By Austin VornhagenHere is a fact that should be far more famous than it is. You can explain why July afternoons are getting hotter with four numbers, one law of physics from the 1870s, and the calculator on your phone. No supercomputer. No “trust the experts.” Just arithmetic.
I know because I asked an AI to walk me through every step of it, then checked the numbers against NASA, NOAA, and the Intergovernmental Panel on Climate Change. What came out was not the argument I usually hear on the news or at a barbecue. It was an accounting problem.
Earth runs a budget. Energy arrives as sunlight and leaves as heat. Global warming is what happens when leaving gets a little harder.
That really is the whole mechanism. Melting ice, record heat, drier soil in August: all of it sits downstream of one ledger. Which leaves one question worth answering properly:
Carbon dioxide is about 430 molecules out of every million in the air. How can something that rare change the temperature of an entire planet?
By the end of this page you will be able to answer that on a napkin. You will also see why “summers will keep getting hotter” is both more true and less true than it sounds.
The prediction came first. By about a century.
Climate science usually gets told as if it started in the 1980s. It started with a French mathematician in the 1820s doing arithmetic that refused to come out right.
- 1820sJoseph Fourier
Works out that a planet which simply soaked up sunlight and radiated it away should be far colder than Earth is. Something in the air must be slowing the heat's escape.
- 1856Eunice Foote
Sets glass cylinders of different gases in the sun. The one filled with carbon dioxide heats the most and stays hot longest. She suggests an atmosphere with more of it would give Earth a higher temperature.
- 1859John Tyndall
Measures in the lab that water vapor and carbon dioxide absorb radiant heat, while nitrogen and oxygen, most of the air, barely touch it.
- 1896Svante Arrhenius
Does the first calculation of how much warmer a world with more CO₂ would be. His numbers are rough, the direction is right, and he assumes it will take a very long time.
- 1938Guy Callendar
Collects temperature and CO₂ records and argues that burning fuel is already warming the climate. Most scientists are not convinced.
- 1958Charles David Keeling
Begins precise CO₂ measurements on Mauna Loa. The line he starts has gone up every year since.
- 2024The instrumental record
The warmest year measured, about 1.55°C above the 1850 to 1900 average, according to the World Meteorological Organization.
Read that list again and look at the tense. The idea was proposed, measured in a lab, and calculated before anyone had the instruments to watch it happen. Then people built the instruments and watched it happen.
So the interesting question is no longer whether the machine works. It is how it works, because the how is what tells you what your summers do next. Let's open the books.
Earth's energy budget, step by step
A diagram of sunlight arriving from the left, a disk intercepting it, part reflected back as blue arrows, and infrared heat leaving Earth as orange arrows, with a large readout that steps from 1,361 to 340 to 240 watts per square meter, then to 255 kelvin, and a thermometer comparing minus 18 and plus 15 degrees Celsius.
Point a panel at the Sun
Hold one square meter of panel straight at the Sun, above the atmosphere, and it catches about 1,361 watts. That is the solar constant, and satellites have measured it for decades. Think of it as the planet's paycheck.
Divide by four
Earth is not a flat panel. It catches sunlight on a disk, area πR², but spreads it over a spinning ball, area 4πR², exactly four times bigger. Nights, poles, and winter included, the average square meter gets about 340 watts.
Minus what bounces
Clouds, snow, ice, and bright ground reflect roughly 30 percent straight back to space before it ever becomes heat. What is left, about 240 watts per square meter, is what the planet actually gets to spend.
What comes in must go out
Here is the rule that runs everything. A planet whose temperature is steady has to shed exactly what it absorbs. So Earth must glow about 240 watts per square meter of infrared heat back to space, all day, every day. If it sheds less, it warms until it does not.
Everything warm glows
Every object radiates heat in proportion to its temperature to the fourth power: F = σT⁴, the Stefan-Boltzmann law, first worked out in the 1870s and 1880s. Ask what temperature glows at exactly 240, solve, and you get 255 kelvin. That is −18°C, about 0°F, averaged over the whole planet.
The 33-degree mystery
Earth's real average surface is about 288 kelvin, or +15°C, 59°F. Our tidy calculation just came up 33 degrees short. That gap is Fourier's puzzle from the 1820s, and it has a name: the natural greenhouse effect. Without it you would live on an ice planet.
The ground glows 390. Space only receives 240.
Plug the real surface temperature into the same law. At 288 kelvin, the ground radiates about 390 watts per square meter of infrared. Yet only about 240 leaves the planet.
That sounds like a bookkeeping error. It is actually the clue. Somewhere between your backyard and outer space, about 150 watts per square meter of the ground's glow never makes it out. The air absorbs it and radiates much of it back down, and that returned heat is why the surface can sit 33 degrees warmer than bare arithmetic says it should.
None of this is hypothetical. Instruments on NASA satellites, called CERES, measure both sides of the ledger from orbit: sunlight coming in, sunlight bouncing off, infrared going out. Combine them with the heat piling up in the oceans, measured by thousands of floating robotic probes, and you get Earth's actual balance sheet. A 2021 NASA and NOAA analysis found the planet taking in roughly one watt per square meter more than it gives back, and that the gap had roughly doubled between 2005 and 2019.
So the books are out of balance, and we can watch it happen. The question is what knocked them out. The popular answer is a blanket. It is not wrong. But it hides the part that makes the math work.
Space can't see the ground
Here is where my own mental model broke. I had always pictured CO₂ as a blanket that traps heat near the ground, and that picture invites a fair objection. If the blanket is already thick, why would a little more matter? Scientists call this the saturation question, and the blanket has no good answer to it.
The real mechanism has nothing to do with trapping. It is about where space sees Earth from. Scroll, and watch one column of air.
How more CO2 raises the altitude heat escapes from
A diagram of one column of air, warm orange at the ground and cold violet at the top, with a glowing band marking the altitude where heat escapes to space. As CO2 is added the band rises into colder air, outgoing heat drops below 240 watts per square meter, and then the whole column warms until outgoing heat returns to 240.
The air is a fog, for heat
At the wavelengths water vapor and CO₂ absorb, the lower air is opaque, like fog for infrared light. Heat from the ground gets absorbed and re-emitted again and again, a lot of it back downward. It only escapes to space from the height where the fog finally thins out above it, on average about five kilometers up.
Up is cold
In the lower atmosphere, the higher you go, the colder it gets, about 6.5°C per kilometer on average. Pilots know this. So does anyone who has driven up a mountain in shorts. The escape layer sits around −18°C, which is exactly the 255 kelvin we calculated. Space sees Earth at that temperature because that is the layer it sees.
Add CO₂. The fog thickens.
More CO₂ makes the air more opaque at its wavelengths, so the height where heat can finally slip out to space moves up. Not by much, well under a kilometer for a doubling. But up is the only direction it goes.
Higher is colder. Colder is dimmer.
The new escape layer is colder, and colder things glow less, fourth power again. So Earth is still absorbing 240 but now sending out less. For a doubling of CO₂, the gap is about 4 watts per square meter. That is the forcing. Nothing was trapped. The exit just moved somewhere colder.
The planet warms until the books balance
The surplus piles up, about 90 percent of it in the oceans. The whole column warms until the new, higher layer reaches the old escape temperature and 240 goes out again. Same outgoing heat as before. Warmer everything underneath it, including the air on your porch.
Now look back at the opening film. When the view flipped to infrared, the clouds turned into dark blots. They were not cold because they were blocking sunlight. They were dark because their tops are high up, where it is freezing, and high, freezing things barely glow. You were watching the escape-altitude idea in a single picture.
It also answers the saturation objection. Even where the middle of the CO₂ band is already opaque near the ground, adding more pushes the escape height higher and widens the edges of the band where the air was still partly clear. The effect never stops. It just gets a little weaker with each addition, the way each extra dollar matters a little less to a millionaire.
The greenhouse effect is not a blanket on the ground. It is a chimney that keeps getting taller.
And “a little weaker with each addition” has a precise shape. It is a logarithm, and it means you can run the numbers yourself.
The formula fits on a sticky note
Physicists have run the full line-by-line calculation across thousands of CO₂ absorption lines. For back-of-envelope work, the result boils down to one line, published by Gunnar Myhre and colleagues in 1998 and used in IPCC reports since:
ΔF = 5.35 × ln(C ÷ C0)
C0 is the preindustrial concentration, about 278 parts per million. C is today's, around 430 on Mauna Loa this year. The answer comes out in watts per square meter. Two more numbers turn watts into degrees:
- 3.2 W/m² per °C. How much more a planet at Earth's temperature radiates for each degree it warms. Physicists call this the Planck response. It is the no-feedback, bare-physics answer.
- About 3°C per doubling. The IPCC's best estimate once you add what a warmer planet does to itself: warm air holds more water vapor, itself a greenhouse gas; melting ice exposes darker ground and ocean that absorb more sunlight; clouds shift. The likely range is 2.5 to 4°C.
Drag the slider. It is doing nothing more than the arithmetic above.
CO₂ only, at equilibrium, after the oceans catch up. It leaves out methane and other gases, which add warming, and pollution particles, which hide some. So it is a way to see the physics, not a forecast.
Put the slider back on today and look at the first box. That +2.3 is the two watts in the title: the extra push CO₂ alone has added to every square meter of the planet since the 1800s. It is about 1 percent of the 240 the planet runs on, delivered around the clock, and it is enough.
Two more things jump out once you play with it. First, the logarithm: going from 278 to 556 adds exactly as much forcing as going from 556 to 1,112. Every doubling costs the same 3.7 watts, about 3.9 in the IPCC's more complete accounting. Second, the feedbacks more than double the bare-physics answer. The CO₂ lights the match. The water vapor and ice do much of the burning.
You may also notice the slider says about 1.8°C at today's concentration, while the world has actually warmed a bit over 1.3°C. That gap is expected. The oceans are still soaking up heat and have not caught up yet, and air pollution reflects some sunlight. Methane pushes the other way. The oceans are the reason the imbalance persists: they are a very large pot of water on a very small burner.
Nobody lives in a global average
Everything so far describes the whole planet. You, on the other hand, live somewhere specific, on land, probably in a place that gets a real summer. Two facts make your summer different from the global number.
Land warms faster than ocean. Water has enormous heat capacity and can cool itself by evaporating. Over 2011 to 2020, the IPCC estimates land had warmed about 1.59°C since the late 1800s, versus 0.88°C over the oceans.
The hottest days can warm faster still. In some midlatitude and semi-arid regions, the IPCC finds the hottest days warming at up to about 1.5 to 2 times the rate of global warming. So “the world warmed half a degree” does not mean your worst afternoon got half a degree worse.
But the most important effect is not about the average day at all. It is about the tails. Here is the intuition I had to unlearn: a small shift in the average makes a small change in the extremes. Watch what the IPCC actually calculated.
How the once-a-decade hot extreme changes with global warming
A bell curve of summer temperatures with a fixed line marking the old once-a-decade hot extreme. As global warming increases the curve slides right and the shaded tail beyond the line grows, while a row of ten boxes for ten summers fills up: 1, then 2.8, 4.1, 5.6, and 9.4 per decade.
Once a decade
The world your great-great-grandparents farmed in. A scorching day this hot shows up about once a decade. A family could go years without one.
2.8 times a decade
One degree of global warming. The average barely moved, but that once-a-decade scorcher now arrives almost three times per decade, and it runs 1.2°C hotter when it does.
4.1 times a decade
Half a degree more. The old extreme shows up four times a decade. Your kids would call it a normal bad summer.
5.6 times a decade
The old once-a-decade event now happens more years than not. It is no longer an extreme. It is the weather.
9.4 times a decade
Nearly every year. And the hottest days are now more than 5°C hotter than the old extreme. The tail has become the middle.
Weather rolls the dice. Warming loads them.
Look at the first step again. One degree of global warming, a change you could not feel walking outside, nearly tripled how often the old extreme shows up. That is what happens when you slide a bell curve: the middle barely changes, but the thin tail beyond any fixed line swells fast.
It also clears up the most common misunderstanding about hotter summers. It does not mean every summer is hotter than the last. Weather still rolls the dice every year. A wet, cloudy summer still happens, and it still feels mild. The world has already passed the first row of that chart, and you will still get cool Julys. You will just get them less often, and the bad ones will be worse.
What the math means for a Nebraska summer
Take the central United States, where I live and where summers already have a reputation. Here is roughly how the projections read decade by decade. These are descriptions of the climate, the loaded dice, not forecasts for any particular July.
- 2030s
Recognizable, with a lean
Every major IPCC scenario puts 2021 to 2040 at about 1.5°C of global warming, because this stretch is mostly set by past emissions. Plenty of ordinary summers. Hot nights and hot spells get a little more common, and genuinely cool summers a little rarer.
- 2050s
Noticeably different
University of Nebraska projections suggest the state's days above 95°F, historically about 10 to 20 a year, could roughly double by mid-century. The heat also gets stickier at night, which matters for sleep, livestock, crops, and power bills.
- 2090s
Two very different futures
By late century the spread is driven almost entirely by what gets emitted between now and then. In a very high emissions case, some Nebraska projections exceed 50 days a year above 100°F. In a strongly curbed case, it is hotter than today but still a place you would recognize.
That fork deserves a picture of its own. Here are the IPCC's best estimates of global warming for 2081 to 2100 under its five main scenarios. The physics is identical in every bar. Only the emissions differ.
Best estimates for 2081 to 2100 relative to 1850 to 1900, scenarios SSP1-1.9 through SSP5-8.5. IPCC AR6 Working Group I.
Numbers like “+4°F” are hard to feel, so the University of Nebraska built something smarter: a tool that matches a city's projected climate to a place that has that climate today. Tap through it for Lincoln.
Lincoln's climate is Lincoln's climate. Tap a year to move it.
Annual temperature match from the University of Nebraska–Lincoln climate “sister city” tool, under a scenario in which greenhouse emissions are curbed enough to bend today's trajectory. Map simplified.
That is the curbed scenario, and it still moves Lincoln a couple hundred miles south in climate terms. The useful part is what it enables. You do not have to imagine how to adapt to Wichita's heat. You can go look at what Wichita already does.
There is one more mechanism worth knowing if you live on the Plains, and it is our energy budget again at ground level. When soil is wet, a lot of the sun's energy goes into evaporating water instead of heating the air. When the soil dries out, that outlet closes, and the same sunshine goes straight into hotter ground and hotter air, which dries the soil further. That is why drought and heat waves in the middle of the country feed each other, and why a region can see heavier downpours and worse dry spells at the same time.
Summers do not get hotter forever
Here is the part that surprised me most, and it falls straight out of the budget. There is no rule that says every decade must be hotter than the last until the end of time. The warming continues because CO₂ keeps accumulating, which keeps pushing the escape height up, which keeps the books out of balance.
Stop adding CO₂ on net, and the concentration stops climbing. The IPCC's assessment is that reaching net zero CO₂ emissions would roughly stabilize global temperature from CO₂, rather than committing us to endless warming. Other gases and slow feedbacks complicate the fine print, but the headline is unusually clean:
Summers keep getting hotter for exactly as long as we keep adding CO₂. Not a year longer.
That reframes the fight. And once you see climate as a two-sided ledger, a more radical idea becomes obvious. Everything so far has been about the outgoing side, the infrared escaping. What about the incoming side?
What if you just dimmed the Sun a little?
It sounds like a supervillain plan, and people have proposed it seriously. Park a swarm of shades, reflectors, or fine particles near the L1 point, about 1.5 million kilometers toward the Sun, where they can stay lined up between us and it. A 2006 study by the astronomer Roger Angel estimated that blocking about 1.8 percent of sunlight there would offset the warming from a doubling of CO₂.
The arithmetic is almost suspiciously easy. Block 1 percent of the 340 watts and you remove 3.4. About 70 percent of that would have been absorbed, so the planet loses about 2.4 watts per square meter, roughly the entire CO₂ forcing we have added so far. Try balancing the ledger, then see what happens when the shade fails.
Net = CO₂ forcing (5.35 × ln(C ÷ 278)) minus blocked sunlight (340 × share blocked × 0.7). CO₂ only; illustrative.
That button is the catch, and it has a name: termination shock. A shade holding back a lot of CO₂ forcing is a promise that must be kept every single year. If it breaks, gets switched off in a war, or simply stops being paid for, the warming it was hiding arrives in years instead of decades.
And a shade only fixes the temperature line. The CO₂ is still there, still dissolving into the ocean and making it more acidic. Dimming sunlight also is not an exact mirror of trapping infrared, so rainfall and circulation would not return precisely to where they were. That is why a 2026 U.S. Government Accountability Office brief called the effects on the environment and public health uncertain, and flagged how little oversight exists.
The honest place for the idea, if it has one, is as an emergency brake that buys time while emissions fall and some CO₂ is removed. Then you ease off the brake. What it cannot be is a substitute for fixing the outgoing side.
It is not a prediction anymore
Fourier noticed the missing warmth in the 1820s. Foote and Tyndall measured the gases responsible in the 1850s. Arrhenius did the math in 1896. Keeling started the measurement in 1958. Then we raised CO₂ by roughly half, and the planet warmed in the direction the physics said it would.
Global warming stopped being a prediction a while ago. It is an experiment we are partway through. What is still genuinely uncertain is how far we run it.
About 240 watts per square meter in, and the same must go out. Anything that makes leaving harder warms the planet until leaving is easy again.
CO₂ does not trap heat on the ground. It raises the height heat escapes from into colder air, where it glows less. More CO₂, taller chimney. It does not saturate.
A small shift in the average multiplies the extremes. One degree of global warming nearly tripled the once-a-decade scorcher. That is what “hotter summers” really means.
So the next time someone at a barbecue says it has always been hot in July, you can agree with them. It has. Then you can ask the better question, the one the whole ledger points to:
Not “will summers get hotter?” How long do we keep the books out of balance?
This essay grew out of a long conversation with an AI, with the numbers checked against primary sources. Energy budget figures are NASA's. The CO₂ forcing formula is from Myhre et al. (1998); doubling forcing, climate sensitivity, land and ocean warming, the hot-extreme frequencies, near-term warming, and the late-century scenarios are from the IPCC's Sixth Assessment Report, Working Group I. Current CO₂ is from NOAA's Mauna Loa record. The 2024 temperature is the World Meteorological Organization's. The energy imbalance trend is from Loeb et al. (2021). The Lincoln analogs are from the University of Nebraska–Lincoln climate “sister city” tool, and the Nebraska day counts from University of Nebraska climate assessments. The L1 figure is from Angel (2006, PNAS). The calculators are simplified illustrations of that arithmetic, not forecasts.
If the energy accounting was your favorite part
Gravity Is Free. Sorting Is Not.
The four laws of thermodynamics, and why every clever free-energy machine eventually pays the same bill. The same ledger, one level down.
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Read the essay →Primary source · IPCCAR6 Summary for Policymakers
The source of the hot-extreme frequencies, the land and ocean numbers, and the five late-century scenarios, with every confidence level spelled out.
Read the summary →Primary source · NOAATrends in Atmospheric CO₂
The Keeling Curve, updated monthly from Mauna Loa. Plug the latest number into the slider above.
See the data →
Hard ideas, made obvious.
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