working on acceleration

lectures.tex

@@ -42,41 +42,44 @@
 
 \tableofcontents\label{sec:contents}
 
-\chapter{Radiative Processes in Astroparticle Physics}
-\input{sections/leptonic}
-\input{sections/hadronic}
-\newpage
-
-\chapter{A Primer on Plasma Astrophysics}
-\input{sections/idealmhd}
-\input{sections/mhdwaves}
-\input{sections/qlt}
-\input{sections/transportequation}
-\newpage
+%\chapter{Radiative Processes in Astroparticle Physics}
+%\input{sections/leptonic}
+%\input{sections/hadronic}
+%\newpage
+%
+%\chapter{A Primer on Plasma Astrophysics}
+%\input{sections/idealmhd}
+%\input{sections/mhdwaves}
+%\input{sections/qlt}
+%\input{sections/transportequation}
+%\newpage
 
 \chapter{Particle Acceleration}
-\input{sections/acceleration}
+\input{sections/acceleration_intro.tex}
+\input{sections/acceleration_generalities.tex}
+\input{sections/acceleration_second.tex}
 \newpage
 
-\chapter{Particle Transport in Galactic environments}
-\input{sections/pillar}
-\input{sections/protons}
-\input{sections/implications}
-\newpage
-
-\chapter{The Physics of Galactic Sources}
-\input{sections/sources}
-\newpage
+%\chapter{Particle Transport in Galactic environments}
+%\input{sections/pillar}
+%\input{sections/protons}
+%\input{sections/implications}
+%\newpage
+%
+%\chapter{The Physics of Galactic Sources}
+%\input{sections/sources}
+%\newpage
 
 \appendix
 
 \newpage
 \chapter{Appendix}
-\input{sections/app_larmor}
-\input{sections/app_radtransfer}
-\input{sections/app_intensity}
-\input{sections/app_thermodynamics}
-\input{sections/app_rkconditions}
+%\input{sections/app_larmor}
+%\input{sections/app_radtransfer}
+%\input{sections/app_intensity}
+%\input{sections/app_thermodynamics}
+%\input{sections/app_rkconditions}
+% app_collision
 \newpage
 
 % --- Bibliography ---

sections/acceleration.tex

@@ -1,266 +1,8 @@
-% !TEX root = ../main.tex
-\section{How to accelerate cosmic particles?}
+% !TEX root = ../lectures.tex
 
-{\color{red}To be done}
-
-% The presence of non-thermal particles is very common in the Universe: • Solar wind
-% • Supernova remnants
-% • Active galaxies
-% • Gamma-Ray Bursts
-% • Pulsar Wind Nebulae
-% The presence of magnetized plasma is tightly connected to non-thermal particles.
-%
-%What we need is a system which satisfy these condition:
-%
-%\begin{itemize}
-%\item \textbf{Large energetics}: we must take energy from somewhere! Kinetic energy translational in SNRs, roatitional in Pulsars, Gravitational energy in accretion disks, ...
-%
-%\item {Enough confinement time}: The particle has to stay in the accelerator for the time needed to accelerate it.
-%
-%\item {Lack of significant energy-losses}: Accelerating particles is useless if the loose energy too quickly.
-%
-%\item {A mechanism for energy transfer}: How to transfer energy from macroscopic objects into the (microscopic) acceleration of particles $\rightarrow$ we need to use electromagnetic.
-%
-%\end{itemize}
-%
-%While we have several candidates to supply the needed energy, having large scale, surving long enough, and with sufficiently low density, to solve the first three problems, the actual mechanism is trickier, and it was addressed for the first time by Enrico Fermin in 1949.
-%
-%Remember that all known acceleration mechanisms are electromagnetic in Nature. Since magnetic fields cannot make work on charged particles, one needs electric fields. 
-%
-%However, the only two possibilities are:
-%%
-%\begin{itemize}
-%\item
-%\textit{Regular acceleration}: we have that $\langle \vec{E} \rangle\neq 0$,  so we have to violate the conditions of ideal MHD,  which is very difficult.
-%\item
-%\textit{Stochastic acceleration}: in this case we respect the condition $\langle \vec{E}\rangle=0$,  but we have that $\langle \vec{E}^2\rangle\neq 0$.  This is the so called \textbf{second order Fermi acceleration}
-%\end{itemize}
-
-\section{Generalities of stochastic acceleration}
-
-%%% CGPT
-Consider a cyclic process in which particles gain energy, requiring a time \( \tau \) per cycle. Each cycle has an escape probability \( P_{\text{esc}} \) and an average fractional energy gain per cycle \( \xi \).
-
-At each cycle, a particle with initial energy \( E_n \) has a probability \( 1-P_{\text{esc}} \) of being accelerated to \( E_{n+1} = (1 + \xi) E_n \). Thus, the energy of a cosmic particle after \( n \) acceleration cycles is:
-%
-\begin{equation}
-E_n = E_0 (1 + \xi)^n
-\end{equation}
-
-The number of cycles needed to reach an energy \( E_n \) from an initial energy \( E_0 \) is given by:
-%
-\begin{equation}
-n = \frac{\ln \left( E_n/E_0 \right)}{\ln (1 + \xi)}
-\end{equation}
-
-This implies that attaining higher energies requires a greater number of cycles.
-
-Assuming a constant escape probability per encounter, the probability for a particle to remain in the acceleration region after \( n \) encounters is \( (1-P_{\text{esc}})^n \).
-
-Over time, the cumulative fraction of particles with energies exceeding \( E \) can be computed using the sum of a geometric series with ratio \( x = (1-P_{\text{esc}}) \), leading to:
-%
-\begin{equation}
-f(>E) 
-\propto \sum_{m=n}^\infty (1 - P_{\rm esc})^m
-= \frac{(1-P_{\text{esc}})^n}{P_{\text{esc}}} = \frac{(1-P_{\text{esc}})^{\frac{\ln \left( E_n/E_0 \right)}{\ln (1 + \xi)}}}{P_{\text{esc}}} 
-\end{equation}
-
-By utilizing the identity \( a^{\ln b} = b^{\ln a} \), we arrive at:
-%
-\begin{remark}
-\begin{equation}\label{Eq:slopegeneralized}
-f (>E) \propto \frac{1}{P_{\text{esc}}} \left( \frac{E}{E_0} \right)^{\gamma} \,\,\, \text{where} \,\,\, \gamma = \frac{\ln (1-P_{\text{esc}})}{\ln (1+\xi)} \simeq -\frac{P_{\text{esc}}}{\xi}
-\end{equation}
-\end{remark}
-
-Here we used the approximation, \( \xi \ll 1 \) and \( P_{\text{esc}} \ll 1 \). Notice that this approach results in a power-law distribution for both first- and second-order Fermi mechanisms.
-
-The maximum energy achievable in this statistical model is constrained by the finite lifetime of the accelerator, corresponding to a maximum of \( n \sim T / \tau \) cycles. Another limiting factor could be an increase in the escape probability with energy, such as in scenarios involving energy losses, which eventually counterbalances the energy gain.
-%%% CGPT
-
-\section{Second-Order Fermi Mechanism}
-
-%%% CGPT
-In 1949, Fermi proposed a physical system where this mechanism for particle acceleration can take place. In particular, he postulated the existence of an inhomogeneous interstellar medium, hence the presence of \emph{magnetic clouds} moving in random directions relative to the Galactic frame. These clouds, carrying magnetic fields, can reflect incoming charged particles.
-
-The acceleration mechanism works as follows: \emph{particles gain energy when they encounter a magnetic cloud moving towards them and lose energy in encounters with clouds moving away}. {\color{red}Aggiungi plot.} Due to the greater frequency of head-on encounters compared to tail-on ones, there is an overall increase in energy.
-
-To calculate the energy gain or loss per encounter, we use a double change of reference frame. We denote quantities in the cloud frame with primes and those in the Galactic frame without.
-
-A test particle with initial energy \( E \) encounters a magnetic cloud moving with a velocity factor \( \beta = V/c \) along $x$. An observer on the cloud sees the following\footnote{In this context, we simplify for relativistic particles, thus \( p \simeq E \).
-}:
-%
-\begin{equation}
-E' = \gamma (E - \beta p_x) = \gamma E \left(1 - \beta \frac{p_x}{E} \right) = \gamma E \left(1 - \beta \mu_{\rm in} \right)
-\end{equation}
-%
-where \( -1 \le \mu_{\rm in} \le 1 \) is the cosine of the angle between particle velocity and cloud velocity. 
-
-Upon reflection by the cloud, the particle's energy, as observed externally, becomes:
-%
-\begin{equation}
-E^{\prime\prime} 
-= \gamma E^\prime (1+ \beta \mu^\prime_{\rm out}) 
-= \gamma^2 E \left[ 1 - \beta \mu_{\rm in} + \beta \mu^\prime_{\rm out} - \beta^2 \mu_{\rm in} \mu^\prime_{\rm out} \right]
-\end{equation}
-%
-clearly, if $\beta$ is the cloud velocity in the Galactic frame, $-\beta$ is the Galactic frame velocity with respect to the cloud.
-
-Since magnetic fields do not perform work on the particles, the particle undergoes only elastic scattering within the cloud. This means its energy upon exiting the cloud remains unchanged in the cloud's frame of reference, represented as \( E'_f = E'_i \).
-
-The relative change in energy is:
-%
-\begin{equation}
-\frac{\Delta E}{E} = \frac{E'' - E}{E} =
-\gamma^2  \left[ 1 - \beta \mu_{\rm in} + \beta \mu^\prime_{\rm out} - \beta^2 \mu_{\rm in} \mu^\prime_{\rm out} \right] - 1
-%= \frac{ \beta^2 - \beta \mu_{\rm in} + \beta \mu^\prime_{\rm out} - \beta^2 \mu_{\rm in} \mu^\prime_{\rm out}}{1-\beta^2} 
-%simeq 2\beta^2 + 2\beta \mu
-\end{equation}
-
-This result shows that the energy gain is proportional to the initial energy, meaning \( \Delta E/E \) is independent of \( E \).
-
-It is crucial to recognize that both energy gain and loss are possible in this mechanism. This variability arises because the movements of both particles and magnetic clouds (consequently, the angles of interaction \( \mu \) and \( \mu' \)) are random. 
-%
-However, not all configurations are equally probable.
-
-Given that a particle undergoes multiple scatterings off magnetic irregularities within the cloud, its exit direction becomes randomized, with an average \( \langle \mu'_{\rm out} \rangle = 0 \). Initially, we can average over the exit angle to get:
-%
-\begin{equation}
-\left\langle \frac{\Delta E}{E} \right\rangle_{\mu^\prime} = 
-\gamma^2 \left[ 1 - \beta \mu_{\rm in} \right] - 1
-\end{equation}
-
-Eventually, we must consider averaging over all possible initial angles. The rate at which a particle collides with a cloud is proportional to their relative velocity, leading to:
-%
-\begin{equation}
-P(\mu) \propto v_{\rm{rel}} \propto 1 - \beta \mu_{\rm in} \rightarrow P(\mu) = \frac{1}{2} (1 - \beta \mu_{\rm in})
-\end{equation}
+\end{document}
 
-Here we assumed \( v \approx c \) and we normalized so that the total probability equals one:
-%
-\begin{equation}
-A \int_{-1}^{+1} d\mu (1 - \beta \mu) = 1 \rightarrow A = \frac{1}{2}
-\end{equation}
-
-Notice that \( \int_{\mu < 0} d\mu P(\mu) = 1 + \beta / 2 \) is larger than \( \int_{\mu > 0} d\mu P(\mu) = 1 - \beta/2 \), which means that \emph{head-on collisions} are more frequent compared to \emph{tail-on collisions}, which is the essence of Fermi’s acceleration mechanism.
-
-Consequently, the average change in energy is given by:
-%
-\begin{remark}
-\begin{equation}
-\left\langle \frac{\Delta E}{E} \right\rangle_{\mu\mu'} 
-= \int_{-1}^{+1} d\mu \, P(\mu) \left[ \gamma^2 \left( 1 - \beta \mu \right) - 1 \right] \simeq \frac{4}{3} \beta^2
-\end{equation}
-\end{remark}
-
-Therefore, we have demonstrated that, on average, the energy variation in Fermi’s mechanism is \emph{positive}.
-%
-This confirms that Fermi's mechanism effectively accelerates charged particles. However, the average energy change is proportional only to \( \beta^2 \) underlining the \emph{stochastic nature} of the energy gain process.
-
-Note that in this scheme the magnetic field's primary role is to alter the direction of particle motion, but the magnetic field itself does not provide the energy to increase the particle energy. Instead, the energy is supplied by an induced electric field~{\color{red}approfondisci}.
-
-Considering \( \beta = u/c \), with \( u \sim v_A \sim 10 \) km/s, the fractional energy change per encounter \( \frac{\Delta E}{E} \) turns out to be around \( 10^{-8} \), indicating a rather inefficient acceleration process.
-
-Indeed, for this mechanism to be a viable candidate for accelerating particles to the high energies observed in cosmic rays, it must do so efficiently.
-%
-Let's define the acceleration time, \( \tau_{\rm acc} \), as:
-%
-\begin{equation}
-\tau_{\rm acc} = \left( \frac{1}{E} \frac{dE}{dt} \right)^{-1}
-\end{equation}
-
-Assuming a typical distance \( L \) between two clouds and no magnetic field in between (making our estimate a lower limit), the average time between two encounters is \( \tau_{\rm c} = L/c \).
-
-Neglecting the time particles spend inside the cloud:
-%
-\begin{equation}
-\frac{dE}{dt} \simeq \frac{\Delta E}{\tau_{\rm c}} = \frac{4}{3} \frac{\beta^2 c E}{L} \rightarrow \tau_{\rm acc} = \frac{3}{4} \frac{L}{c} \beta^{-2} 
-\end{equation}
-
-With typical values of \( \beta \sim 10^{-4} \) and \( L \sim 1 \) pc, it becomes evident that this mechanism would require nearly $\tau_{\rm acc} \sim $~Gyrs for a particle to double its energy. 
-%
-This timescale is far too long to explain the very high energies observed in Galactic cosmic rays.
-In fact, energy losses in the ISM, such as ionization losses or spallation, typically occur more rapidly than the acceleration process postulated by Fermi, rendering the process even less efficient.
-
-Moreover, the energy spectrum resulting from Fermi's original acceleration mechanism would depend on the ratio of $\tau_{\rm acc}$ to the energy-independent escape timescale $\tau_{\rm esc}$ (see Eq.~\ref{Eq:slopegeneralized}). {\color{red}Spiega meglio.}
-
-This ratio is inherently unpredictable, as it varies based on the specific properties of the magnetic clouds and the regions where these clouds are densely concentrated. Consequently, different areas within the Galaxy could potentially accelerate cosmic rays with varying power law distributions. When combined, these contributions are unlikely to produce a singular, coherent power law spectrum akin to what is observed for Galactic cosmic rays on Earth.
-
-This inconsistency is another drawback of the Fermi mechanism. In contrast, acceleration at shocks, which we will discuss next, circumvents both of these issues. 
-
-Finally, we notice that although we do not believe that the bulk of cosmic-ray acceleration in our Galaxy is due to this mechanism, in several models of Galactic transport a similar mechanism is responsible of a tiny re-energization of the already accelerated cosmic rays.
-
-%Furthermore, the energy gain does not depend on \( B \), the magnetic field strength. While the magnetic field mediates particle reflection, it does not directly appear in the Lorentz transformations.\todo{Approfondisci}
-
-%As mentioned above, another possible way of working out the acceleration the to calculate the electric field seen in the Galactic frame by Lorentz transformation of the pure B field seen in the cloud frame. Since the two approaches must be equivalent the acceleration and the energy gain of the particle must also be independent of the cloud magnetic field in this case. This result is however far less intuitive with this approach.
-
-\subsection{Second-order Fermi re-acceleration in the Fokker-Planck approach}
-
-{\color{red}To be done}
-
-%\begin{mdframed}
-%\subsection*{Re-acceleration in the Fokker-Planck approach}
-%
-%Widely used for description of stochastic processes.
-%%
-%Let's define the probability that particle with momentum $\vb p$ at time $t$ changes momentum by $\Delta \vb p$ in time $\Delta t$.
-%
-%The phase space distribution function is $f(\vb x, \vb p, t)$ probability to find particle in phase space volume element d3xd3p.
-%
-%Using this defition
-%%
-%\begin{equation}
-%f(\vb p, t+\Delta t) = 
-%\int d^3(\Delta \vb p) \, f(\vb p - \Delta \vb p, t) P(\vb p - \Delta \vb p, \Delta \vb p)  
-%\end{equation}
-%
-%Taylor expansion gives (to 2nd order in small $\Delta p$):
-%%
-%\begin{equation}
-%f(\vb p, t) + \frac{\partial f}{\partial t} \Delta t \simeq \int d^3(\Delta \vb p) \, \left[ f P - \frac{\partial (fP)}{\partial p_i} \Delta p_i + \frac{1}{2} \frac{\partial^2 (fP)}{\partial p_i \partial p_j} \Delta p_i \Delta p_j + \dots \right]
-%\end{equation}
-%
-%We impose now the normalization condition for the probability
-%%
-%\begin{equation}
-%\int P(\vb p, \Delta \vb p) d^3 (\Delta p) = 1
-%\end{equation}
-%%
-%and we define the Fokker-Planck coefficients as
-%%
-%\begin{eqnarray}
-%\langle \Delta p_i \rangle & = & \int d^3 (\Delta p) P(\vb p, \Delta \vb p) \Delta p_i \\
-%\langle \Delta p_i p_j \rangle&  = & \int d^3 (\Delta p) P(\vb p, \Delta \vb p) \Delta p_i \Delta p_j 
-%\end{eqnarray}
-%%
-%which leads to
-%%
-%\begin{equation}
-%\frac{\partial f}{\partial t} = 
-%- \frac{\partial}{\partial p_i} \left( \frac{\langle \Delta p_i \rangle}{\Delta t} f \right) 
-%+ \frac{1}{2} \frac{\partial^2}{\partial p_i \partial p_j} \left( \frac{\langle \Delta p_i \Delta p_j \rangle}{\Delta t} f \right)
-%\end{equation}
-%%
-%with 1st term (rhs) describing systematic energy gain/losses, and 2nd diffu- sion part/dispersion/broadening.
-%
-%If scattering process is reversible in the sense that:
-%%
-%\begin{equation}
-%P(\vb p, \Delta \vb p) = P(\vb p - \Delta \vb p, \Delta \vb p)
-%\end{equation}
-%
-%\dots
-%
-%Fokker-Planck eq. then reduces to a diffusion equation in momentum space:
-%%
-%\begin{remark}
-%\begin{equation}
-%\frac{\partial f}{\partial t} = \frac{\partial}{\partial p_i} \left( D_{ij} \frac{\partial f}{\partial p_j} \right)
-%\end{equation}
-%\end{remark}
-%%
-%where $D_{ij}$ are the components of the diffusion tensor.
-%\end{mdframed}
+\input{sections/app_collisions}
 
 \section{First-Order Fermi Mechanism or Diffusive Shock Acceleration}