bound un-bound gas clouds ??

[Widdekind]

Spherical gas cloud gravitational binding energy

[math]U = -\frac{2 G M^2}{5 R}[/math]

Cloud thermal energy

[math]E = \frac{3}{2}N k_B T[/math]

Hydrostatic equilibrium (one-zone approximation)

[math]\frac{dP}{dr} = - \rho g[/math]

 

[math]\frac{P}{R} \approx \frac{M}{V} \frac{G M}{R^2}[/math]

 

[math]P = \frac{2}{3} \frac{E}{V}[/math]

 

[math]E \approx \frac{3 G M^2}{2 R}[/math]

Force-bound, energy-un-bound, gas ???

 

[math]\frac{2}{5} \times \frac{G M^2}{R} < E < \frac{3}{2} \times \frac{G M^2}{R}[/math]

How could a gas be energetically unbound, yet unable to overcome (gravity) forces involved ?? Inexpertly, the HSE force equation is the more stringent.