# Does Universal Time really track mean solar time?

At innumerable places on the net you can find the claim that leap seconds are inserted into UTC in order to make it track UT1 which again tracks mean solar time at the reference meridian.

However, when I tried to locate an actual definition of UT1, the closest I could find was IAU resolution B1.8 from 2000 which seems to define it implicitly from the following relationship with the Earth Rotation Angle: $$\theta = 2\pi\bigl(0.7790572732640+1.00273781191135448({\it JD}_{\rm UT1}-2451545.0)\bigr)$$

Here the magic constant 1.00273781191135448 appears to be derived from the ratio between the Earth's rotational period and the sidereal year -- but no provision for adjusting it for observed changes in that ratio seems to be made. (The least significant several digits of the constant must be wrong already).

As the Earth's rotation slows down, is UT1 therefore going to drift out of sync with mean solar time at the IERS reference meridian? Or am I missing something?

• Could you link to the IAU resolution? I think 1.00273781191135448 is also the ratio between the sidereal and solar day (which is identical to what you said since there is exactly one more sidereal day per year than there are solar days), but might make more sense in the formula).
– user21
Commented Apr 24, 2018 at 16:31
• @barrycarter: Almost, except that if Wikipedia is to believed, a "sidereal day" commonly denotes the time it takes for the earth make one rotation relative to the (precessing) equinoxes, whereas the rotation period relevant here is that relative to the fixed stars (that is, actually distant quasars). There's a difference of several milliseconds. Commented Apr 24, 2018 at 16:50
• nist.gov/time-and-frequency-services/s-so and astronomy.swin.edu.au/cosmos/S/Sidereal+Day say "fixed stars", but they might be approximating. At one point, I found a URL that gives the exact length of the sidereal day as measured by observatories, but I can't seem to find it now. physics.stackexchange.com/questions/43161/… also discusses this issue.
– user21
Commented Apr 24, 2018 at 18:28

This question asks in effect: Is there a discrepancy, perhaps progressive, between the official time-measure UT1, and some other credible measure of mean solar time? This is harder to answer than it might first seem.

UT1 is the current official representative of mean solar time: it is doubtful that any independent measure of mean solar time (other than direct relatives of UT1 such as UT0 or UT2) has been recently defined or computed to any precision. (The method of the equation of time as normally carried out is only roughly approximate, because it ignores the many solar perturbations discovered since the early 18th-c. and its results may vary from accurate mean solar time by up to about 3 seconds: see Hughes et al. (1989), "The equation of time", http://adsabs.harvard.edu/abs/1989MNRAS.238.1529H .) But as the question indicates, one reason for questioning UT1 is that solar time naturally depends on two independent variables, related to the angular rates of the earth's axial rotation and of the earth's orbital motion around the sun: In contrast, the official UT1 calculation appears to condense these into a single constant ratio, thus losing one of the independent variables of the natural physical model. This may well raise suspicion that UT1 deviates or will deviate from a more physically-defined mean solar time.

There is in fact a (no-longer-official) alternative and physical basis for calculating mean solar time. This can be gathered from the history of 19th-century mean solar time definitions and calculations. At the same time, there is also some documentation from the history to indicate the origin and nature of some of the steps taken in the 20th/21st-century, to change the old 19th-century physical model to the current (and very non-transparent!) basis of calculation for UT1, which is described in: (N Capitaine et al. (2003), "Expressions to implement the IAU 2000 definition of UT1", Astronomy & Astrophysics 406, 1135-1149, http://adsabs.harvard.edu/abs/2003A%26A...406.1135C ).

Classical method for determining mean solar time (Nautical Almanac, 19th-c.): (Not discussed here is the method of the equation of time, which grew more cumbersome as perturbations of the apparent solar motion were successively discovered during the 18th-c.; all of them had to be included in precise calculations.) A simpler equivalent method for mean solar time, to a suitable astronomical precision for its period, was provided by Nevil Maskelyne's tables of 1774 (found at http://docs.lib.noaa.gov/rescue/rarebooks_1600-1800/QB12M31774.pdf), especially his table of the Sun's mean longitude converted from degrees to hours and given to a precision of 0.01 time-second. His instructions showed how to use this to convert observed sidereal time to mean solar time. It may be unclear whether Maskelyne intended this method to make a standard out of Greenwich mean time, but that seems to be what happened. Thus initially, according to J J Lalande's 'Astronomie' (3 editions, Paris, 1764, 1771, 1792) methods for mean time still relied usually on the equation of time, but Lalande's third edition of 1792 (vol.1, https://archive.org/details/astronomielalande01lala , see art.1014-5, p.361) added a mention of Maskelyne's table, commenting that 'in England' it was used to find the mean time of an observation. The Nautical Almanac eventually began (from 1833) to tabulate the relation between sidereal and mean solar time (Sidereal time at (Greenwich) mean noon), and during the rest of the 19th-c. such tabulations provided what became standard observatory methods of finding mean solar from sidereal time (see J C Adams 1884, "On the Definition of Mean Solar Time", Observatory 7, 42-44, http://adsabs.harvard.edu/abs/1884Obs.....7...42A ).

The method has a simple physical basis in spherical astronomy, using a division into two parts of the arc of the celestial equator that extends from the equinoctial point eastwards to the point culminating at the meridian -- shown in the diagram below.

(The upper part of the diagram represents the celestial equator above the horizon (marked E W) for an observer in the northern hemisphere looking south. M is the culminating point where the equator intersects the (south) meridian. Q is the equinoctial point 'first point of Aries'. QS represents the sun's mean longitude reckoned from the equinox: it has been (in concept) transferred from the ecliptic to the equator where it now represents the mean right ascension of the fictitious mean sun. For convenience of arrangement, the diagram represents a position in early afternoon (S somewhat west of M) in about June (S is nearly 90° east of Q). (N) is below the horizon.)

The time-definitions and explanations of J-J de Lalande (1792, 'Astronomie', 3rd ed. vol.1, art.1014, p.361, https://archive.org/details/astronomielalande01lala) state clearly (expressing right ascensions in degrees, according to Lalande's custom):-

"The mean longitude of the sun, plus the mean time converted to degrees, gives the right ascension of the mid-heaven." That translates, in terms of the diagram here, to: QS + MS = QM .

Lalande also pointed out (art.1015) that "The Sun's right ascension, or that of the star used to find the right ascension of the midheaven, should be counted from mean equinox like the mean longitude of the sun ....". Modern readers have to note further that (a) observed and tabulated 'sidereal time' up to about the end of the 19th century always included the effect of nutation in right ascension, i.e. it was 'apparent' sidereal time (see explanation in Nautical Almanac 1864, pp.515, https://babel.hathitrust.org/cgi/pt?id=mdp.39015068159329 ), and (b) also in those days tabular mean longitudes of the sun still included the effect of mean aberration: Newcomb's tables of 1895-8 were the first to attempt to remove it.

Lalande (art.1015) continued: "it follows that the right ascension of the mid-heaven" {i.e. sidereal time} "added to the complement to 24h of the Sun's mean longitude gives the mean time", and "this method is used in England to find the time of an observation: Mr Maskelyne has given in his tables the Sun's movement in time to hundredths of a second"." In terms of the diagram, this translates as: mean time MS = QM + (24h - QS) = QM - QS .

A modern version of the foregoing classical method for defining mean solar time: During the period when methods like the foregoing were current, sources of the solar tables used to compute the annual almanacs were occasionally updated, when newer and more correct theories and tables became available. Thus, for the (UK) Nautical Almanac, Delambre's mean elements for the Sun were used with minor corrections down to 1833, then Bessel's mean elements from 1834 to 1863, and Leverrier's from 1864 to the end of the century. The principle remained unchanged although slight discontinuities in the tabulations occurred at the points of changeover. Accordingly, a modernized version of the same method can be implemented using recent accurate data for the sun's mean longitudes. Two modern sources of mean elements/longitudes for the sun (both in terms of ephemeris/dynamical time TT) are available: in J L Simon et al. (1994) "Numerical expressions for precession formulae and mean elements for the Moon and the planets" (derived from the JPL ephemeris DE200 which was the official basis of the Astronomical Almanac from 1984-2002), Astronomy & Astrophysics 282, 663-683, http://adsabs.harvard.edu/abs/1994A%26A...282..663S ; and in J Chapront et al., "A new determination of lunar orbital parameters ..." (derived from the JPL successor ephemeris DE405 in official use 2003-14), Astronomy & Astrophysics 387, 700-709, http://adsabs.harvard.edu/abs/2002A%26A...387..700C . The solar positions in those two ephemerides hardly differ from each other for present purposes. Mean solar longitudes reckoned from mean equinox of date are given in Simon et al. The data of Chapront et al. are for a fixed equinox of J2000 to which general precession in longitude has to be applied. Both sets of modern mean longitudes are geometric, leaving mean aberration to be applied as -20".4955 in arc. With that adjustment they can then be used like the old mean longitudes, converted to time, treated as right ascensions of the fictitious mean sun, and applied in the formulae given above to relate mean solar time with mean sidereal time. The results may be compared with current values for UT1. {Some example calculations will be added if requested.}

Modern IERS zero meridian: It is worth noting that it appears no allowance is needed for any difference between time at the old transit instrument at Greenwich observatory and time at the new IERS reference (zero) meridian. The ground track of the IERS reference meridian is located about 102m to the east of the meridian line through the old transit instrument. A paper of S Malys et al. (2015), "Why the Greenwich meridian moved" (J Geodesy 89, 1263-1272, at http://adsabs.harvard.edu/abs/2015JGeod..89.1263M ) accounts for the difference of ground track as an effect of local gravitational deflection of the vertical. Thus, while the ground track of the IERS zero meridian is appreciably east of the old Airy transit instrument, by a distance which would make a rotation of about 5.3" arc in longitude or about 0.35 time-seconds if it represented a geocentric rotation on a spheroidal earth, there has in fact been no rotation of the reference frame (within measurement error): the deflection of the vertical makes the difference a parallel displacement. With the nominally meridian (but off-geocentric) plane of the old transit instrument practically parallel to a geocentric meridian plane through the more easterly ground track of the IERS reference meridian, a transit instrument mounted in the IERS geocentric meridian plane would 'see' stellar transits at the same times as those occurring at the Airy transit instrument.

Modern changes in mean solar time/UT1: The 19th-century Nautical Almanac standard for mean solar time was altered in several successive ways towards the current standard for UT1. (Not discussed here are the corrections for observed polar wandering and seasonal effects leading to the differences between UT0, UT1 and UT2.) But there was a history of 19th-c. controversy about one of the changeovers mentioned above from older to newer solar data, worth brief description because it foreshadows and may possibly have influenced some of the 20th/21st-c. departures from the use of mean solar longitudes.

At one of the changeovers (from Bessel's data down to the end of 1863, to LeVerrier's data used from 1864 on), a discontinuity of a little over half a second was generated in the tabular relation between sidereal and mean solar time (shown e.g. by the daily tabular values of sidereal time at mean noon in the Nautical Almanac for '1863 Dec 32' and 1864 Jan 1) (data for Greenwich mean noon 1864 Jan 1 in the 1864 volume, at https://babel.hathitrust.org/cgi/pt?id=mdp.39015068159329 , and for the same physical day in the 1863 volume where it was designated '1863 Dec 32', https://babel.hathitrust.org/cgi/pt?id=mdp.39015068159006 ).

There was a brief explanation and apology in the 1864 preface: the step-change arose because LeVerrier's theory had identified new long-term solar periodic perturbations: as long as these remained undiscovered, their effect had been implicit in the old mean elements, and so they had to be removed from the new; the largest of them due to Mars and Jupiter was near its peak at the start of 1864 and responsible for nearly all the time-discrepancy of the changeover.

This changeover and discontinuity attracted little attention for nearly twenty years; but then E J Stone, an astronomer of some repute and director of a university observatory, was trying in the 1880s to account for growing discrepancies between observed places of the moon and positions calculated by Hansen's 1857 'Tables de la Lune'. Stone seized (quite mistakenly), as the cause of the lunar errors, on the 1864 discontinuity of about half a second in calculated mean solar time arising from the changeover from Bessel's mean solar elements to those of Leverrier (E J Stone, 1883 MNRAS 43, 335-345 and 401-407). The two effects were unrelated and Stone had also made a mistake that exaggerated errors by a factor of about 365. Stone's mistakes were pointed out in print by no less a quartet than John Couch Adams (1883 MNRAS 44, 43-47; 1884 MNRAS 44, 82-84), Arthur Cayley (1883 MNRAS 44, 47-49; 1884 MNRAS 44, 84-85), Simon Newcomb (1884 MNRAS 44, 234-5; 1884 MNRAS 44, 381-3; 1894 MNRAS 54, 286-8) and G B Airy (1883 Observatory 6, 184-5), but Stone stubbornly stuck to his position. He also wrote a vehement paper for the Royal Society, based on his faulty analysis of the lunar errors, in which he insisted that no revision should ever again be allowed in future to create any discontinuity in the tabular measure of mean solar time (E J Stone (1883), Proc. R. Soc. Lond. 35, 135-7, http://rspl.royalsocietypublishing.org/content/35/224-226/135 ). This was of course a demand incompatible with the use of corrected solar mean longitudes whenever new theories and observations would show the need for correction. But strangely enough, Stone's principle of requiring continuity has actually been followed in the alterations to UT1 made explicitly in 1984 and 2003 (and implicitly in 1960). It is hard to tell whether or not this arose from some descendant of Stone's influence.

There have been four main occasions of alteration away from the 19th-c. practice described above: in 1900, with Newcomb's formula for time quantities; in 1960, and the introduction of Ephemeris Time into many calculations but not into the time quantities; in 1984 with a fresh official definition of UT1 adopted from Aoki et al (1982), "The new definition of Universal Time", http://adsabs.harvard.edu/abs/1982A%26A...105..359A ; and in 2003 with the introduction of the current basis of UT1, described briefly in IAU 2000 resolution B1.8 and more fully in N Capitaine (2003) "Expressions to implement the IAU 2000 definition of UT1", http://adsabs.harvard.edu/abs/2003A%26A...406.1135C .

1900: Newcomb's time formula made a very small alteration in the time-equivalent of the sun's mean longitude, by increasing the quadratic term. The motivation and effect go back to the work and practice of LeVerrier, summarised by A Gaillot in 1886: "Sur la mesure du temps", Bulletin Astronomique, Ser. I, v.3, 221-232, http://adsabs.harvard.edu/abs/1886BuAsI...3..221G . LeVerrier's analysis of precession supposed that the earth's axial rotation was perfectly uniform. He also found that his theory of the precession of the equinoxes in right ascension led to a quadratic term discordant with the quadratic term for the sun's secular acceleration in longitude. By incorporating an increment in the quadratic term in the sun's longitude to remove the 2nd-order discrepancy, the effect was to make increments in mean time closely proportional to those in sidereal time (and to transmit the supposed uniformity of earth rotation into a proportional uniformity of the derived (but altered) calculation for mean solar time). Newcomb mentioned the point briefly in his book "The elements of the four inner planets and the fundamental constants of astronomy" (1895, at https://archive.org/details/cihm_16774 , see p.188). This appears to be the historical origin of the still-current practice of making the measure of mean solar time an exact linear function of earth rotation, a point specifically raised by the current question. No-one now supposes that earth- rotation is uniform.

1960: With the introduction of Ephemeris Time (https://en.wikipedia.org/wiki/Ephemeris_time) into the official almanacs from 1960, the solar data was effectively corrected, not as had been usual by change of numbers, but by time-shifting it by a displacement DeltaT, the difference (Ephemeris Time - Universal Time). The numbers were maintained unchanged, but they related now to a different timescale. No alteration at all was made in the calculation of nominally mean solar time. The effect was to retain the buildup of the time errors constituted by the difference between the new best estimate of the sun's mean position and the old.

1984: The definition of UT1 was revised in a way that maintained continuity of value and rate at the moment of changeover, the start of 1984 (see S Aoki et al., (1982), link above).

2003: A further revision to UT1 was made, maintaining continuity at the moment of changeover, the start of 2003 (see N Capitaine et al., (2003), link above).

Summary/conclusion: The description above shows something of how the current calculations for UT1 have departed in principle from classical methods used to define and calculate mean solar time. But it seems no longer to be claimed that UT1 is mean solar time: thus the IAU 2006 NFA glossary calls UT only "a measure of time that conforms, within a close approximation, to the mean diurnal motion of the Sun". Current values of UT1 can be compared with results of a modernized version of an old standard method for mean solar time from sidereal time. It seems that differences between this and current UT1 are still well under a second, though they can be expected to increase with increasing DeltaT, the difference in seconds between ((dynamical) TT - UT1). Thus the differences probably are remaining less than allowable differences for example between UT1 and UTC. The IAU gloss seems broadly justified, although it may be of continuing interest to have a method of calculating a more traditional form of mean solar time to check on any accumulating discrepancies.

{Notice of any mistakes in all of this will be gratefully appreciated!}

• That's a long and detailed explanation, so +1, but I'm not entirely sure which answer it arrives at. No, I'm not missing anything, and in fact there is nothing that actually anchors UTC to the motion of the sun as seen from Greenwich? Commented Apr 28, 2018 at 21:11
• @Henning-Makolm : Well, the answer expressly says (with reasons and sources!) that the principles are now different, so far the tracking is fairly close, but it can be expected to diverge in future with increasing DeltaT. The answer also deals with the point about the constant ratio used between diurnal and annual rotation, and with the point about the IERS reference meridian. It also links to a more explanatory source for the current method than IAU2000 resolution B1.8. What else did you want answered? Commented Apr 28, 2018 at 21:20
• All I was hoping for more was an explicit yes or no. Even though "it seems no longer to be claimed that UT1 is mean solar time" may be true within the astronomical community, the reason we in the general public are given why we need to live with irregular discontinuities in UTC is that it is somehow supremely important that civil timekeeping aligns closely to the mean motion of the sun in Greenwich. I've never really understood why that is important, but it surprised me to learn that the process for adjusting UTC by observations doesn't even aim to achieve it anyway. Commented Apr 28, 2018 at 21:42
• @HenningMakholm : Yes, I agree that the organizations often give out stories at best not quite true, and that is bad practice. An example is the leap-seconds business. The SI second has always been a bit too short for clock/calendar use in general life, but then the astronomers who first proposed it expressly recommended it only for purposes of high precision, and to leave mean solar time in place for general life. The SI-standard setting body ignored that, and adopted what's now the SI second as a general standard anyway, reckless of the inconvenience of so many leap seconds. ../.. Commented Apr 28, 2018 at 21:59
• @mike-g : thanks for that, I've amended to mention now only the 'continuity' stated by Capitaine et al. without particularizing it. Curiously, the Aoki and Capitaine articles do not completely agree about the 1982 UT1 computation: Aoki & al. make it entirely depend on the UT, Capitaine & al., make the higher-order terms depend on TT. Commented Apr 29, 2018 at 11:56

No, not since it has been called Universal Time and calculated with expressions that were internationally agreed has it exactly tracked Mean Solar Time, but it is very close.

Seago and Seidelmann made several simulations of when the mean sun is over the zero meridian, and they published their result as part of American Astronautical Society meeting Advances in the Astronautical Sciences 148:1789-1807 (meeting was AAS/AIAA Spaceflight Mechanics Meeting Feb. 10-14, 2013, Kauai, HI as seen at http://www.univelt.com/book=4186 and paper was AAS 13-486). This paper is available online at http://www.agi.com/resources/white-papers/the-mean-solar-time-origin-of-universal-time-and-u

Seago and Seidelmann found that because of assumptions that Simon Newcomb made when constructing his reference frame, UT and MST deviate by an increasing amount that is roughly equal to 0.002738 ∆T

In 1896 at a meeting Conférence Internationale des Étoiles Fondamentales in Paris the directors of the major observatories all agreed that Newcomb's expressions in Tables of the Sun (Astronomical Papers of the American Ephemeris, Vol 6) and other USNO publications should be adopted by all the national ephemerides starting from 1900, and they continued to be used until the IAU 1976 system was implemented in 1984. Newcomb's expressions are on page 9 of that volume. They were used as the definition of Universal Time, and as noted in other answers all subsequent modifications of the definition of UT1 maintained continuity in value and rate at the date of changeover. Newcomb's expression for Right Ascension of the fictitious mean sun (given as τ) is the same as equation 5 in Seago and Seidelmann.

Unknown to Seago and Seidelmann when they wrote that paper in 2013, this deviation had already been understood at the inception of Ephemeris Time and documented by D.H. Sadler in his 1954 monograph in Occasional Notices of the RAS, 3, p 103 which is transcribed at https://www.ucolick.org/~sla/leapsecs/twokindsoftime.html The number Sadler had obtained was 0.00273 ΔT

Note that 1.00273 is the ratio of mean solar and sidereal days, and Sadler's understanding was rooted in his attendance at the 1950 CNRS meeting Constantes Fondamentales de l'Astronomie held at Observatoire de Paris where G.M. Clemence proposed Ephemeris Time in concord with the Astronomer Royal Sir H. Spencer Jones who had documented the nonuniform rotation of the earth in 1939 (http://adsabs.harvard.edu/abs/1939MNRAS..99..541S).

So, yes, as earth rotation slows the difference between UT=12:00 and mean sun over Greenwich will deviate, but very slowly.

• Surely it can't be correct to say that UT "never has" tracked mean solar time? The 1884 international meridian conference actually defined the universal day as a mean solar day. Commented Apr 30, 2018 at 9:57
• The 1884 conference adopted GMT as a global baseline. IAU didn't call it UT until 1928; they adopted Newcomb's formula for mean solar time in terms of sidereal time in 1896. Commented Apr 30, 2018 at 13:56
• @mike-g : I fear that reference can't be right, the Meridian conference was in 1884, it considered no specific formulae, and Newcomb's time formula in his 1895 book was a form not later used & not that of the 1895-8 v.6 APAE tables. The one for 'tau' on p.9 of the APAE tables was often quoted later on, differing from the (then) best estimate of sun's mean longitude by the quadratic increment shown on p.188 of Newcomb's book and discussed in the answer I offered. After ET was introduced, Newcomb's mean sun longitude in terms of mean solar time was of course no longer best available estimate. Commented Apr 30, 2018 at 18:17
• @terry-s The CNRS 1950 meeting at Observatoire de Paris was to consider changes to constants and expressions. The end recommendations were not to change anything except define Ephemeris Time in accord with Spencer Jones MNRAS paper and undo the Great Empirical Term that had been added into Brown's lunar theory in order to compensate for being based on observations given in UT (rather than ET which did not exist). In particular, the 1950 meeting and 1952 IAU resolutions said "No change is contemplated or recommended in the measure of Universal Time, nor in its definition." Commented Apr 30, 2018 at 20:28
• @terry-s But give a close read to Sadler's 1954 monograph. That does a better job of explaining the characteristics of Ephemeris Time vis à vis Universal Time than most anything else for a lifetime. Many misconceptions would have been avoided if Sadler had been more widely read. Part of Sadler's motivation to write that seems to be that at the 1950 CNRS meeting G.M. Clemence reiterated his 1948 Ephemeris Time idea without understanding how it related to Universal Time, and after the Astronomer Royal pointed out the true meaning Clemence appended a note in proof to correct his misconception. Commented May 1, 2018 at 16:28