Detection Techniques and Experimental Results

Abstract

In the present chapter we first give an overview over the experimental techniques used for the detection of high energy cosmic rays and $\gamma$-rays. This includes the development of the most relevant aspects of air shower physics. The second part then summarizes the current experimental situation and the main open questions, roughly separated for cosmic radiation of Galactic and extragalactic origin. The following chapters will then provide the theoretical and phenomenological fundamentals required to understand the observations and find solutions to the open questions.

Problems

5.1

Acceleration of a Point Charge in the Field of a Moving Ion

Use the Lorentz transformation of electromagnetic fields, Eq. (2.9) to derive Eq. (5.1) for the acceleration in the instantaneous rest frame of a point charge q with mass m moving with velocity v and impact parameter b with respect to an ion of charge Ze.

5.2

Momentum Transfer in Relativistic Bremsstrahlung

Derive Eq. (5.28),

$$ q_\parallel \simeq \frac{m_e^2\,k}{2E(E-k)}+\frac{E-k}{2}\theta _e^2+\frac{k}{2}\theta _\gamma ^2\, $$

for the momentum transfer to the nucleus parallel to the electron momentum in the nuclear bremsstrahlung process by expanding Eq. (5.27) to first order in $m_e^2$ and the squared angles $\theta _e^2$ and $\theta _\gamma ^2$ of outgoing electron and photon relative to the incoming electron.

5.3

Formation Length of Bremsstrahlung in Vacuum

Derive the formation length Eq. (5.29) for forward scattering of an electron of energy E emitting a bremsstrahlung photon of energy k by requiring the phase $\omega t-\mathbf{k}\cdot \mathbf{r}(t)$ to vary by less than unity, with $\omega =k$ for the photon. Hint: Use $\mathbf{r}(t)=v_e\hat{\mathbf{k}}t$ for the electron trajectory along the photon direction $\hat{\mathbf{k}}$, where $v_e$ is the electron velocity.

5.4

Liénard–Wiechert Potentials of a Moving Charge

Derive the Liénard–Wiechert Potentials potentials Eq. (5.44),

$$\begin{aligned} \phi (t,\mathbf{r})= & {} \frac{q}{4\pi \epsilon _0}\frac{1}{\left[ r-\mathbf{v}\cdot \mathbf{r}/c_0\right] (t_\mathrm{ret})}\,,\nonumber \\ \mathbf{A}(t,\mathbf{r})= & {} \frac{\mu _0q}{4\pi }\frac{\mathbf{v}(t_\mathrm{ret})}{\left[ r-\mathbf{v}\cdot \mathbf{r}/c_0\right] (t_\mathrm{ret})}\,, \end{aligned}$$

from the general solution Eqs. (2.211)–(2.213) for a point charge q moving on a world line $\mathbf{l}(t)$ with velocity $\mathbf{v}(t)=\dot{\mathbf{l}}(t)$, such that $\rho _\mathrm{em}(t,\mathbf{r})=q\delta ^3\left[ \mathbf{r}-\mathbf{l}(t)\right]$ and $\mathbf{j}_\mathrm{em}(t,\mathbf{r})=q\mathbf{v}(t)\delta ^3\left[ \mathbf{r}-\mathbf{l}(t)\right]$, where $\mathbf{r}$ is the radius vector from the point charge to the observer, $t_\mathrm{ret}\equiv t-|\mathbf{r}-\mathbf{l}(t_\mathrm{ret})|/c_0$ is the retarded time and the propagation speed of light in the medium is c. Hint: Use the standard property

$$\int _{-\infty }^{+\infty }dxf(x)\delta (h(x))=\sum _i\frac{f(x_i)}{|h^\prime (x_i)|}$$

of the one-dimensional delta function, where $x_i$ are the roots of h(x).

5.5

Timing in Air Shower Fluorescence Technique

Derive Eq. (5.48) for the dependence of detection time and direction of the shower center.