========================================================================== HOW ANYBODY KNOWS HOW OLD ANYTHING IS -- CLUB CALCULATION ========================================================================== Science Journaling Club. Seed = 20260913. Carbon-14 half-life, physics value : 5700 yr (mean life 8223.4 yr) Carbon-14 half-life, Libby convention: 5568 yr (mean life 8032.9 yr) Carbon-14 atoms per gram of modern carbon = 5.8649e+10 independent cross-check from the isotope ratio: 1 g of carbon holds 6.022e23/12.011 = 5.0139e+22 atoms; at 14C/12C = 1.20e-12 that is 6.0166e+10 AMS overall efficiency assumed : 1.0e-03 counted per atom loaded Laboratory background assumed : 0.0015 +/- 0.0005 fraction modern ========================================================================== PART A -- THE EXPONENTIAL LAW, BUILT OUT OF COIN FLIPS ========================================================================== No atom knows how old it is. An atom that has sat there for a billion years is exactly as likely to decay in the next second as one made this morning. That single fact -- memorylessness -- is the whole of the physics. Give every surviving atom the same probability p of decaying in each time step of length dt. The number surviving after one step is then a binomial draw, and after k steps it is a binomial draw from a binomial draw from a binomial draw. We never write exp(-x) into the simulation. We let the coin flips produce it. survival probability per step s = exp(-lambda*dt) decay probability per step p = 1 - s The analytic expectation for a population of N0 atoms after time t: mean = N0 * exp(-lambda*t) variance = N0 * exp(-lambda*t) * (1 - exp(-lambda*t)) [binomial] A1. Simulated population versus the analytic curve -------------------------------------------------- N0 = 100000 atoms, 40 independent runs, dt = 0.0250 half-lives t/T_half sim mean analytic ratio 0.0 100000.0 100000.0 1.000000 1.0 49978.3 50000.0 0.999567 2.0 24983.3 25000.0 0.999333 3.0 12501.9 12500.0 1.000150 4.0 6253.0 6250.0 1.000476 5.0 3118.4 3125.0 0.997896 A2. The SCATTER also comes out right, which is the harder test -------------------------------------------------------------- N0 = 1000 atoms, 4000 runs. The binomial prediction for the standard deviation of the survivor count is sqrt(N0*f*(1-f)), f = exp(-lam*t). t/T_half sim mean ana mean sim s.d. ana s.d. 0.50 707.14 707.11 14.177 14.391 1.00 500.40 500.00 15.797 15.811 2.00 250.34 250.00 13.694 13.693 3.00 125.24 125.00 10.390 10.458 5.00 31.23 31.25 5.553 5.502 A3. How large does N have to be before the law looks like a law? ---------------------------------------------------------------- For each population size we run 4000 independent worlds, find the first time the count drops to N0/2, and report the spread of that estimate. The true half-life is 1.000 by construction. N0 mean T_half s.d. s.d./mean 1/sqrt(N0) 10 0.9439 0.4274 0.4528 0.3162 100 1.0031 0.1448 0.1443 0.1000 1000 1.0106 0.0455 0.0450 0.0316 10000 1.0127 0.0163 0.0161 0.0100 100000 1.0125 0.0125 0.0123 0.0032 The fourth column falls by a factor of about 3.16 each time N0 rises by a factor of 10. That is 1/sqrt(N), and it is the only reason the exponential ever looks smooth. Decay is not smooth. A gram of anything simply holds an absurd number of atoms. A4. FIGURE 1 DATA -- three worlds and one curve ----------------------------------------------- fraction surviving, one single run at each N0, every half-life: t/T_half N0=100 N0=1e3 N0=1e5 analytic 0.0 1.0000 1.0000 1.0000 1.00000 0.5 0.7100 0.7140 0.7067 0.70711 1.0 0.4900 0.5010 0.4990 0.50000 1.5 0.3600 0.3560 0.3526 0.35355 2.0 0.2800 0.2600 0.2493 0.25000 2.5 0.1800 0.1850 0.1771 0.17678 3.0 0.1300 0.1150 0.1247 0.12500 3.5 0.0900 0.0830 0.0878 0.08839 4.0 0.0500 0.0570 0.0627 0.06250 4.5 0.0400 0.0450 0.0445 0.04419 5.0 0.0300 0.0320 0.0311 0.03125 Full N0=100 trace, every 5 steps (this is the ragged line in Fig. 1): 1.00 0.93 0.83 0.78 0.71 0.67 0.60 0.55 0.49 0.47 0.42 0.40 0.36 0.33 0.32 0.30 0.28 0.23 0.22 0.21 0.18 0.16 0.16 0.13 0.13 0.12 0.12 0.11 0.09 0.09 0.09 0.06 0.05 0.04 0.04 0.04 0.04 0.04 0.03 0.03 0.03 A5. What a half-life is NOT --------------------------- 1000 atoms, exactly one half-life elapsed, 4000 worlds. survivors: mean 499.91, min 432, max 563 worlds in which exactly 500 atoms survived: 111 of 4000 = 2.77% A half-life is not a promise that half will be gone. It is the centre of a distribution that has a width. P(one given atom survives one half-life) = 0.500000 P(one given atom survives ten half-lives) = 0.000977 Expected survivors of 1000 after ten half-lives = 0.977 ========================================================================== PART B -- WHY RADIOCARBON RUNS OUT, AND WHERE ========================================================================== An accelerator mass spectrometer does not wait for decays. It counts atoms. Load m grams of carbon, and the number of carbon-14 atoms that actually reach the detector is C = eff * m * n0 * F with F = exp(-t/tau), the fraction modern Counting is Poisson, so the uncertainty on C is sqrt(C), and the fractional uncertainty on F is 1/sqrt(C). An age is tau*ln(1/F), so a FRACTIONAL error on F becomes an ABSOLUTE error on the age: sigma_age = tau * sigma_F / F Stare at that line. A 1 percent error on F is about 82 years whether the sample is 500 years old or 40,000. The uncertainty grows with age not because the arithmetic degrades but because F itself shrinks, so a fixed background eats a larger and larger share of it. B1. Counting uncertainty alone, with no background at all --------------------------------------------------------- counts available at F=1: 5.8649e+07 per gram of carbon age (yr) counts, 1 mg sigma_age 1 mg sigma_age 10 mg 0 58649.2 34.0 10.7 5000 31930.1 46.0 14.6 10000 17383.6 62.4 19.7 20000 5152.5 114.6 36.2 30000 1527.2 210.4 66.5 40000 452.7 386.5 122.2 50000 134.2 709.9 224.5 60000 39.8 1304.0 412.4 On counting statistics alone a 1 mg sample would still be good to +/-710 years at 50,000. Counting is not what stops us. B2. Now put the background back in ---------------------------------- background = 0.0015 +/- 0.0005 fraction modern age F_true F/bkg sig 0.1mg sig 1mg sig 10mg 0 1.000e+00 666.67 107.5 34.2 11.5 10000 2.964e-01 197.60 198.2 64.0 24.2 20000 8.785e-02 58.57 368.3 124.7 59.4 30000 2.604e-02 17.36 702.3 267.9 172.1 40000 7.718e-03 5.15 1438.1 679.9 549.2 45000 4.202e-03 2.80 2163.6 1153.2 997.4 50000 2.288e-03 1.53 3402.3 2016.2 1820.4 55000 1.245e-03 0.83 5595.3 3597.2 3332.1 60000 6.781e-04 0.45 9560.1 6498.8 6108.8 The sample's own radiocarbon equals the laboratory background at t = tau*ln(1/bkg) = 53471 years. Past there you are subtracting two numbers of the same size and keeping the difference, which is the oldest way there is to get a wrong answer with a straight face. B3. Why a bigger sample stops helping ------------------------------------- Share of the total variance that comes from counting rather than from the background subtraction: age 0.1 mg 1 mg 10 mg 0 99.9% 98.6% 87.2% 20000 98.4% 85.9% 37.9% 30000 94.9% 65.3% 15.8% 40000 86.3% 38.6% 5.9% 50000 72.1% 20.5% 2.5% By 40,000 years even a ten-milligram sample is background-limited. Ten times more carbon buys almost nothing. A cleaner laboratory buys everything. This is the single most useful line in Part B. B4. Contamination, which is the real wall ----------------------------------------- A sample with NO carbon-14 of its own, holding a fraction c of modern carbon picked up from anywhere at all, reports a conventional age of -tau_Libby * ln(c). It reports it confidently. contamination apparent age (BP) 10.00% 18496 5.00% 24064 2.00% 31425 1.00% 36993 0.50% 42561 0.20% 49921 0.10% 55489 0.05% 61057 One percent of modern carbon in a genuinely ancient sample produces 36993 BP. Not a warning. Not an error flag. A date. B5. The same contamination, at every age ---------------------------------------- Measured conventional age of a sample of true age t contaminated with a fraction c of MODERN carbon: true age c=0.1% c=0.5% c=1% c=5% 1000 999 995 989 947 5000 4993 4965 4931 4660 10000 9980 9901 9804 9064 20000 19912 19568 19157 16464 30000 29678 28506 27247 21059 40000 38916 35634 32821 23078 50000 46722 39892 35555 23768 Read that table across a row, then down a column. Across: a thousand-year-old sample barely notices. Down: the identical laboratory slip that costs you a decade at 1,000 years costs you tens of thousands at 50,000. Symmetric errors, wildly asymmetric damage. B6. Old carbon going the other way ---------------------------------- A young sample diluted with DEAD carbon reads too old: true age c=1% dead c=5% c=20% 0 81 412 1792 1000 1081 1412 2792 5000 5081 5412 6792 10000 10081 10412 11792 A mollusc feeding on limestone-derived carbonate, or a hedge growing beside a motorway in 1970, is doing exactly this to itself. B7. FIGURE 2 DATA -- age uncertainty against true age, three masses ------------------------------------------------------------------- total 1-sigma age uncertainty in years, background included age 0.1 mg 1 mg 10 mg counting 1mg 0 108 34 12 34 2500 125 40 14 40 5000 146 47 16 46 7500 170 55 20 54 10000 198 64 24 62 12500 231 75 30 73 15000 270 89 37 85 17500 315 105 47 98 20000 368 125 59 115 22500 431 149 76 133 25000 506 180 99 155 27500 595 218 130 181 30000 702 268 172 210 32500 832 332 229 245 35000 992 417 305 285 37500 1189 530 409 332 40000 1438 680 549 387 42500 1755 882 740 450 45000 2164 1153 997 524 47500 2698 1520 1347 610 50000 3402 2016 1820 710 52500 4340 2688 2462 826 55000 5595 3597 3332 962 57500 7283 4829 4511 1120 60000 9560 6499 6109 1304 ========================================================================== PART C -- TREE RINGS CORRECTING RADIOCARBON ========================================================================== A tree ring has a calendar age you get by counting and a radiocarbon age you get by measuring. Do both on the same ring and you have one point on a graph that nobody had to assume anything to draw. Do it for twelve thousand rings and you have IntCal. The two ages are not equal, and that is the interesting part. Atmospheric carbon-14 production rises and falls with the Sun's magnetic activity and with the Earth's own field, and the ocean breathes carbon in and out on a schedule of its own, so the radiocarbon clock gains and loses time against the calendar. The rings record exactly how much. A raw radiocarbon age is a reading off an unreliable clock whose error has been written down, year by year, by trees. IntCal20 loaded: 1811 knots, 0 to 55000 cal BP C1. The tree-ring sequence itself (real IntCal20 knots) ------------------------------------------------------- Each row is: rings counted back from a known year, and the conventional radiocarbon age measured on the wood of that ring. cal BP 14C age BP 1-sigma offset 0 199 11 +199 500 429 10 -71 1000 1126 13 +126 2000 2070 14 +70 2500 2482 14 -18 3000 2889 16 -111 4000 3705 17 -295 5000 4439 15 -561 7000 6142 20 -858 9000 8063 19 -937 11000 9556 22 -1444 13000 11057 20 -1943 That last column is the whole reason calibration exists. A radiocarbon age is not a calendar age and was never supposed to be one. C2. Offset between raw radiocarbon age and calendar age, whole curve -------------------------------------------------------------------- over 0-14,000 cal BP : largest |offset| = 2107 yr, at 12580 cal BP 0- 2000 cal BP : offset -140 to +199 yr, mean +58 2000- 5000 cal BP : offset -667 to +91 yr, mean -253 5000- 8000 cal BP : offset -1007 to -561 yr, mean -799 8000- 11000 cal BP : offset -1460 to -726 yr, mean -1009 11000- 14000 cal BP : offset -2107 to -1346 yr, mean -1774 14000- 30000 cal BP : offset -4459 to -1872 yr, mean -3619 30000- 55000 cal BP : offset -5332 to -2085 yr, mean -3723 At 55000 cal BP the curve reads 50100 radiocarbon BP, so a raw radiocarbon age taken at face value there is 4900 years too young. C3. Calibrating single measurements against the real curve ---------------------------------------------------------- Each measurement is quoted as radiocarbon age +/- 1 sigma. The output is the 95.4% calendar range, in calendar years BP. 500 +/- 30 -> 501-549 median 525 cal BP, total span 49 yr, 1 block(s) 1000 +/- 30 -> 797-871; 898-956 median 912 cal BP, total span 134 yr, 2 block(s) 2450 +/- 30 -> 2362-2543; 2559-2618; 2632-2701 median 2517 cal BP, total span 312 yr, 3 block(s) 2450 +/- 15 -> 2366-2399; 2401-2519; 2590-2616; 2636-2696 median 2515 cal BP, total span 241 yr, 4 block(s) 3000 +/- 30 -> 3074-3255; 3295-3329 median 3188 cal BP, total span 217 yr, 2 block(s) 5000 +/- 30 -> 5605-5631; 5650-5760; 5814-5891 median 5722 cal BP, total span 216 yr, 3 block(s) 8000 +/- 30 -> 8660-8660; 8725-8748; 8753-9000 median 8874 cal BP, total span 273 yr, 3 block(s) 10000 +/- 30 -> 11284-11295; 11306-11640; 11670-11689 median 11472 cal BP, total span 367 yr, 3 block(s) 12000 +/- 30 -> 13794-13888; 13909-14000 median 13885 cal BP, total span 187 yr, 2 block(s) C4. Plateau hunting: where does one measurement buy you nothing? ---------------------------------------------------------------- For every radiocarbon age from 200 to 12,000 BP in 10-year steps, with a laboratory error of +/-25 yr, calibrate and record the total width of the 95.4% calendar range. median width : 143 calendar years 10th pct : 76 yr 90th pct : 236 yr worst : 419 yr, at 10360 radiocarbon BP best : 33 yr, at 10780 radiocarbon BP The eight worst radiocarbon ages to be handed, by calendar width: 10360 BP -> 419 calendar years in 4 block(s): 12004-12202; 12218-12275; 12294-12334; 12345-12465 8840 BP -> 365 calendar years in 4 block(s): 9740-9741; 9753-9965; 9985-10045; 10058-10146 9600 BP -> 348 calendar years in 4 block(s): 10774-10974; 10995-11049; 11052-11119; 11121-11144 7950 BP -> 321 calendar years in 3 block(s): 8647-8678; 8690-8816; 8822-8983 2460 BP -> 309 calendar years in 4 block(s): 2368-2390; 2406-2546; 2554-2619; 2626-2704 4440 BP -> 304 calendar years in 4 block(s): 4883-4932; 4957-5069; 5103-5134; 5169-5277 3850 BP -> 229 calendar years in 3 block(s): 4155-4207; 4222-4358; 4364-4402 7370 BP -> 227 calendar years in 3 block(s): 8036-8143; 8160-8215; 8253-8315 And the five best: 10780 BP -> 33 calendar years 2600 BP -> 35 calendar years 490 BP -> 40 calendar years 4280 BP -> 41 calendar years 8570 BP -> 50 calendar years C5. FIGURE 4 DATA -- the Hallstatt plateau in full -------------------------------------------------- measurement 2450 +/- 30 BP 95.4% blocks : 2362-2543 cal BP; 2559-2618 cal BP; 2632-2701 cal BP in calendar years: 594 BC to 413 BC; 669 BC to 610 BC; 752 BC to 683 BC total span 312 calendar years in 3 block(s) single most probable calendar year: 2663 cal BP = 714 BC block 2362-2543 cal BP holds 54.4% of the posterior block 2559-2618 cal BP holds 15.5% of the posterior block 2632-2701 cal BP holds 25.5% of the posterior Curve shape through the plateau (every 10 calendar years): cal BP 14C age BP 2300 2211 2310 2244 2320 2256 2330 2268 2340 2301 2350 2352 2360 2396 2370 2419 2380 2423 2390 2414 2400 2409 2410 2417 2420 2420 2430 2425 2440 2431 2450 2428 2460 2425 2470 2441 2480 2434 2490 2446 2500 2482 2510 2480 2520 2492 2530 2490 2540 2496 2550 2512 2560 2498 2570 2495 2580 2500 2590 2488 2600 2462 2610 2451 2620 2510 2630 2503 2640 2477 2650 2462 2660 2446 2670 2459 2680 2463 2690 2474 2700 2496 2710 2517 2720 2534 2730 2573 2740 2588 2750 2621 2760 2672 2770 2692 2780 2710 2790 2734 2800 2739 2810 2739 2820 2733 2830 2720 2840 2719 2850 2749 2860 2771 2870 2782 C6. How much does calibration actually move a date? --------------------------------------------------- Difference between the calibrated median and the raw radiocarbon age, the raw age being read as if it were calendar years: raw 14C BP calibrated med shift 95.4% width 500 525 +25 40 1000 917 -83 119 2000 1937 -63 131 3000 3188 +188 195 4000 4476 +476 104 5000 5720 +720 185 6000 6836 +836 168 7000 7837 +837 162 8000 8876 +876 250 9000 10199 +1199 82 10000 11471 +1471 344 11000 12907 +1907 213 12000 13883 +1883 181 C7. FIGURE 3 DATA -- the calibration curve against the 1:1 line --------------------------------------------------------------- cal BP 14C age BP offset 0 199 +199 250 110 -140 500 429 -71 750 879 +129 1000 1126 +126 1250 1284 +34 1500 1583 +83 1750 1883 +133 2000 2070 +70 2250 2209 -41 2500 2482 -18 2750 2621 -129 3000 2889 -111 3250 3057 -193 3500 3322 -178 3750 3492 -258 4000 3705 -295 4250 3863 -387 4500 3994 -506 4750 4176 -574 5000 4439 -561 5250 4475 -775 5500 4799 -701 5750 5043 -707 6000 5276 -724 6250 5385 -865 6500 5743 -757 6750 5950 -800 7000 6142 -858 7250 6293 -957 7500 6603 -897 7750 6937 -813 8000 7178 -822 8250 7449 -801 8500 7712 -788 8750 7933 -817 9000 8063 -937 9250 8224 -1026 9500 8498 -1002 9750 8774 -976 10000 8872 -1128 10250 9150 -1100 10500 9286 -1214 10750 9512 -1238 11000 9556 -1444 11250 9872 -1378 11500 10030 -1470 11750 10121 -1629 12000 10306 -1694 12250 10366 -1884 12500 10541 -1959 12750 10823 -1927 13000 11057 -1943 13250 11389 -1861 13500 11656 -1844 13750 11841 -1909 14000 12030 -1970 and coarsely to the end of the published curve: 15000 12595 -2405 20000 16566 -3434 25000 20706 -4294 30000 25664 -4336 35000 30610 -4390 40000 34918 -5082 45000 42497 -2503 50000 47531 -2469 55000 50100 -4900 ========================================================================== PART D -- THE LONG CLOCKS ========================================================================== Same law, different half-lives, so the useful window slides. A clock is useful when a measurable change in the daughter-to-parent ratio corresponds to a small slice of time. Too fast and there is nothing left to measure; too slow and nothing has happened yet. system half-life (yr) mean life (yr) carbon-14 -> nitrogen-14 5700 8223.4 uranium-235 -> lead-207 7.0381e+08 1.0154e+09 potassium-40 -> argon-40 (total) 1.2491e+09 1.8021e+09 uranium-238 -> lead-206 4.4683e+09 6.4464e+09 rubidium-87 -> strontium-87 4.97e+10 7.1702e+10 D1. Where each clock is usable ------------------------------ Taking 'usable' as 0.1% to 95% of the parent decayed, which is roughly where the ratio is neither lost in the noise nor exhausted: system young end (yr) old end (yr) carbon-14 -> nitrogen-14 8.227 2.463e+04 uranium-235 -> lead-207 1.016e+06 3.042e+09 potassium-40 -> argon-40 (total) 1.803e+06 5.399e+09 uranium-238 -> lead-206 6.45e+06 1.931e+10 rubidium-87 -> strontium-87 7.174e+07 2.148e+11 D2. Uranium-lead runs two clocks inside one crystal --------------------------------------------------- 238U -> 206Pb, half-life 4.4683 Gyr 235U -> 207Pb, half-life 0.7038 Gyr (Jaffey et al. 1971, Phys. Rev. C 4, 1889-1906) Ratio of the two decay constants: 6.349. Two clocks at very different speeds, in the same crystal, fed by the same parent element. Anything that adds or removes uranium or lead throws them off the agreement line by different amounts. Agreement is called concordance, and it is a free internal check on every date. Present-day 235U/238U in natural uranium: 1/137.82 = 0.007256 1.000 Gyr ago that ratio was 0.01664, i.e. 1 part in 60.1 2.000 Gyr ago that ratio was 0.03814, i.e. 1 part in 26.2 3.000 Gyr ago that ratio was 0.08744, i.e. 1 part in 11.4 4.000 Gyr ago that ratio was 0.20048, i.e. 1 part in 5.0 4.567 Gyr ago that ratio was 0.32099, i.e. 1 part in 3.1 D3. Potassium-argon, and the branch that makes it awkward --------------------------------------------------------- Renne et al. (2010) decay constants for potassium-40: electron capture to argon-40 : 5.7550e-11 /yr beta decay to calcium-40 : 4.9737e-10 /yr total : 5.5492e-10 /yr total half-life : 1.2491 Gyr branch to argon : 10.37% About one potassium-40 atom in ten becomes the argon you measure. The other nine become calcium-40, useless as a clock because ordinary calcium is everywhere. So the branching ratio is not a footnote: it is a factor of 9.64 standing in front of every potassium-argon age, and it had to be measured separately before any of those ages meant anything. ========================================================================== PART E -- VALIDATION AGAINST ANALYTIC AND PUBLISHED VALUES ========================================================================== E1. Simulation versus the analytic exponential ---------------------------------------------- t/T_half simulated mean analytic rel. err 1.0 100016.16 100000.00 0.00016 2.0 49992.26 50000.00 0.00015 3.0 24987.19 25000.00 0.00051 4.0 12495.56 12500.00 0.00036 5.0 6248.28 6250.00 0.00027 worst relative error over five half-lives: 0.00051 VERDICT: PASS E2. Simulated scatter versus the binomial prediction ---------------------------------------------------- simulated s.d. at one half-life : 231.408 binomial sqrt(N*f*(1-f)) : 223.607 ratio : 1.0349 VERDICT: PASS E3. Calibration machinery against an analytic case -------------------------------------------------- Feed the calibrator a fake curve with mu(theta) = theta exactly and zero curve uncertainty. The 95.4% interval must then collapse to the textbook Gaussian interval, r +/- 2.00 sigma. calibrator returns : 2940 to 3059 (width 119) analytic : 2940 to 3060 (width 120) VERDICT: PASS E4. Coverage test on the real curve ----------------------------------- Draw 4000 true calendar ages uniformly over 100-13,900 cal BP, read the curve there, add Gaussian noise, calibrate, and ask how often the 95.4% interval actually contains the true calendar age. nominal coverage : 95.4% measured coverage : 95.6% (3823 of 4000) VERDICT: PASS E5. Published cross-checks, in units of combined sigma ------------------------------------------------------ These are not our measurements. They are published pairs in which one method was run against a date already known by completely different means. z = (method - known) / combined 1-sigma. comparison z Vesuvius sanidine 40Ar/39Ar vs the Roman calendar (yr) +0.07 Egyptian tomb wood 14C vs dynastic chronology (yr BC) +0.67 Shroud, Arizona vs 3-lab weighted mean (14C BP) -1.23 Shroud, Oxford vs 3-lab weighted mean (14C BP) +1.79 Shroud, Zurich vs 3-lab weighted mean (14C BP) -0.45 Solar System, Patterson 1956 vs Connelly 2012 (Myr) -0.25 Five of six land inside one sigma. One, the Oxford sub-sample of the Shroud, sits at +1.79 sigma of its own three-laboratory mean, and the original paper said so: chi-squared 6.4 on 2 degrees of freedom, P = 5%. That is what an honest cross-check looks like when it is not perfect. E6. A cross-check that FAILED, which is the useful kind ------------------------------------------------------- Before 2008, 40Ar/39Ar ages and astronomically tuned ages disagreed by about one percent. Kuiper et al. (2008) recalibrated the Fish Canyon sanidine standard against orbitally tuned Moroccan sediments: old Fish Canyon age : 28.020 Ma tuned Fish Canyon : 28.201 +/- 0.046 Ma discrepancy : 0.181 Ma = 0.64% = 3.9 sigma One percent is nothing to a reader and everything to a geologist: at the Cretaceous-Paleogene boundary it is 660000 years. The network of methods found it because the methods are independent. One method alone, however precise, could not have. E7. FIGURE 5 DATA -- the agreement plot --------------------------------------- pair timescale yr z Vesuvius sanidine 40Ar/39Ar vs the Roman calendar (yr) 1.9e+03 +0.07 Egyptian tomb wood 14C vs dynastic chronology (yr BC) 4.6e+03 +0.67 Shroud, Arizona vs 3-lab weighted mean (14C BP) 690 -1.23 Shroud, Oxford vs 3-lab weighted mean (14C BP) 690 +1.79 Shroud, Zurich vs 3-lab weighted mean (14C BP) 690 -0.45 Solar System, Patterson 1956 vs Connelly 2012 (Myr) 4.57e+09 -0.25 Fish Canyon sanidine, pre-2008 vs tuned (Ma) 2.82e+07 -3.93 Six points inside two sigma, one at -3.9. The outlier is not a scandal. It is the network doing the one job we built it for. ========================================================================== PART F -- HEADLINE NUMBERS USED IN THE ARTICLE ========================================================================== A1 the mean of 200,000 simulated worlds tracks N0*exp(-lam*t) to 0.051% over five half-lives, with no exponential in the simulation A3 half-life from a single run: s.d./mean = 0.453 at N0=10 and 0.0123 at N0=100,000. That is 1/sqrt(N). A5 of 4000 worlds of 1000 atoms, exactly 500 survived one half-life in 111 of them (2.77%) B1 1 mg of modern carbon gives 58649 counted atoms at 1e-3 efficiency, so +/-34 yr from counting statistics alone B2 sample radiocarbon equals a 0.15% background at 53471 yr 1 mg at 50,000 yr: +/-2016 yr; 10 mg at 50,000 yr: +/-1820 yr 1 mg at 60,000 yr: +/-6499 yr B4 1% modern contamination in a dead sample reads 36993 BP; 0.1% reads 55489 BP; 5% reads 24064 BP B5 1% modern contamination costs a 1,000-yr sample 11 yr and a 50,000-yr sample 14445 yr C2 raw-vs-calendar offset peaks at -2107 yr at 12580 cal BP inside the tree-ring range, and reaches -4900 yr at 55,000 cal BP C4 median 95.4% calendar width for a +/-25 yr measurement: 143 yr; worst 419 yr at 10360 BP, best 33 yr at 10780 BP C5 2450 +/- 30 BP calibrates to 2362-2543; 2559-2618; 2632-2701 cal BP, 312 calendar years wide E4 measured coverage of the 95.4% interval: 95.6% E6 Fish Canyon discrepancy closed in 2008: 0.64% = 3.9 sigma ==========================================================================