Antiprotons Produced in Supernova Remnants
We present the energy spectrum of antiproton cosmic ray (CR) component calculated on the basis of the nonlinear kinetic model of CR production in supernova remnants (SNR). The model includes reacceleration of already existing in interstellar medium antiprotons as well as creation of antiprotons in nuclear collisions of accelerated protons with gas nuclei and their subsequent acceleration by SNR shock. It is shown that antiprotons production in SNRs produces considerable effect in their resultant energy spectrum making it essentially flatter above 10 GeV so that the spectrum at TeV-energies increases by a factor of five. Calculated antiproton spectrum is well consistent with the PAMELA data, which correspond to energies below 100 GeV. As a consistency check we have also calculated within the same model the energy spectra of secondary nuclei and show that the measured boron-to-carbon ratio is consistent with the significant SNR contribution.
Subject headings:acceleration of particles — cosmic rays — ISM: supernova remnants
There is a great astrophysical interest in cosmic ray (CR) antiprotons. It is believed that most of antiprotons originate in collisions of CR protons with interstellar medium (ISM) gas nuclei. Therefore antiprotons represent a kind of so-called secondary CR component, opposite to the primary CRs, which originate in CR sources, presumably in supernova remnants (SNRs). The same is trough for positrons, which are the other kind of secondary CR component.
The positron energy spectrum measured recently in PAMELA, Fermi and AMS-02 experiments at kinetic energy GeV turned out to be much flatter than it is expected for positrons created in p-p collisions in ISM. This stimulated many kind of assumptions that significant part of positrons originate from new astrophysical sources such as pulsars or the annihilation of dark matter particles (see Serpico, 2012, for a review).
At the same time SNRs are not only the most probable sources of primary CRs with energies below eV (e.g. Berezhko & Völk, 2007), but also contribute significantly in the production of secondary CRs due to reacceleration of already existing in ISM CRs and due to nuclear collisions of primary CR particles with gas nuclei leading to the creation of secondary particles which undergo subsequent acceleration by SNR shock. The detailed study of these processes for the case of secondary nuclei (Berezhko et al., 2003) based on the nonlinear kinetic theory of CR acceleration in SNRs (Berezhko et al., 1996) demonstrated that SNRs are expected to contribute significantly to secondary CR spectra at kinetic energies GeV/nucleon making it significantly flatter. Therefore it is natural to suggest that the observed flattering of positron energy spectrum at GeV can be partly or even predominantly due to SNR contribution. Simple estimate (Blasi, 2009) and the detailed study (Berezhko & Ksenofontov, 2013) have indeed confirmed that the observed flat high energy positron spectrum is consistent with the expected SNR contribution. (Note that both these studies were performed in the leaky-box framework which is a poor approximation for electrons and positrons. More detailed consideration, based on the diffusive model for CR prpopagation in the ISM (Ahlers et al., 2009; Mertsch & Sarkar, 2014), confirmed such a conclusion.) One can therefore expect that SNRs also contribute significantly to antiproton spectrum at high energies.
Here we calculate the energy spectrum of antiprotons produced in SNRs to compare it with the existing data and make the prediction at higher energies GeV where experimental data are not existed yet. As a consistency check (see also Mertsch & Sarkar, 2009, for a similar point) we have also calculated within the same model the energy spectra of secondary nuclei and show that measured boron-to-carbon ratio is consistent with the considerable SNR contribution.
2. Production of secondary CRs in SNRs
Acceleration of CRs in SNRs starts at some relatively low energy when some kind of suprathermal particles begin to cross the SNR shock front. Any mechanism which supply suprathermal particles into the shock acceleration is called injection.
Some small fraction of the postshock thermal particle population are able to recross the shock that means the beginning of their shock acceleration. This is the most general and the most intense injection mechanism. It occurs for all kind of ions and electrons existing in the interstellar medium (ISM) and therefore it is relevant for primary CRs only. The corresponding injection rate is determined by the number of particles involved into the acceleration from each medium volume crossed the shock and can be represented in the form (Berezhko et al., 1996):
where is the gas number density, is the sound speed, the subscripts 1(2) refer to the point just ahead (behind) the shock. Typical values of the dimensionless injection parameters which provide CR production with required efficiency are and . Secondary CRs like secondary nuclei Li, Be, B or positrons and antiprotons by definition are not presented in ISM and therefore they can not be produced due to such an injection.
Kinetic energy of all kind of Galactic cosmic ray (GCR) particles is considerably larger then the energy of gas particles injected from the postshock thermal pool. Therefore all GCRs which meet the expanding SNR shock are naturally involved into the diffusive shock acceleration. CR acceleration due to this second relevant injection mechanism is usually called ”reacceleration”. This term is used to distinguish the process of further increase of energy of already energetic particles due to interactions with SNR shocks during their propagation in ISM. In this regard it is similar to the stochastic acceleration (also called reacceleration) of GCRs due to their interactions with background MHD-turbulence. Since GCR energy spectra are relatively steep and it have a peak at kinetic energy GeV their injection can be represented in the form
where is the total number of GCR species per unit volume and is their mean momentum, that corresponds to .
Primary nuclei during their acceleration inside SNRs produce secondary nuclei in nuclear collisions with the background gas like GCRs do it in the Galactic disk. Essential fraction of these already energetic particles has possibility to be involved in further shock acceleration. This is the third mechanism of secondary CR production inside SNRs. For the first time it was studied to describe the formation of the secondary CR nuclei spectra (Berezhko et al., 2003).
The production rate of secondary CR particles due to the nuclear collisions of primary CRs with the gas nuclei within SNR can be described by the source term
in the transport equation for the distribution function of secondary CRs . Here is inelastic cross-section of secondary CRs production with momentum in the collision of primary (parent) nuclei of momentum with the gas nuclei, is the time interval since the supernova explosion, is the radial distance from the presupernova star. In the case of secondary nuclei Li, Be, B the parent nuclei are heavier nuclei C, N, O, whereas in the case antiprotons all kind of accelerated in SNRs nuclei (predominantly protons) play a role of parent nuclei. Reacceleration and the acceleration of nuclei created in SNRs in nuclear collisions are of the prime importance for the secondary CRs even though relatively small part of primary CRs is also produced due to these processes.
The source term describes the creation of secondaries throughout the remnant, everywhere downstream and upstream of SNR shock up to the distances of the order of the diffusive length of their parent primary CRs. Essential part of these particles are naturally involving in the acceleration at SNR shock. It includes all the particles created upstream and the particles created downstream at distances less then their diffusive length from the shock front. The number of these particles is increasing function of their momentum because for the Bohm type diffusion coefficient which is realized during efficient CR acceleration in SNRs (e.g. Berezhko, 2008). This makes the secondary particle spectra
produced in SNR, harder compared with the spectra of primaries .
The SNR efficiently accelerates CRs up to some maximal age when SNR release all previously accelerated CRs, primaries and secondaries, with the spectra and respectively, into surrounding ISM. Here and are the kinetic energy and speed of particle with momentum .
The number of secondary CRs involved into the reacceleration at the SNR evolutionary epoch is proportional to the SNR volume , therefore . This is not so for primary CRs because the progressively increasing number of injected CRs is accompanied by the decrease of their momentum due to the shock deceleration. As a result the number of primary relativistic CRs remains nearly constant at late Sedov evolutionary phases (e.g. Berezhko et al., 1996). As a consequence the number of secondary CRs created in nuclear collisions increases at late evolutionary phases proportionally to SNR age . Due to above factors secondary CRs are manly produced on the late SNR evolutionary phases .
These CRs released from SNRs together with secondary CRs produced in ISM form the total secondary and primary CR populations. At sufficiently high energies the s/p ratio of nuclear component within simple leaky box model is given by the expression (Berezhko et al., 2003)
where represents the spectrum of secondaries produced in nuclear collisions of primary CRs within the Galactic disk. Within the leaky box model it is approximately given by the expression (Berezhko et al., 2003) , where is the escape length which is the mean matter thickness traversed by GCRs in the course of their random walk in the Galaxy, is the ISM gas density, is the CR escape time from the Galaxy, is the proton mass,
Note that at sufficiently high energies the s/p ratio is determined by the s/p ratio produced in the SNRs independently on the propagation model which influence the ratio .
3. Results and discussion
We have calculated the overall energy spectra of all relevant CR species accelerated in SNRs, within kinetic nonlinear model. The model is based on a fully time-dependent self-consistent solution of the CR transport equation together with the gas dynamic equations in spherical symmetry. It includes the most relevant physical factors, essential for the evolution and CR acceleration in a SNR and it is able to make quantitative predictions of the expected properties of CRs produced in SNRs and their nonthermal radiation. The application of the theory to individual SNRs has demonstrated its capability of explaining the observed SNR nonthermal emission properties (Berezhko, 2008). The theory is able to explain major characteristics of the observed CR spectrum up to an energy of eV (see Berezhko et al., 1996; Berezhko, 2008, for details). Similar approach was developed recently by other authors (Ptuskin et al., 2010; Kang, 2010).
We restrict ourselves by the most simple case of type Ia SN in uniform ISM with corresponding SN parameter values: explosion energy erg and ejecta mass. We use typical values of the dimensionless parameters and , which describes the injection of gas particles into the shock acceleration (Völk et al., 2003). We consider the typical ISM with hydrogen number density cm, temperature K and magnetic field values 5 G, which roughly corresponds to the average ISM within the Galactic disk. We adopt time-independent upstream magnetic field value and ignore magnetic field amplification effect because the secondaries are mainly produced at the late evolutionary phases (Berezhko et al., 2003) when this effect becomes irrelevant.
We perform selfconsistent calculation up to the SNR age when SNR release all previously accelerated CRs into surrounding ISM. We adopt the value yr appropriate for considered range of ISM density (Berezhko et al., 2003).
Calculated antiproton-to-proton ratio as a function of energy together with PAMELA data are shown in Fig. 1. We use the crossection , parametrization of Shibata et al. (2008) for crossection of antiproton production in p-p collisions and correction factor 1.6 which describes the contribution of heavier nuclei. Since the observed antiproton spectrum has a peak at GeV we use cm and the value of which corresponds to the kinetic energy GeV. For the spectrum of antiprotons produced in ISM we use the results of calculations performed in Donato et al. (2001). It is very close to what was used by Blasi & Serpico (2009) in their similar consideration. It is seen that antiprotons at energies GeV are produced in SNRs equally effectively by both mechanisms whereas at GeV the creation of antiprotons in p-p collisions and their subsequent acceleration becomes dominant. In total the antiproton production in SNRs makes the energy dependence of considerably more flatter so that at GeV the ratio becomes larger by a factor of about five. Within the energy range 30 GeV GeV the energy dependence of the ratio is expected to be very flat.
PAMELA data, which well agrees with our calculation, within the energy range 10 GeV GeV provide the evidence that the actual ratio is indeed more flatter than it is expected if antiprotons are created in ISM only.
The production of antiprotons in SNRs estimated by Blasi & Serpico (2009) is considerably larger (by a factor of four at GeV) compared with our calculation even though these authors have neglected the reacceleration process at all. It is due to a number of simplifications made by these authors. For example in actual situation the overlap between the radial profile of protons with the gas density profile which has a peak value at the shock , progressively decreases with increase of energy at high energies GeV, because the radial profile of protons becomes progressively broader. Here is the gas density of ISM. This leads to the decrease of the effective gas density from the value to and to that follows by the decrease of the antiproton production. This factor was neglected by Blasi & Serpico (2009), that is one of the reason which lead them to overestimation of antiproton production.
In order to check the consistency of other types of secondary CR production we have calculated within the same model the boron-to-carbon (B/C) ratio and compare it in Fig. 2 with the existing experimental data.
The boron nuclei represent the example of secondary nuclei. To calculate boron spectrum we use the overall number density of boron nuclei in ISM cm injected at a kinetic energy GeV/n which corresponds to the mean GCR energy for these element. Compared with the previous study (Berezhko et al., 2003) except C, N and O nuclei as parent species following Mertsch & Sarkar (2009) and Tomassetti & Donato (2013) we included also heavier primaries up to Si, which contribute about 10% into the boron production.
Due to boron production in SNRs the expected B/C ratio undergoes considerable flattering which starts at energy GeV/nucleon. As one can see in Fig. 2 this is consistent with the measurements recently performed in balloon (Derbina et al., 2005) and AMS-02 space (Aguilar, 2013) experiments even though for more strict conclusion one needs the measurements with higher statistics at energies above 1 TeV/n.
Calculated SNR contribution into the secondary CR spectra represents the component which is unavoidably expected if SNRs are the main source of GCRs. Comparison with the existing data leads to a conclusion that the observed high energy excess of secondary nuclei can be produced in Galactic SNRs. This enable to expect similar excess in the antiproton energy spectrum. The data expected very soon from AMS-02 experiment will make it clear whether the actual ratio is indeed not less flat at energies GeV then we predict.
- affiliationtext: Yu. G. Shafer Institute of Cosmophysical Research and Aeronomy, 31 Lenin Ave., 677891 Yakutsk, Russia
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