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Update 4/30 (Radiation Pressure)
WASP-12b paper accepted - will be posting to ArXiv today.
Radiation pressure:
\alpha = F_{rad}/F_{grav}
\beta = F_{rad}/F_{ram}
\delta = F_{rad}/F_{press}
All of the following are in computational units:
F_{grav} = \frac{G \rho M_p}{r^2}
F_{ram} = \nabla (\rho |v|^2) \cdot \hat{x}
F_{press} = (\nabla P) \cdot \hat{x}
For \beta and \delta, white is F_{rad} \approx F_{other}.
\alpha
\beta
\delta
Cross-cut
Ly-\alpha flux for HD209458b
From here (Bourrier & Lecavalier des Etangs 2013), flux peaks at J = ~28x10-14 erg cm-2 s-1 AU-1.
Converting to photons cm-2 s-1,
F = \frac{J (1.5 \times 10^{13} \mbox{ cm})}{1.63\times10^{-11} \mbox{ erg phot}^{-1}} = 2.5\times10^11 \mbox{ phot cm}^{-2} \mbox{s}^{-1}
Which is 3 orders of magnitude less than the maximum flux in our simulation. If instead it should be scaled to the orbital radius (0.04747 AU, not 1 AU), that reduces it to 1.2x1010.
Timescale for blowoff vs. replenshiment
Assume torus of axis a = 0.04747 AU, radius of 12 R_p = 12(1.5\times10^{10}) cm, and uniform density \rho = 4.11\times10^{-19} g/cm3. Given the mass loss rate \dot{M} of the planet, the timescale for replenishment is:
\tau_{rep} = \frac{M}{\dot{M}} = \frac{\rho V}{\dot {M}} = \frac{\rho \pi^2 (12 R_p)^2 a}{\dot{M}} = 6.2\times10^6 s
The timescale for blowoff is dependent upon the gravitational binding energy and the energy deposited by photons on the edge of the torus per unit time:
E_G = 2 G \rho \pi^2 (12 R_p)^2 a M_p = 8.7\times10^{27} erg
P_{phot} = F A e_\gamma = 1.45\times10^{17} erg/s
\tau_{blowoff} = E_G/P_{phot} = 6\times10^{10} s
\tau_{blowoff} \gg \tau_{rep}
- Posted: 7 years ago
- Author: adebrech
- Categories: (none)
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