Closed-form expression for the phase-averaged absorption integral

Derive a closed-form expression for the angular integral I(τ_perp) = ∫_{−π/2}^{π/2} |cos θ| (1 − exp(−τ_perp |sec θ|)) dθ that appears in the phase-averaged absorbed thermal power of a TARS paddle under the transmission model T_angle(θ) = exp(−τ_perp |sec θ|). This would replace the numerical approximation 2 − (4/3) exp(−τ_perp) used to evaluate the effective transmission factor in the absorbed/emitted power expression.

Background

In the flux calculations for TARS, the phase-averaged absorbed thermal power requires evaluating angular integrals of the form ∫ |cos θ| (1 − exp(−τ_perp |sec θ|)) dθ over half-rotations. The authors introduce a transmission model where opacity scales with path length as |sec θ|, leading to these integrals.

The paper reports that a closed-form solution was not found and instead provides a numerical fit, approximating the integral by 2 − (4/3) exp(−τ_perp). A closed-form expression would remove reliance on this approximation, improving analytical precision in subsequent force and torque calculations and in design optimization over τ_perp.