 Solution

Power is energy per time. In geometrized units, energy has the dimension of length and time is converted to length using $c$, so the dimensionless geometrized luminosity isEquivalently, $c^5/G$ is the corresponding SI power unit. The lifetime for emitting the entire rest energy at the given constant luminosity isThe same result follows from $M_{\odot }^{(\mathrm{time})}/L_{\odot ,\mathrm{geom}}$. The stipulated full-mass radiation time is of order $10^{13}$ years; this is the constant-luminosity energy budget asked for, not a stellar-evolution calculation.