9. Atmospheres of the Terrestrial Planets#
9.1. Atmosphere Retention#
Planets hold onto their atmosphere through their own gravity. We can estimate a planet’s ability to hold onto its atmosphere by comparing the average speed of particles in a gas to the escape velocity from the planet (see 4.2.2). The escape velocity is given as
where \(M\) is the mass of the planet and \(R\) is the distance from the center (often just the planetary radius). The units of \(M\) and \(R\) depend on the choice of \(G\) (the Universal Gravitational constant). However, we can rewrite the above equation in terms of parameters for the Earth as
where we just need to know the escape velocity of the Earth (\(v_{\rm esc}^{\rm Earth} \sim 11\ {\rm km/s}\)) if we need more standard units.
The temperature \(T\) of a gas is proportional to the kinetic energy of the particles \((KE=1/2\ mv^2)\). The average kinetic energy \(\overline{KE}\) for a gas is
where \(k\) represents the Boltzmann constant. Using the average squared velocity \(\overline{v^2}\) for the kinetic energy \(KE\), we can equate these two energies through
We can then arrive at an expression for the velocity of a gas molecule as
The temperature \(T\) is measured in Kelvin, where the mass \(m\) of the molecule is measured in kilograms (kg). There are no tricks here because of the Bolztmann constant \(k = 1.38 \times 10^{-23}\ {\rm J/K}\).
The atomic mass of the molecule measures the mass relative to carbon-12. The molecular mass can be calculated by summing the individual atomic masses. For example, the molecular mass of methane \(CH_4\) can be determined by adding 1 carbon to 4 hydrogens. See the table below.
Standard Atomic Weight |
Number |
Molecular Mass |
|
|---|---|---|---|
\(C\) |
12.011 |
1 |
12.011 |
\(H\) |
1.008 |
4 |
4.032 |
\(CH_4\) |
16.043 |
The mass of a hydrogen atom is \(m_H = 1.67 \times 10^{-27}\ {\rm kg}\), where we can simply multiply the molecular mass by \(m_H\) to arrive at the mass of the molecule. However, we don’t need to do this for every calculation, instead we can write
This relation shows that the speed of a molecule changes depending on the temperature of the gas or the molecular mass. The molecular mass is on the bottom, which means it is inversely proportional (or lower mass \(\rightarrow\) faster speeds).
Let’s calculate the molecular speed \(v_{\rm molecule}\) for molecular hydrogen gas \(H_2\) and methane \(CH_4\) at room temperature (\(288\ \rm K\)):
For \(H_2\): \( v_{\rm molecule} = (0.157\ {\rm km/s}) \times \sqrt{\frac{288}{2.016}} = 1.88\ {\rm km/s}\)
For \(CH_4\): \( v_{\rm molecule} = (0.157\ {\rm km/s}) \times \sqrt{\frac{288}{16.043}} = 0.665\ {\rm km/s}\)
We can see that at room temperature the molecules in the molecular hydrogen gas \(H_2\) are moving faster (on average) than those in methane \(CH_4\).
Earth’s exosphere can vary in temperature, but the range is from \(700-1400\ {\rm K}\) depending on the solar activity. This means that we can expect the average speed of molecular hydrogen gas can be at least
and up to
The Earth’s escape velocity at the surface is \(v_{\rm esc} \sim 11\ {\rm km/s}\) and a general rule is that the average speed needs to be less than \(1/6\ v_{\rm esc}\), or
to remain bound to the Earth over the lifetime of the Solar System. This limit on gas speed is \(1.83\ {\rm km/s}\) for the Earth. At either temperature, the average speed of molecular hydrogen gas is larger than the limiting speed. Therefore, molecular hydrogen gas can escape the Earth after it reaches the upper atmosphere.
If methane reaches the upper atmosphere and can be heated to \(700\ {\rm K}\), then its average speed can be calculated as
which is less than the limiting speed. Methane would not be able to escape the Earth as a molecule. A process called photochemistry can break the molecular bond on the methane which frees some of the hydrogen into molecular hydrogen gas.