Tap anywhere on the globe to stand there. Then watch the year run and the daylight pile up — with the solar disc, refraction and orbital eccentricity switchable in real time.
Drag to spin the globe · tap to plant the marker · the shaded half is night right now in the simulation.
Lit hours in local solar time, equinox to equinox. Scrubbing never re-runs the simulation — the year is solved once into a list of sunrise/sunset instants and a running total, so any moment is a lookup. The sim only re-solves when you move the marker or flip a switch (about 45 ms).
When I think ellipse, I erroneously picture an egg. An egg is lopsided — one end blunt, one end pointed. An ellipse never is. Reflect it across its short axis and you get back the identical curve, so the two ends of the long axis are exactly the same shape as one another. Nothing about the curve tells them apart.
This is the question that finally made it click. If the two ends are identical, how does an orbit end up with one perihelion and one aphelion, instead of two of each? An ellipse has two foci, spaced evenly either side of the centre — and the Sun can only occupy one of them. That single fact is what breaks the tie. The end nearer the occupied focus is perihelion; the end further from it is aphelion. The curve is symmetric. The Sun's position in it is not, and the calendar follows the Sun, not the curve.
The other focus just stays empty — a bare point with nothing at it and nothing orbiting it. Mirror the whole diagram and swap the labels and no physics changes, so as a pair the two foci genuinely are interchangeable. But you only get to make that choice once, and everything else follows from it. Nor is the empty one a spare: it's load-bearing geometry, because the distances from the two foci out to any point on the curve always sum to exactly 2a. That is what being an ellipse means.
Which brings up the thing that genuinely surprised me. Eccentricity does two separate jobs, and it does them at completely different rates:
by a·e — plain, straight-line proportional to e. At Earth's e = 0.0167 that is 1.67% of the orbital radius. About 2.5 million km.
by a − b ≈ a·e²⁄2 — proportional to e squared. At the same e that is 0.014%, thinner than the line this diagram is drawn with.
Divide one by the other and the squashing loses by a factor of 2/e — call it 120 to 1 for Earth. Read that ratio again, because it runs backwards to intuition: the smaller the eccentricity, the more totally the offset wins. Halve e and the offset merely halves, while the squashing drops to a quarter. Push e toward zero and both effects die, but the orbit turns into a circle far faster than the Sun returns to the middle.
So a slight eccentricity does not give you a slightly oval orbit with a slightly off-centre Sun. It gives you a circle you cannot tell from a drawn one, with the Sun a frank 1.67% off the middle — and 1.67% is enough to hand the northern hemisphere an extra week of orbit.
Switch that off for the honest picture: a circle with the Sun apparently dead centre. Nothing to see — which is the point. On, the offset is drawn fourteen times bigger so the mechanism is visible. Either way the day counts and the distance readout are the real, unexaggerated numbers.
The straight edge splitting the two tinted halves is the equinox line, and it runs through the Sun, not through the centre of the orbit. That is the whole asymmetry in one picture: a chord through an off-centre point cuts the disc into unequal pieces, equal areas take equal times, so the bigger piece takes longer. Northern spring and summer get 186.4 days; autumn and winter get 178.8. Flip Circular orbit and the Sun slides to the centre, the chord becomes a diameter, and the halves come out identical.
One last word for that empty focus, which looks so useless sitting there. Stand on it and the planet's angular motion is almost perfectly steady. Seen from the Sun, Earth's angular rate swings by 6.9% across the year — quick at perihelion, slow at aphelion. Seen from the empty focus it swings by 0.03%. That near-uniformity is Ptolemy's equant, rigged up centuries before anyone knew what an ellipse was, and it works for the same reason the squashing is invisible: the error is second order in e.
The ticking counter is the raw accumulation over this one exact window. The full-year figure above averages the window start across a whole day, which removes the ±2.8 h artefact of where the year boundary happens to fall in the diurnal cycle — so at the end of the year the two can differ by up to about three hours. The report explains why.
Draw the Sun as a point. Take away the air. Now sunrise is a clean little bit of geometry — a dot crosses a line — and on the equinox every latitude on Earth gets exactly twelve hours of day and twelve of night. No exceptions. No rounding.
It's a beautiful result. It's also wrong.
Two things ruin it, and here is the part that got me curious in the first place: they both ruin it in the same direction. Neither one has ever shortened anybody's day.
It's a disc, about 32′ across — half a degree, roughly your little fingernail held out at arm's length. And we don't start the clock when the centre arrives, we start it when the first sliver shows. So the day opens with the Sun's centre still 16′ below the horizon, and closes with it 16′ below on the other side. No atmosphere required. This one is pure geometry and it would happen on a dead airless rock.
Air thickens as it goes down, and light grazing the horizon bends downward through that gradient, lifting the Sun's apparent position by about 34′ when it's sitting on the horizon. Sit with that number a second. It is bigger than the Sun is wide. The Sun you watch touching the sea at sunset has already set. You're looking at a picture of it.
Stack the two and you get 50′ of grace at each end of the day. The Sun's centre can sit fifty arcminutes under the true horizon and the day still counts as running.
Fifty arcminutes is nothing. So why does it add up to anything?
Because you aren't banking angle. You're banking time — and the exchange rate between the two is not the same everywhere you stand.
That's the whole answer, and it turned out simpler than I expected it to be. On the equinox the Sun runs along the celestial equator, and the celestial equator meets your horizon at an angle of exactly 90° minus your latitude. At the equator the Sun comes straight up, dead vertical, and clears those fifty arcminutes in about three and a half minutes. At 60° it arrives at a 30° slant and has to walk twice as far to climb the same height. At the pole the path is parallel to the horizon — the Sun doesn't rise in any daily sense at all. It gets lifted by the slow seasonal drift of its declination, and crossing 50′ takes it about two days.
Same fifty arcminutes in every panel, measured at the equinox. The gold stretch is how far the Sun has to walk to cross it — that length is the time.
So the hunch was right, and it's worth being precise about how big it is. The equinox day at the equator isn't 12 h 00 m, it's about 12 h 07 m. Seven free minutes, twice a day, three hundred and sixty-five days a year — call it 42 extra hours. Move up to 60°, where that same 50′ costs the Sun two and a half times as long to cross, and it swells to about 103 hours. Every latitude on the planet finishes the year with more day than night, and the further you get from the equator the fatter the margin.
Then, somewhere around 60°, it stops growing. Past there you stop trading fast sunrises for slow ones and start trading many sunrises for few, and the two very nearly cancel.
Which leaves the other half of the question. Earth's orbit is an ellipse, so it doesn't travel at a constant rate, and the two halves of the year that the equinoxes cut are not the same length. That certainly sounds like it ought to hand somebody extra daylight too.
It doesn't. It only moves it around. The third switch above lets you watch it do exactly that.
Each switch takes one piece of physics away. Whatever the number does when you flip it is what that piece was worth. That's the entire method, and it's the reason there are three switches here instead of one.
What it changes. On, the day runs from first sliver to last sliver. Off, the Sun collapses to a point and its centre has to do the crossing itself. Watch the rise threshold in the readout move by 16′.
Why you'd flip it. This is the half of the bonus that owes the atmosphere nothing at all. It's here because the Sun is an object and not a coordinate. Turn it off and you're asking a clean question: how much of this would survive on an airless planet?
What to watch. The year loses about 13.5 h at the equator and about 33 h at 60°. Now notice that the ratio between those two is the same ratio refraction gives you. Both effects buy time at the identical exchange rate, because they're buying the same thing.
What it changes. On, the horizon threshold carries 34′ of refraction — the standard IAU number for an ordinary atmosphere. Off, light travels in straight lines like a textbook diagram.
Why you'd flip it. Refraction is the bigger of the two, worth about 2.13× the disc. It's also the only part of this that isn't a fixed fact about the solar system. Real horizontal refraction runs from something like 28′ in warm thin air to well past 42′ over polar ice — so if you want to know where the uncertainty in all of this lives, it lives right here.
What to watch. Go stand at the South Pole and switch it off. The year goes negative — more dark than light. The South Pole's entire surplus is on loan from the atmosphere.
What it changes. Off — the default — Earth runs its real ellipse, e = 0.0167, moving fastest at perihelion in early January. On, the orbit gets circularised: constant speed, constant distance, same year length, same tilt. Nothing else touched.
Why you'd flip it. Aphelion lands in early July, so Earth dawdles through the northern summer and hurries through the southern one. 186.4 days from the March equinox to the September one; 178.8 days to come back. That lopsidedness is worth ±90 hours at the poles, which is bigger than everything else on this page put together.
What to watch. Flip it on and the planet goes symmetric. The North Pole hands over 90.6 h and the South Pole picks up 90.6 h — the same number, to the minute. Nobody gains. The hemispheres are splitting one fixed pot, and the ellipse only decides who gets the bigger half.
Surplus over the day = night line, in hours per year. The outlined bar is where the switches are set right now.
The bottom bar is the control — point Sun, no air, circular orbit. It lands on zero at every latitude on Earth, and that is the only reason to trust anything above it. If the machinery were lying to me, that bar is where it would show.
Each one jumps to the end of the year so you can read the total straight off, then hit ⟲ Restart and Play to watch it accumulate.