1/ AD-A099 338 AIR FORCE GEOPHYSICS LAB HANSCOM AFB MA F/6 20/?ON THE FIRST AND A RELATED ADIABATIC INVARIANT WITH A FORCE IN -ETCCU)

UNCLASSIFIED AFOL-TR-80-0193 NL

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AFGL-TR-80-0193ENVIRONMENTAL RESEARCH PAPERS, NO. 708

On the First and a Related Adiabatic

Invariant With a Force in theDirection of the Velocity

CHARLES W. DUBS

25 June 1980

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1IS UPPLEMENoTARY NOTES

19 wEv wSOes (c-II.-.S. .... V d fS~IIIbt .A.,

First adiabatic Slowing downRlelativistic adiabatic Charged particleProtonsi Magnetic fieldInhomugeneous Velocity forceInvariant Field-geometric

so ABSTRACT (C..I,. 7 ." of..... .'"'. -4 i,.s..b' SY d 6l'I b#I.'# -l6

Is the well known firut adiabatic invariant p,/2mB. pi the perpendicularcomponent of momenturr. (if a charged particle in a magnetic field 11 still anadiabatic invariant if the particle ex~periences a force in the v*7direction(for example, if it is being rilowed down)' No. Is there a related quantityIthat is) Both a simple and a rigorous proof are given that the closely relatedquantity C 0 sin2 a/B, a the pitch angle, is an adiabatic invariant for a par-ticle of any energy in any magnetic field experiencing a force with any v

00*~'~ 1473 toliVoNso OP no asy Sis OBsoLEE UnclassifiedsECuarIT Ca.rb,cd'.OA 0' TMTI PaGE(A. e ,.E

UnclassifiedSECUftIYY CLASStPICATION Or T"IS PAGK~nwS, D.' Entered)

20. Abstract (Continued)

dependence in the ±V direction. dC /dt is independent of this force, and isthe same derived relativistically as not. For the simplest field %% ith thefollowing gradients, the rigorous proof shows !Q to be constant to secondorder of the gradient of BEt in the direction of B, and to first order of thegradient of B 1 perpendicula r to il. The adiabatic invariance of C is usefulfor analytical treatments of trapped and auroral protons slowing downbetween nonforward scatte rings. 0 but not K is also adiabatically invariantwith slowing down.

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Contents

5

1. INTRODUCTION

2. SIMPLE DERIVATION

3. RIGOROUS DERIVATION 7

3. 1 Simple Magnetic Field 7

3.2 General Magnetic Field 10

4. OTHER ADIABATIC INVARIANTS 15

5. CONCLUSIONS 18

ACKNOWLEDGEMENTS 19

REFERENCES 20

3

On the First and a Related Adiabatic InvariantWith a Force in the Direction of the Velocity

. INTIKOlI (:TI( '

"l'P . adiabatiL' invarianc, of " 2mo 3. the first adiabatic invariant. p) the ',i ,-

ponent of nirrinenturn perpendicular to the nmagneti" field Bi of a 'hariged partic'le (if

rest mass ni has been known for three decades. Is this quantity an adiabatic

invariant if the - .irticle experiences a force f(v), 1 the p-. rticle's velocitx for

examnple, if it is slo-,ing down" No. ls there a related cuantit that is ., Yes. To

the best of the author's knowledge, Dom Inii 1 \,,,Si lhe ritst jie I-sn to point cut that

the closely related qeaitity

2sin e (1)

the. pitch angle, is an adiabatic invariant for this casi. (n lni a\ r'isn irl -

tively that a force in the t%'directio, will rnot ,hang e, and th, .for, no C, hut

will cause the guiding centeor to 0,,ve across thc. field lines sin, cn the radius it'

gyration is proportional to v 1.in 2 and Dubs 3 ind eperidntlyV dr('ived the, ad tabli(

invariance if C nonri-ativisticall- f,r this 1).s N w :i r,.- %vY simplh, but 11r! v ir

-, 4rigorous method, a modificatitmn (if ihat by Alfv,'1 irid l.' lttrmmin r. A ri ,,r'(,is

(Received for publication 24 June 19801

Because of the large number of references cited above, theY xwill not be listed here.See References, page 21.

5_

nonrelativistic niethod has also been found. 3Is C an adiabatic invariant for

relativistic particles slowing down? Yes. A miodification Of OW in ethod of A lfveni

and F751thanimar 4 that shows this is prIesen~ted in the next sect ion. This is fol -

lowed by a rigorous, longer method that results in the timne de rivat ive of C. other

adiabatic invariants are then examrined briefly. A conidensed version of this report

has been submitted to a journal.5

2. SIMPLE DERIVATION

At any time to, choose the origin of a cylindrical coordinate system at the

center of gyration of the particle of charge e and rest mass m with the z axis inthe direction of B. Assume a constant magnetic field gradient 3B From

-B BR 11 aB 3z'Iff = 0,neglecting 2P and- B~ r--- I ev, and thusep are

negative. Let 80 g R 2I

dX (2)

p 1 + (3)

-1 -2(4)

2 4 2 2 me2 24 2 2 2 2(5E m C +pc C my mc + pc +p c(5

ppc2 3. 6E =-y-- my vvY V V 11V + v (p)

=m-Y3 v inc 3Y . (7)

The component of force in the z direction is

ii av B + ii Vi LL 2_B + Y3 vil

-e BR+ v z v 2 B caz v (8

5. Dubs, C.W. (1980) Adiabatic invariants for a charged particle slowing down,submitted to Journal of Geophysical Research.

Pi!

IP

since R nearly -- the radius of gyration. Substituting this into the expression

for t:

23 v dz a13 3 V2ril vv ni + vzD i

213 dt az -v- v+ p V (9)

2 2V -V 11 V Ip 1 .B (10)3n- v m r +t vl

Multiplying by -

v vi

"2

v_.,2.L ,+-__, 1 i 0(11

Substituting p, I m v sin a:

2-t 2 d(lv) 2 d sina I dB 2 d sina 1 dl 0v -v dt sin'3 dt Sno dt-- B dt

(12)

d F sin2ald- L----j -0 (13)

S2

510 cI - constant (14)

:1. RIG;OROUTS IDERIATVrON

3.1 Simple Mag:efiv Field

('h oose a cvlimdrical coordinate svstem as above. Assume a vector potential

^ 8 T t + 2 + R ( sA 6 1)." (15)

Let bli be the ma~et mic rield. The normalized magnetic field then is

1 II (?1-

1 3 -~ A - I ? + 7 i-H ~ f I +(-14- L

(1i is normahzeld from here through Eq. (63). ) The constants b, h, and 11 are seen

t, be the field :it the orig inr and the 13 doubling distances respectively.

7

1,-mc 2 RVA2 /C'2 +eX'. , + (17)

dt (3i L l + Qi (18)

Q1 represents the generalized component of the drag force f(v)v. (See Section 3. 2.)

dE =33x =- 3 v AV) =fvv (19) !dt

So,

f(v) = m - (20)

(The latter equals p, as it must.) Only the R and z Lagrange equations are

needed:

dt m 3' k) = my R; 2 +ebR ( ) + m Y3 vk (21a)

d [m'yz]= ebR2 j+ my3, (21b)t 2 h -v

which yield

R R 2 + m + ir (22a)

= ebR 2 + (22b)2myh

R RThe equations are now partially normalized with c - E 0 R R P ,

z= R , v = R 0 u, where R0 is the initial value of R and of the gyroradius. (Note

that E from here to the end of Section 3. 1 meanq this instead of energy.

a 1 + L + = (1 + 4 + Ep) (23)

_ ;2 + p j ()+ . (24a)my u

4ep -- 124h)2 ) 11

2

1) 5 ) - K - ---- - - 2

W d I W

1) ( 1) , (27

d it B2 (it ]1 -2/2

1 - 1 l - [2[)2

B3 ( ) (29E) )P u ) _P

Let M be u B3 times the middle two of these four terms. Use the two equations of

motion to eliminae p and .

-2 2 21 F c b p U - 2

+ + ) + - -.

u Bi

+~b 2~1 + p P)

u 2m 2u u 2u JAl 2 I( ) - f ( )'p +u + 3E2 -7 2E-,T 3b

C - .... " (3" 1 aI.L

S V,,t all or hIi rtlat ivi t tt' t ':IS a1d all oftilt' I C , hlt ttltlg tih hangt, in

pat rticle's s i, ,I rant'. S, D is tilt sante w itt %iti.aut she,,ing down and

w ith ot without iing driv,.d c.lat ivistically. The rcsult lthen is:

13 4 ' 4- +E. + e + L4)+- -- U- (P)(+ + + E (

u2 +BL2,± _2E (p ,24 + O L (b2 + p 2 )(l + + + Ep) 3

9

2Epp (1 + r + tEp)

2 + 0( - p2 _ -3 2)(1 + E + Ep)

+ 4E 3 } (32)

2 3 4Note that, for E 0 (H = o), D contains terms only in E , f and c divided25 2 3 '4by u B, and, for E 0 (h oo), D contains terms only in E, E, I., and E

2 5. - 1) (divided by u B

5. Alternatively, D 0 (€ E ) + 0 (Co E).

3.2 General Magnetic Field

Which of these results hold for any magnetic field? Repeat the last section,

but with

2R + c ba(R, , z) - I + 7- (33)

where al is an arbitrary function of R, 6, and z except that 7 X a, vanishes at

R - z - 0. Then bl - 7 X< A is completely arbitrary,, regardless of what function is

chosen for the scalac U. To simplify the algebra, choose

U - aij dR , (34)

holding and z constant. Then a R 0, and the normalized field is

~~~i~a )~E x a )+ 1__

C- fc31 + 3j3 + ( +E) . (35)

10

- -(3 6)11 R 5q Z

aa o

a= R (37)

Sa(Ra¢) ,(8

i3z 8 R (38)

As in the last section,

dE 3 + R + z R + .z (39)

From mechanics (for example, Goldstein 6), the generalized drag force is

Q f T -. so

QR=m-y3 ik (40)

Q mY 2 (41)--€ v

3 V

Qz m-Y (42)

2L =-mc 22

c

A 2 R - + + Era) ' + ca (43)

d- - 2 + -- + QR . (44

d (L (44)dt aR + R

6. Goldstein, H. (1950) Classical Mechanics, Addison-Wesle y Press, Inc.,p. 22.

11

d (fil C,)

filfl t. b R ( W -'

2 (47)H2

. '

Q ,/ 0

(it,

H .lljj: ZcI~. - v.~I(

i eneral, the tequat i( ri , fr' H is& .n(,d (d.

(49)

d 1. l . + (5d(Ra.

3 v ,) -

Cd

R-r + +fi l (50)

H6 -2Ro~~ (1u H a 1+) (1

R.,) -2A i3.c(,- (52)

d a . "I (53)

d + ~ ) + +,) Till

t(m, 7 +1 v )fa hcZ(4

V

4 *da (aan .+cvbdt eb onz + *v (55)

12

eb (aa R a z a a a OamY az az Z R Z +-v (56)

- eb (57)

B 0 2 v+ 2++(+2R%+R3+ ) (58)

D B (B2)- 1/2 - + (I +R$.)+ z] 2 -2 (B2)-3/2 }

(1+ E3 ) z+ E(3 + ) 2[+ + R +

B 3 v2 B 3

_ 2 [ ] f (l + 1z ) + c R R + IER B 12(_ 2 {0

2 B3 3 B 3vB vB

]2(-3){(1 + ) 3 + (3R R + 203

v 2 B 5 (9

where is z + (O R + R ,f3 + Z 0z). Using Eqs. (47), (52), and (57), v2 B3

times the third and fourth terms is

-[ ~ ~ 0 + M-- (- 3+ 3)I+ + -E - (+ + 0 z) + c R 213

feb .. v *2O

eb- RO(_ R + c e R _ z 13) +e L3R _cv

-E' (iRRn

-"v O R + v ; +--z

R -[ ]-{R 2 3R-2i (3} (60)

13

Note two things: First, the , terms cancel, so 1) is unchange.d by tt i.s il t hc

tv direction and thus of any processes producing only slowing down. Second, til.

terms containing - cancel, so D is the same with a relativistic derivatio)n as it is

with a nonrelativistic one. So

(1 - E3z) ' ? 2( W R l -

D =- B3

139 3+ 2[" c(I 1 tR + It 3 + zlz3 7 l"l + fj ;2 13 - ie t 3 I?0,, ;z t

2 1 3

+ 2 2

v 1

3-2 z ' 0 (E2) (62)

Thus, in .gneral, 1) is of first order in c.

Perhaps the simplest way to se(, that I) is indepndent 0,f shit w ( jidwn p ,,- -

I's is as ldlows.

d I 13 131 dt it

14

Ill 2 Tt

.Si nce t- (untiormalized f'romn here" on) is a f'unc(tion onlyv (if R, o, an z iiv ,il.

thiniz in this expre'ssion ithich could (Jcpi,-d onl sl,,wing do wn is d-

I),,tting v" into this:

14

dvsince 17 must be perpendicular to v. So

dt nv-yv

the same as it would without slowing down. Since this is the only term in I) ctn-

taining -y but it contributes nothing to D, the latter is seen again to be independent

of whether or not it is derived relativistically.

If there be a component of drag force perpendicular to -;, write the drag force

asf f + f n, n v = 0. Carrying out the same procedure,dv n

my n (67)

and

f.d fnd~ n Bn (68)

instead of zero in the expression for D.

4. OTHER ADIABATIC INVARIANTS

More generally, consider a purely field-geometric quantity, g(B,0,), j cos a.

dj g fdlB a~g~ . + dv1

ddt + + vB T

By the same reasoning (p. 14, last paragraph), g is seen to be independent of

slowing down and of relativity. Further, consider

SIm

G =f g(B, ) ds

m

where s is the distance along a field line quite close to the locus of guiding centers,

and sm and s' are mirror point values. For C -I-= const., B the mirrorm B mm

point value of B, sm and s' are constant. Then

15

A I,

I i l . 1 . 1 3 I I

I rl

tfj 'Y I-

f F -- I

+ +.'- '2!' ,' if7\.' .2' k . ' ' -L . ' 'i. .

S[ 21

th,. par i l ' Is ,,,A -''71-'uL.. n'ug ' ' ' L;i , . , ff,., I,. I i 'hi t L ,.- , ., .

vl i ;i t. h :

B -H Itz +

h"

A treatnent like. that in Section 3. I leads to

,b R

nih2

Supp use that at t 0: v v, z 0, -o, and R H, the radius of gyration. Fr , i- - ,,i ,

so ut

,1 1 , -I t L

ivi7 2 - - 1 ,,,v

"Fh.r,,F, ,r,.,}

I/ k kh"

fhe' .- ,lut in i,:

j 1. .2 2 Ii' 1

. . In I ) 1 , =

1 In

k '2

Then

k k iJ

12

771ift 2k-

kVT~ 2kI

AI I

present. D -b----, b a consta, is shown to be ind jch, : ric ,.f f :ind thus ,f slv, , -(iting down fc a magnetic fields. It ;., also shoxwn 1) b , sa11 , %hictl;.r d(.r'iv.d

el: i i- vi,:lx 1 0. Tn .;r, l B i / ,., b. i-, .. d. , .I , Dh, rS' n( .

i"r. th. sijl&A _ __ netic fi.ld, with a parz: P ! 8-Ai.nt -f B , -, and p 'rpthn-b D (

2 ',, r r 'h'dicular gradient of Bz, ]-], D 0 (E E ° ) + 0 (c '), wher' €- R 'h, I lo 11,

1 is the initial gyroradius, and B bz at the origin.0

The adiabatic invariance of C N ith a force on the particle in the ifv direction,

among other cases, is applicable to trapped and auroral protons between nonfor-

ward scatterings. Examples of processes which may cause charged particles to

slow down without large angle scattering are elastic and inelastic bcattering, ioni-

zation, Bremsstrahlung, and C erenkov radiation.

In general, field geometric quantities: C, half bounce path length, I, and 4

remain adiabatically invariant with slowing down, but M, J, and K do not.

Acknowledgments

I am gratel, to Dong Lin for the idea that C is invariant with slowing down and

for comnr.nts on the criticism of the referee of the version for J. G. 1. I thank all

who read a preprint and made suggestions, particularly Tom Chang, Pradip Bakshi,

and the- ala vye-mentioned referee. To the latter, among other things, is ow ed corn-

sideration (,F adiabatic invariants other than C and the invariance with slowing down

(f pure field geometric quantities.

References

1. 1ll, D. I.. (1980) SetiIa r qI beam -plasna interact ions in spact, Air ForceGeophysics I.aboratory, 18 March 1980.

2. Iin,. I). .. (1080) Adiabatic invariance itt the pireserice of a slowing-derl force,21i March 1980, unpublished. I.in inforted rite that he first carritd out thisderivation 25 oct. 1979. It will be included in a paper on a Monte ('arocode.

3. Dubs, c.w. (1980) The first adiabatic invariant for a particle slowing down,26 March 1980, unpublished.

4. Alfven, If., and Falthammar, ('.G. (1963) (osiical Hlhctrodvnanlies,Oxford University Press, london, pp. 29,30.

r. Dubs, ('.W. (1980) Adiabktic invariants for a charged partitcle slow ing dowi,stubmritted to Journal of Gclhysica-- 6l -esearch.

6(ldstein, If. (1950) Classical NlMchatmcs., Addison-Wesley Press, Inc.p. 22.

"0.

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