A&A 486, 837-841 (2008)
DOI: 10.1051/0004-6361:200809669
Jacco Vink
Astronomical Institute Utrecht, Utrecht University, PO Box 80000 3508TA, Utrecht, The Netherlands
Received 27 February 2008 / Accepted 28 May 2008
Abstract
Aims. We investigate the shape of the electron cosmic ray spectrum in the range up to 1000 keV, assuming that the acceleration process at the shock results in a power law in momentum, and that downstream of the shock the spectrum is affected by Coulomb interactions with background electrons only.
Methods. In the non-relativistic regime one can analytically determine the energy of an electron starting with a certain energy, and use this result to produce an electron cosmic ray spectrum, modified by Coulomb losses.
Results. An analytic expression for the electron spectrum is obtained that depends on the parameter
,
which can be estimated from a similar parameter used to characterize the line spectra of supernova remnants.
Conclusions. For the brightest supernova remnants
cm-3 s, and most of the electrons accelerated to <100 keV have lost their energy. Because of its high radio flux, Cas A is the most likely candidate for non-thermal bremsstrahlung. Although it has
1011 cm-3 s, one may expect to pick up non-thermal bremsstrahlung above 100 keV with current hard X-ray detectors.
Key words: ISM: cosmic rays - acceleration of particles - ISM: supernova remnants - X-rays: ISM - shock waves - radiation mechanisms: non-thermal
Over the last five years there has been considerable progress in
our understanding of high energy cosmic ray acceleration by supernova
remnants (SNRs).
The spatial distribution and spectral shape of
X-ray synchrotron emission from all historical SNRs and some
other SNRs has made it clear that electron can be accelerated to
energies up to 100 TeV (Koyama et al. 1995),
that diffusive shock acceleration works very efficiently, with diffusion
close to the Bohm limit (Vink et al. 2006; Vink 2005; Stage et al. 2006; Parizot et al. 2006), and that
magnetic fields are amplified by cosmic ray streaming
(Vink & Laming 2003; Bamba et al. 2005; Völk et al. 2005; Berezhko et al. 2003). Moreover, the
morphology of Tycho's SNR shows that at least in this remnant, but
also probably in Kepler's SNR, cosmic ray acceleration is so efficient that
it gives rise to enhanced compression ratios (Warren et al. 2005; Ellison et al. 2004).
In addition, considerable progress has been made in the field of
TeV astronomy, with many SNRs being established as sources
of TeV
-rays (e.g. Albert et al. 2007; Aharonian et al. 2005,2004).
However, it has not yet been firmly established whether the
-ray emission is dominated by inverse Compton scattering from TeV electrons,
or from neutral pion decay, caused by TeV ion cosmic rays.
In contrast, little has happened concerning observations of low energy cosmic rays. This is unfortunate, since from a theoretical perspective, the initial stages of the acceleration process, from thermal energies up to energies where Fermi acceleration efficiently operates, is complex and not well understood (see e.g. Malkov & Drury 2001, for a review). In fact, the processes by which particles are injected into the Fermi acceleration process, are likely to be different for electrons and ions, unlike Fermi acceleration itself. Our current understanding of the initial stages of acceleration is largely based on either in situ observations of inter-planetary shocks, and on computer modeling using hybrid or particle in cell codes (e.g. Bykov & Uvarov 1999; Lee et al. 2004; Schmitz et al. 2002). There was some hope that hard X-ray observations of SNRs might provide observational information on at least the electron component of low energy cosmic rays (Vink & Laming 2003; Favata et al. 1997; Asvarov et al. 1990; Vink et al. 1997). However, it is generally agreed that the non-thermal hard X-ray emission from SNRs such as Cas A (Allen et al. 1997; Vink & Laming 2003; Renaud et al. 2006b; The et al. 1996; Favata et al. 1997), SN1006 (Kalemci et al. 2006; Allen et al. 1999), Tycho (Allen et al. 1999), Kepler and RCW 86 (Allen et al. 1999) is due to X-ray synchrotron from the highest energy electron cosmic rays, rather than from bremsstrahlung from low energy electrons. Nevertheless, the issues remains of interest as current and future hard X-ray detectors push the detection of hard X-ray emission to higher photon energies. At high enough energies, the steepening X-ray synchrotron spectrum is likely to be overtaken by non-thermal bremsstrahlung.
In this paper, I discuss the shape of the low energy cosmic ray electron spectra, given the importance of Coulomb-losses. These Coulomb losses in general alter the shape of the low electron cosmic ray spectrum as generated near the shock front. Note that a similar situation was investigated by Sarazin (1999) for higher energy electrons in clusters of galaxies.
Supra-thermal electrons loose energy through various processes:
bremsstrahlung losses, Coulomb losses (collisions with electrons/ions),
and ionization losses.
In the hot plasmas inside SNR shells Coulomb
losses are likely to be the dominant source of energy losses of
electrons, in particular through electron-electron collisions
(e.g. Haug 2004). Electron-electron collisions are most efficient
at low electron energies. For SNRs we are interested in the bremsstrahlung
emission of supra-thermal electron with energies in the range of
10-1000 keV. The problem with this energy range is that neither
the non-relativistic, nor the relativistic approximations for Coulomb
losses are completely valid. However, I will show that for most SNRs the
non-relativistic approximation is sufficient.
The energy loss rate for an electron with
is given by
(e.g. Haug 2004; Huba 2002):
![]() |
(2) |
The solution to this differential equation is:
In fact, the correct equation, Eq. (1), can also be solved, but only gives
an implicit function for E(t):
![]() |
Figure 1:
The electron energy as a function of
![]() ![]() |
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The advantage of an explicit function (Eq. (4)) is that
it can be easily used to calculate the evolution of the electron spectrum,
using the transformation
,
if an analytic approximation is known. This can be done
by inverting Eq. (4) and insert the result in N(E),
taking into account that
![]() |
Figure 2:
Low energy electron cosmic rays as expected from first order
Fermi acceleration, including the effects of Coulomb losses (Eq. (9)). The solid line gives the expected spectrum near the shock front,
whereas the other lines show the models for different values of
![]() |
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Figure 2 shows the electron distributions for
values of
covering the relevant range for SNRs,
109-1012 cm-3 s. Figure 1 shows that for young SNRs
the non-relativistic approximation is expected to be valid, as large
errors only appear for
1011 cm-3 s.
Moreover, the errors mostly affect the spectral shapes above 500 keV.
As explained above the shape of the non-thermal electron spectrum below
1000 keV depends on the parameter
,
which is the same
parameter that governs the ionization state of the SNR. However,
it depends on the details of the shock acceleration history, whether one can
safely assume that the average
of the hot plasma corresponds
to the
governing the shape of the low energy electron cosmic
rays. A close correspondence is likely if
shock heating results always in a fixed ratio of heated and accelerated
electrons at the shock front. Realistically, one should
take into account the variation of
throughout the SNR, and
the evolution with time. However, in practise
one or two values for
characterizes the overall plasma characteristics reasonably well
(e.g. Vink et al. 1996).
For young SNRs like Cas A, Tycho, Kepler and SN1006, the
X-ray spectra are dominated by line emission from ejecta, shock heated by
the reverse shock. This means that also
is determined
mostly by the plasma heated by the reverse shock, rather than by the
forward shock heated plasma.
It is often assumed that the reverse shock has
a low magnetic field and therefore does not efficiently accelerate
cosmic rays. However, Helder & Vink (2008)
showed recently that the
reverse shock of Cas A is responsible for most of the X-ray synchrotron
emission, this implies that the reverse shock is capable of accelerating
particles as well. This is a confirmation of recent theoretical calculations
by Ellison et al. (2005). Nevertheless, it is not quite clear what the
ratio is of the electrons accelerated by the forward shock, as opposed to
the reverse shock. Moreover, the initial acceleration process for electrons
may be different: the reverse shocked plasma is
dominated by metals.
In that case, a sizable fraction of the accelerated electrons
may originate from ionization of (mildly) accelerated ions.
Apart from the plasma
there is another reason why
the
for the accelerated electrons is expected to be high
in Cas A: most of the electrons seem to
have been accelerated in the early life of the SNR. The best
evidence for this is the
1%/yr decline in its radio flux,
which is best explained by adiabatic expansion of the relativistic electron
gas (Shklovsky 1968). Cas A may be a special case since it is evolving
into a red super giant wind, which has a density falling of with radius as r-2. This means that the particle flux entering the forward shock at any
radius is more or less constant, whereas the shock velocity is dropping
with radius. As a result particle acceleration was more efficient early on,
suggesting a high
.
Unfortunately, this also means that it is quite unlikely that the
non-thermal X-ray emission of Cas A below 100 keV is dominated by
non-thermal bremsstrahlung, either from the forward shock, or
from internal shocks, as assumed by (Laming 2001a; Vink & Laming 2003; Laming 2001b).
The reason is that all the emission must then come from a limited fraction
of the total shock heated plasma for which
cm-3 s
(see Willingale et al. 2002; Yang et al. 2007, for the spatial distribution of
).
If I simply assume a linear relation between age and
,
less than 5% of the plasma has
cm-3 s.
The lower hybrid wave model propagated by
Laming (2001a) remains an attractive
option for the injection of electrons into the Fermi acceleration process
itself (see also Ghavamian et al. 2007).
![]() |
Figure 3:
Left: spectral energy distribution of Cas A, as measured with
BeppoSAX-PDS (Vink et al. 2001, red) and INTEGRAL-IBIS (Renaud et al. 2006b, green),
assuming a magnetic field of ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Nevertheless, Cas A remains one of the most important targets for
searching for non-thermal bremsstrahlung, but only above 100 keV, where
the spectrum is less affected by Coulomb losses. The reason is
that Cas A is by far the brightest Galactic SNR in the radio.
This implies the ample presence of relativistic electrons, in combination
with a relatively high magnetic field of
G
(Vink & Laming 2003; Shklovsky 1968; Berezhko & Völk 2004). This is illustrated in
Fig. 3, which suggests that with INTEGRAL-ISGRI we may be close
to a detection, or even that part of the flux density around
100 keV is caused by non-thermal bremsstrahlung (cf. the CGRO-OSSE
detection, The et al. 1996). No deep Suzaku Hard X-ray Detector (HXD)
(Takahashi et al. 2007)
of Cas A exist, but a deep observation with this sensitive instrument, may
also be able to detect emission above 100 keV.
The normalization of the spectrum in Fig. 3 is determined from
the radio flux, and assuming a mean magnetic field of G.
The bremsstrahlung also depends on the emission measure:
,
with
being,
here, the density of the accelerated electrons.
For Cas A the accelerated electrons are present in both
the forward shock heated plasma, with
cm-3,
and in the reverse shock heated plasma, which consists
of pure metals, probably dominated by almost completely ionized oxygen
(e.g. Vink et al. 1996). The emitting volume of the forward shock heated plasma
is
1057 cm3, based on a
shock radius of 2.6 pc and a contact discontinuity at
2 pc.
The reverse shock radius is
1.9 pc (Helder & Vink 2008), thus
1056 cm3.
Conservatively assuming that there is
of
ionized oxygen (Vink et al. 1996) and that the other metals contribute
little to the bremsstrahlung, one finds for the shocked ejecta
cm-3. For the volume averaged
normalization one finds
cm-3. Note that both the bremsstrahlung and synchrotron luminosity
scale with the emitting volume.
It is not unreasonable to assume that the synchrotron emitting
volume is equal to the volume of the shock heated plasma.
However, it is uncertain how the accelerated electron density
and magnetic field is distributed over this volume.
So the normalization of
cm-3 used in Fig. 3
should be considered as a reasonable guess.
For a constant non-thermal bremsstrahlung emissivity a higher value
of
implies a higher magnetic field.
In contrast, for Tycho's SNR, due to its much lower radio luminosity,
one only expects a detection of the non-thermal
bremsstrahlung, if the magnetic field turns out to be unexpectedly low:
G. This is roughly the value
of the compressed average ISM magnetic field, whereas
estimates for the magnetic field in Tycho
indicates
G (Völk et al. 2005).
Finally, one could also consider SNRs in which the densities and
values are modest, and for which also the magnetic fields
are low; for example SN1006 (
109 cm-3 s, Vink et al. 2003)
and the Northeast of RCW 86
(
109 cm-3 s, Vink et al. 2006).
The advantage is that although the bremsstrahlung may be weaker, the low
ensures non-thermal bremsstrahlung at lower photon energies.
Unfortunately, both these SNRs have also X-ray synchrotron radiation,
which may be hard to distinguish from non-thermal bremsstrahlung. However,
with imaging spectroscopy above 10 keV one may isolate regions where
non-thermal bremsstrahlung is important and X-ray synchrotron radiation
absent. Moreover, high spectral resolution spectra
may reveal departures from Maxwellian electron distributions.
This type of research will have to wait till the emergence of
high resolution, high throughput, X-ray imaging spectroscopy with
XEUS (Parmar et al. 2006).
I have derived an analytic expression for the low energy electron
cosmic ray spectrum, assuming that the cosmic ray spectrum, as produced
by the diffusive acceleration process,
is a power law in momentum, and is only affected by
Coulomb losses downstream of the shock.
As may be expected, the parameter
governing the Coulomb losses depends on
,
the product
of (background) electron density and the age of the accelerated
electron population. This parameter is similar to the ionization
time scale of the hot, shock heated plasma, and can be easily estimated
from the line spectra of SNRs.
For the SNRs with high densities, and large
,
one does not expect a large
population of accelerated electrons below 100 keV. This is in particular
the case for the very bright radio source Cas A. Nevertheless, Cas A
remains a good candidate for searching for non-thermal bremsstrahlung,
because of its high ion density and because its radio brightness indicates
a large amount of accelerated electrons.
Based on reasonable, but somewhat uncertain, assumptions,
the results presented here
imply that only above
100 keV one expects to pick up the non-thermal
bremsstrahlung component from Cas A.
This may be accomplished with future deep Suzaku HXD
observations, or additional deep INTEGRAL observations.
Acknowledgements
I am grateful to Prof. Ishida and Dr Bamba for inviting me for a short stay at ISAS, Japan in November 2007. The discussion on Suzaku HXD data of Cas A led to the results reported here. J.V. is financially supported by a Vidi grant from the Netherlands Organization for Scientific Research (NWO).