As recently advocated in [H. Fischer et al. , Phys. Rev. D 98 , 043013 (2018)], there is a fundamentally new mechanism for the axion production in the Sun and Earth. However, the role of very slow axions in previous studies were neglected because of its negligible contribution to the total axion production by this new mechanism. In the present work we specifically focus on analysis of the nonrelativistic axions which will be trapped by the Sun and Earth due to the gravitational forces. The corresponding emission rate of these low energy axions (below the escape velocity) is very tiny. However, these axions will be accumulated by the Sun and Earth during their life-times, i.e., 4.5?billion of years, which greatly enhances the discovery potential. The computations are based on the so-called axion quark nugget (AQN) dark matter (DM) model. This model was originally invented as a natural explanation of the observed ratio Ω dark ～ Ω visible when the DM and visible matter densities assume the same order of magnitude values, irrespectively to the axion mass m a or initial misalignment angle θ 0 . This model, without adjustment of any parameters, gives a very reasonable intensity of the extreme UV (EUV) radiation from the solar corona as a result of the AQN annihilation events with the solar material. This extra energy released in corona represents a resolution, within AQN framework, a long standing puzzle known in the literature as the “solar corona heating mystery”. The same annihilation events also produce the axions. The flux of these axions is unambiguously fixed in this model and expressed in terms of the EUV luminosity from solar corona. We make few comments on the potential discovery of these gravitationally bound axions.
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