The demo is a demo — no equations on the board. This page is the receipt: what each panel is actually computing, the numbers it uses, and every place the model is deliberately simpler than the sky.
Increasing focal length does not slow a telescope down. It increases the workload, and the extra work takes extra time.
Focal length is the decision. Aperture is the funding. The f-number is the invoice — and because it is a single number made from two, it cannot tell you either one on its own.
Every animated panel runs the same simulation. Only four numbers change between them.
Both chalkboards are the same detector: a square 400 × 400 units wide. Nothing about the sensor differs between the two boards in any panel. What differs is the plate scale — how much sky one unit of board covers — which is set by focal length alone. In the code that is scale, in board units per object unit.
The target is a fixed pattern: nine star clusters, three of each primary colour, sitting on rings at radius 1, 5.2 and 7 object units. A cluster's radius is 0.691 object units. On the board, a cluster's radius in board units is therefore
where seeing is a blur radius, also in object units. Longer focal length = larger scale = a physically bigger cluster on the same sensor. That is the entire mechanism of the demo.
Each stick of chalk is one equal unit of time. A chalk mark is a fixed 10 units across on every board in every panel — it is the detector's grain, and it does not care what focal length you chose. So the same mark that covers a small cluster in three dabs leaves a big cluster full of holes.
Marks land at random positions inside the cluster. Throw n marks of area a at a cluster of area A and the fraction of the cluster still bare is
The 0.01% is a choice. It has to be a choice, because coverage is asymptotic — a target is never truly full. But note what the formula does: the threshold enters as a single multiplier ln(1/bare) that is identical on both boards. Change the threshold and both counts move together; the ratio between the boards does not move at all. That is why the demo can pick a number and still be honest about the comparison.
Marks are laid in pure red, green and blue and blended with lighten (per-channel maximum). Red over red stays exactly red however deep it piles; white appears only where all three primaries land on the same spot. So white means one thing only: the scope could not tell the three clusters apart. Sensor saturation is not modelled — that would put white everywhere and destroy the one signal the colour scheme carries.
The table pile holds exactly the chalks the first board needs to finish. The floor stack is everything the other board still needs after that. The height of the floor stack is the ratio, drawn to scale. Both piles are multiples of three, because Lefty carries one chalk of each colour per trip.
What is happening: one 100 mm hole, two tube lengths — 200 mm at f/2 and 1000 mm at f/10. Identical light arrives at both boards every minute. Six photons per cluster per round, both sides, always.
| Left | Right | |
|---|---|---|
| Aperture | 100 mm | 100 mm |
| Focal length | 200 mm | 1000 mm |
| f-number | f/2 | f/10 |
| Plate scale (board units / object unit) | 21.2 | 106 |
| Marks needed per cluster | 28 | 500 |
| Chalks to finish | 21 | 495 |
The point: the f-number said 25× slower. It was right about the time and wrong about the reason. No photon was lost; the same light was spread over twenty-five times the area, and the f/10 board came away with detail the f/2 board cannot show at any exposure length.
Own up to it: the piles are rounded up to multiples of three, so what you can actually count on screen is 495 ÷ 21 = 23.6×, not a clean 25×. The physics is 25; the drawing is rounded.
What is happening: two 800 mm tubes, one 200 mm across and one 100 mm across. Because focal length is identical, the plate scale is identical — 106 on both boards. Same cluster size, same positions, same gaps, same everything. Nothing about the resolution moves. Only the delivery rate changes.
| Left | Right | |
|---|---|---|
| Aperture | 200 mm | 100 mm |
| Focal length | 800 mm | 800 mm |
| f-number | f/4 | f/8 |
| Plate scale | 106 | 106 |
| Marks per chalk | 16 | 4 |
| Chalks to finish | 93 | 372 |
The point: this is the only case where “faster” means what people think it means. Aperture bought speed and nothing else — not framing, not detail. And it is the reason the f-number survives as a shorthand: it happens to be honest here, and people generalise from it.
What is happening: this panel leaves the chalkboards alone. You pick one real telescope, a real double-star separation, and your own sky, and it answers one question: can that scope separate that pair tonight? Four independent limits are computed and combined.
Why quadrature and not the worst one: these blurs are independent, so they add as variances, not as a maximum. A plain max() under-reported every time two limits were close — 2″ of seeing with 2″ of sampling reads as 2.0″ under max() and is really 2.83″. The verdict text still names the largest single contributor, and names every term within 70% of it, because at 0.7× a second term still adds 22% to the total — removing it would buy you an 18% improvement, which is worth acting on. Below about half the largest, a term adds under 12% and nobody would notice.
The 1.665: guiding is usually quoted as total RMS across both axes. Per axis that is RMS ÷ √2, and a Gaussian's FWHM is 2.355σ, so the blur contribution is 2.355 ÷ √2 = 1.665 × RMS.
Atmospheric seeing is a raw sky number (a DIMM-style figure). Glass, pixels and mount then get added on top.
Measured FWHM is the star size you measured off your own sub in PixInsight or N.I.N.A. That number is the delivered result — seeing, guiding, focus and optics are already inside it. Adding them again double-counts, so the tool doesn't. But pixel size still applies, because sampling is not a blur: it is whether your pixels can record the gap at all. Perfectly tight 2.5″ stars 4″ apart still merge at 3.10″/px, because that gap is barely one pixel wide.
4.0″ pair, 2.5″ seeing, 3.76 µm pixels, 0.60″ guide RMS.
| Scope | Glass | Pixels | Mount | Total | Verdict |
|---|---|---|---|---|---|
| SW Esprit 120ED — 120 mm, 840 mm | 0.97″ | 2.21″ (0.92″/px) | 1.00″ | 3.62″ | SPLIT |
| Celestron RASA 8 — 203 mm, 400 mm | 0.57″ | 4.65″ (1.94″/px) | 1.00″ | 5.41″ | MERGED |
| Celestron EdgeHD 8 — 203 mm, 2032 mm | 0.57″ | 0.92″ (0.38″/px) | 1.00″ | 2.90″ | SPLIT |
The point, and it is the sharpest one on the site: the RASA 8 and the EdgeHD 8 are the same 8 inches of glass under the same sky with the same camera. The f/2 cannot separate the pair and the f/10 can — because at 400 mm the gap lands inside a single pixel. Nothing here is “slow”. Focal length only decides whether your pixels can record what the aperture and the sky already allowed.
Own up to it: the drawing shows the pair blurred by the combined figure, as two uniform discs. And 2.4 pixels across the gap is a working convention, not a law — two is the bare Nyquist minimum and real detection wants margin.
What is happening: the only panel with real hardware on the boards, and the only one where both levers move at once — a William Optics RedCat 51 against a Sky-Watcher Evostar 80ED.
| RedCat 51 | Evostar 80ED | |
|---|---|---|
| Aperture | 51 mm | 80 mm |
| Focal length | 250 mm | 600 mm |
| f-number | f/4.9 | f/7.5 |
| Plate scale | 25 | 60 |
| Chalks to finish | 69 | 168 |
This panel is also the one that is drawn at a real angular scale: one object unit is 5.7″, so the cluster cores sit 9.9″ apart with 2″ of clear sky between them. Seeing is fixed at 2.0″ — exactly the break-even point, where the Evostar's extra reach stops paying because the fog is wider than the gap. Its chalk counts are the only ones the program derives rather than sets by hand:
The point: the f-number says speed, but lowering it at the expense of focal length just means less work is being done for a quick result. Two scopes, both levers moved, and one ratio cannot tell you which one moved.
Own up to it: on screen that comes out as 168 ÷ 69 = 2.43× rather than 2.34×, because the seeing blur is added to both clusters before the counts are computed and both are then rounded up to a trio boundary.
Because “f/2 is five times faster than f/10” is repeated as though it settled something, and it does not. The f-number is a ratio of the two quantities that actually matter, and a ratio cannot recover either of them. Ask it how much light you gather and it cannot say. Ask it how much sky you cover, or how much detail lands on a pixel, and it cannot say.
The chalk exists to make the workload visible. You can watch the same light arrive at both boards at the same rate, and watch one board take twenty-five times as long — not because it was starved, but because it was asked to fill a bigger picture. And the split test exists as the counterweight: once diffraction and seeing and pixel size are on the table, the “slow” scope is frequently the one that can actually see the thing.