I Plotted 12 Stars From Catalogue Data — A Magnitude Accuracy Test

Six stars, one sheet of A3, and a magnitude column that runs from −1.44 to 0.08. That is the entire test. We pulled Sirius, Canopus, Arcturus, Rigil Kentaurus, Vega and Capella from HYG v41, plotted them at scale, and asked a single question: does the chart, held at arm's length, match the sky the eye actually reports? The answer came back no, twice, before it came back yes — and the reason it failed the first two times is the quiet scandal of every star map ever printed, ours included.

The Assignment: Six Anchor Stars, One Sheet of Paper

We picked the six brightest single points in the working catalogue because a test set has to fail visibly or it does not test anything. Sirius in Canis Major at magnitude −1.44. Canopus in Carina at −0.62. Arcturus in Boötes at −0.05. Rigil Kentaurus in Centaurus at −0.01. Vega in Lyra at 0.03. Capella in Auriga at 0.08. Every value from HYG v41, no rounding, no fudge, no "adjusted for what looks nice."

The spread matters. From −1.44 to 0.08 is a range of 1.52 magnitudes across the whole set — a narrow window on paper, an enormous window in the eye. Sirius is not slightly brighter than Capella. It is roughly four times brighter as photons hit the cornea, but we are drawing black dots on white paper, so the eye is not the customer. Ink coverage is.

The plotting rig was deliberately dumb: right ascension mapped to horizontal position, declination to vertical, single-panel equatorial projection, no attempt to preserve angular truth across a hemisphere because we were not testing projection today. We were testing whether the dot size we chose for each star, chosen from its magnitude, produced a page that felt right when held up against a memory of the sky. If you cannot get six stars right in isolation, you have no business drawing eighty-eight constellations.

What HYG v41 Actually Hands You

The HYG catalogue is a compiled dataset — Hipparcos astrometry, Yale Bright Star spectral notes, Gliese proximity data — stitched into a single working file that anyone drawing a chart tends to use because the alternative is loading three catalogues and reconciling them by hand. Version 41 hands you, per star, a name where a name exists, a constellation abbreviation (CMa, Car, Boo, Cen, Lyr, Aur in our set), right ascension in decimal hours, declination in decimal degrees, and an apparent magnitude.

Apparent magnitude is the number the eye would report if the eye were calibrated. It is not luminosity. It is not distance-corrected. It is what reaches us here, through atmosphere and glass and whatever else the photons had to survive. Sirius at −1.44 is not intrinsically the brightest star in our set — Canopus outshines it by an order of magnitude in absolute terms — but Sirius is closer, so on the page marked "how it looks from Earth," Sirius wins.

What the catalogue does not hand you is the drawing instruction. There is no column labelled "dot diameter in millimetres." The magnitude value is a physical measurement expressed on a scale that predates every modern printing constraint by roughly two thousand years, and turning it into ink is entirely the cartographer's problem. This is where every star atlas — Bayer's Uranometria in 1603, Bode's Uranographia in 1801, the Norton's every schoolchild used in the twentieth century — had to make a choice and defend it. Most did not defend it. Most just chose.

The First Plot: Where Sirius Landed and Why It Looked Wrong

The first attempt used the naive rule you find in half the online tutorials: dot diameter proportional to magnitude, inverted. Brighter star, bigger dot, scaled linearly. We set Sirius at 6mm and let the arithmetic distribute the rest. Sirius at 6.0mm. Canopus at 5.2mm. Arcturus at 4.6mm. Rigil Kentaurus at 4.6mm. Vega at 4.5mm. Capella at 4.4mm.

Held at arm's length in daylight, the sheet looked like six stars of nearly identical importance with Sirius as a mild outlier. Held under a desk lamp at night, same story. The eye read six anchors, one slightly larger, five essentially equal. That is not what the sky reports. On any clear night with Sirius above the horizon, Sirius does not read as a mild outlier. Sirius reads as the brightest thing that is not the Moon or a planet. You look up and your eye goes to it, unmistakably. The chart was wrong in a way that a five-year-old with no astronomy would have flagged.

We knew the arithmetic was going to fail — that was the point of running it — but the size of the failure was still instructive. The linear-magnitude rule compresses everything. Between Sirius and Capella there is a magnitude gap of 1.52, which the linear rule rendered as a 1.6mm difference in dot diameter, or about a 27% difference in area. In photons, that same gap represents a brightness ratio of roughly four. The chart was under-reporting the reality by a factor of three, and that error was baked into the arithmetic before ink touched paper.

Magnitude Is a Logarithm Pretending to Be a Number

Here is the scandal, and it is old. The magnitude scale is logarithmic, but it does not look logarithmic. It looks like a plain number. Hipparchus, working in the second century BCE, sorted the stars he could see into six classes — first magnitude for the brightest, sixth for the faintest naked-eye stars — and the scheme survived because the eye's response to light is itself logarithmic. He was fitting a scale to human perception, without knowing that was what he was doing.

Norman Pogson formalised it in 1856 by fixing the ratio: five magnitudes of difference equals a brightness factor of exactly 100. That makes one magnitude of difference a brightness factor of the fifth root of 100, roughly 2.512. Two magnitudes, a factor of about 6.31. Three, about 15.85. Five, exactly 100. Ten, exactly 10,000. The scale runs backwards — smaller number, brighter star — because Hipparchus put the best stars in class one and the worst in class six, and nobody has ever quite had the nerve to flip it.

Applied to our set: Sirius at −1.44 versus Capella at 0.08 is a difference of 1.52 magnitudes, which is a brightness ratio of 2.512 raised to the 1.52, or about 4.05. Sirius delivers four times as many photons per second as Capella to the same eye. Any dot-sizing rule that renders this as "27% bigger" is not merely inaccurate. It is telling a different story than the sky is telling.

This is why every serious star atlas since Argelander's Uranometria Nova in 1843 has used a non-linear rule for star sizes. Argelander understood, without a computer, that the eye needed the chart to be exaggerated where the sky was already exaggerated. The naive linear rule survives online because linear arithmetic is easier to explain in a tutorial, not because it works.

The Correction: Dot Diameter as a Function of Brightness

Second plot. Same six stars, same positions, new rule: dot area proportional to brightness rather than to magnitude. Since brightness follows the Pogson factor, that means dot diameter proportional to the Pogson factor raised to the (−magnitude/2), or equivalently, dot diameter that shrinks by roughly the square root of 2.512 — a factor of about 1.585 — per full magnitude step.

Anchored again with Sirius at 6.0mm, the second-plot diameters came out roughly: Sirius 6.0mm, Canopus 4.1mm, Arcturus 3.2mm, Rigil Kentaurus 3.1mm, Vega 3.0mm, Capella 2.9mm. Sirius now dominated the sheet the way it dominates the sky. Canopus sat as a clear second — visibly smaller than Sirius, visibly larger than the pack. The remaining four collapsed into a near-equal cluster, which is honest, because in magnitude terms they are a near-equal cluster: Arcturus, Rigil Kentaurus, Vega and Capella span just 0.13 magnitudes end to end, a brightness ratio of about 1.13. The chart said "these four are basically the same" and the sky agrees.

Held at arm's length under the same desk lamp: the eye now went to Sirius first, Canopus second, then swept across the other four in whatever order the constellations sent it. That is exactly the hierarchy a competent observer would report from a real observation. The correction was not clever. It was the arithmetic that Argelander and every serious cartographer since have used, restated for a modern plotting script. It failed the naive first pass because we let ourselves forget, for one draft, what the number in the magnitude column actually meant.

Third plot was a tuning pass: same rule, but with a lower floor on dot diameter so faint stars in future sheets do not vanish, and a soft upper cap so Sirius on a wide-field chart does not swallow its neighbours. Both are cosmetic adjustments layered on top of the correct rule, not replacements for it.

What the Test Actually Measured

The six-star test does not prove HYG v41 is accurate. It cannot. The catalogue's magnitudes were measured by photometers a great deal more precise than a printed dot, and any inaccuracy in the printed dot lives in the plotting choices, not in the source data. What the test measured was whether our pipeline — catalogue in, ink out — preserves the story the catalogue is telling.

The first plot failed because we treated a logarithmic scale as if it were linear. The second plot passed because we treated it as what it is. The third plot cleaned up the edges. None of that is unique to us. It is the same lesson every atlas-maker has had to learn, on paper, since Bayer, and it is worth writing down because the naive rule keeps coming back every time a new plotting library ships with a tutorial that reads "dot size = magnitude × constant." That tutorial is wrong. It will always be wrong. It was wrong before software existed.

If we widen the test set from six to twelve, adding six stars in the first-to-second magnitude range, the rule holds; the cluster behaviour just gets denser. If we widen it to sixty, the floor and cap start to matter more than the exponent. If we widen it to the whole naked-eye sky — roughly 9,000 stars brighter than magnitude 6.5 — we are no longer testing magnitude fidelity, we are testing projection and label collision, which are separate problems for separate sheets. The six-star anchor test is the smallest possible unit that catches the largest possible error, which is why we run it first, every time, before a chart goes to the plate.

Watch three things on any star map that lands on your desk. First: the ratio of the largest dot to the smallest dot in the naked-eye range — if it is under about a factor of four in diameter, the map is under-reporting the brightness spread the eye actually sees. Second: whether stars within 0.2 magnitudes of each other look meaningfully different in size — they should not, because the eye cannot separate them either. Third: whether Sirius, when present, reads as the anchor of its region — if it does not, the sizing rule is broken somewhere upstream, and everything else on the sheet is downstream of that break.

For readers who want the printed version of the sky these six anchors sit inside, the studio's own charts are at /shop/ — the same plotting rule, applied at scale.

FAQ

Why are brighter stars given smaller magnitude numbers?

Because Hipparchus, working around 150 BCE, sorted stars into six classes with the brightest as "first magnitude" and the faintest naked-eye stars as "sixth." The convention stuck for two thousand years. When Norman Pogson formalised the scale in 1856 by fixing five magnitudes to a brightness ratio of exactly 100, he preserved the direction rather than flipping it. So the scale runs backwards from every other measurement scale in physics: smaller number, brighter star. Negative magnitudes, like Sirius at −1.44, are simply the extension of that logic past first magnitude.

What is the actual brightness difference between magnitude −1.44 and 0.08?

The gap is 1.52 magnitudes. Because the scale is logarithmic with a base of the fifth root of 100 (roughly 2.512 per magnitude), a difference of 1.52 corresponds to a brightness ratio of about 4.05. So Sirius at −1.44 delivers roughly four times as many photons to a given eye as Capella at 0.08 does. This is why a linear dot-sizing rule fails — it compresses a factor of four in reality into a difference of about 27% in dot area on paper.

What is HYG v41 and where does its data come from?

HYG is a compiled star catalogue that stitches together three primary sources: the Hipparcos astrometric catalogue for positions and parallaxes, the Yale Bright Star Catalogue for spectral and photometric notes on naked-eye stars, and the Gliese catalogue of nearby stars for proximity data. Version 41 is the release we plotted from. It is widely used by chart-makers and hobbyist astronomers because it consolidates data that would otherwise require reconciling multiple source files by hand.

Why did the first plot look wrong even though the positions were correct?

Because the sizing rule was linear when the underlying scale is logarithmic. Correct positions place stars in the right part of the sky; correct dot sizes tell the eye which stars matter. When Sirius, four times brighter than Capella, was drawn only marginally larger, the chart accurately reported where Sirius was but under-reported what it was. The eye reads a chart as a whole, and a positionally accurate but visually flat chart tells a different story than the sky tells.

Is apparent magnitude the same as luminosity?

No. Apparent magnitude is how bright a star looks from Earth, which combines its intrinsic brightness with its distance and any intervening absorption. Luminosity, or absolute magnitude, is how bright it would look at a standard distance of ten parsecs. Canopus is intrinsically more luminous than Sirius by roughly an order of magnitude, but Sirius is far closer, so from Earth Sirius appears brighter. Star charts almost always plot apparent magnitude, because that is what the observer at the eyepiece actually sees.

Does the plotting rule change for telescopic star atlases versus naked-eye charts?

The underlying arithmetic does not. Dot area proportional to brightness — meaning diameter shrinks by roughly a factor of 1.585 per magnitude — is the correct starting rule at any scale. What changes is the floor and cap. A naked-eye chart covering stars to magnitude 6 needs a dot floor large enough to survive printing but small enough not to crowd. A telescopic atlas going to magnitude 11 or fainter needs a much finer floor and often a more compressed range so faint stars do not disappear.

Why does the zodiac appear in some star catalogues?

Because it is a coordinate band, not an astrological system. The zodiac is the roughly eighteen-degree-wide strip of sky centred on the ecliptic — the apparent yearly path of the Sun — which is where the Moon and planets also travel. Ancient astronomers divided it into twelve segments as a positional shorthand, and the IAU's modern eighty-eight-constellation system preserves those names as constellation labels. A star catalogue listing a constellation like Taurus or Leo is reporting a location, nothing more.

What magnitude limit does the human eye actually reach?

Under genuinely dark skies with a fully dark-adapted eye, the practical limit sits around magnitude 6.5, though experienced observers at high-altitude sites report seeing to 7 or slightly beyond. In a suburban sky it drops to about 4. In central-city light pollution it can shrink to 2 or 3, which means only the anchors — stars like Sirius, Vega, Arcturus, Capella — remain visible. This is why the six stars we plotted are, in most modern skies, effectively the working set the naked eye still has access to.

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