diff --git a/lectures.tex b/lectures.tex index 94f9e11..99cb1d6 100644 --- a/lectures.tex +++ b/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 --- diff --git a/sections/acceleration.tex b/sections/acceleration.tex index 15df024..81002cc 100644 --- a/sections/acceleration.tex +++ b/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} diff --git a/sections/acceleration_generalities.tex b/sections/acceleration_generalities.tex new file mode 100644 index 0000000..ce759f1 --- /dev/null +++ b/sections/acceleration_generalities.tex @@ -0,0 +1,57 @@ +% !TEX root = ../lectures.tex +\section{Generalities of Stochastic Acceleration} + +Stochastic acceleration, a cornerstone of high-energy astroparticle physics, describes the random and iterative energy gains experienced by particles within astrophysical accelerators. Let us explore the underlying principles in a structured manner. + +In a stochastic acceleration process: +% +\begin{itemize} +\item Particles gain energy in \emph{cyclic encounters}, with each cycle requiring a characteristic time \( \tau \). +\item During each cycle, there is a probability \( P_{\text{esc}} \) that a particle escapes the acceleration region, and a probability \( 1 - P_{\text{esc}} \) that it remains for further acceleration. +\item The fractional energy gain per cycle is denoted by \( \xi \), implying that after one cycle, a particle with initial energy \( E_n \) becomes \( E_{n+1} = (1 + \xi) E_n \)~. +\end{itemize} + +Over \( n \) cycles, a particle's energy evolves geometrically, leading to: +\[ +E_n = E_0 (1 + \xi)^n, +\] +where \( E_0 \) is the initial energy. Consequently, the number of cycles required to reach a final energy \( E_n \) from \( E_0 \) is: +\[ +n = \frac{\ln \left( E_n/E_0 \right)}{\ln (1 + \xi)}. +\] + +Therefore, in this model, attaining significantly higher energies demands a \emph{larger number of acceleration cycles}. + +Now consider the cumulative probability of a particle remaining in the acceleration region after \( n \) cycles: +\[ +P_{\text{survive}} = (1 - P_{\text{esc}})^n. +\] +s +The fraction of particles that reach energies exceeding a threshold \( E \) can be determined by summing the probabilities for particles that survive \( m \geq n \) cycles. Using the geometric series with ratio \( x = (1 - P_{\text{esc}}) \), we obtain: +\[ +f(>E) \propto \sum_{m=n}^\infty (1 - P_{\text{esc}})^m = \frac{(1-P_{\text{esc}})^n}{P_{\text{esc}}}. +\] +Substituting for \( n \) using the energy relation: +\[ +f(>E) \propto \frac{(1-P_{\text{esc}})^{\frac{\ln \left( E/E_0 \right)}{\ln (1 + \xi)}}}{P_{\text{esc}}}. +\] + +Utilizing the logarithmic identity \( a^{\ln b} = b^{\ln a} \), this expression simplifies to a power-law distribution: +\[ +f(>E) \propto \frac{1}{P_{\text{esc}}} \left( \frac{E}{E_0} \right)^{\gamma}, \quad \text{where} \quad \gamma = \frac{\ln (1-P_{\text{esc}})}{\ln (1+\xi)}. +\] + +The parameter \( \gamma \) governs the spectral slope and can be approximated under the assumptions \( \xi \ll 1 \) and \( P_{\text{esc}} \ll 1 \): +\[ +\gamma \simeq -\frac{P_{\text{esc}}}{\xi}. +\] + +Despite its simplicity, this framework highlights a universal feature of stochastic acceleration: it naturally produces a \emph{power-law energy distribution}, which is ubiquitous in astrophysical observations. + +The maximum energy achievable by this stochastic process is constrained by two primary factors: +\begin{itemize} +\item Finite Lifetime of the Accelerator: The total time available for acceleration, \( T \), limits the number of cycles \( n \), with \( n_{\text{max}} \sim T / \tau \), +\item Energy-Dependent Escape Probability: In realistic scenarios, the escape probability \( P_{\text{esc}} \) often increases with energy due to competing processes such as energy losses or dynamic changes in the accelerator. These factors eventually counteract the energy gains, capping the achievable energy. +\end{itemize} + +Stochastic acceleration serves as a fundamental model for understanding cosmic particle spectra. Future sections will explore specific implementations, including Fermi second-order acceleration. \ No newline at end of file diff --git a/sections/acceleration_intro.tex b/sections/acceleration_intro.tex new file mode 100644 index 0000000..58c6709 --- /dev/null +++ b/sections/acceleration_intro.tex @@ -0,0 +1,38 @@ +% !TEX root = ../lectures.tex +\section{How to accelerate cosmic particles?} + +{\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} + diff --git a/sections/acceleration_second.tex b/sections/acceleration_second.tex new file mode 100644 index 0000000..7454300 --- /dev/null +++ b/sections/acceleration_second.tex @@ -0,0 +1,203 @@ +% !TEX root = ../lectures.tex +\section{Second-Order Fermi Mechanism} + +{\color{red}Aggiungi plot.} + +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 fundamental principle of the second-order Fermi mechanism is straightforward: \emph{particles gain energy when they encounter a magnetic cloud moving towards them and lose energy in encounters with clouds moving away}\footnote{This behavior mirrors a general property of elastic collisions in mechanics (see ~\cref{sec:app_collisions} for details on Newtonian collisions).}. Due to the greater frequency of head-on encounters compared to tail-on ones, there is an overall \emph{increase} in energy. + +To quantify the energy change during a single interaction, we employ a double reference frame transformation: quantities measured in the magnetic cloud rest frame are denoted by primes, while those in the Galactic frame remain unprimed. + +A particle with initial energy \( E_{\rm i} \) and momentum \( p_{\rm i} \) in the Galactic frame encounters a magnetic cloud moving along the \( x \)-axis with a velocity factor \( \beta = V / c \). We transform to the cloud rest frame, where the particle energy is given by\footnote{For simplicity, we have assumed the particle is relativistic, where \( p \simeq E \) (using units where \( c = 1 \))}: +\begin{equation} +E_i^\prime = \gamma (E_i - \beta p_{i,x}) = \gamma E_i \left( 1 - \beta \frac{p_{i,x}}{E_i} \right) = \gamma E_i \left( 1 - \beta \mu_i \right) +\end{equation} +where \( p_{i,x} \) is the component of the particle momentum along the cloud motion direction, and \( \mu_i = p_{i,x} / p_i = \cos\theta_i \) is the cosine of the angle between the particle velocity and the cloud velocity in the Galactic frame. + +%- \( -1 \leq \mu_{\rm in} \leq 1 \), corresponding to all possible directions of particle motion relative to the cloud. + +Upon reflection by the cloud, the particle energy, as observed externally, becomes: +\begin{equation} +E_f = \gamma E_f^\prime \left(1+ \beta \mu^\prime_f \right) +\end{equation} +where \( \mu^\prime_f \) is the cosine of the angle \emph{after} reflection in the cloud frame. 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, so that \( E^\prime_f = E^\prime_i \), and thus +% +\begin{equation} +E_f = \gamma^2 E_i \left(1 - \beta \mu_i + \beta \mu^\prime_f - \beta^2 \mu_i \mu^\prime_f \right) +\end{equation} + +The relative change in energy after one encounter is: +% +\begin{equation} +\frac{\Delta E}{E} = \frac{E_f - E_i}{E_i} = +\gamma^2 \left(1 - \beta \mu_i + \beta \mu^\prime_f - \beta^2 \mu_i \mu^\prime_f \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 \( \mu_i \) and \( \mu_f \)) 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 \( \langle \mu^\prime_f \rangle = 0 \). +% +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_i \right) - 1 +\end{equation} + +Next, we must consider averaging over all possible initial angles. Since the frequency of particle-cloud collisions depends on their relative velocity, the probability distribution for the incoming angle is: +\begin{equation} +P(\mu) \propto v_{\rm{rel}} \propto 1 - \beta \mu_i +\end{equation} + +Normalizing this probability over \( \mu_i \in [-1,1] \), we get: +\begin{equation} +P(\mu) = \frac{1 - \beta \mu_i}{2} +\end{equation} + +Notice that \( \int_{\mu < 0} d\mu P(\mu) = 1 + \beta / 2 \) is larger than \( \int_{\mu > 0} d\mu P(\mu) = 1 - \beta/2 \), implying 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} + +This confirms that Fermi's mechanism effectively accelerates charged particles: while individual interactions may result in energy gains or losses, the \emph{average energy change is positive}. However, the average energy gain is proportional to \( \beta^2 \), highlighting the fundamentally \emph{stochastic nature} of the process. + +For the second-order Fermi mechanism to account for the high-energy particles observed in cosmic rays, it must accelerate particles efficiently. Let us assess its viability within the ISM, where the typical cloud velocity is of the order of the Alfvén speed, \( v_A \sim 10 \, \text{km/s} \). In terms of \( \beta = v / c \), this translates to \( \beta \sim 10^{-4} \). + +The fractional energy change per encounter is therefore: +\[ +\frac{\Delta E}{E} \sim \frac{4}{3} \beta^2 \sim 10^{-8}, +\] +indicating a highly inefficient energy gain mechanism. + +To further quantify this inefficiency, we define the \emph{acceleration time}, \( \tau_{\rm acc} \), as the characteristic timescale for a particle to significantly increase its energy: +\[ +\tau_{\rm acc} = \left( \frac{1}{E} \frac{dE}{dt} \right)^{-1}. +\] + +Assuming the average distance between magnetic clouds is \( L \), and neglecting the magnetic field in the space between clouds (providing a lower limit), the average time between two encounters is: +\[ +\tau_{\rm c} = \frac{L}{c}. +\] + +The rate of energy gain is then: +\[ +\frac{dE}{dt} \simeq \frac{\Delta E}{\tau_{\rm c}} = \frac{4}{3} \frac{\beta^2 c E}{L}. +\] + +From this, the acceleration time can be expressed as: +\[ +\tau_{\rm acc} = \frac{3}{4} \frac{L}{c \beta^2}. +\] + +Using typical ISM values, \( \beta \sim 10^{-4} \) and \( L \sim 1 \, \text{pc} \), we find: +\[ +\tau_{\rm acc} \gtrsim \text{Gyr}~, +\] +just for the particle to \emph{double its energy}. This timescale is vastly longer than the lifetimes of most astrophysical accelerators and is far too slow to explain the high energies observed in Galactic cosmic rays. In fact, particles in the ISM are subject to various energy loss mechanisms, such as ionization losses, that occur on timescales much shorter than \( \tau_{\rm acc} \), further diminishing the effectiveness of the second-order Fermi mechanism. + +Another issue lies in the resulting energy spectrum. From previous discussions (see equation~\ref{Eq:slopegeneralized}), the spectrum slope depends on the ratio of the acceleration timescale \( \tau_{\rm acc} \) to the escape timescale \( \tau_{\rm esc} \). This ratio is highly variable across different regions of the Galaxy, depending on the density and velocity of magnetic clouds. As a result, the energy spectrum varies significantly between regions, thereby, when summed over the Galaxy, the contributions from different regions are unlikely to produce a coherent, universal power-law spectrum as observed for Galactic cosmic rays. + +These limitations point to significant drawbacks of the second-order Fermi mechanism. +% +In contrast, \emph{diffusive shock acceleration} (discussed in the next section) circumvent these challenges. Shock fronts provide more efficient energy gain and yield spectra with consistent power-law behavior, making them far better candidates for explaining the acceleration of Galactic cosmic rays. + +While it is unlikely to account for the bulk of cosmic-ray acceleration, however, the second-order Fermi mechanism may still play a role in cosmic-ray physics as in certain models of Galactic transport, interactions with turbulent magnetic fields driven by second-order Fermi-like processes can slightly \emph{re-energize} the spectrum of already-accelerated cosmic rays while they propagate through the Galaxy. + +\subsection{Second-order Fermi re-acceleration} + +{\color{red}To be done} + +\subsection{Where do the cosmic ray get their energy?} + +{\color{red}To be done} + +\end{document} + +%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}. + +%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. + + + +%\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} + + diff --git a/sections/app_collisions.tex b/sections/app_collisions.tex new file mode 100644 index 0000000..2435802 --- /dev/null +++ b/sections/app_collisions.tex @@ -0,0 +1,75 @@ +% !TEX root = ../lectures.tex +\section{Kinematics of Head-On and Tail-On Collisions in Newtonian Elastic Scattering} + +Elastic scattering describes collisions in which the total kinetic energy of the system is conserved. This appendix focuses on energy changes during elastic collisions between a particle and a larger moving target, specifically for two distinct scenarios: +% +\begin{itemize} +\item Head-on collisions: The particle and target move toward each other. +\item Tail-on collisions: The particle moves in the same direction as the target, trailing behind it. +\end{itemize} + +These cases illustrate fundamental principles of energy transfer in Newtonian mechanics, which underpin phenomena like cosmic ray acceleration. + +We begin with a particle of mass \( m_1 \) and velocity \( v_1 \), colliding elastically with a target of mass \( m_2 \) and velocity \( v_2 \). For simplicity, we restrict the analysis to one-dimensional (1D) motion along a straight line. + +In elastic collisions, \emph{momentum} and \emph{kinetic energy} are conserved. +% +The conservation laws are expressed as: +\[ +m_1 v_1 + m_2 v_2 = m_1 v_1^\prime + m_2 v_2^\prime~\quad\text{(momentum conservation)}~, +\] +and +\[ +\frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 = \frac{1}{2} m_1 {v_1^\prime}^2 + \frac{1}{2} m_2 {v_2^\prime}^2~\quad\text{(energy conservation)}~. +\] + +From these principles, the post-collision velocities \( v_1^\prime \) and \( v_2^\prime \) are derived: +\[ +v_1^\prime = \frac{(m_1 - m_2)v_1 + 2 m_2 v_2}{m_1 + m_2}~, \quad v_2^\prime = \frac{(m_2 - m_1)v_2 + 2 m_1 v_1}{m_1 + m_2}~. +\] + +These expressions encapsulate the dynamics of the collision, where the final velocities depend on the masses \( m_1, m_2 \) and the initial relative velocity \( v_1 - v_2 \). + +The energy change for the particle, \( \Delta K_1 \), is defined as the difference in the particle's kinetic energy before and after the collision: +\[ +\Delta K_1 = \frac{1}{2} m_1 {v_1^\prime}^2 - \frac{1}{2} m_1 v_1^2~. +\] + +Substituting \( v_1^\prime \) from the velocity expression, we obtain: +\[ +\Delta K_1 = \frac{1}{2} m_1 \left[ \left( \frac{(m_1 - m_2)v_1 + 2 m_2 v_2}{m_1 + m_2} \right)^2 - v_1^2 \right]~. +\] + +This result shows how the energy transfer depends on the configuration of the system. +% +Let’s examine two important scenarios in the limit \( m_1 \ll m_2 \), where the target is significantly more massive than the particle. + +In a \emph{head-on collision}, the particle and target move toward each other, meaning \( v_1 > 0 \) and \( v_2 < 0 \). + +When \( m_1 \ll m_2 \), the target’s velocity remains approximately constant, and the particle’s post-collision velocity simplifies to: +\[ +v_1^\prime \approx -v_1 + 2 v_2 <0 ~. +\] +This result indicates that the particle's direction is reversed. + +In the same approximation, the energy gained by the particle can be approximated as: +\[ +\Delta K_1 \approx 2 m_1 v_2 (v_2 - v_1) > 0~. +\] +Since \( v_2 < 0 \) (opposite to \( v_1 \)), the particle \emph{gains energy} in the collision. + +In a \emph{tail-on collision}, the particle trails the target, meaning \( v_1 > v_2 > 0 \). + +For \( m_1 \ll m_2 \), the particle's velocity after the collision is approximately: +\[ +v_1^\prime \approx -v_1 + 2 v_2~, +\] +with its direction determined by the relative velocity \( v_1 - v_2 \). + +The kinetic energy change is approximately: +\[ +\Delta K_1 \approx 2 m_1 v_2 (v_2 - v_1) < 0~. +\] +Since \( v_2 - v_1 < 0 \), the particle \emph{loses energy} during the collision. + +These principles of head-on and tail-on collisions are pivotal in understanding energy transfer mechanisms in high-energy astrophysics. For example, head-on collisions dominate energy gain in second-order Fermi acceleration, where particles interact with moving magnetic turbulence.