P A R T I C L E L I F E T I M E S IN S T R O N G D I F F U S I O N

MICHAEL SCHULZ Space Physics Laboratory, The Aerospace Corporation, El Segundo, Calif., U.S.A.

(Received 15 February, 1974)

Abstract. The mean lifetime r of a particle distribution, driven to isotropy by intense pitch-angle diffusion, is calculated by analytical means for conditions applicable to the Earth's magnetosphere. The resulting algebraic expressions reduce to z~ [64La/35ve~(1 -r/)] in the limit of a small equatorial loss cone (half-angle c~), where v is the particle speed, L is the magnetic shell parameter, a is the radius of the Earth, and ~/is the particle albedo from the atmosphere at either foot of the field line. Distinction is made in the full expressions for z between complete isotropy (caused by strong pitch-angle diffusion all along the field line) and incomplete isotropy (caused by strong diffusion that is localized at the magnetic equator) over the upward hemisphere in velocity space.

1. Introduction

As described by Kennel (1969), strong diffusion is the consequence of having a bounce- averaged pitch-angle diffusion coefficient Dxx (x -cos ine of equatorial pitch angle) much larger than c~t22, where f22/2~ is a particle's energy-dependent bounce frequency in the geomagnetic field and ~c is the half-angle of the equatorial loss cone. In this limit the mean lifetime of a particle against pitch-angle diffusion into the atmosphere approaches a minimum value z that is independent of the magnitude of Dx~, but sensi- tive to the magnitude of e2. This is in contrast to the weak-diffusion limit (Dxx ~ ~s in which the particle lifetime would he proportional to 1/D~x.

Kennel (1969) estimated the magnitude of z as n/f2ze~ for e~ ~ 1, by reasoning that a particle in either loss cone (each having solid angle ne~ out ot its respective hemisphere, 2~) will be lost within a quarter bounce period after traversing the equator. Lyons (1973) has recently refined the calculation of z (at least for e ~ 1) by more carefully specifying the probability that a particle whose guiding center lies within a given magnetic field tube is actually in the loss cone. Lyons evaluated this probability as 1.1 sin z ~c (rather than Kennel's ce~/2) by calculating the relative amounts of phase space inside and outside the loss cone seen by the equatorial pitch-angle distribution of particles having the same speed v. By assigning a loss time of ~/~2 (rather than Kennel's rc/2g?2) and taking a properly weighted (by the factor x/•z) average of g22 over the equatorial pitch-angle distribution, Lyons (1973) thus Obtained ~= 2La/1.1 v sin 2 ~c, where L is the magnetic shell parameter and a is the radius of the Earth. The weighting factor x/02 enters because it is proportional (at constant v a n d L) to the Jacobian of the transformation from canonical phase space to the variables v, x, and L (e.g., Roederer, 1970). Procedures for averaging over the trapped-particle distribution are reasonably straightforward in the limit D~x>> ~ 2 , since the effect of

Astrophysics and Space Science 31 (1974) 37-42. All Rights Reserved Copyright �9 1974 by D. Reidel Publishing Company, Dordreeht-Holland

38 MICHAEL SCHULZ

strong diffusion is to make the equatorial pitch-angle distribution isotropic (Kennel, 1969).

The approach adopted in the present investigation differs somewhat in outlook from that used by Lyons (1973). In our work, the particle content of a magnetic field tube is calculated explicitly, and is divided by the particle current through the feet of the field tube to determine the strong-diffusion lifetime v. In this case it is the unweighted local pitch-angle distribution which enters, rather than the weighted equatorial pitch-angle distribution. The two approaches are equivalent in principle, and yield the same result in practice for eg ~ 1. However, the present approach leads naturally to certain con- ceptual refinements, which are quantitatively significant for e~ ~ 1 but cannot readily be introduced in the formulation used by Lyons (1973).

The most important advantage of dealing with the local pitch-angle distribution is the ability to identify (and thereby exclude from averages) those points in phase space that are unoccupied by particles. The purely directional averages performed by Lyons (1973) have the effect of including, at any given time, particle coordinates actually located beneath the surface of the Earth. While the contamination of such averages by unoccupied particle coordinates is not significant for e ~ 1, the present calculation imposes no such limit on ec, and therefore provides at least a modest generalization of the strong-diffusion result obtained by Lyons (1973).

2. Basic Equations

Evaluation of 1: is made tractable by assuming a dipolar magnetic field (which, how- ever, need not be centered within the Earth). Along a given field line (L), the equatorial (minimum) field intensity is denoted Bo. The field intensity at the northern foot (where the field line enters the dense atmosphere) is denoted B~, and the field intensity at the southern foot is denoted B~. The details of particle interaction with the dense atmos- phere are thereby suppressed (a simplifying approximation). However, it is permissible within the framework of the present analysis to allow for a specular albedo (preserving the incident isotropy) of magnitude 0.

The differential particle flux per unit solid angle, incident on a surface normal to B at either foot of the field tube, is equal to (1/2zc) J2~(E), where J2~(E) is the differential flux over the entire downward hemisphere (2rc sr) in velocity space. The particle flux across the foot surface is therefore given by

1

t-cos e d(cos e) = �89 - ~/)J2,~(E), (1) Js(E) (1 t/)Y2~(E)

o

where e is the local pitch angle. If dA is the equatorial area of the field tube, then [(Bo/Bn) + (Bo/Bs)]J~, (E) dA is the rate of particle loss (per unit energy) from the field tube, since the cross-sectional area of any magnetic field tube is inversely proportional to B.

PARTICLE LIFETIMES IN STRONG DIFFUSION 39

It remains to calculate the particle content of the same field tube. There is a slight subtlety related to the question of how the intense pitch-angle diffusion is distributed along the field line. If conditions of strong diffusion hold all along the field line, so that the particle flux is completely isotropic (even over the upward hemisphere in velocity space), then the particle density is (2/v)J2~(E) per unit energy, and the field-tube con- tent is given by

0 s t o

C(E) = (2La/v)J2~(E) [sin7 0 dO dA (2) On

per unit energy, where s is the coordinate of arc length along the field line and 0 is the colatitude measured from the northern magnetic pole. The particular colatitudes 0,, and Os correspond to the feet of the field line. Evaluation of the integral in (2) yields

C(E) = (2/35v)LaJ2,~(E){[1 - (Bo/Bn)]l/2[16

+ 8(Bo/B.) + 6(Bo/B.) z + 5(Bo/B.) 31

+ [1 - (Bo/B311/2[~6 + 8(Bo/Bs)

+ 6(Bo/B~) 2 + 5(Bo/Bs)a]} dA, (3)

where a is the radius of the Earth. It proves convenient to introduce the abbreviated notation Bo/B,,=y,,- 2 =- 1 - x,2 and

Bo/B~ =-y~ = - 1 - x~. The strong-diffusion lifetime v is thereupon given by

"c = C(E) + [(y2 + y2)j+(E) dA]

= [4La/35v(y ] + y2)(1 - I/)]

x {(16 + 8y, z + @4 + 5y6)x,

+ (16 + 8y 2 + 6y 4 + 5y6)x~}. (4)

This expression for v scales, as it should, like 4La/v. The bounce period of an individual particle is given by 2re/f22 = (4La/v)T(y), where yZ= Bo/B,,, and B,, is the mirror-field intensity (attained at 0= Ore). However, the function

~[2

T(y ) ~ ~ [1 - (B/Bm)]-~/2(1 + 3 cos 2 0) 5/2 sin 0 dO, (5) I /

0 m

which contains the explicit dependence of 02 on equatorial pitch angle, does not appear in (4) at all. This is as it should be, since v is the mean lifetime of a whole distribution of pitch angles. More significantly, and in contrast to prior derivations of z, the bounce period 2~/f2 2 has not entered at any intermediate step between (1) and (4). Given the condition of pitch-angle isotropy, which requires D~,~>>o~v/4La, it is unnecessary to invoke either the bounce frequency or the 'probability that a particle is in the loss cone', in deriving a valid expression for v. Applied to (4), the limit y 2 = y ~ l (whence 2 9 y . ~ eg) yields T~ [64La/35vct~(1-~/)]. This result agrees (for t/=0) with the expression v = 2La/1. lv sin z e~ given by Lyons (I973).

40 MICHAEL SCHULZ

3 . R e f i n e m e n t s

The assumption of complete pitch-angle isotropy, invoked above in obtaining (2)-(4), seems dubious in reality (see, however, Koons et al., 1972). It requires, in the absence of a unit albedo t/, that pitch-angle diffusion low on the field line equalize the upward and downward particle fluxes - i.e., immediately replenish the particle trajectories depleted by entry into the dense atmosphere.

It is perhaps more natural to assume that such particle trajectories are replenished only by the intense pitch-angle diffusion that occurs in the equatorial region, idealized as the point where B = Bo. In this limit the upward hemisphere in velocity space would remain depleted along trajectories connecting the equator with the atmosphere. The result is a somewhat shorter lifetime ~r than given by (4), since the particle density along the field line is somewhat smaller than the value (2/v)J2,~(E) assumed in deriving (2). The particle content of a field tube is given instead by

Os r t~

C(E) = (1/v)LaJz~(E)~(1 + rl) l s in7 0 dO \ I L l

On

~/2

- r/) ~ [1 - (B/B,)] ~/2 sin 7 0 dO + (I On

Os tll N

+ (1 - - t/) [ [1 -- (B/Bs)] '/z sin 7 0 d0~ dA. (6)

n/2

The first integral has already been performed in obtaining (3). The second integral in (6), to be denoted Z(y,) , is evaluated by observing [cf. (5)]

that

dZ/dB,, = (Bo/2B~)T(y,), (7a)

o r

dZ/dy = - y T ( y ) . (7b)

An excellent approximation for T(y), accurate within 1% for all values of y between 0 and 1, is given by the expression

T(y) ~ T(O) + �89 - T(0)l(y + yl/2), (8a)

due to Lenchek et aL (1961). The end-point values are expressible in closed form, viz.,

T(0) = 1 + (1/2~/3) In (2 + ~/3) ~ 1.380 173 (8b)

and

T(1) = (n/6)X/2 ~ 0.740 480 5. (8c)

PARTICLE LIFETIMES IN STRONG DIFFUSION 41

Since (6) implies that Z(1)=0, it follows from (7b) that

1

30Z(y) = 30 j uT(u ) du

y

( 4 - 15y 2 + 6y -'/~ + 5y3)T(0)

+ (11 - 6y 5/z - 5y3)T(1). (9)

Since direct evaluation of (6) yields Z(0)= 16/35~0.457 14, it would be a good check on the accuracy of (9) to evaluate Z(O)~(4/30)T(O)+(ll/30)T(1). Although this approximate expression for Z(0) does not look much like 16/35, it reduces in fact to about 0.455 53 and therefore yields an error of much less than 1%.

Thus, the strong-diffusion lifetime 7: under conditions of incomplete isotropy (Dxx>>~g22 for all x, but with pitch-angle diffusion localized at B=Bo) is given by

7: = [2La/35v(y~ + y~)(1 - t/)]

• {(16 + 8y z + 6y 4 + 5y6)(1 + r/)x,

+ (16 + 8y~ + 6y 4 + 5y6)(1 + r/)xs

+ 35(1 - r/)[Z(y,) + Z(y,)]}, (10)

with Z(y,,) and Z(y.O obtained from (9). Of course, the strong-diffusion lifetime is minimized with respect to r/by the limit of vanishing albedo (r/= 0), and approaches infinity in the limit of perfect reflection (r/= 1) from the top of the atmosphere. Since Z(0) = 16/35, the limit 2_ ,2 y,,-),s 41 yields 7:~ [64La/35wz~(1-~/)] when applied to (10). As noted above, the same limiting expression for z follows from (4).

4. Discussion

The foregoing results enable the strong-diffusion lifetime v for a given field line to be computed, directly from simple algebraic expressions, in terms of the field intensities at the foot points where the field line enters the dense atmosphere. Although a field line in general enters the atmosphere obliquely, the foot of the field tube (across which particle precipitation occurs) is a surface normal to B. The logic of this contention, invoked above in calculating the particle loss rate, is that the opportunity for precipita- tion is decided by the location of a particle's guiding center. Precipitation is inevitable (up to a factor of 1 - I/) once the guiding center has passed a point of no return, such that a particle will lose its energy through ionizing collisions within the next gyro- period. Thus, the flux J , (E) calculated in (1) is really a flux of guiding centers as well as a flux of particles. This last interpretation could not be made if the foot of the field tube were not defined as being a surface normal to B.

In the limit of small gyroradii there is little difference between the minimum altitude of a particle and the 'perigee' of its guiding center. A particle having appreciable magnetic rigidity, however, can experience a much larger atmospheric density (averaged

4 2 MICHAEL SCHULZ

over gyration) than its guiding center experiences. On the other hand, the gyration- averaged atmospheric density required for precipitation increases with particle energy, as the ionization cross section decreases. Thus, it is not immediately obvious whether the guiding-center altitude that locates the foot of a field line (and thereby determines the parameters B, and Bs) should be treated as a function (either increasing or de- creasing) of the particle energy. Any such energy dependence would be quite weak, and its evaluation would exceed the intended scope of the present work.

In summary, the strong-diffusion lifetime z has been calculated, for an offset-dipole model of the geomagnetic field, under conditions of complete pitch-angle isotropy (strong diffusion at all values of B/Bo) and incomplete isotropy (strong diffusion localized at B/Bo = 1). The calculation allows for an arbitrary specular albedo ~/from the foot of the field line (top of the atmosphere). Conditions of incomplete isotropy and vanishing albedo are found to yield the minimum possible particle lifetimes under strong diffusion.

Acknowledgements

The author is pleased to thank Dr L. R. Lyons for an interesting discussion that helped to stimulate interest in the strong-diffusion problem, and Dr A. L. Vampola for clari- fying the distinction between complete and incomplete pitch-angle isotropy under con- ditions of strong diffusion.

This work was conducted in part under U.S. Air Force Space and Missile Systems Organization (SAMSO) contract F04701-73-C-0074, and in part under Battelle Pacific-Northwest Laboratories special agreement ]3-758. Special agreement B-758 is a provision of U.S. Navy Office of Naval Research (ONR) contract N00014-73-C-0333.

References

Kennel, C. F.: 1969, Rev. Geophys. 7, 379. Koons, H. C., Vampola, A. L., and McPherson, D. A. : 1972, Y. Geophys. Res. 77, 1771. Lenchek, A. M., Singer, S. F., and Wentworth, R. C. : 1961, J. Geophys. Res. 66, 4027. Lyons, L. R.: 1973, J. Geophys. Res. 78, 6793. Roederer, J. G.: 1970, Dynamics of Geomagnetically Trapped Radiation, Springer-Verlag, Heidelberg,

p. 124.