Gravitational microlensing

Definition

Consider a compact lens of mass $M$ with Einstein radius $\theta_E$, and a point source at angular separation $\beta$ from it. Write $u = \beta / \theta_E$. Gravitational microlensing is lensing in which the images of the lens equation are closer than the angular resolution of the observation. So only the total magnification of the source is observed. For a point lens and a point source this total magnification is $$A(u) = \frac{u^2 + 2}{u \sqrt{u^2 + 4}}.$$ Let the lens and source move with constant relative proper motion $\mu_{rel}$. Then $$u(t) = \sqrt{u_0^2 + \left( \frac{t - t_0}{t_E} \right)^2}, \qquad t_E = \frac{\theta_E}{\mu_{rel}},$$ where $u_0$ is the minimal separation, reached at time $t_0$, and $t_E$ is the Einstein crossing time.