The evolution of the energy spectrum of cosmic rays accelerated by the first order Fermi mechanism, by a supernova remnant shock wave, including adiabatic deceleration effects behind the front, is carried out by means of a time-dependent numerical code. The calculations apply to the adiabatic stage (or Sedov stage) of the supernova explosion, and the energetic particle spectrum is calculated in the test particle limit (i.e., the back reaction of the cosmic rays on the flow is not included). The particles are injected mono-energetically at the shock. The radial distribution, The radial distribution, and the spectrum of the accelerated and decelerated particles is shown

Jokipii, J. R.

Ko, C. M.

English

NASA Technical Reports Server

:156 OG 8.2-2
A NUMERICAL STUDY OF DIFFUSIVE SHOCK ACCELERATION
OF COSMIC RAYS IN SUPERNOVA SHOCKS
C. M. Ko
J. R. Jokipii
University of Arizona
Tucson, Arizona 85721 U.S.A.
Abstract. The evolution of the energy spectrum of cosmic rays accelerated
by the first order Fermi mechanism, by a supernova remnant shock wave,
including adiabatic deceleration effects behind the front, is carried out
by means of a time-dependent numerical code. The calculations apply to
the adiabatic stage (or Sedov stage) of the supernova explosion, and the
energetic particle spectrum is calculated in the test particle limit
(i.e., the back reaction of the cosmic rays on the flow is not included).
The particles are injected mono-energetically at the shock. We show the
radial distribution, and the spectrum of the accelerated and decelerated
particles.
I. Introduction. Diffusive acceleration of cosmic rays at collisionless
astr_s-hock waves, where the particles pass (diffuse) across a
plane shock repeatedly is a promising candidate for the origin of cosmic
rays (Axford, Leer and Skadron, 1977; Krymskii, 1977; Bell, 1978;
Blandford and Ostriker, 1978).
Supernova explosions have long been supposed to be the origin of
cosmic rays, primarily because of the energy they liberate. The
diffusive shock acceleration mechanism has been applied to spherically
symmetric supernova shocks (Krymskii and Petukhov, 1980, Prishchep and
Ptuskin, 1981; Drury, 1983) and analytical solutions can be obtained with
the assumption that K/RR is a constant (see also Blandford and Ostriker,
1980 Bogdan and VDIk, 1983; Moraal and Axford, 1983).
In this paper, we solve the time dependent problem of cosmic rays
diffusively accelerated in a spherical shock by numerical integration
II. The Model. Under the assumption that there is sufficient scattering
that the pitch-angle distribution is nearly isotropic (diffusion limit)
the cosmic ray transport equation (Parker_ 1965) may be written (for
spherical symmetry)
where _t_?_) is the phase space distribution function, U is the flow
velocity, and S is the source.
If there is a shock at radius R, we require f to be continuous at
the shock (Toptygin, 1980), and the jump in streaming flux can be
obtained by integrating (I) across the shock. Subscript I corresponds to
the region outside the shock (upstream region), while subscript 2
corresponds to the region inside the shock (downstream region), the
boundary conditions at the shock are:
- Li_ _.m+E-
,4
III Supernova Shock If the shock front is moving, i.e., R = R(t), then
it is useful to use the following dimensionless quantities:
g = VR+.<_) , "g = K°+-/R'(.+-} an<:t _ --t,-,CP/p.) , (3)
where K_ and P_ are constantsto be defined later. This transformation
breaks _own if_tR/R=t because the Jacobian of the transformation is then
zero. Note that if K is constant and so is RR/K, then R _ %,/L •
Assume a velocity profile U=RV(%)_I-%) (e.g. Sedov's blast wave;
Sedov, 1959) and consider particle injection at the shock_ i.e.,
https://ntrs.nasa.gov/search.jsp?R=19850026590 2020-03-20T16:48:48+00:00Z
OG 8.2-2
157.
S =Q%(_-R_(P-?o)= _o_(%-I)_%_ , (4)
where the units of Q are #/sec/Iength2/momentum 2. If V(E)goes to zero
faster than 5 k , _ > i (which is true fdr Sedov's blast wave
approximation), then _(0)--o _& %[_V)/_o=O (i.e., the boundary condition
at %=o is %_/_%=o). If we assume self-similar solutions to the shock, R
= A h a then by equation (3) we have:)
_°tC''_) /A'__/cI-_)
Substituting (3), (4) and (5), together withK--_,_ -[K_-K,_'_)0(_ -l)
into (I) and (2), the equations for the transport of cosmic rays in a
moving spherical shock can be obtained. In order to have a forward in
time equation, 4 has to be smaller than 1/2 (e.g. Sedov's blast wave,
_= 2/5).
/
IV. Results and Discussion. We employ _ finite differencing scheme
implicit in the spatial variable but explicit in the momentum variable;
it is also second order in spatial variable and first order in the
momentum variable. To satisfy causality, we use downstream differencing
for momentum except at the shock, where we use upstream differencing. We
replace $(_) by a Gaussian with a spread of the order of the step
size in q during computation.
In the case of a moving spherical shock, we use the supernova shock
as an example. We concentrate on the adiabatic stage (or Sedov's stage)
of a supernova explosion only.
The self-similar solution (Sedov, 1959) to the blast wave equations
is R = At _ where A = (E/@,)Y_ and o_ = 2/5. The energy, E, is
proportional to the explosion energy, Eo • If _" = 5/3, then E = 2.02 F_
• p, is the density of the undisturbed (i.e. upstream) gas. The
vel6cityV(%) is given by V(E)=_/z%v(%)where lt(_)is a complicated but
monotonically increasing function and o %4(_[%)_< 0.5 .
The adiabatic stage of a supernova remnant begins roughly when the
mass of interstellar gas swept by the shock is equal to the expelled mass
of the supernova (e.g. Spitzer, 1978). At the beginning of this stage
the shock radius is:
4-'_ _',
where _ is the expelled mass in solar mass units (M_. Substituting (4)
and R_= At_ into (6), we get:
% = K° ( (7)I _,/
The adiabatic stage will last roughly until the temperature behind the
shock drops below the recombination temperature (i.e. the shock becomes
radiative) (e.g. Spitzer, 1978). Let subscript f represent this. The
relation between upstream and downstream temperature for a strong shock
is (e.g. Landau and Lifshitz, 1959):l-
_c'e-O e',k?-
T. = (8)
_--7- (_+0_
Assume an ideal gas law in the undisturbed (upstream) region
p_/_=_T,//_where_is mean molecular weight• This, together with R_= o_E_£_-')
and T2= Tr (the recombination temperature), gives:
Both %; and %_ of the adiabatic stag.e are proportional to Ro_,'/_• We have
studied cases with _'-s/_ (which g_ves E = 2.02 Eo(Sedov, 1959)) and the
ratio of the density of helium to that of hydrogen @_/_,,= o.4 .
158. OG 8.2-2
9.E-O6
8.E-06 In(P/Po) = 0.26
7.E-06
6.E-06
_.E-06
4,E-06
3.E-05
2.E-OG
•t.E-O6 LO.E-O0 • .. , .... t .... , .... =• • •
.4 .S .6 .P .8 .9 1,0 1.1 t.2 1.3 1.4
r/R
I.E-OE ..........."I '_.....'..... _. Spatial distribution of 20. ,
partlcles accelerated by a moving .......' _.' ' ......_ ......
t.E-OE r • R spherical shock. Particles are 16. - ' ' r - Rinjected at the shock. ..
12. - I|
1.E-O;
8.. ii
o=
r 1.c-o_- "- _ o. •
I.E-I( om -4. -.................... ,===¢=_ ........
t.E-tt
-12.
1.E-12 (_
I -iS.
#/
,.__,_......J_.,.J........... ....................
-2_°'r_o2.._ J . , ,,,t.E-02 t .E*OO 1.E+02 t.E+OO I.E+02
P/Po P/Po
Figure 2. Spectrusl of _. The spectral
particles accelerated index with small ,,6;,
by a mov}ng spherical and T_.
shock. RR/K is large.
60 ........ • v "r..', ..... _ ..... "4
1.E+02_ 48. "_ I r = R
I.E+Ot r • R 36.
t
i .E+OC _. 2'1. ]
=_ ,2. -]I,E-OI
o. ................................................t
f t.E-o; _ 1
o= -12. 3
t.E-O: _ "
-24. _ =
-60. ' "'"_ ........I 1.E-02 t.E+O0 1.E+02
1.E-06 ..... J .... I ...... = .....
l. E'02 1. E+O0 1 • E+02 P/Po
P/Po Figure 5. The spedtral
_. Spectrum of index with large _
particles accelerated and T4:,
by a moving spherical
shock. RR/K is small.
159 OG 8.2-2
In_Figure 1 to 3 we used E0 = 3 x 1050 erg, _ = 0.3, K 1 = K 2 = K^ = 1026
cm 2 sec-I, aI = a2= 0, Tr=1026 K (the recombination temperatuKe 0_oproton
and electron) and the number density of hydrogen n, ffi3 x i0-° cm-° (hot
interstellar medium); i.e.,_;= l_RxtO "_ _d _ = 5 _xt0 -3 . An absorb-
ing boundary is set up at a radius 2R.
By studying Figure I, we see that only a very small portion of
particle (high or low energies) stays outside the shock, the absorbing
boundary can be considered as being at infinity where the particle
distribution, f, is supposed to be zero. This can be understood by the
fact that there is no convection outside the shock and the diffusion time
scale (R_/Ko) is much larger than the accelerating time scale (KdU_);
and the adiabatic stage ends before the steady state is attained.
The low energy spectrum is very steep and process a cut-off (see
Figures 2 and 3), which indicates the adiabatic deceleration is not very
effective. The spectral index for accelerated particles (p > po) starts
from values larger than -4 and then rela_s to pass -4 (see Figure 3). o_
In Figures 4 to 5 we used Eo= 3 x 10)U^erg and _=0.3,K,=K_=Ko= I0 _
cm_sec "I, a-=a_--0, and T =i0 ° K, n_=0.1cm-_; i.e._ = 0.572 and q_._= 1.79.
The absorb_ng_oundary _s at a radius 5R. The spectrum (see Figures 4
to 5) is very steep due to the large curvature of the shock and the
escape of particles out of the system.
We have studied the case of momentum dependent diffusion
coefficient. The basic features are the same; some detailed changes
are consistent with results of a steady state plane shock.
V. Conclusion. The results discussed above may be understood as a
consequence of the length scale or time scale involved (e.g. Prishchep
and Ptuskin, 1980). There are three time scales involved: diffusion
time scale R=/K, convection time scale R/R and acceleration time scale
K/R _ . If the radius of curvature parameter RR/K is large, then the
acceleration is very efficient and the spectrum will approach that from
plane shock and vice versa.
Acknowledgements.
This work was supported in part by NASA under grant NsG7101. The
authors are grateful to Garry Webb for helpful discussions.
References
Axford, W.I., Leer, E. and Skadron, G.: 1977, Proc. 15th International
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Blandford, R.D. and Ostriker, J.P.: 1978, Ap. J. (Letters), 221, 129.
Blandford, R.D. and Ostriker, J.P.: 1980, Ap. J., 237, 793.
Bogdan, T.J. and V_ik, H.J.: 1983, Astron. Astrophys., 122, 129.
Drury, L. O'C.: 1983, Rep. Prog. Phys., 46, 973.
Krymskii. G.F.: 1977, Dokl. Akad. Nauk-.-SSSR, 234, 1306 [1977, Soy.
Phys. Dokl., 22(6), 327].
Krymsklio G.F. an-d-Petukhov, S.I.: 1980, Pisma Astron. zh., 6, 227
_ [1980, Soy. Astron. Lett., 6(2), 124].
Landau, L.D. and Lifshitz, E.M.: 1959, Course of Theoretical Physics,
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Moraal, H. and Axford, W.I.: 1983, 125, 204.
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Prishchep, V.L. and Ptuskin, V.S.: 1981, Astron. Zh., 58, 779 [1981,
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A numerical study of diffusive shock acceleration of cosmic rays in supernova shocks

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