The spectrum of cosmic ray electrons, accelerated in the shock front of a supernova remnant (SNR), is calculated in the test-particle approximation using an onion-shell model. Particle diffusion within the evolving remnant is explicity taken into account. The particle spectrum becomes steeper with increasing radius as well as SNR age. Simple models of the magnetic field distribution allow a prediction of the intensity and spectrum of radio synchrotron emission and their radial variation. The agreement with existing observations is satisfactory in several SNR's but fails in other cases. Radiative cooling may be an important effect, especially in SNR's exploding in a dense interstellar medium

Beck, R.

Bogdan, T. J.

Drury, L. O.

Voelk, H. J.

English

NASA Technical Reports Server

140 OG 8. I-i0
ONION-SHELL MODEL FOR COSMIC RAY ELECTRONS AND RADIO
SYNCHROTRON EMISSION IN SUPERNOVA REMNANTS
by
R. Beck, L. O'C. Drury, H. V_lk, Nax-Planck-Znstttut for Kernphyslk, Postfach lO 39 80,
6900 Heidelberg, W. Germany,
T.3. Bogdan, National Center for Atmospheric Research*, Hlgh Altitude Observatory, P.O.
Box 3000, Boulder, CO. 80307, USA.
ABSTRACT
The spectrum of cosmic ray electrons, accelerated in the shock front of a supernova
remnant (SNR),is calculated in the test-partlcle approximation uslng an "onion-shell"
model. Particle diffusion within the evolving remnant is explicitly taken into
account.The particle spectrum becomes steeper with increasing radius as well as SNR
age. Simple models of the magnetic field distribution allow a prediction of the
intensity and spectrum of radio synchrotron emission and their radial variation. The
agreement with existing observations is satisfactory in several SNR's but fails in
other cases. Radiative cooling may be an important effect, especially in SNR's
exploding in a dense interstellar medium.
i. Introduction.Diffusive acceleration of relatiyistic particles in the shock fronts of
supernova remnants (SNR's)is thought to be an important mechanism to replenish cosmic
rays in the interstellar medium. In case of strong, steady, plane shocks, the particle
energy spectrum N(E)_E -q becomes a power law wlth a slope of q = 2. In reality,
intensity and slope of the spectrum are changed by various effects:
(i) The acceleration time is limited by the SNR age, leading to a hlgh-energy cutoff. (il)
The SNR Mach number decreases wlth time. (ili) The particle energy decreases due to
(adiabatic) expansion. (iv) Radiative cooling sets in below,JlO6 K temperature. (v) The
back-reaction of the accelerated particles modifies strength and structure of the shock.
(vi) The accelerated particles excite magnetohydrodynamlc waves which produce a shock
precursor. (vii) The accelerated particles (particularly the electrons) are subject to
energy loss processes due to collisions and radiation.
The full particle transport equation can be solved only for rather special cases (1,2,3),
which we cannot use here. Numerical studies of the hydrodynamlcal version of the time-
dependent problem have been started only recently (4), but do not provide information
about the particle spectrum. A test-partlcle model for the cosmic ray production by
recurrent passages of SNR shock fronts has been discussed (5). Models for cosmic ray
acceleration in a _ SNR have been calculated (6,7),in which effects (i), (il), (ili)
and - schematically (vi) are taken into account. The lifetime of the SNR is
* NCAR is sponsored by the NSF
https://ntrs.nasa.gov/search.jsp?R=19850026587 2020-03-20T16:48:37+00:00Z
141 OG 8.i-i0
(technically) split into equally spaced intervals corresponding to a series of shells,
characterized by the Mach number of the bounding shock. Suprathermal particles with 1 keV
energy are continuously injected into the shock front, accelerated up to the cutoff,
according to the instantaneous shock Math number in a shell, and adiabatically expanded
with the gas until the gas pressure reaches the external pressure. Particles remain
confined within their shell until being released at the end of the SNR evolution. The sum
of the particle spectra from all these '_nion shells" yields a power-law with q = 2.1-2.3
between lO5 and several lO13 eV, only slightly varying with different choices of the
injection energy and the other input parameters. Allowing for energy-dependentsubsequent
escape from the galaxy, the spectral index of the galactic radio emission is consistent
with this model.
2. The Model. We have extended the "onion-shell"model to study the particle spectrum as a
function of time and position. Instead of confining the particles to a shell we take
account of both the intrinsic width of the acceleration region and of the subsequent
diffusion by introducing a smearing length scale'given by:
I (t,E) -- + K(E) (t-t)
R(t) o
Here R(t) is the expansion velocity of the shock front; (t-to) is the time since the
formation of the shell at the shock.K is the energy-dependent diffusion coefficient; it
roughly increases proportional to particle energy.Due to wave generation by accelerated
nucleons N is taken to be given by the gyroradlus limit; for 1 MeV electrons and 3_G
magnetic field strength K becomes_lO 19 cm2s-1. At low energies (< 1 GeV) the electrons
still remain concentrated in their original shell while at the highest energies (> 1000
GeV) electrons essentially diffuse across the whole SNR.
The SNR is assumed to expand freely until the ejecta have swept up an equal mass of
interstellar material. This initial phase is followed by an adiabatic Sedov phase, where
R_ t2/5. After passage of the shock each shell expands with the downstream flow (8) until
the time of observation tF. Any point in the spectrum follows from the integral over the
contributions from all shells at a fixed observation radius.
Hence it is possible to approximately compute the electron spectrum at any given SNR age
and at any given radius.
The accelerated electrons emit synchrotron radio waves in the magnetic field of the SNR.
Two extreme field models have been investigated, which bracket the actual situation: (I)
constant field strength, (II) field compression B_ _ . The distribution of relativistic
electrons and of the magnetic field allows a prediction of the radio synchrotron emissiv-
ity. For comparison with radio observations,the emission from all shells along the line
of sight has to be summed.
3. Results. Models have been calculated for different SNR ages (representedby the Mach
number MF at the time of observation) and for different external gas densities. The
explosion was assumed to eject 1 Me with lO 000 km/s velocity.A field strength of 3_G
and a pressure of 1.5 10-12 dyne/cm2 in the interstellar medium has been adopted.
| 42 OG 8.1-10
Particles with low energies (< i GeV) are supplied by all shells created at different SNR
ages. High-energy particles, however, are produced only during the early stages of theSNR
evolution when the Mach number is high. This result still holds after extensive adiabatic
expansion, i.e.for old adiabatic SNR's.Therefore, a steepening of the particle spectrum
with increasing distance from the center is expected, except for the highest energies
where diffusion dominates.
The spectral index for the electron energy spectrum between 106 and lO12 eV varies varies
with radius. In young SNR's (final Mach number MFI 5) almost no variation occurs. A
spectral steepening occurs in the outer region of old adiabatic SNR's (MF & 2). The
corresponding radio spectral index between I00 MHz and lO GHz frequency for Bm_ (Fig.l)
refers to the emission along the line of slight radius R so that variations with R are
somewhat smoothed out. The assumption B = const does not change the curves of Fig. 1
significantly.The spectral index of the integral radio emission varies with SNR age, from
(_£_ 0.60 in young SNR (MF > 5) toO_q 0_5 in old SNR (MF < 1.5),which is consistent
with the result given (6).
These general conclusions hold if the SNR remains adiabatic during its evolution.In fact,
for an external hot interstellarmedium gas density of 5xlO-3cm-3 radiative cooling of the
outer shell becomes important only in the latest phases (MF & 1.1) when the downstream
RADIO SPECTRAL INDEX
ALPHA "_ ""'"_' ''.... '"_'"'"'-'.... '.... '.... ''"' 'I'"' ''....
f:
1.5
1"@" i
= ................ , i _
b
,i¢llili,|.,.,l,ilil,..,lilliliilil..,ii,l.lli,.l@
0 0.2 0.4 0.6 0.8 1,0
R/RF
Fig. l: The variation of radio synchrotron spectral index_with radius relative to the
final radius RF of the shock front for three models:
(1) Final Mach number MF = 3 (RF = 90 pc), (2) MF = 2 (RF = ll7 pc),
(3) MF = 1.5 (RF = 142 pc). The maximum radio intensity is marked. At fixed external
pressure the curves are independent of the external density.
OG 8. i-i0
14@
temperature drops to 106 K. For larger densities, say n_3xlO-2cm -_, shell cooling sets
in earlier (MF _3). Then the outer shock tends to become isothermal. The increased
compression ratio then yields a harder spectrum, at least for high energies, leading to
a radial decrease of the spectral index. Although this effect is not explicitly treated
here, it is an obvious diagnostic possibility regarding the evolutionary state of a
given SNR, with radial increase (decrease)indicating an adiabatic (radiative)phase of
the compressed outer shell.
4. Comparison with observations. The mean spectral index of SNR's, including also flat-
spectrum (Crab-type) SNRs, is oc = 0.45 _0.15 (9) which is not inconsistent with the
prediction of the "onion-shell"model. The same data do not indicate a correlationbetween
SNR spectral index and diameter(lO),while in a more recent catalogue (ll) a statistical-
ly weak flattening with increasing diameter is visible.Since the diameter depends mainly
on the initial velocity and the external gas density and is far from being a unique
measure of SNR age, these observationsdo not conflict with our model.
The determination of radio spectral index within a SNR requires an accurate subtractionof
the background emission which is difficult especially for large SNRs in the galactic
plane. The remnant of Tycho's supernova is a favourite object because it is small and
radio-bright.The spectral index of the integrated emission is oc= 0.61 ± 0.03 without a
significant variation across the remnant (12).Thisagrees with the prediction of our model
for young SNR's. The large, probably old, SNR G65.2+5.7 in Cygnus reveals a spectral
steepening from _ % 0.4 in the inner part to _ _ 0.6 in the main shell of emission,
followed by an increase to o£_ 1.O beyond, (13), as expected in view of our model.
A larger sample of SNRs with radial spectra has now become available (F0rst, private
communication).In most cases, no spectral index variation across the SNR is detected. In
some sources, however, the spectrum seems to flatten with radius, indicating the limits of
the model presented here.
References
1. Drury, L. O'C.,Axford, W.I.,Summers, D.: 1982, MNRAS _ 833.
2. Bogdan, T.&, Lerche, I: 1985, MNRAS _ 413.
3. Webb, G.M., Bogdan, T.J.,Lee, M.A.,Lerche, I.: 1985, NNRAS (submitted).
4. Dorfi, E.: 1984, Adv. Space Res. _, No. 2-3, 205.
5. Blandford, R.D.,Ostriker, &P.: 1980, Astrophys.J.237, 793.
6. Bogdan, T.J.,V61k, H.J.:1983, Astron.Astrophys.122, 129.
7. Moraal, H., Axford, W.I.:1983, Astron.Astrophys._ 204.
8. Kahn, F.D.:1975, Proc. 14th Int. Cosmic Ray Conf., MOnchen, Vol. ll, 3566.
9. Clark, D.H.,Caswell, J.L.:1976, NNRAS 174, 267.
lO. Lerche, I.: 1980, Astron.Astrophys._ 141.
ll. Green, D.A.:1984, MNRAS 209, 449.
12. Klein, U., Emerson, D.T.,Haslam, C.G.T.,Salter, C.3.:1979, Astron.Astrophys. 76,
120.
13. Reich, W., Berkhuijsen, E.M.,Sofue, Y.: 1979, Astron.Astrophys._ 270.


Onion-shell model for cosmic ray electrons and radio synchrotron emission in supernova remnants

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server