INTERACTIVE COMPANION · ORBITAL GEOMETRY · EARTH'S ENERGY BUDGET

The Insolation Bench

A live model accompanying “Computing the Sunlight That Started and Ended the Ice Ages”

Four numbers decide how much sunlight falls anywhere on Earth on any day: the latitude, where the planet is in its orbit, how elliptical that orbit is, and how far the spin axis is tipped. Everything below comes out of one equation with those four inputs and no physics at all. Not a single measurement of the sky is involved.

Two benches sit below. The first lets you retune the orbit and watch the whole map of sunlight change. The second runs the real orbit backwards for a million years and shows which of the three orbital parameters is driving at any given moment. Both compute the same formula the article does, in your browser, and both work with the network unplugged.

Model 1. Retune the orbit

The daily-mean insolation at latitude φ when the Sun sits at longitude λ around the year is

Q = (S₀ / π) · (a/r)² · [ H₀ sinφ sinδ + cosφ cosδ sin H₀ ]

with declination sinδ = sinε sinλ, sunset hour angle H₀ = arccos(−tanφ tanδ) clipped to the range 0 to π, and distance r/a = (1 − e²) / (1 − e cos(λ − ω)). The solar constant is S₀ = 1361 W/m². The clipping is what produces polar night and the midnight sun: push the argument past ±1 and the day either never starts or never ends.

Eccentricity 0.0167 Today's value is 0.0167. Over the past million years it ran from 0.0042 to 0.0578. Drag it to zero and watch the two hemispheres become mirror images of each other, because a circular orbit has no near side and no far side. Obliquity 23.44° The real range over a million years is 22.08° to 24.45°, which looks tiny. Push it out to 20 and 26 anyway to see what the tilt is doing: the polar night wedge opens and closes, and the summer pole gains or loses its place as the sunniest spot on Earth. Perihelion longitude 102.9° This is the one to play with. At today's value the Earth is near aphelion in northern June, so northern summers are mild. Turn it to about 283° and the Earth passes perihelion in northern June instead. At the eccentricity slider's maximum that single change is worth 115 W/m² at 65°N. Readout latitude 65°N 65°N is the conventional Milankovitch latitude, roughly the southern edge of the old northern ice sheets. Slide it to 90 to see the pole, where a twenty-four-hour day beats the equator, or to the equator to find the flattest seasonal cycle on the map. Day marker June solstice These are astronomical positions in the orbit, not calendar dates, which is why they stay meaningful when precession changes the length of the seasons.

Q at marker: – vs today: – annual mean here: – day length: –

–

Why the summer pole beats the equator

Set the latitude slider to 90 and leave the day marker on the June solstice. The readout says 524.2 W/m². Now set it to 0. The equator gets 384.9 on the same day, which is 36% less.

The Sun at the north pole on the solstice never rises more than 23.4° above the horizon, so every square metre of ground is catching light at a glancing angle and collecting only sin(23.4°) = 0.40 of what a perpendicular surface would. But it does that for twenty-four hours without a break. The equator gets a much better angle and only twelve hours of it, and the twenty-four hours win. Drop the obliquity slider to 20° and the polar value falls to about 448 while the equator barely moves, because the equator does not care much about the tilt and the pole cares about nothing else.

Model 2. Run the past million years

The second bench takes the real orbit. Eccentricity, obliquity and perihelion longitude are read from the La2004 solution of Laskar and colleagues, one value per thousand years for the past million, and the insolation is recomputed from them at every step.

The three lower strips are the point of it. Each one holds two orbital parameters frozen at their million-year mean and lets the third vary, so you can see which parameter is responsible for any particular wiggle in the top curve. The three do not add up to the top curve exactly, because precession's amplitude is eccentricity and the two are multiplied rather than summed.

Time 0 kyr before present Drag to scrub. Stop on 231 for the dimmest northern summer of the past million years and on 958 for the brightest. Termination II sits at 130, right after a steep climb. Speed 8 kyr per second Slow it to 1 and the 23,000-year precession beat becomes countable. Run it at 40 and the beat blurs while the 100,000-year envelope of eccentricity becomes the obvious thing on the screen, which is exactly the illusion the article warns about: that envelope carries almost no power of its own. Window 300 kyr wide Narrow it to 60 to read individual precession cycles. Widen it to 1000 to see the whole record at once and count the maxima. There are 46 of them, and nine terminations. Summer measured as Switch to the caloric half year, which is Milankovitch's own definition, and watch the fast precession beat mostly vanish while the 41,000-year obliquity rhythm takes over. Same sunlight, different integral, different answer about what runs the ice ages.

at 0 kyr BP Q: – e: – obliquity: – e sinω: – driver: –

What this does not do

There is no atmosphere here, no cloud, no albedo, no ocean and no ice. The number in the readout is sunlight arriving at the top of a transparent atmosphere, and it says nothing at all about temperature. Nothing on this page lags anything else, whereas a real ice sheet takes thousands of years to build and thousands to fall apart. And the 100,000-year rhythm that dominates the actual ice record is almost absent from every curve on this page, which is the honest and unresolved ending the article gives it.

The orbital elements are quantised to six decimal places to keep the page small, which shifts the insolation by at most 0.0009 W/m² anywhere in the record. Everything else is computed live from the equation printed above.