Is Radiocarbon Dating Possible on Titan?

 

  Would the equilibrium condition for C-14 in the atmosphere work on Titan? The solar wind extends out to Saturn and we know that Saturn has aurorae. Titan has a nitrogen atmosphere, there is a source of carbon in the form of methane, CH₄, and there appears to be hydrocarbon lakes on the surface. So we would expect a source of C-14 and some carbon sinks. But is this enough?

 

  The lighter gravity causes the atmosphere to extend farther out from the surface than on Earth and as a result its surface pressure ends up slightly greater than that of the Earth. The solar wind particle density at Saturn is 0.1 per cm² while on Earth the value is somewhat larger at 6 per cm². As a result the production rate of C-14 is probably less on Titan. On the other hand, like the Moon, Titan's rational rate is the same as its orbital rate, 16 days, with a small inclination of its axis and its year is 29.5 Earth years. As a result one would expect the rate associated with carbon sinks to be less than that of the Earth. So it is possible that there is a measurable quantity of C-14 in Titan's atmosphere. But it may not be as stable as the quantity on the Earth since there are longer time scales associated with the fluctuations for the source and drain. The deviation between C-14 times and calendar times may as a result be much larger on Titan.

 

  So if Titan could support life that utilizes atmospheric methane one might be able to obtain a C-14 date for some remnant and compare this with dates for layers caused by annual deposits. For more accurate carbon dating an ocean world such as Earth might be necessary.

 

Equilibrium Test Kit

 

  One can put together a little kit to experiment with the equilibrium condition and see how it is affected by changes using the simplified circuit below.

 

 

  It's easier to deal with resistances and one can derive similar formulas for the equilibrium condition and its changes. The relative change formula is approximately true also. The slight difference in formulas is due to the fact that in the electrical circuit both resistances affect the flow of electrons.

 

 

  Pushing the button approximately doubles V_B which rapidly returns to normal when it is released. Note the difference in the RC time constants for charging and draining the capacitor. The subscripts for the resistors stand for "source" and "drain".

 

Equilibrium Changes

 

  The equilibrium condition, αA = λB, allows us to derive a formula for the changes in B. The production rate, α, is dependent on the flux of cosmic rays, A is the quantity of nitrogen, λ is the removal rate for C-14 associated with carbon sinks and B is the equilibrium quantity of C-14. Taking partial derivatives of the equation allows us to determine the partial derivatives of B which can be inserted into the derivative chain rule. The resulting equation only involves relative changes. Note that increases in α produce increases in B while increases in λ result in decreases in B which may help in identifying the source of the fluctuations in B. However the observation of an increase in the C-14 concentration in Summer might be due to more than one cause such as an increase in the number of protons from the solar wind due to the tilt of the magnetic pole towards the Sun in summer or a decrease in the solubility of CO₂ in the oceans with increasing temperature.

 

 

  Supplemental (26 Jul): Solubility of CO₂. A similar equilibrium condition (11-5) applies.

 

An Electrical Analog

 

  There are electrical circuits which behave similarly to the C-14 model. In the circuit below there is a battery which is a large source of electrons that can be used to represent the supply of nitrogen in the air. The production of C-14 is represented by a current flowing through the first conductance which is I = GV = αA approximately. The amount of C-14 in the air is represented by the charge stored in the capacitor which is proportional to the voltage across it, B. The removal of C-14 is represented by the currents through the two remaining resistors. If α is much smaller than the other conductances then the combination α-A acts like a current source which determines the flow through the circuit. If the voltage B is initially zero it will rise to reach an equilibrium value and no current will flow to the capacitor. Again letting λ = γ + β the equilibrium condition is that for which the flow through this equivalent conductance is equal to that of the source. But then one also has I = GV = λB and the equilibrium condition is αA = λB.

 

 

  In the water bucket model the air gaps and "waterfalls" are necessary to isolate the buckets so that the flow from them is proportional to the amount of water in the bucket. In the electrical circuit is isolation isn't as good but if the other voltage drops are negligible compared to A the model will be a reasonably good analog for the C-14 system.

 

A Water Bucket Analogy for the C-14 Relaxation

 

  The for C-14 excess relaxation can be modeled using water buckets and valves which gives similar solutions. The buckets are set at different heights and the valves can be used to adjust the flow rates. The flow rste is proportional to the opening of the valve and the relative pressure at the height of the valve which is dependent on the amount of water in the bucket. The air gap between buckets ensures that the flow is oneway.

 

 

  The equations for the water bucket model are the same as those for the approximation used and the system mimics the excess C-14 relaxation with the flow approching an equilibrium state.

 

 

  The analogy can also be used to show how seasonal changes in the removal rate can alter the observed amount of water in the intermediate state.

 

 

Annual Variations in C-14 Excess

 

  One can find seasonal variations in the C-14 excess as seen in data for Antartica below. Long term changes in the amount of C-14 have been correlated with sunspot activity but this also affects weather patterns so changes in the cosmic ray flux would not be the only explanation for changes in C-14. An alternative explanation might be changes the relaxation rate.

 

src: CDIAC

 

History of Radiocarbon Dating

 

  Willard Libby and a group of chemists at the University of Chicago were responsible for the development of radiocarbon dating. He wrote two papers on the history of the subject.

 

 History of Radiocarbon Dating, 1967 (.pdf)

 

 Radiocarbon Dating, Memories and Hopes, 1972 (.pdf)

 

  The first paper is more detailed. He talks about the production of Carbon-14 by cosmic rays, the neutron absorption cross section, the equilibrium concentration of C-14, and mentions the relative amounts in the oceans in the form of carbonates and dissolved, on land and in the air. His group was responsible for the development of early detectors which proved the existence of C-14 in nature. One of his students showed that elemental carbon obtained from natural sources produced about 15 decays per minute per gram of carbon. It was also shown from the characteristic decay length on passing its radiation through matter that the natural radioactive carbon behaved like that produced by atomic reactors.

 

  The radiocarbon dates showed some variation from the known historical dates and the possibility that cosmic rays varied with time was considered. Libby mentions the increase in atmospheric C-14 due to nuclear testing does not appear to be aware of the relaxation time or the derived connection between the production rate and the equilibrium amount.

 

  To understand the absorption of radiation by matter a 1902 paper by Rutherford and Brooks might prove useful.

 

   Supplemental (21 Jul): In fairness to Libby I would have to say that he appeared to be working with a model involving carbon sinks. It would have been premature to discuss a work in progress prior to publication. At the time his histories were published he may not have had a good value for a relaxation rate and the subject would have gone beyond the bounds of a history of radiocarbon dating. It would seem more appropriate to treat it as a multidisciplinary study.

 

Questions About Radiocarbon Dating - Part 3

 

  We are now in a position to consider how to do radiocarbon dating on an alien planet. First of all the planet would need a good amount of nitrogen in its atmosphere to produce a measurable quantities of C-14. We can measure both of these and if someone hasn't been setting off atmospheric nuclear explosions we can assume that the amount of C-14 is close to the equilibrium value. Since the C-14 is in equilibrium we can't measure the C-14 removal rate, λ, but we could conceivably calculate the neutron capture cross section based the neutron flux in cosmic radiation in order to get the rate constant for the production of C-14, α. The formula relating α and the equilibrium value of C-14, B_eq, can then be used to determine the relaxation time, λ. So we would have all that we need to know how to do everything from scratch. Just doing radiocarbon dating would require the C-14 equilibrium value and the C-14 decay rate, β.

 

  Mars is not a likely candidate planet since there is not much nitrogen in its atmosphere (2.7%). Saturn's moon Titan would be a better choice (98.4%). The presence of carbon sinks on a planet might indicate the existence of life but there are purely physical sinks like acid rain that need to be considered.

 

An Approximation for the Production Rate of C-14

 

  The C-14 model that was used can't be easily solved for the production rate of C-14, α. If however we assume that the quantity of nitrogen in the atmosphere is approximately constant then we can get a solution if the other rates and the C-14 equilibrium value are known. We have to use the following state diagram.

 

 

  It turns out that the solution is α = λB_eq/A where A is the amount of nitrogen and B_eq the equilibrium value for the C-14 and λ = γ + β. The equilibrium value is then approximately proportional to the production rate and dependent of the rate of removal from the atmosphere.

 

 

  Supplemental (18 Jul): αA = λB is a steady state condition and is the connection between the two models. Note that the flows along the branches for both models are the same to a high degree of approximation when e^-μt is approximately one. One can think of D as a necessary dump.

 

The C-14 Model Gives an Incomplete Picture of What's Happening

 

  The simple mathematical model used for the relaxation response only applies to the ¹⁴C in the atmosphere and over short periods of time. The ¹⁴C behaves chemically and biologically like ordinary CO₂. But there are time lags involved and the carbon that is released back into the atmosphere may be depleted in ¹⁴C.

 

 

  If αA > βB in the state diagram above the end result would be that all of the nitrogen in the atmosphere, A, would disappear and end up being transferred to the storage state, C. This result is also likely for γ > β. The ¹⁴C that ends in storage also undergoes decay and is transmuted into ¹⁴N. This nitrogen could end up being released back into the atmosphere through some chemical or biological decay process that is not included in the model. Some of the processes that affect ordinary CO₂ are referred to in the following video.

 

src: Royal Society

 

Questions About Radiocarbon Dating - Part 2

 

  To determine the relaxation time for the atmospheric excess of Carbon-14 I downloaded the data for Fruholmen, Norway from CDIAC. Exponential decay applies to other processes besides radioactive decay. The Michaelis-Menten process published in 1903 for the action of a catalyst is one example. P. Curie and J. Danne also published the exponential laws governing radioactive decay and induced radioactivity in 1903. One can fit the excess C-14 curves to a simple model involving the rates of transmutation of N-14 and C-14, α and β respectively, and a net rate for the removal of C-14 from the atmosphere, γ,. In the model below A is the amount of N-14 and B is that of C-14 in the atmosphere. The amount of C-14 removed from the atmosphere is C.

 

 

  The first three lines are the rates of change for the quantities. Note that the first equations involve only A and B and can be solved separately. The sum of the rates of change is zero which tells us that the sum is a constant and this can be used to find C. Since α is much smaller than the other two rates we can simplify the fit by assuming the exponential terms involving it are effectively constant. The equation for C-14 (B) results in a simple exponential curve. The excess C-14 in the data is based on the "modern fraction" and expressed in mills (thousandths of unity). The reference amount of C-14 in the atmosphere is taken to be the value for 1950. The data points deviate from the expected mean values since the actual number of decays are determined by a Poisson process. Since the decay involves an exponential curve one can convert it to one that is approximately linear by taking the natural log of the relative number of atoms and then smoothing it before doing the actual fit. In this case the average of a point and 21 points to each side were used to smooth the data. The resulting fit turns out quite well showing that the reference value is fairly close the the equilibrium C-14 value. I found that subtracting 0.75 from the excess C-14 values gave the least error.

 

 

  The first 6 years in which there were fairly large fluctuations in the data were excluded from the fit. The relaxation time found was 15.67 years. In this simple model Δα is the difference between two quantities approximately equal to α and so B'₀ is the sum of two terms which makes the it difficult to determine this rate for the formation of C-14 from the equilibrium value. The equilibrium value of C-14 in the atmosphere may be slightly off from the standard 1950 value used for reference purposes but there may also be a systematic error in data due to relatively smaller counts as time progresses.

 

Questions about Radiocarbon Dating - Part 1

 

  About a week ago an actress that I occasionally follow on YouTube did a vlog which touched on CERN's announcement on the activity at 125 GeV and also raised a question about the movie Prometheus and how one could do radiocarbon dating on an alien planet (see 4:43-6:11 in the vlog). The process which determines the amount of Carbon-14 in the atmosphere involves more than just the rate of decay of Carbon-14. First of all the Carbon-14 is produced by the interaction of cosmic rays with Nitrogen-14 in the stratosphere. It is then oxidized to form CO₂ which is removed by absorption in the oceans and by plant life through the process of photosynthesis. Excess Carbon-14 was introduced into the atmosphere by nuclear testing and monitoring has shown that CO₂ is removed from the atmosphere at a faster rate than could be explained by ratioactive decay.

 

src: Wikipedia

 

  The decrease of Carbon-14 in the atmosphere is an example of a storage process with a relaxation time, τ, of about 16 years. The emission of CO₂ by burning fossil fuels has also helped to reduce the relative amount of atmospheric Carbon-14. Radioactive decay of Carbon-14 in fossil fuels is responsible for the small fraction found in them. Burning wood can return Carbon-14 to the atmosphere.

 

More on the July 4th CERN Announcement

 

src: CERNTV

 

  Another video was posted on YouTube a couple fo days ago by CERNTV which has scenes from the physicist conference in which the announcement of the latest data analysis was made. There is also a CMS press release available at CERN.

 

These books provide some interesting reading on the Higgs Boson:

 

Gerard 't Hooft, In search of the ultimate building blocks, pp. 68-75

 

Martinus Veltman, Facts and Mysteries in Elementary Particle Physics, pp. 279-292

 

Latest Evidence for the Higgs Boson

  Earlier today researchers at CERN commented in a press conference on the latest evidence for the existence of the Higgs boson. The analysis of the data from collision experiments in 2011 and 2012 indicated a "5 sigma" deviation at about 125-6 GeV from the expected number of events due to known processes. This suggests the existence of a new boson whose characteristics need to be further studied. The physicists now know where they need to focus their efforts in order to prove the existence of the Higgs boson and expect results by the end of the year.

  Higgs and others commented today on the announcement.