7 Chronology & Math Basics
Part 1: Dating & radioactive decay (7.1 - 7.5)
As an entrance to dating and chronology, it is critical to familiarise oneself with the fundamentals of the radioactive decay. Inform yourself with the modes of radioactive decay, how the decay proceeds, the meaning of the decay constant/half life, and how the radioactive decay is presented in plots. Describe the important fundamentals of the radioactive decay.
Learning Goals
Describe the varies modes of radioaktive decay. Describe and explain the radioactive decay, as well as the radiogenic evolution. Relate the decay constant to the half life.
Part 2: The isochron equation and the isochron diagram (7.6 - 7.9)
Age dating is done using isochron diagrams from which – and in conjunction with the isochron equation – the absolute or relative age of a sample is determined, depending on whether it is a short lived or long lived decay system. The isochron equation can be directly derived from the decay equation. Understand the connection between the decay and isochron equations, and learn the difference between long-lived decay system/absolute dating and short-lived decay system/relative dating.
Learning Goals
Recall the decay equation. Relate the decay to the isochron equation. Name the plot-type of the isochron plot. Explain and describe the difference between long-lived and short lived decay system and plot, respectively.
Part 3: The age of meteorites and planetary bodies (7.10 - 7.11)
Dating of solar system objects such as meteorites or IDPs allows us to produce a chronology of events such as chondrule formation or planetary accretion and differentiation in the early solar system. Various and appropriately chosen short- and long-lived radioactive isotope decay systems are required to reconstruct these events.
Learning Goals
Name important decay systems to reconstruct the chronology of early solar system events. Describe the succession of events in the early solar system.
Bonus 1: Mathematical basics for isotopes and dating (7.12 - 7.15)
A few mathematical basics are critical to understand the equations for dating and chronology. Explain what exponents, logarithms, and differential equations are, and how these work. Learning Goals
Confidently recall and apply exponents and logarithms. Explain what a differential equation is. Relate the function e^x to is derivative e^x.
Bonus 2: Special Appearance: Mojo (7.16 - 7.26)
A. Morbidelli and S. Raymond guide through this special series entitled: ›MOJO - Modeling the Origin of JOvian planets‹. The series consists of 11 parts, covering a variety of topics related to the formation of the planets, with a focus on the gas planets.
You find the 11 part series here.
Learning Goals
For guest series such as this, no specific learning goals are currently defined. Summarise and present the important topics of this series.
7.1 Modes of Radioactive Decay
Radioactive elements decay along 3 main paths: (i) Release of a positron. Thereby, a neutron converts into a proton, i.e., the nucleus gains a proton and changes to an element with an atomic number that is 1 higher. It further looses one neutron. (i) Release of an electron. Thereby, a proton converts into a neutron, i.e., the nucleus looses a proton and changes to an element with an atomic number that is 1 lower. It further gains a neutron. (iii) Emission of a 4He nucleus. No conversions occur, but the nucleus looses 2 protons and becomes an element with a by 2 lower atomic number. Further, 2 neutrons are lost. Additional decay modes exist that are, however, by far more rare, e.g., spontaneous spallation.
beta-: an electron is lost and a neutron changes into a proton, i.e., a chemical element with a +1 higher atomic number is formed; beta+: a positron is lost and a proton changes into a neutron, i.e., a chemical element with a -1 lower atomic number is formed; alpha: a 4He nucleus is lost, i.e., a chemical element with a -2 lower atomic number is formed.
To understand nucleosynthetic processes. To understand the basis of chronology.
✓True
✗False
✓True
✗False
✗True
✓False
7.2 Coin Tosser – Illustrating the De- and Increase of the Radioactive Elements
The radioactive decay can be illustrated by a coin toss: Imagine 1000 coins that are tossed. About half will show head, the others tail. Now assume the coins showing head are the decayed radioactive parent nuclides, then these can be removed. The remaining are tossed again, and again about half will show head, the others tail – and again those showing head are removed, and so on. Th decrease of coins showing heads is equivalent to the radioactive decay. Then imagine dices with n sides instead of coins. After each toss, only dices showing 1 will be removed, i.e., these will be fewer compared to the coin tosses (except when n=2). The number of cube sides n is the equivalent to the decay constant.
In analogy to a coin or dice toss. Such a toss is purely statistical. After a toss, only those coins with either heads or tails, or dices with only a certain number, respectively, are removed. If the remaining coins/dices are plotted against the number of tosses, the exact same exponential curve emerges, as known from the radioactive decay. This nicely illustrates the statistical nature of the radioactive decay.
The inverse number of sides a hypothetical dice has.
✓10
✗20
✗50
✗100
✓… an exponential curve
✗… a sinus curve
✗… a linear curve
✗… a quadratic curve
✗Half a half-life
✗One half-life
✓Two half-lifes
✗Three half-lifes
✗About 10 half-lifes
7.3 Radiogenic Evolution
The radiogenic evolution of various reservoirs depend on their initial ratio of parent to daughter isotope. High ratios result in large amounts of daughter nuclides in a reservoir, and vice versa. The later reservoirs separate from each other, the smaller is the amount of radiogenic daughter nuclides in a reservoir. These various scenarios are readily illustrated in an evolution plot.
To study the radiogenic evolution of various reservoirs that separate at a certain time from a common parental reservoir.
There is none. An evolution plot is essentially a decay plot, but only the ingrowth of the daughter nuclide is shown. Further, the evolution plot typically has the normalised daughter element on the y-axis. Such a normalising isotope is required for analytical reasons.
✗… is a line with a certain slope.
✓… is a vertical line.
✗… cannot be shown.
✓… connect the endpoints of all evolution lines.
✓… their parent/daughter isotope ratios are different.
✗… they have various abundances of the radioactive parent isotope.
✗… they have various initial abundances of the daughter isotope.
✗… radioactive decay of the parent isotope speeds up over time.
✓… bigger …
✗… about the same …
✗… smaller …
7.4 Radiogenic Decay Plot
The simple decay diagram plots the decay of the radioactive parent nuclide, as well as the increase of the radiogenic daughter nuclide versus time. The decay equation is: e^(-lambda*t), with lambda being the decay constant and t the time.
N = N0e^(-lambdat), with N: amount of the radioactive element still present N0: initial amount of the radioactive element e: Euler-Number (2.71828) lambda: decay constant t: time
N = N0-N0e^(-lambdat), with N: amount of the radioactive element still present N0: initial amount of the radioactive element e: Euler-Number (2.71828) lambda: decay constant t: time
✓lambda = Ln(2)/hl
✗lambda = Ln(2)*hl
✗hl = lambda/Ln(2)
✓hl = Ln(2)/lambda
✓… 2.2 kg
✗… 1.1 kg
✗… 0.4 kg
✗… 4.4 kg
✓… 1/s
✗… s
✓… 1/Ga
✗… Ga
7.5 Relationship Decay Constant <-> Half-Life
The decay constant (lambda) and the half-life (hl) are related by the equation lambda = Ln(2)/hl. This relationship is readily derived from the decay equation, when the remaining amount of the radioactive parent is set to 0.5, and the equation is then solved for the time t.
This is the time after which half the amount of the initially present radioactive parent nuclides is decayed.
✓lambda = Ln(2)/hl
✓hl = Ln(2)/lambda
✗lambda = Log(2)/hl
✗hl = Log(2)/lambda
✗s
✓1/s
✗a
✓1/a
7.6 Deriving the Decay Equation
At each time step a certain fraction of the available parent nuclides – N(t), with N amount of available parent nuclides, and t time – decays, i.e., the amount decayed is: DN/Dt, with DN being the amount of parent nuclides that decayed, and Dt the time interval during which this decay happens. In the next time step, the amount of available parent nuclides is correspondingly smaller, as part of it already decayed, i.e., N(t) is smaller in the next step. The fraction of parent nuclides that decay in this next step is, however, the same. Therefore the proportionality is N(t) ~ DN/Dt. The exact fraction of decaying parent nuclides at each time step depends on the specific parent nuclide. This fraction (lambda) is multiplied with DN/Dt, which then becomes the equation N(t) = lambda DN/Dt, and lambda is called the decay constant. This equation is a differential equation that, if solved, results in N(t) = N0 e^(-lamda*t).
The amount of nuclides decaying in a certain time interval is proportional to the amount of nuclides available.
The rate of nuclides decaying at each time depends on the amount of nuclides present at each time. In this setting, the rate of nuclides decaying at each time is the derivative of the amount of nuclides present at each time. This relationship of a derivative to a function is what is called a differential equation.
✓… of parent nuclide remaining after a certain time.
✗… of parent nuclide decayed after a certain time.
✗… of daughter nuclide produced after a certain time.
✓… decay constant
✗… half-life
✗… parent nuclide
✗… N(t) = e^(-lambda*t)
✓… N(t) = N0 e^(-lambda*t)
✗… N(t) = e^(lambda*t)
✗… N(t) = N0 e^(lambda*t)
7.7 Deriving the Isochron Equation
The isochron plot is a parametric plot. The y-value of the decay equation y = e^(-lambdat), with lambda: decay constant and t: time, is plotted along the x-axis, i.e., the amount of remaining parent nuclide is plotted along the x axis. The y-value of the decay equation, but indicating the increase of the daughter nuclide y = 1-e(-lambdat) is plotted along the y-axis, i.e., the amount of increasing daughter nuclide is plotted along the y axis. The resulting plot is the isochron plot. The isochron itself is the connection of the evolution lines produced by the parameter plot, when various initial amounts of the parent nuclide are plotted. The slope of the isochron is then y/x, i.e., m = (1-e^(-lambdat))/e(-lambdat), which is m = e^(lambda*t)-1. The isochron equation is then y = mx+b, with y being the daughter nuclide on the y-axis, x the parent nuclide on the x-axis, and b being the initial amount of the daughter nuclide – as b denotes the intercept on the y-axis.
The amount of the parent nuclide is displayed along the x-axis, and over time decreasing to lower values. The amount of the daughter nuclide is displayed along the y-axis, and over time increasing towards higher values. The nuclides are further reported as ratios, not absolute values.
It is the time t. Over time, the amount of the parent nuclide decreases and the amount of the daughter nuclide increases – in combination evolving to the upper left in the plot.
✗True
✓False
✓True
✗False
✗D = D0 + P * (1-e^(-lambda * t))
✗D = P0 + P * (e^(-lambda * t)-1)
✓D = D0 + P * (e^(lambda * t)-1)
✗D = P0 + P * (e^(lambda * t)-1)
7.8 Isochron-Plot of Long-Lived Decay Systems
Long-lived decay systems are those in which the half-life of the parent is significantly larger than 100 Ma. An example is the 87Rb -> 87Sr system, with a long half-life of 48.1 Ga. For an isochron plot it is required that a number of components, e.g., various minerals, had variable 87Rb/86Sr ratios, but the same 87Sr/86Sr ratios. An example would be a melt, in which these minerals crystallise at about the same time. 87Rb then decays to 87Sr. Thereby, the 87Rb/86Sr ratio in the various minerals decrease, while their 87Sr/86Sr ratios increase. As a result, the compositions of the minerals move along lines with a negative slope. The connection of the endpoints of these lines define the isochron. The slope of this isochron can then be used to back-calculate when the various minerals crystallised, i.e., how old the rock is. The isochron plot itself is a parametric plot.
The radioactive parent nuclide (e.g., 87Rb) is plotted along the x-axis and the radiogenic daughter nuclide (e.g., 87Sr) is plotted along the y-axis. Both nuclides are normalised to one and the same stable nuclide of the daughter element (e.g., 86Sr). Hence, e.g., the 87Rb/86Sr ratios is plotted along the x-axis, and the 87Sr/86Sr ratio along the y-axis.
The various components – e.g., minerals – are characterised by various initial ratios along the x-axis, i.e. for example various 87Rb/86Sr ratios. The composition of these components changed during the decay of 87Rb to 87Sr. The composition changes along the x-axis towards lower 87Rb/86Sr ratios, and along the y-axis towards higher 87Sr/86Sr ratios. Effectively, a component changes along a line with a negative slope.
✗ca. 1 Ma
✗ca. 10 Ma
✓ca. 100 Ma
✗ca. 1000 Ma
✓parameter plot
✗x-y plot
✗a function
✗category plot
✓True
✗False
7.9 Isochron-Plot of Short-Lived Decay Systems
Short-lived decay systems are those in which the half-life of the parent is smaller than about 100 Ma. An example is the 182Hf -> 182W system, with a half-life of 8.9 Ma. For an isochron plot it is required that a number of components, e.g., various minerals, had variable 180Hf/184W ratios, but the same 182W/184W ratios. An example would be a melt, in which these minerals crystallise at about the same time. 182Hf then decays to 182W. This, of course, does not change the 180Hf/184W ratios of the minerals, however their 182W/184W ratios increase. As a result, the compositions of the minerals move along lines parallel to the y-axis. The connection of the endpoints of these lines define the isochron. The slope of this isochron can then be used to back-calculate when the various minerals crystallised, i.e., how old the rock is – but only, and this is important to note, relative to another rock of which also an isochron has been determined.
On the x-axis, a stabile nuclide (e.g., 180Hf) of the radioactive parent element is plotted, and the radiogenic daughter nuclide (e.g., 182W) of the actual radioactive parent (i.e., 182Hf in the example) is plotted along the y-axis. Both nuclides are normalised to one and the same stable nuclide of the daughter element (e.g., 184W). Hence, e.g., the 180Hf/184W ratios is plotted along the x-axis, and the 182W/184W ratio along the y-axis.
The various components – e.g., minerals – are characterised by various initial ratios along the x-axis, i.e. for example various 180Hf/184W. Initially, the 182Hf/184W ratio changed during the decay of 182Hf to 182W. However, as no more 182Hf is present, the 180Hf/184W ratio is plotted along the x-axis. As 180Hf is stable, the 180Hf/184W ratio on the x-axis remains constant. Nonetheless, the composition of 182W/184W changes along the y-axis towards higher ratios. Effectively, a component changes along a vertical line upwards.
✗ca. 0.1 Ma
✗ca. 1 Ma
✗ca. 10 Ma
✓ca. 100 Ma
✗From the slope of the isochrone of the object
✗From the difference of the isochron slopes of min. three objects.
✗From the difference of the 182W/184W ratios of two objects.
✓From the difference of the isochron slopes of two objects.
✗True
✓False
7.10 Chronology of Events in the Early Solar System
A brief chronological overview of the major events in the early solar system shows that the starting point (t0 – t zero) is defined by the formation of CAIs. Then iron meteorite parent bodies formed in the first million years. Next is chondrule and matrix formation during about 2-5 Ma after t0. These then aggregate to form the chondrite parent bodies, about 3-7 Ma after t0. This is followed by the formation of the Earth, Moon and the other terrestrial planets during a few tens of millions of years after CAIs formed. Jupiter and the other gas planets likely formed earlier at around 5 Ma.
7.11 Choosing Appropriate Decay Systems for Age Dating
The various events throughout the evolution of our solar system are dated using various decay systems, appropriately chosen for the specific process that is studied. Short-lived decay system with half-lifes of a few hundred thousands to few millions of years are used for relative dating of the earliest processes such as chondrule formation. Examples are 26Al -> 26Mg, 60Fe -> 60Ni, and 53Mn -> 53Cr. Short-lived decay systems with half lives of a few millions to tens of millions of years or used for asteroid and planetary differentiation processes. Examples are: 53Mn -> 53Cr, and 182Hf -> 182W. Long-lived decay systems with half-lifes billions to hundreds of billions of years are used for absolute ages. Examples are 87Rb -> 87Sr, 176Lu -> 176Hf, and the various various U-series.
7.12 Basic Exponential and Logarithmic Laws
Knowing a number of basic exponential/logarithmic laws is beneficial to understand some equations when working with isotopes. The few most important are recapitulated here in their respective contexts.
Ln(alpha12)/Ln(alpha23) = beta -> Ln(alpha12) = beta * Ln(alpha23) -> Ln(alpha12) = Ln(alpha23^beta) -> e^Ln(alpha12) = eLn(alpha23beta) -> alpha12 = ealpha23beta
✓b^(n+m)
✗(bn)m
✗m * b^n
✗1/b^-n
✓1/b^n
✗1/-b^n
✗x* Log(n)
✗Log(x)+n
✓n* Log(x)
✗Log(x)+Log(y)
✓Log(x)-Log(y)
✗Log(y)+Log(x)
7.13 Why the derivative of e^(x) equals e^(x)
The derivative of e^(x) is also e^(x). The function e^(x) is an exponential function with a specific number (e) as the basis. The general exponential function is: y = a^x. If the basis a is large, y’ is always larger than y for all x. If the basis a is small, y’ is always smaller than y for all x. Then there exists one number for the basis x, for which y’ equals y for all x. This specific number is e. And if y’ equals y for all x, then y’ = y, i.e., the derivative of e^(x) is also e^(x).
7.14 What is a Differential Equation
A differential equation contains a function and the derivative of this function in the same equation, e.g., y - y’ = 0. As the derivate of e^x is again e^x, the function e^x is used to solve differential equations.
Any equation, in which a function and the derivative of this function occur together.
f(x) = f’(x), y = y’
✓g(x) = g’(x) + f(x)
✓Sin(x) = d(Sin(x))
✓Sin(x) = Cos(x)
✓e(x) = e(x)
✓e(x) = e’(x)
✓f(x) - f’(x) = 0
✗True
✓False
✗… 1.414
✓… 2.718
✗… 3.141
7.15 Solving (Differential) Equations
Ordinary, as well as differential equines are best solved using a mathematics software. The square-root of a number is calculated using a calculator, similarly, equations are best solved using mathematics software. This is shwon here.