2.13.2 Pion Production

Home , Charge (physics), Gamma ray, Photon, Pion, Proton (rocket family), Relativistic particle

2.13. INTERACTIONS OF HIGH-ENERGY PROTONS 51

From this expression it is clear that particles accelerated in a similar way emit comparable power if their gamma factors are comparable. Thus, if we compare the energy loss of electrons and protons of the same energy, the energy loss of protons will be suppressed, because their gamma factor is by a 3 factor of mp/me ∼ 2 × 10 lower. This applies for all the main classical radiative losses, including curvature, synchrotron, inverse Compton and Bremsstrahlung radiation.

2.13.2 Pion production Energy losses for protons are more important for the quantum processes of production of new particles in interactions with other protons or with photons. The most astrophysically important example is production and decays of pions, two-quark particles with masses in the mπ ≃ 100 MeV range. The threshold for the p + γ → p + π reaction could be found from the kinematics considera- tions. Introducing the proton and photon four- momena and transforming to the center-of-mass ˜ ˜ ~ ˜ ~ frame where P p =(EP , p˜p) and P γ = (˜ǫ, p˜p), we could write an expression for the absolute value of the total four-momentum before and after the reaction p + γ → p(n)+ π. This gives

2 2 P p + P γ = mp + 2mpγǫ(1 − v cos θ) (2.169) before the reaction (in the lab frame) and Figure 2.24: Cross-section of pion production in pγ collisions. 2 ˜ ˜ 2 2 P p + P π = mp + 2mpmπ + mπ (2.170) after the reaction (the expression is in the center-of mass frame and both proton and pion are at rest after the reaction). Equating the two expressions for the conserved quantity we ﬁnd the condition for the possibility of the pion production Ep >Ep,thr,

−1 mpmπ(1 + mπ/(2mp)) 17 ǫ Ep,thr = ≃ 10 eV (2.171) 2ǫ h1 eVi The proton looses a small fraction of its energy in each pion production event. Indeed, consider the pion production near the threshold. In the reference system comoving with the proton, both the proton and the newly produced pion are almost at rest. Their energies are mp and mπ, respectively. Transforming to the lab frame, both the proton and the election energy is boosted by the proton gamma factor, to mpγ and mπγ, respectively. This means that the pion carries away only a fraction −1 κ = mπ/mp ≃ 10 of the proton energy. This means that proton looses a signiﬁcant fraction of its energy in several collisions. The cross-section of this process is determined by the physics of strong interactions. Close to the −28 −28 threshold is is as large as σpγ ≃ 6 × 10 cm and drops to ≃ 10 cm much above the threshold, see Fig. 2.24.

Example: GZK cutoﬀ in the spectrum of cosmic rays

2.14. EXCERCISES 53 this gives numerically Ekin = Ep,thr − mp ≃ 280 MeV (2.175) The cross-section of this reaction is also determined by the strong interactions and is about the geometrical cross-section of the proton,

−26 −25 σpp ≃ 4 × 10 cm....10 cm (2.176) depending on (growing with) the proton energy, see Fig. 2.26. As an example,we could consider propaga- tion of cosmic ray protons in the interstellar medium. Typical density of the interstellar −3 medium around us is nISM ∼ 1 cm . The mean free path of the proton is 1 λpp ≃ ≃ 3 − 10 Mpc (2.177) σppnISM Inelasticity of the reaction in the case of proton- proton collisions is quite high, κ ≃ 0.5, so that single collision takes away a sizeable fraction of the proton energy. The cooling time due to the pion production process is about (1−3)×108 yr. This is somewhat longer than the residence time of cosmic rays in the Galaxy, but still, a fraction of cosmic rays interacts in the Galactic Disk before escaping from it. Neutral and charged pions π0, π± are unstable particles which decay into γ-rays, π0 → 2γ, ± ± neutrinos and muons π → µ + νµ. Muons, in Figure 2.26: Cross-section of proton-proton colli- turn , are also unstable and decay into electrons sions. ± ± and neutrinos µ → e + νe + νµ. Thus, pion production in pp collisions results in production of γ-rays, neutrinos and high-energy electron / positrons. Gamma-ray emission induced by interactions of cosmic rays with the protons from the interstellar medium is the main source of high-energy γ-rays from the Milky Way galaxy, see Fig. 2.27. The spectrum of the pion decay emission has a characteristic ”bump” at the energy Eγ = mπ/2 = 67.5 MeV (the energy carried by each photon from the neutral pion decay at rest). This bump serves as the identiﬁcation feature for the pion decay component of the γ-ray spectra of astronomical sources, including the γ-ray emission from the interstellar medium.

2.14 Excercises

Excercise 2.1. Use the expression for the intensity of radiation from a relativistically moving charge (2.16) for an order-of-magnitude comparison of contributions due to the ﬁrst and second term for an ultra-relativistic particle moving with veolcity v = 1 − ǫ, ǫ ≪ 1, experiencing acceleration on a time scale t = 1/ω0. Which term dominates: the one proportional to a||, or the one proportional to a⊥?

Excercise 2.2. By analogy with the astrophysical example of curvature radiation from pulsar magnetosphere, consider the possibility of curvature radiation from a magnetosphere of a black hole in 9 an Active Galactic Nucleus (AGN) of M87 galaxy. The black hole mass is M ∼ 4 × 10 M⊙ and the 54 CHAPTER 2. RADIATIVE PROCESSES

Figure 2.27: Fermi/LAT map of the sky in the energy range above 1 GeV. Diﬀuse emission from the Galactic Plane is dominated by the signal from interactions of cosmic ray protons with the interstellar medium with production of pions.

characteristic distance scale in the source is the gravitational radius Rg (see. Excercise 1.4). γ-rays with energies up to 10 TeV are detected from M87 (see Fig. 2.28). Using this information ﬁnd an upper bound on energies of electrons and protons present in the source. Find a characteristic curvature radiation energy loss time scale tcurv = E/Icurv for electrons and protons of energy E. Is it shorter or longer than the ”light-crossing” time scale tlc = Rg/c?

Excercise 2.3. Find steady-state spectrum of high-energy particles injected at a constant rate with −Γinj a powerlaw spectrum Qe(E) ∼ E and cooling due to a radiative energy loss with the energy loss rate which scales as powerlaw of particle energy: E˙ ∼ Eγcool .

Excercise 2.4. ”Landau levels”. In quantum mechanical settings, ﬁnd the spectrum of energy levels for a particle of charge e moving in a plane perpendicular to the direction of a homogeneous magnetic ﬁeld with strength B.

Excercise 2.5. Find the slope of the powerlaw spectrum of high-energy electrons in Crab nebula, by measuring the slope of its emission spectrum in 10−108 eV photon energy range (Fig. 2.6).

Excercise 2.6. Estimate the mass of objects powering active galactic nuclei assuming that these sources with typical 45 luminosity 10 erg/s radiate at the Eddington luminosity Figure 2.28: Spectrum of γ-ray emission from vicinity of supermassive black hole in M87 galaxy, from Ref. [9]. 2.14. EXCERCISES 55 limit.

Excercise 2.7. Estimate angular resolution of Compton telescope based on given precision of measurement of energy deposits ∆E by the recoil electron and at the absorption point of the scattered photon.

Excercise 2.7. Calculate the steady state spectrum of electrons injected with the powerlaw spectrum dN/dE ∝ E−Γinj and cooling under the inﬂuence of inverse Compton emission in Thomson regime.

Excercise 2.9. Estimate the energy density of synchrotron radiation inside the Crab Nebula, taking into account that its distance is DCrab = 2 kpc, and its emission spectrum is that shown in Fig. 2.6. The size of the Nebula is about 1 pc. Estimate which low energy photon background provides dominant low-energy photon population for inverse Compton scattering: CMB, interstellar radiation ﬁeld or the synchrotron radiation from the Nebula itself?

Excercise 2.10. Photons with energies above MeV are not able penetrate through the Earth atmosphere because of the Bethe-Heitler pair production process. Using a numerical estimate for the pair production cross-section, estimate the altitude at which high-energy photons incident vertically on the atmosphere are converted into electron positron pairs. Assume that the atmosphere has an exponential density proﬁle ρ ≃ ρ0 exp(−H/H0) with the scale height H0 = 7 km and the density at −3 3 the sea level ρ0 ≃ 10 g/cm .

Excercise 2.11. Mergers of neutron stars which give rise to gravitational wave bursts also result in injection of large amounts of mechanical energy into a very compact region of the size R ∼ 30−100 km. This energy heats the material of the merged stars to the temperatures about T ∼ 10 MeV. Could we see the black body electromagnetic emission from this hot blob of material? Estimate the optical depth of the black body photon ﬁeld with respect to gamma-gamma pair production. 56 CHAPTER 2. RADIATIVE PROCESSES Chapter 3

Particle acceleration mechanisms

Up to now we have considered the mechanisms of interactions of high-energy particles in astronomical sources, on their way from the source to the Earth and ﬁnally in the telescopes detecting the particles and high-energy photons here on Earth. We have never asked the question ”How do these high-energy particles appear in the source at the ﬁrst place?”. In this chapter we will consider possible mechanisms of acceleration of charged particles in astronomical environments and try to estimate the limitations of potential ”cosmic particle accelerators” in terms of maximal attainable energies and power. To start with, it is useful to understand what methods do we use to produce high-energy particles here on Earth and what are the limitations of the laboratory setups in terms of maximal attainable energies and power.

3.0.1 The problem of injection of low energy particles The history of accelerator physics started roughly a century ago and the first accelerator machines were the Cockroft- Wolton and van den Graaf generators of high DC voltage. The voltage of these machines has reached U ∼ 100 kV and up to MV, enough to boost electrons to relativistic velocities. Particles with charge are accelerated by the electric field E~ = dU/dx. Having crossed the full potential drop they should gain the energy E = eU. The main obstacle for efficient particle acceleration from Figure 3.1: Time dependence of electron the non-relativistic low energy states is a severe Coulomb and proton fluxes from solar flares. energy loss counteracting the accelerating force. The ”friction” force of Coulomb collisions counteracting the accelerating force could be expressed as

∆p F = = mvν (3.1) Coul ∆t coll where we have introduced the collision rate νcoll and assumed that each collision signiﬁcantly modiﬁes particle velocity.

57 58 CHAPTER 3. PARTICLE ACCELERATION MECHANISMS

From Eqs. (2.176), (2.177) (see also Fig. 2.21) we recall that the ionisation energy loss scales with particle velocity as dE/dx =∼ 1/v2 and it is much higher for the lower velocity particles. Let us consider a particle propagating through a medium of the density n. From the Bethe-Bloch formula (2.177) we ﬁnd that the energy loss rate is

4πnZ2e4 2γ2mv2 −dE ∆(mv2/2) ln − v2 = = − ∼ ν mv (3.2) mv2 I dx v∆t coll

3 The collision rate νcoll = 1/∆t is therefore proportional to 1/v . Substituting the expression for the collision rate to the friction force and equating it to the acceleration force F~ = eE~ ,

eE = νcollmv (3.3) we get an expression for the minimal particle velocity needed for the acceleration to be eﬃcient:

4πnZe3 lnΛ 1/2 v = (3.4) min Em

Particles with v>vmin are efficiently accelerated by the electric field and could finally reach the maximal energies E ∼ eU after crossing the full potential drop, see Fig. 3.2. To the contrary, particles with low velocities vleads to the reduction of the critical velocity and facilitates the injection of low energy particles for further acceleration. Similar problem is typically encountered in the case of acceleration of particles in astrophysical environments. Successful acceleration is possible only if the acceleration time scale tacc is shorter than −1 the collision time scale νcoll.

3.0.2 The problem of discharge in linear accelerators Creating a low-density environment in the acceleration volume is beneficial because it reduces the minimal energy above which particles are efficiently accelerated, thus re- moving the injection problem. At the same time, it creates another problem, known as the problem of discharge or ”vacuum breakdown” in the presence of high voltage. The energy gained by each particle between the subse- quent collisions, on the distance scale ∆x = v/νcoll ∼ 1/n is ∆E ≃ eU∆x/R, where we have introduced the size of the acceleration volume R. Decrease of the density n leads to the increase of ∆x and, as a consequence, to the increase of the typical energies of particle collisions. If the Figure 3.2: Conventional assumption residual medium inside the acceleration volume is initially about injection of low-energy particles not ionised, collisions of the accelerated particles with the into cosmic accelerator: only small atoms with energy transfer exceeding the ionisation energy amount of particles from the high- start to ”kick out” electrons from atoms. The ”kicked out” velocity tail of Maxwellian distribution electrons are also accelerated by the electric field and at the could be efficiently accelerated. next collision they also ionise atoms, further increasing the number of free charged particles As a result, the number of particles in the acceleration volume increases in an exponential way. Charge redistribution in this 59 volume proceeds in the direction of minimisation of the external electric field. This ”discharge” process leads to the ”short circuit” which results in the drop of the voltage and reduction of efficiency of acceleration. To neutralise an external electric field E the medium should be filled with a sufficiently dense distribution of charged particles. A first estimate of the plasma density n needed for the neutralisation of an electric field of the strength E over the volume of the size R is given by the condition that the Coulomb field of the plasma, with the overall charge Q ∼ enR3 should be at least comparable to the external electric field. This gives a condition

enR3 E~ ∼ = enR ∼ E~ (3.5) Coulomb R2 Thus, the minimal plasma density suﬃcient for the neutralisation of the external ﬁeld is

E n = (3.6) crit eR We usually do not have direct information on electric fields in the astrophysical sources. Instead, we have techniques for measurement / estimates of magnetic fields. Contrary to electric fields, magnetic fields are omni-present in astrophysical environments. Electric and magnetic fields together form the tensor of electromagnetic field. As a first approximation for the estimates of possible values of electric fields in astronomical sources one adopts an assumption that the electric field could be ”at most” as strong as the magnetic field (in any case, a variable electric field would generate a measurable magnetic field of comparable strength). Adopting an assumption E . B one could rewrite the Eq. (3.7) as

B B R −1 n>n ∼ ≃ 0.1 cm−3 (3.7) crit eR 1 G1010 cm

The discharge problem prevents the existence of strong large scale electric ﬁelds in astrophysical conditions. Charge redistribution in the cosmic plasma, either already present in the source, or created in result of the avalanche process, would always neutralise the external large scale electric ﬁeld.

3.0.3 Acceleration in small increments: Cyclotron and Synchrotron machines Acceleration of particles by strong large scale electric field is problematic because it is difficult to support this strong field in a rarified medium over a large enough volume. Be- cause of this, the possibilities of high-voltage generators as particle accelerators are limited of several hundred keV energies. To accelerate particles to higher energies, laboratory setups use a different approach, in which particles are systematically returned to the acceleration site, see Fig. 3.3. One of the first examples of this approach is given by the cyclotron machines, Fig. 3.3. In these machines, particle trajectories are affected by both electric and magnetic fields. Particles of fixed energy would move in circles in Figure 3.3: The principle of operation of the magnetic field. However, the trajectories of the par- Cyclotron. ticles systematically pass through a small volume with a potential drop, so that at each drop the energies of particles increase by a small amount ∆E ≪ E. The Larmor radius scales proportionally to the particle energy, RL = E/eB. so that the particles with gradually increasing energy move along an unfolding spiral, as it is shown in the figure. 60 CHAPTER 3. PARTICLE ACCELERATION MECHANISMS

As long as the size of the spiral is smaller than the overall size of the accelerator, particle still returns to the accelerating volume with the potential drop and its energy is increased at each step. However, as soon as the energy reaches the value at which the Larmor radius becomes comparable to the size of the accelerator, R, i.e. R B E = eBR ≃ 3 × 1013 eV (3.8) max 10 km105 G particles escape form the accelerator and no further increase of the particle energy is possible. Synchrotron machines work using the same principle as the cyclotron, with the main difference that the strength of the magnetic field is always adjusted in such a way that particle always moves along a circle, rather than along an unfolding spiral. To achieve this, the magnetic field grows together with the particle energy, B ∼E and the size of the Larmor circle stays always the same. This is convenient because very large sizes of accelerator machines could be reached in this way. Indeed, the cyclotron machines use the whole volume of the size R for their operation. Because of this, their sizes usually do not exceed the sizes of single laboratory rooms. To the contrary, in the synchrotron machines, particles could be put in a narrow ring shaped tube, with the radius of the ring adjusted to the size of the Larmor circle. The largest existing synchrotron type accelerator, the LHC at CERN uses a tube of several kilometres in radius in the form of an underground tunnel. Its linear size is about RLHC ∼ 10 km and the maximal attainable magnetic field in the accelerator ring is B ∼ 10 kG (this is nearly the strongest magnetic field which could be created in laboratory conditions). From Eq. (3.10) we find that the limiting energies attainable in the LHC ring are about 10 TeV. This is indeed the energy scale to which protons and atomic nuclei are boosted by this accelerator machine. If is useful to note that any future development of accelerator particle physics, aimed at achieving still higher energies for particle collisions would involve construction of an accelerator machine of still larger size scale, perhaps up to hindred(s) of kilometres. This important statement could be derived from the simple qualitative Eq. (3.10). Similar acceleration principle certainly operates in astrophysical conditions. As it was discussed in the previous section, the astrophysical particle accelerators could hardly support strong large scale electric fields (except perhaps for a special case of pulsars, particle accelerators near neutron stars). In most of the environments, particle acceleration proceeds in small incremental steps, with particles systematically returning to the acceleration volume. The limiting energies of particles accelerated in astronomical sources of the size R possessing magnetic field B are naturally limited by the condition (3.10). This condition is known in high- energy astrophysics under the name of Hillas condition and is usually visualised in terms of a ”Hillas plot”, shown in Fig. 3.4. The energies of charged particles in the cosmic rays Figure 3.4: The Hillas plot. reach some 1020 eV, some seven orders of magnitude higher than the energies of particles produced at LHC. Accelera- tion to such extremely high energies is possible only in the sources large enough and/or possessing strong enough magnetic field, R B E = eBR ≃ 1021 eV (3.9) max 1 pc1 G The qualitative analysis of astronomical sources classes expressed by the Hillas plot shows that there are not so many possible ”candidate” astronomical sources which could even potentially be considered

62 CHAPTER 3. PARTICLE ACCELERATION MECHANISMS

3.0.5 Fermi acceleration mechanism(s) Similarly to the Sun, most of the astrophysical environments are filled with charged particles and the large scale magnetic fields could not exist in such conditions1. Taking this fact into account, an efficient acceleration mechanism which might operate in cosmic particle accelerators has to be based on assumption of weak or small scale and possibly transient electric fields, which gradually increase the energies of particles in small increments, continuously returning the accelerated particles to the acceleration site, very much the same was as in the cyclotron and synchrotron machines.

Second order Fermi acceleration Such a mechanism was proposed for the first time by E.Fermi in 1949. Fermi considered a hypothetical set of magnetised ”clouds” with typical masses M moving with velocities vM ≪ c and a set of particles of mass m ≪ M propagating between the clouds with velocities vp and Lorentz-factors γp. Particles captured by the cloud turbulent magnetic fields are ejected after a short ”collision time” in a random direction. The cloud-particle collision satisfies the usual energy and momentum conservation laws

2 2 MvM,i/2+ mγp,i = MvM,f /2+ mγp,f M~vM,i + mγp,i~vp,i = M~vM,f + mγp,f~vp,f (3.11)

Taking the square of the momentum conservation law and re-arranging the terms one ﬁnds, for head-on collisions

2 2 2 2 2 2 2 2 Mm(γp,f − γp,i)= M (vM,i − vM,f )= m γp,f vp,f − γp,ivp,i − 2Mm (γp,f vp,f vM,f + γp,ivp,ivM,i) (3.12) (we have also assumed that the particle coming out of the cloud escapes in the direction opposite to its initial velocity). If the change of the particle energy in collision is small, γp,i ≃ γp,f ≃ γp we could approximate the above expression as

2 Mm∆γp ≃ 2m γp∆γp − 4MmγpvM v (3.13) so that ∆γp ≃−4vM v (3.14) γp

For a head-on collision, vM is directed oppositely to v and particle energy grows after the bounce from the cloud. If the collision is tail-in, particle velocity and cloud velocity directions coincide and particle decelerates after the bounce. A particle moving in a medium ﬁlled with moving magnetised clouds would experience both head- on and tail-in collisions. The rate of collisions at a given angle

R = σnvrel (3.15)

(σ is the collision cross-section, n is the density of the clouds) depends on the relative velocity of the clouds and particles, which is vrel = |vM | + |v| for the head-on collisions and vrel = |vM |−|v| for the tail-in collisions. Thus, the head-on collisions are more often and particle on average gains energy more often than it looses it. All in all, the time average energy gain scales as

∆E = σn [(|v | + |v|)Ev − (|v|−|v |)Ev ] ≃ 2σnEv2 (3.16) ∆t M M M M M 1An important exception are the magnetosphere of neutron stars and black holes where large scale electric ﬁelds could be supported 63

It is proportional to the second order of the cloud velocity vM . Because of this, the process of acceleration due to the scattering of particles oﬀ moving magnetised clouds is called ”second-order Fermi acceleration”. From Eq. 3.16) one could see that the energy gain is exponential in time, E(t) ∼ exp(t/tacc), where the acceleration time 1 tacc = 2 (3.17) 2σnvM The longer particle bounces oﬀ the clouds, the higher is its energy. This run-away growth of particle energy stops if particle escapes from the region occupied by the magnetised clouds. Assuming that the escape happens within a certain escape time, tesc we could describe the acceleration process following an episode of injection of low energy particles, as a competition between the growth of particle energy and decrease of the number of particles, dN N ∼ (3.18) dt tesc As a result , the number of particles escaping from the acceleration volume is a function of energy:

t −tacc/tesc N(E > E0)= N0 exp − ∼E (3.19) tesc

This corresponds to a diﬀerential particle spectrum

dN ∼E−(1+tacc/tesc) (3.20) dE

If the radio tacc/tesc is energy-independent, the above spectrum is a powerlaw-type spectrum, as observed in a range of high-energy astronomical sources. Second order Fermi acceleration mechanism is a ”universal” phenomenon expected to take place in a variety of astrophysical environments. Indeed, most of the astronomical objects possess turbulent magnetic fields, which readily provide a range of inhomogeneities or ”clouds” on which charged particles could scatter. These turbulent ”eddies”, or ”clouds” are dynamical entities, moving with random velocities. Thus, the conditions for operation of the second-order Fermi acceleration mechanism are most probably present in most of the astronomical sources. However, in spite of the universally present moving magnetic inhomogeneities, this acceleration mechanism is rarely considered to be of any importance because of its low efficiency. As an example, we could consider the possibility of particle acceleration in the interstellar medium in our Galaxy. The interstellar medium is known to be turbulent, and the power spectrum of turbulence is measured on a wide range of scales, from the maximal scale of the order of 100 pc down to some 109 cm, i.e down to the size scale about the Earth radius. All over this large dynamic range the power spectrum follows a powerlaw, determined by the general laws of turbulence (Kolmogorov spectrum), see Fig. 3.7. Turbulent structures in the Galaxy are magnetised and we could assume that the turbulent eddies serve as scattering centres for charged particles. At the integral scale of λturb ∼ 100 pc, typical turbulent velocity scales are −4.5 Figure 3.7: Power spectrum of interstel- vturb ∼ 10 km/s ∼ 10 c. Particles scattering off tur- lar turbulence. bulent eddies experience energy gain ∆E/E ∼ 10−9. An 64 CHAPTER 3. PARTICLE ACCELERATION MECHANISMS

average mean free path of particles scattering oﬀ the eddies −1 could be estimated as (σn) ∼ λturb. The acceleration time is

λturb 19 tacc ∼ 2 ∼ 10 s (3.21) cvturb which exceeds the age of the Universe. Thus, second-order Fermi acceleration in the interstellar medium of the Galaxy is an extremely slow and inefficient process. Turbulence is a phenomenon which involves a transfer of energy over a range of distance scales, from the largest ”integral scale” (∼ 100 pc in the case of interstellar turbulence) down to the shortest ”damping scale” where the viscosity forces are able to dissipate kinetic energy of the fluid. Typical Kolmogorov power spectrum of the turbulence describes the dirstibution of energies over the distance 2 2 5/3 scales. The kinetic energy at a scale λ<λturb scales as v (λ)= vturb(λ/λturb) . The typical velocities of inhomogeneities of the size λ scale as λ5/6. Consideration of the details of the scattering on smaller scale inhomogeneities is rather inefficient and the full scattering takes place 2 2 in about (vturb/v(λ)) Larmor cycles. The the scattering length scales as λscat ≃ λ(vturb/v(λ)) . As a result, the acceleration time scales with particle energy:

−7/3 λscat 19 λ tacc ∼ 2 ∼ 10 s (3.22) cv (λ) λturb Thus, the time scale of acceleration on the smaller scale eddies is still longer. There is, however, a situation in which the second order Fermi process could play an important role. Consider a region of massive star formation. The massive stars usually form in large groups (hundreds or thousands of them) nearly simultaneously (on the time scale of 107 yr) in the dense environments of molecular clouds. One of the best known examples of large nearby star forming region is Cygnux X region situated at the distance about 1 kpc from the Sular system. All the massive stars drive strong stellar winds and blow the stellar wind bubble around themselves all through their life time. Since all the stars are close to each other, the stellar wind bubbles of different stars start to overlap soon after the onset of star formation. As a result, the motions of the interstellar medium in the star forming region become highly turbulent. The collective pressure of the winds of all the massive stars exceeds the pressure of the interstellar medium and the star forming region starts to blow a ”superbubble” in the interstellar medium. Parameters of the turbulence inside the suppressible could be roughly estimated from the known 3 parameters of the stellar winds. The winds of the massive stars have velocity about vw ∼ 10 km/s∼ 10−2.5c. The distance between the massive stars is about several parsecs, so that the integral scale of −2.5 the turbulence is λturb ∼ 10 pc and the velocity scale is vturb ∼ 10 c. Estimating the acceleration time as before we find λturb 14 6.5 tacc ∼ 2 ∼ 10 s ∼ 10 yr (3.23) cvturb This time scale is comparable to the lifetime of the superbubble. Thus, the superbubble environment provides favourable conditions for particle acceleration via the second order Fermi acceleration process. In the particular case of Cygnus X superbubble, we know that there is an acceleration process operating inside it, because the γ-ray spectrum of the superbubble is much harder than the typical interstellar medium spectrum. It has the slope γ ≃ 2.2, see Fig. 3.8. The γ-ray emission is most probably produced by the interactions of accelerated protons with the interstellar medium of the bubble. In this case the spectrum of γ-ray emission nearly repeats the spectrum of the accelerated protons. Thus, the spectrum of accelerated particles is a powerlaw with the slope γ ≃ 2.2. One could estimate the maximal energies of particles attainable for the acceleration in the suppressible environment. Taking into account that the particles of maximal energies scatter on the inhomogeneities of the size comparable to the intogral scale of the turbulence R ∼ 10 pc, one finds 16 −5 that the Hillas condition gives an estimate Emax ∼ 10 [R/10 pc] B/10 G eV.

66 CHAPTER 3. PARTICLE ACCELERATION MECHANISMS could be written as

[ρv] = 0 P + ρv2 = 0 v(u + P + ρv2/2) = 0 (3.24) where P = P1,2 is the pressure of the medium on the two sides of the shock and u = u1,2 is the internal energy per unit volume. If we consider an medium with an equation of state P = (γ − 1)u, with an adiabatic index γ, the energy and mass conservation equations give together

γP v2 + = 0 (3.25) (γ − 1)ρ 2

In the shock reference frame the junction conditions could be written as

ρ1v1 = ρ2v2 2 2 P1 + ρ1v1 = P2 + ρ2v2 γ P v2 γ P v2 1 + 1 = 2 + 2 (3.26) γ − 1 ρ1 2 γ − 1 ρ2 2

We consider a ”strong shock” in which the upstream medium has no pressure, P1 = 0. Introducing the ”compression ratio” R = v1/v2, we can write the junction conditions as ρ 2 = R ρ1 P2 2 R − 1 = v1 2 ρ2 R 2γ R − 1 1 + = 1 (3.27) γ − 1 R2 R2 which gives γ + 1 R = (3.28) γ − 1 For an ideal gas, γ = 5/3 and R = 4. Strong shock always leads to an increase of the medium density by a factor of 4. The downstream medium is heated to the temperature

2 2 P2 2 R − 1 3 mv1 7 v1 T2 = m = mv1 2 = ≃ 10 3 K (3.29) ρ2 R 8 2 10 km/s

Typical velocity scales of the stellar wind bubbles blown by the massive stars and of the expanding supernova shells are in the v ∼ 103 km/s range. The above equation shows that all these objects should be X-ray sources. Fig. ?? shows some examples of X-ray thermal (Bremsstrahlung plus atomic lines) emission from the SNR Cas A. In the reference frame of the shock the upstream and downstream media are approaching each other at the speed v 3 v = v − 2 = v (3.30) rel 1 R 4 1 If we assume that on each side of the medium there are magnetised inhomogeneities which could deviate trajectories of high-energy particles, we ﬁnd that particles bouncing between the upstream and downstream media are accelerated.

68 CHAPTER 3. PARTICLE ACCELERATION MECHANISMS

n The number of particles escaping from the shock after n crossings is Nn = N0(1−P ) . The energy n of the particles escaping after n crossings is En = E0(1 + κ) . The energy spectrum of the escaping particles is, therefore, P 2+ R N(E ≥ E ) ∝ E−P/κ = E1−p, p = +1= = 2 (3.36) n κ R − 1 The spectrum of particles ejected from the shock is a powerlaw with the slope which depends only on the compression ratio of the shock. The compression ratio of the strong shocks is determined by the equation of state of the medium. For the ideal gas it is R = 4, so that the resulting spectrum is a dN/dE ∼ E−2 powerlaw. For an astrophysical example, let us consider acceleration at the shock front of an expanding supernova remnant shell. The acceleration time scale is the time of the upstream-downstream bounce, which is determined by the scattering length of particles or, equivalently, by the scale of turbulence of the shocked medium. The turbulence is driven by the hydrodynamical instabilities. The characteristic turbulence scale is given by the thickness of the SNR shell. From the image of Cas A SNR, one could estimate the shell thickness as being a fraction (0.01 to 0.1) of the shell size. The shell sizes of SNR shell are in the ∼ 10 pc scale range, so that the scale of the turbulence is estimated to be λturb ∼ 0.1 − 1 pc. 3 The acceleration time is estimated as tacc = λturb(E/∆E) ∼ λturbκ ≃ 10 yr, for the velocity scale v ≃ 103 km/s. The maximal energies of particle which could be produced by the acceleration process 14 15 are found from the Hillas condition, Emax ∼ eλturbB ≃ 10 ...10 eV. Right panel of Fig. 3.9 shows the spectrum γ-ray emission from Cas A SNR. Detection of γ-rays with energies up to 1-10 TeV provides a proof of the operation of the first-order Fermi process in this source. Case A is a young SNR with the age of just 330 years, comparable to the accederation time scale of particles in the SNR shell. Supernova remnant shells are produced in explosions of massive stars, which typically form in the suppressible environments. particles acceleration by the first order Fermi process operating in the SNR shells could serve as an efficient ”pre-acceleration” mechanism for the injection of particles for the second-odrer Fermi acceleration in the suppressible environment. 69

Abbreviations AGN Active Galactic Nucleus FR I, FR II Fanaroﬀ-Riley radio galaxy, type I or type II GRB Gamma-Ray Burst HMXRB High-Mass X-ray Binary LMXRB Low-Mass X-ray Binary PSR Pulsar (name in astronomical catalogues) PWN Pulsar Wind Nebula QSO Quasi-Stellar Object (name in astronomical catalogues) SNR Supernova Remnant Sy I, Sy II Seyfert galaxy, type I or II XRB X-Ray Binary 70 CHAPTER 3. PARTICLE ACCELERATION MECHANISMS Bibliography

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