Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 57 / 1 / c

Positive $F_{\mathrm{acc}}$ denotes inward flow. In steady state, mass conservation gives $d{F}_{\mathrm{acc}}/dr=2\pi rW$. In the remote-feeding approximation, take the incoming flux at large radius inside the feeding region to approach ${\dot{M}}_d$. The cumulative loss exterior to $r\ge r_g$ is ${\dot{M}}_w\sqrt{r_g/r}$, so the steady photoevaporating-disk mass flux isIt is constant inside $r_g$, continuous at $r_g$, and increases outwards towards ${\dot{M}}_d$. Its right derivative is positive and decreases as $r^{-3/2}$. The curve has a change of slope at $r_g$ because the wind turns on there. Physically less mass survives to cross successively smaller radii, while the inner disk has no wind sink. The star accretes at ${\dot{M}}_d-{\dot{M}}_w$.

For constant kinematic viscosity $\nu$, set $Y=\nu \Sigma \sqrt{r}$. The supplied viscous evolution of an accretion disk relation becomes $Y^{\prime }=F_{\mathrm{acc}}/(6\pi \sqrt{r})$. The viscous torque in an accretion disk is proportional to $Y$, so the zero-stress condition at the idealized origin sets $Y(0)=0$. ThusFor $r<r_g$ the integral is $2({\dot{M}}_d-{\dot{M}}_w)\sqrt{r}$. For $r\ge r_g$, separating the two intervals givesConsequently the surface density of a steady photoevaporating disk isBoth the density and its first derivative match at $r_g$. Let $x=r/r_g$ and take the special case ${\dot{M}}_d={\dot{M}}_w$. Then $\Sigma =0$ throughout the inner disk, while outsideThe profile leaves zero with zero slope at $x=1$, rising as $(x-1{)}^2/8$ locally, and approaches ${\dot{M}}_w/(3\pi \nu )$ from below at large radius. The requested sketches are:

The analytic profiles describe the region interior to a distant feeding boundary. A finite outer radius and the disk outside the feeding point need additional boundary conditions. For example, if the wind is truncated at $R\gg r_g$ and $F_{\mathrm{acc}}(R)={\dot{M}}_d$, the exact interior formulas above replace ${\dot{M}}_d$ by ${\dot{M}}_d+{\dot{M}}_w\sqrt{r_g/R}$; the total wind inside $R$ is ${\dot{M}}_w(1-\sqrt{r_g/R})$. The displayed profiles are their remote-boundary limit. Extending the prescribed wind to infinity is useful for estimating mass loss but is not a complete global angular-momentum boundary condition for a finite feeding radius.

As the viscous supply decreases towards the wind rate, the inner accretion flux approaches zero. Wind removal near $r_g$ can cut off replenishment of the inner disk, which then drains on its own viscous timescale, leaving a gap or inner hole. This is photoevaporative gap opening. If supply falls below wind loss, the formal steady profile would require negative inner density and is unphysical; time-dependent depletion must replace it. The exposed outer edge may then be removed rapidly. Equality of the two rates indicates the onset of rapid clearing, rather than a negative steady surface density.