Thermal Escape


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Contact: F.X.Timmes
my one page vitae,
full vitae,
research statement, and
teaching statement.

Given a planet's mass ${\rm M}$, radius ${\rm R}$, and the height of its exosphere ${\rm h}$ at temperature ${\rm T}$, the thermal escape timescale $\tau_{\rm escape}$ for an atom/molecule of mass ${\rm m}$ can be estimated from $$ {\rm \tau_{escape} = \dfrac{H}{v_{jeans}} } \label{eq1} \tag{1} $$ where the scale height ${\rm H}$ and acceleration due to gravity ${\rm g}$ are $$ {\rm H =\dfrac{k \ T}{m \ g} } \qquad \qquad {\rm g = \dfrac{G \ M} {(R + h)^2} } \label{eq2} \tag{2} $$ and the Jeans speed ${\rm v_{Jeans}}$ is $$ {\rm v_{jeans} = v_{{\rm peak}} \dfrac{(1+ \lambda) \ e^{-\lambda}}{\sqrt{4 \pi}} } \ . \label{eq3} \tag{3} $$ The peak speed ${\rm v_{peak}}$, escape speed ${\rm v_{escape}}$, and their ratio $\lambda$ are $$ {\rm v_{peak} = \sqrt{\dfrac{2 k T}{m}} } \qquad \qquad {\rm v_{escape} = \sqrt {\dfrac{2 G M}{R + h}} } \qquad \qquad {\rm \lambda = \left ( \dfrac{ v_{escape}}{v_{peak}} \right )^2 } \label{eq4} \tag{4} $$
The tool contained in public_jeans_escape.tbz implements these simple analytic formulas along with comparing their results to numerical integrations of the Maxwell-Boltzmann distribution. Here are the speed distributions for hydrogen helium, carbon, nitrogen, and oxygen for Earth's exosphere and their associated thermal escape timescales.



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