Jetted and non-jetted astroparticle sources in the universe

1Multi-messenger emissions on cosmic scales¶

Energy density, surface brightness, flux density and luminosity¶

An observable directly related to the energy density of an isotropic photon field is the sky surface brightness, from which all the foregrounds have been subtracted. Following the notation of Rybicki & Lightman (1986), the bolometric surface brightness or bolometric intensity, $I$, is defined as the energy passing through a surface $\dd A$ during a time $\dd t$ and from a solid angle $\dd \Omega$:

with $[I] = \mathrm{W}\,\mathrm{m}^{-2}\,\mathrm{sr}^{-1}$.

The radiative energy density, $\varepsilon$, in a volume $\dd V = c \dd t \dd A$ is such as $dE = \varepsilon \dd V$, so that the average intensity integrated over the sphere is

Following again Rybicki & Lightman (1986), the energy density in a field of particles with momentum between $p$ and $p + \dd p$ depends on the number of particles per phase volume, $\dd \mathcal{N}/\dd^3 x \dd^3 p$ as

$\dd \mathcal{N}/\dd^3 x \dd^3 p$ is invariant under a Lorentz transformation. Indeed $\dd \mathcal{N}$ is countable and thus invariant. Under a boost $(\beta, \gamma)$ along the x-axis from the coomving frame (K’) towards the observer’s frame (K), one finds $\dd x = \gamma^{-1} \dd x'$ (length contraction) and $\dd p_x = \gamma (\dd p_{x'} + \beta \dd E') = \gamma \dd p_{x}'$ for particles with fixed energy (total momentum fixed between $p$ and $p + \dd p$). Thus $\dd^3 x \dd^3 p$ is invariant, quod erat demonstrandum.

One finds

so that

The spectrum of the universe¶

The broadband emission from all galaxy populations is responsible for the spectrum of the universe shown in Figure 1. In particular, electromagnetic radiations include the cosmic radio background (CRB) from both active and star-forming galaxies, the cosmic infrared and optical backgrounds (CIB and COB) mostly from nucleosynthesis and dust emission, the cosmic X-ray background (CXB) from active galaxies and the cosmic gamma-ray background (CGB) from jetted active galaxies. The differential measurements of these cosmic backgrounds are of fundamental value: they reflect our knowledge of the distribution of light emitted by star formation, accretion and ejection integrated since the formation of the first astrophysical sources. Although these emissions are only a negligible part of the cosmic energy inventory, they provide us with a cosmological consistency test that is essential for understanding the content and evolution of the post-recombination universe Fukugita & Peebles, 2004.

The values indicated by vertical text in Figure 1 correspond to the energy density of each component, i.e. the integral of the specific intensity multiplied by $4\pi/c$. In particular, it can be verified that the expected energy density from the nucleosynthesis and accretion processes calculated in the previous chapter, i.e. ${\sim}\,(13+1.5) \times 10^3\,\mathrm{eV}\,\mathrm{m^{-3}}$, is found in its entirety in the COB and CIB. The energy density from ejection processes around supermassive black holes, expected at ${\sim}\,3 \,\mathrm{eV}\,\mathrm{m^{-3}}$, is found in the CGB.

The measurements shown in Figure 1 also quantify the degree of darkness of the night sky once the foregrounds are subtracted. It is the history of their emission that provides the solution to the Olbers paradox and the present intensity of these backgrounds that determines the darkness (or rather grayness) of the night sky. We can understand the extragalactic backgrounds at all electromagnetic wavelengths, the so-called extragalactic background light, using synthetic models of galaxy populations. Some unknowns in the extraglactic background light, including tensions between measurements, are nonetheless still the subject of active research. Light emission is fundamentally the result of decay, heating and acceleration of matter. We will explore in the following lessons the knowledge established with photons, in particular through multi-wavelength observations of gamma-ray sources, and to which extent this multi-wavelength knowledge allows us to understand extragalactic backgrounds observed today with other messengers, in particular the extragalactic neutrino background (ENB) between $30\,$TeV and $3\,$PeV and the extragalactic cosmic-ray background (ECRB) between $200\,$PeV and $200\,$EeV.

The spectrum, composition and arrival directions of cosmic rays¶

The top panel in Figure 3 shows the same cosmic ray spectrum as in Figure 2 multiplied by a power law of index 2.7 in order to better see the different spectral breaks: $E^{2.7}J(E)$ is shown as a function of $E$. Between the hip at a few hundred GeV and the knee at $E \approx 3-5\,$PeV, the cosmic-ray flux is well described to first order by a power law $J(E) \propto E^{-2.7}$, followed by a break in slope around $E \approx 150\,$PeV corresponding to a softening of the spectrum (intensity decreasing more rapidly with energy), called the second knee. We will return to the link between the first and second knees in section 2.2. At higher energies, around $5\,$EeV, we observe a hardening of the spectrum (intensity decreasing less rapidly with energy) followed finally by a softening of the spectrum around $50\,$EeV in the cut-off region.

The second panel in Figure 3 shows the evolution of the mean logarithm of the atomic mass, $A$, of the observed cosmic rays as a function of energy. This logarithmic quantity is close to the observables reconstructed with the dedicated instruments. We observe that, on average, $\ln A \approx 0$ up to a few tens of GeV, i.e. the composition is dominated by protons. The composition is proton and helium up to the knee, then becomes heavier, possibly containing some iron at the second knee. The measurements between the second knee and the ankle are too sparse to be shown in the figure. Beyond the ankle, the composition becomes heavier again, ranging from helium to a mass close to that of the nuclei of carbon, nitrogen and oxygen.

The third and fourth panels show the amplitude and right-ascension direction (see Figure 4) of the dipolar component of the cosmic-ray flux as a function of energy. As shown in the third panel and in Figure 5, the amplitude of the dipole around $10\,$TeV relative to that of the monopole (isotropic component) is of the order of 10^-3. This amplitude increases with energy in the range in which it is measured, reaching around ten per cent above the ankle.

Observations of cosmic rays, dissected in terms of flux, composition and arrival direction, suggest the following paradigm. Cosmic rays are mainly of Galactic origin (i.e. from the Milky Way) up to the second knee. This is corroborated by the mean right ascension of their arrival directions aligned with that of the Galactic Centre around PeV energies. At lower energies, around $10\,$TeV, these cosmic rays are affected by local magnetic fields, in particular those of the Local Bubble that extends to a few hundred pc around the Sun and is thought to have originated in a past supernova explosion.

Beyond the ankle, cosmic rays are too energetic to be confined by the Milky Way’s magnetic field. These cosmic rays are extragalactic, i.e. they come from galaxies other than our own. This is supported by the large and increasing amplitude of the dipole above $5\,$EeV and by their arrival directions, which are in relatively good agreement with the direction expected from the distribution of galaxies within a few hundred Mpc.

The origin of cosmic rays between the second knee and the ankle, whether Galactic or extragalactic, remains an unsolved problem to this day.

2Candidate sources for cosmic-ray production¶

Known gamma-ray emitters with and without jets¶

Bestiary of sources¶

It is difficult to see the sources of cosmic rays directly on the sky, as cosmic rays are charged nuclei and are therefore deflected by the magnetic fields they pass through. Only anisotropies on angular scales of more than ten degrees have been evidenced to date with cosmic rays. However, we can search for such sources using the neutral messengers that are photons. Gamma rays are now observed up to the PeV energy range. The most complete gamma-ray sky map to date is that obtained by the Fermi-LAT satellite in the GeV range, which is shown in Galactic coordinates in Figure 6.

In this spherical coordinate system, the Galactic centre is in the middle of the map and the Galactic plane separates the northern and southern hemispheres. Most of the sources observed at GeV energies are located outside the Galactic Plane. They are mainly jetted active galactic nuclei, the brightest of which are blazars whose axis of emission is aligned with the line of sight, while the others are radio galaxies. The extragalactic universe in gamma rays is also populated by a dozen or so detected starburst galaxies, whose rate of star formation per unit stellar mass exceeds that of our Galaxy: more short-lived massive stars are formed there and end their lives in supernovae. Several hundred gamma-ray bursts have finally been detected in the extragalctic sky, mainly long gamma-ray bursts from explosions of massive stars, up to redshift $z>4$.

Diffuse emissions from our Galaxy can be seen firstly along the Galactic plane and secondly as peanut-shaped emissions on either side of the Galactic centre: the Fermi bubbles. These bubbles have a similar morphology to those observed in microwaves by the WMAP and Planck satellites and in X-rays by the eROSITA satellite. These large-scale structures are evidence of the past acceleration of cosmic rays in the Milky Way and of the diffusion of these charged particles in the Galactic magnetic field.

Finally, our Galaxy is populated by a myriad of stellar-sized sources. Hollow shells are the remains of supernovae (of which there are two types: core-collapse and thermonuclear, also known as SN1a). Core-collapse supernovae can leave a highly magnetised neutron star in their core after their explosion, which is known as a pulsar. The winds from these pulsars, known as pulsar wind nebulae, fill the space left by the supernova explosion and accelerate electrons and positrons, which re-radiate gamma rays. In more advanced stages of their lives, the diffusion of particles around the pulsar can even lead to extended emission over several degrees, known as TeV halos. Our galaxies also contain numerous gamma-ray sources arising from binary systems: recurrent nova, X-ray binaries and even microquasars, which are the stellar-scale analogue of the jetted active galactic nuclei.

Relativistic beaming¶

The existence of emission zones with a bulk velocity (relative to the central compact object) approaching the speed of light can be illustrated by the apparent superluminal motion of plasma blobs in astrophysical jets. This apparent velocity exceeding the speed of light is an effect that can be explained by purely geometrical arguments based on classical physics, as the following exercise illustrates.

Solution to Exercise 2

1. Following the notations in figure 8, the apparent velocity, $v_\mathrm{app}$, of the plasma blob is given by

$$\begin{align} v_\mathrm{app} &= \frac{L}{t_2-t_1} \\ &= \frac{L}{t_{i,2}-t_{i,1} - \frac{d_1-d_2}{c}} \\ &= \frac{H\sin\theta}{H/v - H\cos\theta/c} \\ &= \frac{v\sin\theta}{1 - \frac{v}{c}\cos\theta}, \end{align}$$

(14)

where $v$ is the physical speed of the plasma blob along the jet in the observer’s frame.

2. Using $\beta = \frac{v}{c}$ and $t = \tan(\theta/2)$, together with standard trigonometry gives:
   $$v_\mathrm{app} = \frac{2t\beta }{1+t^2 - (1-t^2)\beta}$$

   (15)
   So that after a bit of algebra, an apparent speed that is equal to a fraction $k$ of the speed of light reads
   $$v_\mathrm{app} = kc \Leftrightarrow \Big[(1+\beta)t - \beta/k\Big]^2 = (\beta/k - 1/\Gamma) \times (\beta/k + 1/\Gamma),$$

   (16)
   where $\Gamma = 1/\sqrt{1-\beta^2}$ is the Lorentz factor of the plasma blob.

The right-hand-side equation as a solution if and only if $\beta/k - 1/\Gamma \geq 0$ i.e. $k \leq \Gamma\beta$. As $k = v_\mathrm{app}/c$ and $\beta < 1$, one gets $\Gamma > v_\mathrm{app}/c$ i.e. a Lorentz factor larger than 4 for the jet of the active galactic nucleus 3C 279.

The relativistic motion of the plasma blob in jetted astrophysical sources results in anisotropic emission. The motion also increases the energy of the photons and the intensity of the radiation as the region moves towards the observer. The textbook derivation of this relativistic Doppler effect is based on velocity transformations. Alternatively, we use here a more straightforward approach based on the transformation of the energy of an emitted photon.

Assume an isotropic emission of energy $E'$ in the frame that is comoving with the emitting region, with corresponding four-momentum $[E' , p_x'c, p_y'c, p_z'c]$. The observer receives photons of energy $E$ in the lab frame along the $x$ direction, so the observed four-momentum is $[E, p_xc = E, p_yc = 0, p_zc = 0]$. We can get from one frame to the other by a Lorentz-boost $(\Gamma, \Gamma \vec{\beta})$ of the emitting region and a rotation by an angle θ from the direction of motion, i.e.

The first space-like component reads $p_x'c = \Gamma E(\cos\theta-\beta)$. Photons emitted in the forward direction $p_x'>0$ thus reach the observer within a cone defined by $\cos\theta > \beta$. In the ultra-relativistic limit, this inequality reads $1-\theta^2/2 > 1 - 1/(2\Gamma^2)$, which defines a cone of half-opening angle $\theta < 1 /\Gamma$. This corresponds to a half-opening angle $\theta_\mathrm{max} = \frac{1}{\Gamma} \approx 6^\circ \times \big( \frac{\Gamma}{10} \big)^{-1}$.

The time-like component equation reads $E' = \Gamma E (1-\beta\cos\theta)$. Thus, in the observer’s frame, the energy is enhanced by a Doppler factor $\delta = E / E'$ i.e.

This increase in energy $E=h\nu$ corresponds to an increase in frequency and therefore to a shortening of the observed time scales by a factor δ. In the ultra-relativistic limit and for a jet nearly aligned with the line of site (blazar-like source), this correspond to an energy enhancement or a time shortening by a factor $\delta \approx \frac{1}{\Gamma\big(1-[1-1/(2\Gamma^2)]\big)} \approx 2 \Gamma$ that is $\delta \approx 10 \times \big( \frac{\Gamma}{5} \big)$.

The specific intensity of the source, i.e. its brightness, is also enhanced. As $I_\nu/\nu^3$ is Lorentz invariant, the specific intensity is enhanced by a factor $\delta^3$ and the bolometric intensity, $I = \int_0^{+\infty} I_\nu \dd \nu$, is enhanced by a factor $\delta^4 \approx 10,000 \times \big( \frac{\Gamma}{5} \big)^4$.