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 |
| 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 | |
| 36 | === Including Ablation === |
| 37 | |
| 38 | As a first attempt, I've taken the energy required to unbind a thick parabola (seen in the steady state at the end of [http://www.pas.rochester.edu/~adebrech/PlanetIonization/rad_press_2d14_side.gif this movie]) compared to the time required to refill it. This needs to be modified still to account for the fact that the material being pushed around the torus pulls new material in behind it, increasing the fill rate compared to the mass loss rate of the planet alone. |
| 39 | |
| 40 | The edge of the torus is approximately $4 R_P$ from the planet, so the area exposed to Lyman-$\alpha$ radiation is |
| 41 | |
| 42 | $A_P = 2 \pi (a_0 - 4 R_P) [\sqrt{a_1^2 + 4h_1^2} + \frac{a_1}{2 h_1} \mbox{arcsinh}{\frac{2 h_1}{a_1}}]$, |
| 43 | |
| 44 | where the term in square brackets is the arc length of the outside edge of the thick parabola (with a,,1,, the half-width (approximately from planet to edge in -z direction) and h,,1,, the height (approximately from planet to edge in -x direction). Additionally, we can calculate the volume of the thick portion: |
| 45 | |
| 46 | $V_P = 2 \pi (a_0 - 4 R_P) [\frac43 (a_1 h_1 - a_2 h_2)]$, |
| 47 | |
| 48 | where the term in square brackets here is the difference in area between the outer and inner bounding parabolas. |
| 49 | |
| 50 | We can calculate the power delivered by the Lyman-$\alpha$ photons as before. |
| 51 | |
| 52 | $P_{gamma} = F_0 (\frac{a_0}{a_0 - 4 R_P})^2 A_P e_{\gamma}$ |
| 53 | |
| 54 | And the ablation timescale is then |
| 55 | |
| 56 | $\tau_{abl} = \frac{G M_{\star} \rho V_P}{2(a_0 - 4R_P)}/P_{\gamma} = 6800$ s |
| 57 | |
| 58 | And the timescale for replenishing (taking into account only the mass loss rate of the planet) is the same: |
| 59 | |
| 60 | $\tau_{rep,P} = \frac{\rho V_P}{\dot{M}} = 320000$ s |
| 61 | |
| 62 | Both of these timescales are linearly related to $\rho$, so an increase in density would not explain the factor of 50 difference (though we don't see an increase in density, in any case). However, material appears to accelerate to the outer edge as it reaches steady state. It seems as though the ablated material is pulling new material in to fill the vacuum left by it, which could set up a steady state (although it seems like this would require extra material from somewhere?). |