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This entry is largely based on Chiang & Goldreich 1997.

Let us consider a disc that is illuminated and heated by the host star. The scale height of the disc is given by

where is the Boltzmann constant, is the temperature, is the semi major axis, is the gravitation constant, is the mass of the star and is the mass of particles in the disc. A compact object in blackbody equilibrium with the radiation from the host star will have a temperature

where is the radius of the star and is the temperature at the photosphere. In the case of a disc, due to the oblique illumination, the temperature at the same distance from the star will be lower by a factor that depends on the angle between the tangent to the surface of the disc and the photon direction .

Plugging this in the expression for the scale height gives

This expression can be further simplified by introducing the scale height on the star

and we get

The disc flares up when , which happens at

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