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I understand that the number of days per year has changed throughout the history of the Earth. Apparently there were once over 400 days per trip around the sun.

How long will it take approximately for the Earth rotation to slow some more to 365 days a year, with no extra time to make leap years?

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    $\begingroup$ Do you mean "how long will it be before the revolutionary period of the Earth around the Sun matches an integer multiple of the rotational period of the Earth?" Or are you talking about calendar days? Currently, the year and the day are defined such that there will not be an integer match in your or my lifetime. $\endgroup$ – Bill N Jul 30 '15 at 20:51
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    $\begingroup$ According to the [Washington Post][1]: "For Jurassic-era stegosauruses 200 million years ago, the day was perhaps 23 hours long and each year had about 385 days" Assuming the rate of change stays the same, It slows by 20 days (385-365) in 200 million years, which make one day every 10 million year. Leap years are caused by approximately 1/4 day extra every year. So that extra should be eliminated by slowdown in about 2.5 million years. Melting the polar icecaps shortens the wait by 20 000 years. $\endgroup$ – babou Jul 30 '15 at 21:21
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    $\begingroup$ @babou According to this resource the rate of change is non-linear. spacemath.gsfc.nasa.gov/earth/6Page58.pdf $\endgroup$ – Myles Jul 30 '15 at 21:26
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    $\begingroup$ These figures differ from mine. But why don't you do the computation yourself. There is no precise answer anyway. I am editing your question so that I has a chance of being reopened. What your table shows is that any prediction will be extremely approximate, though the variations in rate are surprising. $\endgroup$ – babou Jul 30 '15 at 21:37
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    $\begingroup$ The rate of change depend on too many factors. Any alteration in Earth's moment of inertia is enough to change how long is the day. Also, you still have tide dampening which is making the Moon move away from Earth and making a day bigger. $\endgroup$ – Andre Maizel Jul 30 '15 at 21:58
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With the data you linked to: http://spacemath.gsfc.nasa.gov/earth/6Page58.pdf

A linear fit seems fairly reasonable for the given data, but using it to predict the near future is not really possible, since the data deviates from a linear trend a fair amount.

I used linear regression to get that the change in the number of days in a year versus the number of years from now (in millions) is given by $$\Delta D = -0.13\Delta T$$

So if we let $\Delta D = -0.2425$, we get that $\Delta T = 1.87$. So we estimate it would take 1.87 million years to have exactly 365 days in a year, which is on the same order as the estimation made by babou (since we both assumed linearity).

However, as I said above, the data does deviate from the linear regression a fair amount. I found that the mean absolute difference between the linear regression and the data points was about 8 days (max = 19 days, min = 0.4 days), which is a lot more than the 0.2425 days that we are dealing with.

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  • $\begingroup$ And it only takes a couple major meteorite strikes on either the Earth or the Moon to mess up the extrapolation :-) $\endgroup$ – Carl Witthoft Jul 31 '15 at 11:48
  • $\begingroup$ "A linear fit seems fairly reasonable for the given data" and "the data does deviate from the linear regression a fair amount" seem to contradict each other. By the data the length has decreased 7 days in the past 220 million years and 39 in the 220 before that. Also the length decreased by 69 days between 900 and 600 million years ago but only 42 days in the 600 million since then. $\endgroup$ – Myles Aug 4 '15 at 18:12
  • $\begingroup$ Yeah, I was a bit unclear on that part. What I meant with the first statement was that the data seems to follow a fairly linear trend, and by the second statement I meant that the data points still deviate from the linear regression line somewhat (basically like noise around the line). I get $R^{2} = 0.91$ for a linear fit, so it's not too bad. I think the change in the number of days from 900 to 600 million years ago compared to 600 million years ago until now is within noise bounds. Also note that we only have two data points for the 900 to 600 million years ago time span. $\endgroup$ – Santeri Aug 4 '15 at 18:37
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I've been out of school too many years to relearn Lagrange interpolation by hand but Wolfram Alpha will interpolate 5 data points before it breaks. Using the 5 most recent data points in http://spacemath.gsfc.nasa.gov/earth/6Page58.pdf this gives an aproximation of

y=-3.2957×10^-9 x^4-5.02018×10^-6 x^3-0.00147727 x^2-0.147798 x+365.25

where y is the number of days and x is millions of years (today being x=0). Solving this equation for y=365.0 gives x=1.66 million years.

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