Calculate the escape speed from (a) the surface of the presentday Sun and (b) the surface of the Sun when it becomes a red giant, with essentially the same mass as today but with a radius that is 100 times larger. (c) Explain how your results show that a red-giant star can lose mass more easily than a main- sequence star.

Astrophysics is the branch of astronomy that deals with the physics of celestial bodies and the phenomena that occur within the universe. One of the fundamental concepts in astrophysics is escape velocity—the minimum speed needed for an object to 'break free' from the gravitational pull of a celestial body without further propulsion.

To comprehend the escape velocity, imagine throwing a ball upwards. The higher the speed, the higher it goes before falling back down. If you could throw it fast enough, it would leave Earth’s gravitational field entirely. The escape velocity from Earth is about 11.2 kilometers per second. In the case of the Sun, its massive gravity requires an even higher escape speed.

The formula for calculating escape velocity is: \[ v = \sqrt{2GM/R} \] where \( v \) is the escape velocity, \( G \) is the gravitational constant, \( M \) is the mass of the celestial body, and \( R \) is the radius from the center of mass to the point of escape. In astrophysics, understanding escape velocity helps us grasp how different celestial events, like the shedding of a star's outer layers or the launch of spacecraft, are influenced by the gravitational characteristics of celestial bodies.

A red giant star represents a late stage in the life of a star. As a star exhausts the hydrogen in its core, nuclear reactions begin to occur in a shell around the core, causing the outer layers to expand and cool, thus taking on a reddish hue. A red giant star can be many times larger in volume than it was during the main sequence stage of its life.

The increase in size of a red giant leads to a decrease in surface gravity, which according to the escape velocity formula means that a lighter escape velocity is required to overcome its gravitational pull. For the Sun, a current main-sequence star, its transition to a red giant will dramatically increase its radius, specifically, up to 100 times its current size. Despite maintaining a similar mass, the surface of the red giant Sun will be far less gravitationally binding than it is today.

Stellar evolution is the process of change that a star undergoes over its lifetime. Beginning with nebular gas and dust collapse, a star goes through various phases based primarily on its mass. The main sequence is the longest and most stable period of a star’s life, after which it may become a red giant, as is expected for our Sun.

Post-main sequence, stars like the Sun will swell into red giants. This transformation sees a reduction in escape velocity, making mass loss more feasible due to lower gravitational binding on the outer layers. Stellar winds and radiation can then strip away these layers, often contributing to the creation of nebulae, and thus recycling stellar material into the interstellar environment. This mass-shedding phase greatly influences the star's evolution, eventually leading to its demise as a white dwarf, or if massive enough, a supernova event and possibly a neutron star or black hole.