Protoplanetary Discs

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

$H\approx {\sqrt {\frac {kTa^{3}}{GM\mu }}}$

where $k$ is the Boltzmann constant, $T$ is the temperature, $a$ is the semi major axis, $G$ is the gravitation constant, $M$ is the mass of the star and $\mu$ 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

$T_{BB}\approx T_{s}{\sqrt {\frac {R_{s}}{a}}}$

where $R_{s}$ is the radius of the star and $T_{s}$ 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 $H/a$.

$T\approx \left({\frac {H}{a}}\right)^{1/4}T_{BB}\approx \left({\frac {H}{a}}\right)^{1/4}{\sqrt {\frac {R_{s}}{a}}}T_{s}$

Plugging this in the expression for the scale height gives

${\frac {H}{a}}\approx \left({\frac {kT_{s}R_{s}}{GM\mu }}\right)^{4/7}\left({\frac {a}{R_{s}}}\right)^{2/7}$

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

$H_{s}\approx {\frac {kT_{s}R_{s}^{2}}{GM\mu }}$

and we get

${\frac {H}{a}}\approx \left({\frac {H_{s}}{R_{s}}}\right)^{4/7}\left({\frac {a}{R_{s}}}\right)^{2/7}$

The disc flares up when $H\approx a$, which happens at

$a\approx {\frac {R_{s}^{3}}{H_{s}^{2}}}$