The Sun does radiate light in all directions, but its brightness does not appear greater in the center of the disc due to a phenomenon known as limb darkening. Limb darkening refers to the gradual decrease in brightness from the center of a celestial object, such as the Sun or a planet, towards its outer edges or limb.
Limb darkening occurs due to the variation in temperature and density across the Sun's surface. The Sun is not a uniform, solid object but is composed of hot gases undergoing various processes such as nuclear fusion. The temperature and density of the gases are highest in the Sun's core and gradually decrease towards the surface.
When we observe the Sun from a distance, the light we receive is a combination of photons emitted from different layers of the Sun's atmosphere. The photons emitted from the deeper, hotter regions of the Sun's interior have a higher energy and are more likely to be absorbed or scattered by the outer layers of the atmosphere. This absorption and scattering process reduces the intensity of the light coming from the Sun's center.
In contrast, the photons emitted from the cooler, outer regions of the Sun's atmosphere have a lower energy and are less likely to be absorbed or scattered. Therefore, the outer regions contribute more to the overall brightness we observe.
As a result, the combination of limb darkening and the variation in temperature and density across the Sun's surface causes the brightness to be lower towards the edges or limb of the solar disc compared to the center. This effect is more pronounced when observing the Sun directly without proper solar filters, as the intense brightness of the solar center can make the limb darkening less apparent. However, when observing the Sun with appropriate instruments or during a solar eclipse, the limb darkening becomes more evident.