A comment on the properties of the matter flow through the first Langrangian point

We analyse properties of the mass outflow from the Roche-lobe filling component of a semi-detached binary system. We follow the approaches published by Paczy\'nski \& Sienkiewicz and by Lubow \& Shu, which we compare with other simplified approaches. We find that the density of the flow at $L_1$ is orders of magnitude lower than the density on the same equipotential but away from $L_1$. Furthermore, the effective cross section of the flow, after averaging over its profile of the momentum density, is much lower than some published estimates done without accounting for the averaging. Thus, the use of some simplified formulae for the density and the flow cross section can lead to overestimates of the accretion rate and of the mass contained in the $L_1$ regions by very large factors unless they are supported by simultaneous integrations of the equations of stellar structure for the outer layers of the donor.