# Re: The real problem with leap seconds

```   On 2006-01-13, Mark Calabretta wrote:

>  I have two time scales, TAI and UT1, that tick at very slightly
>  different rates.  I want to make TAI the basis for civil time keeping
>  but I need to make adjustments occasionally to keep it in step with
>  UT1.  How do I do it?
>
>  The answer provided by CCIR was to represent TAI in a variable-radix
>  notation that matches (or appears to match), to within 0.9s, that of
>  UT1 expressed in the usual calendar/clock format.  This is done by
>  varying the radix of the seconds field in a pseudo-sexagesimal clock
>  format from 60 to 61 (or in principle 59) on occasions announced 6
>
>  So if asked for a definition I would say that "UTC (post 1972) is a
>  representation of TAI such that ... (you know the rest)".
>
>  The point is that UTC is simply a representation of TAI.  "Writing UTC
>  as a real" reveals it to be TAI.```
```
I believe I'm now grasping what you mean: the rate of UTC is the same
as the rate of TAI (since 1972), that is, the derivative
d( UTC )/d( TAI ) = 1. Hence, when I integrate the "ticks" of UTC
I must get TAI, up to an integration constant. This is correct.
The integral of d( UTC ) is TAI (up to an integration constant),
but this integral is UTC only where UTC is a continuous function
of TAI.

Astronomers who "write UTC as a real" (eg, in JD or MJD notation)
want an approximation of UT1 to point their telescopes, they do
_not_ want TAI. They use UTC as a timescale whose values are
close to UT1, but whose rate nevertheless is d( UTC ) = d( TAI )
and not d( UT1 ). Such a function cannot be continuous (and it
cannot be differentiable everywhere). At the latest discontinuity
of UTC, it jumped from a little bit after UT1 to a little bit before
UT1.

Michael Deckers
```