---
title: "Planetary atmospheres from thermal tails to winds"
description: "Derive Jeans escape and critical winds, run an independent Python benchmark, and read the primary papers behind atmospheric loss and the radius valley."
canonical_url: "https://sandboxphysics.com/stories/atmospheric-escape/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/atmospheric-escape/"
last_modified: "2026-10-28"
---

AC13

# Planetary atmospheres from thermal tails to winds

Sandbox Physics · 2026-10-28

A few fast particles can leak away, or gas can flow out collectively. Follow thermal motion through a critical wind, test its heat supply, and read how atmospheric loss reshapes a planet population.

1. L1

   ## [How a planet loses its atmosphere](https://sandboxphysics.com/stories/atmospheric-escape/leaving-a-planet/)

   A warm atmosphere can leak a fast particle tail or flow outward as a wind. Gravity and heating decide which calculation is useful.

2. L2

   ## [Calculate the escaping thermal tail](https://sandboxphysics.com/stories/atmospheric-escape/thermal-tails/)

   Derive the Jeans flux from a velocity-weighted Maxwellian, locate an illustrative exobase and distinguish particle fractions from surface fluxes.

3. L3

   ## [Build a wind through its sonic point](https://sandboxphysics.com/stories/atmospheric-escape/critical-wind/)

   Derive the isothermal Parker equation, choose its accelerating critical branch and compare the finite-radius heat requirement with stellar XUV power.

4. L3

   ## [Solve and check a critical atmospheric wind](https://sandboxphysics.com/stories/atmospheric-escape/solve-a-wind/)

   Run a standalone Python critical-wind benchmark, compare a regular ODE integration with Lambert-W branches and diagnose failed physical assumptions.

5. L4

   ## [Read the competing explanations of the radius valley](https://sandboxphysics.com/stories/atmospheric-escape/radius-valley/)

   Read fixed primary papers on photoevaporation, interior cooling and coupled winds, distinguishing observed populations from simulations.

[Let the atmosphere flow](https://sandboxphysics.com/labs/planetary-atmospheric-escape/)

## Sources and calculation scope

1. [Cranmer 2004, New views of the solar wind with the Lambert W function, v1](https://arxiv.org/pdf/astro-ph/0406176v1)

   7June2004; AJP 72,1397–1403. Full text, SectionsIII–IV, equations12–16: isothermal momentum equation, critical point and the two real Lambert-W branches. The solar ionized-hydrogen sound-speed convention is replaced by our explicitly fixed single-particle mass.

2. [Volkov et al. 2011, Thermally driven atmospheric escape, arXiv v1](https://arxiv.org/pdf/1009.5110v1)

   26September2010 preprint, published ApJL729,L24(2011). Full text, Section 2 and Figures1–5: hard-sphere mean free path, radius-based versus local scale-height Knudsen numbers, kinetic escape and conditional Jeans corrections. Their heating below the lower boundary differs from a globally maintained isothermal wind. The adopted Lab collision cross section is a teaching input, not their molecular-nitrogen value.

3. [Murray-Clay, Chiang & Murray2009, Atmospheric escape from hot Jupiters, v2](https://arxiv.org/pdf/0811.0006v2)

   18November2008 revision, ApJ 693,23(2009). Full text, Section 3.2, equations19/22 and Figure 7: absorbing area, energy-limited scaling and the cooling/recombination limitations of a fixed efficiency. This source develops hot-Jupiter radiation hydrodynamics; the small-planet illustration uses only the explicitly stated global power comparison.

4. [Owen & Wu 2017, The evaporation valley in the Kepler planets, v2](https://arxiv.org/pdf/1705.10810v2)

   4October2017 revision; ApJ 847,29. Full text consulted, Section 2.2 erosion timescale and Figure 6 synthetic population distributions. The predicted valley is a model outcome conditioned on initial envelopes and irradiation histories, not a directly imaged escaping atmosphere.

5. [Ginzburg, Schlichting & Sari 2018, Core-powered mass loss, v2](https://arxiv.org/pdf/1708.01621v2)

   6March2018 revision; MNRAS 476,759–765. Full text, Section 2.1, equations5–7, Figures1/3/5: the cooling reservoir, energy and transport limits, synthetic radius distributions and their comparison with the observed population.

6. [Misener et al. 2026, Bolometric–photoevaporative transition, v2](https://arxiv.org/pdf/2605.02766v2)

   arXiv version 2 submitted6August2026; PDF draft header7August2026, archive marks accepted ApJ, related DOI10.3847/1538-4357/ae969c. Full PDF consulted, Sections2 and3.2, Figures2–5. These are AIOLOS radiation-hydrodynamic simulation profiles and rates, not observed planet profiles. This course reads the fixed figures; it does not reproduce the code or its evolution calculation.