Cosmic Ray Helium Hardening
Recent observations by CREAM and ATIC-2 experiments suggest that (1) the spectrum of cosmic ray (CR) helium is harder than that of CR proton below the knee and (2) all CR spectra become hard at . We propose a new picture that higher energy CRs are generated in more helium-rich region to explain the hardening (1) without introducing different sources for CR helium. The helium to proton ratio at TeV exceeds the Big Bang abundance by several times, and the different spectrum is not reproduced within the diffusive shock acceleration theory. We argue that CRs are produced in the chemically enriched region, such as a superbubble, and the outward-decreasing abundance naturally leads to the hard spectrum of CR helium if CRs escape from the supernova remnant (SNR) shock in an energy-dependent way. We provide a simple analytical spectrum that also fits well the hardening (2) because of the decreasing Mach number in the hot superbubble with K. Our model predicts hard and concave spectra for heavier CR elements.
Subject headings:acceleration of particles — cosmic rays — shock waves — supernova remnants
Recently, the Cosmic Ray Energetics And Mass (CREAM) has directly observed the CR compositions with high statistics in the wide energy range up to about . Interestingly, CREAM shows for CR proton and for CR helium in the energy region –, that is, the spectrum of CR helium is harder than that of CR proton (Ahn et al., 2010). Although the difference of the spectral index appears small, the implications are of great importance as shown below. In addition, the spectral index becomes hard by for CR proton and by for CR helium at because the Alpha Magnet Spectrometer (AMS) shows for the CR proton (Alcaraz et al., 2000a) and for CR helium (Alcaraz et al., 2000b) in the low-energy range –. These results have been already obtained by the Advanced Thin Ionization Calorimeter-2 (ATIC-2) (Panov et al., 2009).
For CR electrons, the Fermi gamma-ray space telescope has recently observed the spectrum of CR electrons in the wide energy range from to (Ackermann et al., 2010). Fermi shows that the observed date can be fitted by a power law with spectral index in the interval and the spectral hardening at about , which may have the same origin as that of the CR nuclei. (For other models, see, e.g., Kashiyama et al., 2010; Kawanaka et al., 2010; Ioka, 2010, and references therein). Note that we do not discuss CR positrons in this letter.
Supernova remnants (SNRs) are thought as the origin of the Galactic CRs. The most popular acceleration mechanism at SNRs is the diffusive shock acceleration (DSA) (Axford et al., 1977; Krymsky, 1977; Bell, 1978; Blandford & Ostriker, 1978). In fact, Fermi and AGILE show that middle-age SNRs interacting with molecular clouds emit gamma-rays (e.g. Abdo et al., 2009; Tavani et al., 2010) and the gamma-ray observations support that SNRs produce the bulk of Galactic CRs (e.g., Ohira et al., 2011; Li & Chen, 2010).
According to DSA theory, the spectrum of accelerated particles at a shock does not depend on CR elements, but depends only on the velocity profile of the shock. Thus, naively, recent CR observations seem to show that the acceleration site of CR helium is different from that of CR proton (Biermann et al., 2010). However, in this different site scenario, it should be by chance that the observed ratio of CR helium and proton, , at is similar to the cosmic abundance (). Furthermore, the difference of the spectral index means that at is about 3 times higher than that at . This enhancement is amazing since the mean helium abundance in the universe is virtually maintained constant. The stellar nucleosynthesis never enhances the mean helium abundance by a factor, which is the essential reason that the big bang nucleosynthesis is indispensable for the cosmic helium abundance. To make the enhancement, we should consider inhomogeneous abundance regions. We show that this leads to the different spectrum of CR proton and helium when escaping from SNRs.
In this letter, considering the inhomogeneous abundance region, we provide a new explanation about (1) the different spectrum of CR proton and helium, even if CR proton and helium are accelerated simultaneously. Our idea uses the fact that CRs escaping from SNRs generally have a different spectrum than that of the acceleration site (Ptsuskin & Zirakashvili, 2005; Ohira et al., 2010; Caprioli et al., 2010). The runaway CR spectrum depends on not only the acceleration spectrum at shocks but also the evolution of the maximum energy and the number of accelerated CRs (Ohira et al., 2010). We also suggest that (2) the spectral hardening of CRs is caused by the decreasing Mach number in the high temperature medium. Both the inhomogeneous abundance and the high temperature can be realized in the superbubbles with multiple supernovae. Our conclusions are summarized as follows.
Runway CR spectra depend on not only CR spectra inside the SNR but also the evolution of the maximum energy and the number of accelerated CRs. Therefore, taking account of the inhomogeneous abundance region, runaway CR spectra of different CR elements have different spectra (section 2 and 3.1).
Our model is in excellent agreement with observed spectra of CR proton and helium. Harder spectrum of CR helium is due to the enhancement of the helium abundance around the explosion center. On the other hand, the concave spectra of all CR elements are due to the decreasing Mach number in the hot gas with . The concave spectra may be also produced by the CR nonlinear effect, the energy dependent effects on the accelerated CRs (on or ), the propagation effect (), and/or multi components with different spectral indices (section 3.2 and 4).
Within the single component scenario, the hard helium spectrum suggests that the origin of the Galactic CR is SNRs in superbubbles, although we are not excluding the multi component scenario (section 5).
Our model predicts that heavier (at least volatile) CR elements also have harder spectra than that of CR proton and have concave spectra (section 5).
2. Runaway CR spectrum
In this section, we briefly review the runaway CR spectrum (see Appendix of Ohira et al. (2010)). We here use a variable (for example the shock radius or the SNR age) to describe the evolution of an SNR. Let and be the CR momentum spectrum [(] and the maximum four momentum of CR inside the SNR at a certain epoch labeled by , respectively. CRs escape in order, from the maximum energy CR because the diffusion length of high-energy CRs is larger than that of low-energy CRs. Then, the number of runaway CRs between and is
which corresponds to the number of runaway CRs between and , . Hence, is
where is the inverse function of . Assuming and , we obtain the runaway CR spectrum as
where and are parameters to describe the evolution of maximum energy and the number of accelerated CRs, respectively. (We use and later.) Therefore, the runaway CR spectrum is different from that in the SNR . Figure 1 shows the schematic picture of the runaway CR spectrum. In this Letter, we use the shock radius, , as .
The evolution of the maximum energy of CRs at the SNR has not been understood. This strongly depends on the evolution of the magnetic field around the shock (e.g. Ptsuskin & Zirakashvili, 2003). Although some magnetic field amplifications have been proposed (e.g., Lucek & Bell, 2000; Bell, 2004; Giacalone & Jokipii, 2007; Ohira et al., 2009b) and investigated by simulations (e.g., Niemiec et al., 2008; Riquelme & Spitkovsky, 2009; Ohira et al., 2009a; Gargaté et al., 2010), the evolution of the magnetic field has not been completely understood yet. Here we assume that CRs with the knee energy escape at , where is the shock radius at the beginning of the Sedov phase. Furthermore, we use the phenomenological approach with the power-law dependence (Gabici et al., 2009; Ohira et al., 2010),
where is the four momentum of the knee energy. Note that does not depend on the CR composition because the evolution of the maximum energy depends only on the evolution of the magnetic field and the shock velocity.
The evolution of the number of CRs inside the SNR has not been also understood. This depends on the injection mechanism (Ohira et al., 2010) and the density profile around the SNR. We here adopt the thermal leakage model (Malkov & Völk, 1995) as an injection model. For the total density profile, where and are the number density of proton and helium and is the proton mass, the shock velocity of the Sedov phase is
In the thermal leakage model, the injection momentum of element is proportional to the shock velocity, , and the number density of CR with momentum is proportional to the density, , where is the distribution function of CR element . Hence, the number of CR element with a reference momentum , is
where and is the spectral index in the non-relativistic energy region. For the nonlinear DSA, the spectral index in the non-relativistic energy region is different from that in the relativistic energy region (Berezhko & Ellison, 1999). To understand the essential feature of the runaway CR spectrum, we here consider only the test-particle DSA, that is, . Because is not always a single power-law form, the evolution of the number of accelerated CRs can not be always described by a constant .
3. Basic idea
3.1. Different spectrum of CR proton and helium
According to the test particle DSA theory, the index of relativistic CR energy spectrum depends only on the velocity jump at the shock,
where we use the Rankine-Hugoniot relation at the second equation and is the Mach number. Then, the index of the runaway CR spectrum, , is
in equation (3). Therefore, if (in particular , the index for the accelerated CR number evolution) is different, the runaway CR spectrum is different between the CR compositions. This is our main idea to explain the helium hardening observed by CREAM and ATIC-2. From equation (6), depends on the ambient number density . Therefore, different density profiles make different runaway CR spectra (See Section 4 for more details). Figure 2 shows the schematic picture of our idea.
3.2. Spectral hardening of all CRs at the same energy per nucleon
In this subsection, we discuss the spectral hardening of the observed CRs. The Galactic CR spectrum observed at the Earth, , is obtained by the simple leaky box model
where is the diffusion coefficient (e.g. Strong et al., 2007). Hence, the index of the observed spectrum is
The deviation from a single power law means that at least one of , , , and has an energy dependence or that the origin of low energy CRs below is different from that of high energy CRs above . Although the multi component scenario may be the case because there are many types of SNRs, we discuss the single component scenario in this letter.
Firstly, we discuss the energy dependence of . From equation (7), depends on the shock radius because the Mach number decreases with the shock radius. Then we can expect the spectral harding of all CR compositions at the same rigidity , that is, at approximately the same energy per nucleon. From equation (5), the Mach number is
where is the surrounding temperature and we assume that the ejecta mass and the energy of supernova explosion are and , respectively. From equations (4), (7) and (11), we can obtain as a function of (see § 4).
Next, we discuss the energy dependence of that is the parameter to describe the evolution of the number of accelerated CRs. In Section 3.1, we consider different power-law forms for and to make the different spectrum of the CR proton and helium. Therefore, is not a single power law form, and has an energy dependence (see § 4).
The energy dependence of will be soon precisely determined by AMS-02 (Pato et al., 2010). We do not discuss the energy dependence of because the complete physics of the CR escape and magnetic turbulence is beyond the scope of this Letter.
4. Comparison of our model with observations
In this section, specifying model parameters, we calculate the Galactic CR spectrum. For simplicity, we here assume the number densities of proton and helium as follows,
where is the number density of proton at , and is the normalization factor of the helium density. We set so that the helium abundance is that of the solar abundance, (i.e., ), when with equation (4). Note that the power-law dependence is a first step approximation for the mean value. Then, from equations (2), (4), (6), (9), observed spectra of CR proton and helium are
Figure 3 shows the comparison of our model with observations. We take into account the solar modulation effects with the modulation potential (Gleeson & Axford, 1968). Our model is in excellent agreement with the observed spectra, with , , , , . The different spectra of CR proton and helium originate from the different density profiles in equations (12). Figure 4 shows the evolution of the maximum energy of CRs and the spectral index of CRs inside the SNR. In the early phase, the spectral index is 2 and after then, the spectral index decreases with the shock radius because the Mach number decreases with shock radius. The change of spectral index is about 0.1 which is almost the same as the observed hardening. The observed hardening is not the result of the change of the injection history, , but the result of spectral change of CRs inside the SNR. The high temperature is necessary for the spectral hardening .
In addition, our model also makes a concave spectrum of CR electrons as observed (Ackermann et al., 2010). However, the evolution of injection efficiency of CR electrons has not been understood well. So we need further studies to discuss the CR electron spectrum in detail.
To make the different spectrum, our model requires that the helium abundance around the explosion center is higher than that of the solar abundance. SNRs in superbubbles are one of candidates. Higdon et al. (1998) show that supernova ejecta can dominate the superbubble mass within a core radius of one third of the superbubble radius. In the stellar wind and the supernova explosion, the stellar hydrogen envelope has lower density and higher velocity than that of helium. Then we expect that the helium fraction in the center of superbubbles is higher than that in the outer region. Furthermore, to make the concave spectrum, our model requires an ambient medium with high temperature, . This is also consistent with superbubbles. According to the CR composition study, SNRs in superbubbles have been considered as the origin of Galactic CRs (e.g., Lingenfelter & Higdon, 2007; Ogliore et al., 2009). Particle accelerations in superbubbles have been also investigated by intensive studies (e.g., Bykov & Fleishman, 1992; Parizot et al., 2004; Dar & De Rújula, 2008; Ferrand & Marcowith, 2010). We here considered a spherically symmetric system. The off-center effects may be important for the initial phase and thereby for the high energy spectrum, because the shock radius at the beginning of the Sedov phase is about which is comparable to the typical size of OB association, (Parizot et al., 2004), and the shock radius is about at the end of the Sedov phase. This is an interesting future problem.
Note that the spectral hardening can be also made by the nonlinear model, the energy dependence of the CR diffusion coefficient and/or multi components with different spectral indices. So the high temperature may not be absolutely necessary. The stellar wind of red giants is one of candidates for the cold and helium rich ambient. Still, the dominant core-collapse supernovae is type II (e.g., Smartt et al., 2009) which has no helium rich wind, so that the superbubble scenario looks more likely as the origin of the Galactic CRs above . For the CRs below , the spectral difference between CR proton and helium may be caused by the solar modulation and the inelastic interactions (Putze et al., 2010).
The spatial variation of the helium ionization degree can also change the injection history. The injection efficiency of the large rigidity is thought to be higher than that of low rigidity since particles with large rigidity can easily penetrate through the shock front from the downstream region. If the ionization degree increases with the SNR radius, the CR helium spectrum becomes harder than the CR proton one, . However, the rigidity dependence of the injection efficiency has not been understood completely. Moreover, the injection from neutral particles should also be understood (Ohira et al., 2009b, 2010).
According to our model, CR spectra of heavier volatile elements than helium is also harder than that of proton. Low-energy CRs of refractory elements are thought to result from suprathermal injection by sputtering off preaccelerated, high-velocity grains (Ellision et al., 1997). To be accelerated to the relativistic energy, the refractory elements should be sputtered because the grains can not be accelerated to the relativistic energy. The SNR shock velocity is not fast enough to accelerate refractory elements to the knee energy when the refractory elements are injected because the sputtering time scale is too long. Therefore, refractory CRs around the knee energy should be injected by the standard manner similar to volatile CRs. In this case, the refractory CRs also have harder spectra than protons, although we need further studies of the injection of refractory CRs at the knee energy.
If CRs trapped inside the SNR and released at the end of the SNR’s life outnumber runaway CRs (see figure 3 in Caprioli et al. (2010)), our scenario does not work for producing hard and concave spectra. In our model with in Eq. (4), trapped CRs have energy below when they are released, that is, when , and are not relevant for our interest. Higher energy CRs escape from the SNR even after advected to the downstream since the CR diffusion is faster than the expansion of the SNR. Our case is similar to the right figure 7 in Caprioli et al. (2010) where trapped CRs are released below . The energy boundary between trapped CRs and runaway CRs depends on the evolution of the maximum energy ().
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