Question

If the sun expanded to a radius 200 times it's present radius, what would be it's average density be in g/cm^3

Answer 1

If the Sun expands, its mass remains constant. We assume the initial and final Suns to be spherically symmetric, so the mass is $M=\langle\rho\rangle V = \langle\rho\rangle \dfrac43 \pi R^3.$

$M= \langle\rho_{\text{init}}\rangle \dfrac43 \pi R_{\text{init}}^3 = \langle\rho_{\text{fin}}\rangle \dfrac43 \pi R_{\text{fin}}^3$ , $\dfrac{ \langle\rho_{\text{fin}}\rangle}{ \langle\rho_{\text{init}}\rangle} = \left(\dfrac{R_{\text{init}}}{R_{\text{fin}}}\right)^3 = \left(\dfrac{1}{200}\right)^3 = 1.25\cdot10^{-7}.$

The average density of modern Sun is 1.4 g/sm³, so the final density will be

$1.4\,\mathrm{g/sm}^3\cdot1.25\cdot10^{-7} = 1.75\cdot10^{-7}\,\mathrm{g/sm^3}.$