I had another brief Twitter discussion with Ned Nikolov, whose paper I discussed in this post. Ned seems to think that there is no atmospheric greenhouse effect and that the enhanced surface temperature is due to atmospheric pressure somehow enhancing the energy provided by the Sun. Well, this is wrong, and I thought I would try to illustrate why by explaining something that I find interesting.
I’m currently teaching our core Astrophysics course. A big part of what I’m doing in this course is deriving the equations of stellar structure, which includes the equation of hydrostatic equilibrium. A star, like the Sun, will settle into a state of hydrostatic equilibrium, in which the inward gravitational force is balanced by an outward pressure force. Any self-gravitating system (by which I mean something for which its own gravity is important) in a state of hydrostatic equilibrium satisfies something called the Virial Theorem. This is essentially that the gravitational potential energy of the system is about the same as its thermal, or kinetic, energy (in fact, it is that the magnitude of the gravitational potential energy is twice the thermal/kinetic energy).
We know the mass, , and radius, , of the Sun and – hence – can estimate its gravitational potential energy; it will be of order . From the Virial Theorem we also then know the total thermal energy of the Sun. We also know the Sun’s luminosity (how much energy it is losing per second). This means that we can estimate how long it would live if it was simply radiating its thermal energy into space. The answer is that it would live for only a few tens of millions of years.
The idea that the Sun might simply be radiating thermal energy into space was first suggested by Kelvin and Helmholtz in the 19th century. However, at that time it was also known that the Earth (and, hence, the Sun) was probably billions of years old, rather than only a few tens of millions of years old. This meant that the Sun’s energy source could not simply be gravitational potential energy being converted into thermal energy as it slowly contracted, because that would imply a much, much younger Sun than geological, and fossil, evidence suggested.
This paradox was resolved with the discovery of nuclear reactions, specifically nuclear fusion. In the core of the Sun, protons combine to form Helium, and this process releases energy (Helium has a lower mass than the total mass of 4 protons, and this mass deficit is released as energy – ). It is this that allows the Sun to remain in a roughly steady state for billions of years, rather than for only a few tens of millions of years.
The above isn’t strictly relevant to the Earth’s atmosphere, because the gravity of the atmosphere itself isn’t all that important; the Earth’s atmosphere is in hydrostatic equilibrium because the outward pressure force is balancing the gravitational force from the central, rocky planet. It is, however, relevant for big gas giant planets, like Jupiter and Saturn. However, we can still consider much of the same basic physics.
If the Earth’s atmospheric pressure is to contribute to the enhanced surface temperature, then that would mean that the atmosphere would need to continually provide energy to the surface. It could only do this through the conversion of gravitational potential energy to thermal energy. This would then require the continual contraction of the Earth’s atmosphere. However, we can work out how much energy is available in the Earth’s atmosphere and there is far, far too little to explain the enhanced surface temperature.
As many already know, the enhanced surface temperature is due to radiatively active gases in the atmosphere that act to reduce the outgoing energy flux, causing the surface to warm until the amount of energy we’re losing into space matches the amount we’re receiving from the Sun. It is not simply a consequence of atmospheric pressure. Those who argue that it is due to atmospheric compression are essentially failing to understand something that was well understood by physicists in the 19th century.
As pointed out in this comment I’ve probably somewhat over-stated the discrepancy, in the 19th century, between geological and fossil evidence for the Earth’s age, and how long the Sun could live if it were simply radiating away thermal energy (10s of millions of years). At the time of Kelvin the estimated age of the Earth was probably more like 100s of millions of years, rather than billions. Today, however, we would estimate the Earth to be about 4.56 billion years old.