sist sults

GEOMAGNETIC FIELD PERTURBATIONS DUE TO TRAPPED PARTICLES

J. R. Apel, The Johns Hopkins University

Applied Physics Laboratory, Silver Spring, Maryland S. P. Singer, University of Maryland

College Park, Maryland

Abstract

The charged particles trapped in the earth's magnetic field give rise to currents flowing in the space about the earth. These currents, which cause perturbations to the geo­

magnetic field, are of two types: a magnetization current arising from the spiraling motion of the particles about the lines of force; and a westward ring current having its origin in the azimuthal drift motion of the charges.

The theory derived here permits the calculation of mag­

netic perturbations as seen by a space vehicle proceeding radially outward in the equatorial plane of the earth. Two radiation belt models are used as examples. For the outer belt, which peaks at 25,000 km geocentric, a maximum current density of 10" amp/m flows. The total current of 5x10 amp leads to a field perturbation of the order of -100 gamma at the center of the belt.

The calculations appear capable of explaining the mag­

netometer measurements of four space vehicles. The gross be­

havior and the order of magnitude of the perturbations are reproduced, but their detailed structure cannot be deduced without more detailed knowledge of the distribution of the particles in the belt.

Introduction

The discovery of the two systems of radiation belts surrounding the earth (.1) was one of the most important re­

sults of the International Geophysical Year. These belts con­

sist chiefly of protons and electrons trapped in the earth's magnetic field, moving approximately as shown in Figure 1.

The protons of the inner belt are of about 200 mev energy (2) and their concentration peaks at an altitude of some 3500 km.

The shape of the belt is similar to a napkin ring surrounding

Fig. 1. Motion of a Trapped Particle in the Earth1 s Field.

248

the earth about the geomagnetic equator. The outer belt con­

tains electrons whose average energy is roughly 50 kev; its maximum intensity occurs near 25,000 km geocentric in the equatorial plane. The particles are confined by the geomag­

netic field and hence the general shape of the belt is similar to the earth's magnetic lines of force.

Prior to the discovery of the trapped particle b e l t s , Singer advanced the theory that ionized solar gas could be trapped about the earth through a distortion of the dipole- like field where the particles were incident upon it (3). The distortion allows particles coming from infinity to have ac­

cess to regions of bound orbits (6); after the passage of the solar gas, the field then relaxes back into its original un- distorted shape, thus trapping a portion of the electrons and protons. This calculation placed the position of maximum par­

ticle concentration near six to eight earth radii (3).

After trapping, the protons drift to the west and the electrons to the east, to form a westward flowing ring current.

Chapman and Ferraro in 1931 had postulated such a ring current to explain the main phase decrease of the earth's field during magnetic storms (4). The exact mechanism for stabilizing this current was unclear, however. Stormer's earlier work on bound orbits of the type shown on Figure 1 provided the clue to the stability ( 6 ), and Singer invoked s.uch orbits in his explana­

tion of the main phase decrease.

The f i r s t direct experimental indication of the exist­

ence of the ring current came from the flight of the Soviet moon rocket Mechta I (J? ) . In the vicinity of the outer radia­

tion b e l t , the magnetometer sensed a large departure from the theoretical value of the geomagnetic f i e l d , thus indicating a region of current flow. Subsequent flights by Mechta I I , Ex­

plorer VI ( 7 ), and Pioneer V (8) have shown similar but less striking perturbations at various positions near the earth.

The present paper sets forth the mechanisms which drive currents in the radiation belt. Two models of the belt are constructed and their magnetic fields are computed. These fields are superimposed upon the earth's field to obtain the perturbed form which would be seen by a space vehicle proceed­

ing outward in the magnetic equatorial plane. Some experiments are suggested which would help in resolving certain problems remaining in the interpretation of the experimental results.

Trapped Particle Motion

The approximate motion of a trapped particle is shown in Fig. 1. As the particle oscillates from hemisphere to hemis­

phere, it drifts slowly in azimuth; this drift is not shown on the figure. As long as the earth's f i e l d , ^B, changes slightly

over the radius of gyration of the particle, the magnetic mom­

ent μ remains constant. The d r i f t velocity of a positive charge q, which is given by v^, is directed to the west in the case of the earth1s dipole f i e l d . By using the MKS expression for a centered dipole of moment Mg, the earth's field may be approximated by (in spherical coordinates)

Β = ^

(2

(1)

4*r^

|B|= (1+3 cos² Θ) . (2)

The unit vectors ί , and are in the directions indicat­

ed by their subscripts.

Spitzer ^x(9) has shown that, in the absence of appreci­

able currents in the field B, the drift velocity of the par­

t i c l e is given by

qB²

1 2 2

- m^Vj_ + mv^j( (3)

Here m is the mass of the particle. The factor ^S7±B is the component of the gradient of Β which is perpendicular to the vector fie Id B^. The f i r s t term in the perpendicular velocity, v^x , arises from the particle's reactions to the geomagnetic field gradient, while the second term, in vu , is the result of drift due to curvature of the lines of force in the north- south direction.

The pitch angle α is defined as the angle between the field line and the particle's velocity vector^. For the pur­

poses of calculating the magnetic effects in the equatorial plane, it is sufficient to assume an isotropic distribution of velocities everywhere (1£). One may then substitute v„ = ν cos α and v^± = ν sin α in Eq. (3) and average over the iso­

tropic distribution (1/2) sin ada. The kinetic energy Ε of a particle is also constant in the static f i e l d . With these r e ­ placements, the drift velocity becomes

r ²s i n⁵0 (l+cos²9

Z P

" E S *

Here we have used the equation for the line of force

r = r sine ²e (5)

250

where r^e is the distance at which a line of force intersects the equatorial plane (θ=π/2).

Similarly, the magnetic moment μ (which is an axial vec­

tor directed antiparallel to the field B) can be averaged over an isotropic pitch angle distribution. From

1 Β

-Β

Β (6)

one obtains for the dipole f i e l d ,

μ ^8jtE r ^;e 3 6 s i n θ

2cos9i +3ΐηθ1r θ ^η

l+3cos θ ( 7 )

The magnitude of the moment remains constant for a particular particle as it moves along the line of force; particles of the

same energy Ε have larger moments as their distance from the center of the earth increases.

Currents and Fields

To see that net currents result from the d r i f t velocity and magnetic moment of the trapped particles, refer to Fig. 2.

The gyration of the particle about the line of force is equiv­

alent to a magnetic loop, or dipole, which tends to exclude the driving f i e l d Β from the interior of the loop; that i s , the particles tend to be diamagnetic The spiraling is of opposite sense for particles of unlike charge, as is the d r i f t motion. Both protons and electrons contribute to the diamag­

netic effect as well as to the ring current. Thus the current arising from trapped particles is not simply a ring current but has an induced magnetization current superimposed.

There is no magnetic field arising from the north-south motion for in a steady state there are as many particles going one way as the other. Thus the fields seen in space can be qualitatively described as follows. The ring current w i l l tend to depress the earth's field at altitudes below the trap­

ping region and increase it at radial distances beyond the radiation belts. The spiraling motion tends to expel the lines of force from the trapping region and crowd them together outside the b e l t . The net effect w i l l be shown to be a de­

crease in the field at the surface of the earth, becoming (negatively) greatest at the point of maximum particle concen­

tration, and recovering to a slight positive increase beyond the belt. The total magnetic flux through the equatorial plane is zero, as it must be to satisfy div B==0.

In order to describe the perturbation field B^p which

arises from the particles, we have calculated the d r i f t cur­

rent density arising from the westward ring current, and the induced magnetization JV^, defined as the magnetic dipole moment per unit volume. I f Ν( Γ ,Θ) is the particle number den­

sity, the total current density can be written as

Here

and

j = J^D + V x M

(8)

JD = NevD (9)

Μ = Νμ . (10)

The field is derivable from these sources by the Biot-Savart law :

s

x ^ d V . (11)

The vector is the directed distance from source point

( r , θ, φ) to observation point ( r^Q, π/2, φ^). The integration is over the volume V available to the current.

It turns out that the properties of the magnetic field perturbation depend chiefly on the form of the number density function N ( r , 0 ) . We have chosen several functions for N; the one most physically acceptable is determined by assuming pro­

portionality between magnetic pressure Β / 2 μ⁰ and particle pressure NE. Ê is defined as the average energy per particle.

If the ratio of these pressures (particle to magnetic) be de- noted by β, Eq. (12) is obtained for the number density in the equatorial plane.

32* Er e

It has also been assumed that the particle concentration away from the plane is determined by the reflection of particles from the converging field at north-south points, and by loss of particles into the atmosphere. The relative concentration along the line of force has been calculated (10) using

Liouville's theorem in a model which assumes complete loss (to the atmosphere) of particles mirroring below an altitude of 1500 km. The relative concentration along a line of force, F, is rather complicated and is given here only in graphical form

(Pig. 3). It is seen that the density is very nearly the same as in the equatorial plane up to latitudes like + 45°. For purposes of computing equatorial fields, F=l is a sufficiently good approximation.

Herlofson (11) has shown that diffusion of particles across the lines of force, due to scattering from magnetic in- homogeneities, can lead to density functions similar to those seen by the space vehicles. Therefore, we modify Eq. (12) by multiplying it by an exponentially decreasing factor roughly describing the diffusion process. The final equation for num­

ber density is then given by Eq. (13).

N ( rE, 0 ) = 2 - 32π Ε

exp m' e F(r

,9)

This gives about 5 electrons/cm*^ of energy E

point of maximum particle concentration r

10 2

implies a flux of Nv 5x10 particles/cm -sec, in good agreement with Van Allen's measurements from Pioneer IV (2).

Using Eqs. (4), (7) and (13) in (8) and performing the differentiations in the appropriate coordinate system, we may derive the net current density in the outer belt.

- ^ r~

exp ( - p ;

) ( l - p ^

) sin

(14) Here

ρ = — and 6 r

m

Now j is independent of particle energy or energy distribution;

this occurs because the number density is inversely propor­

tional to the average energy Ε while both the drift velocity and magnetic moment depend linearly on particle energy E. By introducing an energy distribution function and integrating over all energies present, one may replace Ε by Ε and cancel out energy dependent terms. Thus the particles f i l l the field to the point of equal pressures and no more; penetration into more intense field regions is then diffusion controlled as in­

dicated by the exponential factor in Eq. (14),

An interesting result of the calculation of the current is that the drift term has been exactly cancelled by one term in the expansion of c u r l ( ^ ) .

That is , from "~

νχ(Νμ) = Ν ν χ μ + (VN) X (15) 254

•θοαιο,ί jo saun ^σοτν ^Q-îsuaa .iaqranfl ΘΑΤ^ΙΘΗ ·£ ·3τ,£

and from the equations for Ν and μ, we find that

Nevⁿ = -Ν \7χμ (16)

(17) Physically, the only magnetic perturbation arising from the radiation belt is due to a gradient in the number density or more generally, in the energy density. It is readily seen that this is due to incomplete cancellation of particle spiral orbits when the density varies. Pig. 2 suggests a larger num­

ber of particles by the heavier orbit and shows the magnetiza­

tion current 1 = SJ xM flowing along the line where the orbits are tangent.

Pig. 4 shows the density, the current density (in shaded lines) and the magnetic field perturbation calculated from this model of the outer b e l t , as viewed in the equatorial plane. ( I t should be mentioned that in the calculation, azi- muthal symmetry and time-independence have been assumed.) The field is about -70 gamma near the earth's surface (and on the geomagnetic equator), dips to -100 gamma at the point of max­

imum concentration, goes through zero at 50,000 km and has a slight positive value of the order of 3 gamma out to some 80,000 km, whence it f a l l s slowly to zero at infinity. The net current flows eastward on the inner boundary of the b e l t , is zero at 25,000 km and is westward beyond.

The field is quite different from one due to a simple ring current. The latter field would have a large minimum at the inner edge of the b e l t , would be about zero in the center, and reach its maximum positive value at the outer edge of the belt. Such a field has been calculated by Smith et al (8), who assume a 5,000,000 ampere torodial ring current at 10 earth r a d i i , of minor radius equal to 3 earth r a d i i . Their field matches the Explorer VI data well. In view of the de­

velopment here, the current could have been located closer in at about 7 Rg, the diamagnetism taken into account, and a somewhat similar field would have resulted.

The total current flowing in the system shown in Fig. 4 is about 5,600,000 amperes. Such a current is based entirely on theoretical considerations (except for the location of its maximum) and is in excellent agreement with Smith's assumption and with the current needed to experimentally account for large magnetic storms. It is our view, however, that magnetic

Results

256

h 80 h 60 he CM 2 ν* OC g Ul

%

0 ζ Ο ω ^-2

0 ho

CL < Ό 3-0 -20 UJ tr­

oc 3

-40 h-60 h-ΘΟ

he H 8

70 80 90 100 re and r0 , GEOCENTRIC DISTANCE IN I03 KM Number and Current Densities and Magnetic Field for the Smooth Distribution

storms are not chiefly due to the outer belt but instead are due to a third, transient radiation belt at 6 to 10 earth r a ­ d i i composed of 20 kev solar protons. Such particles are too soft to have been seen on the detectors used so f a r .

Another possible field has been calculated and is shown on Fig. 5. The figure represents a radiation belt cut off sharply at an inner radius of r^m=25,00g km; at greater dis­

tances, the concentration goes like r~ . At the inner face an eastward surface current must flow, due to lack of cancel­

lation of spiral orbits at the discontinuity. The field drops from a value of -79 7 at the surface of the earth to about -250 7 at this boundary. At greater distances, its behavior is very similar to that in Fig. 4.

Figure 6 shows the sum of the dipole and the perturba­

tion fields for both particle configurations above, in units of 10"9 weber/m², or gamma. The graph does not extend to 50,000 lan and so does not show the crossing of the two fields at this distance (c.f. Figures 4 and 5 ) .

Comparison With Experimental Results

None of the calculations presented here are directly comparable with the measurements made from the space vehicles, because the model computations are confined to the equatorial plane for simplicity while the experiments have a l l been made away from the plane. The Soviet data (Fig. 7- a ) give the total field value at latitudes near 20° to 30° Ν magnetic.

The American measurements for Explorer VI are for an arbitrary (and unspecified) field component, due to failure of the equipment intended to measure the remaining component. The Pioneer V curve was taken nearer to the equatorial plane than the others, at about 10° N.

Figure 7-b illustrates the magnetic fields of three ra­

diation belts whose number densities are zero inside a magnet­

ic shell of force r = r^ms i n²0 , and which follow Eq. (12) at greater distances. The cut-off values for the shells are r^m = 18,000, 24,000 and 40,000 km. We have chosen to compare these somewhat a r t i f i c i a l models with the measurements because of the ease with which the calculations can be made in this case.

A comparison of Fig. 7-a with 7-b shows that the orders of magnitude of the calculated and measured fields are in very good agreement at the points of maximum depression. In addi­

tion, the model fields have the same general behavior as the measured ones in that the perturbations are negative near the earth, go through zero at some point and are slightly positive at larger distances. The experiments and theory d i f f e r con­

siderably in even the largest details, however. We have not 258

2 0 0 0

E A R T H S D I P O L E F I E L D P E R T U R B E D FIELD IN

S M O O T H D E N S I T Y M O D E L

— P E R T U R B E D F I E L D IN

D I S C O N T I N U O U S D E N S I T Y M O D E L

1 0 0 U _ L

2 0 2 5 3 0 3 5 rQ1 E Q U A T O R I A L D I S T A N C E ,

T H O U S A N D S O F K M

Fig. 6. Dipole and Perturbed Fields in the Equatorial Plane.

260

10 0 -10 0 <

-20 0

S "30 0

LU > GC IU </> ω ο

-40 0 _ -50 0

-60 0 -70 0 -80 0

Η

1 1

\ - Υ

I

/

I

Δ MECHTA I - 1/2/59 ~ 25°N • MECHTA Π - 9/17/59 ~25e N Ο EXPLORER 3DL - 8/9/59 D PIONEER 3£ -3/11/60- I0°S

ι I

'V /

J

I L J L

16 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2 3 4 3 6 3 8 4 0 4 2 4 4 4 6 GEOCENTRI C OISTANCE , I0

K M

I I I I I I I I -U I I"

-Γ Ί / V

/ / _I

I

I

N(re). J 1 I I I 1

"DISCONTINOUS" MODEL CALCULATION ' β m 32TT2 Ê x\

r

m'> Ρ*

16 1 8 2 0 2 2 2 4 2 6 2 8 3 0 3 2 3 4 3 6 3 8 4 0 4 2 4 4 4 6 r

, GEOCENTRI C DISTANCE , I0

KM

262

tried to reproduce the detailed structure because to do so would involve a knowledge of the detailed form of the particle energy density function everywhere, and this is not available.

It is felt that the general agreement between theory and experiment over a large region of space is very suggestive that the theory proposed here is essentially correct. It ap­

pears that the peak value of the perturbation is controlled by equality of magnetic and particle pressures , while its shape is determined by the particle energy density gradient. It also appears that the perturbations are time dependent and show up in different regions of space at different times. The complete theory must take this into account.

Possible Experiments

One interesting experiment which could be performed is a simultaneous measurement of the magnetic perturbation and the particle energy density as a function of position and tine.

Such an experiment must include a l l particles, not just the energetic ones and would probably involve the use of plasma probes for the low energy component as well as detectors for the high energy particles. If one could reproduce the observ­

ed field from the energy density function by calculating with the theory developed here, it could be considered a partial verification of the theory.

It would also be of interest to observe low energy (20 kev) solar protons in the vicinity of 6 to. 8 earth r a d i i . The presence of a transient radiation belt there is indicated by magnetic storm theories (3) and perhaps by the small mag­

netic perturbations observed by Pioneer V and Explorer VI (see Fig. 7- a ) .

Summary

The motion of geomagnetically trapped radiation is used to explain how non-vanishing currents can flow in a dipole f i e l d . Current densities are derived and it is shown that the net current density arises from a gradient in particle energy density.

The magnetic fields of two model radiation belts are calculated and these fields are added to the earth's dipole f i e l d . The results are compared with four experiments and general, but not detailed agreement is obtained between them.

A possible experiment is sketched out which would aid in con­

firming the origin of the magnetic perturbations.

The authors gratefully acknowledge the assistance of Miss Gari Borror in some of the numerical calculations, and Miss Cathy Thomson for aid in preparing the drawings.

References

1. J. A. Van Allen et a l , "Observations of High Intensity Radiation by Satellites 1958 Alpha and Gamma", Jet Propulsion 28, 588-592 (1958).

2 . J. A. Van Allen and L. A. Frank, "Radiation Measurements to 658,300 km with Pioneer IV", State University of Iowa (1959).

3. S. F, Singer, "A New Model of Magnetic Storms and

Aurorae", Trans. Am. Geophys. Union 38, 175-190 ( 1 9 5 7 ) . k. S. Chapman and V. C. A. Ferraro, "A New Theory of Mag­

netic Storms", Terr. Mag. 36, 7 7 , 1 7 1 ( 1 9 3 1 ) .

5. S. Sh. Dolginov e t^va l , "Measuring the Magnetic Fields of the Earth and Moon by Means of Sputnik I I I and Space Rockets I and I I " , Space Research, pp 836-868, North- Holland ( i 9 6 0 ) .

6. C. Stormer, The Polar Aurora, Oxford University Press, ( 1 9 5 5 ) .

7. C. P. Sonett e_t a l , "Current Systems in the Vestigial Geomagnetic Field; Explorer V I " , Phys. Rev. Letters 4, l 6 l ( i 9 6 0 ) .

8. E. J. Smith et a l , "Characteristics of the Extraterres­

t r i a l Current System: Explorer VI and Pioneer V", J. Geophys. Research 6 5 , 1858 ( i 9 6 0 ) .

9. L, Spitzer, J r . , Physics of Fully Ionized Gases, Inter- sçience ( I 9 5 6 ) .

1 0 . J. R. Apel, "Geomagnetic Field Perturbations Due to Trapped Particles", University of Maryland Physics Department, ( 1 9 6 1 ) .

1 1 . N. Herlofson, "Diffusion of Particles in the Earth's Radiation Belts", Phys. Rev. Letters 5, 414 ( i 9 6 0 ) .

264