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 impor- tant 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 frac- tion of cosmic rays interacts in the Galactic Disk before escaping from it. Neutral and charged pions π0, π± are unsta- ble 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 mag- netosphere, 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 ac- tive 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 at- mosphere 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 interac- tions 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?”.