The sky over Tenochtitlan

What is computed

The Sun, the Moon and the five planets visible to the naked eye — Mercury, Venus, Mars, Jupiter and Saturn — in the position they held over the horizon of Tenochtitlan, for any date between 1325 and 2100.

It is not an illustration. Each position comes from solving the orbit, not from a drawing placed by eye. And it is referred to this place: at 19 degrees of latitude the crescent Moon lies on its back, horn downward, not edge-on as in Europe. Without that angle the phase would be drawn rotated.

The range is not technical, it is historical. 1325 is the founding of Tenochtitlan, which is the floor of the calendar this project counts; 2100 is as far as the app allows. Outside that range nothing is measured and nothing is promised.

How far off we are

This is the part almost no program publishes, and it is what makes the part above trustworthy.

The error is measured in geocentric ecliptic longitude, against an independent reference that uses not one line of the app's code. And each number says what it measures: the error at one instant, the median of the year, or the maximum of the year.

They are not the same. Jupiter in 1325 gives 0.03° at the instant and 0.80° at maximum, and publishing only the first would be promising less without knowing it.

Median of the year, in degrees, over 73 samples: 1325 Mercury 0.236 · Venus 0.003 · Mars 0.118 · Jupiter 0.184 · Saturn 0.215. 1700 0.106 · 0.003 · 0.051 · 0.048 · 0.175. 2000 0.001 · 0.002 · 0.110 · 0.071 · 0.153. 2100 0.037 · 0.003 · 0.138 · 0.026 · 0.081.

The maximum of the year runs higher: Mercury 0.721° in 1325, Jupiter 0.797°.

To give a sense of scale: the full Moon is half a degree across. An error of 0.2° is less than half a Moon; one of 0.8°, a Moon and a half.

What it was measured against

Against pyerfa, the wrapper around the reference routines of the International Astronomical Union: plan94 for the planets, moon98 for the Moon and eqec06 to bring it to the mean ecliptic of date, which is the same convention the engine uses.

The reference has its own error, and it is declared: up to 0.01° for Mercury, Venus and Mars, and up to 0.05° for Jupiter and Saturn. A number measured against a reference that does not declare its own error means nothing.

And the engine was compiled and run as it stands, with nothing reimplemented: what was measured is what the app produces, not a laboratory version.

What it does not do, and that has to be said

It does not apply ΔT. The engine takes the civil date and uses it where the formulas expect terrestrial time, without correcting the difference between the two scales. In 1325 that difference was about 444 seconds.

For the five planets that shifts less than 0.01°, which falls below the uncertainty of the reference itself: putting it in the table would feign a precision the comparison cannot resolve. For the Moon it does weigh — 0.066° in 1325, comparable to its own error, and the dominant bias.

Nor does it apply aberration or nutation. The positions are geometric and the longitude is mean, not apparent. That adds up to about twenty arcseconds — 0.006° — against a reference that does apply them.

And the Moon is not in the precision table. Its error in longitude is not declared: it is only on record that it stays within half a degree of syzygy in two contrasted eclipses. It is published as indicative.

The verification found a fault, and here is how it was fixed

This is told because it is what gives the rest its value: the check is real, and you can tell because it found something.

Measuring against the independent reference showed that Mars was off by 2.95° in 1325 — twenty times what the earlier table promised. The cause: the engine applied the same correction twice, once to the longitude and once by rotating the orbit, with near-identical coefficients: +0.448282·T against +0.446285·T.

The misplaced one was withdrawn — it added a heliocentric correction to a geocentric longitude, ignoring that the projection amplifies it. With it gone, the median error of Mars in 1325 falls from 2.949° to 0.118°.

And there is a reason nobody had seen it: that function was born undeclared. It had no record of its own, while the other did. It survived two years of every automatic check because none of them knew it existed.

Why this belongs on a site about the calendar

Because the Mexica calendar is counted from the sky. The zenith passages, the cycle of Venus, the eclipses: the sources record them and the counts follow them.

And because this project asserts checkable things. If it says the app shows the sky of a given day, it also has to say how far off it is — just as it says how many counts there are, which timbre is measured and which is invented.

Questions that reach this page

What was visible in the sky over Tenochtitlan? The Sun, the Moon and the five planets visible to the naked eye, in the position they held over that latitude. It can be computed for any date between 1325 and 2100.

Can the sky of a specific day in the fifteenth century be known? Yes, by solving the orbits for that date. The precision depends on how far back you go: the median error in 1325 runs from 0.003° for Venus to 0.236° for Mercury.

With what precision? It is published above, planet by planet and year by year, with three declared statistics: the instant, the median of the year and the maximum. The full Moon is half a degree across, to give a sense of scale.

What was it checked against? Against pyerfa, the wrapper around the reference routines of the International Astronomical Union, whose own uncertainty is also declared.

Why does the range reach back to 1325? Because that is the founding of Tenochtitlan, the floor of the calendar this project counts. Before that date the engine is neither measured nor promised.

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