# Hydrodynamic escape

Schematic of hydrodynamic escape. Energy from solar radiation is deposited in a thin shell. This energy heats the atmosphere, which then begins to expand. This expansion continues into the vacuum of space, accelerating as it goes until it escapes.

Hydrodynamic escape refers to a thermal atmospheric escape mechanism that can lead to the escape of heavier atoms of a planetary atmosphere through numerous collisions with lighter atoms.

## Description

Hydrodynamic escape occurs if there is a strong thermally driven atmospheric escape of light atoms which, through drag effects (collisions), also drive off heavier atoms.[1] The heaviest species of atom that can be removed in this manner is called the cross-over mass.[2]

In order to maintain a significant hydrodynamic escape, a large source of energy at a certain altitude is required. Soft X-ray or extreme ultraviolet radiation, momentum transfer from impacting meteoroids or asteroids, or the heat input from planetary accretion processes[3] may provide the requisite energy for hydrodynamic escape.

## Calculations

Estimating the rate of hydrodynamic escape is important in analyzing both the history and current state of a planet's atmosphere. In 1981, Watson et al. published[4] calculations that describe energy-limited escape, where all incoming energy is balanced by escape to space. Recent numerical simulations on exoplanets have suggested that this calculation overestimates the hydrodynamic flux by 20 - 100 times.[30] However, as an special case and upper limit approximation on the atmospheric escape, it is worth noting here.

Hydrodynamic escape flux (${\displaystyle \Phi }$ , [m${\displaystyle ^{-2}}$ s${\displaystyle ^{-1}}$ ]) in an energy-limited escape can calculated, assuming (1) an atmosphere composed of non-viscous, (2) constant molecular weight gas, with (3) isotropic pressure, (4) fixed temperature, (5) perfect XUV absorption, and that (6) pressure decreases to zero as distance from the planet increases.[4]

${\displaystyle \Phi ={\frac {F_{XUV}R_{p}R_{XUV}^{2}}{GM_{p}}}}$

where ${\displaystyle F_{XUV}}$  is the photon flux [J m${\displaystyle ^{-2}}$ s${\displaystyle ^{-1}}$ ] over the wavelengths of interest, ${\displaystyle R_{p}}$  is the radius of the planet, ${\displaystyle G}$ is the gravitational constant, ${\displaystyle M_{p}}$  is the mass of the planet, and ${\displaystyle R_{XUV}}$  is the effective radius where the XUV absorption occurs. Corrections to this model have been proposed over the years to account for the Roche lobe of a planet and efficiency in absorbing photon flux.[5][6][7]

However, as computational power has improved, increasingly sophisticated models have emerged, incorporating radiative transfer, photochemistry, and hydrodynamics that provide better estimates of hydrodynamic escape.[8]

## Isotope fractionation as evidence

The root mean square thermal velocity (${\displaystyle v_{th}}$ ) of an atomic species is

${\displaystyle v_{th}={\sqrt {\frac {3kT}{m}}}}$

where ${\displaystyle k}$  is the Boltzmann constant, ${\displaystyle T}$ is the temperature, and ${\displaystyle m}$ is the mass of the species. Lighter molecules or atoms will therefore be moving faster than heavier molecules or atoms at the same temperature. This is why atomic hydrogen escapes preferentially from an atmosphere and also explains why the ratio of lighter to heavier isotopes of atmospheric particles can indicate hydrodynamic escape.

Specifically, the ratio of different noble gas isotopes (20Ne/22Ne, 36Ar/38Ar, 78,80,82,83,86Kr/84Kr, 124,126,128,129,131,132,134,136Xe/130Xe) or hydrogen isotopes (D/H) can be compared to solar levels to indicate likelihood of hydrodynamic escape in the atmospheric evolution. Ratios larger or smaller than compared with that in the sun or CI chondrites, which are used as proxy for the sun, indicate that significant hydrodynamic escape has occurred since the formation of the planet. Since lighter atoms preferentially escape, we expect smaller ratios for the noble gas isotopes (or a larger D/H) correspond to a greater likelihood of hydrodynamic escape, as indicated in the table.

Isotopic fractionation in Venus, Earth, and Mars [9]
Source 36Ar/38Ar 20Ne/22Ne 82Kr/84Kr 128Xe/130Xe
Sun 5.8 13.7 20.501 50.873
CI chondrites 5.3±0.05 8.9±1.3 20.149±0.080 50.73±0.38
Venus 5.56±0.62 11.8±0.7 -- --
Earth 5.320±0.002 9.800±0.08 20.217±0.021 47.146±0.047
Mars 4.1±0.2 10.1±0.7 20.54±0.20 47.67±1.03

Matching these ratios can also be used to validate or verify computational models seeking to describe atmospheric evolution. This method has also been used to determine the escape of oxygen relative to hydrogen in early atmospheres.[10]

## Examples

Exoplanets that are extremely close to their parent star, such as hot Jupiters can experience significant hydrodynamic escape[11][12] to the point where the star "burns off" their atmosphere upon which they cease to be gas giants and are left with just the core, at which point they would be called Chthonian planets. Hydrodynamic escape has been observed for exoplanets close to their host star, including the hot Jupiters HD 209458b.[13]

Within a stellar lifetime, the solar flux may change. Younger stars produce more EUV, and the early protoatmospheres of Earth, Mars, and Venus likely underwent hydrodynamic escape, which accounts for the noble gas isotope fractionation present in their atmospheres.[14]

## References

1. ^ Irwin, Patrick G. J. (2006). Giant planets of our solar system: an introduction. Birkhäuser. p. 58. ISBN 3-540-31317-6. Retrieved 22 Dec 2009.
2. ^ Hunten, Donald M.; Pepin, Robert O.; Walker, James C. G. (1987-03-01). "Mass fractionation in hydrodynamic escape". Icarus. 69 (3): 532–549. doi:10.1016/0019-1035(87)90022-4. hdl:2027.42/26796. ISSN 0019-1035.
3. ^ Pater, Imke De; Jack Jonathan Lissauer (2001). Planetary sciences. Cambridge University Press. p. 129. ISBN 0-521-48219-4.
4. ^ a b Watson, Andrew J.; Donahue, Thomas M.; Walker, James C.G. (November 1981). "The dynamics of a rapidly escaping atmosphere: Applications to the evolution of Earth and Venus" (PDF). Icarus. 48 (2): 150–166. doi:10.1016/0019-1035(81)90101-9. hdl:2027.42/24204.
5. ^ Erkaev, N. V.; Kulikov, Yu. N.; Lammer, H.; Selsis, F.; Langmayr, D.; Jaritz, G. F.; Biernat, H. K. (September 2007). "Roche lobe effects on the atmospheric loss from "Hot Jupiters"". Astronomy & Astrophysics. 472 (1): 329–334. doi:10.1051/0004-6361:20066929. ISSN 0004-6361.
6. ^ Lecavelier des Etangs, A. (January 2007). "A diagram to determine the evaporation status of extrasolar planets". Astronomy & Astrophysics. 461 (3): 1185–1193. arXiv:astro-ph/0609744. doi:10.1051/0004-6361:20065014. ISSN 0004-6361.
7. ^ Tian, Feng; Güdel, Manuel; Johnstone, Colin P.; Lammer, Helmut; Luger, Rodrigo; Odert, Petra (April 2018). "Water Loss from Young Planets". Space Science Reviews. 214 (3). doi:10.1007/s11214-018-0490-9. ISSN 0038-6308.
8. ^ Owen, James E. (2019-05-30). "Atmospheric Escape and the Evolution of Close-In Exoplanets". Annual Review of Earth and Planetary Sciences. 47 (1): 67–90. arXiv:1807.07609. doi:10.1146/annurev-earth-053018-060246. ISSN 0084-6597.
9. ^ Pepin, Robert O. (1991-07-01). "On the origin and early evolution of terrestrial planet atmospheres and meteoritic volatiles". Icarus. 92 (1): 2–79. doi:10.1016/0019-1035(91)90036-S. ISSN 0019-1035.
10. ^ Hunten, Donald M.; Pepin, Robert O.; Walker, James C. G. (1987-03-01). "Mass fractionation in hydrodynamic escape". Icarus. 69 (3): 532–549. doi:10.1016/0019-1035(87)90022-4. hdl:2027.42/26796. ISSN 0019-1035.
11. ^ Tian, Feng; Toon, Owen B.; Pavlov, Alexander A.; de Sterck, H. (March 10, 2005). "Transonic Hydrodynamic Escape of Hydrogen from Extrasolar Planetary Atmospheres". The Astrophysical Journal. 621 (2): 1049–1060. Bibcode:2005ApJ...621.1049T. CiteSeerX 10.1.1.122.9085. doi:10.1086/427204.
12. ^ Swift, Damian C.; Eggert, Jon; Hicks, Damien G.; Hamel, Sebastien; Caspersen, Kyle; Schwegler, Eric; Collins, Gilbert W. (2012). "Mass-radius relationships for exoplanets". The Astrophysical Journal. 744 (1): 59. arXiv:1001.4851. Bibcode:2012ApJ...744...59S. doi:10.1088/0004-637X/744/1/59.
13. ^ "Vidal-Madjar et al., Oxygen and Carbon in HD 209458b". doi:10.1086/383347. Cite journal requires |journal= (help)
14. ^ Gillmann, Cédric; Chassefière, Eric; Lognonné, Philippe (2009-09-15). "A consistent picture of early hydrodynamic escape of Venus atmosphere explaining present Ne and Ar isotopic ratios and low oxygen atmospheric content". Earth and Planetary Science Letters. 286 (3): 503–513. doi:10.1016/j.epsl.2009.07.016. ISSN 0012-821X.