\magnification=\magstep1

$$
P \quad = \quad n k T 
$$

$$
{ {P_{\rm in}}\over{P_{\rm out}} } \quad = \quad 
    {  {n_{\rm in} T_{\rm in}} \over {n_{\rm out} T_{\rm out}} }
$$

$$
{\rm speed\ of\ sound\ } c_s \quad = \quad \sqrt{ {2 k T}\over{m} }
$$


$$
{\rm time\ to\ double\ in\ size\ } t_{\rm double} \quad = \quad 
     { {R_{\rm initial}}\over{c_s} }
$$

$$
{\rm time\ to\ expand\ fully\ } t_{\rm expand} \quad = \quad 
     { {R_{\rm final}}\over{c_s} }
$$

$$
{\rm lifetime\ on\ main\ sequence\ } \quad \sim  \quad
    { {\rm mass\ of\ fuel}\over{\rm rate\ of\ burning\ fuel} } 
         \quad \sim \quad
    { {M} \over {L} }
$$


\bye

