Lots of fun (and useful to the OP and even to scientists lacking access to a TARDIS) stuff in that paper, by the way. For instance, T billion years ago the length of the (sidereal) day was 23.934468 + 7.432167 T - 0.727046 T² + 0.409572 T³ - 0.589692 T⁴. So you could really start to narrow things down without much more than a chronometer. If the day is 22 hours long, you know it’s not 2017.
Hm, true. That’s the same process that’s behind the Moon’s period changing, but since the Earth’s period is shorter than the Moon’s, you could gather the relevant data much quicker.
I don’t get this equation. It’s supposed to give the length of the sidereal day in hours, when T equals the number of years into the future or past? That can’t be right.
eta: Or is T in billions of years?
Billions of years into the past. There is a different polynomial if you want to go into the future. I don’t remember the domain of validity, but for a few hundred million years back there should be no problem.
But if T=1, a billion years in the past, then the length of the day would be 23.934468 + 7.432167 - 0.727046 + 0.409572 - 0.589692, or ~30 hours.
Or would a billion years in the past mean T= -1?
Then the day would be ~18 hours, which makes more sense.
Frustratingly, that paper does not appear to be on arXiv.org, and I no longer have access to institutional subscriptions, or I’d check myself. I suspect, though, that a polynomial was never the right sort of functional form to use there in the first place.
My bad; try https://www.aanda.org/articles/aa/pdf/2004/46/aa1335.pdf
I suspect the polynomial is merely given for the convenience of the geologists applying the results, and that it has no significance beyond being a valid approximation over a definite period of time. What do you think?
The Moon is currently moving away at a steady rate, but it hasn’t always been steady. Over geologic time, the rate changes, depending to a large extent on the configuration of the continents. As I understand it, the more easily tides can move around the Earth, the less the Moon moves away. Right now, with the Americas and Eurasia/Africa forming big roadblocks to tides, the rate of recession is fairly high (3.8 cm/year). But go back some 3 million years to the time before Central America connected Americas and the rate should have been less. Other continental movements over the eons would also change the rate. This would also change the rate the Earth’s rotation changes in a non-linear fashion, so you can’t use length of day for this purpose either.
Of course it’s not linear. It only has to be predictable.
Here’s an article from a couple of years ago which gives animations of how the constellations of the Big Dipper, Orion, Leo and Crux have changed and are likely to look in the future:
I seem to recall this is your field. Or am I mistaken?
Yes, but it’s not going to be done with a polynomial. Nor is it likely to be very exact.
Of course, with a time machine and the opportunity to make a broad assortment of measurements at many different times, we could very precisely calibrate the formula for the recession of the Moon (as well as refining our ephemeris for the planets, and refine our cosmological models for the CMB, and so on). In other words, the TARDIS itself could come with reference books, even very simple reference books easily understood by us mere humans, that could enable us to measure our time much better than we can with our data.
Again, sorry for the confusion. T is indeed negative when computing the length of the day in the past, and positive when computing it for the future, except in that case the polynomial is 23.934468 + 7.444649 T - 0.715049 T² + 0.458097 T³. So the approximation is composed of two polynomials glued together. Also, it is only valid for T between -0.250 and 0.250, ie 250 million years before and after now, so you cannot substitute T= -1.
Those of you suggesting it is inappropriate to use polynomials for this purpose, what would you suggest?
This thread reminded me of the xkcd “Time” comic. At one point in the narrative, several strips are used to show the stars in the sky as the night progresses. xkcd readers being what they are, the latitude and approximate date are estimated from the stars as portrayed.
I haven’t looked into the problem of tidal recession specifically, but for something monotonic and never negative, and for where the domain has no significance to “0”, my blind first pass would generally be an exponential of some sort, and then maybe use trig functions to deal with any structure left over.
Really, though, what I would do is start by modeling the recession assuming some constant tidal coupling, get a formula for that, and then fit the time variance of that coupling “constant” in the solution. And I’d probably use a combination of trig functions to fit the coupling, because that’s also going to be bounded between some extreme possible values. I might also do something to revert the constant to the mean, for extreme times where I’m not sure what it’s actually doing, so that even in extrapolation, my model would still give a reasonable approximation.
Oh, yeah, Gyrate, that was a classic. Especially how Munroe deliberately shifted the star positions to show that, with the understanding and expectation that his readership would figure it out.
I’ve multiplied together the orbital periods of Mars to Neptune and came up with 9.1 million years … does that mean any configuration of these planets is unique in any 9 million year period? …
Also, how sure are we about the ages of stars, could we narrow down our time just by looking for specific stars … “there’s star ABC but we can’t find star XYZ … we must be between 275m years and 265m years before present” …
ETA: As I remember, we’ll need at least an 8" telescope to see the disk of Neptune … which gives us positive identification of that planet …
No, you multiplied together the orbital periods of Mars and Neptune and got 9.1 million square years. What’s a square year? You also need to account for how precisely we can determine where they are in their cycles.
You can if you can identify specific stars. For short timescales, like XKCD’s Time and Antares being missing, this is doable. But over millions of years, both the positions and the spectra of stars will be radically different.
If by “star” you are willing to include “shooting star” and if by “shooting star” are willing to include meteorites and if you are able to find a carbonaceous chonodrite* meteorite and if you remembered to pack a mass spectrometer, then you would be able to date your time period within a few million years at worst, maybe as good a few thousand years at best.
*Some other types of meteorites would also work, but the best components to test would be a type of Calcium and Aluminum rich inclusion found in carbonaceous chondrites that were the first minerals to condense out of the solar nebula.