Quantum scattering cross sections of O($^3P$) + N$_2$ collisions for planetary aeronomy

"Hot atoms", which are atoms in their excited states, transfer their energy to the surrounding atmosphere through collisions. This process of energy transfer is known as thermalization, and it plays a crucial role in various astrophysical and atmospheric processes. Thermalization of hot atoms is mainly governed by the amount of species present in the surrounding atmosphere and the collision cross-section between the hot atoms and surrounding species. In this work, we investigated the elastic and inelastic collisions between hot oxygen atoms and neutral N$_2$ molecules, relevant to oxygen gas escape from the martian atmosphere and for characterizing the chemical reactions in hypersonic flows. We conducted a series of quantum scattering calculations between various isotopes of O($^3P$) atoms and N$_2$ molecules across a range of collision energies (0.3 to 4 eV), and computed both their differential and collision cross-sections using quantum time$-$independent coupled-channel approach. Our differential cross-section results indicate a strong preference for forward scattering over sideways or backward scattering, and this anisotropy in scattering is further pronounced at higher collision energies. By comparing the cross-sections of three oxygen isotopes, we find that the heavier isotopes consistently have larger collision cross-sections than the lighter isotopes over the entire collision energy range examined. However, for all the isotopes, the variation of collision cross-section with respect to collision energy is the same. As a whole, the present study contributes to a better understanding of the energy distribution and thermalization processes of hot atoms within atmospheric environments. Specifically, the cross$-$sectional data presented in this work is directly useful in improving the accuracy of energy relaxation modeling of O and N$_2$ collisions over Mars and Venus atmospheres.