Microlensing optical depth ($\tau =\int n_l\pi R_E^{2}d{D}_l$)

The microlensing optical depth is the probability, when small, that a source lies within one Einstein radius of a foreground lens. For point lenses and compact-object mass density ${\rho }_l$ in nearby Euclidean geometry,The individual lens mass cancels because $n_l={\rho }_l/m$ and $R_E^2\propto m$. Detection efficiencies and event durations still depend on individual masses and survey cadence.