### Re: The real problem with leap seconds

On 2006-01-10, Mark Calabretta wrote: I can't let this one pass - UTC is continuous and monotonic. In fact, ignoring differences in origin, UTC = TAI. Surprised? If so then you're confusing a quantity with its representation (though in good company in doing so). I do not understand. As a function of TAI, UTC is neither continuous nor monotone increasing in the mathematical sense. In the defining document, [ITU-R TF.460-6], UTC is given as UTC = TAI - DTAI where DTAI is a step function of TAI assuming as values only integral multiples of 1 s. Unless constant, such a function is not continuous on a connected domain in the mathematical sense. Hence, UTC is not a continuous function of TAI. At some instant when TAI took a value in the positive leap second between 2006-01-01 + 00 h + 00 min + 32 s and 2006-01-01 + 00 h + 00 min + 33 s (the exact instant is not clear from [ITU-R TF.460-6 2002]), DTAI jumped from 32 s to 33 s; thus, UTC is not a monotone increasing function of TAI either. Perhaps you consider UTC as a function of something other than TAI where it may well be continuous or monotone (eg, as a function of itself, UTC trivially is both continuous and monotone). Reference: [ITU-R TF.460-6] Recommendation ITU-R TF.460-6 Standard-frequency and time-signal emissions. 2002 Geneva. Michael Deckers

### Re: The real problem with leap seconds

[A lot of discussion on this list seem to revolve around people understanding terms in different ways. In an impractical example of that spirit...] I do not understand. As a function of TAI, UTC is neither continuous nor monotone increasing in the mathematical sense. To say if TAI is a monotone function of UTC, you need to put an order on the set of possible TAI and UTC values. To say if UTC is a continious function of TAI, you need to put a topology on both. To me, TAI seems to be a union of copies of [0,1) labelled by YEAR-MM-DD HH:MM:SS where you glue the ends together in the obvious way and SS runs from 00-59. You then put the obvious order on it that makes it look like the real numbers. OTOH, UTC seems to be a union of copies of [0,1) labelled by YEAR-MM-DD HH:MM:SS where SS runs from 00-60. You glue both the end of second 59 and 60 to the start of the next minute, in adition to the obvious glueing. I haven't checked all the details, but seems to me that you can put a reasonable topology and order on the set of UTC values that will make UTC a continious monotone function of TAI. The topology is unlikely to be Hausdorf, but you can't have everything. DTAI jumped from 32 s to 33 s; thus, UTC is not a monotone increasing function of TAI either. Since DTAI involves subtracting quantities that aren't real numbers, you can't conclude that a discontinuity in DTAI results in a discontinuity in UTC. David.

### Re: The real problem with leap seconds

On 11 Jan 2006 at 0:08, Tim Shepard wrote: If humans spread out to other places besides the earth, an earth-centric time scale might begin to seem somewhat quaint. Distributing leap second information to a Mars colony seems kind of silly. As I recall, the NASA Mars missions are using Mars-centric time scales, which include a Martian second that has a different length from the SI second in order for the different-length Martian day (called a Sol) to be subdivided into a familiar 24 hours composed of 60 minutes each with 60 seconds. If, however, this Martian second is actually defined as a particular multiple of the SI second, then the use of leap seconds on Mars would ultimately be necessary to account for any future changes in the length of the Martian day. -- == Dan == Dan's Mail Format Site: http://mailformat.dan.info/ Dan's Web Tips: http://webtips.dan.info/ Dan's Domain Site: http://domains.dan.info/

### Re: The real problem with leap seconds

On Wed 2006-01-11T09:01:07 -0500, Daniel R. Tobias hath writ: If, however, this Martian second is actually defined as a particular multiple of the SI second, then the use of leap seconds on Mars would ultimately be necessary to account for any future changes in the length of the Martian day. The martian second in their case is defined purely for the sake of the local solar time of the solar powered rovers which are effectively stationary on the planet. Something akin to the long-used Newcomb expression for mean solar time is more than sufficient. In that respect the rovers are sundials. The mission clock on the lander has a more uniform time scale used for the sake of such things as the cool photos of the martian moons transiting the sun. Leap seconds for Mars seem irrelevant until there are VLBI antennae on Mars performing routine observations. The atomic clocks used by such VLBI antennae will not, however, keep synchronized with earth using anything less than a fully general relativistic expression for the intercomparisons. -- Steve Allen [EMAIL PROTECTED]WGS-84 (GPS) UCO/Lick ObservatoryNatural Sciences II, Room 165Lat +36.99858 University of CaliforniaVoice: +1 831 459 3046 Lng -122.06014 Santa Cruz, CA 95064http://www.ucolick.org/~sla/ Hgt +250 m

### War of the Worlds

I see Steve Allen has already supplied a thorough answer. Interested individuals might also scrounge through the list archives (http:// rom.usno.navy.mil/archives/leapsecs.html) since the topic has come up before. In fact, Demetrios Matsakis speculated on solar system wide timescales even before this list was started. My own skepticism over more extreme flights of fancy is expressed under Future Directions in http://iraf.noao.edu/~seaman/leap. On Jan 11, 2006, at 7:01 AM, Daniel R. Tobias wrote: If, however, this Martian second is actually defined as a particular multiple of the SI second, then the use of leap seconds on Mars would ultimately be necessary to account for any future changes in the length of the Martian day. That is indeed the issue. Is there anybody from the geophysics side who can comment on long term trends in Martian length of day? I don't have an envelope large enough, but there are various issues to consider. The Hurtling Moons of Barsoom are much smaller than our own and should have a negligible tidal breaking effect. (See http:// www.freemars.org/mars/marssys.html, for instance, for their interesting history.) And do the Earth's oceans mediate our Moon's breaking or is that a crustal phenomenon? (The Earth-Moon system should better be regarded as a double planet, than planet and satellite.) On the other hand, Mars passes much closer to Jupiter, the 800 pound gorilla of the solar system, but then it is further from King Kong - the Sun, that is - and tides are an inverse cube effect. But Mars is much smaller and has a smaller moment of inertia in the first place - but then Mars is much smaller and the lever arm to grapple with it is less pronounced. Taken all together, one suspects that LOD(Mars) is many orders of magnitude more constant than LOD(Earth). One would not be flabbergasted to be utterly wrong, however. Hopeful news for John Carter fans: http://filmforce.ign.com/articles/679/679046p1.html Rob

### Re: The real problem with leap seconds

On 2006-01-11, David Malone wrote: [A lot of discussion on this list seem to revolve around people understanding terms in different ways. In an impractical example of that spirit...] Anyway: excuse me for repeating some basics of classical mechanics; but I believe it to be necessary. To say if TAI is a monotone function of UTC, you need to put an order on the set of possible TAI and UTC values. To say if UTC is a continuous function of TAI, you need to put a topology on both. Yes: there is an order on the set of values of timescales - it is a basic property of spacetime models that one can distinguish past and present, at least locally. Spacetime is a differentiable 4-dimensional manifold, its coordinate functions are usually two times differentiable or more. In particular, the set of values of timescales does indeed have a topology (which is Hausdorff). To me, TAI seems to be a union of copies of [0,1) labelled by YEAR-MM-DD HH:MM:SS where you glue the ends together in the obvious way and SS runs from 00-59. You then put the obvious order on it that makes it look like the real numbers. TAI is determined as a weighted mean of the (scaled) proper times measured by an ensemble of clocks close to the geoid - so the values of TAI must belong to the same space as these proper times, which (being line integrals of a 1-form) take their values in the same space as the time coordinates of spacetime (such as TCB and TCG). No gluing is needed. And yes: this space is diffeomorphic to the real line. All of this is completely independent from the choice of a particular calendar or of the time units to be used for expressing timescale values. OTOH, UTC seems to be a union of copies of [0,1) labelled by YEAR-MM-DD HH:MM:SS where SS runs from 00-60. You glue both the end of second 59 and 60 to the start of the next minute, in adition to the obvious glueing. I haven't checked all the details, but seems to me that you can put a reasonable topology and order on the set of UTC values that will make UTC a continious monotone function of TAI. The topology is unlikely to be Hausdorf, but you can't have everything. If you subtract a time from a timescale value, you get another timescale value. If you mean to say that UTC takes its values in a different space than TAI then you cannot agree with UTC = TAI - DTAI, as in the official definition of UTC. And if you say that UTC - TAI can be discontinuous (as a function of whatever) with both UTC and TAI continuous then you must have a subtraction that is not continuous. Strange indeed. Where did I misinterpret your post? And can you give some reference for your assertions? Michael

### Re: The real problem with leap seconds

On Wed 2006/01/11 10:47:25 -, Michael Deckers wrote in a message to: LEAPSECS@ROM.USNO.NAVY.MIL At some instant when TAI took a value in the positive leap second between 2006-01-01 + 00 h + 00 min + 32 s and 2006-01-01 + 00 h + 00 min + 33 s (the exact instant is not clear from [ITU-R TF.460-6 2002]), DTAI jumped from 32 s to 33 s; thus, UTC is not a monotone increasing function of TAI either. Here in a topology-free way is what the axis labels of my graph look like during the said leap second insertion: UTC axisTAI axis DTAI 2005/12/31 23:59:58 2006/01/01 00:00:3032 2005/12/31 23:59:59 2006/01/01 00:00:3132 2005/12/31 23:59:60 2006/01/01 00:00:3232 60.932.9 32 60.99 32.99 32 60.999... 32.999... 32 2006/01/01 00:00:00 2006/01/01 00:00:3333 2006/01/01 00:00:01 2006/01/01 00:00:3433 The seconds keep step and the graph has no gaps, jumps or kinks. The only difference between UTC and TAI is one of representation, like the current year which may be represented as 2006 or MMVI but it's still the same year. Mark Calabretta ATNF

### Re: The real problem with leap seconds

In message: [EMAIL PROTECTED] Mark Calabretta [EMAIL PROTECTED] writes: : On Wed 2006/01/11 10:47:25 -, Michael Deckers wrote : in a message to: LEAPSECS@ROM.USNO.NAVY.MIL : :At some instant when TAI took a value in the positive leap second between :2006-01-01 + 00 h + 00 min + 32 s and 2006-01-01 + 00 h + 00 min + 33 s :(the exact instant is not clear from [ITU-R TF.460-6 2002]), DTAI jumped :from 32 s to 33 s; thus, UTC is not a monotone increasing function of :TAI either. : : Here in a topology-free way is what the axis labels of my graph look : like during the said leap second insertion: : : UTC axisTAI axis DTAI :2005/12/31 23:59:58 2006/01/01 00:00:3032 :2005/12/31 23:59:59 2006/01/01 00:00:3132 :2005/12/31 23:59:60 2006/01/01 00:00:3232 : 60.932.9 32 : 60.99 32.99 32 : 60.999... 32.999... 32 :2006/01/01 00:00:00 2006/01/01 00:00:3333 :2006/01/01 00:00:01 2006/01/01 00:00:3433 : : The seconds keep step and the graph has no gaps, jumps or kinks. Shouldn't the DTAI increment for the leap second? That's the point of the leap second... Or is that only needed when you do the POSIX time_t thing of repeating 59? In computer science, regular things are easy. Irregular ones are hard and prone to errors. The :60 kludge makes a regular 1-1 mapping function for time into a table driven nightmare :-(. : The only difference between UTC and TAI is one of representation, like : the current year which may be represented as 2006 or MMVI but it's still : the same year. The point isn't that one can construct a 'second labelling' that is unambiguous, monotonic and increasing, but rather that when one distills the UTC time down to a single number, you necessarily have ambiguities and discontinuties in that function. Of course there's an implicit assumption that the conversion function is the same as POSIX's time_t, otherwise you would be using TAI or a count of actual seconds. We call it PPSes or pulses at work to distinguish it. We label UTC time as having leap seconds swizzled in, and TAI time as unswizzled. Warner

### Re: War of the Worlds

Rob Seaman scripsit: I don't have an envelope large enough, but there are various issues to consider. The Hurtling Moons of Barsoom are much smaller than our own and should have a negligible tidal breaking effect. (See http:// www.freemars.org/mars/marssys.html, for instance, for their interesting history.) And do the Earth's oceans mediate our Moon's breaking or is that a crustal phenomenon? (The Earth-Moon system should better be regarded as a double planet, than planet and satellite.) On the other hand, Mars passes much closer to Jupiter, the 800 pound gorilla of the solar system, but then it is further from King Kong - the Sun, that is - and tides are an inverse cube effect. But Mars is much smaller and has a smaller moment of inertia in the first place - but then Mars is much smaller and the lever arm to grapple with it is less pronounced. Taken all together, one suspects that LOD(Mars) is many orders of magnitude more constant than LOD(Earth). One would not be flabbergasted to be utterly wrong, however. I wouldn't be too quick to dismiss tidal braking from Phobos. It's awfully close to Mars, and tidal braking is as you say an inverse-cube effect. The formula (kai Wikipedia) is (2GMmr)/R^3, where M and m are the masses, r is the radius of the primary, and R is the orbital radius of the secondary. The mass of the Earth-Moon system is eight orders of magnitude larger than the Mars-Phobos system, and the radius of Earth is only twice the radius of Mars, but the ratio of the cubed orbital radii is five orders larger for Phobos than for the Moon. So the tidal acceleration of the Moon toward the Earth is only some three orders larger than Phobos's toward Mars. That puts the effect in the same ballpark. How much difference in actual slowing can be attributed to Earth's ocean and Mars's lack of one I don't know. (See http://www.madsci.org/posts/archives/oct98/908453811.As.r.html for the relevant masses and radii.) -- Eric Raymond is the Margaret Mead John Cowan of the Open Source movement.[EMAIL PROTECTED] --Bruce Perens, http://www.ccil.org/~cowan some years agohttp://www.reutershealth.com

### Re: The real problem with leap seconds

On Wed 2006/01/11 20:58:25 PDT, M. Warner Losh wrote in a message to: [EMAIL PROTECTED] and copied to: LEAPSECS@ROM.USNO.NAVY.MIL : 60.999... 32.999... 32 :2006/01/01 00:00:00 2006/01/01 00:00:3333 :2006/01/01 00:00:01 2006/01/01 00:00:3433 : : The seconds keep step and the graph has no gaps, jumps or kinks. Shouldn't the DTAI increment for the leap second? It goes from 32 to 33 at precisely 2006/01/01 00:00:00 (UTC) (or at precisely 2006/01/01 00:00:33 (TAI) if you prefer), as shown in the above table. Refer to the last page of the current IERS Bulletin A, Vol. XIX No. 1, ftp://maia.usno.navy.mil/ser7/ser7.dat, which says the same as above though in a little less detail. That's the point of the leap second... Or is that only needed when you do the POSIX time_t thing of repeating 59? My comments on discontinuity etc. do not apply to time_t which is not the same thing as UTC. In computer science, regular things are easy. Irregular ones are hard and prone to errors. The :60 kludge makes a regular 1-1 mapping function for time into a table driven nightmare :-(. It sure does. The point isn't that one can construct a 'second labelling' that is unambiguous, monotonic and increasing, but rather that when one distills the UTC time down to a single number, you necessarily have ambiguities and discontinuties in that function. time_t has ambiguities and discontinuties, but that is not a necessary consequence of leap seconds. To their credit the CCIR did not stuff up in specifying the clock notation to be used for UTC. Mark Calabretta ATNF

### Monsters from the id

What now, Dr. Moebius? Prepare your minds for a new scale... of physical scientific values, gentlemen.Mark Calabretta takes the lazy man's way out and appeals to facts: Here in a topology-free way is what the axis labels of my graph looklike during the said leap second insertion: UTC axis TAI axis DTAI 2005/12/31 23:59:58 2006/01/01 00:00:30 32 2005/12/31 23:59:59 2006/01/01 00:00:31 32 2005/12/31 23:59:60 2006/01/01 00:00:32 32 60.9 32.9 32 60.99 32.99 32 60.999... 32.999... 32 2006/01/01 00:00:00 2006/01/01 00:00:33 33 2006/01/01 00:00:01 2006/01/01 00:00:34 33The seconds keep step and the graph has no gaps, jumps or kinks.Now let's look at a leap hour introduced as an extra "fall back" hour: UTCTAI 2600-12-31T23:59:58 2601-01-01T00:00:31 33 2600-12-31T23:59:59 2601-01-01T00:00:32 33 2600-12-31T23:00:00 2601-01-01T00:00:33 33 2600-12-31T23:00:01 2601-01-01T00:00:34 33 (?) ... ... 2600-12-31T23:59:58 2601-01-01T01:00:31 33 (?) 2600-12-31T23:59:59 2601-01-01T01:00:32 33 (?) 2601-01-01T00:00:00 2601-01-01T01:00:33 3633I chose to introduce the leap hour on December 31 - I don't believe the proposal indicates the date for doing so. Folks have been tossing around the notion of aligning this with daylight saving time - but DST in what locality? Does anyone really believe that a leap hour would be introduced on different calendar dates worldwide? (It seems to me that the one time it is guaranteed NOT to occur is during a daylight saving transition.)Not satisfied with the ITU position that UTC should merely be emasculated to correspond to TAI - 33s - Nx3600s (which, of course, really has the effect of ensuring that TAI itself will remain a completely irrelevant mystery to the public), some would completely eliminate UTC from the equation (or is it that they would eliminate TAI?) Something like: GMT TAI 2600-12-31T23:59:58 2601-01-01T00:00:31 2600-12-31T23:59:59 2601-01-01T00:00:32 2600-12-31T23:00:00 2601-01-01T00:00:33 2600-12-31T23:00:01 2601-01-01T00:00:34 ... ... 2600-12-31T23:59:58 2601-01-01T01:00:31 2600-12-31T23:59:59 2601-01-01T01:00:32 2601-01-01T00:00:00 2601-01-01T01:00:33 But we're to believe that this would be implemented as an omitted "spring forward" hour - ignoring the fact that many localities don't currently have this option because they don't use DST at all - can't omit what you don't have in the first place. Well - fine, a "spring forward" event might look like: GMT TAI 2600-12-31T23:59:58 2601-01-01T00:00:31 2600-12-31T23:59:59 2601-01-01T00:00:32 2601-01-01T01:00:002601-01-01T00:00:33 2601-01-01T01:00:01 2601-01-01T00:00:34 2601-01-01T01:00:02 2601-01-01T00:00:35 But under this interpretation we're to believe that the very notion of international civil time is anathema (except perhaps for TAI with some oddball persistent 33s offset and either a one hour gap or one hour repetition every few hundred years). What this means is that *local* civil/business/legal time contains this gap or this repetition. I suspect we can agree that the civilians/businesspersons/lawyers won't care whether the issue is local or not, all they are going to see is a repeated time sequence or a gap - and with no possibility of appeal to standard time, because standard time as we know it simply won't exist anymore.And historical time? Well, historians will simply have to get with the program. Suck it up. Perhaps loudspeakers will announce the arrival of the leap hour (or leap timezone migration event) with the admonition to refrain from historically significant activity for the space of one hour. (This announcement would be unnecessary in the Washington, D.C. city limits, of course.)And more to the point, since international time is a fiction, this gap/overlap in civil/business/legal/historical time would occur twice

### Re: War of the Worlds

I referenced this page, but missed the most interesting part of it: http://www.exo.net/~pauld/physics/tides/tidalevolution.htm The height of a tidal bulge on a planet is proportional to the inverse cube of the distance between the planet and the object causing the tidal bulge. The torque which slows down the planet is proportional to the inverse sixth power of the distance. presumably because the the same third power works both on the size of the bulge and the differential pull on the bulge. That suggests that Phobos might raise a lower tide than the sun, but yet have a greater tidal braking effect. But I expect the speed of response of the planet to the tidal force could still play a role in comparing the Phobos and solar effects. Data on the speed of change of the orbit of Phobos combined with the conservation of angular momentum should give a good handle on the size of the effect from Phobox. http://en.wikipedia.org/wiki/Phobos_%28moon%29 tidal forces are lowering its orbit, currently at the rate of about 1.8 metres per century, and in about 50 million years it will either impact the surface of Mars or (more likely) break up into a planetary ring. But its too late to do that math tonight -Neal On Wed, Jan 11, 2006 at 11:41:27PM -0700, Neal McBurnett wrote: On Wed, Jan 11, 2006 at 11:44:13PM -0500, John Cowan wrote: I wouldn't be too quick to dismiss tidal braking from Phobos. It's awfully close to Mars, and tidal braking is as you say an inverse-cube effect. The formula (kai Wikipedia) is (2GMmr)/R^3, where M and m are the masses, r is the radius of the primary, and R is the orbital radius of the secondary. The mass of the Earth-Moon system is eight orders of magnitude larger than the Mars-Phobos system, and the radius of Earth I assume you mean the mass of phobos vs the mass of the moon, not the systems, since that is what fits in the raw numbers and equations you provide. But that is less than 7 orders of magnitude different, as I read your reference. is only twice the radius of Mars, but the ratio of the cubed orbital radii is five orders larger for Phobos than for the Moon. So the tidal acceleration of the Moon toward the Earth is only some three orders larger than Phobos's toward Mars. That puts the effect in the same ballpark. But the tides from the sun are very significant on earth, and much more pronounced on mars. (See http://www.madsci.org/posts/archives/oct98/908453811.As.r.html for the relevant masses and radii.) A quick google search for mars tides yields much more useful and interesting answers. http://www.findarticles.com/p/articles/mi_m1134/is_5_112/ai_102275148 the solid-body tides on Mars--caused by the Sun, not by a Martian satellite--are large enough to indicate that at least part of that planet's core is liquid. ... Early in the study, the investigators realized only a liquid core could give rise to a tidal bulge capable of having the observed gravitational effect on the spacecraft. And how much bulge is that? About a third of an inch. Fluid core size of Mars from detection of the solar tide, Science 300:299-303, April 11,2003) But of course we need to treat the web with some skepticism. I doubt this tidbit got it right about what causes the tides (Phobos vs the sun): http://ganymede.ipgp.jussieu.fr/GB/projets/netlander/ Another way to proceed will be to measure tides produced by Phobos, one of Mars' moons. Those tides are 10 times lower than the tides produced by the Earth' Moon. As for changes in the length of the day, we have to look at the mechanism by which tides relate to the slowing of the day: http://www.exo.net/~pauld/physics/tides/tidalevolution.htm There are also tides in the solid earth. The tidal bulge is about 1 meter high. The moon pulls up this tidal bulge on the earth, there is a time delay between the pull of the moon and the time when the tidal bulge reaches its maximum height. During this time the rotation of the earth carries this tidal bulge around the planet in the direction of rotation. The moon then pulls on the mass of the tidal bulge and slows the rotation of the earth. So the degree of slowing is affected by both the size of the bulge, how delayed the bulge is, and the angular velocity of the body giving rise to the tides, making it harder to compare the effects of the sun with rapidly-moving phobos. That is what would relate to this aspect of your question: How much difference in actual slowing can be attributed to Earth's ocean and Mars's lack of one I don't know. I also note that the axial orientation of Mars changes widely back and forth, which would clearly affect the long term effects due to the sun: While Earth's tilt varies from 23 to 25 degrees, the Red Planet's actually shifts from 15 to 40 degrees over a 100 million year period I don't see a handy reference to pull all that together

### Re: War of the Worlds

Neal McBurnett scripsit: I wouldn't be too quick to dismiss tidal braking from Phobos. It's awfully close to Mars, and tidal braking is as you say an inverse-cube effect. The formula (kai Wikipedia) is (2GMmr)/R^3, where M and m are the masses, r is the radius of the primary, and R is the orbital radius of the secondary. The mass of the Earth-Moon system is eight orders of magnitude larger than the Mars-Phobos system, and the radius of Earth I assume you mean the mass of phobos vs the mass of the moon, not the systems, since that is what fits in the raw numbers and equations you provide. But that is less than 7 orders of magnitude different, as I read your reference. Actually I didn't mean either one: I meant the mass of the primary times (not plus) the mass of the secondary, the Mm in the formula. So the mass of Mars times the mass of the Phobos is ~ 10^39 kg, whereas the mass of Earth times the mass of the Moon is ~ 10^47 kg: eight orders of magnitude, as I said. Sorry for the misstatement. -- Even a refrigerator can conform to the XML John Cowan Infoset, as long as it has a door sticker [EMAIL PROTECTED] saying No information items inside. http://www.reutershealth.com --Eve Maler http://www.ccil.org/~cowan