ON THE ORIGIN OF FAST RADIO BURSTS
Almog Yalinewich
29.10.18
Compton Catastrophe
the challenge is not to produce radio
...but to produce just radio
photon
compton
electron
Compton Catastrophe
Synchrotron luminosity
L_{s} \approx r_e^2 c \gamma^2 B^2
L_{c} \approx r_e^2 c \gamma^2 u
Inverse Compson
Photon field
Inverse Compton dominates when
u \approx B^2
u \approx k T\frac{\nu^3}{c^3}, \, \nu \approx \frac{q B}{m_e c}\gamma^2, \, k T \approx m_e c^2 \gamma^2
u < B^2 \Rightarrow T < 10^{12} \, {\rm K} \left(\nu / 1 \, {\rm GHz} \right)^{1/5}
Coherent Emission
Larmor Formula for a single particle
L_{1} \approx \frac{q^2 a^2}{c^3}
Number of particles
N
Coherent
Incoherent
L_{N} \approx N \frac{q^2 a^2}{c^3}
L_{N} \approx N^2 \frac{q^2 a^2}{c^3}
Synchrotron Emission
\frac{1}{\gamma}
\vec{R} = R \hat{x} \cos \Omega t+R\hat{y} \sin \Omega t
R \Omega = c
Synchrotron Emission
A \left(r,t\right) \approx \frac{1}{c} \int \frac{J \left(r',t'\right)}{\Delta r} \delta \left(\Delta t- \Delta r/c\right) d t' d^3r'
A \left(r,t\right) \approx \frac{q}{c r} \int \dot{R} \left(t'\right) \delta \left(\Delta t- \Delta r/c\right) d t'
\hat{r} \cdot R\left(t\right) = R \sin \Omega t \approx R \left(\Omega t-\frac{1}{6} \Omega^3 t^3\right)
\Delta r \approx \left(\hat{r} \cdot R - r\right)/c
\Delta t + \Delta r /c \approx \Omega^2 t^3
Fourier transform
\tilde{A} \left(\omega\right) \propto \int dt\cdot t \cdot \exp \left(\omega \Omega^2 t^3 \right)
x = \omega \Omega^2 t^3
\tilde{A} \propto \omega^{-2/3}
\tilde{E} \propto \omega \tilde{A} \propto \omega^{1/3}
I \propto \tilde{E}^2 \propto \omega^{2/3}
Geometric Effects
\alpha
\alpha
\Delta t \propto \alpha - \sin \alpha \propto \alpha^3
\omega \propto \frac{1}{\Delta t} \propto \alpha^{-3} \Rightarrow \alpha \propto \omega^{-1/3}
Radiation from multiple electrons
N I \propto \alpha I \propto \omega^{1/3}
Razin Suppression
\theta
\beta c
c
\beta c = c \cos \theta \Rightarrow \theta \approx \frac{1}{\gamma}
Relativistic beaming
Vacuum
Dielectric medium
\beta c = c \cos \theta / n
\theta \approx \sqrt{\gamma^{-2} + 1-n}
\gamma^{-2} \gg 1-n
\gamma^{-2} \ll 1-n
\theta \approx 1/\gamma
\theta \approx \sqrt{1-n}
Razin Suppression
Razin Suppression
\Delta \theta \approx \sqrt{\left(\frac{\omega_s}{\omega}\right)^{2/3} \frac{1}{\gamma^2} + 1-n}
n = 1-\frac{\omega_p^2}{\omega^2}
Critical frequency
\omega_R \approx \sqrt{\omega_b \omega_p}
\omega_s = \gamma^2 \omega_b
\omega_b = \frac{q B}{m_e c}
Detailed balance
Equilibrium for blackbody
\mathcal{B}_{\omega}\approx \frac{k T}{ \omega^2} c^2
Razin Suppression
\frac{d \ln n_e}{d \ln \gamma_e} > -2