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In atmospheric science, 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, typically hydrogen. This mechanism may explain why some planetary atmospheres are depleted in oxygen, nitrogen, and heavier noble gases, such as xenon.^[1]

Particles in the atmosphere need to achieve sufficiently high velocity (higher than the escape velocity) to escape from the planetary gravity field. There are different ways to achieve this velocity. Those processes in which the high velocity is related to the temperature are called thermal escape. The root mean square thermal velocity (v_th) of an atomic species is ${\displaystyle v_{\mathrm {th} }={\sqrt {\frac {3kT}{m}}}}$

where k is the Boltzmann constant, T is the temperature, and 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. Thus they are easier to escape from planetary gravity field. This is why atomic hydrogen escapes preferentially from an atmosphere.

If there is a strong thermally driven atmospheric escape of light atoms, heavier atoms can achieve the escape velocity through viscous drag by those escaping lighter atoms.^[2] This is another way of thermal escape, called hydrodynamic escape. The heaviest species of atom that can be removed in this manner is called the cross-over mass.^[3]

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 (solar EUV heating), momentum transfer from impacting meteoroids or asteroids, or the heat input from planetary accretion processes^[4] may provide the requisite energy for hydrodynamic escape. Such conditions may have been reached in H- or He-rich thermospheres heated by the strong extreme ultraviolet radiation flux of the young Sun.^[5] Thus hydrodynamic escape is more likely to occur in the early atmosphere of planets.

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^[6] 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 a special case and upper limit approximation on the atmospheric escape, it is worth noting here.

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

Hydrodynamic escape flux of hydrogen ${\displaystyle \Phi _{H}}$ can be expressed as:

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

where (in SI units):

• F_XUV is the photon flux [J m^-2s^-1] over the wavelengths of interest,
• R_p is the radius of the planet [m],
• G is the gravitational constant [ms^-2],
• M_p is the mass of the planet [kg],
• R_XUV is the effective radius where the XUV absorption occurs [m].

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.^[7][8][9]

However, as computational power has improved, increasingly sophisticated models have emerged, incorporating radiative transfer, photochemistry, and hydrodynamics that provide better estimates of hydrodynamic escape.^[10]

On the other hand, the hydrodynamic escape flux of heavier species ${\displaystyle \Phi _{i}}$ can be expressed as:^[11]

${\displaystyle \Phi _{i}=(\Phi _{H}-{\frac {(m_{i}-m_{H})gb(i,H)}{kT}})f_{i}}$

where

• ${\displaystyle m_{H},m_{i}}$ are the masses of hydrogen and of heavier atoms i,
• ${\displaystyle g}$ is the acceleration due to the gravitational field,
• ${\displaystyle k}$ is the Boltzmann constant,
• ${\displaystyle T}$ is the temperature,
• ${\displaystyle b(i,H)}$ is the binary diffusion coefficient,
• ${\displaystyle f_{i}}$ is the mixing ratio of heavier atoms i divided by the mixing ratio of hydrogen.

It can be observed from this formula that the hydrodynamic escape flux of heavier species is higher for less heavier atoms, which is discussed in detail in the next section.

Hydrodynamic escape is a mass fractionating process since all isotopes are dragged by protons with the same force but heavy isotopes are more gravitationally bound compared to light ones.^[11] Therefore, hydrogen preferentially drags lighter isotopes to space, leaving the residual atmosphere enriched in heavier isotopes.^[12] This is 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, ^{78,80,82,83,86}Kr/⁸⁴Kr, ^{124,126,128,129,131,132,134,136}Xe/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.

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.^[14]

Exoplanets that are extremely close to their parent star, such as hot Jupiters can experience significant hydrodynamic escape^[15][16] 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.^[17]

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.^[18]

It can be observed from the above table that atmospheric Xe experiences more fractionation than Kr, which seems unreasonable since Xe is heavier than Kr and should be less influenced by hydrodynamic escape than Kr. Actually, according to the formula of hydrodynamic escape flux ${\displaystyle \Phi _{i}}$ above, it requires extreme high ${\displaystyle \Phi _{H}}$, which can only be achieved during the first 100 Ma of Earth’s history when the EUV flux from the young Sun was sufficiently strong.^[19] However, from the analysis of ancient atmospheric gases trapped in fluid inclusions contained in minerals of Archean (3.3 Ga) to Paleozoic (404 Ma) rocks, it has been observed that the fractionation of atmospheric Xe was still ongoing at about 2.1 Ga before.

One possible explanation is that Xe may be the only noble gas which escapes as an ion as it is the only noble gas more easily ionized than hydrogen.^[20] Ionized Xe⁺ can interact with H⁺ protons via the strong Coulomb force, which effectively decreases the binary diffusion coefficient b(Xe⁺, H⁺) by several orders of magnitude compared to the case of neutral Xe.^[11] That means it needs lower hydrogen escape fluxes ${\displaystyle \Phi _{H}}$ compared with neutral Xe. Actually, its requisite ${\displaystyle \Phi _{H}}$ is lower enough to be met during Archean eon,^[21] which means the mass-fractionated hydrodynamic escape of Xe can persist during Archean.