Several unexpected astrophysical observations can be explained by gravitationally captured massive axions or axion-like particles, which are produced inside the Sun or other stars and are accumulated over cosmic times. Their radiative decay in solar outer space would give rise to a 'self-irradiation' of the whole star, providing the time-independent component of the corona heating source (we do not address here the flaring Sun). In analogy with the Sun-irradiated Earth atmosphere, the temperature and density gradient in the corona−chromosphere transition region is suggestive for an omnipresent irradiation of the Sun, which is the strongest evidence for the generic axion-like scenario. The same mechanism is compatible with phenomena like the solar wind, the X-rays from the dark-side of the Moon, the X-Ray Background Radiation, the diffuse X-ray excesses (below ∼ 1 keV), the non-cooling of oldest Stars, etc. A temperature of ∼ 10 6 K is observed in various places, while the radiative decay of a population of such elusive particles mimics a hot gas, which fits unexpected astrophysical X-ray observations. Furthermore, the recently reconstructed quiet solar X-ray spectrum during solar minimum supports this work, since it covers the expected energy range, and it is consistent with the result of a simulation based on Kaluza-Klein axions above ∼ 1 keV. The derived axion luminosity (L a ≈ 0.16L ) fits the cosmic energy density spectrum and is compatible within 2σ with the recent SNO result, showing the important interplay between any exotic energy loss mechanism and neutrino production. At lower energies, using also a ROSAT observation, only ∼ 3% of the X-ray intensity is explained. Data from orbiting X-ray Telescopes provide upper limits for particle decay rates 1 AU from the Sun, and suggest new types of searches on Earth or in space. In particular, X-ray observatories, with an unrivalled equivalent fiducial volume of ∼ 10 3 m 3 for the 0.1 -10 keV range, can search for the radiative decay of new particles even from existing data. This work introduces the elongation angle of the X-ray Telescope relative to the Sun as a relevant new parameter.