AC13 · From thermal tails to a wind

Planetary Atmospheric Escape

When does a warm atmosphere become an escaping wind? A strongly bound atmosphere can lose a tiny high-speed tail after collisions become rare. A hotter, sufficiently collisional atmosphere can instead carry gas outward collectively through a sonic point. Gravity, particle mass and temperature set the binding; an independent heat budget must still supply the escaping flow. A mathematical wind alone does not establish that heating is available.

Drag the planet and its irradiating star. Compare a Maxwellian escape tail, a transonic isothermal wind and the power needed to sustain it.

Base binding parameter $3.79$

Sonic radius / base radius $1.90$

Displayed model rate · kg/s $7.70\times10^6$

Wind power / available XUV power $0.228$

Let the atmosphere go

Planet mass

4

M/M_\oplus

Prescribed gas temperature

4000

T\;[\mathrm{K}]

XUV irradiance, logarithm

2

\log_{10}(F_{\rm XUV}/[\mathrm{W\,m^{-2}}])

Drag the planet vertically to change its gravity; drag the star vertically to change its XUV power. Temperature stays independently prescribed.

Boundary and model limits

Reservoir radius

2

R_{\rm b}/R_\oplus

Base number density, logarithm

15

\log_{10}(n_{\rm b}/[\mathrm{m^{-3}}])

Usable XUV fraction

0.15

\eta

The reservoir boundary is held fixed. Neutral-H-like particles have a fixed effective collision cross section. Sonic Knudsen below 0.1 is a necessary continuum check; temperature, ionization and cooling are not solved. XUV power is compared only out to twelve base radii. A formal wind outside these limits is labeled as a reference.

Follow the escape course and offline wind calculation

Model assumptions

Synthetic single-component spherical models with prescribed temperature and a fixed reservoir boundary. Jeans flux is evaluated only at a located hydrostatic local-Knudsen exobase with strong binding; importance-sampled ballistic tracers exaggerate rare escape paths. Parker transport has independently fixed base density and requires a collisional sonic region and a subsonic lower boundary. XUV power is a separate finite-radius budget, not a temperature or heating solution. No ionization, chemistry, cooling, magnetic field, tides, stellar-wind confinement or planet evolution is solved. The apparent planet is the gas reservoir boundary, not a measured rocky surface. The tutorial supplies independent numerical synthetic benchmarks, not observed atmosphere profiles.

Prescribed-temperature Parker wind; collisional sonic region.

Cyan and amber tracers show subsonic and supersonic model flow. A dotted sphere marks the sonic radius for the wind, or the located exobase for Jeans escape. In the Jeans view, returning paths are cyan and rare unbound paths are amber; eight escape paths are deliberately oversampled and carry their correct small flux weights. These are synthetic trajectories, not measured images.

Formal accelerating Parker solution

Wind density and local continuum check

Numerical and display conventions

The planet edge represents the gas-reservoir radius. The scene extends to four base radii; profiles and power extend to twelve. The ambient glow is decorative, not a calibrated optical image. Wind tracer positions follow the computed travel time; time is accelerated. Ballistic paths follow central gravity above the located exobase; their display speeds are enlarged. No atmosphere depletion or radius evolution is integrated.

Start with the essentials

Focus question: Which assumptions support this escape rate?
One-sentence intuition: Binding, transport and heating answer different questions.

Core mathematical model

Binding

\lambda=\frac{GMm}{rkT}

Binding energy compared with the thermal energy per particle.

Continuity

\dot M=4\pi r^2\rho u

A steady spherical flow carries the same mass through every sphere.

Critical point

r_{\rm s}=\frac{GM}{2c^2},\qquad c^2=\frac{kT}{m}

The accelerating isothermal wind crosses the sonic point at this radius.

Common difficulties

Power does not prescribe temperature

Typical misconceptionMore XUV must change this prescribed-temperature wind rate.

Better mental modelThe wind has a fixed temperature and base density. XUV changes a separate available-power comparison; a full energy equation would couple them.

Run the experiment

01
Move gravity
Drag the planet upward. Watch the dotted radius and the speed profile.
What to observe: At fixed temperature, stronger gravity moves the sonic point outward and reduces the base flow.
02
Separate a tail from a wind
Select A rare escaping tail, then Collisions run out. Read the local continuum warning.
What to observe: A hydrostatic exobase can support an illustrative Jeans calculation while the formal Parker sonic region is collisionless.
03
Ask who pays for the wind
Choose The missing heating and drag the star upward. Keep temperature fixed.
What to observe: The available power changes; the independently prescribed-temperature transport does not. A real atmosphere would need an energy equation to decide its temperature and rate.

Planetary Atmospheric Escape

Three questions before calling it a wind

Core mathematical model

Binding

Continuity

Critical point

Common difficulties

Power does not prescribe temperature

Run the experiment

Move gravity

Separate a tail from a wind

Ask who pays for the wind

Related experiments and articles

Planetary atmospheres from thermal tails to winds