Analytic model for the statistics of ultra-high magnification events

Highly magnified individual stars such as Icarus and Earendel have recently been observed near critical curves of galaxy clusters with Hubble Space Telescope (HST) observations. These stars are estimated to be magnified with a factor of more than a few thousands. In addition to the smooth mass distribution in the macro-lens model, the distribution of microlenses originating from, for instance, intracluster stars affects the event rate and the peak magnification significantly. We propose an analytic model of the high-magnification tail of the probability distribution function (PDF) in which the probability is assumed to be proportional to the number of independent microlens critical curves. Our model can explain the parameter dependence of the PDF on the mass fraction of the microlenses and the background magnification seen in ray-tracing simulations. The effect of a finite source size is also studied to derive a fitting formula for the suppression factor. For an application of our model, we calculate the event rate of the Icarus-like system and the probability distribution of observed positions of such system, showing good agreement with the HST observations. Our model predicts a complicated dependence of the probability distribution of observed positions of highly magnified events on the magnification threshold.