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Since our planet's magnetic poles are not usually, precisely, lined up with the geographic poles of Earth's axis, and since they move around a lot, can paleomagneticists (is that a word?) determine the dates of ancient rock formation more precisely using this extra information?

Or vice versa? (I.e., does knowing, somehow, the precise date of formation of a rock(s), then measuring its exact direction of magnetism, not just polarity, tell you where the magnetic poles were at that time?)

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An important distinction is what kind of magnetic pole we're talking about. The "exact direction of Earth's magnetic poles", would be the dip poles, and no, we can't determine these as we have too little information. The rough approximation of the field in terms of the geomagnetic dipoles though we can do.

With a single measurement alone the problem is non-unique (more than one shape of magnetic field could produce any given measurement at a single point in space and time).

That said, a lot of paleomagnetism relies on assuming that Earth's field has always been predominantly a dipole. So with this assumption you can use paleomagnetic field direction measurements to estimate the magnetic dipole latitude of the sampled rock, and thus the location of the geomagnetic dipole at that time.

So you can estimate the geomagnetic dipole location from a dated paleomagnetic rock sample. The other way round is possible but it's a chicken-egg situation - you don't know what the paleomagnetic field should look like at a particular time without having measurements from dated rocks to infer this already, and there may be multiple times and places that a measurement would fit.

And paleomagneticist is a valid (hard to spell) term, though some may prefer the way cooler sounding paleomagician...

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