A general formula for the sec/prim ratio, independently of any details of the propagation and acceleration model is presented. In the limit of equal fragmentation paths for primaries and secondaries, this ratio at a given momentum nucleon is proportional only to the mean path of the observed primaries at that moment. It is shown that it is unlikely to get a decreasing sec/prim ratio with energy if an acceleration process takes place during particle propagation in the interstellar medium (ISM)

Giler, M.

Szabelska, B.

Wdowczyk, J.

Wolfendale, A. W.

English

NASA Technical Reports Server

234
O_ 8.3-4
SEGONDARY TO PRII,<%RYRATIO
AND TI_ CONTINUOUS ACCELERATION
M.Giler 4 , B.Szabelska & , J.Wdowczy_ and A.W.Wolfendale Z
4 Institute of Physics, University of _6d{
90-236 _,6d_, Nowotki 149 153, Poland
I Institute of Nuclear Studies
90-950 _6dE I, _ox 4_7, Poland
$ University of Dt_rham, Durham DHI 3LE, England
1:,In%roduction. It is w&ll known that the ratio of secon-
dary to primary nuclei in the oosmic radiation is a de-
creasing function of energy for E_ 2 GeV/n. This fact has
to be interpreted in terms of c._. propagation and acceler-
ation model, An important problem is whether these two
processes are separated in time (and in space) or ca_ occur
simultaneously. As sumin_ the leaky box model, Cowsi_showed
•that the decreasing sec/prim ratio is in a strong disagree-
merit with an effec:tive acceleration takimg place in the ISI_It
predictin_ an increasing sec/prim ratio with energy. Howev-
r _it seems tha-t"t-h-_ Ts st-ill some confusion whether this
" conclusion is applicable to other models of c.r.. propaga-/
tiom or it is true for the leaky box model only.(:_
In this paper we give a general formula for the sec/prim
ratio, independently of _ details of the propagation and
acceleration model. In the limit of equal fragmentation
paths for primaries and secondaries, this ratio at a given
momentum nucleon is proportional only to the mean path of
the observed primaries at that momentum. We shall show
(basing parily on this formula) that it is unlikely to get
a decreasing sec/prim ratio with energy if an acceleration
• p_ocess takes place during particle propagation in the ISM.
2.General formula. Let us denote by f(p,t) the vacuum time
distribution of.primaries observed at the Earth with the
momentum/nucleon p;_The nu_b.er of the observed p_imaries is
of course _):_ _(pn_)e-_4_ , where T is their mean
life time agi_nst_ffagmentation. ( We shall keep in mind
that "time" means "path length" in g/cruZ). Particles arriv-
ing with age t have produced secondaries, which must propa-
gate and be accelerated in the same way as their parent
panticles_ if we adopt_ a _easonable assumption that these
processes depend on p only(which is not changed by fragmen-
tation). So they come to the observation point with the
same mome_tum/n and their number is _, --- ._ - i
-- "- "i's "
The total number of secondari=_ 41L[_)=f _=._(p)_and for
the sec/prim rat io_e have_ _
0.+'ll"('p)l!_4'%4[p) - '-- where (2)
https://ntrs.nasa.gov/search.jsp?R=19850026610 2020-03-20T16:48:28+00:00Z
286
OG 8,3-4
Of course it has been well _ known for a long time bi,at the
sec/prim ratio depends on the path length distribution but
we would 1J/_e to stress here that eq. (2) holds for any
model of the Galaxy and for any assumptions about the ac-
ce:leration or deceleration processes (provided they depend
on p only). For O<(TI-T4)/T4Z_I__._e q. (2) gives
_ +4A- (r,- 3 (3)
where _(p)=/_p,%_e'%fT'_#J°{(_)_'_ (and similarly for
t (p)) iS _heJmJean time (_at_l length) of the observed _pri_
mary particles at a given p. Thus the see/prim -_e-_y de-
pendence is _practically equivalent to the energy dependence
of the mean pro___ga_ion time of the observed particles (for
T_-T4<_t and _) ,_but not of the mean vacuum time.
_2_.Exam_lest First we shall consider _a situation when o is
a unique, function of t. This could o_cu_ if, for example,
c,r, were produced with a constant p and then accelerated
according to dp/dt = h(p)> O. _._. f(p,t) is reduced to
F(t) where a4@}dp-F . )]4{CD}eT P' 4. see/prim
ratio we have from" (I) _(_)_@_P)/A.4)with _(_)%J _/_)- (_)
so n./ru is a _owin_. function of p. For the '-first order
_e.rmi process, when dp/d.t = _p (_>0), we get
If particles are produced with a _istribution of nrim_.r_
moment_ p then f(p,t)dt = f(p_,t_)dt_ where tl= t + _ with
so "[(_)'4"[_¢_),4]e _;n % hence r(_)t" %(p). o l_fHere _ is independent of t but it i_ not a necessary condi-
tion for r(_) to grow. r(p) will also grow if f(p,t) for
highe_ pl is effectively shifted to longer times so that,
for example, f(p,t)dt = f(#,t_dt_with t = t +_(p,p_,t) and
_ O. One would expect that to be rather natural when
the acceler%tign t_-es place. Growing of r(p). is seen from(6) since _''_ _ for any t for T%:_ T4 which is the
case for secondaries being lighter.
Let us next consider a second order Fermi acceleration
- when p is not a unique function of time. In particular we
shall assu_ne that its behaviour with time correm)onds to a
uniform diffusion along the Io<_ p axis . i4oreover we adoot
o, l-dim, model of the Galaxy, the dimension x bein_ per oe_%-
dicular tO the Galactic plane. C.r. nuclei are produced in
the region 0< x<l at a constant rate q (per unit length)
with a single momentum p,. They diffuse, are accelerated
and fragment at the same time, leaking out of the Galaxy at
x = 0 and x = !. First we shall eonside_ a case of a con-
stant spatial diffusion coefficient D. 7for that case it is
since particles with age_t have a g&ussian distribution, of
:_,-!nCglp.). Denoting J-f(p,t)_"Pr, m I_ we find
_•SQ
0G 8,3-4
(9)
It is evident that r(p) increases with momentmm for T_>T 4 •
In the limit p_ _ , when the first term in the series dom-
inates, we set q.(p)__ [_/n._-__ .I (10)k,r-j
_.z_Ot_qersolut.igns_ It is sometimes difficult.to find an_.-
finding f(p,t) in the above descrioea moae± ii _ne spa_laA
diffusion depends on p but this does not seem to be, an easy
task. So we shall treat this problem considering the equi-
librium equations. We assume a second order acceleration oc-
curing, for exampTe, as a result of particle collisions with
Alfven waves in the ISM. This corresponds to a particle
difftusion in the 3-dimensional momentum space with a momen-
tum deoendent coefficient 1_(p)__The_e_uation for primary
-.oarticiedensity in the phase •space- ( F4(x,P)-4_P_dP dx =
(11)
Let us first negIect the te2m with fra6mentation _'_/_.x;.
As it is usually _one we look for oolutions in _he _form
F_(;_,p) =.F!(p).F_(x). Assuming further K(p)- B.p _ and
_[_ _ _,e _et for p_ p__
' -
r _p_ "x "_ _t' _' "
',]eshall lo0k for power l_w spectra and this implies that
_= 2 +_. (However, for a consistent picture of acceleza_::
%ion and soatial diffusion due to Alfven wave_ both diffu-
sion coefficients are related by K(p)-D(p)'_p for relativ-
istic -oartic3es, but this does no% lead_to experimentally
-- = _ " _ ,, • -_ -_ _ ° 1 2
observed -oower law soectro.). ,l_th-(p)_-p we h_ve _rom I._,)
-_il(&)_ sin(_._:), with" _L=(4_l[/g) _ and n = 1,3,5,..; Ztence
where 8 (p - p,) is a step function. Substituting (14) to
(11) (with no fragmentation) we get
=Zov -'.-Po
'At high p the morrten_um s_:_ec_rtumbehaves _s p-_- 4"_ P', so
independently of the spatia! distribution' of the sources _..
_,,,hic]_. influences only G_.
_;e o_al], find now the seeondsmy psmtic.le spectmtuu. Let
12_(x.,x,p) denote the phase space density of secondaries
observed at x with n, oroduced at x.. Then we have
where l,_e,(:zo,:.c,:p) is the solution to the ee. (1"1) (m__3
• . .... .............or ....... __om (14-) on.q_, by different coe_{-T],:<s so3 ut.i.o_. '" 4-_'-r" ,-;-_ .... ,, _ -.....
237
OG 8.3-4
ciento G_(x@), the_depen¢_ence bezng the same. Solving (16)
and r_ _
. _ , _-,._ (I
For sil_licit_ w9 have put P_v = Po. I_ (I_) corresponds to
the secondaries that have been produced with momenta small-
er_(larger) than p. To find F_(x,p) we have to integrate
# FL(_o,_,p)d_o but its momentum depe_p_ce i_._&_eady seen
°_rom (17). The terms with p-_ and _l_ominate for p_p_,
so the sec/prim ratio increases with p as _-(_/_)-_, practi-
cally indeoendently of the gas density distribulion _(x).
hot @(x) =-const. andg= 0.6 F_/F_ (for p/p_= 10) reaches
,,,,80_4 of its maximum value. .
Taking now into account the fragmentation term in (II)
_e look, as before, for solutions in _he fo_m F_(x,p) =
#(p)-_x), if T_(x) = const. For _(p)_p-_ we get
This can only be fulfilled at p_ _ and it can be sffen that
then the fragmentation term does not play any r_le. In par-
ticular if _- 2 =K, _ has the same form as in (13). If we
asoume _hat the secondary spectrum has a form _-°" for
p -_ _ then we get
;:r, r, p"- + o (.19)
This can only be fulfilled at high momenta if _ = _ . So at
p-_ the sec/prim ratio--_ const, even if we take into ac-
count fragmentation.
5.Conclusions. _^Teconclude that, contrary to some sugges-
tions , a simultaneous acceleration and propagation in the
ISI'Iwould lead to the sec/prim ratio increasing with momen-
tum (tending in some cases to a constant for p-_ ). That
is in a strong discrepancy with observation. The logarithm-
ic rise, stressed by Cowsik _, is obtained for some partic-
ul_r cases only. _ioreover the si_ape of the particle spectra
:Jr p _ does not depend on the spatial distribution of
their sources.
i{eferences
_._T_, _., (_9191,_p._.2_7,85_
• (19_0), A_.J. 2_, 1195(19oi), in "Origin of Cosmic Rays", IAU Syrup.
no. 94-_Bologna
2. Schlickeiser, R., (1984.), lectures given at Int. School
of C.R. Astrophysics, Erice


Secondary to primary ratio and the continuous acceleration

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