Cosmic Ray Propagation

Abstract

One of the main goals of cosmic ray physics is to identify the sources and production/acceleration mechanisms of high energy cosmic particles. However, we can only observe the spectra and composition of the cosmic particle flux as it arrives at Earth. In general, flux and composition of the particles injected at the sources will be modified during the propagation from these sources to the detector at Earth. Therefore, it is important to understand and model the interactions of high energy particles propagating on cosmic scales ranging from our Galactic environment to intergalactic space up to cosmological distances and even within the sources, including the deflection of charged particles in cosmic magnetic fields. In the present chapter we will discuss the propagation of charged hadronic cosmic rays. After setting up the notation and providing the most important formulae for interaction rates and propagation, the first part focuses on propagation within our Galaxy which is dominated by diffusive processes. Interactions and energy losses are generally more prominent for propagation over extragalactic distances, which is the subject of the second part. The propagation of high energy photons will be the subject of Chap. 8 and high energy neutrinos will be discussed in Chaps. 9–12.

Appendices

Appendix 7

Some General Properties of Fokker–Planck and Diffusion Equations

Let us consider a simplified version of the Fokker–Planck equation (7.32) by only considering the momentum dependent part for ${\varvec{\nabla }}\cdot \mathbf{v}_c=0$ without the interactions. In three-dimensional notation for the distribution function $f(t,\mathbf{p})$ this reads

$$\begin{aligned} \frac{\partial f}{\partial t}={\varvec{\nabla }}_\mathbf{p}\cdot \left( D_{pp}{\varvec{\nabla }}_\mathbf{p}f-\dot{\mathbf{p}}f\right) \,, \end{aligned}$$

(7.126)

see also the discussion in Ref. [10]. This equation describes diffusion in momentum space with an additional force $\mathbf{F}=\dot{\mathbf{p}}$. The latter often has the nature of a drag force similar to the one encountered in Eq. (3.223). Let us thus assume

$$\begin{aligned} \mathbf{F}=\dot{\mathbf{p}}\sim -\frac{\mathbf{p}}{\tau _f(p)}\,, \end{aligned}$$

(7.127)

which is also known as dynamical friction , with a friction or relaxation time scale $\tau _f(p)$ which can also be momentum dependent. Such forces can arise, for example, due to scattering in the medium, as discussed in the context of charge carriers in a plasma discussed in Sect. 3.6.1 for which $\tau _f(p)$ is given by Eq. (3.235), due to gravitational forces, see Eq. (3.237), or due to radiative energy losses that will be the subject of Sect. 8.1. If $\tau _f(p)$ and $D_{pp}$ are momentum independent, as for an ordinary drag force and momentum independent diffusion, Eq. (7.126) has the stationary solution

$$\begin{aligned} f(\mathbf{p})\propto \exp \left( -\frac{\mathbf{p}^2}{2D_{pp}\tau _f}\right) \,. \end{aligned}$$

(7.128)

It is interesting to note that this is essentially obtained by substituting $t\rightarrow \tau _f/2$ in the standard time dependent solution of the diffusion equation  (7.25). More generally, one can find a stationary spherically symmetric solution of Eq. (7.126) by equating the expression in braces with zero. Separation of variables then gives

$$\begin{aligned} f(\mathbf{p})\propto \exp \left( -\int ^p dp^\prime \frac{p}{2D_{pp}(p)\tau _f(p)}\right) \,. \end{aligned}$$

(7.129)

Eq. (7.34) suggests that as long as $\tau _f(p)$ does not strongly grow with p the integral in the exponent increases with p. Therefore, momenta above a characteristic scale at which the exponent is of order unity will tend to be exponentially suppressed. This is in contrast to pure diffusion which would lead to unlimited growth in time, $\langle p^2\rangle \propto t$. As a consequence, dynamical friction limits the momenta at large time.

Problems

7.1

The Interaction Rate for a Given Cross Section

(a) Assume that a cosmic ray particle interacts with a mono-energetic beam of target particles that comes from a fixed direction, with a cross section $\sigma (s)$ which depends on the squared center of mass energy s, also known as the first Mandelstam-variable , see Eq. (1.119). Following the general definition of a cross section by Eq. (2.36) in Sect. 2.2.2, in the rest frame of the cosmic ray particle (unprimed quantities) its interaction rate is given by

$$R=j\sigma (s)\,,$$

where $\mathbf{j}=c_0{\varvec{\beta }}_b n$ is the flux density of the beam of target particles whose velocity is $c_0{\varvec{\beta }}_b$, corresponding to a target particle density n, and $j=|\mathbf{j}|=c_0\beta _bn$. Using Lorentz transformations, show that the interaction rate in the cosmic ray frame in which the cosmic ray moves with velocity $\mathbf{v}=c_0{\varvec{\beta }}$ (primed quantities) is given by

$$\begin{aligned} R^\prime =v(\beta ,\beta _b,\cos \theta )\,c_0n^\prime \sigma (s)\,, \end{aligned}$$

(7.130)

where $\theta$ is the angle between the CR velocity ${\varvec{\beta }}$ and the direction of the target particle beam ${\varvec{\beta }}_b$ and the relative velocity $v(\beta ,\beta _b,\cos \theta )$ between the cosmic ray and the target particle is given by Eq. (7.3). Use the fact that the components of the flux densities $(c_0n,\mathbf{j})$ and $(c_0n^\prime ,\mathbf{j}^\prime )$, respectively, in these two frames transform as a four-vector $j^\mu$.

(b) Use Eq. (7.130) to derive the general formula Eq. (7.1) for the interaction length of a CR in a background of particles that is isotropic in the cosmic ray frame,

$$ l(E)^{-1}=\frac{1}{\beta }\int d\varepsilon n_b(\varepsilon )\int _{-1}^{+1}d\mu \frac{v(\beta ,\beta _b,\mu )}{2}\,\sigma (s)\,, $$

where $\mu =\cos \theta$. Hints: Substitute $n^\prime \rightarrow n_b(\varepsilon )d\varOmega /(4\pi )=n_b(\varepsilon )d\cos \theta /2=n_b(\varepsilon )d\mu /2$.

7.2

Different Forms for the Mean Free Path Formula

Consider Eq. (7.1) for the mean free path for the interaction of a high energy particle with an isotropically distributed low energy photon background with an energy distribution $n_b(\varepsilon )$, and a cross section $\sigma \left[ s=m_\mathrm{CR}^2+2E\varepsilon (1-\mu \beta )\right]$. Express the integral over $\mu$ in this equation through an integration over the energy $\varepsilon _0$ of the target photons in the rest frame of the high energy particles (cosmic radiation). What are the integration limits for $\varepsilon$ if the interaction has a threshold $\varepsilon _\mathrm{th}^0$ in the rest frame of the high energy particle? For example the GZK effect /pion production has a threshold of $\varepsilon _\mathrm{th}^0\simeq 150\,$MeV. Show that the answer is

$$\begin{aligned} l(E)^{-1}=\frac{1}{2\varGamma ^2\beta ^2}\int _{\varepsilon _\mathrm{th}^0}^\infty d\varepsilon _0\sigma (\varepsilon _0)\varepsilon _0 \int _{\varepsilon _\mathrm{th}^0/[(1+\beta )\varGamma ]}^\infty \frac{d\varepsilon }{\varepsilon ^2} n_b(\varepsilon )\,, \end{aligned}$$

(7.131)

where $\varGamma =E/m$ is the Lorentz factor of the cosmic ray.

7.3

Energy Loss and Diffusion

Show that the rate of change of the total energy $U\equiv \int dE\,E\int d^3\mathbf{r}\,n(\mathbf{r},E)$ of a system described by the diffusion-energy loss equation  (7.12) is given by

$$\frac{dU}{dt}=\int dE\int \,d^3\mathbf{r}\,b(E)n(\mathbf{r},E)+\int dE\int \,d^3\mathbf{r}\,E\,\varPhi (\mathbf{r},E)\,,$$

provided $n(\mathbf{r},E)$ goes to zero sufficiently fast at the integration boundaries in energy and space.

7.4

The Syrovatskii Solution to the Homogeneous Stationary Diffusion-Energy Loss Equation

(a) Show that for a diffusion coefficient D(E) that does not depend on location and a stationary point source injecting the spectrum $Q(E)=\int d^3\mathbf{r}\,\varPhi (\mathbf{r},E)$ the stationary solution of the diffusion-energy loss equation  (7.12) for the number density spectrum at a distance r from the source with $\varPhi (\mathbf{r},E)=Q(E)\delta ^3(\mathbf{r})$ can be written as

$$\begin{aligned} n(r,E)=\frac{1}{[-b(E)]}\int _E^\infty \,dE^\prime \,\frac{Q(E^\prime )}{\left[ 4\pi \lambda (E,E^\prime )^2\right] ^{3/2}} \,\exp \left[ -\frac{r^2}{4\lambda (E,E^\prime )^2}\right] \,, \end{aligned}$$

(7.132)

where the Syrovatskii variable

$$\begin{aligned} \lambda (E_1,E_2)\equiv \left[ \int _{E_1}^{E_2}\,dE^\prime \,\frac{D(E^\prime )}{[-b(E^\prime )]}\right] ^{1/2}={\tilde{\lambda }}(E_1,E_2)^{1/2} \end{aligned}$$

(7.133)

is the average propagation distance traveled during the time in which the energy of the particle decreased from $E_2$ to $E_1$ which is the square root of the variable ${\tilde{\lambda }}$ introduced in Eq. (7.13). Note that the energy loss time is given by $t_E(E)=-b(E)/E$. The solution Eq. (7.132) is also known as the Syrovatskii solution  [445]. Hint: Use the fact that the function

$$G(t,\mathbf{r})=\frac{\exp \left( -\frac{r^2}{4Dt}\right) }{\left( 4\pi Dt\right) ^{3/2}}$$

solves the diffusion equation

$$\frac{\partial G(t,\mathbf{r})}{\partial t}-D\varDelta G(t,\mathbf{r})=0\,,$$

with the boundary condition $G(t\rightarrow 0,\mathbf{r})=\delta ^3(\mathbf{r})$. This is a special case of Eq. (7.25) when the diffusion tensor is a scalar, $D_{ab}=D\delta _{ab}$. Also note that differentiating Eq. (7.132) with respect to E involves differentiating both the integrand and the integration boundary. One can also first solve Eq. (7.14) in terms of the variables defined in Eq. (7.13).

(b) Show that in the CEL approximation the injection energy $E_i(E,t)$ which satisfies the differential equation $dE_i/dt=b(E)$ with initial condition $E_i(t_0)=E$ for a fixed propagation time $t_0-t$ satisfies

$$\frac{dE_i}{dE}=\frac{b(E_i)}{b(E)}\,.$$

Using this show that the Syrovatskii solution Eq. (7.132) can also be written as

$$\begin{aligned} n(r,E,t)=\int _{t_\mathrm{min}}^{t_0}\,dt\,\frac{Q\left[ E_i(E,t),t\right] }{\left[ 4\pi \lambda (E,t)^2\right] ^{3/2}}\, \frac{dE_i}{dE}\,\exp \left[ -\frac{r^2}{4\lambda (E,t)^2}\right] \,, \end{aligned}$$

(7.134)

where the time $t_\mathrm{min}$ is chosen sufficiently small that $E_i(E,t_\mathrm{min})$ is larger than the maximal injection energy such that the time integral becomes independent of $t_\mathrm{min}$ and where we have included the obvious generalization to a non-stationary, time-dependent source Q(E, t). Furthermore, the Syrovatskii variable can be expressed as

$$\begin{aligned} \lambda (E,t)\equiv \left[ \int _t^{t_0}\,dt^\prime \,D\left[ E_i(E,t^\prime )\right] \right] ^{1/2}\,. \end{aligned}$$

(7.135)

(c) Eq. (7.132) is the solution of the diffusion-energy loss equation which vanishes at infinity, $r\rightarrow \infty$. Show that if instead there is a boundary condition which requires the solution to vanish at some two-dimensional surface at finite distance Eq. (7.132) generalizes to

$$\begin{aligned} n(\mathbf{r},E)=\frac{1}{[-b(E)]}\int _E^\infty \,dE^\prime \,\frac{Q(E^\prime )}{\left[ 4\pi \lambda (E,E^\prime )^2\right] ^{3/2}} \,\sum _{i=-\infty }^{i=+\infty }(-1)^i\exp \left[ -\frac{(\mathbf{r}-\mathbf{r}_i)^2}{4\lambda (E,E^\prime )^2}\right] \,, \end{aligned}$$

(7.136)

where $\mathbf{r}_i$ are the positions of suitably chosen image charges where $\mathbf{r}_0=(x_0,y_0,z_0)$ is the position of the real source. For example, in a slab with boundary at $z=\pm L$ one has $\mathbf{r}_i=(x_0,y_0,(-1)^iz_0+2iL)$. This case is relevant for the distribution of electrons and positrons within the leaky box model of the Galaxy. As we will see in Sects. 8.1.2 and 8.1.3 electrons and positrons above ${\sim }100\,$GeV undergo significant energy loss during diffusion in the Galaxy.

7.5

The Propagation Theorem

(a) Show that in the limit of high source density $n_s$, corresponding to distances $d_s\sim n_s^{-1/3}$ between neighboring sources satisfying $d_s\ll \lambda (E_1,E_2)\sim [D(E)t_E(E)]^{1/2}$, the Syrovatskii solution Eqs. (7.132) and (7.134) integrated over all sources tends to the spectrum

$$\begin{aligned} n(E)=\frac{1}{[-b(E)]}\int _E^\infty \,dE^\prime \,\varPhi (E^\prime )=\int _{t_\mathrm{min}}^{t_0}\,dt\,\varPhi \left[ E_i(E,t)\right] \frac{dE_i}{dE}\,, \end{aligned}$$

(7.137)

where $\varPhi (E)=n_sQ(E)$ and in the last expression we have again generalized to a time-dependent source injection $\varPhi (E,t)$. Show that because this solution is location independent, it can also more directly be obtained from the continuous energy loss equation (7.9). The solution Eq. (7.137) is manifestly independent of the diffusion coefficient D (E) and thus of the propagation mode. This is known as the propagation theorem  [446] and Eq. (7.137) is called the universal cosmic ray spectrum .

(b) Use Eq. (7.137) to derive Eq. (7.10),

$$\begin{aligned} j(E)=&\frac{1}{4\pi H_0}\int _0^{z_{i,\mathrm max}}\frac{dz_i}{(1+z_i)^4 \left[ \varOmega _m(1+z)^3+\varOmega _r(1+z)^4+ \varOmega _k(1+z)^2+\varOmega _v\right] ^{1/2}}\nonumber \\&\qquad \qquad \quad \times \frac{dE_i(E,z_i)}{dE}\,\varPhi \left[ z_i,E_i(E,z_i)\right] \nonumber \end{aligned}$$

for the flux from a cosmological source distribution $\varPhi (z,E)$ by transforming from time to redshift applying the relevant cosmological relation.

7.6

The Liouville Theorem

The Liouville theorem in the absence of scattering states that the phase space density $f(t,\mathbf{r},\mathbf{p})$ of a given particle species is conserved,

$$ \frac{\partial f}{\partial t}+\dot{\mathbf{r}}\cdot {\varvec{\nabla }}_\mathbf{r}f+\dot{\mathbf{p}}\cdot {\varvec{\nabla }}_\mathbf{p}f=0\,,$$

see Eq. (7.15). Prove this by using the continuity equation in phase space,

$$\begin{aligned} \frac{\partial f}{\partial t}+{\varvec{\nabla }}_\mathbf{r}\left( \dot{\mathbf{r}}f\right) +{\varvec{\nabla }}_\mathbf{p}\left( \dot{\mathbf{p}}f\right) =0\,, \end{aligned}$$

(7.138)

and the Hamilton equations of classical mechanics, analogous to Eq. (2.63),

$$\begin{aligned} \frac{dx^i}{dt}=\frac{\partial H}{\partial p^i}\,,\quad \frac{dp^i}{dt}=-\frac{\partial H}{\partial x^i}\,, \end{aligned}$$

(7.139)

where $H(t,x^i,p^j)$ is the Hamilton function or Hamiltonian of the system characterized by the position vector $\mathbf{r}$ with components $x^i$ and the momentum vector $\mathbf{p}$ with components $p^j$. As is well known form the Lagrange and Hamilton formalisms for systems with a finite number of degrees of freedom N, the latter is constructed out of the Lagrange function $L(t,x^i,\dot{x}^j)$ by

$$\begin{aligned} H(t,x^i,p^j)=\sum _{i=1}^N p^i\dot{x}^i-L(t,x^i,p^j)\,, \end{aligned}$$

(7.140)

where $p^i\equiv (\partial L/\partial \dot{x}^i)$ are the canonically conjugated momenta . Equation (7.139) is equivalent to the equations of motion in the Lagrange formalism,

$$\begin{aligned} \frac{d}{dt}\frac{\partial L}{\partial \dot{x}^i}=\frac{\partial L}{\partial x^i}\,, \end{aligned}$$

(7.141)

which for a massive particle is essentially the spatial part of Eq. (2.148) in the presence of external forces $f^\mu$.

7.7

Pitch Angle Scattering

(a) Derive Eq. (7.43),

$$ \delta \alpha \simeq \frac{p_\perp }{2p_\parallel }\frac{\delta B}{B}\,, $$

for the change of the pitch angle $\alpha =\arctan (p_\perp /p_\parallel )$ under a change $\delta B$ of the magnetic field. Note that $\delta B$ and $p_\parallel$ can be negative, whereas $p_\perp$ and B are non-negative quantities.

(b) Derive Eq. (7.62) from Eq. (7.60) by using the definitions of the density n and the current j in Eq. (7.61).

(c) Derive Eq. (7.63) from the definition of the flux in Eq. (7.61). Hints: First perform an integration by parts in the definition of j, then integrate the Fokker–Planck equation Eq. (7.60) over $\mu$ from −1 to a given $\mu _0$ to obtain an equation for $\partial f(t,z,\mu _0)/\partial \mu$.

7.8

The Taylor-Green-Kubo Formulation of Diffusion

Derive Eq. (7.52) by performing suitable integrations by parts.

7.9

Drift Motions in a Magnetic Field

(a) Show that Eq. (7.84),

$$ \frac{dE}{dt}=\,\mathrm{const.}\,,\quad {\varvec{\beta }}_d=\frac{q\mathbf{F}\times \mathbf{B}+\frac{dE}{dt}{} \mathbf{F}}{q^2B^2+\frac{dE}{dt}}=\,\mathrm{const.} $$

is a solution of the equations of motion Eq. (7.83),

$$ \frac{d\mathbf{p}_d}{dt}=\frac{dE}{dt}{\varvec{\beta }}_d=\mathbf{F}+q\frac{\mathbf{p}_d}{E}\times \mathbf{B}\,. $$

(b) Demonstrate that for $F\ll |q|B$, the drift velocity is non-relativistic, $\beta _d\sim F/(|q|B)\ll 1$, and $dE/dt\sim \beta _d\beta _g|q|B\ll |q|B$ can be neglected in the equations above. Hint: Use

$$\begin{aligned} E\frac{dE}{dt}= & {} (\mathbf{p}_g+\mathbf{p}_d)\cdot \left( \frac{d\mathbf{p}_g}{dt}+\frac{d\mathbf{p}_d}{dt}\right) = \mathbf{p}_d\cdot \mathbf{F}+\mathbf{p}_c\cdot {\varvec{\beta }}_d\frac{dE}{dt}+\mathbf{p}_d\cdot \frac{d\mathbf{p}_c}{dt}\nonumber \\\sim & {} \mathbf{p}_d\cdot \mathbf{F}+E\beta _d\beta _c|q|B+\beta _d\beta _c E\frac{dE}{dt}\,, \end{aligned}$$

and the expression for ${\varvec{\beta }}_d$.

7.10

Cosmic Ray Interactions with the Galactic Gas

The interstellar density in our Galaxy is about 1 (mostly hydrogen) atom per cm$^3$ and the residence time of a high energy cosmic ray is of the order of $10^7$ years or smaller. Estimate the interaction probability of a cosmic ray during its propagation within the Galaxy.

7.11

The Z-burst Effect

Assume that primordial neutrinos today are non-relativistic and have a mass $m_\nu$. Calculate the threshold for production of a $Z^0$ boson through high energy neutrinos $\nu +\bar{\nu }\rightarrow Z^0$ and express it in terms of the mass $m_\nu$ of the primordial neutrinos. This process is also known as the Z-burst effect.