the role of ionization in the shock acceleration theory

Mon. Not. R. Astron. Soc. 412, 2333–2344 (2011) doi:10.1111/j.1365-2966.2010.18054.x

The role of ionization in the shock acceleration theory

Giovanni Morlino�

INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi, 5, 50125 Firenze, Italy

Accepted 2010 November 22. Received 2010 October 22

ABSTRACTWe study the acceleration of heavy nuclei at supernova remnant (SNR) shocks, taking intoaccount the process of ionization. In the interstellar medium, atoms heavier than hydrogen,which start the diffusive shock acceleration (DSA), are never fully ionized at the moment ofinjection. We will show that electrons in the atomic shells are stripped during the accelerationprocess, when the atoms already move relativistically. For typical environment around SNRs,the dominant ionization process is the photoionization due to the background galactic radiation.The ionization has two interesting consequences. First, because the total photoionization timeis comparable to the beginning of the Sedov–Taylor phase, the maximum energy which ionscan achieve is smaller than the standard result of the DSA, which predicts Emax ∝ ZN . Asa consequence, the structure of the cosmic-ray spectrum in the knee region can be affected.The second consequence is that electrons are stripped from atoms when they already moverelativistically; hence, they can start the DSA without any pre-acceleration mechanism. We usethe linear quasi-stationary approach to compute the spectrum of ions and electrons acceleratedafter being stripped. We show that the number of these secondary electrons is enough toaccount for the synchrotron radiation observed from young SNRs, if the amplification of themagnetic field occurs.

Key words: acceleration of particles – shock waves – ISM: supernova remnants – cosmicrays.

1 IN T RO D U C T I O N

The bulk of Galactic cosmic rays (CRs) is largely thought to beaccelerated at the shock waves associated with supernova rem-nants (SNRs) through the mechanism of diffusive shock accelera-tion (DSA). A key feature of this mechanism is that the accelerationrate is proportional to the particle charge. This property is especiallyappealing if one tries to explain the structure of the knee in the CRspectrum: if we assume that the maximum energy of acceleratedparticles scales with the charge of the particles involved, a kneearises naturally as a superposition of spectra of chemicals with dif-ferent nuclear charges ZNe (Horandel 2003). This result is basedon the assumption that nuclei are completely ionized during the ac-celeration. Conversely, when ions are injected into the accelerationprocess, they are unlikely to be fully stripped, especially if they areof high nuclear charge.

The atoms relevant for the injection are those present in thecircumstellar medium where the forward shock propagates. Thetemperature of this plasma varies from 104 K, if the SNR expandsinto the regular interstellar medium (ISM), up to 106 K, if theexpansion occurs into the bubble created by the progenitor’s wind. IfT ∼ 104 K, then even hydrogen is not fully ionized, as demonstrated

�E-mail: [email protected]

by the presence of Balmer lines associated with shocks in someyoung SNRs (Chevalier, Kirshner & Raymond 1980; Sollerman etal. 2003; Heng 2009). For T ∼ 106, only atoms up to ZN = 5 can becompletely ionized (Porquet, Arnaud & Decourchelle 2001).

The typical assumption made in the literature is that atoms loseall electrons in the atomic orbitals soon after the beginning of theacceleration process, that is, the ionization time needed to stripall the electrons is much smaller than the acceleration time. Inspite of this assumption in Morlino (2009), we showed that, fora typical SNR shock, the ionization time is comparable with theacceleration time; hence, electrons are stripped when ions alreadymove relativistically. This fact has two important consequences:(1) the maximum energy of ions can be reduced with respect tothe standard prediction of DSA; and (2) the ejected electrons caneasily start the acceleration process because they already moverelativistically.

The possibility that ionization can provide a source of relativis-tic electrons is especially relevant, because the question of howelectrons are injected into the DSA is still an unsolved issue. DSAapplies only for particles with a Larmor radius larger than the typ-ical shock thickness, which is of the order of the Larmor radiusof the shocked downstream thermal ions. The injection conditioncan be easily fulfilled for suprathermal protons, which reside in thehighest-energy tail of the Maxwellian distribution. The injection ofheavier ions is, in principle, even simpler than that of protons in that

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2334 G. Morlino

their Larmor radius is larger, assuming they have the same protontemperature (which could be not the case). Conversely, the sameargument tells that electrons cannot be injected from the thermalbath, because, even if we assume equilibration between electronsand protons, the Larmor radius of electrons is a factor of (me/mp)1/2

smaller than that of protons. Only electrons which are already rela-tivistic can cross the shock and start the DSA.

Most proposed solutions to the electron injection problem in-volve some kind of pre-acceleration mechanism, driven by plasmainstabilities able to accelerate thermal electrons up to mildly rel-ativistic energy. For instance, Galeev (1984) showed that magne-tosonic turbulence excited by a beam of reflected ions ahead ofthe shock can accelerate electrons along the magnetic field lines,thanks to their Cerenkov interaction with excited waves. Other stud-ies predict that electrons can be effectively pre-accelerated by self-generated whistler waves (Levinson 1992, 1996). More recently,Amano & Hoshino (2009) showed that efficient electron acceler-ation can happen in perpendicular shocks, due to ‘shock surfing’of electrons on electrostatic waves excited by Buneman instabilityat the leading edge of the shock foot. A different mechanism pro-posed by Riquelme & Spitkovsky (2010) can take place in quasi-perpendicular, shocks, and is driven by the presence of obliquewhistler waves excited by the returning ions in the shock foot. Thestudy of these electromagnetic pre-acceleration mechanisms canonly be performed using numerical techniques, which show thatthe fraction of injected electrons strongly depends on the values ofinitial conditions (like magnetic field strength and orientation), andare difficult to apply to realistic cases.

In this paper, we investigate in detail the process of ionizationapplied to the acceleration of heavy ions, presenting the full steady-state solution for shock acceleration in the test particle approach.

The paper is organized as follows. In Section 2, we comparethe acceleration time of ions to the ionization time due both tophotoionization and to Coulomb scattering, showing that the for-mer dominates the latter. In Section 3, we use the linear accelerationtheory to compute the maximum energy achieved by different chem-ical species when the ionization process is taken into account. InSection 4, we compute the distribution function of both ions andelectrons in the framework of linear acceleration theory, includingthe term due to ionization. We conclude in Section 5.

2 IONIZATION V ERSUS ACCELERATIONTIME

In this section, we show that the ionization of different chemicalspecies occurs during the acceleration process on a time-scale whichis comparable with the acceleration time needed to achieve rela-tivistic energies. We also show that the ionization due to Coulombcollision is generally negligible compared to the photoionizationdue to the interstellar radiation field (ISRF).

Let us consider atoms of a single chemical species N with nuclearcharge ZN and mass mN = Amp, which start the DSA with initialcharge Z < ZN and momentum pinj. We want to compute the ioniza-tion time needed to lose one electron, changing the net charge fromZ to Z + 1, and the momentum p that ions reach when ionizationoccurs.

For simplicity, we compute the acceleration time in the frame-work of linear shock acceleration theory for plane shock geometry,that is, we assume that during the ionization time needed to stripone single electron, the shock structure does not change. If a particlewith momentum p diffuses with a diffusion coefficient D(p), then

the well-known expression for the acceleration time is

τacc(pinj, p) =∫ p

pinj

3

u1 − u2

[D1(p)

u1+ D2(p)

u2

]dp

p, (1)

where u is the plasma speed in the shock rest frame and the sub-script 1 (2) refers to the upstream (downstream) quantities (notethat ushock ≡ u1). The downstream plasma speed is related to theupstream one through the compression factor u2 = u1/r. We limitour considerations to strong shocks, which have compression factorr = 4, and we assume the Bohm diffusion coefficient DB = rLβc/3,where β c is the particle speed and rL = pc/ZeB is the Larmorradius. The turbulent magnetic field responsible for the particle dif-fusion is assumed to be compressed downstream according to B2 =rB1. Even if this relation applies only for the magnetic componentparallel to the shock plane, such assumption does not affect ourresults strongly. It is useful to define the instantaneous accelerationtime as tacc ≡ dτ/(dp/p). Using all previous assumptions, tacc canbe expressed as follows:

tacc(p) = 0.85βp

mNcB−1

µG u−28

(A

Z

)yr . (2)

Here the upstream magnetic field is expressed in μG and the shockspeed is u1 = 108u8 cm s−1. Notice that, when B and u are assumedconstant, τ acc reduces to tacc for pinj � p and D ∝ p. To compute theenergy reached by particles when the ionization event occurs, equa-tion (2) has to be compared to the ionization time-scale. Ionizationcan occur either via Coulomb collisions with thermal particles orvia photoionization with background photons. Whether the formerprocess dominates the latter depends on the ISM number densitycompared to the ionizing photon density. In the following, we showthat photoionization dominates when SNRs are young and expandin a typical ISM.

As soon as the ions start the DSA, the photoionization can occuronly when the energy of background photons, ε ′, as seen in the ionrest frame, is larger than the ionization energy I. Atoms movingrelativistically with a Lorentz factor γ see a distribution of photonspeaked in the forward direction of motion, with a mean photonenergy ε ′ = γ ε. The photoionization cross-section can be estimatedusing the simplest approximation for the K-shell cross-section ofhydrogen-like atoms with effective nuclear charge Z (Heitler 1954):

σph(ε ′) = 64α−3σTZ−2

(IN,Z

ε ′

)7/2

, (3)

where σ T is the Thompson cross-section, α is the fine structureconstant and IN,Z is the ionization energy threshold for the groundstate of the chemical species N with ZN − Z electrons. The numericalvalues of IN,Z can be found in the literature (see e.g. Allen 1973 forelements up to ZN = 30). To get the full photoionization time, weneed to integrate over the total photon energy spectrum:

τ−1ph (γ ) =

∫dε

dnph(ε)

dεcσph(γ ε) , (4)

where d nph/dε is the photon spectrum as seen in the plasma restframe. Because the photoionization cross-section decreases rapidlywith increasing photon energy, for a fixed ion speed, the relevantionizing photons are only those with energy close to the threshold,that is, ε � IN,Z/γ , measured in the plasma frame. The correspond-ing numerical value is

τph(γ ) � 0.01Z2[nph(IN,Z/γ )/cm−3

]−1yr . (5)

We assume that the maximum possible acceleration time is equalto the Sedov–Taylor time, tST, corresponding to the end of the free-expansion phase. Comparing tST with equation (5), we see that the

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Ionization in the shock acceleration theory 2335

photons which can be relevant for ionization are those with a numberdensity nph > 10−5Z2(tST/103 yr)−1 cm−3. For this reason, we canneglect the high-energy radiation coming from the remnant itself,because the typical number density of X-ray photons is less than10−7 photons cm−3.

An important consequence of the photoionization is that elec-trons are stripped from the atoms with a kinetic energy �mec2 asmeasured in the atom rest frame. Hence, if the parent atoms moverelativistically in the plasma rest frame, then electrons move with thesame Lorentz factor and in the same direction as the parent atoms.This property will be used in Section 4 to simplify the calculationof the electron spectrum.

Now we consider the ionization of accelerated ions due toCoulomb collisions with protons and electrons of the thermalplasma. We neglect the thermal energy of those particles and weconsider the process in the ion’s rest frame. The role of collisions ismainly relevant during the very initial stage of acceleration, namely,when the kinetic energy of the incident particle, Ekin, is close to theionization energy, I, as seen in the ion’s rest frame. This is a con-sequence of the shape of ionization cross-section, which has a peakvalue for Ekin ∼ I and rapidly decreases for larger energies. A goodupper limit for the peak value of the cross-section is provided by theclassical (non-relativistic) Thomson approach (Allen 1973), whichgives a maximum for Ekin = 2 I equal to

σ maxcoll = πa2

0NeI−2Ryd, (6)

where a0 is the Bohr radius, IRyd is the ionization energy expressedin Rydberg unity and Ne is the number of electrons in the atomicorbital considered. For larger energies, the Thomson cross-sectiondecreases like E−2

kin, while the asymptotic behaviour predicted bythe quantum mechanics is σ ∼ log(Ekin)/Ekin. This implies thatthe collisional ionization time has a minimum σ = σ max

coll and thenincreases. In order to get this minimum value, one must average thecontribution of collisions in the upstream and downstream regions.Assuming equal density for thermal protons and electrons ne = np

≡ n1, the result is (Morlino 2009):

τmincoll ≈ [

cσ maxcoll n1(1 + r)

]−1 = 0.0024I 2Ryd

( n1

cm−3

)−1yr . (7)

Equation (7) is valid in the non-relativistic regime. In this regime,the acceleration time increases only like tacc ∼ p ∼ E1/2

kin . As a conse-quence, the collisional ionization can occur only if the accelerationtime needed to have Ekin = 2 I is larger than τmin

coll . Let us call γ ∗ theLorentz factor, which corresponds to this condition, namely, γ ∗ =1 + 2I/mec2. Comparing equation (7) to equation (2), the conditiontacc(γ ∗) = τmin

coll provides a lower limit for the upstream density:

n1,min = 27IRydBµGu28

(Z

ZN

)cm−3. (8)

Hence, for n1 < n1,min, ions are accelerated up to relativistic ener-gies before being ionized by collisions. Notice that we neglectedcollisions with atoms heavier than hydrogen: such contributionscould reduce the ionization time by less than a factor of 2, henceour main conclusions remain valid.

Now, in order to understand the relevance of collisions in therelativistic regime, we can use the asymptotic Bethe cross-sectionvalid in the limit Ekin � mec2 (Bethe 1930). The relativistic formu-lation of this cross-section for the total ionization is often writtenas (Kim, Santos & Parente 2000):

σRBethe(γ ) = 4πa20α

2

β2

{M2

[ln(γ 2β2) − β2

] + CR

}. (9)

The two constant M2 and CR are related to the atomic form factors ofthe target and are independent of the incident particle energy. Theirexact numerical values are very difficult to compute with somenotable exception like H-like atoms. In principle, they can also beinferred from photoionization experiments, but, in practice, theyare known only for a bunch of targets. Nevertheless, from theory,we know that M2 ∼ Ne/IRyd, while CR ∼ 10M2 (Inokuti 1971;Kim at al. 2000). Using these estimates in the ultrarelativistic limit,we have σ RBethe/σ

maxcoll � 10−3. This result implies that Coulomb

collisions are much less important in the relativistic regime than inthe non-relativistic one.

In order to compare the photoionization to the collisional ion-ization times, in Fig. 1, they have been plotted together with theacceleration time for three different H-like ions: He+, C5+ andFe25+. The characteristic times are plotted as functions of the ion’sLorentz factor. The acceleration time, tacc, plotted with solid lines,is shown for two different choices of parameters: u1 = 1000 km s−1,B1 = 3 μG (upper line) and u1 = 104 km s−1, B1 = 20 μG (lowerline). The photoionization time τ ph (dashed lines) is computed ac-cording to equation (4) using the cosmic microwave backgroundplus the Galactic ISRF as reported by Porter & Strong (2005).The ISRF strongly changes going from the centre towards the pe-riphery of the Galaxy; hence, the photoionization depends on theSNR location. In order to evaluate this variation, we show τ ph

using the ISRF in the Galactic Centre (lower dashed line) andthe one at 12 kpc away from the Galactic Centre in the Galac-tic plane (upper dashed lines). Finally, with the dot–dashed lineswe plot the collisional ionization time for the upstream densityn1 = 1 cm−3: the thin horizontal line corresponds to the non-relativistic lower limit, τmin

coll expressed in equation (7), while thethick line is calculated using the relativistic Bethe cross-section forH-like atoms. In this case, the values of the parameters used in theBethe cross-section are M2 = 0.30/Z4

N and CR = 4.30/Z4N (Bethe

1930). Fig. 1 clearly shows that the collisional ionization can beneglected for the case of young SNRs expanding into a mediumwith a number density less than or equal to a few cm−3. The pho-toionization is the dominant process even for SNRs located far awayfrom the Galactic Centre. This consideration is strengthened in thecase of core-collapse SNRs, which typically expand into the bubblecreated by the progenitor wind, whose density is usually assumedto be ∼10−2 cm−3. The only phase where the collisional ionizationcould dominate over photoionization is the very initial stage, whenthe shock is expanding into the progenitor’s wind, whose densityis much higher. This phase typically lasts few years and will beneglected in this work.

In Fig. 1, the value of γ , where tacc crosses τ ph, identifies theLorentz factor of the ejected electrons. Looking at the upper plot, itis clear that even electrons coming from the ionization of He+ can, inprinciple, start the DSA. In fact, their typical Lorentz factor is �10.The minimum Lorentz factor required for electrons to be injectedis γ inj ≈ 3–30 for typical parameters of SNR shocks (Morlino2009).

3 MA X I M U M EN E R G Y O F IO N S

The process of ionization can affect the maximum energy that nucleiachieve during the acceleration, especially those with large nuclearcharge. It is worth stressing that the knowledge of the maximumenergy is intimately connected with two important aspects of the CRspectrum: the interpretation of the knee structure and the transitionregion from the Galactic to extragalactic component. The knee iscommonly interpreted as due to the superposition of the spectra of



2336 G. Morlino

He

0 1 2 3 4 5 63

2

1

0

1

2

Log

Log

tyr

C5

0 1 2 3 4 5 61

0

1

2

3

4

Log

Log

tyr

Fe25

0 1 2 3 4 5 61

2

3

4

5

6

Log

Log

tyr

Figure 1. Comparison between acceleration time (solid lines), photoioniza-tion time (dashed lines) and collisional ionization time (dot–dashed lines)for three H-like ions: He+, C5+ and Fe25+ (from top to bottom). Accel-eration and photoionization times are shown with two curves representingtwo different sets of parameters, as explained in the text. The horizontal thindot–dashed line represents the lower limit for the collisional ionization time,equation (6), while the thick dot–dashed line is computed using the Bethecross-section with a plasma density n1 = 1 cm−3.

all chemicals with different cut-off energies. Using the flux of dif-ferent components measured at low energies, it has been shown thatthe knee structure is well reproduced if one assumes that the max-imum energy of each species Emax,N is proportional to the nuclearcharge ZN (Horandel 2003). Nevertheless, the superposition of sub-sequent cut-off seems to be confirmed by the measurements of thespectrum of single components in the knee region: data presented bythe KASKADE experiments show that the maximum energy of Heis approximately two times larger than that of the protons (Antoni

et al. 2005). From the theoretical point of view, the relation Emax,N ∝ZN is clearly predicted by DSA if one assumes that the diffusion co-efficient is rigidity-dependent. This is correct, provided that duringthe acceleration process nuclei are completely ionized; otherwise,the maximum energy is proportional to the effective charge of theions rather to their nuclear charge.

Now, the maximum energy is thought to be achieved at the be-ginning of the Sedov–Taylor phase (tST) (Blasi, Amato & Caprioli2007). For later times, t > tST, the shock speed decreases fasterthan the diffusion velocity; hence, particles at the maximum energycan escape from the accelerator and the maximum energy cannotincrease further. It is worth noting that the process of escaping isnot completely understood in that it strongly depends on the turbu-lence produced by the same escaping particles and on the dampingmechanisms of the turbulence itself. Anyway, here, we assume thatthe maximum energy, Emax, is achieved at t = tST. If the total ioniza-tion time is comparable, or even larger than tST, we do expect thatEmax < E0

max, where we call E0max the maximum energy achieved by

ions which are completely ionized since the beginning of accelera-tion.

A consistent treatment of the ionization effects would require theuse of time-dependent calculation. However, our aim is to get a first-order approximation of the effect of ionization. This can be achievedusing the quasi-stationary version of the linear acceleration theory.

In order to compute Emax, the first piece of information we needis the initial ionization level of ions. In the case of volatile elements,which exist mainly in the gas phase, the level of ionization is easyto estimate, being a function of only the plasma temperature: ionsare ionized up to a level such that the ionization energy is of thesame order as the kinetic energy of thermal electrons as seen in theion’s rest frame. Conversely, refractory elements are largely lockedinto solid dust grains. This occurs both in the regular ISM and instellar winds. As claimed by Ellison, Drury & Meyer (1997), thereis evidence that elements condensed in dust grains are efficientlyinjected into the DSA, thanks to the fact that dust grains can be easilyaccelerated up to energies where the grains start to be sputtered.A non-negligible fraction of the atoms expelled by sputtering areenergetic enough to start the DSA. In this scenario, the ionizationlevel of ejected atoms is very difficult to predict in that it dependson the structure of the grains. However, a reasonable assumptioncould be that ions preserve all the electrons in the inner shells,which are not shared in the orbitals of the crystal structure of thegrains. Moreover, as noted by Ellison et al. (1997), if the ejectedion is highly ionized, then its electron affinity is strong and electronexchange with the atoms of thermal plasma could reduce the levelof ionization up to the equilibrium one. For these reasons, in thefollowing calculation, we neglect the complication arising from thedust sputtering process and we assume that the level of ionizationis only determined by the plasma temperature.

Let us consider a single ion injected with momentum p0 andtotal charge Z1. The ion undergoes acceleration at a constant rate∝ Z1 during a time equal to the ionization time needed to lose oneelectron, τ ph,1, when it achieves the momentum p1. Notice that weinclude only the photoionization, because, as we saw in the previousparagraph, for typical density of the external medium around theSNR (n0 � 1 cm−3), Coulomb collisions can be neglected. In orderto compute simultaneously τ ph,1 and p1, we have to equate theacceleration time with the ionization time:

τph,1

(p1

mNc

)= τacc(p0, p1) = tacc(p1) − tacc(p0) . (10)




The last equality holds because we use Bohm diffusion and assumethat the shock velocity and the magnetic field both remain constantduring τ ph,1. Moreover, we saw that the photoionization occurswhen ions already move relativistically; hence, using equation (2)we set β = 1. Equation (10) gives

p1

mpc= p0

mpc+ Z1B1u

21

1.7ZN

τph,1

(p1

mNc

), (11)

where the subscript 1 labels the quantities during the time τ ph,1. Inequation (11), B and u are expressed in units of μG and u8, respec-tively, while τ ph,1 is expressed in yr. Once the background photondistribution is known, equation (11) can be solved numerically inorder to get p1 and τ ph,1. After the first ionization, the accelerationproceeds at a rate proportional to Z2 ≡ Z1 + 1 during a time τ ph,2,which is the time needed to lose the second electron. Applyingequation (11) repeatedly for all subsequent ionization steps, we getthe momentum when the ionization is complete. If the total ion-ization time, τ tot

ion, is smaller than the Sedov–Taylor time, then, inorder to get the maximum momentum, we need to add the furtheracceleration during the time (tST − τ tot

ion). The final expression forthe maximum momentum can be written as follows:

pmax

mpc=

ZN −Z0∑k=1

ZkBku2kτph,k

1.7ZN

+ θ(tST − τ tot

ion

) M∑i=1

Biu2i

1.7M, (12)

where we neglected the contribution of p0. The last term has beenwritten as a sum over M time-steps in order to handle the case wheremagnetic field and shock speed change with time.

Now we can quantify the effect of ionization in the determinationof pmax. The simplest approximation we can do is to assume ush andB constant during the free-expansion phase, that is, up to t = tST. Inorder to estimate tST, we consider two different situations, which canrepresent a typical Type Ia SN and a core-collapse SN. In the firstcase, the SN explodes in the regular ISM with typical density andtemperature n1 = 1 cm−3 and T0 = 104 K, respectively. Conversely,SNRs generated by very massive stars expand into medium withhigher temperature. This could be either the bubble generated bythe progenitor’s wind or a so-called superbubble, a region wherean elevated rate of SN explosions produces a diluted and hot gas(Higdon & Lingenfelter 2005). In both cases, typical values for thedensity and temperature inside the bubble are n1 = 10−2 cm−3 andT0 = 106 K, respectively. For both the Type Ia and core-collapseSNe, we assume the same value for the explosion energy ESN =1051 erg and mass ejecta Mej = 1.4 M�. The resulting Sedov–Taylortimes are tST = 470 and 2185 yr, respectively. The average shockspeed during the free-expansion phase is ush � 6000 km s−1 in bothcases.

It is worth stressing that if one would consider a more realisticscenario for core-collapse SNe, then it is important to take intoaccount the type of the progenitor and its wind, in order to providea better estimate of the ejected mass, and of the bubble’s dimensionand density. These parameters strongly affect the value of tST, whichdetermines how effective the ionization is in reducing the maximumenergy: indeed for tST � 103 yr, the role of ionization becomesnegligible, as we see below.

The last parameter we need is the magnetic field strength. Westress that our aim is only to estimate the effect of ionization in thescenario where the SNRs indeed produce the observed CR spec-trum and the knee is interpreted as the superposition of a rigidity-dependent cut-off of different species. For this reason, we assumethat both scenarios are able to accelerate protons up to the knee en-ergy, that is, Eknee = 3 × 1015 eV. This condition gives B1 = 160 and35 μG for the Type Ia and core-collapse cases, respectively. We note

0 5 10 15 20 25 300

20

40

60

80

ZN

Et S

ed10

6G

eV

d 12 kpc

d 4 kpc

d 0 kpc

fully ionized

n0 1 cm 3

B1 160 G

tST 470 yr

0 5 10 15 20 25 300

20

40

60

80

ZN

Et S

ed10

6G

eV

d 12 kpc

d 4 kpc

d 0 kpc

fully ionized

n0 0.01 cm 3

B1 35 G

tST 2185 yr

Figure 2. Maximum energy achieved by chemical species with nuclearcharge ZN (ranging from H to Zn) at the beginning of the Sedov–Taylorphase. The upper panel shows the case of Type Ia SNe, expanding into amedium with density ρ0 = 1 cm−3, while the lower panel shows the case ofcore-collapse SNe expanding into a diluted bubble (ρ0 = 0.01 cm−3). Thethin solid line shows the maximum energy achieved by atoms which arefully ionized since the beginning of the acceleration, while the other curvesare computed including the photoionization due to ISRF for SNRs locatedat three different distances d from the Galactic Centre, as specified in thecaption.

that the chosen values of magnetic field strength are consistent withthose predicted by the CR-induced magnetic filed amplification.

In Fig. 2, we plot the maximum energy achieved at the beginningof the Sedov–Taylor phase by different chemical species, from H upto Zn (ZN = 30). The two panels show the cases of the Type Ia (upperpanel) and core-collapse SNe (lower panel) previously described.Each panel contains four lines: the thin solid lines are the maximumenergy achieved by ions, which start the acceleration completelystripped, E0

max, while the remaining lines show Emax computed ac-cording to equation (12) for three different locations of the SNR inthe Galactic plane: at the centre of the Galaxy, and at 4 and 12 kpcaway from the Galactic Centre. Looking at the upper panel, we seethat the maximum energy achieved by different nuclei in Type IaSNe does not increase linearly with ZN ; instead, it reaches a plateaufor ZN � 25 at a distance d = 4 kpc and for ZN � 15 at d =12 kpc. Only for SNRs located in the Galactic bulge the reductionin the maximum energy is negligible, at least for elements up toZN = 30. The effect of ionization is much less relevant for core-collapse SNe: only those remnants located at a distance of 12 kpcshow a notable reduction in Emax.

The numerical results for Fe nuclei corresponding to all casesshown in Fig. 2 are summarized in Table 1. Here, we report thetotal ionization time, and the ratio between the energy achieved attST and the maximum theoretical energy achieved by bare nuclei,



2338 G. Morlino

Table 1. Total photoionization time and normalized maximum energyachieved by Fe nuclei for the two idealized cases of Type Ia and core-collapsed SNe, which are located at a distance d = 0, 4 and 12 kpc from theGalactic Centre. For comparison, the Sedov–Taylor time is also listed. Thebare numbers represent the case where the shock speed and the magneticfield are both constant during the free-expansion phase (as in Fig. 2), whilethe numbers in round brackets are computed assuming ush and B1, whichevolve in time according to the model explained in the text (see also Fig. 3).

tST (yr) d (kpc) τ totion (yr) Emax/E0

max

Type Ia SNe 470 0 76 (137) 0.98 (0.52)’’ 4 356 (385) 0.80 (0.39)’’ 12 1446 (2932) 0.48 (0.24)

Core-collapse SNe 2185 0 113 (142) 0.99 (0.58)’’ 4 264 (374) 0.97 (0.48)’’ 12 2860 (2780) 0.65 (0.35)

0 5 10 15 20 25 300

20

40

60

80

ZN

Et S

ed10

6G

eV

d 12 kpc

d 4 kpc

d 0 kpc

fully ionized

n0 1 cm 3

tST 470 yr

0 5 10 15 20 25 300

20

40

60

80

ZN

Et S

ed10

6G

eV

d 12 kpc

d 4 kpc

d 0 kpc

fully ionized

n0 0.01 cm 3

tST 2185 yr

Figure 3. The same as in Fig. 2, but for a scenario where both the shockspeed and the magnetic field strength evolve with time, as explained in thetext.

E0max (note that the numbers in brackets refer to Fig. 3 as explained

below).The results presented in Fig. 2 are inferred using a simple steady-

state approach. This approach could be too reductive and one canguess how the effect due to the ionization process changes when thefull evolution history of the SNR is taken into account. Needless tosay, a correct calculation of the maximum energy requires not onlythe treatment of the evolution of the SNR, but also the inclusionof non-linear effects which consistently describe the magnetic fieldamplification and the back reaction of CRs on to the shock dynam-ics. A fully consistent treatment of the problem is beyond the scopeof this work. However, we would understand whether the evolutionand the non-linearity can reduce or exacerbate the effect induced

by ionization. In order to reach this goal, we approximate the con-tinuum evolution into time-steps equal to the ionization times, τ ph,k

and we assume that the stationary approximation is valid for eachtime-step. Under this approximation, equation (12) can be used toinclude the effect of evolution if we make reasonable assumptionson how the shock speed and the magnetic field change with time.We follow Truelove & McKee (1999) for the description of thetime-evolution of the SNR. Specifically, we adopt the solution fora remnant characterized by a power-law profile of the ejecta (in thevelocity domain) with index n = 7 expanding into an homogeneousmedium. For this specific case, Truelove & McKee (1999) providethe following expression for the velocity of the forward shock (seetheir table 7):

u1(t)/uch = 0.606 (t/tch)−3/7 (for t < tST), (13)

where uch = (ESN/Mej)1/2, tch = E−1/2SN M5/6

ej ρ−1/31 and tST = 0.732tch.

We adopt the same values of ESN, Mej and ρ1 used in the two casesabove.

For what concern the magnetic field we assume that the amplifi-cation mechanism is at work, converting a fraction of the incomingkinetic energy flux into magnetic energy density downstream of theshock. The simplest way to write this relation is

B22 (t)/(8π) = αBρ1u

21(t). (14)

We always assume B2 = 4B1. The parameter αB hides all the com-plex physics of magnetic amplification and particle acceleration. Forthe sake of completeness, we mention here that in the case of reso-nant streaming instability, αB ∝ ξ crvA/u1, where ξ cr is the efficiencyin CRs and vA is the Alfven velocity, while in the case of resonantamplification, αB ∝ ξ cru1/4c (Bell 2004). However, such relationscannot be applied in a straightforward way, since they require usinga non-linear theory. Here, we prefer to make the simplest assump-tion, taking αB as a constant. We determine its value from the samecondition previously used to fix the value of the constant magneticfield: namely, we assume that for t = tST, the energy of protons is 3× 1015 eV. This condition gives αB = 3.25 × 10−3 and 6.80 × 10−3

for the Type Ia and core-collapse cases, respectively. It is worth not-ing that in the case of some young SNRs, the value of αB has beenestimated from the measurement of both the shock speed and themagnetic field strength. However, we stress that the magnetic fieldcannot be determined in a completely model-independent way. Apossible method requires the measurement of the spatial thicknessof the X-ray filaments: assuming that this thickness is determinedby the rapid synchrotron losses of radiating electrons, we can getthe average magnetic field (see Table 2 and the listed references and

Table 2. Estimated value for the magnetic field and Kep for some youngSNRs.

SNR Age (yr) d (kpc) B2 (µG) Kep/10−4 Reference

Kepler 400 3.4–7.0 440–500 0.7–2.8 (a)Tycho 430 3.1–4.5 368–420 4.2–15 (b)

SN 1006 1000 1.8 90 1.3 (c)’’ ’’ 2.2 120 4.1 (d)

G347.3-0.5 1600(?) 1.0 100 0.6 (e)’’ ’’ 1.0 130 1.0 (f)

References: (a) – Berezhko, Ksenofontov & Volk (2006); (b) – Volk,Berezhko & Ksenofontov (2008); (c) – Morlino et al. (2010); (d) – Berezhko,Ksenofontov & Volk (2009); (e) – Morlino, Amato & Blasi (2009); and (f)– Berezhko & Volk (2006).




also table 1 from Caprioli et al. 2009). Notably the values estimatedfor αB are only a factor of 5–10 larger than the values we use here.

Now equation (12) can be used to compute the maximum mo-mentum at t = tST. For each time-step, τ ph,k, the values of uk andBk are computed according to equations (13) and (14), respectively,evaluated at the beginning of the time-step. In Fig. 3, we reportthe results for the maximum energy for the cases of Type Ia (up-per panel) and core-collapse SNe (lower panel). The lines have thesame meaning as in Fig. 2. As it is clear from both the panels, thereduction in the maximum energy is now more pronounced withrespect to the stationary case shown in Fig. 2. Even in the scenarioof a core-collapse SN located in the Galactic bulge, Fe nuclei areaccelerated only up to about one-half of the maximum theoreticalenergy. The numerical values of τ tot

ion and Emax/E0max for Fe are re-

ported in Table 1 in round brackets. One can see that even in thecases where τ tot

ion does not change with respect to the stationary case,the maximum energy results smaller.

The reason why ionization is less effective when the evolutionis taken into account is a consequence of the fact that accelerationoccurs mainly during the first stage of the SNR expansion. In factboth the shock speed and the magnetic field strength are larger atsmaller times. Conversely, the effective ions’ charge is small duringthe very initial phase of the expansion; hence, the acceleration rateis smaller than its maximum possible value. In order to clarify thispoint in Fig. 4, we compare the energy achieved by Fe nuclei as afunction of the time, for the case of core-collapse SNe located inthe Galactic bulge, with and without the evolution. The dashed linesshow the case of completely ionized nuclei, while the solid linestake into account the ionization process. When the acceleration rateis constant (thin lines), the bulk of the acceleration occurs closeto tST, when the ions are almost completely ionized; hence, Emax

and E0max are practically the same. Conversely, when the evolution

is taken into account (thick lines), the acceleration mainly occursduring the first 200 yr when the ionization is still not completeand for t = tST, we have Emax/E0

max = 0.58. Besides the numericalvalue, this exercise shows that the inclusion of the time-evolutionand non-linear effects can enhance the reduction in the maximumenergy due to the ionization process.

10 20 50 100 200 500 1000 20000.01

0.1

1

10

100

t yr

Et

106

GeV

Figure 4. Behaviour of maximum energy of Fe ions as a function of thetime in the case of core-collapse SNe located in the Galactic bulge. Twodifferent scenarios are shown: (thin lines) acceleration at a constant rate(B1 = constant and u1 = constant); and (thick lines) acceleration includingthe time-evolution of the remnant [B1 = B1(t) and u1 = u1(t)]. The dashedlines represent the case of completely ionized ions, while the solid linesinclude the effect of photoionization.

Some comments are in order. The most relevant consequenceof the ionization mechanism concerns the shape of the CR spec-trum in the knee region. As we have already discussed, the relationEmax,N ∝ ZN is needed in order to fit the data. Even a small deviationof the cut-off energy from the direct proportionality can affect theslope. Let us assume that the measured slope, s = 3.1, indeed re-sults from this proportionality relation. We have Emax,Fe = 26 Emax,p.Now if the maximum energy of iron decreases to 80 per cent of itsmaximum value, the slope changes to ∼3.33, while for a decreaseof 50 per cent, the slope becomes ∼3.94.

According to our results, we can say that in the context of theSNR paradigm, the primary sources of CRs above the knee energyare most probably the core-collapse SNe with Mej � 1 M�. Type IaSNe seem to be unable to accelerate ions up to an energy ZN timesthe proton energy, due to their small Sedov–Taylor age. This consid-eration is strengthened by the fact that the CR flux observed on theEarth is mostly due to the SNRs located in the solar neighbourhood,rather than those located in the bulge. In fact the escaping lengthfrom the Galaxy is determined by the thickness of the Galactic halo,which is ∼3–5 kpc, while the Galactic bulge is at 8 kpc from us.

A second comment concerns the acceleration of elements beyondthe Fe group. Even if the contribution of such ultraheavy elementsto the CR spectrum is totally negligible at low energies, at higherenergies, it can hardly be measured. In principle, the contributionof ultraheavy elements could be significant in the 100-PeV regime,which is the energy region where the transition between Galacticand extragalactic CRs occurs. Indeed several authors pointed outthat a new component is needed to fit this transition region, be-yond the elements up to Fe accelerated in ‘standard’ SNRs (seee.g. the discussion in Caprioli, Blasi & Amato 2011). As inferredby Horandel (2003), stable elements heavier than Fe can signifi-cantly contribute to the CR spectrum in the 100-PeV regime if oneassumes that their maximum energy scales like ZN . This assump-tion is especially appealing also for a second reason: in principle,it could explain the presence of the second knee in the CR spec-trum (Horandel 2003); in fact, the ratio Esecond knee/Eknee � 90 isvery close to the nuclear charge of the last stable nucleus, uranium,which should have Emax,U = 92Emax,p = 414 PeV. Conversely, theionization mechanism discussed here provides a strong constrainton the role of ultraheavy elements. If the acceleration of elementsheavier than Fe occurs at SNRs like those considered above, thenit is easy to show that they cannot achieve energies much largerthan Fe itself, because the total ionization time increases rapidlywith the nuclear charge. A contribution from ultraheavy elementsmay be possible only if they are accelerated in very massive SNRs,with Mej � 1 M�. A second possibility that would be interestingto investigate is the acceleration during the very initial stage of theremnant, when the expansion occurs into the progenitor’s wind. Inthis case, the high density of the wind could strongly reduce theionization time, thanks to the Coulomb collisions, and nuclei couldbe totally stripped in a short time.

4 SO L U T I O N O F TH E T R A N S P O RT EQUATI O N

In this section, we present the solution of the stationary transportequation for ions and electrons accelerated at shocks when theionization process is taken into account. We first get the solutionfor the distribution function of ions, which can be used, in turn, asthe source term responsible for the injection of electrons. We willconsider only the test-particle solution, neglecting all the non-lineareffects coming from the back reaction of the accelerated particleson to the shock dynamics.



2340 G. Morlino

4.1 Spectrum of ions

In order to compute the distribution function of a single chemicalspecies, we consider separately each ionization state. Let f Z

N (x, p)be the distribution function of the species N with effective chargeZ. The equation that describes the diffusive transport of ions in onedimension, including the ionization process, reads

u∂f Z

N

∂x= D(p)

∂2f ZN

∂x2+ 1

3

du

dxp

∂f ZN

∂p− f Z

N

τZN (p)

+ QZN (x, p) ,

(15)

where u(x) is the fluid velocity and D(p) is the diffusion coefficient,which is a function of Z, and it is assumed constant in space. τ Z

N(p)is the ionization time for the losses of one electron, namely, for theprocess NZ → NZ+1 + e−. We neglect multiple ionization processes.In order to solve equation (15), we need to specify the injection termQZ

N(x, p). We assume that ions with the lower degree of ionization,Z0, are injected only at the shock position (x = 0) and at a fixedmomentum pinj. Hence, the source term for f

Z0N is

QZ0N (x, p) = K

Z0N δ(x) δ(p − pinj) . (16)

The normalization constant KZ0N is determined from the total num-

ber of particles injected per unit time. For the subsequent ionizationstates, Z > Z0, the injection comes from the ionization of atomswith charge equal to Z − 1, that is,

QZ+1N (x, p) = f Z

N (x, p)/τZN (p) . (17)

Equation (15) can be solved separately in the upstream and down-stream regions, where the term du/dx vanishes and the equation re-duces to an ordinary differential equation of second order. We labelthe quantities in the upstream (downstream) with a subscript 1 (2)and we adopt the convention x < (>) 0 in the upstream (down-stream). We fix the boundary conditions at the shock position, suchthat f (x = 0, p) ≡ f 0(p), where f 0(p) has to be determined. Theboundary condition at infinity is ∂f /∂x = 0 for x → ±∞. For thesake of simplicity, we now drop the label characterizing the ionspecies, such that f Z

N ≡ f , and we write the solution in a compactform as follows:

fi(x, p) = f0(p)e∓λi∓x ±∫ ±∞

x

Qi(x ′, p)

ui

√1 + �i

e±λi±(x−x′)dx ′

±∫ x

0

Qi(x ′, p)

ui

√1 + �i

e∓λi∓(x−x′)dx ′

∓ e−λi∓x

∫ ±∞

0

Qi(x ′, p)

ui

√1 + �i

e−λi±x′dx ′ . (18)

The upper and lower signs refer to the downstream (i = 2) andupstream (i = 1) solutions, respectively. The dimensionless function�(p) is the ratio between the diffusion time and the ionization time,that is, �i(p) ≡ 4Di (p)

u2i τ (p)

. λi+ and λi− are the inverse of the propagation

lengths along the counter-streaming and the streaming directionswith respect to the fluid motion, respectively, that is,

λ−1i± (p) = 2Di(p)

ui[√

1 + �i(p) ± 1]. (19)

The quantity λ takes into account the diffusion, the advection andthe ionization processes. One can see that in the limit where theionization is negligible, namely, when τ → ∞, the streaming prop-agation length diverges, while the counter-streaming one reduces toD/u.

The three integral terms which contribute to the distribution func-tion in equation (18) have a clear physical interpretation. Let usconsider the solution in the upstream: in this case, the first inte-gral represents the contribution due to particles advected with the

fluid from −∞, the second accounts for the particles diffusing inthe counter-streaming direction, while the last integral takes intoaccount the flux escaping across the shock surface. For the down-stream solution, the meaning of the first and second integrals isreversed, while the third one keeps the same meaning. This inter-pretation becomes clear if one takes the limit τ → ∞.

From equation (18), we can easily recover the standard solutionwhere the ionization is not taken into account, assuming that theinjection occurs only at the shock position and using �i = 0: inthis case, the integral terms vanish and we find f 1(x, p) = f 0(p)exp[−u1x/D1] and f 2(x, p) = f 0(p).

In order to get the complete solution, we need to compute theboundary distribution function at the shock position, f 0(p). We fol-low a standard technique, which consists in integrating equation (15)around the shock discontinuity (Blasi 2002). We got the followingdifferential equation for f 0(p):(

u1 − u2

3

)p

∂f0

∂p=

(D2

∂f2

∂x

)0+

+(

D1∂f1

∂x

)0−

, (20)

where 0+ and 0− indicate the positions immediately downstreamand upstream of the shock. The quantities D ∂f /∂x can be easilycomputed deriving f with respect to x from equation (18). We get(

D2∂f2

∂x

)0+

= f0u2

2

(1 +

√1 + �2

)+

∫ ∞

0Q2e−λ2+x′

dx ′

(21)

and(D1

∂f1

∂x

)0−

= f0u1

2

(1 −

√1 + �1

)−

∫ 0

−∞Q1eλ1−x′

dx ′ .(22)

After a little algebra, the solution of equation (20) can be expressedin the following form:

f0(p) = s p−s

∫ p

pinj

dp′

p′ p′s G(p′)u1

exp

[−s

∫ p

p′

dp′′

p′′ h(p′′)]

,

(23)

where the usual definition of the power-law index is s = 3u1/(u1 −u2). The injection term G is the sum of the upstream and downstreamcontributions:

G(p) =∫ 0

−∞Q1eλ1−x′

dx ′ +∫ ∞

0Q2e−λ2+x′

dx ′ (24)

while the function h in the exponential is

h(p) = 1

2

[√1 + �1(p) − 1

]+ 1

2r

[√1 + �2(p) − 1

].

(25)

The solution in equation (23) shows the typical power-law behaviour∝ p−s in the region of momentum where neither injection nor ioniza-tion is important; in fact, when �(p) � 1, h(p) vanishes. Conversely,when the ionization length becomes comparable with the diffusionlength, that is, �(p) � 1, the function h(p) produces an exponentialcut-off in the distribution function.

Now that we have the formal solution, we can show some re-sults. Let us start considering the acceleration of He. We assumethat the initial ionization state at the moment of injection is He+.The distribution function fHe+ is computed with the procedure de-scribed above, using the injection term in equation (16). We fixpinj = 10−3 mpc and the shock velocity equal to 3000 km s−1. Thephotoionization time can be an arbitrary function of p. We use theapproximate expression given in equation (5). Equations (18) and(23) can be integrated numerically. Once fHe+ is known, we cancompute fHe++ using the injection term in equation (17), that is,




He total

He He

e 104

3 2 1 0 1 2 3 4

0.10

1.00

0.50

0.20

0.30

0.15

0.70

Log p mpc

p4f 0

p

Figure 5. Distribution function of He at the shock position, f He,0(p). Thethin solid lines represent the distributions of He+ and He++, while the thickline is the sum of both the contributions. The dotted lines represent thedistribution of electrons (×104) ejected in the process He+ → He++ + e−and subsequently accelerated.

e total 104

C

C C3 C4 C5

C6

C total

2 0 2 4 6

0.10

1.00

0.50

0.20

0.30

0.15

0.70

Log p mpc

p4f 0

p

Figure 6. Distribution function of C in all the ionization states from C+up to C6+. The thick solid line is the sum of all contributions. The spectraof electrons ejected from each ionization process are plotted with the thindashed lines, while the thick dashed line represents the total sum (×104).

with QHe++ = fHe+/τHe+ . The resulting distributions are plotted inFig. 5 with the thin solid lines. We note that fHe+ ∝ p−4 up to p �10mpc; above such value, the distribution of He+ drops exponen-tially, while the distribution of He++ starts rising. It is worth notingthat the total distribution (plotted with the thick solid line) is slightlysteeper than p−4 in the transition region. This effect is due to the dif-ferent probability which He+ and He++ have to return at the shock,once they are downstream. In fact, an ion with charge Z located atthe position x in the downstream has a probability to return at theshock ∝ e−u2x/D(Z); after the ionization, the charge becomes Z + 1and the return probability is reduced by a factor of e −(Z+1)/Z . Thistranslates into a net loss of particles from downstream. This effectof losses is more pronounced when considering heavier elements.In Fig. 6, we show the case of carbon atoms, which start the accel-eration as singly ionized. The distribution functions of all ionizedstates are plotted with the thin solid lines, from C+ up to C6+. In theenergy region where ionization occurs, 1.5 < γ < 4.5, the slope ofthe total distribution function is s ∼ 4.1.

4.2 Spectrum of electrons

Now that we know how to compute the distribution function of ions,the solution for the electron distribution is straightforward. We haveto solve the same transport equation (15) by writing down the rightexpression for the electron injection, Qe, and dropping the term dueto ionization. In the present calculation, we neglect energy losses.

Let us consider a single species N. From each photoionizationprocess of the type NZ → NZ+1 + e−, we have the followingcontribution to the electron injection:

QN,Ze (x, p) =

∫ ∞

p

dp′cf z

N (x, p′)τ zN (p′)

δ(3)(p − ξNp′)

= ξ−3N

f zN,1(x, p/ξN )

τ zN (p/ξN )

. (26)

The δ-Dirac function in momentum is present because electronsare ejected with the same Lorentz factor of the parent ions, as wehave discussed in Section 2. Hence, the electron momentum is pe =pNme/mN ≡ pN ξN . Following the procedure used in the previoussection, we can write the distribution function for electrons at theshock position as follows:

f N,Ze,0 (p) = sp−sξ s−3

N

∫ p/ξN

pinj

dy

yys GN,z

e (y)

u1. (27)

Equation (27) is similar to equation (23) with the exception ofthe multiplying factor ξ s−3

N and absence of the exponential cut-off.We note that the contribution of injection from the downstreamis negligible with respect to the upstream. This occurs becauseelectrons produced in the downstream have a negligible probabilityto come back to the shock. The return probability for one electronstripped downstream at a position x is a factor of e−mN /ZN me smallerthan the return probability of the parent ion. Hence, in the injectionterm, we consider only the contribution from upstream, which reads

GN,Ze (p) =

∫ 0

−∞QN,Z

e (x ′, p) dx ′. (28)

In Figs 5 and 6, besides the distribution of ions, we plot the distri-butions of the electrons due to each single ionization step (dashedlines). Comparing equations (23) and (27) we see that the ratiobetween electron and ion distribution functions is

KeN ≡ f N,Ze,0 (p)/f Z

N,0(p) = ξNIN,Z(p) . (29)

The function IN,Z(p) represents the ratio between the injection in-tegrals in equations (23) and (27). It can be computed numericallyand in the asymptotic limit p � p∗ [where p∗ is such that �(p∗) =1], it is IN,Z ∼ Z/(2Z − 1). This result is not surprising: it reflects thefact that for each ionization event, electrons can be injected only ifthey are released upstream; hence, their number is roughly one-halfof the injected ions.

Now we are interested in the total number of accelerated electronsthat can be produced via the ionization process. In the literature, thenumber of electrons is usually compared to that of accelerated pro-tons: DSA operates in the same way for both kinds of particles;hence, a proportionality relation between their distribution func-tions is usually assumed, that is, f e(p) = Kepf p(p). From the ob-servational point of view, the value of Kep in the source stronglydepends on the assumption for the magnetic field strength in the re-gion where electrons radiate and, in the context of the DSA theory,it can be determined for those SNRs where both non-thermal X-rayand TeV radiation are observed. Two possible scenarios have beenproposed. In the first one, electrons produce both the X-ray andthe TeV components via the synchrotron emission and the inverse



2342 G. Morlino

3.7 3.8 3.9 4.0 4.1 4.2

0.1

0.2

0.5

1.0

2.0

5.0

10.0

s

Kep

104

Figure 7. Value of Kep computed from equation (30) as a function of theslope s and for δ = 0.6 (solid line) and δ = 0.3 (dashed line).

Compton effect, respectively; this scenario requires a downstreammagnetic field of around 20 μG and Kep ∼ 10−2–10−3. The secondscenario assumes that the number of accelerated protons is largeenough to explain the TeV emission as due to the decay of neutralpions produced in hadronic collisions. In this case, the DSA requiresa magnetic field strength of few hundreds μG and Kep ∼ 10−4–10−5.Such a large magnetic field is consistently predicted by the theoryas a result of the magnetic amplification mechanisms which operatewhen a strong CR current is present. In Table 2, we report someexamples of young SNRs where the non-linear theory of accelera-tion has been applied to fit the multiwavelength spectrum. The forthand fifth columns report the estimated values of the downstreammagnetic field strength and the corresponding electron/proton ratio.

In order to give an approximated estimate for Kep, we need tomultiply equation (29) by the total number of ejected electrons,that is, ZN − ZN,0, and sum over all atomic species present in theaccelerator, that is, Kep � ∑

N KNp (ZN − ZN0) KeN . KNp are theabundances of ions measured at the source in the range of energywhere the ionization occurs. Even if the values of KNp are widelyunknown, we can estimate them using the abundances measured onthe Earth and adding a correction factor to compensate for propaga-tion effects, namely, the fact that particles with different Z diffuse ina different way. The diffusion time in the Galaxy is usually assumedto be τ diff ∝ (p/Z)−δ , with δ ≈ 0.3–0.6 (see Blasi 2007 for a reviewon recent CR experiments). If KNp,0 is the ion-to-proton ratio mea-sured on the Earth, then the same quantity measured at the source isKNp = KNp,0Z−δ

N . Hence, the final expression for the electron-to-proton ratio at the source is

Kep =∑

N

KNp,0 Z−δN (ZN − ZN,0)

ZN

2ZN − 1

(me

mN

)s−3

. (30)

In Fig. 7, we report the value of Kep computed according to equa-tion (30) as a function of the spectral slope s, and for δ = 0.3 and0.6. For KNp,0 we use the abundances of nuclei in the CR spec-trum measured at 1 TeV (Wiebel-Sooth, Biermann & Meyer 1998).Moreover, we determine the initial charge of different species, ZN,0,adopting the thermal equilibrium in a plasma with a temperatureT ∼ 105. Remarkably, using the slope predicted by linear theoryfor strong shock, s = 4, we have Kep ∼ 10−4: this number givesthe right order of magnitude required to explain the values of Kep

reported in Table 2. We stress here that in order to provide a bet-ter estimate of Kep for a specific SNR, we should know the localchemical composition and have a realistic model for the injectionof heavy elements into the DSA, which can provide the right valueof ZN,0.

The result in equation (30) has been obtained using the test-particle approach. A better estimate requires the use of non-lineartheory which is, indeed, much more complicated to develop. Nev-ertheless, it is easy to realize that the main effect that non-linearitycan produce on Kep is due to the different prediction of the slopes. In fact, while the test-particle approach gives the universal slopes = 4, the non-linear theory predicts concave spectra with s =s(p). The basic formulation of the non-linear theory usually leadsto s > 4 for momenta lower that ∼10Z−1 GeV/c and s < 4 forlarger momenta (Amato & Blasi 2005). As discussed in Section 2,ionization typically occurs for γ � 10, where the slope is s < 4.Hence, we can conclude that the non-linear effects tend to increasethe value of Kep. For instance, from Fig. 7, we see that for s = 3.7,Kep rises up to ∼10−3.

It is worth noting that non-linear effects can also result in theopposite situation with Kep < 10−4, as we will show below. A keyingredient of the DSA is the speed of the magnetic turbulence, whichis responsible for particle scattering. The speed of the scatteringcentres contributes to the determination of the effective compressionratio felt by particles which, in turns, determines the spectral slope.The typical speed of the turbulence is of the order of Alfven speedwhich, when computed in the background magnetic field, is of theorder of a few tens km s−1 and it is negligible with respect to theshock speed. Conversely, if one assumes that turbulence moveswith the Alfven speed computed in the amplified magnetic field,as shown by Caprioli et al. (2011), it cannot be neglected and theresulting slope can be >4. If this happens, then the value of Kep

drops and the number of electrons produced via ionization may beinsufficient to explain the synchrotron emission.

A final comment concerns the mechanism of electron injection incore-collapse SNe. The value of Kep shown in Fig. 7 has been com-puted using the ionization level of chemical elements in a plasmawith T ∼ 105. We recall that Type Ia SNe expand in the warm ISM,where the typical temperature is around 104 K; hence, we expectthe injection of electrons to be relevant. Conversely, core-collapseSNe expand into a diluted bubble whose temperature reaches 106 K(Higdon & Lingenfelter 2005); hence, the degree of ionization ofatoms is higher and the number of electrons available for the injec-tion is smaller (see e.g. Porquet et al. 2001). Using T = 106, weestimate that Kep is a factor of ∼4 smaller than the values shown inFig. 7. Conversely, in those bubbles, the metallicity is estimated tobe greater than the ISM mean value (Higdon & Lingenfelter 2005).Hence, even for core-collapse SNe, the electron injection throughionization could play a relevant role.

5 D I SCUSSI ON AND C ONCLUSI ONS

In this work, we include for the first time the ionization process ofheavy ions in the theory of DSA at SNR shocks. We showed that,for the typical environments where SNRs propagate, the photoion-ization due to the ISRF dominates the collisional ionization andproduces two important effects: (1) the reduction in the maximumenergy achieved by ions; and (2) the production of a relativisticpopulation of electrons.

(1) The first effect is especially important for the interpretation ofthe knee structure in the all-particle spectrum of CRs. The changein slope on the two sides of the knee is generally interpreted as dueto the superposition of spectra of chemicals with different nuclearcharges combined with their abundances and convolved with therigidity-dependent diffusion in the Galaxy. An important require-ment for a good fit to the knee is that the cut-off energy of each




species, from H up to Fe, should be proportional to the nuclearcharge, ZN .

It is generally assumed that DSA can accelerate ions up to a max-imum energy proportional to ZN . This assumption is valid only if theionization time needed to strip all electrons from atomic orbitals,τ tot

ion, is much shorter than the acceleration time. We assume that themaximum possible acceleration time is equal to the Sedov–Taylortime, tST, corresponding to the end of the free-expansion phase.Using a simple steady-state approach, we showed that, especiallyfor very massive ions, τ tot

ion can be comparable to or even largerthan tST, depending on the type of the remnant and on its locationin the Galaxy. In fact, the ISRF responsible for the photoioniza-tion decreases for increasing distance from the Galactic Centre. Weanalyse two different cases, which are representative of a Type IaSN and a core-collapse SN. Our main results are as follows. Type IaSNe are generally unable to accelerate ions according to Emax,N ∝ZN , because tST does not exceed the typical value of 500 yr. The onlyexception is the remnants located in the Galactic bulge where thephoton field is strong enough to reduce the photoionization time toa value much shorter than tST. The situation for core-collapse SNeis reversed because their typical Sedov–Taylor time is �2000 yr;hence, the ionization time for chemicals up to Fe can be neglectedand the maximum energy is indeed proportional to the nuclearcharge, with the exception of those remnants located very far awayfrom the Galactic Centre where the ISRF is very low.

Previous conclusions are based on the assumption that duringthe free-expansion phase both the magnetic field and the shockvelocity remain constant. This is indeed a poor approximation. Infact during the free-expansion phase, the shock speed can vary by afactor of a few (depending on the velocity profile of the ejecta andon the density profile of the external medium). Now for DSA, theacceleration rate is proportional to u2

sh δB(ush), where δB(ush) is theturbulent magnetic field generated by the CR-induced instabilities(resonant or non-resonant), which is an increasing function of theshock velocity. Hence, the acceleration rate can significantly changeduring the free-expansion phase even if ush varies only by a factorof a few. Always in the framework of steady-state approximation,we develop a simple toy model able to include the magnetic fieldamplification and the time-evolution of the remnant. Using thistoy model, we showed that the magnetic amplification and time-evolution can significantly reduce the acceleration time with respectto tST and, as a consequence, the maximum energy achieved byheavy ions becomes smaller than the previous estimate: the relationEmax,N ∝ ZN can be achieved up to Fe nuclei only by SNRs witha Sedov–Taylor time �103 yr, which means that the mass ejectashould be �1 M�.

Clearly the same mechanism also applies to ions heavier them Fe.Their maximum energy cannot be much larger than than achievedby Fe itself, unless the acceleration occurs in very massive SNRs.Hence, the ionization mechanism puts severe limitations on the pos-sibility that ultra heavier ions can contribute to the CR spectrum inthe transition region between galactic and extragalactic component.

(2) The second effect concerning the production of relativisticelectrons was already put forward in Morlino (2009). The ioniza-tion mechanism provides a source for the injection of relativisticelectrons into the DSA mechanism. We investigate the possibilitythat those electrons can be responsible for the synchrotron radiationobserved from young SNRs. In order to estimate the total number ofinjected electrons, we use a semi-analytical technique to solve thesteady-state transport equation, which describes the electron dis-tribution function in the shock region. Summing the contributionscoming from the ionization of all chemicals, we showed that the

ratio between accelerated electrons and protons is Kep ∼ 10−4. Thisvalue is especially appealing, because it corresponds to the rightorder of magnitude required in order to explain the synchrotronemission, if the magnetic field is amplified up to a few hundredsμG. Interestingly, such magnetic field amplification is consistentlypredicted by non-linear acceleration theory as a consequence of theback reaction of accelerated particles on to the shock dynamics.

Several effects can modify our prediction for the number of elec-trons. The estimate Kep ∼ 10−4 is obtained using test-particle ap-proximation and adopting the abundances of CR spectra observed onthe Earth, corrected for the effect of propagation in the Galaxy. Evenif we did not develop a fully non-linear approach, we showed hownon-linear effects can substantially modify the electron-to-protonratio. Indeed the value of Kep strongly depends on the spectral slopeas one can see from equation (30). Non-linear DSA usually pre-dicts steeper spectra than the test-particle result, s = 4, and thistranslates into an enhancement of Kep. Conversely, the opposite sit-uation can also be realized. Indeed some authors pointed out that,if the magnetic field amplification occurs, the velocity of the mag-netic turbulence with respect to the plasma can be much larger thanthe Alfven speed corresponding to the background magnetic field.If this occurs, then the spectral slope of the accelerated particleswill be appreciably softer than 4 and the electron-to-proton ratiowould drop to a value which is insufficient to explain the observedsynchrotron radiation.

In order to get a correct determination of Kep for a single SNR,other effects should be taken into account: (i) the chemical compo-sition of the environment where the SNR expands; (ii) the initialionization state of each chemical; and (iii) a realistic model for theions injection. A further complication directly related to the pre-vious points is the fact that many heavy elements are condensedin solid dust grains both in the ISM and in the stellar wind. It hasbeen pointed out that the injection of refractory elements into theDSA can be dominated by the sputtering of these dust grains ratherthan by the injection of single atoms from the thermal bath (Ellisonet al. 1997). In this picture, a correct computation of the number ofinjected ions and their initial ionization state is very challenging.

As a final remark, we want to stress that the electron-to-protonratio in a single source, what we call Kep, is different from thesame ratio measured from the CR spectrum on the Earth, whichis 5 × 10−3 at 1 TeV (Blasi 2007). Sometimes these quantitiesare assumed to be the same. Conversely, they could be different,because the latter is the sum of the contribution coming from allsources integrated during the source age and also reflects transport tothe Earth and losses in transport. Especially relevant is the fact thatother sources, like pulsar wind nebulae, can significantly contributeto the electron flux measured on the Earth, but not to the proton flux.Moreover, the value of Kep in a single source can vary during thesource age. Different mechanisms of electron injection, other thanthe ionization, can be relevant in different phases of the remnant.

AC K N OW L E D G M E N T S

I am grateful to P. Blasi, E. Amato and D. Caprioli for useful andcontinuous discussions and exciting collaboration. I also wish tothank the Kavli Institute for Theoretical Physics in Santa Barbarawhere part of this work was done during the programme ‘ParticleAcceleration in Astrophysical Plasmas’, 2009 July 26–October 3.This research was funded through the contract ASI-INAF I/088/06/0(grant TH-037).



2344 G. Morlino

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the role of ionization in the shock acceleration theory

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