Cherenkov radiation in a strong magnetic field

In order to discover new physics from astronomical observations, it is important to understand the mechanisms in which electromagnetic radiation are produced by the astrophysical sources. In most cases, synchrotron radiation produced by accelerating charges provide an adequate account of observations from which astronomers are able to obtain information on the electromagnetic fields in the vicinity of sources such as pulsars. However, these are not the only possible mechanisms of radiation production. According to quantum electrodynamics (QED), as the electromagnetic fields approach the critical value $|\mathbf{B}|_{c}=m^{2}_{e}/e=4.42\times 10^{13}\,\mbox{Gauss}$ where $m_{e}$ is the electron mass and $e$ the electric charge ¹¹1We use the SI units with $c=\hbar=1$ so that 1 Gauss = $1.95\times 10^{-2}$ eV². The Cherenkov and synchrotron spectrum given in Schwinger (1949); Macleod et al. (2019) are in Gaussian units so we need to introduce a factor of $1/(4\pi)$ in their respective power spectra., the vacuum becomes polarized with a refractive index greater than unity Bialynicka-Birula and Bialynicki-Birula (1970); Adler (1971); Brezin and Itzykson (1971); Tsai and Erber (1975); Latorre et al. (1995); Dittrich and Gies (1998); Gies and Dittrich (1998); Dittrich and Gies (2000); Marklund and Shukla (2006). As a result, for ultra-relativistic charged particles travelling in such media, the emission of Cherenkov radiation becomes a real possibility. Taking into account the effects of non-linear QED, in the presence of a constant and spatially uniform magnetic field, the Cherenkov power spectrum is given by Macleod et al. (2019)

where $\beta$ is the velocity of the charged particle which we will take to be electron, $n_{\pm}$ the refractive index and $\omega_{\pm}$ the modified photon frequency Bialynicka-Birula and Bialynicki-Birula (1970); Adler (1971)

with $\mathbf{k}$ being the photon momentum. The coefficients $\lambda_{\pm}$ are derived from the Euler-Heisenberg Lagrangian in the weak field limit Schwinger (1951)

In eq. (3), $\boldsymbol{\epsilon}_{0}$ and $\boldsymbol{\epsilon}_{\pm}$ are the polarization vectors for the Cherenkov radiation and linearly polarized photons in the medium respectively. Their expressions can be found in Bialynicka-Birula and Bialynicki-Birula (1970) but they are not important to us here. Since the refractive index is constant, the spectrum is directly proportional to the photon frequency.

For electrons travelling parallel or anti-parallel to the photon trajectory, the condition for Cherenkov radiation is $\beta>1/n_{\pm}$. For ultra-relativistic electrons, this condition is equivalent to $\gamma_{e}^{2}>1/(2\lambda_{\pm}|\mathbf{Q}|^{2})$. Therefore, in the presence of a strong magnetic field and with the electrons satisfying the stated condition, they will emit both Cherenkov and synchrotron radiation; for convenience, we write the associated $\gamma_{e}$ factor as

The power spectrum of the synchrotron radiation is given by Schwinger (1949)

where $K_{5/3}(\eta)$ is the modified Bessel function, $R$ the radius of curvature and

where $E_{e}$ is the energy of the electron. For ultra-relativistic electrons, the radius of curvature can be approximated as $R\approx 1/\omega_{0}$ so that

Substituting eq. (7) into $\omega_{c}$ and taking $\lambda_{\pm}\approx\lambda\sim 10^{-32}\,\mbox{Gauss}^{-2}$, we obtain

Above the critical frequency $\omega_{c}$, the synchrotron power spectrum exhibits approximate exponential decay. Therefore, if one extrapolates the photon frequency beyond $\omega_{c}$, the Cherenkov radiation given in eqs. (1-3) will eventually dominate over its synchrotron counterpart. However, as we will show below, such an extrapolation is problematic because the photon dispersion and refractive index are derived from an effective field theory. Specifically, the extrapolation is constrained by the following inequality Narozhny (1969); Ritus (1985)

where $F$ is the electromagnetic field strength tensor and $k^{\mu}$ is the photon momentum. In terms of the critical magnetic field $|\mathbf{B}_{c}|$ and the presence of a spatially constant and uniform magnetic field, we have $\chi_{\gamma}\sim(|\mathbf{B}|/|\mathbf{B}_{c}|)(\omega/m_{e})$ thus yielding an upper-bound to the photon frequency

Going beyond the bound, the effective field theory becomes unreliable and quantum effects become important. For instance, when $|\mathbf{B}|\sim|\mathbf{B}_{c}|$ and $\omega\gtrsim 2m_{e}$, photo-pair production $\gamma\rightarrow e^{+}e^{-}$ dominates (Adler, 1971).

Comparing eqs. (11) to (13), we find that $\omega_{\tiny{\mbox{max}}}\ll\omega_{c}$. Therefore, we cannot extend the photon frequency associated with Cherenkov radiation beyond $\omega_{c}$ thus verifying our earlier assertion. Therefore, we may use the following approximation for the synchrotron spectrum Rybicki and Lightman (2004)

To approximate the maximum Cherenkov power spectrum, we take $|\boldsymbol{\epsilon}_{0}\cdot\boldsymbol{\epsilon}_{\pm}|=1$, set $\beta=1$ and take $\omega_{\pm}\approx\omega$, $\lambda\sim 10^{-32}\,\mbox{Gauss}^{-2}$. These approximations are possible because we are working under the condition $\lambda|\mathbf{B}|^{2}\ll 1$. To the leading order in $\lambda|\mathbf{B}|^{2}$, with electrons and photons travelling normal to the magnetic field and $\gamma_{e}^{2}>1/(2\lambda|\mathbf{B}|^{2})$, we get

At $\omega=\omega_{\tiny{\mbox{max}}}$ and with $\gamma^{2}_{e}=X/(2\lambda|\mathbf{B}|^{2})$, after some manipulations, we find

Therefore, the Cherenkov radiation becomes dominant when $\gamma_{e}\gtrsim 4.32\times 10^{19}\,(\mbox{Gauss}/|\mathbf{B}|)$.

Even though the Cherenkov radiation dominates for certain ranges of $\gamma_{e}$, actual detections of the Cherenkov radiation from astrophysical sources remains difficult. A common feature of Cherenkov and synchrotron radiation is that both emissions are concentrated in a narrow cone about the path of the electrons. Let $\theta_{\pm}\approx\theta_{\tiny{\mbox{Cher}}}$ and $\theta_{\tiny{\mbox{sync}}}$ be the opening angles for the Cherenkov and synchrotron radiation respectively. For ultra-relativistic electrons, they are given by

Defining their spectral flux densities to be

where $D$ is the distance between the source and the observer. At $\omega=\omega_{\tiny{\mbox{max}}}$, the ratio of the flux density is given by

Since $X>1$, the spectral flux density of Cherenkov radiation is at least three orders of magnitude smaller than its synchrotron counterpart.

In summary, the production of Cherenkov radiation in strong magnetic fields is a real and fascinating possibility. However, because $\omega_{\tiny{\mbox{max}}}\ll\omega_{c}$, actual detection of Cherenkov radiation from astrophysical sources remains challenging. It may also be instructive to study photon propagation in strong electric fields or other configurations but it seems to us that the pure magnetic field configuration is the most interesting and physically relevant.

Acknowledgement - I would like to thank Chris Gordon and Lance Labun for discussions and reading the manuscript. I would also like to thank the referee for valuable comments. I am grateful to the generous hospitality offered by the Department of Physics and Astronomy at the University of Canterbury where part of this work was completed.