r~ t. 9 Some care is necessary in drawing this conclusion, as is evidenced by the fact t h a t a system which viewed from S appears as a uniformly magnetized sphere, appears in the other system as having in addition equal and opposite charge densities displaced along a line perpendicular to the magnetic induction in the sphere and to the velocity, v. It will, in fact, be recalled t h a t if U is the vector potential, and ~ the scalar potential, U, io constitute a 4-vector, so t h a t the zero value of ~ in one system does not guarantee ~' = 0 for the other system. The apparent paradox in relation to the invariance of total charge becomes resolved as follows : The densities p and p' for a given sign of electricity, let us say negative, are related in the two systems by
,, where u== is the velocity in the system x. Suppose now we superpose a positive density have ,,
p÷
in S, such t h a t p+ -k p-
=
O. We shall
=
Then
[
= ~ p+ + p _ -
v
~ (p+u=~ + p-u=3
] •
Hence, if, and only if (p+u~ 4- p_u,1) = O, can we conclude t h a t p+' 4- p_' == 0 follows from (p+ 4. p_) -- O. In our present problem, since H = 0 in S, curl H = 0 and the total current density (p.u=.~ 4. p-u=1) is zero, and the desired result follows. 10¢, will be sensibly constant except near the edges of the slab,
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[J. F. I.
If Xl is the c o n d u c t i v i t y within the slab, the y c o m p o n e n t of the current density at x = h/2 will be - 4 r z ' X ~ . T h e corresponding component
outside
w i l l b e 11
Eyo' + Vc ~oH f~ ] X2. T h u s we m u s t have
-- 4"tra'Xl =
( 4rca'e + ~,v Hzo' ) X2.
(12)
N o w since H is zero a t all points outside the slab as measured in S, we have for a point outside the slab 0 -- H~. -- ~ ( H /
~~E ~ ' ) .
-
(13)
Thus v H
' =
, =
v
T h u s from (12) - 47r¢'Xl = 47tea'
1 - ~
X..,.
(14)
However, this is a relation b e t w e e n Xl and X~ which d e p e n d s upon e and v only, and these quantities have nothing to do with the cond u c t i v i t y of the media. 12 T h e only possibility is represented b y a' = 0, which permits (14) to hold with a n y assignment of the ratio Xl/X2. If ~' = 0, we have, of course, E~' = E,f = E~' = 0 at all points outside the slab. As a consequence, in the same region H / = 0 on a c c o u n t of (13), and Hy' = 0 because o=H.=
3
(
v)
.
Moreover, H~' = H , = 0. T h e fact t h a t the state of no electric or magnetic field outside of the slab in either S or S' constitutes a solution of the problem is obvious b y direct inspection, since it calls for no currents outside the slab in S' resulting from the motion of t h a t slab. Inside the slab itself there is nothing b u t a magnetic field, and the whole problem reverts completely to the problem we have already considered when we a t t r i b u t e d zero c o n d u c t i v i t y to the m e d i u m outside of the slab. T h e conclusions are exactly the same.
How Is the Condition of Zero External Field Realized in S'? It is of interest to inquire how it can come a b o u t t h a t the " m a g netized" slab can exist in the external m e d i u m which itself shows no 11 O b s e r v e t h a t v. = - v. ~*W e again r e m a r k t h a t s u c h a relation as (12) h a s sense as Xl a n d X2 b o t h t e n d to infinity.
Mar., I954.]
ALFVt~N'S T H E O R Y
OF M A G N E T I C
STORMS
I99
magnetic field in either S or S'. The fact is that, in the simple case, the magnetic field in the slab d e m a n d s a current sheet circulating around its boundary in planes perpendicular to the axis of z. This sheet in itself would call for a return magnetic flux outside the slab, a flux which was small in a m o u n t at any point b u t widely spread out so t h a t its total integrated value was equal to the total flux through the slab. Now the p h e n o m e n o n which provides for the cancellation of the magnetic field outside the slab is another current sheet flowing in the reverse sense outside the first one and again in planes perpendicular to the axes of z. If these two current sheets have different crosssectional areas, 13 they can cancel as regards production of magnetic field outside the outermost, but give a resultant uniform field inside the smaller. In the limit, when the currents are infinitely large, their cross-sections can afford to differ by only infinitesimal amounts, and this condition represents the limiting one applicable for infinite conduction in the m e d i u m external to the slab. In case this above picture presents any difficulty, we m a y cite the particular simple example of a uniformly charged rotating sphere. We shall not trace the details of calculation for this case, which are elementary, and generally known. Such a sphere gives, at points outside, a magnetic potential ft, where M cos 0 y~
Here M = ~o054, where a is the surface density in e.m.u., o~ is the angular velocity and a is the radius. T h e field inside is uniform and is given by H, where H=
2M 53
It will thus be seen t h a t if we have two spheres of radii al, and as with ~'s adjusted so t h a t M1 = - Ms these spheres will cancel as regards magnetic field outside of the larger, while inside the smaller they will have a resultant magnetic field given by 1 )
HI_H~=2M(1 s
a2S
"
If al -- a2 is small ( = ~a) we m a y write - 6M~a H I -- H 2 = - -5 4 1~We refer here to the cross-section of the whole "solenoid structure" which consthutes such a sheet. This cross-sectional area is, of course, sensibly equal to that of the slab.
200
W.F.G.
SWANN
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Case Where Conductivities Xl and X2 Are Finite ~4 The complete problem here involved can become one of considerable complexity. The currents arising from the motion will affect the magnetic field, moreover the charge distributions arising on the slab to provide for continuity of current flow across the slab boundaries will bring about x-components of the current density at any rate in regions remote from the central regions and 9 will destroy even the conclusion that the volume charge density shall be zero in S if it is zero in S' and vice-versa. We shall not a t t e m p t an exhaustive solution of the problem, but shall content ourselves with a kind of model founded upon the considerations represented in Eq. 13, but without the restriction that the magnetic field shall be zero in S for points external to the slab. We have in S a slab with face A at potential V / 2 and face B at potential - V/2, respectively, A and B being at x = h / 2 and x = - h/2, respectively. Outside the slab, the current density is determined entirely by the field arising from this potential difference. Inside the slab we have a force per unit charge given by E~ - vHz. The line integral of the external field from A to B is V. The line integral of the internal total force per unit charge from B to A is - V - vHh/C. Suppose R, is the ordinary external resistance which would be measured between the two surfaces of the slab, if the slab itself were of zero conductivity, and R~ the resistance which would be measured if the conductivity of the external medium were zero. Then we have
V I = -- = R,
V - vHh/C R~
(15)
Hence, 1
1 )
v
-vHh =
V=-
R, Ri+R,
cR, ) vHh C
The field E inside the slab i s
E =
R, R~+R,
)vii --C"
(16)
This has its largest value E = v H / C when R, = ~ which represents the case first discussed in this paper and the result given by Alfv6n. u The effect of finite conductivity has been discussed in the writer's paper "Cosmic Rays," Jova. FRANKLIN INST., Vol. 2.51, pp. 120-155 (1951). The conclusions drawn at the bottom of p. 134 and the top of p. 135 require correction in the light of the development in the present paper.
Mar., I954.]
ALFVI~N'STHEORY OF MAGNETIC STORMS
2OI
We have already studied the other extreme where R, -- 0 and where there is no magnetic field external to the slab as observed in S. In this case the result E = v H / C also evolves. We can form a quasiq u a n t a t i v e picture of the intermediate as follows: It is easy to see t h a t the induced currents external to the slab tend to reduce the field which would have been present in their absence. T h e y reduce the field n o t only in the external region, but in the interior of the slab. T h u s if the field H in (16) represents the field actually in the slab, it should be multiplied by a factor greater than unity. While we have not traced everything in detail, it is reasonably obvious t h a t the net effect will result in a value of E which is never greater than the value v H / C where H is the actual field in the slab, this result being attained exactly in the two limiting cases where R, --oo and where R, -- 0, respectively.
H. Alfv~n's theory envisages a stream of gas shot out from a place on the sun where there is a large magnetic field. The gas, supposed highly conducting, carries the magnetic field H with it, and the field is supposed to be perpendicular to the velocity v. A stationary observer observes an electric polarization perpendicular to H and v, and the potential change across the beam of width h is calculated as Hvh/C. A cosmic ray crossing the beam is supposed to change its energy by Hohe/C, and by making H or h large this change of energy can be made as great as is desired. Alfvdn calculates a 10 per cent change for the energy of a 3 X 10 ~°e.v. ray on crossing a beam of width 5 X 10TM cm. carrying a magnetic field of 3 X 10-4 gauss and travelling with a velocity of 2 X 10 s cm./sec. The present paper recognizes t h a t an observer moving with the beam observes no change of energy, and by a relativity transformation, it is shown t h a t a fixed observer cannot observe a change of energy greater than 2vW/C, where W is the energy of the cosmic ray. This change amounts to only about 1.4 per cent. The discrepancy between the conclusions from relativity and those drawn by AlfvSn are shown to arise from the fact t h a t the cosmic ray path in the example considered by Alfv~n would have, in the beam, a radius of curvature 15 times smaller than the beam width and so could never cross it. Thus relativity gives for the change of energy of the beam an upper limit of the order 2v/C times the energy of the beam, and this limit can never be exceeded, however large H or h may be chosen. An analysis of the considerations resulting from the assumption of a finite conductivity for the space external to the beam shows t h a t the existence of such a conductivity does not change the conclusions. INTRODUCTION
H. Alfv6n ~ has proposed an ingenious theory to explain the effect of magnetic storms on cosmic ray intensity. It is founded upon the assumption t h a t a mass of highly conducting gas originating in a region where there is a large magnetic field emerges a s a stream, carrying the magnetic field with it as a result of its high conductivity. T h e case is considered in which the magnetization is perpendicular to the velocity of the stream, in which case the motion of the matter, in its own magnetic field, brings about a condition in which a stationary observer sees a charge displacement in a direction perpendicular to the velocity and to the magnetic field: Alfv6n's theory, as presented, is confined to * Assisted in part b y the Joint Program of the U. S. Office of Naval Research and the U. S. Atomic Energy Commission. 1 Director, Bartol Research Foundation of The Franklin Institute, Swarthmore, Pa. 2 H. ALFV~N, Phys. Rev., Vol. 75, p. 1732(1949). a In all t h a t follows, we shall denote by S the so-called laboratory system with respect to which the motion is observed. S ' shall denote a system moving with the gas. The velocity, v, shall be parallel to the axis of x . The magnetic field, H, shall be along the axis of g which shall be perpendicular to the plane of the paper, so t h a t the axis of y shall be in the plane of the paper and perpendicular to the velocity. I9I
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[J. F. I.
the case where the stream of gas moves in a non-conducting medium, and our discussion will be limited, in the first instance, to this case. T h e a r g u m e n t of Alfv6n is to the effect t h a t the field along the axis of y is - vH/C, and the potential change of a cosmic r a y in crossing the b e a m is vhH/C, where h is the width of the beam and C is the velocity of light. It is thus concluded t h a t if T is the kinetic energy, the change in kinetic energy in crossing the beam is ~T, where ~T = vhHe/C,
(1)
where e is the charge on the particle. T h e change of kinetic energy leads to an expression for the change of cosmic r a y intensity, in a m a n n e r which does not concern us here, 4 since our problem involves the change in kinetic energy, and in particular the relation (I). DISCUSSION OF T ~ m TrmORY
Relativistic Considerations In connection with the derivation and implications of Eq. 1, one m u s t observe t h a t we are here dealing with a problem in which the magnetic field moves with the m a t t e r . It will be convenient to conc e n t r a t e our a t t e n t i o n upon a finite slab, as represented in Fig. 1.
X
J f f f J
J
I
I
J
J
J
f I !
f
J
J J Jr J J Jr
Jfff
J Jr f J J J J ~
i
~
FIG. 1. 4 Incidentally, there appears to be an error in Alfv~n's calculation of the relation between change of intensity and change of energy. This matter is discussed in the present writer's paper "Cosmic Rays," JouR. FRANKLIN INST., Vol. 251, pp. 120-155 (1951). See in particular pages 146-147.
Mar., 1 9 5 4 . 1
ALFVI~N'S THEORY OF MAGNETIC STORMS
193
N o w we m u s t observe t h a t in the s y s t e m S' which moves with the slab, there is no electric field, b u t only a magnetic field. As a consequence, the cosmic r a y particle never gains a n y e n e r g y in t e r m s of m e a s u r e m e n t s m a d e in S'. If W is the e n e r g y of the particle ~ and M the m o m e n t u m , we have t h a t Mx, M~, M,, i W / C constitutes a 4-vector. T h u s e WU~, WU~, WU~, iWC constitutes a 4-vector where U is the velocity of the cosmic r a y particle in S. T h u s if dashed letters refer to the s y s t e m S', and if 3 = (1 - v2/C2)-i, W=
3
vU~'W' )
W'÷
C---7 -
.
(2)
If, in the s y s t e m S', the particle experiences a change in Ux', from Uxl' to Ux~', then since W' does not change in S', we shall have a change in W, given b y W2 - W1, where W2 -- Wl
= ~W'(
Ux2 ! - - U z l t ) V
C2
(3)
Now the greatest possible value of U . , ' - U.I' is ± 2C. Thus, if [ W~ - W1 [ is the absolute value of the change of W, we have IW2-
23W'v
W,I < ~
(4)
F r o m (2) VUxl !
Wl = 3W'
1÷ ~
]
so t h a t
IW - Wll
2 Wlv
< c
VUxlt )"
N o w the smallest possible value of (1 + vU, I'/C 2) is (1 - v/C). Hence
IW
-W,I <
ITs - TI[ < ~
2v
2Wiv (T, + mock),
(s)
and unless v is comparable with C 2v
ITs - T,I < U (T, + m0C2).
(6)
5 It is to be observed that T ffi W -- m06n. e The essentials of the derivation immediately following are given in my paper cited under footnote 4. See page 134.
I94
W.F.G.
SWhNN
[J. F. L
A lfv£n' s Numerical Conclusions N o w Alfv~n ~ considers a cosmic r a y of e n e r g y 3 X 10 l° e.v., and demands, t h r o u g h his formula (1) a ten per cent change, with h = 5 X 10 TM, v = 2 X 108 cm./sec. He finds H = 3 X 10 -4. W i t h the above value of v, the q u a n t i t y / 3 is practically unity, and our formula (6) gives T1
1 + 3 X 101°]
< ___~
< 1.37 X 10 -~. Thus, Alfvfin, from the aforesaid data, calculates a 10 per cent change in T, and we have shown t h a t the m a x i m u m possible change is 1.37 per cent. Moreover, in calculating the value we h a v e not had occasion to utilize a n y q u a n t i t y b u t v. W h a t is the origin of the disc r e p a n c y in this m a t t e r ?
Origin of the Discrepancy It will be shown t h a t the aforesaid discrepancy lies chiefly in the fact t h a t Alfv6n's m a g n e t i c field H = 3 X 10-* would never allow a cosmic r a y particle to pass t h r o u g h the slab. T h e easiest w a y to see this is from consideration of radius of c u r v a t u r e . Let us consider m a t t e r s in S'. If p is the radius of c u r v a t u r e of our cosmic r a y in the field H', we have, if m is the relativistic mass, m U '2
H ' e U'
p C mU'C mC~(U') W' ( ~ ) O - H'e - H'e --C = ~ee
(7)
p < W'/H'e. For 7 W' = 3 X 10 l ° e . v . a n d H =
3 X 10 -4
3 × 101°
101~ P < 300 X 3 X 10-' = --3-" T h u s the radius of c u r v a t u r e is fifteen times less t h a n the thickness (5 X 101~ cm.) assumed for the slab. A particle entering the slab in S' could n o t possibly travel a distance greater t h a n 2p parallel to the axis of y before r e t u r n i n g and emerging from the slab. Thus, no particle can p e n e t r a t e right through the slab u n d e r the conditions cited. N o second or third " t r y " in the e v e n t of the particle's seeking r e - e n t r y can be a n y more successful. Thus, the example cited b y Alfv6n represents a situation impossible of realization. We need not concern ourselves with the difference b e t w e e n IV' and W in the calculation.
Mar., I 9 5 4 . ]
I95
ALFV~N'S THEORY OF MAGNETIC STORMS
I t is of interest to trace the above story by a n o t h e r p a t h s t a r t i n g w i t h Alfv6n's expression (1), viz. vhHe
6T=~
C
which we m u s t assume to certainly fail if h > 20.
Since (8)
6T -
OvhH' e
C N o w we can s u b s t i t u t e for He' in t e r m s of p by using (7).
6T=
Thus
W' U'vh pC2
~T increases with increase of h, and its m a x i m u m value is reached w h e n h = 2p, t h u s 2/3v ~T <-~--( ~ ) W' < ~ W' (9) which is the s a m e as the result (4) o b t a i n e d directly from the relativity transformation.
Connection between Relativistic Formula and Alfv~n's Formula when the Latter is Applicable On the o t h e r hand, we can show t h a t if the conditions of smallness of change of energy are satisfied, t h e n Alfv6n's result follows from the relativity t r a n s f o r m a t i o n . T h u s if 01' is the angle m a d e by the cosmic ray with the plane of the slab a t e n t r a n c e in S' a n d 02' is the angle m a d e a t emergence, w h e n emergence occurs, we have, referring to (3) UJ
-
Uxl' = U' cos 02' -
U' cos 01'.
Now, if ds' is an e l e m e n t of p a t h in the slab
dy = sin 0' ds' = p sin 8' dO' where p is the radius of curvature, which, in this case, is constant. Thus -
P
h = p ( c o s 0~' -
c o s 01') = ~
Now m U '~
H'e U'
p
C
(U~'
-
Uz,').
(10)
196
W.F.G.
SWANN
[J. F. I.
Hence, from (10) Ux2
~
Uxl ! --
H'eh mC
H'ehC W'
so t h a t (3), which is derived from relativity, yields W2-
W1 --
3H' ehv C
or, since, from (8), Hz = BH' IT2-
T1]=IW2-
Wll-
Hehv C
(11)
which is Alfv6n's result limited however, as aforesaid, to an upper limit defined by (5), a limit which a m o u n t s to a fraction 2v/C of the energy itself, and which is in general much less than 10 per cent. No assumption of large magnetic field or large value of h can enable (W2 - W1) to assume a value larger than that determined by (5). It is of some interest to inquire how the relativity transformation symbolized by its results in (5) seems to " k n o w " of the impossibility of penetrating the slab under the conditions stated. T h e fact is t h a t the transformation from the pure magnetic field in S' to the combined electric and magnetic fields E and H in S is one obeying relativity. Also the force equation of electrodynamics which tells us t h a t there is no gain of energy of the particle in S' while there m a y be a loss or gain in S is also an equation obeying relativity; and if these equations had permitted the passage of the particle through the slab under conditions in which (11) had given a greater value of W2 - WI than was permitted by (5), an algebraical inconsistency would have been involved. Case of a Continuous Beam In Alfv6n's paper, he assumes a continuous stream of gas emerging from the sun. If such a gas spreads on its journey, the value of v will vary. This appears at first sight to introduce a complication. However, the complication is obviously trivial as regards its practical effects. Thus, consider any portion of the beam through which the p a t h of a cosmic ray particle is contemplated. If S' applies to this portion of the beam, and if v is its velocity, the motion in S' will be determined by the magnetic field in S' and by an electric field which comes essentially from the charge displacement in other parts of the beam which do not move with the velocity v.8 The field produced in S' by these displacements, a m o u n t i n g as it does to the field produced by the charges on a "condenser" when evaluated outside of the region between the plates of the condenser, will be negligible for our purposes. A s T h e r e is, of course, no charge d i s p l a c e m e n t in S' a t t h e portion m o v i n g with velocity v.
Mar., 1954.]
ALFVEN'S THEORY
OF M A G N E T I C
]97
STORMS
formal, exact, analysis of all t h a t is here involved would result in a lengthy presentation whose end point would, in the nature of things, involve b u t a small correction on the conclusions which would be drawn by ignoring it. CASE WHERE THE SLAB MOVES IN A CONDUCTING MEDIUM
Limiting Case of Infinite Conductivity T h e fact t h a t we have supposed the slab to carry its internal magnetic field with it demands, for logical consistency, t h a t the external medium, if infinitely conducting, shall resist change of magnetic field within itself to the extent t h a t the magnetic field H remains zero everywhere outside of the slab as viewed in the system S. If the conductivity of the region external to the slab is ohmic as viewed in S, then even though in the limit it tends to infinity, no space charge will arise in S, and consequently 9 by the relativity transformation, there will be no space density in S'. T h e only charge of any kind will be on the surface of the slab. F r o m s y m m e t r y the only possible surface charge will be a density ~° ~' at y = h/2 and - ~' at y = - hi2. T h e field c o m p o n e n t Eyo' in S' just outside the slab at its mid-point will be a small negative fraction - , of the field inside the slab which is - 4 ~ ' . T h u s Evo' = 4*r~ t. 9 Some care is necessary in drawing this conclusion, as is evidenced by the fact t h a t a system which viewed from S appears as a uniformly magnetized sphere, appears in the other system as having in addition equal and opposite charge densities displaced along a line perpendicular to the magnetic induction in the sphere and to the velocity, v. It will, in fact, be recalled t h a t if U is the vector potential, and ~ the scalar potential, U, io constitute a 4-vector, so t h a t the zero value of ~ in one system does not guarantee ~' = 0 for the other system. The apparent paradox in relation to the invariance of total charge becomes resolved as follows : The densities p and p' for a given sign of electricity, let us say negative, are related in the two systems by
,, where u== is the velocity in the system x. Suppose now we superpose a positive density have ,,
p÷
in S, such t h a t p+ -k p-
=
O. We shall
=
Then
[
= ~ p+ + p _ -
v
~ (p+u=~ + p-u=3
] •
Hence, if, and only if (p+u~ 4- p_u,1) = O, can we conclude t h a t p+' 4- p_' == 0 follows from (p+ 4. p_) -- O. In our present problem, since H = 0 in S, curl H = 0 and the total current density (p.u=.~ 4. p-u=1) is zero, and the desired result follows. 10¢, will be sensibly constant except near the edges of the slab,
I98
W.
I7. G .
SWANN
[J. F. I.
If Xl is the c o n d u c t i v i t y within the slab, the y c o m p o n e n t of the current density at x = h/2 will be - 4 r z ' X ~ . T h e corresponding component
outside
w i l l b e 11
Eyo' + Vc ~oH f~ ] X2. T h u s we m u s t have
-- 4"tra'Xl =
( 4rca'e + ~,v Hzo' ) X2.
(12)
N o w since H is zero a t all points outside the slab as measured in S, we have for a point outside the slab 0 -- H~. -- ~ ( H /
~~E ~ ' ) .
-
(13)
Thus v H
' =
, =
v
T h u s from (12) - 47r¢'Xl = 47tea'
1 - ~
X..,.
(14)
However, this is a relation b e t w e e n Xl and X~ which d e p e n d s upon e and v only, and these quantities have nothing to do with the cond u c t i v i t y of the media. 12 T h e only possibility is represented b y a' = 0, which permits (14) to hold with a n y assignment of the ratio Xl/X2. If ~' = 0, we have, of course, E~' = E,f = E~' = 0 at all points outside the slab. As a consequence, in the same region H / = 0 on a c c o u n t of (13), and Hy' = 0 because o=H.=
3
(
v)
.
Moreover, H~' = H , = 0. T h e fact t h a t the state of no electric or magnetic field outside of the slab in either S or S' constitutes a solution of the problem is obvious b y direct inspection, since it calls for no currents outside the slab in S' resulting from the motion of t h a t slab. Inside the slab itself there is nothing b u t a magnetic field, and the whole problem reverts completely to the problem we have already considered when we a t t r i b u t e d zero c o n d u c t i v i t y to the m e d i u m outside of the slab. T h e conclusions are exactly the same.
How Is the Condition of Zero External Field Realized in S'? It is of interest to inquire how it can come a b o u t t h a t the " m a g netized" slab can exist in the external m e d i u m which itself shows no 11 O b s e r v e t h a t v. = - v. ~*W e again r e m a r k t h a t s u c h a relation as (12) h a s sense as Xl a n d X2 b o t h t e n d to infinity.
Mar., I954.]
ALFVt~N'S T H E O R Y
OF M A G N E T I C
STORMS
I99
magnetic field in either S or S'. The fact is that, in the simple case, the magnetic field in the slab d e m a n d s a current sheet circulating around its boundary in planes perpendicular to the axis of z. This sheet in itself would call for a return magnetic flux outside the slab, a flux which was small in a m o u n t at any point b u t widely spread out so t h a t its total integrated value was equal to the total flux through the slab. Now the p h e n o m e n o n which provides for the cancellation of the magnetic field outside the slab is another current sheet flowing in the reverse sense outside the first one and again in planes perpendicular to the axes of z. If these two current sheets have different crosssectional areas, 13 they can cancel as regards production of magnetic field outside the outermost, but give a resultant uniform field inside the smaller. In the limit, when the currents are infinitely large, their cross-sections can afford to differ by only infinitesimal amounts, and this condition represents the limiting one applicable for infinite conduction in the m e d i u m external to the slab. In case this above picture presents any difficulty, we m a y cite the particular simple example of a uniformly charged rotating sphere. We shall not trace the details of calculation for this case, which are elementary, and generally known. Such a sphere gives, at points outside, a magnetic potential ft, where M cos 0 y~
Here M = ~o054, where a is the surface density in e.m.u., o~ is the angular velocity and a is the radius. T h e field inside is uniform and is given by H, where H=
2M 53
It will thus be seen t h a t if we have two spheres of radii al, and as with ~'s adjusted so t h a t M1 = - Ms these spheres will cancel as regards magnetic field outside of the larger, while inside the smaller they will have a resultant magnetic field given by 1 )
HI_H~=2M(1 s
a2S
"
If al -- a2 is small ( = ~a) we m a y write - 6M~a H I -- H 2 = - -5 4 1~We refer here to the cross-section of the whole "solenoid structure" which consthutes such a sheet. This cross-sectional area is, of course, sensibly equal to that of the slab.
200
W.F.G.
SWANN
IJ. F. I.
Case Where Conductivities Xl and X2 Are Finite ~4 The complete problem here involved can become one of considerable complexity. The currents arising from the motion will affect the magnetic field, moreover the charge distributions arising on the slab to provide for continuity of current flow across the slab boundaries will bring about x-components of the current density at any rate in regions remote from the central regions and 9 will destroy even the conclusion that the volume charge density shall be zero in S if it is zero in S' and vice-versa. We shall not a t t e m p t an exhaustive solution of the problem, but shall content ourselves with a kind of model founded upon the considerations represented in Eq. 13, but without the restriction that the magnetic field shall be zero in S for points external to the slab. We have in S a slab with face A at potential V / 2 and face B at potential - V/2, respectively, A and B being at x = h / 2 and x = - h/2, respectively. Outside the slab, the current density is determined entirely by the field arising from this potential difference. Inside the slab we have a force per unit charge given by E~ - vHz. The line integral of the external field from A to B is V. The line integral of the internal total force per unit charge from B to A is - V - vHh/C. Suppose R, is the ordinary external resistance which would be measured between the two surfaces of the slab, if the slab itself were of zero conductivity, and R~ the resistance which would be measured if the conductivity of the external medium were zero. Then we have
V I = -- = R,
V - vHh/C R~
(15)
Hence, 1
1 )
v
-vHh =
V=-
R, Ri+R,
cR, ) vHh C
The field E inside the slab i s
E =
R, R~+R,
)vii --C"
(16)
This has its largest value E = v H / C when R, = ~ which represents the case first discussed in this paper and the result given by Alfv6n. u The effect of finite conductivity has been discussed in the writer's paper "Cosmic Rays," Jova. FRANKLIN INST., Vol. 2.51, pp. 120-155 (1951). The conclusions drawn at the bottom of p. 134 and the top of p. 135 require correction in the light of the development in the present paper.
Mar., I954.]
ALFVI~N'STHEORY OF MAGNETIC STORMS
2OI
We have already studied the other extreme where R, -- 0 and where there is no magnetic field external to the slab as observed in S. In this case the result E = v H / C also evolves. We can form a quasiq u a n t a t i v e picture of the intermediate as follows: It is easy to see t h a t the induced currents external to the slab tend to reduce the field which would have been present in their absence. T h e y reduce the field n o t only in the external region, but in the interior of the slab. T h u s if the field H in (16) represents the field actually in the slab, it should be multiplied by a factor greater than unity. While we have not traced everything in detail, it is reasonably obvious t h a t the net effect will result in a value of E which is never greater than the value v H / C where H is the actual field in the slab, this result being attained exactly in the two limiting cases where R, --oo and where R, -- 0, respectively.