# [LEAPSECS] the big artillery

Michael Deckers michael.deckers at yahoo.com
Sun Nov 2 16:42:59 EST 2014

```   On 2014-11-02 19:04, Warner Losh wrote:

> On Nov 2, 2014, at 11:21 AM, Michael Deckers via LEAPSECS <leapsecs at leapsecond.com> wrote:
>
>>    For instance, the differential rate d(TAI - UT1)/d(UT1) is
>>    published as LOD by the IERS as a "dimensionless" number
>>    with unit ms/d. To compute this, one must be able to
>>    subtract the reading of UT1 from that of TAI, and to
>>    compute the difference numerically one has to convert to
>>    equal units. The rate is computed correctly /only/ if
>>    one assumes that a second of TAI equals a second of UT1.
>
> This isn’t entirely true. You have to compute the length of the
> different time scales to the same seconds. You can compute the
> difference by comparing the clock readings at a fixed point
> in time after interpolation to a common grid. This will give you the
> difference in terms of the units of the common grid. If you select UT1
> as the common grid, then you can also get a rate to come up
> with the unit less number.

Thanks for your reply. If I understand you right you are saying
that comparisons require the same time unit being used in the
expression of the time scale values. I agree.

But I must confess that I do not understand your use of "grid".

Time scales are quantities whose values can always be expressed
as a sum fundamental epoch + a time value, the latter expressed
in a common time unit. The difference between the values
(= phases) of two time scales at the same point in spacetime
thus is just the sum of the difference of their fundamental
epochs plus the difference of their time values (both
differences are again time values).

And if I compare the rates of the two time scales, then the
fundamental epochs used to express the values of either become
irrelevant because they are fixed for each time scale.

I am not sure which common grid is needed here.

> You can also compute the frequency ticking of each time scale
> in terms of one or the other (or a third independent one) to compute
> the frequency error of one or both of the time scales. Once you have
> a frequency error (or difference), conversion of units is trivial. This is
> more likely how the LOD drift number is computed. It’s how you compare
> different atomic clocks to say this one is slow, that one is fast and assign
> a frequency error to each one (and a similar construct to assign the
> phase error of the PPS each one is producing). ...................

Yes, measuring the differential quotient d(TAI)/d(UT1) and
measuring the "drift rate" LOD = d(TAI - UT1)/d(UT1)
= d(TAI)/d(UT1) - 1 are obviously equivalent.

> .............................................. There are a variety
> of ways to measure these differences (though UT1 something has to
> involve astronomy since it is an observational time base) and compute
> these numbers.

Well, most time scales are observed, directly or indirectly -- just
the relationships TCB <-> TDB and TCG <-> TT are fixed, and UTC
and the many civil time scales are determined by fiat.

> Also, UT1 were ticking in SI seconds, there would be no rate difference. :)

No. The unit used to express the values of a time scale does not
determine the rate of the time scale.

UT1 is a timescale that ticks 1 SI second when the Earth Rotation Angle
increases by exactly (2·π rad)/86 636.546 949 141 027 072, and TCB
ticks 1 SI second when proper time at the barycenter of the solar
system increases by 1 SI second. Each of these time scales is
defined or extended to the geoid where their rates differ from
that of TAI.

Michael Deckers.

```