User:KurtBlueSky/sandbox2

The gravitational radius of an irradiated disk is:

$$r_g = \frac{(\gamma - 1)}{2 \gamma}\frac{GM_*m}{kT}$$                                                                     ,

If we denote the coefficient in the above equation by the Greek letter $\kappa$ then

$$\kappa = \frac{(\gamma - 1)}{2 \gamma} = \frac{1}{(2+f)}$$  ,                                           .

where $f$ is the number of degrees of freedom and we have used the formula: $\gamma = 1 + \frac{2}{f}$.

For an atom, such as a hydrogen atom, then $f = 3$, because an atom can move in three different, orthogonal directions. Consequently, $\kappa = 0.2 $. If the hydrogen atom is ionized, i.e., it is a proton, and is in a strong magnetic field then $f = 2 $, because the proton can move along the magnetic field and rotate around the field lines. In this case, $\kappa = 0.25  $. A diatomic molecule, e.g., a hydrogen molecule, has $f = 5 $ and $\kappa = 1/7 \approx 0.143   $. For a non-linear triatomic molecule, such as water, $f = 6 $ and $\kappa = 0.125   $. If $f$  becomes very large, then $\kappa$ approaches zero. This is summarised in the Table 1, where we see that different gases may have different gravitational radii.

Table 1: Gravitational radius coefficient as a function of the degrees of freedom.