A Fast Concentric-disk Contour Integration Method For Microlensing Limb-darkening Effect

Incorporating limb darkening is a computationally demanding step in contour-integration-based microlensing modeling. Conventional concentric-ring integration is incompatible with high-order quadrature schemes, limiting efficiency. We develop a new concentric-disk method, which reformulates the limb-darkening integral and enables the application of high-order adaptive quadrature, significantly accelerating the computation. While mainly demonstrated using a linear limb-darkening profile, the method readily extends to more general limb-darkening profiles. As a source effect, it applies to lens systems of any complexity. The method achieves a convergence rate scaling faster than $N_{\mathrm{u}\mathrm{n}\mathrm{i}}^{-4}$ with the number of uniform-source magnification evaluations $N_{\mathrm{u}\mathrm{n}\mathrm{i}}$, a significant improvement over the concentric-ring $N_{\mathrm{u}\mathrm{n}\mathrm{i}}^{-2}$ scaling. For a relative accuracy of ${10}^{-6}$, the concentric-disk approach typically requires only $30\mathrm{\%}$ or less of the computational cost of traditional algorithms. This method has been implemented in the binary-lens contour integration code $\mathtt{\text{Twinkle}}$, providing an efficient and precise tool for analyzing current and future high-precision microlensing observations.