Monday, January 02, 2006

The Leap Second

My wife and I were watching Sky News on New Year's Eve when we saw a brief interview with Mike Hapgood about the so-called Leap Second.

It really intrigued me. According to the professor, it's becoming a more serious issue since even a one second discrepancy can cause an error of quite a few meters on very sensitive GPS systems. So a 1 second adjustment was made to the sensitive atomic clocks at the stroke of midnight on last day of 2005..

I did some research and found what looks like a U.S. Navy web site which explains the phenomenon. It says :

"This leap second can be either positive or negative depending on the Earth's rotation. Since the first leap second in 1972, all leap seconds have been positive and there were 22 leap seconds in the 27 years to January, 1999. This pattern reflects the general slowing trend of the Earth due to tidal braking."

Tidal braking ? "Braking" suggests the revolution of the earth is slowing but that and other web sites were saying things that gave me the impression the calibration could in principle go the other way. I still don't understand how this is possible although the portion I quoted above specifically says that all 22 leap seconds were "positive" and that this reflects a "slowing trend".

So what is the amount of the slowing ? Going back to the web site we read:
"The Earth is constantly undergoing a deceleration caused by the braking action of the tides. Through the use of ancient observations of eclipses, it is possible to determine the average deceleration of the Earth to be roughly 1.4 milliseconds per day per century. [...] The length of the mean solar day has increased by roughly 2 milliseconds since it was exactly 86,400 seconds of atomic time about 1.79 centuries ago (i.e. the 179 year difference between 1999 and 1820). That is, the length of the mean solar day is at present about 86,400.002 seconds instead of exactly 86,400 seconds."

Putting it in scientific notation 1.4 milliseconds per century is 1.4 x (10^-3)second/century. If the Earth is 4.55 billion years old (4.55 x 10^9) or 4.55 x (10^7) centuries old then the amount of the slowing in seconds is given as : (1.4 x 4.55) x ( 10^7 x 10^-3) = 6.37 x 10^4 = 63,700 seconds = 1,061.67 minutes=17.69 hours which means that initially an Earth day was only about 6.5 hours long.

This represents a rate of rotation four times as large as the present rate. I ask myself : Is a complete terrestrial rotation in 6.5 hours reasonable ? Currently the Earth has an equatorial bulge. What kind of bulge or deformations would one expect if the Earth were revolving four times as fast ? I then found this which refers to Lord Kelvin's calculations and discusses the bulge issue assuming the days were initially 12 hours (I don't know why my calculations yielded a day half as long) :

"Kelvin investigated the deceleration of the earth's rate of rotation due to the energy lost through tidal currents. He showed that, if the earth had been here for 7.2 billion years, its initial rate of rotation would have been twice its present rate (the days being only 12 hours long). That would have yielded four times as much centrifugal force as at present. If, as historical geologists claim, the earth was molten in its initial state, the centrifugal force would have bulged out the mass in the equatorial region, making the earth's radius 86 kilometers greater at the equator than at the poles (the radius of the earth's sea level is presently only 21.5 kilometers greater at the equator due to the centrifugal force with its present rate of spin). Kelvin reasoned that if the earth had consolidated at that time, the land masses would have retained most of that greatly oblated shape, four times its present oblateness. As the years passed the centrifugal force would have been reduced and the oceans would have settled into two very deep basins, one at the north polar region and the other at the south polar region. The continents would in that case now be extremely high in the equatorial regions, 40 miles higher than they actually are!"

I may have made some conceptual errors in my assumptions and calculations above: but, assuming I'm mostly correct the question in my mind is whether a mass the size of the Earth could have sustained those kinds of rates of rotation and still manage to end up looking like the Earth of today ? If not, then that puts a question mark over long age being attributed to the Earth, doesn't it ?

I would appreciate any comments that would clarify this to me and others reading this.

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