6  Extra-Galactic Cosmic Rays: The Knee and Beyond

The Knee

At energies of about 5 \(\times 10^{15}\; \mathrm{eV}\) a steepening in the spectrum from \(\gamma \sim 1.7 \rightarrow \gamma \sim 2\) known as the knee takes place. Already Peters in 1959 concluded that it could be due to:

  • Consequence of the breakdown of an acceleration mechanism.
  • Increased rate of escape from the galaxy at high energies.
Code 
import crdb
import matplotlib.pylab as plt
plt.rcParams['font.family'] = "STIXGeneral"
plt.rcParams.update({'axes.labelsize': 20})
plt.rcParams.update({'legend.fontsize': 20})
plt.rcParams.update({'figure.figsize': [8, 6]})
plt.rcParams['xtick.labelsize'] = 18
plt.rcParams['ytick.labelsize'] = 18
plt.rcParams['xtick.major.width'] = 1.5
plt.rcParams['xtick.minor.width'] = 1
plt.rcParams['xtick.major.pad'] = 8
plt.rcParams['xtick.direction'] = 'in'
plt.rcParams['ytick.major.size'] = 10
plt.rcParams['ytick.minor.size'] = 5
plt.rcParams['ytick.major.width'] = 1.5
plt.rcParams['ytick.minor.width'] = 1
plt.rcParams['ytick.major.pad'] = 8
plt.rcParams['ytick.direction'] = 'in'
plt.rcParams['legend.frameon'] = False
plt.rcParams['lines.linewidth'] = 1.5
plt.rcParams['axes.linewidth'] = 1.5

import numpy as np

from crdb.experimental import convert_energy

elements = ("H", "He", "C", "N", "O", "Si", "Fe")
elements += ("1H-bar", "e-+e+", "AllParticles")

tabs = []
for energy_type in ("EKN", "ETOT"):
    for elem in elements:
        tab = crdb.query(
            elem,
            energy_type=energy_type,
            energy_convert_level=1,
        )
        if energy_type == "EKN":
            tab = convert_energy(tab, "EK")
        tabs.append(tab)
tab = np.concatenate(tabs).view(np.recarray)

with np.errstate(divide="ignore"):
    mask = (tab.err_sys[:, 0] > 0) & (tab.err_sta[:, 0] / tab.value < 0.5)
tab = tab[mask]

xlim = 1e4, 1e11

fig, ax = plt.subplots(1, 1, figsize=(6, 5))
for elem in elements:
    tab = tab[(xlim[0] < tab.e) & (tab.e < xlim[1])]
    ma = tab.quantity == elem
    if len(tab) == 0:
        continue  
    color = "k" if elem == "AllParticles" else None
    t = tab[ma]
    if len(t) == 0:
        continue
    f = t.e**2.7
    sta = np.transpose(t.err_sta)
    color = "k" if elem == "AllParticles" else None
    ax.errorbar(t.e, t.value * f, sta*f, fmt=".", color=color, label=elem)

ax.loglog()
ax.set_ylabel(r"$E_k^{2.7}$ d$J$/d$E_k$ [GeV$^{1.7}$ / (m$^2$ s sr)]")
ax.set_xlabel(r"$E_k$ [GeV]")
ax.grid()
ax.legend(frameon=False, loc="lower center", ncol=2)

The Knee of the CR spectrum

The Knee of the CR spectrum

A third explanation could be a change in CR interactions at \({\sqrt s}\sim\) few TeV. The first two explanantions produce a rigidity dependent knee, ie the position of the knee for different nuclei depends on \(Z\), while the third explanation will depend on \(A\). Experimentally the rigidity dependence is favored.

Experimentally at these energies we cannot observe cosmic-rays in a direct way. We need to start looking at their interactions with the atmosphere (see lecture 3 on air-shower physics). This imposses limitation on the precision of the cosmic ray composition. In particular different models of hadronic interactions have to be assumed.

The Ankle and the End of the Spectrum

  • A proton of energies \(10^{18}\) eV has a gyroradious of a kpc in a typical magnetic field which hints at an extra-Galactic origin for these energies.

  • Greissen-Zatsepin and Kuz-min predicted that at energies of \(\sim 10^{19}\) eV will interact with the low energy photons of the CMB. This interaction leads to a supression of flux above \(5\times10^{19}\) eV unless the sources are within a few tens of Mpc. This supression is referred as GZK cutoff.

High Energy Cosmic Ray Composition

  • Composition of the high energy CR spectrum involves only two archetypes: light nuclei (protons) and heavy nuclei (iron).
  • The plots above show Auger / HiRes measurements near GZK cutoff, all favoring at least a mixed composition tending toward heavy at the higher energies.

Sources of Extra Galactic Cosmic Rays

As we shaw, CR in supernova remmants or blast waves can only accelerate CR up to 100 Z TeV. In order to explain CR beyond this energy, one has to invoke other processes such as Non-Linear Diffusion Acceleration, or extremely high magnetic fields (as suggested in Hillas plot).

Binary systems in which a compact object (black hole, neutrino star, pulsar) is permanentely dragging material for an accompaying object (normal star or galaxy) and whirled into an accreation disk can generate enourmouse plasma motions with very strong electromagnetic fields. The image belowe shows an artistic representation of 4U 0614+091, a X-ray binary.

The Disk Dynamo

Black holes or neutrino stars will have matter acretting around them. Due to the gravitational pull, matter will be ripped of in molecules, atoms, and ultimately elementary charge particles. The energy gain of infalling protons will correspond to the variation in the gravitational potential. If we equal the variation of gravitational potential to the kinetic energy of the accretting matter we have in the classical approach:

\[\frac{1}{2}m_p v^2 = \Delta E = -\int^{R}_\infty G\frac{m_p M}{r^2} = G\frac{m_p M}{R} \rightarrow v =\sqrt{\frac{2 GM}{R}},\]

where \(M\), \(R\), are the mass and radius of the central compact object.

  • For a neutron star (\(M\approx 2\times10^{30}\) kg, \(R= 10\) km): \(\frac{\Delta E}{m_p} \sim 1.32\times 10^{20} \;\mathrm{erg/g}\)

  • For a black hole (\(M\approx 10^8 M_\odot\), \(R = R_S = 2\frac{GM}{c^2}\)): \(\frac{\Delta E}{m_p} \sim 5 \times 10^{20} \;\mathrm{erg/g}\)

The variable magnetic field of the neutron stars or black holes are perpendicular to the direction of the accreation disk generating a Lorentz force:

\[\vec{F} = e (\vec{v} \times \vec{B}) = e\vec{E}\]

So the energy obtained is

\[E=\int \vec{F}d\vec{s} = e v B\Delta s,\]

where \(\Delta s\) is the distance over which the force acts. Under plausible assumptions (\(v \sim c\), \(B= 10^6\) T, \(\Delta s = 10^5\) m) energies of \(3\times 10^{19}\) eV are possible.

Candidates of Extra Galactic Cosmic Rays Sources

The two main candidates for ExtraGalactic Cosmic Rays are:

  • Active Galactic Nuclei (AGN)
  • Gamma Ray Bursts

Active Galactic Nuclei (AGNs)

  • Discovered in 1932 by K. Jansky looking for noise in transatlantic radio transmission for the Bell Telephone Labs. He found a persistent noise in the radio from the centre of the Galaxy too loud to be due to thermal black body radiation.

  • 1953 Ginzburg & Shklovski suggested it was due to synchrotron radiation from highly relativistic electrons, confirmed with discovery of predicted polarization in M87 light.

  • Sandange labeled 3C48 a quasar or quasi-stellar object (it appeared pointlike).

  • In 1962 3C273 radio source position was found with precision of 1 arcsec, which allowed to find the optical counterpart at z = 0.158 (not 1 star but a galaxy).

  • In 1963 Hoyle and Fowler speculated that the tremendous emitted energy is due to the gravitational collapse of a very massive object.

AGN Classification
  • There is two broad classes: Radio quiet (90%) and Radio Loud (10%) depending on the pressence of jets or not.
  • The unified model of ANGs suggests that different AGNs are infact the same object seen from different angles.

Gamma-Ray Bursts

  • GRBs are short bursts lasting a few seconds of \(\gamma\)-ray photons from 0.1 - 1 MeV.

  • They were discovered in the 60s by the U.S. Vela satellites, which were built to detect gamma radiation pulses emitted by nuclear weapons tested in space as the US suspected the URSS might carry on secret nuclear tests despite the Nuclear Test Ban Treaty.

  • They have been hypothesed (given their occurence) to have caused mass extintions events (thousand times since life begun), in particular they are associated with the Ordovician–Silurian extinction.

  • There is some observational evidence suggesting that progenitor of a GRB are stars undergoing a catastrophic energy release by the end of their lifes \(\rightarrow\) Hypernovas

The accepted phenomenological pictured of GRBs is of an expanding relativistic wind fireball dissipating kinetic energy. The observed afterglow on some GRBs result from the collision of the expanding fireball and the surroundings.

In the fireball, the observed radiation is produced by synchrotron emission of shock accelerated electrons, similar to SNRs. Hence, it is likely that protons will be also shock accelerated. The two conditition for GRBs to be sources of UHECR are:

  1. The proton acceleration time must be smaller that the wind expansion time, ie, you need to accelerate protons during the burst duration.

  2. The proton synchrotron loss time must exceed the acceleration time, otherwise you loose energy before you accelerate them.

These two conditions lead to a constrain in the Lorentz boost factor for GRBs:

\[\gamma \geq 130\left(\frac{E}{10^{20}\mathrm{\; eV}}\right)^{3/4}\left(\frac{0.01\mathrm{\; s}}{\Delta t}\right)^{1/4}\]

which matches what we see from GRBs. However IceCube has not seen any neutrino associated with GRBs which puts in tension the idea that GRBs can be the only sources of UHECR.

References