Diffusion-limited escape occurs when the rate of
atmospheric escape
Atmospheric escape is the loss of planetary atmospheric gases to outer space. A number of different mechanisms can be responsible for atmospheric escape; these processes can be divided into thermal escape, non-thermal (or suprathermal) escape, and ...
to space is limited by the upward diffusion of escaping gases through the upper atmosphere, and not by escape mechanisms at the top of the atmosphere (the
exobase
The thermopause is the atmospheric boundary of Earth's energy system, located at the top of the thermosphere. The temperature of the thermopause could range from nearly absolute zero to .
Below this, the atmosphere is defined to be active on the i ...
). The escape of any atmospheric gas can be diffusion-limited, but only diffusion-limited escape of hydrogen has been observed in our solar system, on
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
,
Mars
Mars is the fourth planet from the Sun and the second-smallest planet in the Solar System, only being larger than Mercury (planet), Mercury. In the English language, Mars is named for the Mars (mythology), Roman god of war. Mars is a terr ...
,
Venus
Venus is the second planet from the Sun. It is sometimes called Earth's "sister" or "twin" planet as it is almost as large and has a similar composition. As an interior planet to Earth, Venus (like Mercury) appears in Earth's sky never fa ...
and
Titan.
Diffusion-limited hydrogen escape was likely important for the rise of oxygen in Earth's atmosphere (
the Great Oxidation Event) and can be used to estimate the oxygen and hydrogen content of Earth's prebiotic atmosphere.
Diffusion-limited escape theory was first used by Donald Hunten in 1973 to describe hydrogen escape on one of Saturn's moons, Titan. The following year, in 1974, Hunten found that the diffusion-limited escape theory agreed with observations of hydrogen escape on Earth.
Diffusion-limited escape theory is now used widely to model the composition of exoplanet atmospheres and Earth's ancient atmosphere.
Diffusion-Limited Escape of Hydrogen on Earth
![Diffusion_limited_escape3](https://upload.wikimedia.org/wikipedia/commons/f/f5/Diffusion_limited_escape3.svg)
Hydrogen escape on Earth occurs at ~500 km altitude at the
exobase
The thermopause is the atmospheric boundary of Earth's energy system, located at the top of the thermosphere. The temperature of the thermopause could range from nearly absolute zero to .
Below this, the atmosphere is defined to be active on the i ...
(the lower border of the
exosphere
The exosphere ( grc, ἔξω "outside, external, beyond", grc, σφαῖρα "sphere") is a thin, atmosphere-like volume surrounding a planet or natural satellite where molecules are gravitationally bound to that body, but where the densit ...
) where gases are collisionless. Hydrogen atoms at the exobase exceeding the
escape velocity
In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non- propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically ...
escape to space without colliding into another gas particle.
For a hydrogen atom to escape from the exobase, it must first travel upward through the atmosphere from the
troposphere
The troposphere is the first and lowest layer of the atmosphere of the Earth, and contains 75% of the total mass of the planetary atmosphere, 99% of the total mass of water vapour and aerosols, and is where most weather phenomena occur. From ...
. Near ground level, hydrogen in the form of H
2O, H
2, and CH
4 travels upward in the
homosphere
The homosphere is the layer of an atmosphere where the bulk gases are homogeneously mixed due to turbulent mixing or eddy diffusion. The bulk composition of the air is mostly uniform so the concentrations of molecules are the same throughout the ho ...
through turbulent mixing, which dominates up to the
homopause. At about 17 km altitude, the cold
tropopause
The tropopause is the atmospheric boundary that demarcates the troposphere from the stratosphere; which are two of the five layers of the atmosphere of Earth. The tropopause is a thermodynamic gradient-stratification layer, that marks the end of ...
(known as the "cold trap") freezes out most of the H
2O vapor that travels through it, preventing the upward mixing of some hydrogen. In the upper homosphere, hydrogen bearing molecules are split by
ultraviolet
Ultraviolet (UV) is a form of electromagnetic radiation with wavelength from 10 nanometer, nm (with a corresponding frequency around 30 Hertz, PHz) to 400 nm (750 Hertz, THz), shorter than that of visible light, but longer than ...
photons leaving only H and H
2 behind. The H and H
2 diffuse upward through the
heterosphere The heterosphere is the layer of an atmosphere where the gases are separated out by molecular diffusion with increasing altitude such that lighter species become more abundant relative to heavier species. The heavier molecules and atoms tend to be p ...
to the exobase where they escape the atmosphere by
Jeans thermal escape and/or a number of
suprathermal mechanisms. On Earth, the rate-limiting step or "bottleneck" for hydrogen escape is diffusion through the heterosphere. Therefore, hydrogen escape on Earth is diffusion-limited.
By considering one dimensional molecular diffusion of H
2 through a heavier background atmosphere, you can derive a formula for the upward diffusion-limited flux of hydrogen (
):
is a constant for a particular background atmosphere and planet, and
is the total hydrogen mixing ratio in all its forms above the tropopause. You can calculate
by summing all hydrogen bearing species weighted by the number of hydrogen atoms each species contains:
For Earth's atmosphere,
cm
−2 s
−1,
and, the concentration of hydrogen bearing gases above the tropopause is 1.8 ppmv (
parts per million by volume) CH
4, 3 ppmv H
2O, and 0.55 ppmv H
2.
Plugging these numbers into the formulas above gives a predicted diffusion-limited hydrogen escape rate of
H atoms cm
−2 s
−1. This calculated hydrogen flux agrees with measurements of hydrogen escape.
Note that hydrogen is the only gas in Earth's atmosphere that escapes at the diffusion-limit. Helium escape is not diffusion-limited and instead escapes by a suprathermal process known as the
polar wind
The polar wind or plasma fountain is a permanent outflow of plasma from the polar regions of Earth's magnetosphere, caused by the interaction between the solar wind and the Earth's atmosphere. The solar wind ionizes gas molecules in the upper at ...
.
Derivation
Transport of gas molecules in the atmosphere occurs by two mechanisms: molecular and eddy diffusion. Molecular diffusion is the transport of molecules from an area of higher concentration to lower concentration due to thermal motion. Eddy diffusion is the transport of molecules by the turbulent mixing of a gas. The sum of molecular and eddy diffusion fluxes give the total flux of a gas
through the atmosphere:
The vertical eddy diffusion flux is given by
is the eddy diffusion coefficient,
is the number density of the atmosphere (molecules cm
−3), and
is the volume mixing ratio of gas
. The above formula for eddy diffusion is a simplification for how gases actually mix in the atmosphere. The eddy diffusion coefficient can only be empirically derived from atmospheric tracer studies.
The molecular diffusion flux, on the other hand, can be derived from theory. The general formula for the diffusion of gas 1 relative to gas 2 is given by
Each variable is defined in table on right. The terms on the right hand side of the formula account for diffusion due to molecular concentration, pressure, temperature, and force gradients respectively. The expression above ultimately comes from the
Boltzmann transport equation
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of Thermodynamic equilibrium, equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics ( ...
. We can simplify the above equation considerably with several assumptions. We will consider only vertical diffusion, and a neutral gas such that the accelerations are both equal to gravity (
) so the last term cancels. We are left with
We are interested in the diffusion of a lighter molecule (e.g. hydrogen) through a stationary heavier background gas (air). Therefore, we can take velocity of the heavy background gas to be zero:
. We can also use the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
and the
hydrostatic equation
Fluid statics or hydrostatics is the branch of fluid mechanics that studies the condition of the equilibrium of a floating body and submerged body " fluids at hydrostatic equilibrium and the pressure in a fluid, or exerted by a fluid, on an imm ...
to rewrite the derivative in the second term.
The chain rule can also be used to simplify the derivative in the third term.
Making these substitutions gives
Note, that we have also made the substitution
. The flux of molecular diffusion is given by
By adding the molecular diffusion flux and the eddy diffusion flux, we get the total flux of molecule 1 through the background gas
Temperature gradients are fairly small in the heterosphere, so
, which leaves us with
The maximum flux of gas 1 occurs when
. Qualitatively, this is because
must decrease with altitude in order to contribute to the upward flux of gas 1. If
decreases with altitude, then
must decrease rapidly with altitude (recall that
). Rapidly decreasing
would require rapidly increasing
in order to drive a constant upward flux of gas 1 (recall
). Rapidly increasing
isn't physically possible. For a mathematical explanation for why
, see Walker 1977, p. 160.
The maximum flux of gas 1 relative to gas 2 (
, which occurs when
) is therefore
Since
,
or
This is the diffusion-limited flux of a molecule. For any particular atmosphere,
is a constant. For hydrogen (gas 1) diffusion through air (gas 2) in the heterosphere on Earth
,
m s
−2 ,and
K. Both H and H
2 diffuse through the heterosphere, so we will use a diffusion parameter that is the weighted sum of H and H
2 number densities at the tropopause.
For
molecules cm
−3,
molecules cm
−3,
cm
−1s
−1, and
cm
−1s
−1, the binary diffusion parameter is
. These numbers give
molecules cm
−2 s
−1. In more detailed calculations the constant is
molecules cm
−2 s
−1.
The above formula can be used to calculate the diffusion-limited flux of gases other than hydrogen.
Diffusion-Limited Escape in the Solar System
Every rocky body in the solar system with a substantial atmosphere, including Earth, Mars, Venus, and Titan, loses hydrogen at the diffusion-limited rate.
For Mars, the constant governing diffusion-limited escape of hydrogen is
molecules cm
−2 s
−1.
Spectroscopic measurements of Mars' atmosphere suggest that
. Multiplying these numbers together gives the diffusion-limited rate escape of hydrogen:
H atoms cm
−2 s
−1
''Mariner'' 6 and 7 spacecraft indirectly observed hydrogen escape flux on Mars between
and
H atoms cm
−2 s
−1. These observations suggest that Mars' atmosphere is losing hydrogen at roughly the diffusion limited value.
Observations of hydrogen escape on Venus and Titan are also at the diffusion-limit. On Venus, hydrogen escape was measured to be about
H atoms cm
−2 s
−1, while the calculated diffusion limited rate is about
H atoms cm
−2 s
−1, which are in reasonable agreement.
On Titan, hydrogen escape was measured by the
''Cassini'' spacecraft to be
H atoms cm
−2 s
−1, and the calculated diffusion-limited rate is
H atoms cm
−2 s
−1.
Applications to Earth's Ancient Atmosphere
Oxygen Content of the Prebiotic Atmosphere
We can use diffusion-limited hydrogen escape to estimate the amount of O
2 on the Earth's atmosphere before the rise of life (the prebiotic atmosphere). The O
2 content of the prebiotic atmosphere was controlled by its sources and sinks. If the potential sinks of O
2 greatly outweighed the sources, then the atmosphere would have been nearly devoid of O
2.
In the prebiotic atmosphere, O
2 was produced by the
photolysis
Photodissociation, photolysis, photodecomposition, or photofragmentation is a chemical reaction in which molecules of a chemical compound are broken down by photons. It is defined as the interaction of one or more photons with one target molecule. ...
of CO
2 and H
2O in the atmosphere:
CO_2 + h\nu -> CO + O
H_2O + h\nu -> 1/2O_2 + 2H
These reactions aren't necessarily a net source of O
2. If the CO and O produced from CO
2 photolysis remain in the atmosphere, then they will eventually recombine to make CO
2. Likewise, if the H and O
2 from H
2O photolysis remain in the atmosphere, then they will eventually react to form H
2O. The photolysis of H
2O is a net source of O
2 only if the hydrogen escapes to space.
If we assume that hydrogen escape occurred at the diffusion-limit in the prebiotic atmosphere, then we can estimate the amount of H
2 that escaped due to water photolysis. If the prebiotic atmosphere had a modern stratospheric H
2O mixing ratio of 3 ppmv which is equivalent to 6 ppmv of H after photolysis, then
H atoms cm
−2 s
−1
Stoichiometry
Stoichiometry refers to the relationship between the quantities of reactants and products before, during, and following chemical reactions.
Stoichiometry is founded on the law of conservation of mass where the total mass of the reactants equal ...
says that every mol of H escape produced 0.25 mol of O
2 (i.e.
2H_2O -> O_2 +4H), so the abiotic net production of O
2 from H
2O photolysis was
O
2 molecules cm
−2 s
−1. The main sinks of O
2 would have been reactions with volcanic hydrogen. The modern volcanic H flux is about
H atoms cm
−2 s
−1.
If the prebiotic atmosphere had a similar volcanic hydrogen flux, then the potential O
2 sink would have been a fourth of the hydrogen volcanism, or
O
2 molecules cm
−2 s
−1. These calculated values predict that potential O
2 sinks were ~50 times greater than the abiotic source. Therefore, O
2 must have been nearly absent in the prebiotic atmosphere. Photochemical models, which do more complicated versions of the calculations above, predict prebiotic O
2 mixing ratios below 10
−11, which is extremely low compared to the modern O
2 mixing ratio of 0.21.
Hydrogen Content of the Prebiotic Atmosphere
H
2 concentrations in the prebiotic atmosphere were also controlled by its sources and sinks. In the prebiotic atmosphere, the main source of H
2 was volcanic outgassing, and the main sink of outgassing H
2 would have been escape to space. Some outgassed H
2 would have reacted with atmospheric O
2 to form water, but this was very likely a negligible sink of H
2 because of scarce O
2 (see the previous section). This is not the case in the modern atmosphere where the main sink of volcanic H
2 is its reaction with plentiful atmospheric O
2 to form H
2O.
If we assume that the prebiotic H
2 concentration was at a steady-state, then the volcanic H
2 flux was approximately equal to the escape flux of H
2.
Additionally, if we assume that H
2 was escaping at the diffusion-limited rate as it is on the modern Earth then
If the volcanic H
2 flux was the modern value of
H atoms cm
−2 s
−1, then we can estimate the total hydrogen content of the prebiotic atmosphere.
ppmv
By comparison, H
2 concentration in the modern atmosphere is 0.55 ppmv, so prebiotic H
2 was likely several hundred times higher than today's value.
This estimate should be considered as a lower bound on the actual prebiotic H
2 concentration. There are several important factors that we neglected in this calculation. The Earth likely had higher rates of hydrogen outgassing because the interior of the Earth was much warmer ~4 billion years ago. Additionally, there is geologic evidence that the mantle was more reducing in the distant past, meaning that even more reduced gases (e.g. H
2) would have been outgassed by volcanos relative to oxidized volcanic gases.
Other reduced volcanic gases, like CH
4 and H
2S should also contribute to this calculation.
References
{{Reflist
Atmosphere
Hydrogen
Meteorological hypotheses
Origin of life
Oxygen
Proterozoic