Changes between Version 4 and Version 5 of u/adebrech/Matlab/RadiationPressure

Timestamp:

08/28/18 12:32:07 (8 years ago)

Author:

adebrech

Comment:

—

u/adebrech/Matlab/RadiationPressure

@@ -36 +36 @@
 === Including Ablation ===
 
-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.
+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, as well as the fact that the material being blown out takes longer to flow out of the way of the radiation than it does to absorb sufficient energy to unbind it.
 
 The edge of the torus is approximately $4 R_P$ from the planet, so the area exposed to Lyman-$\alpha$ radiation is
@@ -61 +61 @@
 
 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?).
+
+How long does it take for a parabola of material to blow around the torus, to the point where it is no longer blocking Lyman-$\alpha$ from the lower material? Taking the pressure times the area to get force, we can calculate the acceleration of (half of) the parabola in the approximately "up-slope" direction:
+
+$P = F_0 h \lambda$
+
+$a = \frac{P A_P}{V_P \rho}\cos{(\arctan{\frac{a_2}{h_2}})}$
+
+The time to traverse half the arclength is then
+
+$t = \sqrt{\frac{[\sqrt{a_1^2 + 4h_1^2} + \frac{a_1}{2 h_1} \mbox{arcsinh}{\frac{2 h_1}{a_1}}] V_P \rho}{P A_P \cos{(\arctan{\frac{a_2}{h_2}})}}} \approx 6.3\times10^{10}$ s,
+
+or seven orders of magnitude longer than required to absorb sufficient Lyman-$\alpha$ radiation to be blown off. This would seem to explain most of the difference that we see, then -- this is a very crude calculation, though, and could use some refining.
 
 = Lyman-$\alpha$ Line Transfer =