\magnification=\magstep1 

$$
GPE \quad = \quad  - \thinspace {{G M_1 M_2}\over{R}}
$$

$$
dE \quad = \quad {{G M dm}\over{R}}
$$

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

$$
dm \quad = \quad ({\rm volume}) ({\rm density}) \quad = \quad 
            \left( 4 \pi r^2 dr \right) \left( \rho \right) 
$$

$$
GPE \quad = \quad - \thinspace {{G M dm}\over{r}}
$$

$$
GPE \quad = \quad \int_{r=0}^{r=R} \ - \  
           { {G  \ \left( {{4}\over{3}} \pi r^3 \rho \right) \  
              \left( 4 \pi r^2 dr \rho \right) }\over{r} }
$$

$$
GPE \quad = \quad - \  {{3}\over{5}}  \thinspace \left( {{G M^2}\over{R}} \right) 
$$

$$
GPE \quad = \quad {{G M_{\rm Sun} M_{\rm Moon}}\over{R}}
$$

$$
{\rm system\ is\ unbound\ when\ } \quad \quad KE \ + \ GPE \quad = \quad 0
$$





\bye
