PX282 - J7d - atmospheric escape

this happens when the upper end of the maxwell-boltzmann distribution extends above the escape velocity of the planet
due to equipartition in thermal equilibrium, the lighter elements are more vulnerable to escape - jeans escape
the escape velocity:

v_{e s c} = \sqrt{\frac{2 G M}{R}}

\frac{1}{2} m_{H} {\bar{v}}^{2} = \frac{3}{2} k T_{p}

where, $\bar{v}$ is the average thermal velocity

$\bar{v} ≃ 2.4$ km s $^{- 1}$ at $230$ K
so, the escape rate is very low
escape is driven by solar x-ray heating of the upper atmosphere $⟹ T ≃ 1000$ K $⟹ \bar{v} ≃ 5$ km s $^{- 1}$
this leads to a more efficient mass loss
escape rate is energy-limited

L_{i n, X} = \underset{energy in per unit time}{\underset{⏟}{\frac{L_{X} π R_{p}^{2}}{4 π a^{2}}}} \geq \underset{GPE overcome per unit time}{\underset{⏟}{\frac{G M_{p} \dot{m}}{R_{p}}}}

where, $X$ stands for x-rays

∴ \dot{m} \leq \frac{L_{X} R_{p}^{3}}{4 G M_{p} a^{2}}

sun today, $L_{X} ≃ 10^{- 6} L_{⊙}$
young sun, $L_{X} ≃ 10^{- 3} L_{⊙}$
for young venus: $\dot{m} = 5.7 \times 10^{- 6}$ kg s $^{- 1}$
water on earth $≃ 10^{21}$ kg $⟹ 10^{20}$ kg of $H$ ( $2 / 18 \times m_{w a t e r}$ )
to remove all water in venus: $10^{6}$ years at $L_{X} = 10^{- 3} L_{⊙}$ / $10^{9}$ years at $L_{X} = 10^{- 6} L_{⊙}$
water on earth is protected from UV photolysis by ozone layer, and trapped lower in atmosphere as it condenses into clouds
in runaway greenhouse, water would be in vapour from throughout atmosphere and vulnerable to escape
escape is also driven by solar wind for planets without a global magnetic field: ${\dot{m}}_{w i n d} = 4 π r^{2} n_{H} m_{H} v$
at $r = 1$ AU, $n \sim 7 \times 10^{6}$ m $^{- 3}$ , $v = 500$ km s $^{- 1}$
the KE flux is:

f_{K E} = \frac{1}{2} n_{H} m_{H} v \times v^{2} ≃ 7 \times 10^{- 4} {W m}^{- 2}

where $n_{H} m_{H} v$ is the mass per unit time per unit area

c.f: xray flux at $1$ AU is: $f_{X} = 10^{- 6} L_{⊙} / 4 π r^{2} ≃ 1.3 \times 10^{- 3}$ W m $^{- 2}$ , which is very similar to the KE flux