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Since the earth is slowing its rotation, and as far as I know, each day is 1 second longer every about 1.5 years, how long was an earth day near the formation of earth (4.5 billion years ago)?

I wouldn't assume to just do 4.5b/1.5 and subtract, because you would think the rate of change is changing itself, as seen here from wikimedia. It is a graphical representation of data from INTERNATIONAL EARTH ROTATION AND REFERENCE SYSTEMS SERVICE. They decide when its time for a leap second (the last one being on Jun 30, 2012) The data can be found here.

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I think that BB1 is right that your figures are probably to large by several orders of magnitude. Notice that the green line does not show a steady increase (which would be expected if your figures wee right), but is instead dominated by noise. –  dmckee Jul 6 '12 at 4:41
    
The green line is a moving average of change, and is always positive (which it doesnt have to be, for days to get longer over time). At times, the slowing slows (as seen in the early 2000's) and was exceptionally fast in the 70's. If you notice the red line is generally increasing over time, which is the time of day lengthening. From the early 70s to today, they day IS 25 seconds longer. –  Cameron Aziz Jul 6 '12 at 4:48
    
@Cameron Aziz Are you sure this info is accurate? Where is this from? –  MadScientist Jul 6 '12 at 4:54
    
updated question. –  Cameron Aziz Jul 6 '12 at 5:00
    
@Cameron Aziz, There isn't enough information here to give a much better estimate. Do you have more? Is it safe to assume that the fluctuations in the rate is random? Or does the rate seem to be changing slowly in one direction (because even if it is changing in one direction over the last 30 years- 30 years is a very short sample relatively speaking). That requires statistical analysis and more data. –  MadScientist Jul 6 '12 at 5:00
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3 Answers

up vote 9 down vote accepted

According to this Scientific American article, the Earth's rotation rate just after the collision that formed the Moon was about once every 6 hours. At that time, the Moon would have been about 25,000 kilometers away. The tidal effect of the Moon is the major reason the day has been lengthening, and the Moon's orbit has been widening.

The collision is believed to have taken place about 4.5 billion years, not long after the formation of the proto-Earth.

There are still some open questions about the impact hypothesis (see the linked Wikipedia article), so this is uncertain.

I strongly suspect that the impact would have erased any information about the Earth's rotation rate before the impact. (It might be possible to estimate the pre-collision rotation rate by modelling the initial formation of the Earth; I don't know whether there's been any research in this area.)

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Actually we only gain 1.3 milliseconds every 96-100 years, not 1 second every 1.5 years! :) the shortest known Earth day was 6 hours and the longest is 24 hours & 2.5 milliseconds (today's current day), in 1820 the day was exactly 24 hours, but since it's been nearly 200 years we've gained 2.5 milliseconds to our day. So the days get longer just very shortly, I believe it'll be 15 minutes longer in 50 million years.

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Shouldn't each day be one second longer every 1.5 years, if the Earth is rotating slower? Assuming your info is accurate, (1 sec)/(1.5 years) * (4.5 billion years) = The Number of Seconds Shorter The Day Was 4.5 billion years ago. Subtract that from the number of seconds in a day now. I would convert the final answer to minutes.

But your numbers are wrong and lead to an absurd result. "The average day has grown longer by between 15 millionths and 25 millionths of a second every year" http://www.popsci.com/jessica-cheng/article/2008-09/ive-heard-earths-rotation-slowing-how-long-until-days-last-25-hours

So, do the same calculation with these numbers instead.

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shorter = typo. also, as I stated, that assumption is inheritly false. The rate of change isn't constant. I have added a link to the rate of change for the last 50 years. –  Cameron Aziz Jul 6 '12 at 4:02
    
The numbers don't make sense btw. A day cannot be more than 60*60*24 seconds shorter. This calculation implies that a day in the past was 3 billion seconds shorter than a day today (days were negative length?). –  MadScientist Jul 6 '12 at 4:08
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precisely why its not linear... –  Cameron Aziz Jul 6 '12 at 4:10
    
Although the rate is changing, you can take the average and this should give you a good estimate: "The average day has grown longer by between 15 millionths and 25 millionths of a second every year" (popsci.com/jessica-cheng/article/2008-09/…) –  MadScientist Jul 6 '12 at 4:19
    
You have failed to understand the question. The value given is the rate of slowing in this epoch, not some permanently fixed global value. –  dmckee Jul 6 '12 at 4:20
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protected by Qmechanic Mar 16 at 8:25

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