Gravitational Bounded or Not Bounded

q Measures the ratio of kinetic energy to gravitational potential. In stellar evolution theory, I should take enthalpy instead of just kinetic energy but since I use isothermal temperature, the enthalpy is not conserved so I did not take it.

By Virial Theorem, if h=enthalpy

then the material is stable on its orbit, if it is larger than, it will go to higher orbit, if it is less than, it will go to lower orbit.

is the condition that material will go to infinity (unbounded)

So in my case, q=1 absolutely means unbounded, but gas is not bounded even q is close to 1 because I did not take temperature into account.

The figure shows that the gas in polar direction is not gravitational bounded while it is in equatorial plane. (q<0.5)

At this moment, the circumbinary disk has not reached steady state yet.

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