poles. The choice of this last function will determine the details of the problem, and can be thought of as something of the type

$$g(\theta)=\bar{\rho}_{0}e^{-\left( \frac{\theta- \pi/2}{\sqrt{2} \theta_{0}}   \right)^{2}}$$

with $\bar{\rho}_{0}$ a normalisation constant and $\theta_{0}$ a form constant describing the flattening of the disk of infalling material. However, we shall mostly leave results indicated in terms of $g(\theta)$. The continuity equation (14) now fixes $f(r)$ through:

$$-\frac{g(\theta)}{r^{2}} \frac{d}{d r} \left[ (2 G M)^{1/2} r^{3/2} f(r) \right]=0,$$

and hence $f(r) r^{3/2} =cte.$, which completes the description of the background state through:

$$\rho_{0}(r,\theta)=\left( \frac{\bar{r}}{r} \right)^{3/2} g(\theta),$$

with $\bar{r}$ a constant which determines the point at which $g(\theta=\pi/2)$ $\Rightarrow$ $\rho_{0}=\bar{\rho}_{0}$.

We shall now analyse the behaviour of a fraction of somewhat hotter material, through a first order perturbation analysis of the above solution. This small fraction could represent the last of the material to fully cool, or result from the irradiation of the central star onto the upper and lower surfaces of the radially infalling disk, as described by e.g. Hollenbach et al. (1994) or Alexander et al. (2006) for the standard case of Keplerian accretion disks, in connection with the problem of disk photoevaporation. In the above it is shown that 