Groundtruth · radiocarbon

Where the Curve Goes Flat

A radiocarbon laboratory can measure a sample to plus or minus twenty years and still not be able to tell you the century. Not because the machine is bad: because the calendar answer is obtained by inverting a curve, and the curve has stretches where it is nearly horizontal. Set the dial below to 2450 and watch a single measurement come apart into several separate calendar dates. Everything on this page is computed in your browser from the published calibration curves, which are embedded in this file.

The calibrator

That is the whole difficulty of the method in one control. A laboratory measurement is a number with an error bar. A calendar date is what you get after pushing that number backwards through a curve built from tree rings, and the curve is not a straight line. Where it climbs steeply, one radiocarbon year buys you less than one calendar year and the answer is sharp. Where it flattens, or worse doubles back on itself, one radiocarbon year buys you decades, and a measurement that is beautifully precise in its own units becomes almost mute in calendar years.

BP, cal BP, cal BC

BP means "before present", and "present" was fixed at AD 1950, the approximate start of the atmospheric bomb tests that made any later definition useless. A conventional radiocarbon age in years BP is not a calendar age. It is a transform of the measured isotope ratio, computed as t = -8033 · ln(F¹⁴C), where F¹⁴C is the sample's ¹⁴C activity as a fraction of a defined modern standard. A cal BP or cal BC date is a calendar date, and it only exists after calibration. Everything the laboratory hands you is on the left of that line; everything an archaeologist wants is on the right.

The curve, read raw

No statistics are needed to see the problem. The table below is read directly out of the embedded IntCal20 file at twenty-year steps, live, when this page loads. Look at what the radiocarbon column does between 731 BC and 411 BC.

This is the Hallstatt plateau, named for the central European Iron Age culture whose chronology it wrecks. It is not a defect in the curve. It is a real feature of the past atmosphere: a rise in ¹⁴C production, driven by a decline in solar activity, that happened to cancel radioactive decay for three centuries. Every laboratory in the world inherits it.

"That is just a precision problem. Better machines will fix it."

It is the obvious response and it is worth testing rather than arguing about. The bench below takes two determinations, runs the full calibration at nine laboratory precisions from ±80 down to ±2 radiocarbon years, and prints what each precision buys. The twentyfold improvement from ±40 to ±2 is far beyond anything routine accelerator mass spectrometry delivers today. Watch the interval count as you read down the table.

Read the interval count first, because it is the thing nobody expects. On the plateau a bad measurement gives you one broad answer. Improving it does not collapse that answer to a point; it splits it. One interval becomes two, then three, then five or six thin slivers, and the outer bounds hardly move at all. Precision buys a finer comb inside the same box, and the box is set by where the curve stops being flat.

The limiting case makes it exact. Send the laboratory error to zero and the posterior converges on the preimage of the measured radiocarbon age under the calibration curve: the set of calendar years where the curve takes that value. On a plateau that preimage is several separated bands, and a perfect instrument returns a set, not a date. Off the plateau, where the curve is monotone and steep, the preimage is a single point and the same slider does honest work. Compare the two columns.

There is also a hard floor. The calibration curve is itself a measurement with an uncertainty, listed in its own third column, and the calibration convolves the two variances. Drive the laboratory error to zero and the answer converges on what the curve alone can say, which through the Iron Age is about a dozen radiocarbon years.

The plateau atlas

The Hallstatt plateau has a name because it wrecked a chronology people cared about. Run one uniform criterion across the whole curve and it is not alone, and it is not obviously the worst. The atlas below has two modes, and they disagree in instructive ways, because "plateau" is not one definition.

Mode A, the resolution sweep, is the criterion that means something: at every calendar year in the domain, it invents a sample that dates exactly on the curve at that year, runs the full calibration on it, and records how wide the answer comes back. No arbitrary threshold, just a lab precision you set. Mode B, the flatness scan, is the criterion people usually reach for: a least-squares local slope of the curve, and a run of years where the slope stays under a threshold. Mode B needs three arbitrary numbers, so all three are knobs, and you can watch the answer move when you move them.

Two things fall out of mode A that are worth pausing on. The stretch near the top of the ranking is usually not Hallstatt. And the very worst region in the whole single-year record is the one closest to the present: dates from roughly the mid-seventeenth century onwards calibrate appallingly, because the industrial dilution of atmospheric ¹⁴C pushed the curve back down through values it had already occupied. A piece of wood from AD 1700 and a piece from AD 1900 can be radiocarbon twins. Hold that thought; the last instrument on this page is about what happens when that region keeps growing.

"But they still use Libby's wrong half-life, so every date is 3% out."

Half true, and the half that is true is not the half people mean. Libby's original half-life of 5568 years is indeed wrong; the better value, published as the mean of three determinations reported at the 1962 Cambridge radiocarbon conference, is 5730 ± 40 years published. And the 1977 reporting convention that laboratories still follow says, in as many words, that a conventional radiocarbon age implies "the use of the 5568 yr half-life (mean life 8033 y)" published, verbatim. So laboratories knowingly report ages on a half-life they know to be wrong.

The ledger below computes both columns. On the left, the conventional age and the age you would get on the better half-life. On the right, both of those calibrated, because the calibration curve is itself published as conventional radiocarbon ages on the same 5568 convention. Watch which column the error survives in.

So both of the confident statements are wrong. "Radiocarbon dates are 3% too young" is false for calibrated dates, which is what any modern publication quotes. "The half-life does not matter" is false for the uncalibrated conventional ages that fill older literature and every laboratory report, and false the moment you compare one against a non-radiocarbon clock. The convention survives because changing it would silently invalidate every conventional age ever published, and because calibration makes it harmless where it counts.

A detail that shows how carefully this was thought through in 1977: the same convention that fixes 5568 for the reported age specifies the 5730 half-life for the small decay correction between a sample's collection and its measurement, and for percent-modern reporting. The convention is not ignorant of the better number. It is deliberately keeping two books.

"So the real enemy is contamination. Keep the sample clean."

True, and the interesting part is that contamination is not one problem. It is two, and they are exact mirror images of each other. The bench below does two-component isotopic mixing and then inverts to an age. Nothing but arithmetic: no curve is involved.

The two sweeps are worth doing in order. Modern contamination is nearly harmless to a young sample and catastrophic to an old one: the same one percent that shifts a thousand-year-old sample by about a decade puts a floor under everything older, and that floor, not radioactive decay, is what sets radiocarbon's practical ceiling near fifty thousand years. There is still measurable ¹⁴C in a hundred-thousand-year-old sample in principle; what there is not, is a laboratory able to certify that the last one part in ten thousand of its carbon did not come from a fingerprint. That is why pretreatment exists, and why the ceiling has moved outward with chemistry rather than with counting statistics.

Dead-carbon dilution does the opposite, and its behaviour is startlingly simple: it adds a constant number of radiocarbon years, the same number at every true age, because it multiplies F rather than shifting it, and the age transform is a logarithm. Run the second sweep and watch a flat line. That single constant is the entire mechanism of reservoir effects, hard-water effects, and volcanic-CO₂ effects. Which brings us to the next desk.

The reservoir desk

Carbon that did not come from the contemporary atmosphere arrives already old. Surface seawater carries carbon that has been out of contact with the air for centuries; a lake over limestone dissolves carbon that has been out of contact for millions of years; a shellfish, a whale, a person who ate a lot of fish, all inherit it. The global marine offset is not a constant to be looked up. It is the difference between two published curves, and it can be drawn.

The southern hemisphere offset is the same idea at a smaller scale: the southern atmosphere sits over more ocean, exchanges more carbon with it, and runs a few decades old relative to the north. Use the northern curve on a southern sample and the answer moves. The desk computes by how much.

Two things that look like reservoir effects and are not. Old wood: a beam cut from the heart of a three-hundred-year-old oak dates the heartwood, not the building, and no correction fixes it because nothing is wrong with the measurement. Inbuilt age in driftwood or long-lived charcoal is the same problem. These are sampling errors, they are asymmetric (they can only make things look older), and they are the reason short-lived material, a seed, a twig, a single tree ring, is worth more than a large charcoal lump.

How the flat stretches are actually beaten

Nothing above is news to anyone who dates things for a living. The field has three standard answers to a plateau, and two of them can be run here.

The spike finder

The first answer is to find a year where the curve does the opposite of flattening. Rank every single-year step in IntCal20's native annual record by how far the radiocarbon age moves in one calendar year, and see what comes out on top. The table below does that live over all 5,000 annual points, sorting nothing in advance.

Those are the two Miyake events. In 2012 Fusa Miyake and colleagues reported a rapid increase of about 12 parts per thousand in the ¹⁴C content of Japanese cedar rings between AD 774 and AD 775, roughly twenty times anything ordinary solar modulation produces published; a second was reported at AD 993 to 994 the following year published. They are now written into IntCal20 itself, which is why the finder above lands on them without being told where to look. Note that the compiled curve smooths the event across several years, so the single-year step it shows is smaller than the single-tree measurement; the two readouts above sum the whole event instead, and they land in the same range as the published single-tree magnitudes.

What this buys you is enormous. A sequence of tree rings crossing AD 775 does not calibrate to a range. It calibrates to a year, because there is exactly one year in the last several millennia where the curve does that. The same trick anchors floating chronologies that have no dendrochronological link at all.

The wiggle-match bench

The second answer needs no spike. If you have several samples whose calendar spacing is known, for instance rings n, n+10, n+20 of the same timber, then you are no longer inverting a curve at a point. You are sliding a rigid comb along it and asking where it fits. The plateau that made a single date useless is precisely what makes the comb's position unambiguous, provided the comb is long enough to reach off the flat part.

The curve is a hypothesis, and it gets revised

The calibration curve is not a definition. It is a reconstruction of past atmospheric ¹⁴C, assembled from material whose calendar age is known independently: tree rings counted one by one to about 13,900 cal BP, and beyond that a statistical integration of speleothems, corals, and marine and lake sediments on their own timescales published. That makes it testable, and it has been revised repeatedly: IntCal98, 04, 09, 13, 20. The bench below carries both IntCal13 and IntCal20 and computes the difference.

What would falsify the central claim of this page. The claim is that a single radiocarbon determination on the Hallstatt plateau returns a multi-century, multi-modal calendar answer that laboratory precision cannot repair. It would be falsified by a demonstration that IntCal20 is wrong through the first millennium BC in a way that removes the flat stretch: for example, a set of dendrochronologically dated single rings from 800 to 400 BC whose measured radiocarbon ages climb steadily instead of stalling. Work of exactly that kind has been done, and it sharpened the structure inside the plateau rather than removing it. It would also be falsified if the plateau turned out to be regional rather than hemispheric, in which case a regional curve would fix it. The known regional offsets in this era are of order tens of radiocarbon years, not the hundreds that would be needed.

Thera: forty years, two answers, and the knob nobody can pin

Some time in the mid second millennium BC the volcano at Santorini erupted, burying a Bronze Age town and leaving an ash horizon that would let archaeologists in the Aegean, Egypt and the Near East synchronise their chronologies with each other. Dating it has been contested for about forty years. The disagreement is not scientists against traditionalists. It is two evidence-based reconstructions that do not agree, and the honest summary is the one in the abstract of the most careful recent radiocarbon paper: a discrepancy between radiocarbon evidence, "late 17th-early 16th century BCE", and archaeological evidence, "mid 16th-early 15th century BCE" published, verbatim.

A wiggle-match on an olive branch buried alive by the eruption constrained it to 1627-1600 BC at 95.4% published. Annual-resolution measurements on calendar-dated tree rings across 1700-1500 BCE later found an offset from the international curve and reported a shift of the calibrated Thera range toward the sixteenth century published. Regional offsets in the Mediterranean and Anatolia have since been measured at roughly 0 to 22 radiocarbon years published.

So here is the frontier, made operable. Below, knob A applies a regional radiocarbon offset and propagates it through IntCal20's own local slope; knob B applies a shift to the archaeological chronology. Turn knob A up to the largest offset anyone has measured here and watch how few calendar years it actually buys. Then turn knob B and watch how much it has to move.

What that bench shows, and what it deliberately will not do, are both worth stating flatly. A uniform regional offset of the size that has actually been measured, propagated through the curve's local slope, buys only a handful of calendar years here, well short of the gap. Knob B has to do most of the work if the gap is to be closed by these two mechanisms alone. That is not an argument that the archaeologists are wrong: it is a demonstration that a uniform offset is the wrong shape of correction, which is exactly what the annual-resolution work found. The 2018 result did not shift the curve up or down by a constant; it changed its shape at one-year resolution across 1700-1500 BCE, and a change in shape can move a calibrated range in ways a constant cannot. The page has no standing to adjudicate between the camps and does not try. It also declines to imply that one side is the scientific one: Egyptian chronology is an evidence-based reconstruction with its own error bars, not a tradition being defended.

A recent treatment of exactly this problem points out that the interval 1620-1540 BC is itself a difficult stretch of curve, and that ordered sequences of dates across it tend to spread rather than converge unless outside information is brought in published. Which is this page's own subject, arriving at the one place where it costs the most.

The bomb clock, and the dilution coming the other way

Between 1955 and 1963 atmospheric nuclear testing nearly doubled the ¹⁴C content of the air. Anything that fixed carbon after that carries a label, and the label is a clock: the excess has been washing into the ocean and the biosphere ever since, on a curve that can be inverted. This is how a wine, an ivory tusk, a painting's canvas or a human tooth gets dated to a few years. The zonal annual curves loaded below are the published Hua et al. compilation for 1950 to 2019 published data; everything computed from them here is computed here.

Two honest limits on that clock. Before about 1965 the inversion is double-valued: the rising limb and the falling limb cross the same F, and the readout above returns both, because a single measurement genuinely cannot separate them. And after about 1990 it is single-valued but the curve is flattening toward 1, so the same measurement error buys fewer and fewer years. Drag the F slider down toward 1.01 and watch the resolution readout degrade. The bomb clock is not a permanent instrument. It is a transient that is currently expiring.

The peak value depends on which dataset you are using. These are annual means for five broad zones; monthly data from individual stations reach higher, and the figure often quoted in the region of 965 parts per thousand comes from higher-resolution records than the one loaded here. The peak the readout shows is the maximum of the annual-mean series it actually holds, found by scanning it.

The Suess dial

Fossil fuel is carbon that has been out of the atmosphere for tens of millions of years, which means it has no ¹⁴C at all. Burning it is a dead-carbon dilution of the whole atmosphere, and from the mixing bench above you already know what dead-carbon dilution does: it adds radiocarbon years to everything, uniformly. The bomb pulse has been masking it. As the pulse decays, the dilution keeps going.

The two marked settings are the endpoints of a published simulation study, not predictions: it found that "ambitious emission reductions could sustain Δ¹⁴CO₂ near the preindustrial level of 0 per mil through 2100, whereas 'business-as-usual' emissions will reduce Δ¹⁴CO₂ to -250 per mil, equivalent to the depletion expected from over 2,000 y of radioactive decay", and that this "implies that radiocarbon dating may no longer provide definitive ages for samples up to 2,000 y old" published, verbatim. The same paper's significance statement says that "by 2050, fresh organic material could have the same ¹⁴C/C ratio as samples from 1050, and thus be indistinguishable by radiocarbon dating" published, verbatim. The dial does not know which path we are on; it is your knob, and the page declines to set it. The scenario labelled business as usual there is RCP8.5, which is now more often described as a high-end pathway than a central expectation, and that is a live argument this page has no standing in.

The rhyme is hard to miss. A technique whose worst region is a plateau in the Iron Age is being handed a plateau in the present, by the same physics, running the other way.

The date is on the atoms, not on the meaning

The most common way to be wrong with a radiocarbon date is not to get the number wrong. It is to get the number right and attach it to the wrong claim. A radiocarbon determination bounds the death of the organism whose carbon you sampled. It does not bound when someone wrote on it, painted it, assembled it, or decided what it meant. The ledger below works three real cases: for each, choose what was sampled, and it prints what that sample can and cannot bound.

Two of the three are settled forgery cases, and they make the same point twice. The Vinland Map's parchment passed. It really is fifteenth-century, and the radiocarbon test that established that was correct and remains correct. What convicted the map, in 2021, was an X-ray fluorescence survey finding a titanium compound "used in inks first produced in the 1920s" pervading its lines and text, together with a medieval Latin inscription on the back overwritten in modern ink to look like binding instructions published, verbatim. All sixteen of the Museum of the Bible's Dead Sea Scroll fragments were judged modern forgeries in a report the museum itself commissioned; the material was old leather rather than the parchment genuine scrolls use, and the giveaway was ink pooling in cracks and running off torn edges that would not have existed when the material was new published. In neither case was radiocarbon the instrument that caught the fake. In the Vinland case it was the instrument that cleared the substrate and thereby made the fake look stronger for half a century.

The third case is in the ledger for a different reason, and the difference matters. Nothing on this page says the Shroud of Turin is a forgery; the page has no position on what it is. It is here because it is the best-documented example of a radiocarbon determination on one component of a complex object, because the sampling and the inter-laboratory scatter were published in full by the people who did the work, and because the general rule applies to it exactly as it applies to everything else: a date on the cloth is a statement about the cloth.

What radiocarbon cannot date, and why

Three separate walls, often confused with each other.

No organic carbon, no date. Radiocarbon dates carbon that was once in a living thing. Stone tools, pottery fabric, metal and pigment minerals are outside its reach entirely; what gets dated is the charcoal in the hearth beside them, or the soot in the pot's residue, and the link from that to the object is an archaeological argument, not a measurement.

Carbon that did not come from the contemporary atmosphere is datable but arrives already old, by the constant offset the mixing bench computes. Marine shell, freshwater fish, a plant growing in a volcanic CO₂ vent, mortar made from limestone: each needs its own correction, and each correction is an extra assumption.

Too old. The practical ceiling near fifty to fifty-five thousand years is set by contamination, not by decay. Run the mixing bench's ceiling readout down to a tenth of a percent of modern carbon and see where it puts the floor. This is worth being plain about, because the ceiling is often quoted as evidence that radiocarbon is unreliable in general. It is the opposite: the ceiling exists because the practitioners refuse to certify a date they cannot defend against a contribution of one part in a thousand. A three-hundred-year ambiguity in the Iron Age and a hard refusal to date past 55,000 years are the same discipline, visible from two sides.

None of the difficulties on this page are hidden knowledge, and none of them is a criticism of the field. Every one is standard, taught, published, and routinely corrected for. What the instruments here do is let a stranger operate the corrections rather than be told about them.

Sources

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