ionizing radiation from the star generically creates an ionized layer on the surface of the disk.

Regarding the evolution of this component, we shall also be interested interested in a steady state solution given by:

$$V(r,\theta)=V_{0}(r) + V_{1}(r, \theta),$$

$$U(r,\theta)=0 + U_{1}(r,\theta),$$

$$\rho(r,\theta) = \rho_{0}(r,\theta) +\rho_{1}(r,\theta),$$

where quantities with subscript (1) denote the perturbation on the background solution. Writing eqs.(14), (15) and (16) to first order in the perturbation one obtains after rearranging terms:

$$g(\theta) \frac{\partial \left(r^{1/2} V_{1}\right)}{\partial r} - 
B\frac{\partial \left( r^{3/2} \rho_{1} \right)}{\partial r}=
\frac{-1}{r^{1/2} sin(\theta)} \frac{\partial \left( sin(\theta) g(\theta) U_{1} \right) }{\partial \theta},$$

$$\frac{\partial \left( V_{1}/r^{1/2} \right) }{\partial r} = \frac{A r^{3/2}}{g(\theta)} \frac{\partial \rho_{1}}{\partial r},$$

$$\frac{\partial \left( r U_{1}  \right)}{\partial r} = \frac{A r^{2}}{g(\theta)} \frac{\partial \rho_{1}}{\partial \theta},$$

In the above we have assumed an isothermal equation of state for the perturbation $P_{1}=c^{2}\rho_{1}$, an assumption often used in the modeling of astrophysical jets, e.g. the T Tauri jets observed and modelled by Hartigan et al. (2004). This idealised case serves to illustrate clearly the consequences of the physical setup being considered, as it allows for an analytic solution. The generalisation to more realistic adiabatic, polytropic or otherwise equations of state can be performed numerically, and can be expected 