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Circular polarisation of synchrotron radiation in high magnetic fields

The general model for incoherent synchrotron radiation has long been known, with the first theory being published by Westfold in $1959$ and continued by Westfold and Legg in $1968$. When this model was first developed it was applied to radiation from Jupiter, with a magnetic field of $\approx$ 1 G. Pulsars have a magnetic field of $\approx 10^{12}$ G. The Westfold and Legg model predict a circular polarization which is proportional to the square root of the magnetic field, and consequently predicts greater than 100 per cent circular polarization at high magnetic fields. Here a new model is derived based upon a more detailed analysis of the pitch angle distribution. This model is concerned with the frequency range $f_{B_0}/γ<<f\lesssim f_{B_0}$, noting that $f_{B_0} = 2.7\times10^7B$, which for a relatively high magnetic field ($\sim 10^6-10^8$ Gauss) leaves emission in the optical range. This is much lower than the expected frequency peak for a mono-energetic particle of $0.29\frac{3eB}{4πm_e c}γ^2$. We predict the circular polarization peaks around $10^7$G in the optical regime with the radiation almost $15$ per cent circularly polarized. The linear polarization changes from about $60$ to $80$ per cent in the same regime. We examine implications of this for pulsar studies.