Explain Like I'm Five

Radiocarbon dating works by knowing what portion of carbon-14 (an unstable radioactive isotope of carbon) remains present in an organic sample.

Plants and animals that eat plants take in carbon-14 and normal carbon in measurable proportions, and then when they die they stop taking in more new carbon 14, the ratio of carbon-14 to carbon then changes over time at a measureable rate we can use to determine the age of objects to about 60,000 years ago.

We didn't use carbon dating to determine the age of theia because all traces of carbon-14 would be long gone after 14 billion years.

I'm not familiar enough to speculate further, other methods of radiometric dating would likely have been used alongside other methods to determine the age of theia. Other radiometric methods use similar methods where we know the decay rate of certain radioactive materials, but for different date and age ranges than carbon.

Let's split the question up. First let's talk about carbon dating.

Carbon is an atom and we know an atom consists of protons, electrons and neutrons. For carbon that is six protons and usually the same amount of electrons. This is always the case, otherwise it's not carbon. The interesting thing is the number of neutrons because they can differ. If we have two atoms with a different number of neutrons we call the isotopes. The most common istope of carbon is C-12. This means it has 12 particles in its core. We know we have 6 protons, so we have 6 neutrons.

There is a rare form of carbon that is C-14, which has 8 neutrons. This form is not stable and breaks apart after a while and forms nitrogen. The time it takes for half of the carbon to break apart, or decay as we call it, is called half-time and for C-14 it's about 5000 years. That means every 5000 years the amount of C-14 halves. As long as an organism is alive it will have a pretty constant amount of C-14 in its body. As soon as it dies it can't consume more of it and then the clock ticks. We can measure how much of it is in the sample and with that we can calculate how old it is. This only rally works on a time range between 300 and 60.000 years. Shorter and not enough of the C-14 has decayed and longer and the so much has decayed that we can't measure it anymore.

For longer times we have other methods but I am unfortunately not too versed with too. Some of them also use the decay of elements but others that have longer half times. So they probably used one of these techniques to date your example.

To add something glossed over by other (wise excellent) answers: Where does carbon-14 come from if you

1. Can't get more while dead
2. It decays

The answer is the sun! There's nitrogen in the atmosphere (specifically, nitrogen-14), and it sometimes gets converted into carbon-14 due to radiation from the sun. Living things use the carbon you breathe in to build new cells/parts of their bodies. Dead things stop doing this.

Let me complicate the story more! At long time scales, the amount of nitrogen, carbon, and carbon-14 can vary in the atmosphere (for example, sunspots might change the radiation, or a volcano might release a lot of gasses). So how do we know what ratio of carbon to carbon-14 things started at? The atmosphere is relatively well mixed afaict, and so this happens at a relatively uniform rate. So we just have to find something that we're really sure about how old it is, and use the ratio of carbon to carbon-14 in that as a baseline. This is where ice cores come in! Ice is quite structural, and we can tell when an ice pack was put down up to the ~decade. So if we find organic things (even small ones) in an ice core sample, we can use the ice to date them and then use their carbon to carbon-14 ratio to date other things.

You'd get better answers if you shared where you got the idea that we've found remnants of Theia. (I doubt most folks who would read this have heard of any such Theia-related developments. For that matter, I was completely unfamiliar with Theia until I googled it just now. But if you shared where you got the idea that we'd found "remnants of Theia", even folks who hadn't been familiar with that development could probably read/watch that and maybe provide some insight.)

There are 3 naturally occuring types of carbon, called isotopes, ¹²C, ¹³C, and ¹⁴C. These are all basically the same except for the number of neutrons in the nucleus of the atom. Living things can use all 3 types to build sugars and proteins and all those other organic molecules needed for life. Animals get it from the food they eat, plants get it from the air. These 3 isotopes all exist in nature in a pretty consistent ratio

¹⁴C, however, is unstable (radioactive) and over time decays into more stable elements.

But it also is constantly getting replenished from cosmic rays interacting with other gases in the atmosphere to in the atmosphere, so as long as something is alive the ratio of carbon 14 in that plant or animal to the other types of carbon stays pretty consistent.

But once it dies, it's not taking any new carbon in, so what was there when it died is all it's ever going to have until it decomposes away entirely or gets eaten by scavengers or whatever. The ¹²C and ¹³C are basically going to stay put, while the ¹⁴C slowly decays.

So to date something you just need to look at the ratio of the different types of carbon and you can figure out how old it is.

Let's say ¹⁴C has a half life (the time it takes for half of it to decay) of 1 year (its actually something like 5 or 6 thousand years but this makes the math easy with smaller, easier to digest numbers)

And let's say the ratio is 1:1, ¹²C+¹³C to ¹⁴C (again, it's more like 1% carbon 14 IRL)

So after a year, half of the ¹⁴C will be gone, and the ratio will be 2:1

After another year half of that remaining ¹⁴C will be gone and it'll be a 4:1 ratio, another year and it's 8:1, etc.

This of course has limitations. After a long enough time basically all of the carbon 14 will be gone and you can't really date things that way anymore. I think the limit is something like 50,000 years.

It's also only really useful for plants, animals, and stuff made from them (leather, textiles, the straw in mud brick buildings, food, etc.)

So not really useful for dating rocks from a few billion years ago like we would need to date remnants of Theia.

There's also a few things that can cause more ¹⁴C to form than normal, humans caused one with our nuclear testing, there was another event around the 8th century that we've discovered from measuring the carbon 14 in tree rings from that time that we think may have been caused by a supernova sending more cosmic rays our way than usual. So those events also need to be accounted for.

But there's other similar techniques using elements with longer half lives that we can use to date older or inorganic materials.

For example uranium-lead dating. When the mineral Zircon forms, it tends to include uranium but exclude lead. But over time uranium decays into lead, so if you have a sample of zircon, you can measure how much lead is in it and figure out how long ago it must have formed because that lead must have once been uranium that has decayed at a steady pace over the centuries.

So we can use methods like that to determine when certain rocks formed.

We know pretty well what the earth and moon are made of (X% silica, Y% Iron, Z% aluminum, etc. with certain amounts of different isotopes) so if we were to find some deposit of minerals that deviates a lot from what you'd find elsewhere on earth or the moon, it's pretty safe to say that that material might have originated elsewhere, like on theia.

Carbon Dating would be a good name for an online dating service for elderly people.

https://simple.wikipedia.org/wiki/Radiocarbon_dating