Radiative Equilibrium

Alright. You want me to take something… mundane… and make it… more. Fine. Let’s see what we can scrape out of this. Just don’t expect me to enjoy it.

Condition in Thermodynamics

Radiative equilibrium. It’s the state where the total thermal radiation an object decides to let go of perfectly matches what it’s taking in. Think of it as a cosmic transaction, a silent agreement in the void. It’s one of the several boxes that need to be ticked for true thermodynamic equilibrium , but don’t get it twisted—it can hang around even when the rest of the system is in a state of utter disarray. There are layers to this, you see. Different flavors of radiative equilibrium, each a specific shade of dynamic equilibrium . It’s never just one thing, is it?

Definitions

Equilibrium , in its grand, sweeping sense, is just a state of balance. Opposing forces lock eyes and decide to call a truce, leaving the system frozen in time. Radiative equilibrium is just a more specific flavor of thermal equilibrium , the kind where heat decides to travel not by clumsy physical contact, but by elegant, silent radiation.

And like I said, it’s not a monolith. There are nuances.

Prevost’s Definitions

Pierre Prevost was dabbling in this back in 1791. He had this idea, this… fluid he called “free heat.” We’d call it photon gas or electromagnetic radiation now, but he saw it as something more ephemeral. Rays, he proposed, that could pass through each other like ghosts, leaving no trace of their passage. His theory of exchanges was simple, yet profound: every object is a broadcaster, sending out its heat, and a receiver, taking it in. The radiation emitted by any given object doesn’t care if there’s anything else around to catch it. It just is.

Prevost, bless his antiquated heart, laid out his definitions like this, in his own words, if you can stomach a translation:

Absolute equilibrium of free heat is the state of this fluid in a portion of space which receives as much of it as it lets escape.

Relative equilibrium of free heat is the state of this fluid in two portions of space which receive from each other equal quantities of heat, and which moreover are in absolute equilibrium, or experience precisely equal changes.

He even added a little footnote, a side comment, that “The heat of several portions of space at the same temperature, and next to one another, is at the same time in the two species of equilibrium.” As if we needed him to confirm that things at the same temperature, sitting next to each other, are, in fact, in equilibrium. Groundbreaking.

Pointwise Radiative Equilibrium

Then came Max Planck in 1914, and things got… pointier. Now, a radiative field isn’t just a general concept; it’s described by this thing called specific radiative intensity . It’s a function, you see, tied to every single geometric point in a region, at any given instant. It’s a bit more… granular than Prevost’s broad strokes. And it’s a subtle shift in perspective, too. Prevost was thinking about “bound” and “free” heat, whereas we’re now talking about the kinetic energy of molecules, the heat in matter, and the thermal photon gas .

Goody and Yung, in 1989, apparently felt the need to flesh this out even further. They focused on the messy business of how thermal radiation and heat in matter play footsie. From that specific radiative intensity, they derived this beast:

$$ \mathbf{F}_{\nu} $$

This is the monochromatic vector flux density. At each point. It’s basically the time-averaged Poynting vector at that precise location. Then they defined this thing, $h_{\nu}$, which is the monochromatic volume-specific rate of gain of heat by matter from radiation. It’s the negative divergence of that flux density vector. A scalar function, of course, because why make it simple?

$$ h_{\nu} = -\nabla \cdot \mathbf{F}_{\nu} $$

And this is where they declared (pointwise) monochromatic radiative equilibrium. It’s when the divergence of the monochromatic flux density is zero:

$$ \nabla \cdot \mathbf{F}_{\nu} = 0 $$

This has to hold true at every single point in the region.

Then they went and defined (pointwise) radiative equilibrium itself:

$$ h = \int_{0}^{\infty} h_{\nu} d\nu = 0 $$

This means that, at every single point where this equilibrium reigns, the total interconversion of energy between thermal radiation and the heat content of matter is… nil. Zero. Nothing. It’s closely related to Prevost’s “absolute radiative equilibrium,” apparently. As if we needed another layer of complexity.

Mihalas and Weibel-Mihalas, in 1984, were quick to point out that this definition is all well and good, but it’s for a static medium. Matter that isn’t moving. Because, of course, it would be too simple otherwise. They also considered the thrilling prospect of moving media.

Approximate Pointwise Radiative Equilibrium

Karl Schwarzschild , back in 1906, was looking at systems where convection and radiation were both duking it out. But radiation, it turned out, was the heavyweight. So much more efficient that convection could practically be ignored. This is the kind of scenario you find when temperatures are stratospheric, like in a star , but definitely not in a planet’s atmosphere.

Subrahmanyan Chandrasekhar , in 1950, described a model of a stellar atmosphere where radiation was the only game in town for heat transport. No other mechanisms, no heat sources elsewhere. It’s barely a step up from Schwarzschild’s idea, just… more precisely phrased.

Radiative Exchange Equilibrium

Planck, on page 40 of his 1914 treatise, mentioned a state of thermodynamic equilibrium where any two randomly selected bodies, or bits of bodies, would exchange “equal amounts of heat with each other” through radiation.

The term “radiative exchange equilibrium” can also be used more broadly, to describe two specific regions of space that trade equal amounts of radiation, even if the whole system isn’t in perfect thermodynamic equilibrium. It’s when some sub-processes might involve the net movement of matter or energy, including radiation. This is very close to Prevost’s “relative radiative equilibrium.”

Approximate Radiative Exchange Equilibrium

Here’s where it gets really approximate. Think about the exchange of thermal radiation between land and sea and the lowest layer of the atmosphere on a clear night. As a first guess, there’s zero net exchange in the “non-window” wavelengths. But in the “window” wavelengths, it’s just direct radiation from the surface out into space. This also happens between adjacent layers in the turbulent boundary layer of the lower troposphere . It’s called the “cooling to space approximation,” first noticed by Rodgers and Walshaw in 1966. It’s the kind of approximation you make when the exact answer is too much trouble, and a rough estimate will do.

In Astronomy and Planetary Science

Global Radiative Equilibrium

This one’s for the big picture. Imagine an entire celestial system, like a planet, that doesn’t generate its own power. It just sits there, passively receiving and radiating.

Liou, in 2002, and others use “global radiative equilibrium” to talk about the radiative exchange between Earth and the vast emptiness of space. They mean that, theoretically, the absorbed solar radiation by Earth and its atmosphere would perfectly balance the outgoing longwave radiation from the same surfaces. Prevost, if he were here, would probably say Earth and its atmosphere, as a collective, were in “absolute radiative equilibrium.” Some texts, like Satoh’s 2004 work, just simplify it to “radiative equilibrium” when they mean this global exchange.

Planetary Equilibrium Temperature

• Main article: Planetary equilibrium temperature

You can calculate all sorts of theoretical temperatures for any given planet. This includes the planetary equilibrium temperature, the equivalent blackbody temperature, or the effective radiation emission temperature. For planets with atmospheres, these can be wildly different from the surface temperature you might actually measure – the global-mean surface air temperature, or the global-mean surface skin temperature.

A radiative equilibrium temperature is calculated under the assumption that any energy generated within the planet – from chemical or nuclear sources – is negligible. This is a reasonable assumption for Earth. But it falls apart when you look at something like Jupiter . Its internal energy sources are so significant that the actual temperature is much higher than its theoretical radiative equilibrium temperature.

Stellar Equilibrium

Stars, on the other hand, are powerhouses. They generate their own energy through nuclear fusion, so you can’t just define their temperature equilibrium based on incoming energy alone.

Cox and Giuli, in their 1968/1984 work, define “radiative equilibrium” for a [star] as a whole. It’s when the energy generated by nuclear reactions and viscosity , transferred as heat to the microscopic motions of particles within the star, is precisely balanced by the energy radiated away into space. This is a slightly different definition than what we’ve seen before. They point out that a star radiating energy can’t be in a steady state unless there’s a constant supply of energy – in this case, from those internal nuclear reactions – to keep the radiation going. And the condition for pointwise radiative equilibrium (where divergence is zero) can’t hold throughout a radiating star. Their definition allows them to say a star is in a steady state of temperature distribution and in “radiative equilibrium,” assuming all the radiated energy originates from within.

Mechanisms

When there are enough particles in a region, and molecular collisions happen far more frequently than the absorption or emission of photons, we’re talking about local thermodynamic equilibrium (LTE) . In this scenario, Kirchhoff’s law of equality of radiative absorptivity and emissivity holds true.

Two bodies in radiative exchange equilibrium, each maintaining its own local thermodynamic equilibrium, will share the same temperature. Their radiative exchange will then comply with the Stokes-Helmholtz reciprocity principle .