16.4: Age of the solar system
- Page ID
- 22738
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)So just when did all this happen? An estimate for the age of the solar system can be made using isotopes of the element lead (Pb). There are several isotopes of lead, but for the purposes of figuring out the age of the solar system, consider these four: \(\ce{^{208}Pb}\), \(\ce{^{207}Pb}\), \(\ce{^{206}Pb}\), and \(\ce{^{204}Pb}\).
\(\ce{^{208}Pb}\), \(\ce{^{207}Pb}\), \(\ce{^{206}Pb}\) are all radiogenic: that is to say, they stable “daughter” isotopes that are produced from the radioactive “parent” isotopes. Each is produced from a different parent, at a different rate:
| Parent isotope | Stable daughter | Half-life |
|---|---|---|
| \(\ce{^{232}Th}\) | \(\ce{^{208}Pb}\) | 14.0 b.y. |
| \(\ce{^{238}U}\) | \(\ce{^{206}Pb}\) | 4.5 b.y. |
| \(\ce{^{235}U}\) | \(\ce{^{207}Pb}\) | 0.70 b.y. |
\(\ce{^{204}Pb}\) is, as far as we know, non-radiogenic. It’s relevant to this discussion because it can serve as a ‘standard’ that can allow us to compare the other lead isotopes to one another. Just as if we wanted to compare the currencies of Namibia, Indonesia, and Chile, we might reference all three to the U.S. dollar. The dollar would serve as a standard of comparison, allowing us to better see the value of the Namibian currency relative to the Indonesian currency and the Chilean currency. That’s what \(\ce{^{204}Pb}\) is doing for us here.
This is a plot showing the modeled evolution of our three radiogenic lead isotopes relative to \(\ce{^{204}Pb}\). It is constrained by terrestrial lead samples at the young end, and projected back in time in accordance with our measurements of how quickly these three isotopes of lead are produced by their radioactive parents. Of course, if we go back far enough in time, we run out of samples to evaluate. The Earth’s rock cycle has destroyed all its earliest rocks. They’ve been metamorphosed, or weathered, or melted – perhaps many times over! What would be really nice is to find some rocks from the early end of these curves – some samples that could verify these projections back in time are accurate.
Such samples do exist! But they are not from the Earth so much as “from the Earth’s starting materials.” If the nebular theory is correct, then a few leftover scraps of the planet’s starting materials are found in the solar system’s asteroids. Every now and again, bits of these space rocks fall to earth, and if they survive their passage through the atmosphere, we may be lucky enough to collect them, and analyze them. We call these space rocks “meteors” as they streak through the atmosphere, heating through friction and oxidizing as they fall. Those that make it all the way to Earth’s surface are known as “meteorites.” They can be often be distinguished by their scalloped fusion crust, as with this sample:
Meteorites come in several varieties, including rocky and metallic versions. It is very satisfying that when measurements of these meteorites’ lead isotopes are added to the plot above, they all fall exactly where our understanding of lead isotope production would have them: at the start of each of these model evolution curves. Each lead isotope system tells the same answer for the age of the Earth, acting like three independent witnesses corroborating one another’s testimony. And the answer they all give is 4.6 billion years ago (4.6 Ga). That’s what \(\ce{^{208}Pb}\) says. That’s what \(\ce{^{207}Pb}\) says. And that’s what \(\ce{^{206}Pb}\) says. They all agree, and they agree with the predicted curves based on terrestrial (Earth rock) measurements. This agreement gives us great confidence in this number. The Earth, and meteorites (former asteroids), and the solar system of which they are all a part, began about 4.6 billion years ago...
Did I Get It? - Quiz
How old is the solar system?
a. 6.4 million years.
b. 6.4 billion years.
c. 4.6 million years.
d. 4.6 billion years.
e. 6000 years.
- Answer
-
d. 4.6 billion years.
Isotopes of what element are used to calculate the age of Earth (and meteorites) ((and the rest of the solar system))?
a. Uranium
b. Carbon
c. Hydrogen
d. Lead
- Answer
-
d. Lead
...But what came before that?