wiki:u/johannjc/scratchpad25

Version 3 (modified by Jonathan, 6 years ago) ( diff )

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 fraction

And 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 get

This would be a lower limit as the flux is assumed to not be attenuated

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 , , , and

we have

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