  As recently advocated in \cite{Fischer:2018niu}, 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 non-relativistic 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 Model. This model was originally invented
as a natural explanation of the observed ratio $\Omega_{\rm dark} \sim
\Omega_{\rm visible}$ 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 $\theta_0$.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.
