Chapter 1 Introduction
3.3 Cosmic Ray Transport Equation
Before we start with the modern theory of particle transport in the ICM, we will summarise the original idea for particle acceleration. We will then give a motivation of the modern theory of particle transport in a magnetised plasma.
3.3.1 The Classical Idea
In their seminal paper Fermi (1949) introduce the idea of random scattering of relativistic particles of magnetic irregularities in the interstellar medium. At that time magnetic fields in the ISM were not observed, but Alv´en had introduced his velocity of waves in a magnetised liquid. Still it was unclear how relativistic particles were injected.
3.3 Cosmic Ray Transport Equation 53
From the observations of molecular Calcium lines and their Doppler broadening, it was assumed that there was proper motion of ”magnetic clouds” in the ISM. The authors concluded that weak fields would be adverted with the flow which would not only result in a tangled shape of the field, but also let cosmic-rays scatter off these irregularities, similar to a magnetic bottle (figure 3.9).
For a particle with velocity v =βc and clouds in random motion with velocity V and average separation L, the rate of collision is then:
ν±= v±V
L , (3.44)
where the ν+ is for forward, head-on (a, figure 3.9) collision andν− for backward (b, figure 3.9) collision. Even though the energy gain/loss for a cloud with velocity V is the same, head-on collisions are more probable than backward collisions. This results in a net energy gain of the CR particle by this stochastic process (see also Eilek & Hughes, 1991). With a mean momentum change ∆p=γmV and v ≈c, this energy gain is then
dE
dt = (ν+−ν−)γmcV (3.45)
≈ 2V2
Lc E. (3.46)
This is second order in V and is therefore called second order Fermi acceleration. In the case of only head-on collisions:
dE
dt =ν+γmcV (3.47)
≈ V
LE. (3.48)
we have first order Fermi acceleration, the basis for shock acceleration and subsequent CR injection. In the modern case this process is buried in the collision integral of the Boltzmann equation, as we will see in the next chapter.
3.3.2 The Modern Theory
In what follows we will introduce the basic equations governing our description of the evolution of cosmic-rays in the ICM. There are extensive discussions of this in the literature (e.g. Melrose, 1980; Eilek & Hughes, 1991; Schlickeiser, 2002; Brunetti, 2004a), especially instructive is the article by Eilek & Hughes. We follow Lifshitz & Pitaevskii (1981);
Blandford (1986) in our derivation, which is sloppy in some parts. A more strict derivation from the Vlasov-Maxwell system is found in Schlickeiser (2002), but is far beyond the scope of this work. Specific models, i.e. application of the theory to galaxy clusters and radio halos, are subject of the next sections.
In statistical physics one recognises the inability to describe a complex (classical) system in all details, e.g. the trajectory of every particle of a many body system like a gas. Instead,
54 3. Theoretical Considerations
one turns to a statistical description of the system by a distribution function f(r,p, t)7, usually under the assumption of quasi-ergodicity8.
It follows then, that the total number of particles in phase space Ω is:
Ntotal(t) = Z
Ω
f(r,p, t) drdp (3.49)
The fundamental evolution equation is given by the Boltzmann equation:
∂f
∂t +x˙ · ∂f
∂x +m¨x· ∂f
∂p = ∂f
∂t
collision
+ ∂f
∂t
diffusion
, (3.50)
which describes changes due to drift and external forces (second + third term), effects due to collision (fourth term) and effects due to diffusion (last term). In a Lagrangian description we can neglect the second term. For our application gravity and large scale electric fields do not play a role and we treat magnetic fields as scattering agents9, so we setx¨= 0.
The collision term then carries most of the physics relevant for our application. Fol-lowing Lifshitz & Pitaevskii (1981) the collision term describes the rate of change of the distribution function in a volume element of phase space. Formally this is expressed by the collision integral, which is gains - losses for a 6-dim. volume element10:
∂f
∂t
collision
= Z
(w0f0f10 −wf f1) d3qd3p0d3q0, (3.51) where f(t,p) and f1(t,q) are the distribution functions of the two scattering agents (CR electrons and photon, MHD waves, etc) before and after (primed) scattering and d3q the volume element in momentum space of species 1. w(p,q) marks the probability per unit time to change the momentump→p−qof the particles in the distribution functionf(p).
We can simplify this further by assuming that there is no back-reaction on the scattering agents distribution function f1(q), i.e. it remains unchanged by the collision. Equation 3.51 then simplifies to:
∂f
∂t
collision
= Z
(w(p+q,q)f(t,p+q)−w(p,q)f(t,p)) d3q (3.52)
7which can be understood just as a histogram of all particles in 6 dimensional phase space
8We do not comment on that here, but refer the reader to the textbooks.
9This is where Schlickeiser (2002) does much better, for the price of filling pages with 25 coefficients of a non-linear PDE.
10This requires the concept of detailed balance. We refer the reader to Lifshitz & Pitaevskii (1981)
3.3 Cosmic Ray Transport Equation 55
Assuming that changes of the distribution function due to scattering are small, we can use a Taylor expansion:
w(p+q,q)f(t,p≈w(p,q)f(t,p) +q· ∂
∂pw(p)f(t,p) + 1
2qαqβ ∂2
∂pα∂pβ
w(p,q)f(t,p), (3.53) where α and β refer to the two scattering species. So :
∂f
∂t = ∂
∂pα
Aαf + ∂
∂pβ (Bαβf)
= ∂
∂pα
Aα+ ∂Bαβ
∂pβ
f +Bαβ ∂f
∂pβ
(3.54) Aα =
Z
qαw(p,q) d3q (3.55)
Bαβ = 1 2
Z
qαqβw(p,q) d3q. (3.56)
Equation 3.54 can be understood as the divergence of a vector in α and is a continuity equation in momentum space. We can understand the coefficients Aα and Bαβ by means of time average characteristics:
Aα =X
qα/δt (3.57)
Bαβ =X
qαqβ/(2δt), (3.58)
as a first and second order loss or gain term to the particles or their distribution function f. The sums are to be understood over a large number of collisions during time δt. This way Aα can be identified with the first order systematic energy loss terms from sections 3.1.1, 3.1.2 and 3.1.3. Bαβ describes a second order process that gives a systematic energy change and another non-linear term. This part of equation 3.54 can be identified with a diffusion equation with Bαβ the diffusion coefficient.
This can be understood considering that the first term on the RHS of 3.54 induces a systematic energy gain/loss of the particle distribution. For example a translation of the spectrum to higher or lower momenta (if the term is chosen appropriately), figure 3.10 upper right. The second stochastic term in 3.54 induces abroadening of the spectrum (figure 3.10 lower left). AsBαβ acts on both terms a stochastic momentum gain will always be coupled with a systematic momentum gain. This is the principle of reacceleration (figure 3.10, lower right).
The spatial diffusion term is formally given similar to the second part on the right side of 3.54 :
∂f
∂t
diffusion
= ∂
∂xα
Kαβ ∂f
∂xβ
, (3.59)
56 3. Theoretical Considerations
0 200 400 600 800 1000
p 0.0
0.2 0.4 0.6 0.8 1.0
f(p) Original
0 200 400 600 800 1000
p 0.0
0.2 0.4 0.6 0.8 1.0
f(p) Translated
0 200 400 600 800 1000
p 0.0
0.2 0.4 0.6 0.8 1.0
f(p) Broadened
0 200 400 600 800 1000
p 0.0
0.2 0.4 0.6 0.8 1.0
f(p) Translated
& Broadened
Figure 3.10: Upper left: Gaussian spectrum, Upper right: Translated Gaussian spectrum by a systematic momentum gain. Lower left: Broadened Gaussian spectrum by stochastic momentum gain. Lower right: Translated and broadened spectrum by systematic and stochastic momentum gain.
where Kαβ is the spatial diffusion coefficient. It is often modelled so that:
∂n(E, t)
∂t
diffusion
=Qe(E, t)− Te(E, t)
n(E, t). (3.60)
This is a simple description for particle additionQe(E, t) and particle removalTe(E, t) from a Lagrangian volume element. These two terms are also referred to as injection function and catastrophic loss function.
The Fokker-Planck Equation: The important assumption we made so far is the one of no back-reaction on the scattering agents and the Taylor expansion ( eqn. 3.53). These premises are equivalent with demanding quasi-linear behaviour of the transport process.
We will now make the second assumption of isotropy of the distribution function and write n(E, t) = 4πp2f(p, t) dp/dE. We then combine this with eqn. 3.54, 3.50 and 3.59,