Calculate the escaping thermal tail

Define the particle and the surface

Take a spherical surface of radius $r$ centered on a planet with gravitational parameter $GM$ . Radius increases outward. A particle has mass $m$ ; the local number density $n$ is measured per cubic meter, and temperature $T$ is in kelvin. Introduce $c=\sqrt{kT/m}$ , the one-component thermal speed scale. It is also the isothermal sound speed of this single-component ideal gas.

Assume that immediately below this surface collisions have maintained an isotropic Maxwellian with negligible bulk drift. Above it, approximate motion by collisionless central gravity. Ignore incoming particles from infinity. These assumptions describe an illustrative strong-binding exobase.

f(\mathbf v)=\frac{n}{(2\pi c^2)^{3/2}}\exp\!\left[-\frac{v^2}{2c^2}\right],\qquad v_{\rm esc}=\sqrt{\frac{2GM}{r}}

The velocity density integrates to n over all three-dimensional velocities. Escape requires total speed above the escape speed and a positive outward radial velocity.

Reading: Volkov et al. 2011, Thermally driven atmospheric escape, arXiv v1

Weight each particle by how fast it crosses

In a short time $\Delta t$ , particles with outward velocity $v_r$ cross the surface from a layer of thickness $v_r\Delta t$ . Their contribution per area and time is therefore $v_r f\,d^3v$ . Choose the polar angle from the outward radial direction. Then $v_r=v\cos\theta$ and the velocity-space volume element is $v^2\sin\theta\,dv\,d\theta\,d\phi$ .

Integrate over the outward hemisphere: the azimuth contributes $2\pi$ , and $\int_0^{\pi/2}\cos\theta\sin\theta\,d\theta=1/2$ . The angular factor is therefore $\pi$ . The remaining speed integral contains a third power, because the crossing speed contributes one extra factor of velocity.

\Phi_{\rm J}=\frac{n\pi}{(2\pi c^2)^{3/2}}\int_{v_{\rm esc}}^{\infty}v^3e^{-v^2/(2c^2)}\,dv

This is a number flux per area per time. The angular integral includes only outward trajectories.

z=\frac{v^2}{2c^2},\quad v^3dv=2c^4z\,dz,\quad\lambda=\frac{GM}{rc^2}

The lower integration limit becomes the binding parameter. An energy variable makes the exponential tail easy to integrate.

\Phi_{\rm J}=\frac{nc}{\sqrt{2\pi}}\underbrace{\int_\lambda^\infty ze^{-z}dz}_{(1+\lambda)e^{-\lambda}},\qquad\dot M_{\rm J}=4\pi r^2m\Phi_{\rm J}

Integration by parts gives the last factor. Multiplying by the sphere area and particle mass converts number flux to mass per second.

Reading: Volkov et al. 2011, Thermally driven atmospheric escape, arXiv v1

Locate a surface where collisions run out

An exobase is a local collision criterion. Adopt an effective hard-sphere cross section $\sigma$ and mean free path $\ell=1/(\sqrt2 n\sigma)$ . Compare it with a density scale height $H$ , the distance over which density changes by a substantial fraction. Their ratio $\mathrm{Kn}=\ell/H$ is the local Knudsen number. This differs from a Knudsen number defined using a planet’s radius.

To locate an illustrative exobase, first use a static isothermal atmosphere below it. Balance the pressure gradient against gravity, substitute the ideal-gas pressure, then integrate between the reservoir radius and the candidate surface.

\frac{d\ln n}{dr}=-\frac{GM}{c^2r^2},\quad n(r)=n_{\rm b}\exp\!\left[\lambda_{\rm b}\!\left(\frac{R_{\rm b}}r-1\right)\right],\quad H=\frac{c^2r^2}{GM}

The logarithmic derivative is the relative density change per distance. Integrating its inverse-square dependence gives the density profile and local scale height.

Reading: Volkov et al. 2011, Thermally driven atmospheric escape, arXiv v1

A worked strong-binding example

Use eight adopted Earth-mass units, a reservoir radius of two Earth radii, $T=1000\,\mathrm K$ and $n_{\rm b}=10^{15}\,\mathrm{m^{-3}}$ . The fixed particle mass is $1.6735\times10^{-27}\,\mathrm{kg}$ and the effective cross section is $2\times10^{-19}\,\mathrm{m^2}$ . These are declared synthetic inputs, not a retrieved atmosphere.

The base binding parameter is $30.3342$ . Bracket the first root of the local condition $\mathrm{Kn}=1$ between the reservoir and the formal isothermal sonic radius. It lies at $r_{\rm ex}/R_{\rm b}=1.20447$ , where $\lambda_{\rm ex}=25.1846$ and $n_{\rm ex}=5.80\times10^{12}\,\mathrm{m^{-3}}$ . The flux-weighted escaping fraction is $3.02\times10^{-10}$ . Substituting the local density, radius and thermal speed gives $\dot M_{\rm J}=9.96\times10^{-6}\,\mathrm{kg\,s^{-1}}$ .

The display uses eight unbound paths so that you can see escape at all. Their total weight is $(1+\lambda_{\rm ex})e^{-\lambda_{\rm ex}}$ ; forty-eight bound paths carry the remaining weight. Counting the drawn amber dots would therefore grossly overestimate the real flux fraction. The path shapes follow central gravity, with accelerated display time.

Know when to change the calculation

The hydrostatic profile tends to a nonzero density if extended to infinity. It is only used up to its first local exobase crossing, below the formal critical region. If no such crossing exists above a collisional base, the calculation does not manufacture an exobase.

A strong-binding threshold of six is used here to select a Jeans illustration. It is a teaching restriction on the approximation, not a universal boundary between escaping regimes. Kinetic calculations can include returning particles, non-Maxwellian populations and thermal conduction, and can differ from the simple Jeans value. A collisional bulk outflow instead needs continuity and momentum together; that is the next chapter.

Reading: Volkov et al. 2011, Thermally driven atmospheric escape, arXiv v1 · Cranmer 2004, New views of the solar wind with the Lambert W function, v1

Calculate the escaping thermal tail

Define the particle and the surface

Weight each particle by how fast it crosses

Locate a surface where collisions run out

A worked strong-binding example

Know when to change the calculation

Exercises and explanations

Try the prediction

Locate the rare tail

Sources and calculation scope