| | 1 | = Blowoff Threshold = |
| | 2 | |
| | 3 | Assumptions for below: photon momentum is directed purely radially. Full extent of torus is optically thick (all photons are absorbed). Torus moves uniformly. |
| | 4 | |
| | 5 | The initial gravitational binding energy of the gas torus is given by |
| | 6 | |
| | 7 | $E_0 = \frac{G M_{gas} M_\star}{a_0}$, with $a_0$ the initial average orbital radius (major radius of the torus) and |
| | 8 | $M_{gas} = \pi R^2 \theta a_0 \rho$ the mass of gas contained in a torus that extends for an angle $\theta$ around the star with minor radius $R$. |
| | 9 | |
| | 10 | The area presented to Lyman-alpha radiation is essentially the rectangle of height $2R$ and width $\theta a_0$, so that |
| | 11 | |
| | 12 | $A_0 = 2 \theta a_0 R$. |
| | 13 | |
| | 14 | For some major radius $a$, the flux incident on the torus is |
| | 15 | |
| | 16 | $F = F_0 \left(\frac{a_0}{a}\right)^2$ |
| | 17 | |
| | 18 | (and area is $A = 2 \theta a R$). |
| | 19 | |
| | 20 | At this radius, the power provided by the photons is |
| | 21 | |
| | 22 | $P_{\gamma} = F A e_{\gamma}$. (This part is potentially bothersome - the photons are absorbed as momentum, not energy.) |
| | 23 | |
| | 24 | We can calculate the blowoff timescale by determining the amount of time required for the photons to deposit the binding energy of the torus: |
| | 25 | |
| | 26 | $\tau_{blowoff} = \frac{E_0}{P_{\gamma}}$ |
| | 27 | |
| | 28 | We can also calculate the time required for the torus to be fully replenished: |
| | 29 | |
| | 30 | $\tau_{rep} = \frac{M}{\dot{M}}$ |
| | 31 | |
| | 32 | Taking the ratio of these timescales: |
| | 33 | |
| | 34 | $\frac{\tau_{blowoff}}{\tau_{rep}} = \frac{G \dot{M} M_\star (a_0 + \Delta a)}{2 a_0^3 e_\gamma F_0 R \theta}$, where $\Delta a$ is the change in major radius we consider sufficient for the material to be "blown off." If $\frac{\tau_{blowoff}}{\tau_{rep}} \gg 1$, the torus will remain essentially unaffected. If $\frac{\tau_{blowoff}}{\tau_{rep}} \ll 1$, the torus will be completely blown away. The timescales are equal for a mass loss rate of 3x10^10^ g/s at a flux of ~8.5x10^12^ phot/cm^2^/s |
| | 35 | |
