\magnification=\magstep1

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

$$
M \quad = \quad \int_0^R \thinspace dm \quad = \quad 
             \int_0^R 4 \pi r^2 \rho(r) dr 
$$

$$
M \quad = \quad 4 \pi \rho \ \int_0^R \thinspace r^2  dr 
$$

$$
M \quad = \quad 4 \pi \rho \ \int_0^R \thinspace r^2  dr 
  \quad = \quad
           4 \pi \rho \ \left|{{1}\over{3}} r^3 \right|_0^R 
  \quad = \quad
           {{4}\over{3}} \pi \rho R^3
$$

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

$$
F_{\rm net} \quad = \quad 
        F_{\rm grav}  \ + \ F_{\rm top} \ + \ F_{\rm bot}
$$

$$
F_{\rm net} \quad = \quad 
       - {{G M_r \thinspace dm}\over{r^2}} 
       \ - \ 
       A \thinspace P_{r + dr} \ + \ A \thinspace P_{r}
$$

$$
F_{\rm net} \quad = \quad 
       - {{G M_r \thinspace (\rho A dr) }\over{r^2}} 
       \ - \ 
       A \thinspace \left( P_{r + dr} - P_{r} \right)
$$

$$
F_{\rm net} \quad = \quad 
       - {{G M_r \thinspace (\rho A dr) }\over{r^2}} 
       \ - \ 
       A \thinspace \left (P_{r} + {{dP}\over{dr}} dr - P_{r} \right)
$$

$$
F_{\rm net} \quad = \quad 
       - {{G M_r \thinspace (\rho A) }\over{r^2}} \thinspace dr
       \ - \ 
       A \thinspace \left ({{dP}\over{dr}} dr \right)
$$

$$
{{F_{\rm net}}\over{A}} \quad = \quad 
       - {{G M_r \thinspace \rho  }\over{r^2}} \thinspace dr
       \ - \ 
       {{dP}\over{dr}} dr 
$$

$$
{{dP}\over{dr}} \quad = \quad 
       - {{G M_r \thinspace \rho  }\over{r^2}} 
$$


$$
{{\Delta P}\over{R}} \quad = \quad {{P_{\rm surface} - P_{\rm center}}\over{R}} \quad \sim \quad - {{G M \rho}\over{R^2}}
$$


$$
PV \quad = \quad NkT  
   \qquad \qquad {\rm or\ } \qquad  
   P \quad = \quad nkT
$$

$$
n \quad = \quad {{\rho}\over{m}}
$$

$$
n \quad = \quad {{\rho}\over{\bar{m}}}
$$

$$
P \quad = \quad {{\rho}\over{\bar{m}}} \thinspace k T
$$

$$
\mu \quad = \quad  {{\bar{m}}\over{m_H}} 
$$


$$
P \quad = \quad {{\rho}\over{\mu m_H}} \thinspace k T
$$

$$
P_{\rm rad} \quad = \quad {{4 \sigma}\over{3 c}} \thinspace T^4
                  = \quad {{1}\over{3}} \thinspace a T^4
$$




\bye
