NASA/TP-2002-211930

Radiative Energy Loss by Galactic Cosmic Rays

Sean C. Ahem and John W. Norbuo'

UniversiO_ of Wisconsin-Milwaukee, Milwaukee, Wisconsin

R. K. Tripathi

Langley Research Center, Hampton, Virginia

September 2002

https://ntrs.nasa.gov/search.jsp?R=20020079875 2018-11-02T13:19:10+00:00Z
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NASA/TP-2002-211930

Radiative Energy Loss by Galactic

Sean C. Ahem and John W. Norburv

Universi O, of Wisconsin-Milwaukee, Milwaukee, I_sconsin

R. K. Tripathi

Langley Research Center: Hampton, Iqrginia

Cosmic Rays

National Aeronautics and

Space Administration

Langley Research Center

Hampton, Virginia 23681-2199

September 2002
Acknowledgments

This work was supported in part by NASA grant NCC- 1-354. Sean Ahem gratefully acknowledges funding from

the National Space Grant College and Fellowship Program through the Wisconsin Space Grant Consortium. and useful

input from Sudha Swaminathan. John Norbury gratefully acknowledges the hospitality of the Physics Department

during the summer months at La Trobe University. Bundoora. Australia.

Available from:

NASA Center for AeroSpace Information (CASI)

7121 Standard Drive

Hanover. MD 21076-1320

(301t 621-0390

National Technical Information Service (NTIS)

5285 Port Royal Road

Springfield, VA 22161-217 I

(703) 605-6000
Nomenclature

Note: tho units used in this work are such that h c = 1

5

b

I)min

(.

d:_I

7L4d,,,

d_.A

d_'

dF, l,rem

dX

d_c,,ll

dX

dEra(|ial ion

dX

doAE

Ap±

,.-_l)±min

E

E.f

c

E,M

Eo

C

7]

£

E(x

_.tlv

gz

h

h

[o

4-1)()t,(,ntia, l

v(,ct,()r l)()t(,ntia,l

_illO St, l'llCtlll(' ('()llSta,llt

magnetic fiohl vet:tot

impact paran,('t('rin a ('()llisi()l,

nfininmn, iml)a('ti)a,raxn('t(u cff(,(:t, iv('in a ('()llisi()n

Sl)(,o(lof light

fr(,qu(,ncySl)ectrmn (ofl)()sons)

radiati()n('rosss('(lioll

st()i)i)ingi)()wer(1)r('msstra,hhmg)

st<)I)I)ingI)()wer(('()llisi()nal)

StOl)I)ingl)()wer(general)

sca.tt('ring ('ross section

('n(Tgytransf('rr('(l })ya l)()s()nin a, c()llisi(m

I

a c()nstai,that a,I)I)emsin vari(ms f()rmula,s,(,qua,lt()CM 2 +

a. c()nsta.nt tha.t at)t)('ars in vari()ns f()rmula.s. (,(lua.1 t.() "_'

m()m(mtmn t.ransf('rr('(1 t)y a })()s()n in a collisi()n

Dira(-delta. fimction

a,lll()llllt ()f (,n(u'gy t,ra,llsf('rr(,(l t,()a,n (q('(:tr(m in an a,t()m l)y an in(,i(h,nt Im,rti('h'

energy of th(' Im,rti(:h ' that (,mits a m(,(liat, ing l)()son ()f int(u'est

(,h,('t,r()magneti(" c()ul)lhlg (.()nsta.nt

(,h,ctronmgneti('

initial energy ()f a. cosmic ray as it. (raters a m(,(limn of iilt.erest

a (:harge t)aram(q(,r

a, (:lla,rg(' I)ara, ill('t,(w

electri(' field vect()r

energy of a ('()stoic ray (a.s a fun('tion of d('l)th within a m('tlium ()f interest)

L()r(,ntz fa,ct()r

('i m,rge( I weak (:()Iq )link ('()nstant

neutral weak COUl)ling (:()nstant

Planck's (:(rest,ant

h

971- . .

a c()nst_mt that, a,l)l)ea,rs in tlt(, l)os()n fl(,(lUCn('y Sl)ectrmn, ('(lual to -(27<):_ = l,-c- ....

iii
,It'

,]"

L

A

M

3 L+.,.,,.,>,,

Mf

A7

LI-) 1 llO. X

0

(2

qA

q( 'It '(1 I'1 )lt

qincidvnl

q)a,'g_,t

ql

RCS

P

SCS

Nit

SP

T

t

r:_

0

(9 W

TSCS

lib e

U

F

X

4-('urrellt,

era'rent density

m<)<lified Bessel flmctions of the second kin<l, of orders 0 and 1, resp<,<'tively

h mgit m linal

n()rma.lized h('licity

mass of a bost)n that mediat.es an intera(Mon of int.t,rest

IlItI,SS Of all electron

mass of the particle that emits a. mediating boson :)f int, erest.

mmfl)er density of atoms in a. t.arget material

angular fl(,(tuen('y of a boson

maximum angular frequency of a mediating boson

scalar potential

dEcoll

_1COllSt.allt a.t)t)eariilg in d'-/7-ff-' equal to bmin/kp±

4-111OIllOllt, llllt ()f })OS()ll

a,n ('IIi,('tiv(' charge, O(lllal t,() _ + q_t

axial-vect()r charges

the (:ha, rge Q ()f an ele('tr()n

(lim(,ilsi()nh,ss charge (luailt, Hm Imml)(,r [()r ele(:tromagnetiv int.erat't,i()ns

(h(' ('har_(' Q ()f ;ill in('idell) Imrticle

4-charge

th(' charge Q ()f a target l)article

ve('t()r charge

I)osition vector

ra,(liatioll cross s('(:ti()Ii

charge density

scattering cross se('t,i()n

spin direction

normalized 4-sl)in

st()t)t)ing power

Poynting vector

tr_-i,11sverse

time

third ('Oinl)()nent ()f th(, vector of weak isospin quantum numbers

scattering a.ngle ()f a l)ulse ()f I)reinsst.ralflung radiation emitted

weak-mixing angle

Th()mson s('a.t.teriug ('ross section

4-vel()city

v(,lt)cit.v of a Imls(' of ])remsstrahhmg ra(liation emitted

magnitude ()f vel()citv vector

veh)city vector

distance (wMfin a medium ()f interest) traversed by cosmic ray

iv
radiat, i(m length

Feylmmn scaling Va,lial_h'

a c(mstant, that, at)tmars in various B(,ss(,1 fnncti(ms, (_(tual t,(_ b_l_l

mlit v_,ct(_r t}(_inting in t,h(' +z dir(_ct, i(m

Abstract

I,l_tc'ra,ctio'n._ between galactic cosmic ray,_' arid ,matter a'l_' a prima, r!/.focu._ of the NASA

ra,diatior_ problem. Tlw clcchomognctic forc.,_ i_rvol_,cd arc ,for the _.o,_t part well documented. Building or_ p'r.vi.'u,_ rc,_ca'rct_, thi,_ ,_tu(ly i_r_cstigatcd tlw rclati_! impo'rta'l_cc

of th<_ weak .forces that occur whcT+ a <:osmi<: r<l.q impi'ttWs o_t dff.fcrcnt types .f matcH:

als. For the familiar clcct_vmog'nctic <:ass!. it is k_ww'_ thor c'_wTy.q lo._t in the form of

'radiati.'_ i,_ more..dgTtti[icant tha_t that lo,_t 'vi(l c.ntact colli,_io_m': the rat( at which tb.

cw_:qy is lost is also well u,'rtdcr,_tood. Similar rc,.ult,, wcr. derived .hn tt_. wca, k .force

ca._. It 'wa,_ ,fouml that radi.,tior_ i._' also tb( domir_ant mode _f c'.( rg_] lo._,, m w(_ak

.force: intc'_nction,_', art<t <tat 'weak .force t Jl', ct, <.'_ iud.cd rclativ.l!! weal,: compa, red witt_

cl(:ct,romagnctic cfft cts.

1. Introduction

When a fast-moving cha rgc_l particle passes t hrcmgh matter, it loses energy and slows down.

There arc two ways it can lose elwrgy. It, can either l_)s¢' kilwtic energy during collisions with

other I)articles or h_se energy in the form of ra.diaticm. During any given collision, the incident

partMe will transfer some _)f its kinetic energy t_ other lmrtich's and th('re|)y un_h'rgo an accolera.tion. It. is kn()wn that whenever a charged t)artich' is ac('dora,to_l, it radiates eh,ctroma.gnetic

(EM) energy. This type of radiati(m is referred t(_ as t_remsstrahhmg (braking radiation) l)ecause

it was first obserw,d in experiments where high-energy electrons were st.oI)l)ed in a thick metallic

target. The theory of energy loss 1)y moving (.harges is well understood for EM intemcti(ms. One

imi)orta.nt result ti"om these studies is that for nonrelativistic imrtMes, the energy lost in the form

of bremsstrahhmg is negligilfie compared with the kinetic energy loss, whereas for relativistic particles, bremsstrahhmg can 1)(, the dolninant forln of energy loss. This pat)('r addresses the physics

of energy loss bv high-eIwrgy ('OSnli(' rays as ttwy travel through lmlk ma.t.t('r, such as the wall of

a stmcecraft or th(" Earth's at moSlflwrc. Thus, the t)r(,lnsstrahhmg t)ro('('ss('s were of central c:(m('erlL

For lll()st macroSCol)ic t)arti('le I)heI_onlena, such as the tm.ssage of a. high-energy partich' through

bulk matter, cnorgy transfer via. EM processes is usually t.lw most significa.nt clmnnel, lint c(mt.ributi(ms from w(,ak force proc_'ss('s can sometimes t)lay the dominant roh'. The tmrpose of this I)r(@('t

is t.o gencraliz(" the analysis of energy h_ss 1)v cha,rged parti(:les via EM interactions to that via weak

intera.cti(ms. In tm.rti('t_lar, one quantity was sought afl.er for t)ot.h |)remsstrahhmg and c_)llisional

energy loss nwchanisn_s stot)t)ing l)(_wer (SP), dEra, lia,i,m/dX. The SP is the energy lost by the

in(:ident particle per llllit path h'ngth trav('rsed. The SP gen(,ra.lly (loI)('nds (m the (,llOlgy, Illa.SS aim

charge of the incident im.rtich,, and the density a.nd charge _)f the t.a.lg{'t, material.
2. Method

2.1. Previous research

The meth()(t used here to find the SP via weak interacti()ns for the tw() types ()f energy-h)ss

me('hanisnls was t() follow the well-(h)(:unlente(t ti)Hnalism f()r similar EM t)rocesses. The reason fl)r

this al)t)roa('h was that the equations fl)r weak interactions (lifter from those for EM intera(:ti()ns l)y

()nly two 1)arameters: the charge Q ()f the I)article undergoing the a(x:elerati()n and the mass M of

lh(, t)arti('h' lne(tiating the for('e. F()rmulas Wel(, ai)I)roi)riately generalize(t t()('ases whet(' Q and M

(liffered from the EM case.

For EM imera('ti()ns, Q is the familiar ele('tri(" charge, whi(:h is ahvavs an int('gral mull it)h' ()f

the ('harge -c of the electr(m. For weak interactions, Q is known as the weak charge and is OlflV

slightly more comi)licated than the (qectri(' charge. In all cas(,s, Q is deduced from a kn()wle(tge

()f the 4-current .P' = (p, ,1") that descril)es the charge p a n(l current ,] densities of the t)artMe in

question. The ('orrect form ii)r J" for any given imeraction is provided 1)y the Standard Model. For

all ehwtr()weak interactions ()f interest, the 4-current of a point charge in m()tion is ti)und t,o b(: of the

form .l'(_St) = 6['7(t)]q*', where d[?(t)] is the usual Dirac delt.a f'uIlcti(,n, which vanishes everywhere

ex('et)t at /(t) = (), an(l q' is a new quantity ('alh,d the 4-chm'ge. Generally q' is f'()un(1 t() be ()f

t.]le f()rnl qt, - q_.td'+ qA._ i_, where u" and ,s" are the 4-veh)('it.y and normalized 4-st)in , resi)ectively ,

()f the i)artMe and q_, and qA are c()etficients (the vect()r and axial-vector charges, rest)ectively )

that d('t)end ()n various well-kn()wn t)aramet.ers (viz, (,le('tri(' charge, weak isost)in and weak mixing

ang;le) that al)t)ear in the Standard Model. In a fl'ame ill which the I)arti(:le's velocity is d v;2 and

spin direct.i()n is ._, u' = -?(1, _7) and .s" = "y'(A_,, ._), where 8_- 1/v/i - -_,2 is the Lorentz factor and

A = ._. _ is the nornmlized helMty ()f the I)artMe. Note that if the direction of "spin-uI)" is taken to

|)(' + 5, then A will be either +1 ov -1, and, flH'ther, in the ultrarelativisti(: limit of interest (where

_, -+ 1), .s' = Au', and thus ,/*'(K, t) = 5[_:(t)](q_, + AqA)u'. Using this expression for the 4-(:urrent,

a formula for the 4-t)()tential A" = (q), _4), which des('ril)es the scalar • and vector ,*_ l)oteIltials ()f

the t)()int charge, was (|edu('e(l (a.s a solution to t,he Pr()ca e(luati(m ). \Vith()ut sI)e('ifying the I)os()n

mass 5I at this point, .4" is related t,() J" |)y the ti)llowing e(luatioil:

1 f "J'(J',t) c _ri_-el .4_(a t)= _ 't:b_' ['<-'_'1 "

(1)

The primed quantities in this equation lal)el the (differential) source (:ha,rg(, (qements. By using

the ext)ression fin' ,1' involving the Dirac delta fllncti()n, th(, integral is easily seen to redu(:(' to the

4-ve(%()r exl)r('ssi()n

1 q' _wl,.-(t)l

A,(<t) - (2)

Electri(: _K an(I lnagn('ti(' /_ fields are found fr()m these equations in t,he same way that th('y are in
the IIIH,SS[OSS C}I,SO:

Ot

(4a)

(31))

For a point charg(' m()ving at v _ i ill th(' -+ dire(¢i()n, th(' (rely n(mva.nishing c()nq)()n(mts ()f th('

i)otemia]s and li(q(Is, (,vahmted a.t ilnl)a.('t I)aram('ter b, are

1 "_,(q_. + Aqa) M,. (4a)

@(b, t) = 47r r u

I _,(q_-+ ,Xq,_) -M,. (41))

A- (b, t) = 47r ,'

1 "_(q_, + Aq,_)b(1 + Mr),_ M,. (4(')

e._(b, t) = 4_ r:_

1 ";,'q__,t

e_(b, t) -- (1 + -A[,')(_ -'n' (4(t)

" 47r 'r 3

1 _(q_, + ,XqA)b(1 ÷ Alr)c_M,. (4(')

B_(b, t) = 47r r:_

wh(,r(' r = V//F + (Tv/)". The quantity ()f (:entra,l ('()n(:eri1 t() energy transl)()rt t)y th(' fi('l(Is is th('

P()yntingvect()r,_. F()rinassivofieldstravelingthrough a vacmm,, S can })esh()wn t.()|)egen(:rally

()fthe f()rm S -- E × B + 3I_(1)A'. Ch,arly it.will always d('I)end(lua(haticaJly()n the charg('s.

Furthermore, as seen al)()ve, th(' heli(:ity A ()f the l)()int charg(, inwuia.l)ly ent('rs int() this fa,:tor. As

the usual al)I)lication of a.ny such th('c)ry is to unpolarize(I beams ()f i)artMes, an a.vorage overall

possible helMty sta.t(' of the (:hm'ge is ()f interest. It turns ()ut that the (:harge-relate(l (lua.ntity

tha.t al)l)ears in the fina,1 set of equa.tions (lescril)ing the dist.ril)ut, i()n of })()s()iis ()utsi(le th(' moving

charge is the q,,m,tity q_. + q.._. Q is thus define,| as Q = v/q_ . + q<._. Q is not th(, ,,sual ca.i,(mical

charge (whi(:h hal)l)(Uls t()I)(' simt)ly q_), a.s (lefin(,(l via the N()(,ther l)r('s(:ril)ti()n, s() it can ()nly l)e

referred t() as some s()rt of (L()rentz invaria.nt) effective cha.rg('. F()r ele(:tr()magneti(' intera.(:ti()llS,

qv = Q _:Mc and qA = 0, wh(,r(, e = _ is the ele('tromagn('tic c()ul)ling c()nstant (a -_ 1/137 is the

line stru('tur(' constant) a.nd Q_.:M is th(' familim dimensi()nless charg(' (luantuln lulnll)or f()r ele(:tr()-

nmgnetic int(,ra.(Mons (e.g., QEM = +i for positrons a.n(l ele(:trons, resl)('ctriv('ly). F()r n('utral w(,ak

intera.ct.ions (mediat('d 1)y Z })osons), qv = gz( T:_ - 2Q TM sin_ 0w)/2 and qA -- -gzT:_/2, wher('

0u. __ 28.74 ° is the weak-mixing _mgh', T :_ is the thiM c(mq)()nent of the vector of weak isospin

(luantum imml)ers ()f th(' l)()int (:ha.rg(' (('.g., T 3 = de 1/2 f()r n(,utrinos and (,l(,(:t.r()l_S, r('sl)('t¢iv('ly),

a.n(l .qz - c sin0u.(:()s0u- is th(' neutral w(,ak COul)ling ('()nstant. And finally, f()r ('hmg('(l w('ak

intera(:ti()ns (m(,(liate(l l)y I'I "± l)()s(ms), q_, = .qw/2v/2 an(l qa -- T.qu/2v/_, r('sl )('ctiv('ly, wh('r('

flu = c/sin 0_,_. is the ('harg('(t w('ak c()uI)ling ('(instant.

()n(, gui(ling l)rinuil)le f()r det(u'mining M was (x)nservation ()f energy an(l ill()IneIfl.Uln. S() _l ') =

(SE) 2 - (517) 2, where ,'_E an(t 517 are t.h(' energy all(| lll()lll(?lltlllll t.ra.nsf(wre(1 (lnt,(|ia.t,('(l |)y the
b_)s(m)in tile collision, servesas a._lefiningequation fin this paranwter. The boson massesare

n_t uniquely _lefinedin generalby this equat.i()n,however,and an a.dditi(malpieceof informa.ti_m

is needed. The Lorentz condition in momentum-Slm.Ce, pt,A _' = 0, yields a relation between AE

and Aft (here t_' : (AE, AI_ is the boson's 4-momentum) that can be used to el|in|hate this

a.ndfiguity. It was a,ctuallv one of the more interesting results of this study that AE a.nd AtL (the

z-collll)Oll{'llt ()t' 2__/-;) a,re f'_)lllld t,o b_' related to one another by way of ttw abow'-menti_med charges:

All)':[ = AE(I - _/_ ')::_,, where s = qAA/(q_¢, + AqA) is a lleW charge parameter. After taking the

uttrarelativistic limit, averaging _)vera.ll possible helicity states of the colliding pa.rticle, and making

_)t.her slighl, relm.rameteriza.tions that ensure kl is L(nentz invariant and the concept ()t' ca.usa.lity is

preserw,d (i.e., the boson travels subhmfinally), an equa.tion for ]ll was arriw,d at that is uniqlle t(_

each of the three tyt>es _f int.era.ct.ions of interest. The final mass schem_, is as follows:

V/ d' ['- (r,)] :/? (5)

A new charge tmra.nwter, t/, at)t)ears here. It is very much like e defined above, and. in fa.ct, is

deriw,d flOln e. F_r the photon, _/is f<mnd to t)_, _1= qh/(qT, -q_*) = 0, so tha.t M = 0, as one mig;ht

expect. For t lu, Z boson, _I = qA/(q._ - q_) = T:_(T :_ - Q[,:M sin2 0_.)/2, a.nd 11= 2q.,_/(q_.l :g- ql) = 1

for the I1 "± t)(_son, respectively. Interestingly the masses of the W and Z ])osons d(_ not equal

the masses predicted by the Standard Model (80A19 an(t 91.1882 G(¥, rest)cot|rely), and in fa('t

can be orders _)t magnitude different, yet the distribution functions for the bosons using this mass

scheme agree exa.ctly with the well-known distribution functions found in the patton theory for

electroweak interactions. It is also noteworthy to point out that the ratio of the masses of the Z

and W |)osons here is roughly the sa.me as the ratio found in the Standard Model. Dn example, if"

the colliding particles are either protons ()r electrons, this ratio is 0.9300 here, and cos 0u- = 0.8768

in the Standard Model. The other parameters found in the above equation are as follows. 3,0 is

the mass of the particle (f for fermion) that emits the boson and :r = AE/Ef (Ef is the energy

of the tmrti('le) is the Feynman scaling variable, which is bounded (l)y conservation of energy and

causality) by the inequality string ,],,/[(Tv) 2 + '1] <- :r <_ 1 - 1/7. Ill Sllllllllal'V, a bOSOll lll_-I,SS

scheme was (let|veal from the basic ('oI_('eI)ts of conservation of energy and n_onlentllnl, causality,

and Lorentz invariance that uniquely specifies t.lw lnasses of the equivalent |)osons in the theory

_l_'veloped here. Dest)ite the fact that the boson masses here are n_t constai_t and the values of

the masses of the \V and Z bosons can be quite different than those predicted by the Standard

Model, the t)()s_m distritmtion functions are in perfect agreement with th_)se in the pa.rton theory

for elect.r()weak intera.('ti{ms.

2.2. Bremsstrahlung Energy Loss

The calculation of the SP fl,r bremsstrahhmg processes involved the determination of severa.1

int.ermedia.te quantities. The most difficult one to derive was the frequen( T spect.runi, d'2I/dAd_,

of the field of |)osons _mt.side the charge; this quantity wa.s referred to above as the boson distrilmt.i_m flmcti_m, d2I/dAd_ , represents the total amount of energy ca.rried by the field per unit t)<)son

4
energyper area elenlenton the wavefrontof t,he Lorentz-contract('dE and/3 fields. In short, it. is

the Fourier t.ral_sform of t,he P(_ynting v(,ctor S_ = E × /3 + M_I_,_ ment.i(med above.

A slight subtlety had 1_ lw addressed in deriving d'eI/dAd,:. EM radiation is known t.()})e

polarized in a plane Iserpendicular t(_ the directi(m of propagat.ion, wtfich means that when an EM

wave (i.e., a mva.rm (_f photons) iml)inges (m a. charged particle, the I)art.icle will _mlv move in that

t.ransverse plane. But ra.diati(m ass(wiat.ed with massive 1),)s(ms can exist in a.n a.dditi(ma.1 state

longitudinal 1)olaIizat.ion, which means that. the induced m()t.i(m can ta.k(' place in t)ract.ica.lly a.n.v

direct.ion relative to the (lirecticm ,)f t)ropa.ga.tion of the wave. So there are actually t.w_) different.

fl.e(tnency spectrum flmct.i0ns one corresponding to t.ransversely-pola.rized t)osons (designated with

a T) and the ot.her describing hmgitudinally-I)olarize(1 bos(,ns (designated with _m L); it. can t5('

shown that only the T t_os(m sta.tes c(mt.rilmt, e t.o the bremsstrahhmg t)r_cess. The frequency

spectrum flsr T t)oson sta.t.es is

V_rll(}l'e

_I1(1

_J(t21 _ 7" = I,,_ _h ..,. _(_), (6)

Io - = const (7)

(2_) :_ bZ_,_

- b ill _ + (8)

K1 is a m,)dified Bessel flm('t.ion of the see(rod kind, of order l, b is the impact imramet.er in the

collision, and ,: = AE is the angular frequency of the boson.

Next, the radiation cross s(.('ti(m (RCS), dx/d._', which (:()nveys the l)r()l)al)ility fl)r a l)os(m

with a specified energy t,obe emitte(l from (me ('harge and subse(tuently a,bsorlsed t)y another,

was deterInined. It. in, in a (:rude sense, the i)r(utuct ()f d'2I/dAdo: and another cross section, the

scattering cross section (SCS), dcT/dtL all integrated over a solid angle of 4rr steradial_S and the

(infinite) area of the wavefront of Lorentz contracted /_ and /3 fidds. The SCS can be th(mght of

as the time average of the t)roba.bility fin' the incident charge t.() be deflected int{) a certain specified

direction during the c()llision with a.norhea' charge. For the EM case, this quantity is refl'rred to

a.s the Th, m_son scat.tering cr()ss section (TSCS). A generalized versi(m ()f it. was derived here froIn

first princilsh's:

_ Y + ,.os (,0)

(qi,,,.i+_,_,)u is Q_ fin the incident particle' (e.g., the incident cosmic ray), (q,,_r_,,,)" is (_2 filr the target

part.icle (e.g., a.n atom in the spa.cecraft wall), V is the velocity of the pulse of lwemsstrahhmg radiation emit.ted, and 0 is the scattering angle ,_f the pulse (5t"t)remsstrahhmg radiati(m emitted. The
fornmla reducest() the familiar Thonlsonfi_rmulawhen the chargesareset t,(_the electric charge _,,

an(l 3I is set t() zero.

After these two) quantities (leterlnining d\/&,: were identified, it was fomid tha.t, unlike tit(,

traditional RCS, the generalize(l RCS could not be a simI)le t)ro(lu(:t of t h(, tw(_ a l)()ve-mentione_l

t('rlns. Aft, er ironing out this techlfi('a.lity, a final formula was found for the genetaliz(,(l RCS that

imln(,(lia.t(,ly yMds a generalized version of tit(, SP:

(IX l(J 1 (qitwidem)2 (qtarge_)2 /: @/

bmm

where /_min is tit(' minimum impa,ct parmneter effective in the collision. A detailed analysis shows

that, for any ()f the three types ()f electr()weak intera(:tions ()f int('rest, the ()v(,rall behavior of tit('

RCS is dominat.e(l 1)3' the _., -+ 0 limit. Given that the energy AE ()t the b()s()n that nw(tia.t(,s the

intera(:tion is relate(l t.() ,., via the equation AE = _.,, it. can 1)e said tha.t the n()nrelativisti(: t)()sons

(i.e., those with AE << Ef. or x << 1) contritmte the most to the generation ()f bremsst.rahhmg. The

1)()s()n mass schem(, then lint)lies that th()se t)()sons with A[ << 3/f make th(' grealest ('()ntrilmtion.

As such. tit(' choi('e of brain f()r the lnassive mediator cases sh()ul(t 1)e exactly the same a.s that made

tbr th(, t)h()t()n case. It is easily worked out lhat b,nin -- 1/2M:_, is the al)I)roI)riat(' (:h()i('e. The

generalize(l SP ca ml()t t_e cast. int() a silnt)le corot)act formula and necessarily inv()lves a t.w(_-f()l(l

integral:

_U'IIIax

dE_,,.o,. _ N : (1_: dx ., (:0., (11)

0

where N is tit(' mmd)er density ()f a.t()ms in the target ma.terial, and _',,,_,x is the lnaxilnltln aIlgular

fl'e(luen('y (allowe(t l)y ('onservation of energy) of the mediating boson.

Ill ('()tttrast l.() the above e(lua.tion, the formula, for the traditional (massless mediator) SP is a

()ne-(limensional integral. Nuinerical meth()ds were needed to solve the integrals. Nmneri('al convergence of the integral exl)ression for the RCS was a.ssure(l 15_ c()mt)aring th(, numeri(:a.1 result with

the ext)('('ted analytic result for the photon-mediated l)remsstrahhmg process. Th(, same integra.

ti()n ste I) size was then used t() calculate the RCS ti)r W- and Z-t)oson-mediated intera('tions. As

m(mti()ne(l above, the SP was then (leternfine(1 t)y I)erforming Olte more integral, using the santo

integ_ati()n stel) size as before. F()r ultrarelativist.i(. ('()smi(' rays (in the ('omI)lete screening limit),

the RCS turne(l out t() siml)ly be a ('()nst.ant (indet)endent of the energy ()f the c()smi(: ray) for the

I)h()t()n ca.se, and very nearly a constant for the \V- a.nd Z-t)oson cases. As such, the SP is the

product ()f the RCS, the (h'nsity of the bulk ma.tt, er, a.nd the energy of the (:osnfi(: ray.

Written in another more intuitive way, the energy E(X) ()f tit(' (:Oslni(' ray (:an be shown to be

relate(t to the (listan('e tra.v(wse(1 X a.s: E(X) = E(_c -x/'\°. Here, E0 is the energy of the (:()stoic

ray as it enters the material a.nd X(j is a t)aram(,ter known as the ra.(lia.tion length. The radiation
hmgt.his the dist.anccit takesfin the lmrt.ich"s enorgyto fidl to 1/c of its initial wflue. Acc(nding

t.oan oversimplifiedanalysis(found in the lit('ra.ture) of EM lwomsst.rahhmgelnitt.odl)y _1.11electron

tra.vorsingthrooditfbren[ t)ossiblemat.elia.ls(load, ahmfinum, and air), X0 is a b_mt. 0.4 tin fin' load,

7 cm fin alumilmln, and 270 m fl)r air. These rosults wero vorified, a.ll(I c()rrost)(mtling wdues ()t' X0

for t)_th W- and Z-l)_)s_m-modiatod I)r(wesses wore (lot.ermilwd. For load, X0 was flmnd t.o tw 45

m and 7.9 cm fl)r tli(' W- and Z-tl()S()ll ('_l.S(','.-;,resl)octivoly; for ahmfimum XII was flmn(l to t)(' 20

m and 3.3 m fin tho W- aim Z-t)os_m cases, rost)octively; and for air, X0 was fl)lllld t.{)1)o 20 kin

and 14 km ti)r the W- and Z-1)os(m casos, respectiv(qy. As can 1)(' gh,anod fl<)m tho at_ovo formula,

simfllor values of X0 c<wrost)ond t() groat.m am(milts of Onolgy lost. in the fl)rln of 1)iemsst.rahhmg.

S() clearly, the greatest amount ()f 1)l-Olnsstrahhmg clwrgy is carried away t)y photons aim the h'ast.

alllOllllt, is carried awa.y by W-bosoms. In N.ct, the proportions of blemsst.rahhmg energies carried

away by t.heso three particles can bo easily determined. If an electron is tho incident cosmic ray

(tram'cling at any speed), tho phot.on : Z- 1,oson : W-t,oson ratios of brcmsstralflul_g energy loss are

roughly 1 : 10 2 : 10 _' ibr lead, 1 : 10 '_ : 10 -'1 for alumimml, and 1 : 11)-2 : l0 -_ for a.ir.

The cases considcre_l in this st.udy assume the c_slnic ray t.o be an electron l_eca.usc electroils

are the lnost oflciciollt, of all cosmic rays in emitt.ing bremsstrahhnlg. As a. ruh' of l.huIntL greater

ammmts of bremsstrahhmg will t_e produced t}y COslnic rays with higher chargo-to-mass ratios

traveling through materials with groa.tor charges. The most. officiont, sconario t.urns out to be an

electron tra.voling t.hr(nlgh a dense material such as lead. In figure 1, a. coIntmrison is made l_et,ween

the fra.cti{m ()t cosmic ray energy lost, and breinsstrahhmg associat, ed with the three elect rowoak

bosons fi)r a 500 G('V el['ctIOn traveling through a 0.4 cm slab of lead. As alluded to ab<)ve, tho

energy lost t,o W- and Z-boson bremsstrahhmg is a mero 0.01 and 4 percent, respectively, of that lost

t,o EM t)remsstrahhmg. So, it. can Im roasona|fiy sta.t.ed that the ovorwholming amount of energy

lost in the tbrm of bremsstrahhn_g is carried away by photons.

2.3. Collisional Energy Loss

Tho calcula.tiol_ of the SP fl)r collisional energy loss t)rocessos was easier tha,n that fl)r

1)relnsstra.hhu_g processes. St.a.iMa.rd Methods (in particular, Fourier almlysis) were eml)htved t()

find the _t.lllOllllt of energy/3 transfl'rred t(_ a.ll eh'ct, i'_m orbit, ing an at()ln in a gI'azing collisi(m with

a fast.-moving (incident.) charge,

1 f

E = S---v +

(12)

where Apan,_,, - w/2,_,, Ap± = V/3I 2 + (Ap±,,,i,) 2, and _ = bAp_. It was a simph' matter to

convert, t.his genol'alizod E into a genera.lizod SP. Tho SP flmnd fl_r t.hoso c_fllisi_mal procossos was

mu('h simI)h'r in flwm t.han that fl)r bromsstrahhmg t)rocosses; unliko tho 1)romsstrahhmg case, the
SP11(,1"(' redu('edt()a simple eonqmctformula, albeit quite eOlnt)licated:

,: 2 2{

dE,-,,H 1 (q,,,.,.,.,, ) Oh' (o)lio(O)

(t.¥ 4rr :llvl,,,:tront, _

1 1-- ( Ap±"'i:: 0 z [t£f(0)- Kff(O)]

2 \ -)Ap±

where 0 = t_,,,i,,AP±. When Q is set to the electric charge and M is set t.o zero, the formula reduces

t.(_ the familiar tradition_d w,rsion.

()he of the main goals of this t)a.t)er is t,) cOral)are the amount of energ3, _ lost via this Ine(_h;tnisni

with that lost via bremsstrahhmg for the three diffe, rent types of eleetroweak interaction. It, is

known tlmt the f()rnler energy loss mechanisnl (lonlinates the latter for sl()wer moving electr()ns.

Again, an ele('tr()n is taken t() })e the in('ident c()smi(' ra.y partich, t)eeause it. is the most effi(qent

()f all tmrti('les at radiating energy. St)eeifi('ally, it is kn()wn that (:ollisiolml energy loss (h)nlinates

t)remsstrahhmg enerD, loss if the relativistic factor -). (energy-to-mass ratio) of the cosmic ray is

less than about 20 if the bulk matter through which the electron travels is lead, and 200 if the

nmterial is air. These thresh()lds (fl)r EM I)rocesses) were verified in the I)res('nt analysis, though

the values found were a.bout 17 and 170, respectively, m,l corresponding ones were fi)und for Wand Z-boson I)rocesses. In lead, the threshold value of "y was found to t)e about 8.4 × 10 l and 17

for W- and Z-t)()s(m t)r()('esses, respectively. Ill air, the threshold value of _: was f(mnd t() t)e at)(mt

5 × 10:_ a.nd 1 × l() :_ for W- and Z-1)oson t)ro(:esses, rest)ectively. The results, a.ssuming the ('osmi('

ra.y t.(_ 1)e a.n electr()n and the target t() be made of lead, a.re shown in figure 2, where the ratio ()f

the SP for the bremsstrahhmg t)rocess over that for collisional (mergy loss process is I)h)tted against

":,. The threshohl above wherein t)remsstrahhmg (lominates eollisional energy loss corresponds to 1

along the y-axis. This threshold is clearly suri)assed for all three tyt)es of ele(:troweak intera.ctions

above et)llisi()n energies ('orrest)(mding to 2, = 10'_,. The ('on(:hlsion t.()])e (tl'a.wll froIll these results.

then, is th(, fa.ct that f()r ultrarela.tivistic ('osmi(" rays, where 5' is typically grea.ter than about 1()(_,

the (t()minant energy loss m(wlmnism is most assm'edly bremsstrahhmg, insl('a(t ()f ('()llisions, f()r all

three types ()f el(,('tr()weak intera(:tions.

3. Conclusions

In summary, the stot)ping t)ower fi)r a COSlni(' ray traveling through lmlk nm.tter was derived taking int()a.('('(mnt both eollisional and Bremsstrahlmig energy-loss nlechanisms. Nmneriea.1 r(mtines

were used to evahm.te the stot)t)ing t)ower for different s('enari()s. In all (:ases, the cosnfie ray was

assllnl('d to be all electr()n, and three different possit)le materials were consi(lere(1 a.s the bulk matter: lead, alumimlni, and air. ()no nmjor result ()f the project was tha.t 1)remsstrahhmg (l(mfina.tes

the eollisi(mal energy-h)ss mechanism tbr the ultrarelativistie cosmic rays ()f interest for all three

(lifferent tyi)es of ele('tr()weak intera.ctions. The other major result showed that the brenlsstrahhmg

a.ss()('ia.te(I with \V- an([ Z-t)os(ms only enters as a ('orrection at the 1-percent level t._) ele('tr(mmgneti(" I)remsstrahhmg. This latter result is the more imp()rtant one for the t)url)(>s(,s ()f figuring out

8
the influencethat th(' most energ(,ti("forms(_f'stm.('ermtiat.i(mhave(m hunmns. It. sh(_wst.ha.tweak

fore(' effect.scan 1)esafely ixn(n'edc(mqmre(lwit.hsimilar eh,ct.roina.gn(,tic(,ff(,('t.s.

Bibliography

.

,

Altarelli, (7;." Meh', B.: and Pit()lli, F.

Phw. 13, vol. 287, 1987, pp. 205 224.

Heavy ttiggs Pr(,du('ti(m at Future Colhd_r.'. Nucl.

(atn, 1 1R. N.: Pro(hwti(m of Heavy Higgs B(_s(ms.

1qesult.s. Nucl. Ph.!l.*. /3, vol. 255, 1985, pp. 341 354.

Comtmris(ms (_t' Exa('t and AI)pr(*xilnat('

3. Daws(m, S.: The Effective I1" At)proximation. Nt,:l. Ph.y.s. B, vol. 249, 1985, pp. 42 60.

4. Dawson, S.: Heavy Fermion Piodu(:t.i(m in t.he Efli,ct.ive lI _ Al_proximati(m. N'_,:I. Phys. B,

vol. 284, 1987, pp. 449 462.

5. Griflitlls, D.: h_,t_vd'u(:tio'rl to Eh:mcl_tar.q Partich>. John Wiley & Sons, Inc., 1987.

6. Groom, D. E. ('t a l.: Th._' EtmWea_ Physical ,lour',ml C, v(_l. 15, 2000, p. 1

htt t_://txlg.lbl.g(w. Accessed Aug. 8, 2002.

7..lacks(m, J. D.: Cla._sical Eh:ctrody',l, amics. Secolld e(t., .h)lm Wiley &" Sons, 1975.

8. GuIfiOn, .I.F.; Haber, H. E.; Kane, G.; and Dawson, S.: Th.c Hi99._ H'wl_.tcr'._ Cuid_:. AddisonWesley Publishing Company, 1990.

9. Ha lzen, F.; and Martin, A. D.: ()uar'/,:s _a Leptm_s. AT_ Introdu, ctor!l (:ou.r._(_ i_ Moder',_ Particle

Physics. ,lohn Wiley & Sons, 1984.

10. Kane, G. L.; IRet)ko, W. W.; and 12olnick, \V. B.: The Eflbctive II "e, Z ° Approximati(m tbr

High Energy C()llisions. Phrys. L(ett. 13, vol. 148, 1984, pp. 367 372.

9
o

'5

t,,..

O

e--

ej

r_

10t

10-2

10.3

10-4

10-._

10-f'

_ zJ

/

/

/

/i

,J

J

/

Photon bremsstrahlung (Xo = .32 cm)

Z boson brenlsstrahlung {)," 0 = 7.9 cm)

W boson bremsstrahlung (X0= 450 m)

i I I

0 . I .2 .3

Distance traversed, cm

.4

Figure I. Fraction of energy lost due to Bremsstrahlung by a 500 GeV electron traversing 0.4 em of lead.

10 4

_J

,.,,,,.1

.c;

I0 3

10 2

10 I

l() (I

10-i

I 0-2

10-3

I 0-4

Z boson interactions

_ Wboson interactions

I I i

I 0 i 10 2 10 3 10 4 I 0 $

Gamma

Figure 2. Ratio of SP due to Bremsstrahlung over SP due to collisions versus gamma for an electron traveling

through lead.

10
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4. TITLE AND SUBTITLE

Radiative Energy Loss by Galactic Cosmic Rays

6. AUTHOR(S)

Ahem, Sean C.: Norbury, John W.: and Tripathi, R. K.

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

XlASA Langley Research Center

Hampton, VA 23681-2199

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

5a. CONTRACT NUMBER

5b. GRANT NUMBER

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755-06-00-03

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REPORT NUMBER

L-18209

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,_k31_emand Norbur'g University of Wisconsin Milwaukee Wisconsin. Tripathi, Langley Research Center

An electronic version can be found at http://techreports.larc.nasa.gov/ltl s/or http.,,techreports.larc.nasa.go_/cgl-bm, NTRS

14. ABSTRACT

Interactions between galactic cosmic rays and matter are a primary focus of the NASA radiation problem. The electromagnetic forces inw_lved are for the most part well documented. Building on previous research, this study investigated the

relative importance of the weak tk)rces that occur when a cosmic ray impinges on different types of materials. For the familiar

electromagnetic case. it is known that energy lost in the form of radiation is more significant than that lost via contact

, collisions: the rate at which the energy is lost is also well understood. Similar results were derived for the weak force case. It

was found that radiation is also the dominant mode of energy loss in weak force interactions and that weak force effects are

indeed relatively weak compared to electromagnetic effects.

15. SUBJECT TERMS

Weak interaction energy loss: Cosmic rays: Bremsstrahlung; W- and Z-bosons: Radiation-matter interaction

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