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How Old is The Earth? And How Do We Know This?
The Answers as We Know Them
The Earth is about 4.54 Billion years old (with an factor of error of about .05 Billion years or 50 million years). This has been determined through radiometric dating on igneous rocks which are the result of the solidification of molten materials.
The oldest igneous meteorites have been dated at 4.56 Billion years,this along with the mass of the sun and it's luminosity leads us to believe that our solar system must be at least this old.
Here is a quick primer on Radiometric dating;
When igneous rocks form, or solidify, they contain various elements and among these are radioactive isotopes. At the formation of the rock the isotope amount is 100%. As time passes these radioactive isotopes decay into more stable, lighter end products by 50% within a certain amount of time. This amount of time is called a half-life. Different isotopes have very different half-lives ranging from 23x10^-24 seconds to 2.2x10^24 years. If you're not good with scientific notation that is 0.(24 more zeros here)23 seconds to 22(23 more zeros here) years. That is to say unimaginably short periods of time to periods of time much longer than the universe has existed. There are about 175 various isotopes used to calculate the range of time periods in between these two extremes.
By measuring the more stable by-products of these isotopes in proportion to the remaining isotope and comparing these proportions we can determine how many half lives these isotopes have undergone.
You may be asking at this point but how is the length of a half-life established, particularly one that is millions, billions or trillions of years? By measuring the microscopically small amounts of change from isotope to end product that occurs in observable time periods we can easily extrapolate how long an isotope would take to decay by half. If this seems unclear consider the following hypothetical situation;
If a man can smash 1 pound of rock with a sledge hammer in 10 minutes, considering he smashes at a constant pace, how long will it take him to smash 10 pounds of rock? 100 pounds of Rock? Well of course, 100 minutes and 1,000 minutes respectively. (In this example consider the intact rock the isotope and the smashed rock the end product.) In other words we don't have to watch him work all day to estimate how much work he will do. From this we can see how watching a small amount of change can be used to calculate the time taken for vastly larger amounts of change.
Commonly Used Radiometric Isotopes
Depending on what time frame you would like to measure some Isotopes are better than others. The isotope Carbon-14 for instance has a half life of 5,715 years and so lends itself well to dating the remnants of ancient civilizations since these generally less than 100,000 years ago. (It decays into Nitrogen-14 if you were wondering.) And so by measuring the proportion of Carbon-14 to Nitrogen-14 in the igneous rocks found in the same geological strata as an ancient civilization the age of all things in that strata can be rather accurately approximated.
To determine the age of the Earth we will obviously need something with a more considerable half-life. Uranium-238 which decays into lead has a half live of 4,468,000,000 years and so works well for dating things that are Billions of years old. (It decays into Thorium-231.) And so again by comparing the proportions of Uranium-238 to Lead on a meteorite called the Canyon Diablo meteorite we can determine the number of whole and partial half lives it has under gone and these calculations yield 4.54 Billions years. Meteorites were used because the crust and mantle of the Earth have been mixed and disturbed by plate tectonics and weathering.
This date has been confirmed by tests on hundreds of other meteorites and terrestrial samples, not only with Uranium-238 but other Isotopes such as Uranium-235 which has a half life of 703,800,000 years and these tests converge on the same time. A number of other half-life appropriate isotopes have further verified these results.
Let's Dispel One Popular Viewpoint
I feel a perfunctory duty to say loudly and clearly that the Earth is not less than 10,000 years old as the Abrahamic texts would have us believe. Whatever one's religious affiliation, in order to maintain intellectual honesty the, "Christian Science," or Islamic equivalent that claims to support this must contort themselves into entirely incoherent blather. A telling feature here is that, unlike scientists, the religious answer is presupposed and then evidence is garnered, distorted, invented, etc. to fit the answer that people have already become comfortable with. This is not how science works. Rather than supposing an answer derived from a bronze aged text a priori, Science follows the evidence where is leads however surprising or uncomfortable the resulting answer might be. Does science make mistakes, of course, and the age of the Earth has been mistaken a few times through the use of scientific methodology. The credulous would like to turn this truism against scientific but rather than this tendency toward revision being a foible of the physical sciences I would contend that it is it's ultimate strength and engine of progress. The ability to say with true humility I don't know, or we think we know only to later be proven wrong by better more advanced science is what distinguishes scientific answers to questions from religious answers to questions. The former is an honest inquiry subject to fallibility while the latter is absolutely set in stone no matter what it's dissonance with reality.
But that's Not All
Without going into detail, I'd like to just mention a couple of other ways by which we know the age of the Earth. Continental drift is measurable and confirms the radiometric results. Oscillations in the magnetic polarity of the Earth occurs around every quarter of a million years . This polarity is preserved in magnetic rocks and by observing the number of magnetic switches in these rocks as preserved in the geological strata we can determine the age of the Earth to be at least billions of years.
Further, geological strata observations of fossilized tree cross sections in which rings can be counted and compared then matched to fossilized trees in other geological strata around the world narrate a history of the world of at least tens of thousands of years.
The Plasticity of Science Hypothesis
In 1892 lord Kelvin, using temperature gradients, determined that the age of the Earth was 100 million years. In 1895, John Perry using a similar model estimated the age of the Earth to be 2 or 3 billion years. The fatal flaw in these attempts was the assumption that the Earth had been consistently cooling, with no exceptions to be found, since it's origin. In 1898 Mary and Pierre Curry discovered the radioactive elements Polonium and Radium and showed that they actually produced substantial amounts of heat. The discovery of the process of Radioactive decay showed that heavier elements broke down into lighter ones producing Alpha, Beta, and Gamma rays and by necessity, heat.
Now I can almost hear the keys boards clicking out there of people who would use these past errors to dispute the results derived from radiometric dating. And they would be right to point this out. As I mentioned the beauty of scientific inquiry is it's lack of dogmatism. If a future discovery calls radiometric dating into question, though this is highly unlikely because it produces the same results time and again using a variety of isotopes on a variety of igneous rock samples, then we will throw out our mistaken hypothesis and continue to work on a new answer through better science. This is the beauty, virtue, and utility of science. It does not regard itself as infallible and always leaves open the possibility of revision.
I would only point out that the initial mistaken attempts were cleared up in less than a decade and Radiometric dating has been yielding consistent results and making correct predictions for over a century now.