Constraint on Pulsar Wind Properties from Induced Compton Scattering off Radio Pulses

Pulsar winds have longstanding problems in energy conversion and pair cascade processes which determine the magnetization $σ$, the pair multiplicity $κ$ and the bulk Lorentz factor $γ$ of the wind. We study induced Compton scattering by a relativistically moving cold plasma to constrain wind properties by imposing that radio pulses from the pulsar itself are not scattered by the wind as was first studied by Wilson & Rees. We find that relativistic effects cause a significant increase or decrease of the scattering coefficient depending on scattering geometry. Applying to the Crab, we consider uncertainties of an inclination angle of the wind velocity with respect to the radio beam $θ_{\rm pl}$ and the emission region size $r_{\rm e}$ which determines an opening angle of the radio beam. We obtain the lower limit $γ\gtrsim10^{1.7}r^{1/2}_{\rm e,3}θ^{-1}_{\rm pl}(1+σ)^{-1/4}$ ($r_{\rm e}=10^3r_{\rm e,3}$ cm) at the light cylinder $r_{\rm LC}$ for an inclined wind $θ_{\rm pl}>10^{-2.7}$. For an aligned wind $θ_{\rm pl}<10^{-2.7}$, we require $γ>10^{2.7}$ at $r_{\rm LC}$ and an additional constraint $γ>10^{3.4}r^{1/5}_{\rm e,3}(1+σ)^{-1/10}$ at the characteristic scattering radius $r_{\rm c}=10^{9.6}r^{2/5}_{\rm e,3}$ cm within which the `lack of time' effect prevents scattering. Considering the lower limit $κ\gtrsim10^{6.6}$ suggested by recent studies of the Crab Nebula, for $r_{\rm e}=10^3$ cm, we obtain the most optimistic constraint $10^{1.7}\lesssimγ\lesssim10^{3.9}$ and $10^{6.6}\lesssimκ\lesssim10^{8.8}$ which are independent of $r$ when $θ_{\rm pl}\sim1$ and $1+σ\sim1$ at $r_{\rm LC}$.