[LEAPSECS] the big artillery

Warner Losh imp at bsdimp.com
Wed Nov 5 10:30:24 EST 2014

On Nov 4, 2014, at 2:52 PM, Michael Deckers via LEAPSECS <leapsecs at leapsecond.com> wrote:

>   On 2014-11-04 12:34, Zefram wrote:
>> UT1 always ticks a second for that ERA increase, but Warner's point
>> is that the second of UT1 isn't an *SI* second.  

It is not, and cannot be a SI second, except by accident from time to time.
If it were an SI second, then TAI would never need leap seconds because
there cannot be any divergence, by definition. If two clocks tick at exactly
the same rate, their phase relationship cannot change. This is why TAI
time and GPS time are not diverging.

>> The time taken for
>> that ERA increase, and hence the duration of a UT1 second, very rarely
>> exactly matches an SI second.  The second of UT1 is an angular unit,
>> defined as 1/86400 circle (= 15 arcseconds), not a unit of physical time.
>   Then which unit would that be? When the IERS compute a difference
>   TAI - UT1, how do they do it? Do they convert the UT1 reading in
>   any way before they subtract? Or, if they don't, what is the unit
>   of the difference, SI seconds or "second of UT1"?  The IERS
>   Conventions certainly do not mention any of this. How could they
>   if the units would really differ?

The only practicable way to compute the difference is to compute the phase
of the two time scales, in cycles or radians, at a common point in time. Once
you have a phase difference in cycles, you can get the time difference by
converting cycles to seconds. I imagine they choose SI seconds for this last
conversion. This ‘common point in time’ is what I called the ‘grid’ in an earlier
post. A grid is just a set of common times spaced evenly out.

The cycles are of different size, to be true, but that turns out not to matter. You
are measuring the phase difference in cycles.

So to take an example. Let’s say you have two oscillators. Let’s say for the sake
of argument that you don’t know the true value of either of them. Let’s further say
that the first one happens to be 1.000 000 Hz while the second one is 1.000 001 Hz.
They differ by a part in 10^6. Now, just to run through the math, we’ll ignore the
effects of any frequency drift, flicker, etc. In a real system, you’d have to take those
into account. Let’s also make the job easier by saying that at time t_0 they both start
in phase.

After 10^6 seconds, the phase of the first one is 1 000 000.000 000 000 000 and the
phase of the second one is 999 999.000 000 999 999. The phase difference between
these two would be 0.999 999 000 001 cycles.

If each one of these signals represented a ‘second’ then you’d have a phase
difference of 0.999 999 000 001 seconds (using the first reference) or a difference
of 0.999 998 000 003 using the second one as the reference second. This gives a
difference between the two types of seconds of 0.000 001 seconds or about one part
in 10^6. This is in line frequency error. So unless you are computing the phase
difference to 5 or more places after the decimal, you won’t see a difference.

Now, back to the SI second vs the UT1 second. The UT1 second is 1E-8 or 1E-9 different
from the SI second. Unless they are computing the results to 7 or more digits, the answers
will be identical, no matter which definition of second you use.

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