{'fraction_non_alphanumeric': 0.01592115238817286, 'fraction_numerical': 0.0, 'mean_word_length': 5.082949308755761, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Vertical Distribution of Temperature and Density in a Planetary Ring

We model temperature and density profiles for a di- lute planetary ring, based on the hydrodynamic balance equa- tions for momentum and energy of granular flows. Within our approximation the ring consists of inelastic smooth spheres of unique size and mass, while the fluxes of mass, momentum and energy are linear functions of the gradients of density, velocity and temperature. The resulting system of coupled differential equations leads to temperature and density profiles, which we compare to the results of a triaxial kinetic approach to the dy- namics of a planetary ring. We find that both approaches agree fairly well in the elastic limit. Further, we carry out event driven N-particle simulations of a ring, subject to the conditions of the theoretical model. The simulated profiles are then compared to the theoretical ones: for the density a good agreement is found for both theoretical approaches, but the simulated temperature profiles increase monotonically with vertical distance whereas the theoretical profiles always have a turnover at some distance from the mid plane. This disagreement is likely to be connected to the vertical dependence of the velocity ellipsoid, which is not taken into account in the theoretical treatments.