\magnification=\magstep1

$$
\delta {\rm GPE} \quad  = \quad - {{G M_r dm}\over{r}} 
$$

$$
dm \quad = \quad (4 \pi r^2 dr) \thinspace \rho
$$

$$
M_r \quad = \quad {{4}\over{3}} \pi r^3 \thinspace \rho
$$

$$
{\rm GPE} \quad = \quad \int_0^R - {{16 \pi^2}\over{3}} \thinspace G \rho^2 r^4 dr
$$

$$
{\rm GPE} \quad = \quad - {{16}\over{15}} \thinspace \pi^2 G \rho^2 R^5 
$$

$$
{\rm GPE} \quad = \quad - {{3}\over{5}} \thinspace {{G M^2}\over{R}} 
$$


$$
T_{\rm crit} \quad \sim \quad {{2}\over{3}} {{Z_1 Z_2 e^2}\over{k r}} 
$$

$$
T_{\rm crit} \quad \sim \quad {{4}\over{3}} \thinspace
      {{\mu_m Z_1^2 Z_2^2 e^4}\over{k h^2}}
$$

$$
\mu_m \quad \equiv \quad {{1}\over{ {{1}\over{m_1}} + {{1}\over{m_2}} } }
    \quad = \quad {{m_1 \thinspace m_2 } \over {m_1 + m_2} }
$$


$$
\epsilon_{pp} \quad \sim \quad 1 \thinspace \rho X^2 
                 \left( {{T}\over{10^6 {\rm \thinspace K}} } \right)^4
       \quad \quad {{\rm J}\over{\rm s}} / {\rm kg} / {{\rm{{kg}\over{m^3}} }}
$$

$$
\epsilon_{CNO} \quad \sim \quad 8 \times 10^{-19} \ \rho X X_{\rm CNO}
                 \left( {{T}\over{10^6 {\rm \thinspace K}} } \right)^{19.9}
       \quad \quad {{\rm J}\over{\rm s}} / {\rm kg} / {{\rm{{kg}\over{m^3}} }}
$$

$$
\epsilon_{3\alpha} \quad \sim \quad 5 \times 10^{7} \ \rho^2 Y^2
                 \left( {{T}\over{10^8 {\rm \thinspace K}} } \right)^{41}
       \quad \quad {{\rm J}\over{\rm s}} / {\rm kg} / 
             \left({{\rm{{kg}\over{m^3}} }} \right)^2
$$

$$
dL \quad = \quad \epsilon \thinspace dm
$$

$$
dL \quad = \quad \epsilon \thinspace (4 \pi r^2 dr) \thinspace \rho
$$

$$
{{dL}\over{dr}} \quad = \quad 4 \pi r^2  \rho \thinspace \epsilon
$$


$$
dm \quad = \quad 4 \pi r^2 dr \thinspace \rho 
$$







\bye
