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Gennaro Miele » 13.Cosmic Rays - III

Cosmic Rays III – Outlines

Bottom up or top-down

Results on CR

CR acceleration

Source of CR

Bottom-up or top-down models

The current theories on the origin of CR can be broadly categorized into two distinct scenarios: the bottom-up acceleration scenario and top-down decay scenario. As the name suggests, the two scenarios are, in a sense, exact opposite of each other:

in the bottom-up scenario, charged practicles are accelerated from lower energies to the requisite high energies in certain special astrophysical environments (supernova remnants, rotating neutron stars, AGNs, radio galaxies)
in the top-down scenario, the energetic particles arise simply from decay of certain sufficiently massive particles originating from physical processes in the early universe (supe-heavy retics, topological defects, etc), and no acceleration mechanism is needed.

Bottom-up models for EHECR

Even if it is possible, in principle, to accelerate particles up to EHECR energies (up to 10²⁰ eV) in some astrophysical sources, is generally extremely difficult. In most cases it is hard to get the particles coming out from the dens regions placed inside and/or around the sources without losing much energy.

Moreover, the main problems with extragalactic sources of EHECR is that most of them lie at distances >> 100 Mpc from Earth. This is a problem if EHECR are conventional particles such as nucleons or heavy nuclei, since they lose energy drastically during their propagation from the source to the Earth, due to the Greisen-Zatsepin-Kuzmin (GZK) effect.

In fact, nucleons experience the photo-production of pions in the collisions with photons of the cosmic microwave background (CMB), and heavy nuclei get photo-disintegrated in the CMB and infrared background.

Top-down models for EHECR

The basic idea of a top-down origin of cosmic rays can be traced back to Georges Lemaître and his theory of “Primeval Atom”, which decayed to “atoms” of smaller and smaller atomic weights. Indeed, Lemaître regarded CR as the main evidential relics of the Primeval Atom in the present universe.

The main problem of the top-down models in general is that they are highly model dependent and invariably involve as-yet untested physics beyond the Standard Model of particle physics.
Moreover, a general prediction of top-down theories is a very high energy spectrum dominated by photons and neutrinos which, however, seems not confirmed by experimental observations.

In the following, we will focus on the bottom-up scenario.

General consideration

CR at energies below ~1 GeV are temporally correlated with the solar activity, which is a direct evidence for their origin at the sun.
At higher energies the flux observed at Earth exhibits a temporal anticorrelation with solar activity, indicating an origin outside the solar system.
Several arguments suggest that the bulk of CR between 1 GeV and at least up to the knee region is confined to the Galaxy and is probably produced in supernova remnants (SNR).
The ankle is sometimes interpreted as a crossover from a galactic to an extragalactic component. Beyond 10¹⁹ eV, CR are generally expected to have an extragalactic origin.

Low-energy CR: composition

About 0.25% of primary cosmic rays are light elements (lithium, beryllium and boron), which are essentially absent as final products of stellar nucleosynthesis. This is an evidence for these light elements being produced in collisions of heavier particles with interstellar matter, that is spallation of carbon and oxygen (Li, Be, B), and of iron (Sc, Ti, V, Cr, Mn).
Medium elements (carbon, nitrogen, oxygen and fluorine) are about 10 times their abundance in normal matter and the heavier elements are 100 times more abundant than normal matter. This suggests an origin of cosmic rays in areas of space with greatly enriched amounts of heavy elements.

Differential and integral spectrum

The exponent of the energy in the differential flux of an element is the differential spectral index

$\frac{dN}{dE}\propto E^{-\alpha}$

The integral fluxes are defined as the number of particles above the threshold energy, E

$N(> E)\propto E^{-\gamma}=E^{-(\alpha -1)}$

Low-energy CR: spectrum

Energy distributions of primary elements are well described by an inverse power low in energy with very similar slopes

$\frac{dN}{dE}\propto E^{-(\gamma +1)}$

where γ is the integral spectral index and α=γ+1 is the differential spectral index

with γ∼1.7 up to 10 6 GeV, and γ∼2.0 above this energy.

All secondary nuclei have significantly steeper spectra than the primary nuclei; in fact, the ratio of secondary to primary nuclei is observed to decrease approximately as E^-0.5 with increasing energy. This can be interpreted to mean that the higher energy cosmic rays diffuse faster out of the Galaxy.

Low-energy CR: cosmic clocks

The measurement of the ratio between unstable and stable isotopes of secondary nuclei allows to determine the residence time of CR in the Galaxy. In particular, the most used isotope is ¹⁰Be which has a lifetime of 3.9 10⁶ yr.

CR at the knee

The results of the KASCADE data for the all-particle spectrum do not depend on the considered interaction model (QGSJet or SIBYLL), and indicate that the knee is caused by the decreasing flux of the light primaries (p+He). The fact that the composition becomes heavier at energies of the order of 10¹⁵ eV can indicate that the bulk of CR goes from galactic protons (produced by SNR and trapped by magnetic field) to galactic heavy nuclei (protons are no more trapped at these energies).
SIBYLL suggests a more prominent contribution of heavy primaries than QGSJet at high energy, but there are indications that both models require some adjustment.

High-energy CR

Charged particles above 10¹⁸ eV cannot be trapped by galactic magnetic fields. They can easily escape the host galaxy.
No known acceleration mechanism in the Galaxy (supernova remnants etc.) can accelerate particles to these energies. (Maybe magnetars-pulsars with very high surface magnetic fields).
Particles above 10¹⁹ eV should mainly be of extragalactic origin.
Possible sources for UHECR: jets powered by massive black holes in AGN, gamma-ray bursts (merging neutron stars), topological defects.

CR lifetime in the Galaxy

Estimates show that CR particles cannot pass through more than 50 kg/m² of material, because even the particles with the very highest masses will have broken up. We can use this number to calculate the lifetime of cosmic rays in the disk of our galaxy, assuming that particles are protons which travel to the speed of the light.

$X = \rho L = \rho; c ;\tau_r \longrightarrow \tau_r=\frac X {\rho c}$

$\rho = n m_p = 10^6 m^{-3}1.67\cdot 10^{-27}kg=1.67\cdot 10^{-21}kg ;m^{-3}$

where n = number density in the ISM

$\tau_r=\frac{50 kg; m^{-2}}{1.67\cdot 10^{-21} kg; m^{-3}3\cdot 10^8 ms^{-1}}\cong 10^{14}\scong 3\cdot 10^6 yr$

It could be 10-100 times longer in the halo of the galaxy where the density is lower.
Since the disc of our galaxy is only 1 kpc, it would take only 3000 years for the particles to escape (at v~c), if it were not for the Galaxy magnetic field.

(1 kpc = 3 10⁹ m)

Magnetic confinement

Galactic magnetic field is ~ 3 µG and parallel to the spiral arm. The Larmor radius for a relativistic particle of charge Ze is

$R(cm) = \frac{E(eV)}{300 Z B(G)} \longrightarrow R(kpc)=\frac{E(EeV)}{ZB(mu G)}$

Particles with Z=1 cannot be confined when their energy is > 50 EeV

Power

The numerical density of CR at a given energy, E, is given by

$n(E)=\frac{J(E)}{\beta c}=\frac{4pi}{\beta c}\frac{d\phi}{dE}(E)$

where $\frac{d \phi}{dE}(E)$ is particles per unit area, angle, time, and energy.

Then, the energy density is

$\rho=\int En(E)dE=\frac{4pi}{\beta c}\int E\frac{d\phi}{dE}(E)dE\cong 1 e V cm^{-3}$

where $1eV cm^{-3}$ is local interstellar medium (LIM).

If we assume that this holds also for the galactic disk, the power required to supply all the galactic CR is

$L_{CR}=\frac{V_d;\rho}{\tau_r}=\frac{pi R^2d\rho}{\tau_r}\cong\frac{\pi (15 kpc)^2(200 pc)(1 e V cm^{-3})}{3;10^6 \text{years}}\cong 10^{41} erg s^{-1}$

Is there enough power?

Assuming ~ 10⁸ erg per nova and a rate of about 100 per year, we obtain a CR production rate of 10¹⁰/3 10⁷ erg s^-1, that is ~ 3 10² erg s^-1.
Our Sun emits cosmic rays during solar flares, with energy of 10¹⁰ to 10¹¹ eV. The average rate is only ~ 10²⁴ erg s^-1. Since there are 10¹¹ stars in the Galaxy, we obtain a total of only 10³⁵ erg s^-1.

Synchrotron radiation is observed from supernovae, indicating that particles with very high energies must be involved. The total particle energy in a supernova explosion is estimated to be about 10⁴⁹ erg (per supernova). If one supernova explodes every 100 years, this would release 10⁴⁹/3 10⁹ erg s^-1, that is ~ 3 10³⁹ erg s^-1. Supernovae are also renowned for producing heavy elements of course.

The rotational energy of a neutron star of 1.4 MS with a period of 10 ms and radius of 10⁶ cm is ~ 2 10⁵⁰ erg s^-1.

Acceleration mechanisms

Direct acceleration mechanisms (by an electric field of a rotating neutron star or of an accretion disk threaded by magnetic fields) are not widely favored, for the main reason that it is difficult to obtain the power-law spectrum in any natural way.
The basic idea of the statistical acceleration mechanism originates from a paper by Fermi in 1949: charged particles bounce off moving interstellar magnetic fields and either gain or lose energy, depending on whether the “magnetic mirror” is approaching or receding. In a typical environment, particles would be accelerated on average. This random process is now called 2nd order Fermi acceleration, because the mean energy gain per “bounce” is dependent on the “mirror” velocity squared.
Bell (1978) and Blandford and Ostriker (1978) independently showed that Fermi acceleration by SNR shocks is particularly efficient, because the motions are not random. A charged particle can pass through the shock and then be scattered back by magnetic inhomogeneities behind the shock, in this way gaining energy. Bouncing back and forth again and again, it gains energy each time. This process is now called 1st order Fermi acceleration, because the mean energy gain is dependent on the shock velocity only to the first power

Fermi mechanism

Consider a process in which a test particle increases its energy by an amount proportional to its energy with an “encounter”, gaining ∆E = ξ E per encounter.

After n encounters:

$E_n=E_0(1+xi)^n$

Number of encounters needed to reach the energy E:

$n=\ln\left(\frac E{E_0}\right)\big \ln(1+xi)$

Particles accelerated to energies greater than E are:

$N(\geq E)=\sum_{m=n}^\infty(1-P_{esc})^m=\frac{(1-P_{\esc})^n}{P_{\esc}}\propto \frac 1 {P_{esc}}\left(\frac E{E_0}\right)^{-\gamma}$

where:

$\gamma=\ln\left(\frac 1 {1-P_{esc}}\right)\big\ ln(1+xi))\approx\frac{P_{esc}}xi=\frac 1 xi \frac{T_{cycle}}{T_{esc}}$

After a time t, with n=tT_cycle encounters:

$E \leq E_0(1+xi)^{t/T_{cycle}}$

Average fractional energy gain

After a few scatterings the average motion of the particle coincides with that of the gas.

In the rest frame of the gas:

$E'_1=\gamma E_1(1-\beta \cos \theta_1)$

Back to the lab frame:

$E_2=\gamma E'_1(1+\beta \cos\theta'_2)=\gamma^2E_1(1-\beta \cos\theta_1)(1+\beta\cos\theta'_2)$

where E’₂=E’₁ since scattering is collisionless. Note that we are assuming almost relativistic particles, E ∼ cp.

$\frac{\Delta E}{E_1}=\frac{(1-\beta\cos\theta_1)(1+\beta\cos\theta'_2)}{1-\beta^2}-1$

where 1 - β²is particles can either gain or lose energy depending on the sign of cosθ’₂; particles can only gain energy since cosθ’₂>0.

1^st and 2^nd order Fermi acceleration

Uniform in all directions and normalized in [-1,1]

$\frac{dn}{d\cos \theta'_2}=\frac 1 2$

An isotropic flux projected onto a plane and normalized in [0,1]

$\frac{dn}{d\cos \theta'_2}=2\cos \theta'_2$

Normalized in [-1,1]

$\frac{dn}{d\cos \theta_1}=\frac{c-V\cos\theta_1}{2c}$

Normalized in [-1,0]

$\frac{dn}{d\cos\theta_1}=2\cos\theta_1$

Gas cloud

Non relativistic plane shock front

1^st and 2^nd order Fermi acceleration

Particles can either gain or lose energy, but after many encounters there is a net gain

$xi=\left\langle \frac{\Delta E}{E_1}\right\rangle_{1,2}=\frac{1+1/3\beta^2}{1-\beta^2}-1\approx\frac 4 3 \beta^2$

An encounter always results in an energy gain

$xi=\left\langle \frac{\Delta E}{E_1}\right\rangle_{1,2}=\frac{1+4/3\beta+4/9\beta^2}{1-\beta^2}-1\approx\frac 4 3 \beta$

Galactic candidates?

The standard theory of CR acceleration (Diffusive Shock Acceleration Mechanism) is based on 1^st order Fermi acceleration mechanism. An important feature is that particles emerge out with a power low spectrum with an index depending only on the shock compression ratio.

$\gamma_{2nd;order}=\frac{P_{esc}}{xi}=\frac 1 {xi}\frac{T_{cycle}}{T_{acc}}=\frac 1 {4/3\beta^2T_{acc}}\frac 1 {c\rho\sigma}$

$\gamma_{1st;order}=\frac{P_{esc}}xi=\frac{4u_2/c}{4/3\beta}=\frac 3 {u_1/u_2-1}$

Galactic candidates?

$\gamma_{1st;order}=\frac{u_1/u_2+2}{u_1/u_2-1}$

valid both for relativistic and non-relativistic case.

Shock are frequent in astrophysics: in the interplanetary space, in supernovae, in radio-galaxies. However, SNRs are, maybe, the only serious candidate, both in terms of power and in terms of E_max.

Pulsar, neutron stars in close binary systems, hot spots in the highly relativistic jets from merger and accretion induced collapse of compact stellar objects (microblazars) have also been proposed, but the discussion is open yet.

The accelerated protons in SNRs would produce neutral pions, decaying in γ-rays, while the accelerated electrons would emit synchrotron X-rays.

Supernova as an emitter

Cosmic Ray Production in Supernova Remnants

Hillas plot: its derivation

The maximum energy supplied to a particle with charge Ze is

$E_{max}=\int Ze\varepsilon dx$

If the energy is given by induction

$\nabla \times \vec E=-\frac 1 c\frac{\partial\vec B}{\partial t}\longrightarrow \frac\varepsilon L\cong\frac{B\omega_0}c$

where ε is the electrical field induced in a region of length L.

We obtain

$E_{max}\cong Ze\frac{BL\omega_0}cL=Ze\beta_0BL$

$E_{max}(EeV)\cong Z\beta_0B(\mu G)L(kpc)$

Hillas Plot

Irrespective of the precise acceleration mechanism, there is a simple dimensional argument, given by Hillas, which allows one to restrict attention to only a few classes of astrophysical objects as possible sources capable of accelerating particles to a given energy. In any statistical acceleration mechanism, there must be a magnetic field, B, to keep the particles confined within the acceleration site. The size R of the acceleration region must be larger than the diameter of the orbit of the particle. One gets the general condition

$E(EeV)AGNs\long\rightarrow E(EeV)< Z\beta B(\mu G)R(kpc)$
The above condition also applies to direct acceleration scenarios in which the electric field arises due to a moving magnetic field.

AGNs

An active galactic nucleus (AGN) is a compact region at the centre of a galaxy
which has a much higher than normal luminosity over some or all of the
electromagnetic spectrum (radio, infrared, optical, ultra-violet, X-ray and/or γ-ray).
A galaxy hosting an AGN is called an active galaxy.

AGNs

An active galactic nucleus (AGN) is a compact region at the centre of a galaxy which has a much higher than normal luminosity over some or all of the electromagnetic spectrum (radio, infrared, optical, ultra-violet, X-ray and/or γ-ray).
A galaxy hosting an AGN is called an active galaxy.

The radiation from AGN is believed to be a result of accretion onto the supermassive black hole at the centre of the host galaxy.

γ-rays above 10 TeV have been observed from a certain class of AGN, BL Lacertae.
These can be produced by the decay of pions from the interaction of accelerated protons.
On the other end, the fast variability of flares observed in AGN may favor the acceleration of electrons. Moreover, accelerated protons are severely degraded due to the interactions with the central engine.

Radio-galaxies (hot-spots)

Radio galaxies and their relatives, radio-loud quasars and blazars, are types of active galaxy that are very luminous at radio wavelengths. The radio emission is due to the synchrotron process.

Radio-galaxies (hot-spots)

Radio galaxies and their relatives, radio-loud quasars and blazars, are types of active galaxy that are very luminous at radio wavelengths. The radio emission is due to the synchrotron process.

The observed structure in radio emission is determined by the interaction between twin jets and the external medium, modified by the effects of relativistic beaming.

The maximum energy of even up to ~ 10²¹eV seems to be possible.

The main problem with radio-galaxies is their location: large cosmological distances (> 100 Mpc) from the Earth.

Pulsars

Pulsars are highly magnetized rotating neutron stars which emit a beam of detectable electromagnetic radiation in the form of radio waves. Their observed periods range from 1.5 ms to 8.5 s. The radiation can only be observed when the beam of emission is pointing towards the Earth.

Pulsars

Most of the acceleration scenarios involving pulsars rely upon direct acceleration of particles in the strong electrostatic potential drop at the surface of the neutron star.

The maximum potential drop for typical pulsars can in principle be as high as ~ 10²¹ eV.

However, in realistic models, this potential drop would be significantly short-circuited by electrons and positrons moving in the opposite directions along the field lines.

Magnetars

A magnetar is a neutron star with an extremely powerful magnetic field (about 100 to 1000 times stronger than the Crab Nebula pulsar), the decay of which powers the emission of copious amounts of high-energy electromagnetic radiation, particularly X-rays and gamma-rays.

Magnetars

There are indications of the existence of pulsars with dipole magnetic fields approaching ~ 10¹⁵ G.

It is not clear if the energy loss processes, a generic problem with acceleration around compact object, can be gotten around.

Accretion disks

An accretion disk is a structure formed by diffuse material in orbital motion around a central body which, in our case, can be a neutron star or a black hole.
Material in the disk spirals inward towards the central body. Gravitational energy released in that process is transformed into heat and emitted at the disk surface in form of electromagnetic radiation.

The acceleration is produced by large electric fields produced by induction in the disk.

However, energy loss through interaction of the accelerated particles with the ambient photon field prevents the maximum achievable energy from exiding ~ 10¹⁵ eV.

GRBs

Gamma-ray bursts (GRBs) are the most luminous events occurring in the universe since the Big Bang. They are flashes of gamma rays emanating from seemingly random places in deep space at random times. The duration of GRBs usually ranges between a few milliseconds to minutes, often followed by an afterglow emission at longer wavelengths (X-ray, ultra violet, optical, infrared, and radio).

The majority of observed GRBs appear to be collimated emissions from the collapsing cores of rapidly rotating, highmass stars into black holes. A subclass of GRBs (the “short” bursts) appear to originate from a different process, the leading candidate being the collision of neutron stars orbiting in a binary system.

GRBs most likely contribute a negligible fraction to the low energy CR flux.