Final answer:
The final estimated size of the Sun, through a thought experiment, would be about 20 kilometers in diameter as a neutron star, but in reality, due to electron degeneracy, the Sun will become a red giant and then a white dwarf. Currently, it is in its main-sequence phase. Astronomical observations of other massive objects, like black holes, support the understanding of the density and fate of such astronomical entities.
Step-by-step explanation:
The estimated size of the Sun, if it were to reach the final stage of a stellar life cycle similar to a neutron star, would be about 20 kilometers in diameter based on a theoretical 'thought experiment.' During this stage, the gravitational pull of the Sun would require an escape velocity of about half the speed of light. However, in reality, this transformation cannot occur for the Sun due to electron degeneracy, which results from the mutual repulsion between tightly packed electrons. The transition of the Sun into a red giant and eventual white dwarf is due to its mass—about 1 solar mass—and it will be billions of years before the Sun 'climbs' away from the main-sequence phase of its evolution. In fact, the Sun is currently in the main-sequence phase and is about 4.5 billion years old.
For a better understanding of stellar evolution, astronomers consider the mass of the star and resultant density to determine its ultimate fate. Massive stars may become neutron stars or black holes, while less massive stars like our Sun will expand into a red giant and then contract into a white dwarf. The Sun's current energy output and the rate of contraction are tied to the Kelvin-Helmholtz mechanism, which involves slow contraction over time releasing energy.