\magnification=\magstep1

$$
2K \ + \ U \quad = \quad 0
$$

$$
K \quad = \quad {{3}\over{2}} \thinspace N k T 
$$

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

$$
N \quad = \quad {{M}\over{\mu m_H}} \quad = \quad
   {  { {{4}\over{3}} \pi R^3 \rho}\over{\mu m_H} }
$$

$$
K \quad = \quad {{3}\over{2}} \thinspace k T \thinspace \thinspace
             \left( {{4 \pi R^3 \rho} \over { 3 \mu m_H}}  \right) 
  \quad = \quad
            2 k T \thinspace \left( {{\pi R^3 \rho}\over{\mu m_H}} \right)
$$

$$
U \quad = \quad - \thinspace {{3}\over{5}} \thinspace 
           { { G {\left( {{4}\over{3}} \pi R^3 \rho \right)}^2 } \over{R}} 
$$

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

$$
2 \biggl[ 2 k T \thinspace {{\pi R^3 \rho}\over{\mu m_H}} \biggr]
            \quad = \quad 
    {{16}\over{15}} \thinspace G \pi^2 R^5 \rho^2
$$


$$
{{4 k T }\over{\mu m_H}} \quad = \quad 
    {{16}\over{15}} \thinspace \pi G R^2 \rho
$$

$$
R \quad = \quad \sqrt{ {{15 k T}\over{4 \pi G \rho \thinspace \mu m_H} } }
$$

$$
K \quad = \quad {{3}\over{2}} \thinspace k T \thinspace \thinspace
             \left( {{M}\over{\mu m_H}} \right)
$$

$$
3 k T \thinspace {{M}\over{\mu m_H}}  \quad = \quad
                 {{3}\over{5}} \thinspace {{ G M^2}\over{R}} 
$$

$$
M \quad = \quad {{ k T}\over{\mu m_H}} \ {{5 R}\over{ G}}
$$
        

$$
M \quad = \quad {{4}\over{3}} \pi R^3 \rho  \qquad \Longrightarrow \quad \quad
           R \quad = \quad {\left( {{3 M}\over{4 \pi \rho}} \right)}^{1/3}
$$

$$
M \quad = \quad {\left( {{5 k T}\over{G \mu m_H}} \right)}^{3/2}
           \thinspace \thinspace
                {\left( {{3}\over{4 \pi \rho}} \right)}^{1/2}
$$

$$
\sigma_\lambda \quad \propto \quad {{1}\over{\lambda}}
$$

$$
\sigma_\lambda \quad \simeq \quad {\rm constant}
$$


$$
\sigma_\lambda \quad \propto \quad {{1}\over{\lambda^4}}
$$

\bye
