Radiative processes

1General considerations¶

Radiative processes are crucial to the study of cosmic-ray acceleration for two reasons.

1. As energy losses, they limit the maximum energy reached by astrophysical accelerators (cf. diffusion-loss equation (14));
2. As a mechanism for producing neutral secondary particles (neutrinos, gamma-rays, neutrons) that can more easily escape the magnetized regions where cosmic-ray electrons, protons and nuclei are accelerated.

Stable neutral secondary particles (neutrinos, gamma-rays) are not deflected by intervening magnetic fields (in the interstellar medium, the galactic halo, the cosmic web) and thus trace the direction of the astrophysical accelerators more robustly than the primary charged particles.

Different loss mechanisms should be considered depending on whether the cosmic ray interacts with ambient protons (and helium nuclei), photons, or magnetic fields. For a given cosmic-ray species (e.g. electrons), the comparison of loss time scales, $\left[ \frac{1}{p} \frac{\dd p}{\dd t}\right]^{-1}$, or loss lengths, $\left[ \frac{1}{p} \frac{\dd p}{\dd x}\right]^{-1}$, allows us to identify the main channels of secondary-radiation production and to model them in detail as a function of energy.

As the losses accumulate following $\dot p_\text{loss} = \sum_{i \in \text{processes}} \dot p_i$, the overall loss timescale is

As expected, the fastest losses (i.e. the ones with the shortest $t_i$) dominate the fate of the primary particle.

When multiple cosmic-ray species are present (e.g. co-acceleration of electrons and protons), we have to identify the main losses for each species and account for the relative energy throughput per particle as well as for the number of particles per primary species. Radiative processes are divided in two main categories, hadronic and leptonic, depending on the nature of the primary ($p$ or $^{A}_{Z}X$ versus $e^-$ or $e^+$).

2Hadronic processes¶

2.1Spallation and $p$-$p$ processes¶

The baryonic composition of the interstellar medium and of the cosmic web is dominated by hydrogen (about half molecular, H₂, and half atomic, as neutral HI or ionized HII in the Milky Way) and helium (about $10\,\%$ in number, that is $40\,\%$ in mass in the Milky Way, see Brinks (1990)).

In dense environments, such as molecular clouds near supernova remnants and the circum nuclear medium of starburst galaxies, proton-proton ($p$-$p$) and spallation processes can be dominant (see e.g. timescales in Condorelli et al. (2023)). Nearly all intermediate-mass nuclei such as Li-Be-B, are produced by spallation. Leaving such nuclei aside, the throughput of $p$-$p$ and spallation processes is dominated by neutrons and protons, which in turns mainly interact through:

The three types of pions are produced in a ``democratic’’ fashion by $p$-$p$ interactions, so that the average number of charged pion per neutral pion produced in these interactions is

The subsequent decay of pions leads to the production of neutral secondaries:

Although neutrinos are produced in proportion $(\nu_e : \nu_\mu : \nu_\tau)|_\text{source} = (1 : 2 : 0)$, detection is expected in proportion $(\nu_e : \nu_\mu : \nu_\tau)|_\oplus = (1 : 1 : 1)$ due to neutrino oscillations.

The relative energy loss of the primary nucleon is called the inelasticity of the reaction, κ. The average inelasticity per pion is thus defined as $\langle E_\pi \rangle = \kappa_\pi E_p$, with $\kappa_\pi \approx 20\,\%$ for $p$-$p$ interactions (also valid for $p$-γ interactions discussed in the following, see Ahlers & Halzen (2018)). This means that a population of $10\,\text{PeV}$ protons will produce on average $2\,\text{PeV}$ pions, which will decay in $1\,\text{PeV}$ gamma rays and $500\,\text{TeV}$ neutrinos. Note that while the neutrinos are likely to escape interactions in their environment and on their way to Earth, the $1\,\text{PeV}$ gamma rays could easily interact along their path with lower-energy photons to produce pairs of electrons and positrons (see section dedicated to leptonic processes). While neutrinos and gamma-rays are expected to be produced in about equal proportion by hadronic processes, they are not necessarily expected to arrive at Earth in such proportions!

2.2Photo-erosion and $p$-γ processes¶

Photo-erosion¶

The photo-erosion process, also called photo-disintegration, describes the interaction of a nucleus with a photon:

with $(a,b) \in \mathbb{N}^2$.

The Lorentz boost per nucleon is roughly conserved in such interactions. This is particularly true at the highest energies when the nuclei can be seen as a superposition of independent nucleons.

In the nucleus rest frame, $\mathcal {R}'$, different nuclear/hadronic processes should be considered depending on the target photon energy (see Allard et al. (2006)):

• Giant dipole resonance for $\epsilon_\gamma' \approx 10-20\,\text{MeV}$, i.e. a collective nuclear mode in which the neutrons and protons oscillate in antiphase. The giant dipole resonance can lead, in the final state, to the release of $n$, $p$, $2n$, $2p$, $np$, α;
• Quasi-deuteron process for $\epsilon_\gamma' \approx 30\,\text{MeV}$, i.e. the interaction of the photon with a pair of nucleons that can be released by the parent nucleus;
• Baryon resonance for $\epsilon_\gamma' \gtrsim 150\,\text{MeV}$, i.e. the interaction of the photon with single nucleons. This process is similar to the $p$-γ and $n$-γ processes discussed in the following subsection.

Photo-production of pions and pairs¶

The main interactions of a nucleon in a photon field are:

• the photo-production of pairs:
  • $p+\gamma \rightarrow p + e^+ + e^-$
  • $n+\gamma \rightarrow n + e^+ + e^-$
• the photo-production of pions:
  • $p + \gamma \rightarrow p + \pi^0$ and $p + \gamma \rightarrow n + \pi^+$
  • $n + \gamma \rightarrow n + \pi^0$ and $n + \gamma \rightarrow p + \pi^-$

Note that the three pions are not produced in a ``democratic’’ fashion in $p$-γ processes, rather:

In the nucleon’s rest frame, the threshold for pair and pion production are about $1\,\text{MeV}$ and $140\,\text{MeV}$, respectively. The energy loss-length, $\lambda = \langle (\frac{1}{E}\frac{\dd dE}{\dd X})^{-1} \rangle$, depends on the product of the cross section and of the inelasticity (energy-loss fraction) of the reaction. This product, which is nearly constant at energies larger than a few times the threshold energy, is two orders of magnitude larger for pion production than for pair production, so that the former dominates over the latter at the highest energies.

The various loss processes relevant for the propagation of ultra-high energy cosmic rays on cosmological scales are illustrated in Figure 1.

At the lowest energies, below $1\,\text{EeV}$, the losses are dominated by the so-called adiabatic cooling, which is driven by the expansion of the universe, indeed:

2.3Proton synchrotron¶

A particle of energy $E$, mass $m$ and charge $Ze$ emits synchrotron radiation in a magnetic field $B$. Averaged over the pitch angle θ, such as $\cos\theta = \frac{\vec{\beta}\cdot\vec{B}}{\beta B}$, the synchrotron energy loss reads (see e.g. chapter 14 of Jackson (1999)):

with $(\gamma, \gamma\vec{\beta})$ the four-momentum of the particle and

In the ultra-relativistic limit, $\beta \approx 1$ adopted hereafter, the synchrotron loss rate is thus

Note that the synchrotron loss rate increases with energy and that, for a fixed Lorentz boost $\gamma = E/mc^2$, the rate goes as $1/m^3$. If the loss rate becomes dominant with respect to the acceleration rate, the maximum energy is limited by the losses (so-called synchrotron burn-off limit).

The synchrotron radiation is emitted as photons over a range of frequencies, ν. The power emitted by a single proton per frequency unit reads

where $\int \dd \nu\,f_\text{sync}\left( \frac{\nu}{\nu_\text{c}}\right)=1$ so that $\left.P(\gamma)\right|_\text{sync} \equiv \int \dd \nu\, \left.P_\nu(\nu,\gamma)\right|_\text{sync} = \left.-\frac{\dd E}{\dd t}\right|_\text{sync}$, and where the critical frequency, averaged over the pitch angle, is $\nu_\text{c} = \frac{3}{16} \frac{ZeB}{m} \gamma^2$.

The peak of the photon spectrum is at $E_\text{peak} = \alpha h\nu_\text{c}$, where $\alpha \approx 0.3$ corresponds to the maximum of $f_\text{sync}(x)$.^[1] Using the value of the nuclear magneton, $\mu_N = \frac{e\hbar}{2m_p} \approx \pi \times 10^ {-14}\,\text{MeV\,T}^{-1}$, one finds:

Synchrotron emission from ultra-high energy protons at $10^{18}\,$eV in a magnetic field of $1\,\text{G}$ can thus end up as photons in the MeV range. This process has sometimes been invoked to explain the gamma-ray emission from the jetted active galactic nuclei. In the next section, we will look at alternative leptonic mechanisms.

3Leptonic processes¶

3.1Electron synchrotron¶

For a given Lorentz boost, the synchrotron losses $\left.-\frac{\dd E}{\dd t}\right|_\text{sync}$ go as $m^{-2}$ so that the emission from electrons can be sizeable with respect to that of protons. In what follows, we mostly discuss electrons but synchrotron could also be expected e.g. from positrons if there are no electrons around. Otherwise, the positrons and electrons annihilate resulting, for particles nearly at rest, in a $511\,\text{keV}$ emission that is observed e.g. along the Galactic ridge.

Equation (20) can be used to determine the peak energy of the emission from e.g. $500\,\text{GeV}$ electrons in a $1\,\text{mG}$ field:

So, in a mG field, $0.5\,\text{TeV}$ electrons radiate in the UV, while $5\,\text{TeV}$ electrons radiate in X-rays at $0.4\,\text{keV}$. Synchrotron radiation from electrons thus explains the emission from radio to optical wavelengths (sometimes up to X-rays) of most non-thermal sources.

We can take the calculation one step further and determine the expected spectrum for the synchrotron emission (a similar derivation can of course be made for protons). Consider a number density of electrons, $n_e$, following a power law of energy or Lorentz boost γ, i.e.

with e.g. $s \approx 2$ for diffusive shock acceleration.

The synchrotron luminosity of such electrons filling a spherical region of radius $R$ that is optically thin (transparent) to its own emission reads:

where $P_\nu$ is the power emitted by a single electron, as defined in Equation (19).

To simplify the problem, we assume that the distribution function of emission as a function of photon frequency is sharply peaked around $\alpha\nu_c = \gamma^2 \nu_\text{ref}$, where $\nu_\text{ref} = \frac{3\alpha}{16} \frac {eB}{m_e}$ does not depend on the electron Lorentz boost. The assumption of a sharply peaked function is called the delta approximation:

Then, considering only the ultra-relativistic electrons ($\beta \approx 1$):

For a flat energy spectrum of electrons with $s=2$, neglecting the redshift of the photons, the energy spectrum of synchrotron photons goes as:

i.e. a differential photon spectrum following a power-law, $\frac{\dd N}{\dd E_\gamma} \propto E_\gamma^{-1.5}$, from $\gamma_\text{min} ^2 \nu_\text{ref} \propto \gamma_\text{min}^2 B$ to $\gamma_\text{max}^2 \nu_\text{ref} \propto \gamma_\text{max}^2 B$.

The observed range of frequencies covered by the photons can provide constraints on the range of Lorentz boosts covered by the electrons, provided the magnetic field, and the observed flux normalisation constrains the product of the magnetic field with the number of electrons in the emitting region.

3.2Inverse Compton¶

The inverse Compton process, which is symmetrical to the Compton process, consists of the scattering of an energetic electron onto a photon, resulting in an energy gain for the photon:

where the initial photon and electron have energies $E_{\gamma_1} = h\nu_1$ and $E_{e_1} = \gamma m_e c^2$ in the observer’s frame.

There are two regimes to be considered for this scattering in the rest frame $\mathcal{R}'$ of the electron (see Chapter 9.3 in Longair (2011)):

• Thomson regime: $h\nu_1' \approx \gamma h\nu \ll m_ec^2$. In the Thomson regime, the cross section is independent of the photon energy, $\sigma \approx \sigma_\text{T}$, and the outgoing photon has an energy $h\nu_2' =h\nu_1'$ with an opposite momentum. Back in the observer frame, the energy of the outgoing photon is $h\nu_2 \approx \gamma h\nu_2 \approx \gamma^2 h\nu_2$, which result in a net gain of energy by a factor $\propto \gamma^2$.^[2]
• Klein-Nishina regime: $h\nu_1' \gtrsim m_ec^2$. In the Klein-Nishina regime, the recoil of the electron cannot be neglected and the cross section is suppressed as $\sigma \propto (h\nu_1')^{-1}$. The conservation of energy in the observer’s frame tells us that the energy gain of the photon is at most $h\nu_2 - h\nu_1 = (\gamma-1)m_e c^2$, that is a gain $\propto \gamma$ for an ultrarelativistic electron.

The inverse-Compton energy loss of an electron in an isotropic radiation field of energy density $u_\text{rad}$ reads:

Note that synchrotron radiation can be viewed as the inverse-Compton scattering of virtual photons associated to the magnetic field (so-called method of virtual quanta). This is the fundamental reason behind the similarity of the inverse-Compton and synchrotron energy loss in Equation (16). We can then write

Similarly to synchrotron radiation, the power emitted by a single electron per unit photon frequency can be written:

where $\int \dd \nu f_\text{IC} \left(\frac{\nu}{\gamma^2\nu_1}\right) = 1$.

The inverse-Compton luminosity of ultrarelativistic ($\beta \approx 1$) electrons filling an optically-thin spherical region of radius $R$ is:

In the Thomson regime, the function $f_\text{IC}(x)$ peaks around $x\approx2$. The delta approximation can be roughly adopted so that

Then, the inverse-Compton luminosity of the source reads:

The radiation field can have different origins depending on the environment. We speak of external inverse Compton when the photon field does not come directly from the electron population. For example, it can be an optical/infrared photon field from the environment of the compact object (e.g. accretion disk) or a diffuse photon field such as the CMB for an extended emission region.

A special case, found in many non-thermal sources, is when the scattered photon field comes from the electrons themselves, namely their synchrotron emission. This is known as the synchrotron self-Compton (SSC) model. The energy density in the spherical region of radius $R$ is related to the synchrotron luminosity of this region by the relation (see e.g. Section 6 in Ghisellini (2013)):

where $t_\text{cross} = \frac{3}{4}\frac{R}{c}$ is the average crossing time of the photons through the spherical region.

Using Equations (33) and (34), the SSC losses can then be written as:

The SSC luminosity of an electron population of density $n_0$ and Lorentz factor $\bar \gamma$ is simply its synchrotron luminosity scaled by a factor $n_0 \sigma_\text{T} R/2$ and shifted to frequencies larger by a factor of $2 {\bar \gamma}^2$. The same conclusion can be reached for a power-law distribution of the Lorentz boost of the electrons as in Equation (22):

The SSC spectrum has the same photon index as the synchrotron spectrum. As we have shown in Equation (26), the frequency range covered by the synchrotron photons goes from $\gamma_\text{min}^2 \nu_\text{ref}$ to $\gamma_\text{max}^2 \nu_\text{ref}$, where $\nu_\text{ref} \propto B$. The frequency range covered by the SSC photons should then go from $2\gamma_\text{min}^4 \nu_\text{ref}$ to $2\gamma_\text {max}^4 \nu_\text{ref}$, when we neglect the Klein-Nishina effect. Note though that for $\gamma_\text{max} h\nu_1 \gtrsim m_e c^2$, which is verified e.g. for $\gamma_\text{max}=10^6$ and $h\nu_1 = \left.E_\text{peak}\right|_\text {sync} \approx 4\text{\,eV} \times \left(\frac{B}{1\,\text{mG}}\right) \left(\frac {\gamma}{10^6}\right)^2$, the scattering occurs in the Klein-Nishina regime so that the SSC peak emission can reach an energy of $\left.E_\text{peak}\right|_\text{SSC} \approx \gamma_\text{max} m_e c^2 \approx 0.5 \text{\,TeV}$.

Irrespective of the scattering regime, the ratio of the amplitude of the SSC and synchrotron peaks provides constraints on the product of the density of the region of its size, while the peak energies (or better their ratio in the Thomson regime) constrain the maximum Lorentz factor of the electrons.

3.3Bremsstrahlung¶

The Bremsstrahlung emission, i.e. the radiation of an electron in the electromagnetic field of a nucleus, is relevant in an astrophysical context when the density of matter (neutral or ionised hydrogen/helium) is sufficiently high, as for the $p-p$ process. Through Bremsstrahlung emission, the electron can lose a substantial fraction of its kinetic energy $T=(\gamma-1)m_e c^2$. For a Maxwellian electron velocity distribution with $\beta\ll1$ and $T \approx \frac{1}{2}m_e(\beta c)^2$, we speak of thermal Bremsstrahlung or free-free emission, as the electron is not bound to the nucleus with which it interacts in either the initial or final states.^[3] Thermal Bremsstrahlung emission is found, for example, in the warm-hot plasma of galaxy clusters and explains their X-ray continuum radiation up to $\frac{1}{2}m_e(\beta c)^2 = 2.5\,\text{keV}\times \left(\frac{\beta}{0.1}\right)^2$.

Non-thermal Bremsstrahlung is found in the interstellar medium, particularly in molecular clouds close to supernova remnants. This non-thermal radiation explains some of the gamma-ray emission from molecular clouds up to energies of $(\gamma-1)m_e c^2 \approx 0.5\,\text{TeV}\times \left(\frac{\gamma}{10^6}\right)$, in addition to the emission processes discussed above.

The energy loss of an electron through Bremsstrahlung depends on the radiation length of the electrons in the medium, $X_0$ in $\text{g}\,\text{cm}^{-2}$, so that:

where $n m_p$ is the mass density of protons.

The Bremsstrahlung loss rate, $-\left.\frac{1}{E}\frac{\dd E}{\dd t}\right|_\text{Brem}$, is independent of the electron energy, contrarily to synchrotron and inverse Compton loss rates which go as $\propto E$. As all electrons are affected by Bremsstrahlung in the same manner, the photon index of the Bremsstrahlung spectrum is the same as the electron index.

A comparison of the various emission mechanisms invoked in a lepto-hadronic model for the emission of a supernova remnant interacting with a molecular cloud are shown in Figure 2. The energy flux of the synchrotron and external inverse-Compton components goes as $E_\gamma^2 \frac{\dd N_\gamma}{\dd E_\gamma} \propto E_\gamma^{0.5}$. We can thus infer that electrons are injected in the model with a slope of $s \approx 2$. This confirmed by the Bremsstrahlung component, which is flat ($E_\gamma^2 \frac{\dd N_\gamma}{\dd E_\gamma} \propto E_\gamma^0$ i.e. $\frac{\dd N_\gamma}{\dd E_\gamma} \propto E_\gamma^{-2}$) between ${\sim}\,1\,\text{MeV}$ and ${\sim}\,1\,\text{TeV}$. Finally, while the leptonic emission fully explains the synchrotron peak from radio to X rays, hadronic emission is included to model the high-energy component. This hadronic emission from the $p-p$ process results in $\pi^0$ decay, with a flat spectrum from ${\sim}\,100\,\text{MeV}$ to ${\sim}\,1\,\text{TeV}$ corresponding to the proton index $s \approx 2$.^[4] Note the characteristic feature of the pion bump, with a sharp break close to half the pion mass at ${\sim}\,70\,\text{MeV}$. This break has now been observed in the several bright supernova remnants interacting with molecular clouds in the energy range covered by the Fermi-LAT satellite, demonstrating the co-acceleration of protons and electrons in these environments.