By Joshua Winn, Princeton University
Why are the atmospheres of Mercury, Venus, Earth, and Jupiter so different? Why doesn’t Mercury have any atmosphere? Why is Venus’s atmosphere 100 times more massive than the Earth’s? The answer has three essential physical ingredients: the equilibrium temperature, the Maxwell-Boltzmann distribution, and the escape velocity.
The Maxwell-Boltzmann Distribution
The Maxwell-Boltzmann distribution tells us the distribution of speeds of the gas molecules, and how it depends on temperature. The peak is at a speed of root 2kT over m—that’s the most probable speed. Let’s call it v-thermal, the speed due to thermal fluctuations. The thermal speed is the most probable one, but there is a distribution of velocities, a spread of an order of magnitude, or so. What this means is that for hot gases, and lightweight molecules—high T and low m—there’s a danger that some molecules will be moving faster than the planet’s escape velocity.
Deep in the atmosphere, that doesn’t matter, because after a few nanoseconds the molecule will bash into another one, randomizing its speed again. But at the top of the atmosphere, where the density is low, the fastest moving particles might escape into space and never return.
Let’s consider hydrogen gas, H2, with a mass of 2 proton masses. At the Earth’s average surface temperature of 288 Kelvin, the thermal velocity is 1.5 kilometers per second. Which means a small fraction of hydrogen molecules have speeds as high as 15 kilometers per second. Meanwhile the escape velocity, as we derived a while ago, is the square root of 2GM over R, which for Earth is 11 kilometers per second. So, if there were hydrogen gas in Earth’s atmosphere, the random jostling of all those molecules would cause some of them to leave and by the time a billion years goes by, almost all the hydrogen would be gone.
Evaluating Minimum Mass
To be sure a planet can hold on to a molecule, let’s require the escape velocity to be at least 10 times higher than the thermal velocity. That leads to an inequality that we can solve for m, the mass of the molecule. Running the numbers for the case of the Earth at room temperature, 300 Kelvin, we get a minimum molecular mass of 4. That’s the mass of helium: 2 protons and 2 neutrons. So, this calculation suggests that the Earth can’t retain hydrogen, and helium is a marginal case. But, nitrogen and oxygen are plenty heavy. N2 has a molecular mass of 28, and for O2, it’s 32.
Let’s evaluate that minimum mass for the other planets, too. For Venus, it’s 11. It’s higher, because Venus is hotter. So, Venus can’t retain hydrogen and helium, either, but holding on to CO2, with a molecular mass of 44, is no problem. Mercury is hot and it has a low escape velocity, so it can’t hold anything lighter than 66. That rules out all the common molecules. And for Jupiter, we get a number less than one, meaning Jupiter is massive and cold enough to retain even the lightest gases.
This article comes directly from content in the video series Introduction to Astrophysics. Watch it now, on Wondrium.
Helium
Thus, by comparing thermal velocity to escape velocity, we can understand some of the patterns in planetary atmospheres. However, real atmospheres are complicated. For example, on the Earth, it turns out that at high altitude, where the density is low enough for molecules to escape, the atmosphere isn’t at 300° Kelvin, it’s more like 1000. That’s because the outer atmosphere gets heated not only by the radiation from the Sun, but also the solar wind, a stream of charged particles from the Sun. Because the temperature up there is about 3 times higher, we would need to triple our estimate for the minimum mass, which makes it clear the Earth can’t retain any helium, after all.
But, wait a minute. If the Earth can’t retain helium, where does all the helium come from that we use to fill party balloons? The answer is interesting. Our helium supply comes not from the Earth’s atmosphere, but from the Earth’s crust. It’s from the radioactive decay of uranium and thorium. When those nuclei decay, they release so-called “alpha particles”, which is just another name for helium nuclei. The helium then percolates up through the crust to the surface, gets lofted into the air and eventually escapes into space. But along the way, some of that helium gets trapped in underground cavities, the same kinds of places where we find oil and natural gas.
Venus and Earth
Another interesting question is why are Venus and Earth so different? After all, they’re the same size, basically the same mass. So, why is Venus’s atmosphere 100 times more massive? The current thinking is that Venus and Earth originally did have similar atmospheres but Venus underwent a runaway greenhouse effect. Since Venus is closer to the Sun, water vaporizes more easily. So, if Venus originally did have liquid water oceans, a lot of that water would’ve ended up in the atmosphere. And water, like carbon dioxide, is a potent greenhouse gas, it’s transparent to visible photons and opaque to infrared. That led to a stronger greenhouse effect on Venus than on the Earth. And that, in turn, led to a hotter surface, and even more water being vaporized, which increased the greenhouse effect, and so on. It was a positive feedback loop, which ultimately led to the complete evaporation of all the surface water.
And what’s worse, volcanic activity kept pumping more carbon dioxide into the atmosphere. That happens on the Earth, too, but here, we have a carbon cycle; the CO2 in the air can dissolve in the oceans, and react chemically with hot, wet rocks. But on Venus, the CO2 just piled up in the atmosphere, without any oceans or wet rocks to soak it up. And eventually, even the water vapor disappeared, because in the upper atmosphere, ultraviolet photons from the Sun can break water apart into hydrogen and oxygen, and the hydrogen can escape. Venus was left as a blistering hot, dry planet, smothered in carbon dioxide.
Common Questions about Comparing Planetary Atmospheres
The Maxwell-Boltzmann distribution tells us the distribution of speeds of the gas molecules, and how it depends on temperature.
By comparing thermal velocity to escape velocity, we can understand some of the patterns in planetary atmospheres.
The current thinking is that Venus and Earth originally did have similar atmospheres but Venus underwent a runaway greenhouse effect.
