\magnification=\magstep1


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

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

$$
{{dP_{\rm rad}}\over{dr}} \quad = \quad
       - {{{\bar{\kappa}} \rho}\over{c}} \thinspace F_{\rm rad}
$$

$$
- {{{\bar{\kappa}} \rho}\over{c}} \thinspace F_{\rm rad}
  \quad = \quad
    {{4}\over{3}} \thinspace a T^3 \thinspace {{dT}\over{dr}}
$$

$$
F_{\rm rad} \quad = \quad {{L_r}\over{4 \pi r^2}}
$$


$$
- {{{\bar{\kappa}} \rho}\over{c}} \thinspace  {{L_r}\over{4 \pi r^2}}
  \quad = \quad
    {{4}\over{3}} \thinspace a T^3 \thinspace {{dT}\over{dr}}
$$

$$
{{dT}\over{dr}} \quad = \quad - {{3 \bar{\kappa} \rho L_r}\over
                                 {16 \pi a \thinspace c \thinspace r^2 \thinspace T^3}}
$$

$$
P V \quad = \quad N k T \quad \quad \Rightarrow \quad \quad P \quad = \quad n k T
  \quad  = \quad 
       {{\rho}\over{\bar{m}}} \thinspace k T
$$

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

$$
{{dP}\over{dr}} \quad = \quad 
         {{k}\over{\bar{m}}} 
         \thinspace {{d\rho}\over{dr}} \thinspace T \ + \ 
         {{k}\over{\bar{m}}} 
         \thinspace \rho \thinspace {{dT}\over{dr}}
$$

$$
{{dP}\over{dr}} \quad = \quad 
         {{P}\over{\rho}} \thinspace {{d\rho}\over{dr}}  \ + \ 
         {{P}\over{T}} \thinspace {{dT}\over{dr}} 
$$

$$
P \quad = \quad C \thinspace \rho^\gamma
$$

$$
{{dP}\over{dr}} \quad = 
       \quad C \thinspace \gamma \rho^{\gamma - 1} \thinspace
        {{d \rho}\over{dr}}
  \quad = 
       \quad \gamma C \thinspace {{\rho^{\gamma}}\over{\rho}} \thinspace
        {{d \rho}\over{dr}}
  \quad = 
       \quad \gamma \thinspace  {{P}\over{\rho}} \thinspace
        {{d \rho}\over{dr}}
$$

$$
{{P}\over{\rho}} {{d\rho}\over{dr}} \ + \ {{P}\over{T}} {{dT}\over{dr}} 
       \quad = \quad 
       \gamma \thinspace  {{P}\over{\rho}} \thinspace
        {{d \rho}\over{dr}}
$$

$$
{{dT}\over{dr}} \quad = \quad {{T}\over{P}} \thinspace 
        \left[ (\gamma - 1) \thinspace 
                {{P}\over{\rho}} \thinspace
                {{d\rho}\over{dr}}  \right]
$$

$$
{{dT}\over{dr}} \quad = \quad {{T}\over{P}} \thinspace 
        \left[ \gamma \thinspace 
                {{P}\over{\rho}} \thinspace
                {{d\rho}\over{dr}} 
             \ - \ 
                {{P}\over{\rho}} \thinspace
                {{d\rho}\over{dr}} 
                  \right]
$$

$$
{{dT}\over{dr}} \quad = \quad {{T}\over{P}} \thinspace 
        \left[ {{dP}\over{dr}}
             \ - \ 
                {{1}\over{\gamma}} {{dP}\over{dr}}
                  \right]
$$


$$
{{dT}\over{dr}} \quad = \quad {{T}\over{P}} \thinspace 
                \left( 1 - {{1}\over{\gamma}} \right) \thinspace
                {{dP}\over{dr}} 
$$

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

$$
{{dT}\over{dr}} \quad = \quad 
      - \left( {{\bar{m}}\over{k}} \right)
      \thinspace
          \left( 1 - {{1}\over{\gamma}} \right)
      \thinspace
          {{G M_r}\over{r^2}}
$$ 

$$
{\left| {{dT}\over{dr}} \right|}_{\rm actual} \quad >= \quad 
   {\left| {{dT}\over{dr}} \right|}_{\rm adiabat} \quad = \quad 
      \left( {{\bar{m}}\over{k}} \right)
      \thinspace
          \left( 1 - {{1}\over{\gamma}} \right)
      \thinspace
          {{G M_r}\over{r^2}}
$$

$$
P \quad \propto \quad T^\alpha \qquad \Rightarrow 
            \qquad \ln{P} \quad = \quad \alpha \ln{T}
$$

$$
{{d \ln{P}}\over{d \ln{T}}} \quad = \quad  \alpha
$$

$$
{{d \ln{P}}\over{d \ln{T}}} \quad = \quad {{\gamma}\over{\gamma - 1}}
$$


$$
{{d \ln{P}}\over{d \ln{T}}} \quad < \quad {{\gamma}\over{\gamma - 1}}
$$

\bye
