The non-linear theory of shock acceleration was generalized to include wave dynamics. In the limit of rapid wave damping, it is found that a finite ave velocity tempers the acceleration of high Mach number shocks and limits the maximum compression ratio even when energy loss is important. For a given spectrum, the efficiency of relativistic particle production is essentially independent of v sub Ph. For the three families shown, the percentage of kinetic energy flux going into relativistic particles is (1) 72%, 2) 44%, and (3) 26% (this includes the energy loss at the upper energy cuttoff). Even small v sub ph, typical of the HISM, produce quasi-universal spectra that depend only weakly on the acoustic Mach number. These spectra should be close enough to e(-2) to satisfy cosmic ray source requirements

Eichler, D.

Ellison, D. C.

English

NASA Technical Reports Server

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RELATIVISTIC COSMIC RAY SPECTRA IN THE PULL NON-LINEAR
THEOEY OF SHOCK ACCELERATION
David Eichler 1'2 and Donald C. Ellison 1
_Astronomy Program, University of Maryland, College Park, MD 20742
2Department of Physics, Ben-Gurlon University, Beer Sheva, Israel
i. Introduction. The test partlcle theory of diffusive shock acceleratlon
predicts a power law cosmic ray (CR) spectrum in momentum of the form
N(p) = p-S, s = -(r+2)/(r-l), (1)
where N(p) is the dlfferent_al number density of CRs at momentum p and s
is determined solely by the compression ratio, r = ul/u 2. Here, the shock
is assumed to be planar, of infinite extent, steady state, discontinuous,
plece-wise constant, and u I (u 2) is the flow velocity (measured in the
shock frame) upstream (downstream) of the shock discontinuity (u I is the
shock veloclty in the lab frame) 2,3'4'6.
Since strong, classical hydrodynamic shocks have a compression ratio
of 4, the test particle theory predicts a spectral index of -2, in good
agreement with the source spectrum of galactic CRs that is inferred from
observations. This account of the spectral index is widely viewed as
being superior to previous accounts, which required fine-tuning to get
the correct spectrum I.
The non-llnear theory, however, which allows for the fact that the CR
pressure may be dynamically significant, accordingly allows the
compression ratio to be larger than 4. This follows from the facts that
(a) much of the post shock pressure can be in relativistic CRs, and, for
a relativistic fluid, the classical compression ratio can be as large as
7, and (b) a significant amount of energy can be lost from the shock
front in the form of particles that stream away from the shock before
being convected downstream of it, and, r may then be arbitrarily large 6.
High production efficiencies (>10%) appear to be as widespread in
astrophysical particle acceleration as does a spectral index between -2
and -3, and the possibility that CRs affect the compression ratio needs
to be taken seriously. While one could argue that efficiencies of order
10% are marginally consistent with observation without requiring
significant non-llnear effects, invoking an injection rate that always
gives an efficiency _ 10% in the test particle theory requires a fine-
tuning assumption about the injection, which is probably objectionable.
Here we show tha_ the non-llnear theory gives rise to spectra that are
very close to a p power law, over most of the range of astophyslcally
reasonable shock parameters for supernova blast waves in the hot phase of
the interstellar medium (HISM), even though r may differ significantly
from 4. The universality of the spectrum, in this view, is related to the
nature of the non-llnear feedback of the CRs on their own acceleration
rate, not to the universality of the compression ratio among shocks.
2. The Basic Calculation. The non-llnear, steady-state theory includes an
explicit description of the waves that couple the particles to the fluid.
The full set of equations is
_u = ci, (2)
https://ntrs.nasa.gov/search.jsp?R=19850026583 2020-03-20T16:49:25+00:00Z
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pu 2 + p + p + p = C2, (3)
e w g
dT dP dP
w e
w = u + Vph RP , (4)dx _ dx w
RP = C1T dsw _' (5)
dP
T ds = dh ---£g, (6)
P
du
___ i s 8F 8 8F (7)
- (UsF) + _ d--x--p -_ +-_ D -_ = C3.
Here, p is fluid density, u is fluid velocity, us is the velocity of the
scattering centers, Pe is the pressure in energetic particles, Pw is the
wave pressure, Pg is the pressure in gas that has not yet been shocked,
Tw is the energy flux due to waves, Voh is the phase velocity of the
scattering waves relative to the backgr6und medium, i.e. U-Us, R is the
damping rate, Tds is the heat per unit mass that enters the fluid via the
wave damping, h is the enthalpy per unit mass and CI, C2, and C 3 are
constant. Equations (2) and (3) express mass and momentum conservation,
respectively. Equation (4) is an energy inventory equation for the waves
with the terms on the right hand side representing, in order of
appearance, amplification of existing waves by compression, growth due to
the cyclotron unstable CR gradient, and damping (see 8 and references
therein). Equations (5) and (6) describe the energy deposition into the
thermal fluid by the wave damping, and (7) is the diffusion equation for
F = 4_p3f(dlnp/dlnE), where f(p) is the distribution function and D is
the diffusion coefficient. These equations can all be solved using a non-
perturbative, non-separable, analytic technique that proves to be
extremely accurate in comparison with numerical solutions 5'6. The
numerical solutions referred to are possible only over a limited
dynamical energy range. However, the analytic technique exploits the fact
that in a real situation there is a very large dynamic energy range - ten
decades for galactic CRs - so it should be even more accurate in the
latter case than in the comparison with the numerical results.
We have solved the above eqs. for rapid wave damping (R . _) and for
Vph constant upstream of the shock (Vph is set equal to zero downstream).
Results are shown in Fig. I where we have plotted partial pressure E
(J/3)EF, versus energy, E. Here, j = i for relativistic and = 2 for non-
relativistic particles. Each curve is the partial pressure profile for a
locus of (MI, v_ h) values shown in Fig. 2. Each point on a curve in Fig.
2 produces essegtially the same profile as seen in Fig. I. The range in
compression ratios from v_ h = 0 to MI. _, is as follows: (a) 7.1<r<7.5,
(b) 4.7<r<5.3, and (c) 3._<r<4.6. The dashed lines in Fig. i show energy
spectral indices of -2 (horizontal) and -2.7, representative of galactic
CR source and observed spectra, respectively. Our solutions are
approximately power laws, deviating from the horizontal dashed line by at
most a factor of 3 over 6 decades of energy. (That they are not perfect
power laws results from the slowing of the flow upstream of the shock by
CR pressure). It is clear that a small phase velocity, representative of
the HISM, allows an extremely wide range in shock strengths, all
producing source spectra essentially indistinguishable from that required
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to produce the observed CR spectrum. The uncertainties in the CR source
spectrum caused by energy dependent leakage exceed the small differences
between curves (b), (c), and the horizontal dashed llne. Even curve (a)
is marginally consistent with the inferred source spectrum.
The ratio of Alfv_n to O ' I i "I'_, I ' I _ I ' I'
shock velocity is given by,
VA/U 1 = 2.2 i011B/(uI/n)' hJ /'/'/_'a_7'
where B is the magnetic oi --I a
field and n is the number
density. For the HISM, we GOO0 --2
can use B ~ 3pG, n ~ 0.01 bJ \ c
per cm 3, and shock veloci- Of \
ties ~ I000 km/s. There- O_ --3 - \ --
fore, VA/U I _ 0.07, and, as --J \
seen in Fig. 2, a very wide <f \-- \
range in M 1 will produce _ --4 I- \ -
spectra similar to those _ I \
seen in Fig, i. O_ \
o -5 \-\
3. Scaling Laws. The re- O
suits of the previous see- -] --6 i I I I i J , I , I , I
tion show that the spectrum 2 4 6 8 10 I2 I
of relativistic particles LOG ENERGY (eV)is "quasi-universal" in the
sense that it depends very Fig. i. Postshock partial pressure vs.
weakly on the upstream pa- energy. An energy cutoff = lOlbeV has been
rameters. The reason for used. Each label represents a family of
this, we argue, is that the curves with a particular Ul/U(GeV ).
energy content in relativistic parti- ' ' _"_ '''"I , i , ,lw_[
cles is a very sensitive function of c
the spectrum, so that their back-
pressure on the flow regulates their
spectrum very precisely. This principle 0.10
is illustrated by the fact that the
overall compression ratio of the scat-
terers is always established in a way
such that the compression ratio of
mildly relativistic particles is vir- "_
tually independent of the upstream >_
shock parameters. In Fig. 3 the overall
compression ratio of the shock is plot-
ted as a function of Mach number for
the limiting case of zero phase veloc- 0.00 ' '
ity. It is seen that beyond modest Mach 10 02
numbers, r _ M 3/%, regardless of the MACH NUMBER, M 1
fact that the adlaba_ic index of the Fig. 2. Labels same as Fig. i.
post shock gas (i.e. the relative con-
tribution of relativistic and non-relatlvistlc particle pressure) varies
substantially over this range in M 1 • (The classical result for y = 4/3
is shown by the dotted llne.) This M3/4 law can be derived
straightforwardly by merely assuming that the Math number of the flow
when it encounters Gev particles, M(GeV), and the compression ratio at I
GeV, r(GeV), are _ constant (we find, for 3._bvP[ 0 and for iO < M 1 < I00that 3.3 < M(GeV)_ 3.6 d 3.5 < r(GeV) < . The result then follows
OG 8.1-6
127
from the fact that, because the phase velocity is zero, the upstream
fluid behaves adiabatically, i.e. (using eq. [2]),
P _ 05/3 => [MI/M(GeV)] 2 = [ul/u(GeV)] 8/3, (8)
Here u(GeV) is the upstream flow velocity felt by GeV partlcles 6. Given
that M(Gev) and r(GeV) = u(Gev)/u 2 are about constant for Vph = O, r
scales as M S/b.
As is evident from Fig. 3, the shock compression is dramatically
reduced and stabilized by the introduction of even a small Vph for the
scattering waves. For an infinite upstream acoustic Math number, the
compression ratio, formally infinite when v__ = O, is reduced to only ~ 7H-
when Vph/U I ~ 0.03. This result can be derived analytically to high
accuracy; using the same principle invoked to derive the M 3/_ scaling law
when v_ h vanishes, one can show that ul/u(GeV) ~ 2 when Vph/U I ~
I/[2.7y_-I)M2(GeV)].
I I I I I II I_ I I "_" I ! III
4. Conclusions. The non-linear v_/u,=o/
theory of shock acceleration
has been generalized to in-
elude wave dynamics. In the (_
limit of rapid wave damping, m / 0.0066
it is found that a finite wave _ I0 - /_
velocity tempers the accel- n_
eratlon of high Mach number Z
shocks and limits the maximum O _ _"
60
compression ratio even when 60
energy loss is important. For L_ .C_ v_Ju,---> 1
nl /a given spectrum (see Figs. i- O_2), the efficiency of relati-vistic particle production is Oessentially independent of f-)
v_h. For the three "families" I t , , t i ttJl i I I, , ,,,,
s_own in Figs. 1-2, the per- 10 10 2
centage of kinetic energy flux MACH NUMBER, M1going into relativistic parti-
cles is (a) 72%, (b) 44%, and Fig. 3. Dotted line is classical result.
(c) 26% (this includes the
energy lost at the upper energy cuttoff). Even small Vph, typical of the
HISM, produce quasl-unlversal spectra that depend only weakly on the
acoustic Mach number. These spectra should be close enough to E-2 to
satisfy CR source requirements.
References
i. Axford, W.I. 1981, 17th ICRC (Paris), 12, 155.
2. Axford, W.I. et al. 1977, 15th ICRC (Plovdlv), II, 132.
3. Bell, A.R. 1978, MNRAS, 182, 147.
4. Blandford, R.D. and Ostrlker, J.P. 1978, Ap. J. (Let), 221, L29.
5. Eichler, D. 1985, Ap. J., in press.
6. Elllson, D.C. and Eichler, D. 1984, Ap. J., 286, 691.
7. Krymsky, G.F. 1977, Dokl.Akad.Nauk SSSR, 234, 1306.
8. McKenzle, J.F. and Volk, H.J. 1982, Astr. Ap., 116, 191.


Relativistic cosmic ray spectra in the full non-linear theory of shock acceleration

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