doing the exploration, students benefit from the mathematical activities undertaken and find them both stimulating What is the difference between a mathematical exploration and an extended essay in mathematics? .. Radiocarbon dating. 4School of Mathematics and Statistics, University of Sheffield, . The reliability of this radiocarbon-based date range has been .. Geoarchaeological tsunami deposits at Palaikastro (Crete) and the Late Minoan IA eruption of Santorini. .. to evaluate the conclusions in the paper are present in the paper. It provides an overview and introduction for both the teacher and student to their roles and responsibilities. The internal assessment is based on fieldwork. Field measurements of dissolved (solute) and / or particulate organic carbon carried . All secondary information must be referenced, using a standard author –date.
IB Geography: IA Support
Egypt was the first manifestation of the territorial state, in many respects the forerunner of all modern countries. The rate and direction of the changes that led to this political centralization can only be traced by absolute chronology.
Furthermore, the formation of Egypt was accompanied by profound economic and cultural developments, epitomized by the spread of intensive agriculture and the invention of writing [ 14 ].
The latter is thought to have occurred independently in both Egypt and Mesopotamia, but more dating information is still required to establish exactly when and how this innovation took place.
One of the most effective approaches to absolute chronology currently available uses Bayesian statistical modelling [ 1516 ]. The wealth of relative dating evidence available for Early Egypt makes it highly suitable for this sort of analysis. Here, we use such evidence to refine radiocarbon dates within Bayesian statistical models.
Taken in isolation, radiocarbon dates rarely allow for high-resolution chronological analysis, because of statistical scatter and the imprecision resulting from the calibration process. Experimental set-up We dated organic materials from key sites of the Badarian and Naqada periods and the First Dynasty.
In each case, the samples could be assigned to specific subphases of the relative chronology by means of ceramic typology and horizontal stratigraphy. Our dataset comprised radiocarbon dates—74 from new measurements and sourced from the published literature. Owing to restrictions on the exportation of archaeological material in Egypt, all new samples were obtained from museum collections in Europe and North America with the exception of a group of five freshly excavated seed samples from Tell es-Sakan in the Gaza Strip.
We only selected items supported by the most secure curatorial and excavation records and prioritized short-lived remains such as seeds from granaries, reeds from basketry and fragments of linen [ 18 ]. The results sourced from previous analyses comprised all the dates available for our nominated sites with the exception of organics extracted from mud-based bricks and seals, which have previously been shown to be unreliable [ 18 ]. Of the measurements, 48 were made on short-lived plant and animal remains, 47 on wood, 16 on charcoal and one on shell.
Contextual and analytical details of all the samples are given in table S1 in the electronic supplementary material. The new samples consisted of short-lived plant remains, hair and bone. All were prepared for radiocarbon dating using the Oxford Radiocarbon Accelerator Unit's standard acid—base—acid ABA pre-treatment procedures [ 19 ].
In cases where conservation treatment was suspected, an organic solvent rinse was applied [ 19 ]. For the most fragile hair samples, the base treatment was replaced by five exposures in an ultrasound bath in ultrapure water. The bone collagen obtained for dating was ultrafiltered wherever yields were sufficient. In a handful of cases, the collagen had been extracted by previous work [ 20 ], see the electronic supplementary material, table S1. All samples were combusted in an elemental analyser and reduced to graphite in an excess of H2 over an Fe catalyst [ 19 ].
The radiocarbon data were measured by accelerator mass spectrometry [ 22 ]. Three of our samples showed evidence of contamination; a further three, as well as two published results, were plainly intrusive as they presented dates more than years after our period of study see the electronic supplementary material, table S1.
All eight of these samples 10 dates were excluded from the Bayesian models. Of the further seven previously published replicates, one group of four dates on the same sample of wood failed the test.
In such cases, dates were included in the models as if they had come from separate samples see electronic supplementary material, table S4. Bayesian statistical models a General specifications We constructed Bayesian chronological models using the program OxCal [ 2425 ] and the IntCal09 calibration curve [ 17 ]. Our 10 main models covered the period from the Badarian to the end of the First Dynasty. All the models consisted of sequences of archaeological phases, which enabled us to use the known ordering of such phases as a mathematical constraint—to refine the radiocarbon calibrations.
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Moreover, this approach allowed us to generate estimates for the transition points between phases, often the most important information for socio-political analysis. Each sample was allocated to a phase on the basis of its associated archaeological information.
Where any ambiguity existed, such allocations were broadened to several phases. The radiocarbon dates obtained on each sample were then assigned outlier probabilities. However, wood, charcoal and shell long-lived samples were all treated as if they were likely to predate their contexts.
This was necessary because radiocarbon dates reflect the biological age of the organic material in question—the date it ceases exchanging carbon with its environment.
For inner tree rings and any charcoal derived from themthis can be significantly earlier than the felling date of the tree. In addition, resources such as wood and shell can remain unused for many years before they are incorporated in a given context, widening such age offsets [ 18 ].
To account for these issues, dates on long-lived samples were modelled using an adapted Charcoal Model [ 25 ]. Here, the model was required to select shifts, and the direction of the shift was strongly biased towards younger ages see the electronic supplementary material, table S6. All models ran successfully without limiting the range of possible calendrical solutions. However, for ease of repetition and more rapid convergence, the Naqada models were restricted to solutions between and BCE more than a millennium on either side of archaeological estimates.
Similarly, the Badarian models also all ran without restriction but were constrained to — BCE, more than years earlier and later than current expectation. No such limitations were necessary for the First Dynasty model, because its higher density of data greatly facilitated model convergence.
Along with the main models, various alternatives were run to test the sensitivity of the outputs to the starting assumptions. Such alterations included removing any provision for a localized offset in radiocarbon activity, eliminating all long-lived samples, as well as treating such samples as termini post quos and not applying any outlier analysis. The specifications and outputs of all the alternative models are given in the electronic supplementary material.
Every model was run for at least five million iterations or until completion. This is of course wrong. All it says is that we offsetted the dissolution balance of CO2 in the oceans. If we were to stop burning fossil fuels which is a good thing in general, but totally irrelevant herethen the large CO2 increase would turn into a CO2 decrease, returning back to the pre-industrial level over a century or so.
Think for example on a closed coke bottle. It has coke with dissolved CO2 and it has air with gaseous CO2.
Modelling Radioactive Decay | IB Maths Resources from British International School Phuket
Just like Earth, most of the CO2 is in the dissolved form. If you warm the coke bottle, the coke cannot hold as much CO2, so it releases a little amount and increases the partial pressure of the gaseous CO2, enough to force the rest of the dissolved CO2 to stay dissolved.
Since there is much more dissolved CO2 than gaseous CO2, the amount released from the coke is relatively small. Of course, the comparison can go only so far. The mechanisms governing CO2 in the oceans are much more complicated such that the equilibrium depends on the amount of biological activity, on the complicated chemical reactions in the oceans, and many more interactions I am probably not aware of.
For example, a lower temperature can increase the amount of dust reaching the oceans. This will bring more fertilizing iron which will increase the biological activity since large parts of the ocean's photosynthesis is nutrient limited and with it affect the CO2 dissolution balance.
The bottom line is that the equilibrium is quite complicated to calculate. The change in the amount of atmospheric CO2 is about 80 ppm.
Modelling Radioactive Decay
The main evidence proving that CO2 does not control the climate, but at most can play a second fiddle by just amplifying the variations already present, is that of lags. In all cases where there is a good enough resolution, one finds that the CO2 lags behind the temperature by typically several hundred to a thousand years.
Namely, the basic climate driver which controls the temperature cannot be that of CO2. That driver, whatever it is, affects the climate equilibrium, and the temperature changes accordingly. Once the oceans adjust on time scale of decades to centuriesthe CO2 equilibrium changes as well.
Note that I write "almost anything", because it turns out that the CO2 temperature correlation can be used to say at least one thing about the temperature sensitivity to CO2 variations, as can be seen in the box below. It is interesting to note that the IPCC scientific report e. Instead of pointing it out, they write that in some cases e.
This is of course chaff to cover the fact that when CO2 and temperature are measured with the same cores, or when carefully comparing different cores, a lag of typically several hundred years is found to be present, if the quality and resolution permit. Such an example is found in the figure below. There are many examples of studies finding lags, a few examples include: Science, volp. Science vol, find that the start of the CO2 increase in the beginning of the last interglacial lagged the start of the temperature increase by years.
Clearly, the correlation and lags unequivocally demonstrate that the temperature drives changes in the atmospheric CO2 content.