# The Time

## The SI second

The SI second is (currently) defined as the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom (implicitly: in the absence of any external magnetic field). This means the practical realization of the second is left to atomic clocks, mostly either cesium clocks (using precisely the definition above), rubidium clocks, or hydrogen maser clocks. It is possible that rubidium would, in fact, be a better choice than cesium (to the point that it would be worth redefining SI?).

Various other time units are defined from the second. The SI day is 86400 SI seconds. The Julian year is 365.25 SI days, that is, 31557600 SI seconds. The Gregorian year (rarely used in astronomy, but it is the mean duration of a year of our calendar) is 365.2425 SI days, that is, 31556952 SI seconds. Incidentally, the so-called “light year” (a unit of distance, not time) is the length of the path traveled by light in vacuum in one Julian year, which is therefore (taking into account the definition of the SI meter) exactly 9460730472580800 meters. Naturally, the Julian and Gregorian centuries are one hundred Julian and Gregorian years respectively.

The definition of an SI second prior to 1967 was related to the Earth's rotation around the sun. Precisely: it is the fraction 1/31556925.9747 of the tropical year for 1900 January 0 at 12h ephemeris time (the “ephemeris second”). This definition is probably rather obscure. Of course, the event 1900 January 0 at 12h ephemeris time refers to December 31, 1899 at noon; this epoch (ephemeris Julian date 2415020.0) is generally referred to as “J1900.0”. (The epoch one Julian century later, or 1 January 2000 at 12h ephemeris time (ephemeris Julian date 2451545.0), is referred to as “J2000.0”.) The expression the tropical year at (such a date) means that the second is defined so that the mean geometric longitude of the sun (in other words, its geometric longitude with all periodic perturbative terms canceled out) varies by 129602768.13 seconds of arc per Julian century at the point considered (the former figure is exact according to CGPM 1960, the latter according to the International Astronomers' Union 1952, but they differ by a term too small to be measurable).

Between 1955 and 1958, a joint experiment of the National Physical Laboratory (UK, where the first cesium frequency samples had be obtained) and the US Naval Observatory was conducted to measure the value of the cesium 133 atom's hyperfine transition frequency (in the then-standard ephemeris second). It is in this experiment that the value of 9192631770 transitions per second was measured, which was adopted in 1967 by the Conférence Générale des Poids et Mesures as the new definition of the second (keeping the value as close as possible to the former one, naturally).

Before even the ephemeris second was defined, the second (not “SI” because this is before SI) was defined as the fraction 1/86400 of the mean solar day (this is, after all, the historical definition of the second, the day being divided in 24 hours of 60 minutes of 60 seconds each). But aberrations in the Earth's rotation made this definition inherently inexact. As of J2000, the mean solar day is some 3 milliseconds longer than the SI day (this represents the secular increase of the length of the day since about 1820).

The definition of the ephemeris second in terms of the tropical year 1900 was based on the ephemerides and planetary theories published by Newcomb at the end of the 19th century in England. This explains the choice of 1900 as epoch in that definition. Modern planetary theories tell us that the atomic second adopted in 1967 is in fact slightly longer than the ephemeris second: it is roughly 6 parts per billion longer. (XXX — check this carefully; this seems a bit much: perhaps I am wrong in my reading of said planetary theory.) This also explains why it is around 1820 that the mean solar day was 86400 seconds long: it is because Newcomb had used data that spanned, mostly, the 18th and 19th centuries, in doing his computations (and since the ephemeris second was later adopted as SI second, it is with this portion of the 1820 day that we are now left).

## Julian date

The Julian date counts days in a certain time standard, usually either Universal Time Coordinated or Ephemeris Time. On 2000-01-01 at 12:00:00 (noon), that is, J2000.0, the value of the Julian date was 2451545 exactly. The modified Julian date is the Julian date minus 2400000.5 (so on 2000-01-01 at 00:00:00 (midnight), or 43200 seconds before J2000.0, it was 51544).

This has the advantage of defining a running date that does not have all the bizarre idiosyncrasies of our calendar.

## The calendar

### The Gregorian calendar

The Gregorian calendar is currently our official calendar for just about the entire world. Since no standards body has the necessary authority to alter it (consider, for example, the failed attempt by the French revolutionaries), it is a safe bet that this calendar will remain unmodified for a substantial duration.

The calendar year is divided in 365 or 366 days (of 86400 SI seconds): those with 366 days are called leap years, others are called regular. These days are grouped in twelve months having the following number of days: 31 (January), 28 or 29 (February: 28 for regular years and 29 for leap years), 31 (March), 30 (April), 31 (May), 30 (June), 31 (July), 31 (August), 30 (September), 31 (October), 30 (November) and 31 (December). Months are given numbers going from 1 to 12. Days within a month are given numbers going from 1 to 31 (or less). A calendar day starts at midnight, when the modified Julian date is an integer (thus, each calendar day can be associated precisely one modified Julian date). Years are given (“Common Era”) numbers consecutively so that the year (of the Gregorian calendar) that starts on modified Julian date 51544 is given the number 2000. The rules to determine whether a year is leap are as follows, for the Gregorian calendar: years whose number is divisible by 400 (for example, 2000) are leap years; years whose number is divisible by 100 but not by 400 (for example, 1900) are regular years; years whose number is divisible by 4 but not by 100 (for example, 2004) are leap years; years whose number is not divisible by 4 are regular years. Thus, a period of 400 calendar years contains 97 leap years and 303 regular years, which makes 146097 days in total (or 12622780800 seconds). This is the Gregorian calendar's cycle, and a fraction 1/400 of this duration is one (average) Gregorian year, or 365.2425 days.

(For what it's worth, the duration of the tropical year in J2000 is actually 365.24219 SI days, or 365.24218 mean solar days at that date.)

Calendar days are further grouped in intervals of 7 days (604800 seconds), called weeks. These are named: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday and Sunday, in this order. The day of modified Julian date 51544 is a Saturday (the rest can be deduced from this). Notice that a Gregorian calendar cycle, or 146097 days, contains exactly 20871 weeks. This means that week days will repeat themselves after 400 calendar years; but since 400 is not divisible by 7, it implies that the days are not evenly distributed among the calendar dates (for example, it is a well-known trivia fact that dates which fall on the 13th of a month are more frequently a Friday than any other day of the week).

### The Julian calendar

The Gregorian calendar is the one we use now. An earlier calendar is the Julian calendar. This differs from the Gregorian calendar only in the procedure for computing leap years: a year with a certain number is a leap year in the Julian calendar if and only if the number is divisible by 4. The day with modified Julian date 51544, 2000-01-01 (January 1, 2000) on the Gregorian calendar, is 1999-12-19 (December 19, 1999) on the Julian calendar. Week days are otherwise unaffected: this day is a Saturday for both calendars. The short cycle of the Julian calendar (that is, without counting week days) is 4 years, or 1461 days, or 126230400 seconds. The long cycle of the Julian calendar is 28 years, or 1461 weeks, or 10227 days, or 883612800 seconds. It is worth insisting that the Gregorian calendar only has one cycle of 400 years, with unevenly distributed week days, whereas the Julian calendar has a small and large cycle, and its week days are consequently evenly distributed.

### Official calendar

The Gregorian calendar was adopted over the Julian calendar at various times for various countries. Spain made the change as early as 1582, when the calendar was ordained by pope Gregory XIII, with Thursday 1582-10-04 on the Julian calendar (MJD -100841) being followed by Friday 1582-10-15 on the Gregorian calendar (MJD -100840). England made the change in 1752, and Wednesday 1752-09-02 (MJD −38780) was followed by Thursday 1752-09-14 (MJD −38779). Russia made the change with the revolution, in 1917. Other countries have more complicated stories (the year 1712 in Sweden was a “double leap year” because an error in trying to switch from Julian to Gregorian had left a somewhat confused situation).

Dates in antiquity, “before the Common Era”, that is, before 0001-01-01 on the Julian calendar, are customarily counted backwards starting from ~1 (year 1 before the Common Era, which is the same as year 0 of the Common Era, but the latter is not used; similarly, year ~2, or 2 before the Common Era, is year −1 of the Common Era). As far as months and days go, there is nothing to prevent us from extending the Julian calendar (or, for that matter, the Gregorian calendar) arbitrarily far back in the past. But what we now call the Julian calendar has only been consistently applied since the year 5 of the Common Era. Indeed, years 4, ~1 and ~5 were not leap: this is the “Augustan reform”, taken by Cæsar Augustus in ~8 because the pontiffs had incorrectly inserted one leap year in three (instead of four) between ~45 and ~9. (In other words: though the Julian calendar prescribed that ~45, ~41, ~37, …, ~13, ~9, ~5, ~1, 4, 8 and 12 be leap, in fact, ~45, ~42, ~39, ~36, …, ~12, ~9, 8 and 12 were leap.) Besides, it is evidently anachronistic to use the name “August” for the month named in honor of he who enacted said reform! For what it's worth, before that, the month was called Sextilis. Before the Julian reform, it is about impossible to date anything; “July” was called Quinctilis before it was dedicated to C. Julius Cæsar, but the number of days in a month was very variable, and some years even had 13 months with Mercedonius being the 13th. Also note the the tradition of starting the year on January (and numbering from January) was not kept consistently for dates in the Christian period until the Renaissance (for pre-Christian times, years were designated in Rome by the names of the Consuls, and these indeed took their office in January).

The starting point of the calendar, ~0045-01-01, was Julian date 1704987 (at noon ephemeris time). The assassination of C. Julius Cæsar on the ides of March of year 44 before the Common Era took place on Julian date 1705426.

Remark: The terms “Common Era” and “before the Common Era” are to be preferred over the earlier terms “Anno Domini” (literally, “in the year of the Lord”) and “Before Christ”: first, they are civilization-neutral (the calendar we use is, in any case, completely arbitrary and absurd); moreover, they are more accurate, since the Jesus referred to as “Christ” by the Christians was probably born around ~4 (or year 750 of Rome).

## Ephemeris Time and avatars (astronomical time scales)

Things get a bit complicated when it comes to defining an actual time scale, and not just a unit of time. The reason for this is that, as Einstein's theory of Special Relativity tells us, what we call “time” is simply one dimension of a four-dimensional whole, so its measure depends on the choice of axes, choice which is physically realized as the choice of a reference frame (observers in relative motion to one another do not measure the same parameter); furthermore, General Relativity complicates matters even more by telling us that the four-dimensional space-time continuum is not even flat but is bent by gravity, so that definitions of time cannot simply be extended from one point to another.

All ephemeris time scales are defined to be 32.184 seconds ahead of atomic time on 1977-01-01 at 00:00:00 TAI. There is no real reason behind this number: it is merely due to a lack of sufficient communication between astronomers and physicists.

### Ephemeris Time proper

Ephemeris Time (TE) is defined by the Earth's revolution around the sun. This makes no call to a reference frame except insofar as periodic terms are to be canceled out (and which periodic terms are there depends on the choice of frame). It is useful as a long-term standard, but it is useless for short intervals, as it cannot be realized for such with a satisfactory precision.

The definition of Ephemeris Time is this: the geometric mean longitude of the Sun, as measured from the vernal equinox of date, is given by L0 = 279°41′48.04″ + 129602768.13″T + 1.089″T2, where T is Ephemeris Time measured in Julian centuries since J1900.0. In other words, the Ephemeris second is defined so that there are 31556925.9747 seconds in the tropical year at J1900.0, which is defined as that moment close to 0 January 1900 noon such that the geometric mean longitude of the Sun was 279°41′48.04″ (this is completely theoretical: in practice, J1900.0 would be defined by Universal Time).

This expression is taken from Newcomb's ephemerides, but more accurate theories and the redefinition of the SI second in terms of atomic clocks have made Ephemeris Time proper obsolete.

### Barycentric Coordinate Time

Barycentric Coordinate Time, or TCB is the most fundamental time scale that can be defined. Essentially, it is the proper time measured by an asymptotic observer (i.e. one located infinitely far away) at rest with respect to the solar system. More precisely, it is the time coordinate of a space-time coordinate system that is centered in space on the barycenter of the solar system, that is nonrotating with respect to distant galaxies, and that tends asymptotically to the proper time of an observer at rest with respect to the coordinate system. It was introduced by the International Astronomers' Union in 1991, following the need to clarify the definition of Barycentric Dynamical Time, which is somewhat dubious in principle.

### Barycentric Dynamical Time

Barycentric Dynamical Time, or TDB, is the same as Barycentric Coordinate Time only differing by a constant factor which is chosen so as to make TDB differ from Terrestrial Time by only periodic and Poisson terms. In other words, whereas it is defined by a coordinate system centered at the barycenter of the solar system, TDB is actually chosen to agree “in the mean” (up to periodic and Poisson terms) with a clock located on the Earth's surface.

On the one hand, this is a rather dubious definition. It is not entirely clear what is a periodic (or Poisson) term and what is a secular term (especially when the periodic terms in question are of very long period); this is the reason why the IAU in 1991 decided to recommend the use of TCB over TDB. On the other hand, TDB is very useful because it actually models the Ephemeris Time used in older planetary theories (indeed, planetary theories are calculated in a barycentric coordinate system, and the actual second length was historically that measured on Earth) — such as those of Newcomb. To avoid breaking continuity with earlier works, TDB is still used in constructing planetary theories.

Effectively, the ratio of TCB to TDB is 1 + 1.55051979176×10−8. This constant reflects, approximately, for two thirds the gravitational redshift caused by the Sun's attraction, for one third the time dilation produced by the Earth's movement around the Sun, for around 5% the gravitational redshift caused by the Earth's own attraction, and for yet smaller parts other effects such as the other planets' influence (all these effects are absent in TCB but present in TDB). A different way of stating this is that whereas TCB measures the time seen by an asymptotic observer in SI seconds, on the other hand TDB measures it in TDB seconds, where a TDB second is equal to an SI second on Earth but made longer by 15.5 parts in a billion because of the two effects mentioned (similarly, the TDB meter is longer than the SI meter in the same proportion, since the speed of light is kept constant).

The two scales are synchronized by stating that they coincide on 1977-01-01 at 00:00:00 TAI. It follows that since then TCB has run ahead of TDB by a dozen seconds.

### Geocentric Coordinate Time

Geocentric Coordinate Time, or TCG, is the same as Barycentric Coordinate Time, except that it uses the Earth's center, rather than the barycenter of the solar system (and an “asymptotic” observer is one free from the gravitational field of the Earth rather than that of all the solar system). Thus, TCG coincides with Terrestrial Time except insofar as it omits the gravitational effect of the Earth itself (and the time dilation caused by its rotation, but these are very small indeed). The ratio of TCG to TT is 1 + 6.969290139×10−10. The scales are synchronized in the same way that TCB and TDB are.

The relation of TCG to TCB is truly the scope of General Relativity. In particular, note that their relation depends on the point in space where the comparison is made: this is because a terrestrial observer and one at the center of the solar system might not agree about whether two events are instantaneous. The relation of TCG to TCB has three kinds of terms:

• A secular term: the ratio of TCB to TCG tends to 1 + 1.48082688934×10−8 at infinity. This constant reflects, approximately, for two thirds the gravitational redshift caused by the Sun's attraction, for one third the time dilation produced by the Earth's movement around the Sun, and for yet smaller parts other effects such as the other planets' influence: these effects are present in TCG but not in TCB. Note that the gravitational effect of the Earth itself is, as we have said, not present in TCG, whereas it is in TT.
• Periodic terms: these can amount up to slightly less than 2 milliseconds. The leading periodic term has period one year and amplitude 1657 microseconds; it is currently maximal in the beginning of April (making TCB ahead of TCG by about 1.7 milliseconds more than in the beginning of January: this is due to the ellipticity of the Earth's orbit).
• A term dependent on the observer's position, which cancels if the observer is at the center of the Earth (it just cancels by convention, in the sense that the terms at the center of the Earth are the terms already mentioned above). This term is at most of the order of 2 microseconds for an observer on the Earth's surface (TCB is ahead of TCG by about 2 microseconds more for an observer situated on the equator, at sunrise, on the summer or winter solstice, than for the center of the Earth, because such an observer is ahead of the Earth's revolution).

### Terrestrial Time

Terrestrial Time, or TT (sometime called Terrestrial Dynamical Time, TDT), is, as its name indicates, the time scale on Earth. It is the proper time measured on the surface of the (rotating) geoid. Like all time scales defined in this section, but unlike Atomic Time, Terrestrial Time is an ideal time scale: in other words, it extends arbitrarily far in the past and the future (well, in fact, it doesn't: it only extends as far as the Earth exists in much the same form as it now does; if you need to go further than that, you need barycentric time). Its actual realization is based on Atomic Time: precisely, TT is 32.184 seconds ahead of TAI, to the extend that TAI can be trusted.

TT differs from TDB by the same periodic and local terms as were mentioned for the difference of TCG from TCB (see under TCG); but there is no secular term, by definition of TDB.

## International Atomic Time

International Atomic Time, or TAI, is a practical realization of Terrestrial Time (TT is an abstract scale, whereas TAI is a measured time; further, they differ by that annoying 32.184 seconds). Contrary to Coordinated Universal Time, TAI is a linear standard: to the best precision achieved, it ticks by one second every SI second.

TAI was chosen to coincide with the Universal Time UT1 at midnight on 1958-01-01 (modified Julian date 36204): this is the start of TAI, and it should be emphasized that it makes no sense to refer to a date prior to this in TAI (instead, TT should be used, which can be realized approximately by Ephemeris Time if astronomical data are available). (Actually, TAI has been available since July 1955 or so, but the epoch had not yet been fixed.) TAI received the blessings of the 14th Conférence Générale des Poids et Mesures in 1971.

TAI is the master time standard: not only does it realize TT, but it also serves to define UTC, and, consequently, legal time; it is also used to define certain time standards such as GPS time.

TAI is maintained by the Bureau International des Poids et Mesures which periodically averages various atomic clocks around the world to compute TAI, and publishes the offsets of the various clocks to TAI (the master clock of the Observatoire de Paris is used to centralize the offset computation). The unpleasant consequence of this is that, if very high precision is required, TAI can only be known a posteriori, after the offsets have been published.

## Coordinated Universal Time

Coordinated Universal Time, or UTC is the official standard of time throughout the world, being the basis for legal time. It is to UTC that time zone offsets (generally an integer number of hours) are added to produce local time. UTC has existed since 1961 (prior to this, the basis for legal time was Universal Time, not coordinated to Atomic Time).

UTC does not flow linearly: it is defined with respect to TAI, and is maintained to within 0.9 seconds of UT1. Since 1972, UTC has been defined as offset from TAI by an integer number of seconds. Currently, this offset is 33 (that is, TAI is exactly 33 seconds ahead of UTC): this is true since 2006-01-01 and will remain true at least until 2006-06-30.

The difference between TAI and UTC (since 1972) is increased (or possibly decreased) in the form of leap seconds, during which UTC moves from 23:59:59 to 23:59:60 before becoming 00:00:00 (or skips 23:59:59 if a negative leap second is ordained). Leap seconds can be inserted before January 1 or July 1 on a given year (or theoretically April 1 or October 1); negative leap seconds have never been used.

The decision on when to insert a leap second is made by the International Earth Rotation Service (central bureau, now in Frankfurt), based on observations of the Earth's orientation in space, and is published in the IERS Bulletin C. As of writing, the most recent Bulletin C published the decision to insert a leap second at the end of December 2005, thus increasing the TAIUTC difference from the value 32 (that it has had since January 1, 1999) to 33.

The atomic clocks used to define TAI also compute a local version of UTC. This is, in fact, the time standard generally used to define legal time in the country in question. For example, legal time in the United States of America is defined by adding the time zone offset to the UTC reading of the US Naval Observatory's Master Clock; the difference between UTC(USNO) and UTC(BIPM) is typically of the order of a few nanoseconds (and is, of course, kept as small as possible). The reason why the BIPM's definition of UTC cannot be used directly as a basis for legal time is that it is known only a posteriori, when the various atomic clocks have been read.

Before 1972, the relation between UTC and TAI was more complex, to make UTC agree with Universal Time more closely. Not only were jump discontinuities used occasionally (e.g. a 50 millisecond discontinuity on 1961-08-01), but also the clock frequency (the rate of flow of UTC) was slightly skewed: the function giving UTC from TAI was piecewise affine, and UTC did not always tick by one second every SI second, even after the second was defined in terms of atomic clocks in 1967.

The historical table of TAIUTC differences is available for precise computations of durations, even for dates prior to 1972 (most tables list only the leap seconds, the latter part of the table). The same information is available in a slightly different format from the IERS.

The Time Service Department of the U.S. Naval Observatory issued a questionnaire on December 1999 with the idea of possibly completely doing away with leap seconds. More has not been heard from this.

## Universal Time (GMT, UT0, UT1, UT2)

Universal Time has for basis the rotation of the Earth. The mean solar time on Greenwich meridian is called GMT, for “Greenwich Mean Time”. (Actually, historically, Greenwich Mean Time was measured from mid-day, so that it was twelve hours behind Greenwich civil time. Nowadays, the term is imprecise and should no longer be used.)

The observed mean sidereal time at Greenwich (measured from the position of certain reference stars) is called GMST; it is transformed to give a mean solar time, the first Universal Time standard, or UT0. This in turn gives rise to a time standard, UT1, that corrects for nutation and pole movement. Two further periodic terms (with periods one year and half a year) are removed from UT1 to form UT2. There remains some effects due to tidal forces that slow the Earth's rotation.

Universal Time is rather imprecise, because the rotation of the Earth is subject to so many perturbative effects. The relations between UT1 and UT0, and between UT2 and UT1, are conventional (the latter is simple: UT2UT1 is 0.022 sin(2πT) − 0.012 cos(2πT) − 0.006 sin(4πT) + 0.007 cos(4πT) where T is the time in Besselian years of 365.2422 days; for the former, consult Aoki, Guinot, Kaplan, Kinoshita, McCarthy & Seidelmann, Astron. Astrophys., 105 (1982)).

The International Earth Rotation Service monitors the earth rotation parameters, calculates the value of UT1 and publishes it in the IERS Bulletin B. It is that observation which is used as a basis for deciding when to add a leap second to UTC.

Not only is the flow of UT1 irregular, but its mean rate is not even unity: it increases by one day, on the mean, once every mean solar day, and the mean solar day is presently somewhere around 3 milliseconds longer than the SI day — this is why leap seconds are added quite regularly to UTC (and negative leap seconds will probably never happen).

## SI units

The SI (Système International), comprising the meter, the kilogram, the second, the ampere, the kelvin and the candela as base units, was named by the 11th Conférence Générale des Poids et Mesures in 1960. The 14th Conférence Générale des Poids et Mesures added the mole as a new base unit in 1971.

The metre is the length of the path traveled by light in vacuum during a time interval of 1/299792458 of a second. (17th Conférence Générale des Poids et Mesures, 1983)

(Former definition: The metre is the length equal to 1650763.73 wavelengths in vacuum of the radiation corresponding to the transition between the levels 2p10 and 5d5 of the krypton 86 atom. (11th Conférence Générale des Poids et Mesures, 1960))

(Former definition: The unit of length is the metre, defined by the distance, at 0° [Celsius], between the axes of the two central lines marked on the bar of platinum-iridium kept at the Bureau International des Poids et Mesures and declared Prototype of the metre by the 1st Conférence Générale des Poids et Mesures, this bar being subject to standard atmospheric pressure and supported on two cylinders of at least one centimetre diameter, symmetrically placed in the same horizontal plane at a distance of 571mm from each other. (7th Conférence Générale des Poids et Mesures, 1927))

The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram. (1st Conférence Générale des Poids et Mesures, 1889 & 3rd Conférence Générale des Poids et Mesures, 1901) This prototype is made of an alloy of platinum with 10 per cent iridium in mass (to within 0.0001). It was manufactured in the late 1880's (as a copy of the prior kilogram prototype kept at the French Archives). It is kept in a vault at the the Bureau International des Poids et Mesures in Sèvres, France.

The second is the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cæsium 133 atom. (13th Conférence Générale des Poids et Mesures, 1967)

(Former definition: The second is the fraction 1/31556925.9747 of the tropical year for 1900 January 0 at 12h ephemeris time (Comité International des Poids et Mesures, 1956, ratified by 11th Conférence Générale des Poids et Mesures, 1960))

The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one metre apart in vacuum, would produce between these conductors a force equal to 2×10−7 [newton] per metre of length. (Comité International des Poids et Mesures, 1946 & 9th Conférence Générale des Poids et Mesures, 1948)

The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. (10th Conférence Générale des Poids et Mesures, 1954, made explicit by 13th Conférence Générale des Poids et Mesures, 1967)

The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540*1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. (16th Conférence Générale des Poids et Mesures, 1979)

(Former definition: The candela is the luminous intensity, in the perpendicular direction, of a surface of 1/600000 square metre of a black body at the temperature of freezing platinum under a pressure of 101325 newtons per square metre. (Comité International des Poids et Mesures, 1946, ratified by 9th Conférence Générale des Poids et Mesures, 1948, explicited by 13th Conférence Générale des Poids et Mesures, 1967))

The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12; its symbol is “mol”. When the mole is used, the elementary entities must be specified, and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles. (14th Conférence Générale des Poids et Mesures, 1971)

The Système International is used throughout the world. The United States of America (together with Liberia and Burma) are a notable exception: their system of unit is derived from the SI so as to keep values very close to the former British Imperial system. Thus, the yard was defined in 1959 by the US National Bureau of Standards as 0.9144 meters (with the notable exception of the US survey foot, which is 1200/3937 meters, a value derived from an earlier identification) and the pound as 0.45359237 kilograms (as for the gallon, it is 231 cubic inches, or 3.785411784 liters exactly). Note that a temperature in degrees Celsius is obtained by taking the (absolute thermodynamic) temperature in Kelvins and subtracting 273.15; the temperature in degrees Fahrenheit is obtained as an affine function of the temperature in degrees Celsius: −40 Fahrenheit is −40 Celsius, and the slope is such that 9 degrees Fahrenheit are 5 degrees Celsius.