P = nkT \frac{P_{in}}{P_{out}} = \frac{n_{in}T_{in}}{n_{out}T_{out}}

speed of sound c_{s} = \sqrt{\frac{2kT}{m}} time to double in size t_{double} = \frac{R_{initial}}{c_{s}} time to expand fully t_{expand} = \frac{R_{final}}{c_{s}}

lifetime on main sequence ∼ \frac{mass of fuel}{rate of burning fuel} ∼ \frac{M}{L}