Sunday, July 1, 2012

Where Did That Extra Second Come From?

As most readers are probably aware, yesterday (Saturday) featured 86, 401 seconds rather than the usual 86, 400. One leap second was added to compensate for a slowing Earth rotation (actually - if factored in from 1972, 24 full seconds slower than at that time). A cutesy explanation (on ABC News) last night was that "just as humans slow down when they age so has the Earth." Well, yes...and no.

The actual physical reason has less to do with aging per se and more to do with what we call "tidal friction" resulting from the Moon's cumulative gravitational effects (part of which causes our spring and neap tides as well as regular tides on Earth). Over time those gravitational effects add up....and a leap second must be added to correct it.

Thus did the concept of Ephemeris time (ET) enter the picture and ET thereby became part of the lexicon of astronomers, and integrated within the specialized sub-discipline of astrometry. The job of astrometry is to keep careful quantitative track of any changes that would lead to time corrections, specially related to ET. This isn't so stirring an idea now, but a couple of centuries ago it was almost a given that the Earth's period of rotation about its axis was constant. (Apart from a slow secular increase due to tidal friction). 

Note that "secular changes" are essentially non-reversible, and hence proportional to time passed.

Meanwhile, the value of ET at a given instant is reckoned by extremely accurate observations of sudden variations in the longtiudes of Sun, Moon and planets which of course emanate from variations in Earth's rate of rotation. Hence, if I detect a rather sudden "jump" in the Sun's (heliographic) longitude - say during lengthy observations of a sunspot group, I can tie that in to variation in Earth's rotation. (Generally, however, such changes wouldn't occur that fast.)

To get into the nitty gritty, Simon Newcomb (1895) originally provided us with the detailed expression for the longitude of the Sun (cf. 'Spherical and Practical Astronomy Applied to Geodesy', p. 168):

L= 279 deg 41'48."04 + 129, 602, 768."13 t_e  + 1."089 t_e^2

where t_e is the length of the tropical century from the standard epoch. (Recall the 'tropical year' is the time taken by the fictitious Sun to make one passage between the mean Vernal equinox. Hence, a tropical century refers to 100 such passages.)

Now, differentiating the preceding with respect to time t_e,  the instantaneous rate of change of L per second of ET is obtained, viz.

dL/ dt_e  = 0."0410686389744 + 6."9017 x 10^-10t_e

The last factor featuring the 10 raised to the negative 10 power is an important marker. It allows us to estimate the frequency at which time corrections, say to Earth's rotation, need to be made. Thus, the equation shows us that to define Ephemeris time correctly to 1 part in 10^10 (or one part in ten billion) observations of the Moon are required over at least 5 years.

Another quantity that makes its way into such computations is the Right Ascension of the fictitious mean Sun;

RA = 18 h 38 m 45.s 836  = 8, 640, 184.s 542 t_e  =0.s 0929 t_e^2

What exactly is the fictitious mean Sun? What, for that matter, is the fictitious Sun (referenced earlier in connection with the tropical year, and century)? Basically, the former is that fictitious entity which is presumed to travel always at a uniform rate every hour and every day. This fictional creation enables us to fabricate "time zones" based on the fiction that the Earth turns uniformly through 360 degrees every 24 hours, hence through 15 degrees every hour. Hence, we set out longitude markers to reference the mean time for all locations, say within 15 degrees of longitude.

Hence, in regular standard time (not 'daylight saving') one has the time zone for the Greenwich meridian registering say 12 noon, then at a longitude of 15 deg W. the time is one hour earlier, and at 30 W two hours earlier, and so forth. All locations within the time zone agree (with few exceptions) to keep the same time.

By contrast, the fictititous Sun is not so regularized so its motion is erratic. This is also referenced to what we call "apparent solar time" or "sundial time".  Based on comparisons of the two, one can then compute what is called the "equation of time", see e.g.

Now, if we reckon Ephemeris time into the context of the "fictitious Suns" we will be using two key hour angles: h (the hour angle of the fictitious mean Sun) and h' (the ephemeris hour angle of the fictitious mean Sun. Based on this, Ephemeris time (ET) would be expressed:

ET = h + 12h

While the equation of time (EQN T) is:

EQN T =   h"   -  h'

where h" is the hour angle of the fictititous Sun.(Those who'd like to review hour angles and how to work them out can consult two earlier blogs:


I will also give here, without derivation (though I welcome energized readers attempting it using material from prior blogs) the relation between the corrected time delta T, Ephemeris time and Universal Time (UT):

delta T = ET - UT

Generally, delta T is computed - or at least used to be before the era of atomic clocks - through inverse interpolation in the ephemeris of the Sun, Moon & planets...and deducting the recorded UT time of observation. (Note: An 'Ephemeris' is a manual giving all the changing positions correlated with times, dates for celestial objects).

In the modern atomic clock -time epoch, things are (fortunately!) much more streamlined and we have powerful Cray and other supercomputers to boot. Basically, one associates the atomic time (AT) and Ephemeris time (ET) via the generic expression:

AT - ET = a + bt  + ct^2

where t denotes the time from the epoch when AT = ET + a

Hence, the coefficient 'a' determines the epoch of atomic time in relation to Ephemeris time. The coefficient 'b' is the division ratio adopted for the atomic resonator (as we know, different atomic clocks, e.g. cesium , quartz etc. have different frequencies, hence differing resonator rates). Finally, coefficient c is a cosmic constant which value may be 0 or more likely on the order of 0.s001 per year.

As we can see from the above considerations, for the most recent leap second correction:

ET - UT = 1 s  =  AT  - ET

Just be glad you only have to savor that extra second of time, not figure out when exactly you have to add it!

No comments: