I went into this expecting the Sun to have been more or less constant for the whole history of life, wobbling a bit with sunspot cycles. That is wrong, and it is wrong by a lot.

It is also a question that answers itself slightly sideways. The Sun is getting brighter, dramatically so. It is barely getting hotter. Those are different things, and the gap between them is the interesting part.

Brighter by 48 percent, hotter by 2.8 percent

The standard solar model, the one Bahcall, Pinsonneault and Basu published in 2001, tracks the Sun from the moment it started fusing hydrogen. Two numbers from that paper do most of the work.

The luminosity “has risen monotonically from a zero-age value of 0.677 L☉,” and the radius “increases from 0.87R☉ at the zero age main sequence to 1.0R☉ at the present epoch.” In their words, the solar luminosity “has increased by 48% from the zero main sequence to the present epoch.”

Now put those together. A star’s luminosity is its surface area times how hard each patch of that surface radiates, which is Stefan-Boltzmann: luminosity goes as radius squared times temperature to the fourth. Rearrange, plug in a radius that grew by 15 percent and a luminosity that grew by 48 percent, and you can solve for how much the surface temperature changed.

It went from about 5,613 K to today’s 5,772 K. That is 159 degrees over four and a half billion years, an increase of 2.8 percent.

So the honest answer to “is the Sun getting hotter” is: hardly. It is getting brighter, and it is doing that mostly by getting bigger.

The engine is buried in the core. Hydrogen fuses into helium, four particles become one, and the number of particles bouncing around holding the core up falls. The core contracts slightly, which heats it, which speeds up fusion. So the core really does get hotter, and the extra energy leaks out through a star that has puffed up to accommodate it. The surface just goes along for the ride.

The part where Earth should have been an ice ball

Here is where the brightening stops being trivia.

Work out how warm Earth would be with no atmosphere at all. Balance the sunlight absorbed against the heat radiated away, using today’s solar output and today’s reflectivity, and you get 254.6 K, which is minus 18.6 Celsius. The actual average surface is 288 K, or 15 Celsius. That 33 degree gap is the greenhouse effect, and it is why the planet is habitable rather than a snowball.

Now run the same sum with the young Sun at 0.677 of its current output. Earth’s no-atmosphere temperature drops to 230.9 K, which is minus 42 Celsius. Even if you hand the early Earth exactly the greenhouse effect it has today, all 33 degrees of it, you land at minus 9 Celsius.

Water freezes at zero. The early Earth should have been frozen solid, for a very long time.

It was not. There are sedimentary rocks that formed underwater, and zircons suggesting liquid water within a few hundred million years of the planet’s formation, and by three and a half billion years ago there was life doing complicated things in oceans that were demonstrably not ice.

Carl Sagan and George Mullen wrote this up in Science in 1972, and it has been called the faint young Sun paradox ever since.

Nobody has fixed it

The obvious answer is more greenhouse gas. My own arithmetic says you would need about 57 degrees of greenhouse warming under the young Sun to keep the surface at a pleasant 15 Celsius, against the 33 degrees we get today. So: pile on the carbon dioxide, add methane, maybe some ammonia, and the ice goes away.

That is roughly what Sagan and Mullen proposed, and fifty years later it has not settled. Georg Feulner’s 2012 review in Reviews of Geophysics opens by noting that “models of stellar evolution predict a solar energy input to the climate system which is about 25% lower than today” for the early Earth, walks through the greenhouse proposals and the difficulties with each, and concludes flatly that “the faint young Sun problem cannot be regarded as solved.”

The trouble is that geology pushes back. The carbon dioxide levels required tend to exceed what ancient soils and minerals seem to allow. Methane gets destroyed by ultraviolet light and, later, by oxygen. Lowering the planet’s reflectivity helps, since less land and fewer clouds mean less sunlight bounced away, but not by enough on its own.

The paradox has survived half a century of people who badly wanted to kill it.

The fourth root is the only reason this is merely hard

There is a mercy hidden in the equations, and it is worth pausing on.

Temperature does not scale with sunlight. It scales with the fourth root of sunlight, because radiated heat goes as the fourth power of temperature. So the young Sun delivering 32 percent less energy does not make Earth 32 percent colder. It makes it 9.3 percent colder in absolute terms, and 9.3 percent of 254 K is about 24 degrees.

Twenty-four degrees is a big problem. Eighty degrees would be an unsolvable one. Without that fourth root, no plausible amount of carbon dioxide would rescue the Archean, and the paradox would not be a research topic, it would be a refutation of stellar physics.

Where it goes from here

The brightening has not stopped. Using Gough’s 1981 luminosity law, which is the standard analytic approximation, the Sun is currently gaining about one percent of brightness every 110 million years or so. Slow by any human measure, and relentless.

The way that ends life on Earth is not what I expected. The Sun does not cook us. It starves the plants.

As luminosity rises the planet warms, and warm silicate rock weathers faster, and weathering pulls carbon dioxide out of the atmosphere and locks it into carbonate. Left alone for long enough, that thermostat drives atmospheric carbon dioxide below the level plants need. Ken Caldeira and James Kasting worked through this in Nature in 1992 and found that a biosphere built on C4 plants “could survive for at least another 0.9 Gyr to 1.5 Gyr.”

C4 photosynthesis is the more efficient variant, the one that can operate below 10 parts per million of carbon dioxide, where the more common C3 pathway gives out at about 150. Those same two pathways set the ceiling on how much sunlight any living thing can convert, which is why a photosynthetic human is impossible. Here, the difference between them is worth several hundred million years of biosphere.

The Sun will eventually swell into a red giant and deal with the Earth properly. It will not need to. It will have removed the carbon dioxide a few billion years earlier, and quietly.

A correction, and this time it is mine

I have spent several posts here checking the arithmetic of a language model and finding gaps in it. For balance, the record should show that when I opened this particular conversation I typed, with some confidence, “life evolved 4.54 ± 0.05 billion years ago.”

That is the age of the Earth. Life is younger, probably by four hundred million to a billion years. ChatGPT corrected me straight away, gently and correctly, and then explained the faint young Sun paradox without being asked, and got that right too.

I went looking for something to catch it on and there was nothing. It happens. The interesting error in that conversation was mine.

The short version

The Sun is 48 percent brighter than when it began fusing hydrogen, but its surface is only 159 degrees hotter. It brightens by expanding. That growth is driven by its core slowly contracting and heating as hydrogen turns into helium.

Because the early Sun was so much fainter, the Earth should have been frozen for its first two billion years, and it was not. Nobody has satisfactorily explained why, and the leading candidate, a much thicker greenhouse, keeps running into the rocks that would have recorded it.

The Sun gains roughly one percent every 110 million years. In about a billion years that brightening will have scrubbed enough carbon dioxide out of the air to end plant life, long before the oceans boil.

Not bad for a star everyone assumes is a constant.

Sources