10.9.1 Introduction

Gravitational microlensing is a phenomenon which can be explained in the light of the Einstein’s General Relativity Theory — each compact massive object distorts the space-time forcing the light rays of a background source to change their trajectories. It generally results in an appearance of two images of the source, typically separated by less than one milli-arcsecond. A photometric microlensing event (Microlensing) is defined as a temporal increase of brightness of a background star or other source of light (typically approximated as a point source) due to the emergence of the two magnified images, observed together due to their very small separation. The total observed brightness in the light curve of a source is a combination of the light of a background source of flux $F_{\mathrm{S}}$ amplified $A$-times and possible additional light $F_{\mathrm{B}}$ from unresolved sources within the resolution of the instrument, which might include the light of the lens:

where we defined a baseline magnitude of the event ${\mathrm{mag}}_0$ and a blending parameter, $f_{\mathrm{S}}=F_{\mathrm{S}}/(F_{\mathrm{S}}+F_{\mathrm{B}})$, which defines a fraction of the source light in the total observed flux, i.e., $f_{\mathrm{S}}=1$ when $F_B=0$. The microlensing effect is generally achromatic because the amplification does not depend on the wavelength. However, only the source light is amplified and the spectral energy distribution of the blended light can be different from the one of the source. Hence, the light curve in each Gaia band has to be modelled with a separate set of parameters:

$A(t)$ is the photometric amplification factor for all images(Paczynski 1996), which is a function of the time-varying angular separation between the lens and the source. The formulae for the amplification of single-lens events, including the microlensing parallax, are described in Wyrzykowski et al. (2023). Higher-order effects in Microlensing (e.g., lens or source binarity, finite source size) were not considered as they are way more rare and are not detectable given the sparse observing cadence of Gaia (typically 30 days).

Please note that in our model we ignore the displacement of the observer (Gaia) from the Earth. Typical sizes of the Einstein radius when projected on the Earth’s orbit are of order of 1 au, compared to 1 % of an astronomical unit of Gaia’s distance from Earth. The effect is not be measurable within the Gaia photometric precision, furthermore degraded by a sparse sampling of Gaia data. The effect of parallax could be detected only in case of high magnification and high gradient of magnitude change, with a spectacular example of Gaia16aye (Wyrzykowski et al. 2020). That was, however, only one such event in the entire Gaia sample so far and it was due to a binary lens, whose model was not included in this pipeline.