If the Sun expands to a radius of $100 R_{\odot}$ during the red giant phase, by what factor has its volume increased from its main sequence radius of $1 R_{\odot}$?
Answer
Approximately one million times ($100^3$)
The calculation for the volume change of a spherical object is proportional to the cube of its radius ($V ilde{} R^3$). If the initial radius is $1 R_{\odot}$ and the final radius reaches $100 R_{\odot}$, the volumetric expansion is calculated by cubing the radius ratio: $100^3$. This results in an enormous increase in volume by a factor of one million. This substantial increase in volume is not due to a corresponding increase in mass, but rather is driven by the tremendous puffing up and expansion of the star's outer layers, powered by the efficient shell fusion process deep inside.
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