\magnification=\magstep1

$$
2 KE \ + \ GPE \quad = \quad 0
$$

$$
2 KE \quad = \quad - GPE
$$

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

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

$$
N \quad = \quad {{M}\over{\mu \thinspace m_H}}
$$

$$
M \quad = \quad N \thinspace \mu \thinspace m_H
$$


$$
2 \left[ {{3}\over{2}} \ {{M}\over{\mu \thinspace m_H}} \ kT \right]  \quad = \quad 
     {{3}\over{5}} \ {{G M^2}\over{R}} 
$$

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

$$
R \quad = \quad {\left[ {{3 M}\over{4 \pi \rho}} \right]} ^{1/3}
$$


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

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




\bye

