Question #43117

An RLC circuit with resistance $R$ and inductance $L$ drives an antenna, emitting a plane electromagnetic pulse. Due to the circuit's damping, the emitted pulse is not monochromatic but has a Lorentzian energy fluence spectrum (energy per unit area per unit angular frequency) given by: $J(\omega) = \frac{J_0}{\pi} \frac{\Gamma}{(\omega - \omega_0)^2 + \Gamma^2}$ where $J_0$ is the total energy fluence, $\omega_0$ is the central frequency, and $\Gamma = \frac{R}{2L}$ is the spectral half-width. This pulse is normally incident on a resonant microscopic solar sail. The sail's effective absorption area depends on the frequency of the incident radiation as: $A(\omega) = A_0 \frac{\gamma^2}{(\omega - \omega_0)^2 + \gamma^2}$ where $A_0$ is the peak area and $\gamma$ is the resonance half-width of the sail. Assuming the sail absorbs the radiation completely according to this profile and reflects nothing, what is the total momentum imparted to the sail? (Assume $\omega_0 \gg \Gamma, \gamma$ so the integration limits can be approximated as $-\infty$ to $\infty$).