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Optically thin limit
We can use the ratio of radiation force to gravitational force
For lyman-alpha absorption, we have the radiation force per atom as
where
is the ionization fractionAnd we also have a gravitational force per atom
Combining these we have
For what value of
are kepler orbits escape trajectories?
Setting these equal gives
or a critical flux
where
, ,For nearly fully ionized gas
we getThis would be a lower limit as the flux is assumed to not be attenuated
Non-Optically thick
Now the planet is supplying a torus with bound material at a rate
. The delta v required to liberate that material is of order . We therefore need the star to impart momentum to the torus fast enough to liberate material at the same rate it enters. Setting the rate of momentum injection possible from the stellar lyman alpha flux, to the momentum injection rate needed to liberate material at the same rate it is injected gives
where
is the height of the torus.Solving for
we get
Now again if
where
, , , andwe have
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