# Dynamic time

Relationships between the time scales

The Dynamic Time is the independent variable in the equations of motion for bodies of the solar system and can be determined by observing the body. There are different variants that differ from one another in terms of the reference system used and the scaling. Usually, but not always, in addition to the ephemeris time that is no longer used today , it is considered dynamic time

• the geocentric coordinate time TCG (from French Temps Coordonné Geocentrique , English Geocentric Coordinate Time ) with the unit SI second ; it was introduced in 1991.
• the barycentric coordinate time TCB (from French Temps Coordonné Barycentrique , English Barycentric Coordinate Time ) with the same time unit; it was also introduced in 1991.
• the terrestrial time TT (from the French Temps Terrestrique , English Terrestrial Time ), which differs from TCG only in a scaling and on the geoid corresponds very exactly to TAI  + 32.184 s; it was introduced in 1976, called terrestrial dynamic time ( TDT ) in 1979 , renamed TT in 1991 and redefined for the last time in 2000. Together with the TDB , it replaced the Ephemeris Era in 1984.
• the barycentric dynamic time TDB (from the French Temps Dynamique Barycentrique , English Barycentric Dynamical Time ), which differs from TCB only by one scaling and on the geoid corresponds on average to TT ; it was also introduced in 1976, named TDB in 1979 and redefined for the last time in 2006.

## background

For reliable calculations of astronomical events over several centuries or millennia, astronomy requires a uniform time scale . The time measurements based on the variable earth rotation and orbit ( tropical year ) until 1967 did not meet this requirement. Therefore, in 1960 a uniformly running ephemeris time (ET) was introduced as the basis for astronomical calculations. The ephemeris second was the 31,556,925.9747th part of the tropical year on January 0, 1900 (= December 31, 1899) 12:00  UT .

After the atomic clocks became available with high-precision timepieces, it had to be taken into account that there is no such thing as one time, because relativistic effects also influence the actual time measurement with atomic clocks: Moving clocks run more slowly, as do clocks under the influence of a gravitational field than comparable clocks that are at rest or far from the gravity of a large mass. The IAU therefore introduced two, later four time scales from 1976 onwards; Most recently, it changed the rules in 2006.

Two of the four time scales are linked to the center of the earth, the other two to the barycenter of the solar system. The two geocentric time scales are suitable for investigations in near-earth space, the barycentric time scales for describing the dynamics of the solar system and calculating the orbit of interplanetary probes. One time scale from each of these two pairs is a coordinate time , i.e. the time-like coordinate in the respective space-time coordinate system, which cannot be measured directly because every clock in the solar system tracks the coordinate time, which is based on the SI second , because of the gravitational time dilation . The other time scale from the pair differs from the coordinate time only in a linear mapping, which is chosen so that it corresponds to TAI + 32.184 s as exactly as possible or only with deviations of less than 2 ms .

## Barycentric and geocentric coordinate time

The barycentric coordinate time TCB and the geocentric coordinate time TCG are the time-like coordinates of the Barycentric Celestial Reference System (BCRS) and the Geocentric Celestial Reference System (GCRS). The BCRS agrees with the International Celestial Reference System (ICRS) in the origin (=  barycenter of the solar system) and direction of the space-like axes and supplements it with a metric. The axes of the GCRS are parallel to those of the BCRS, but it is centered in the center of the earth and has a metric adapted to the other origin.

In BCRS, with t  = TCB and neglecting higher powers of, the line element is${\ displaystyle {\ tfrac {1} {c}}}$

${\ displaystyle \ mathrm {d} s ^ {2} = - c ^ {2} \ left (1 - {\ frac {2U} {c ^ {2}}} \ right) \ mathrm {d} t ^ { 2} + \ mathrm {d} {\ vec {r}} ^ {2}}$.

Here and in the following, the potential is calculated positively, as is usual in physical geodesy (the gravitational potential of a point mass is + GM / r ). It consists of two summands: the Newtonian gravitational potential of all bodies in the solar system and a tidal potential generated by the bodies outside of it. The first contribution disappears for , the second in the origin. An analogous relationship applies to the GCRS with t  = TCG . This is composed of the earth's gravitational potential and a tidal potential generated by all other bodies. ${\ displaystyle U}$${\ displaystyle r \ to \ infty}$${\ displaystyle U}$

On both time scales, the unit is the SI second ; the zero point of both scales is set so that the event January 1, 1977, 00: 00: 00,000 TAI in the geocenter corresponds to January 1, 1977, 00: 00: 32,184 for both TCB and TCG. Due to the difference of 32.184 s, the two times seamlessly follow the Ephemeris Time.

### Proper time and coordinate time

A clock does not show the (bary- or geocentric) coordinate time , but its proper time , which is related to the coordinate time via the relationship . With the above line element is then ${\ displaystyle t}$${\ displaystyle \ tau}$${\ displaystyle c ^ {2} \ mathrm {d} \ tau ^ {2} = - \ mathrm {d} s ^ {2}}$

${\ displaystyle c ^ {2} \ mathrm {d} \ tau ^ {2} = c ^ {2} \ mathrm {d} t ^ {2} \ left \ lbrack 1 - {\ frac {2U} {c ^ {2}}} - {\ frac {1} {c ^ {2}}} \ left ({\ frac {\ mathrm {d} {\ vec {r}}} {\ mathrm {d} t}} \ right) ^ {2} \ right \ rbrack}$.

After pulling the roots and Taylor expansion in , it becomes ${\ displaystyle {\ tfrac {1} {c ^ {2}}}}$

${\ displaystyle {\ frac {\ mathrm {d} \ tau} {\ mathrm {d} t}} = 1 - {\ frac {U} {c ^ {2}}} - {\ frac {1} {2 }} {\ frac {{\ vec {v}} ^ {2}} {c ^ {2}}}}$.

The term describes the time dilation due to gravity, the term the time dilation due to movement, which is already known from the special theory of relativity ("moving clocks go slower"). ${\ displaystyle U}$${\ displaystyle v}$

### Proper time on the rotating earth

Both contributions to time dilation are important on the rotating earth. A stationary clock has a fixed position in an earth- fixed coordinate system that rotates with the angular velocity (the earth's rotation) in the GCRS. The speed of the clock in the non-rotating GCRS is then ${\ displaystyle {\ vec {R}}}$${\ displaystyle {\ vec {\ Omega}}}$

${\ displaystyle {\ vec {v}} = {\ vec {\ Omega}} \ times {\ vec {R}}}$.

The following then applies to the proper time of this clock in the GCRS, i.e. with t  = TCG

${\ displaystyle {\ frac {\ mathrm {d} \ tau} {\ mathrm {d} t}} = 1 - {\ frac {U _ {\ mathrm {geo}}} {c ^ {2}}}}$,

in which

${\ displaystyle U _ {\ mathrm {geo}} = U + {\ frac {1} {2}} ({\ vec {\ Omega}} \ times {\ vec {R}}) ^ {2}}$

is the earth's gravity potential, the so-called geopotential . In addition to the gravitational and tidal components, the second summand is the centrifugal potential , which causes the centrifugal force on the rotating earth. For the proper time of a clock rotating with the earth, it is not the value of but that of the geopotential that counts , that is, clocks with the same geopotential elevation (same dynamic altitude ) move at the same speed in relation to TCG . In particular, stationary clocks on the rotating geoid run at the same speed. However, this only applies as long as the temporal variation of the geopotential caused by the tides is not taken into account, which is, however, very small: The main term (98%) of the tidal potential only fluctuates by a maximum of ± 3.8 m² / s², which is a change in altitude of ± 0 .39 m, with some of these fluctuations (around 31%) being compensated for by the deformation of the earth's body due to the tides. The rate of the proper time of a clock fluctuates on the earth's surface by a maximum of ± 3 · 10 −17 . Because of the limited duration of action, the deviations between two clocks on the geoid remain below 1 ps. ${\ displaystyle U}$${\ displaystyle U}$${\ displaystyle U _ {\ mathrm {geo}}}$

### Conversion TCB → TCG

In order to be able to calculate the geocentric coordinate time TCG for an event ( ,  ) in the BCRS, the complete trajectory of the geocenter must be known in the BCRS since (January 1977). Therefore, the calculation is very time-consuming when the accuracy requirements are high. A closed term for TCG is ${\ displaystyle t = {\ mathit {TCB}}}$${\ displaystyle {\ vec {r}}}$${\ displaystyle {\ vec {r}} _ {\ mathrm {e}} (t ')}$${\ displaystyle T_ {0}}$

${\ displaystyle {\ mathit {TCG}} = {\ mathit {TCB}} - {\ frac {1} {c ^ {2}}} \ int \ limits _ {T_ {0}} ^ {t} \ left [U _ {\ mathrm {ext}} (r _ {\ mathrm {e}} (t ')) + {\ frac {{\ vec {v_ {e}}} (t') ^ {2}} {2} } \ right] \ mathrm {d} t '- {\ frac {1} {c ^ {2}}} {\ vec {v_ {e}}} (t) \ cdot ({\ vec {r}} - {\ vec {r}} _ {\ mathrm {e}} (t)) + {\ mathcal {O}} ({\ frac {1} {c ^ {4}}})}$.

Here is the Newtonian gravitational potential created by all bodies in the solar system except the earth. The integral adds the accumulated time dilation due to gravitation and relative movement, the second summand takes into account that two events that are simultaneous in the BCRS are generally not in the GCRS that is moved to do so; see Lorentz transformation at . For an event on the earth's surface, this second term depends on the geographic position on earth and the current angle of rotation of the earth with respect to the sun, i.e. on UT1 , because these two quantities, together with the position of the earth in its orbit , the angle between and determine. He is limited by ${\ displaystyle U _ {\ mathrm {ext}}}$${\ displaystyle T_ {0}}$${\ displaystyle {\ vec {v}} \ nparallel {\ vec {r}}}$${\ displaystyle {\ vec {v _ {\ mathrm {e}}}}}$${\ displaystyle {\ vec {r}} - {\ vec {r}} _ {\ mathrm {e}}}$

${\ displaystyle {\ frac {1} {c ^ {2}}} v _ {\ mathrm {e}} R _ {\ mathrm {e}} \ approx {\ frac {1} {c ^ {2}}} 30 \, {\ frac {\ mathrm {km}} {\ mathrm {s}}} \ cdot 6400 \, \ mathrm {km} \ approx 2 \, \ mathrm {\ mu s}}$.

## Terrestrial time

The proper time of a clock in the earth's gravitational potential passes more slowly than the geocentric coordinate time TCG due to the gravitational time dilation, or vice versa: The TCG passes faster than the proper time measured by a clock. The terrestrial time TT is defined as a modification of the TCG , in which this higher rate is compensated,

${\ displaystyle {\ mathit {TT}} - T_ {0} = (1-L _ {\ mathrm {G}}) ({\ mathit {TCG}} - T_ {0})}$,

or reshaped

${\ displaystyle {\ mathit {TT}} = {\ mathit {TCG}} - L _ {\ mathrm {G}} ({\ mathit {TCG}} - T_ {0})}$.

Here is the time again January 1, 1977, 00: 00: 32,184, which means that TT and TCG are constantly following the ephemeris time that was used earlier. The deviation of the relative gait rate from 1 is ${\ displaystyle T_ {0}}$

${\ displaystyle L _ {\ mathrm {G}} = 6 {,} 969 {.} 290 {.} 134 \ cdot 10 ^ {- 10} \ approx 22 \, \ mathrm {ms} / \ mathrm {a}}$.

Since 1977, TT has lagged behind TCG by around 1 s . The value of L G corresponds to the gravitational time dilation for the geopotential ${\ displaystyle U _ {\ mathrm {geo, 0}} / c ^ {2}}$

${\ displaystyle U _ {\ mathrm {geo, 0}} = 62 {.} 636 {.} 856 {,} 0 {\ frac {\ mathrm {m} ^ {2}} {\ mathrm {s} ^ {2 }}}}$,

which is the best value for the geopotential of the geoid known when the IAU resolution was passed in 2000. This achieved that on the one hand the definition of TT becomes independent of “ the intricacy and temporal changes inherent to the definition and realization of the geoid ” and on the other hand the TT second on the rotating geoid is the SI second with a very high degree of accuracy and TT is still  very well approximated by TAI + 32.184 s. More details on the differences between TT and TAI and a better implementation of TT can be found in the section "International Atomic Time" .

## Barycentric Dynamic Time

The barycentric dynamic time TDB differs from the coordinate time TCB in a similar way as TT differs from TCG ,

${\ displaystyle {\ mathit {TDB}} = {\ mathit {TCB}} - 65 {,} 5 \, \ mathrm {\ mu s} -L _ {\ mathrm {B}} ({\ mathit {TCB}} -T_ {0})}$

with the same as in the definition of TT and ${\ displaystyle T_ {0}}$

${\ displaystyle L _ {\ mathrm {B}} = 1 {,} 550 {.} 519 {.} 768 \ cdot 10 ^ {- 8} \ approx 0 {,} 49 \, \ mathrm {s} / \ mathrm {a}}$.

The value of was chosen so that TDB and TT have the same average gait rate in the geocenter , so that the difference remains limited. On the surface of the earth is before and after for several millennia ${\ displaystyle L _ {\ mathrm {B}}}$

${\ displaystyle \ vert {\ mathit {TDB}} - {\ mathit {TT}} \ vert <2 \, \ mathrm {ms}}$.

The constant offset of 65.5 µs, for which there is no equivalent in the definition of TT , was introduced so that a much-used series development by Fairhead and Bretagnon from 1990 for converting TT to TDB remains unchanged. With the 127 tabulated series elements, TDB can be calculated with an accuracy of 0.1 µs, as is desirable for the observation of millisecond pulsars , for several millennia before and after today. If only the by far dominant term is taken into account, the result is

${\ displaystyle {\ mathit {TDB}} - {\ mathit {TT}} = 1.657 \, \ mathrm {ms} \ cdot \ sin (6 {,} 283 {.} 076 {\ frac {{\ mathit {TT }} - \ mathrm {J2000.0}} {365 {,} 25 \, \ mathrm {d}}} + 6 {,} 240)}$,

where J2000.0 is the default epoch January 1, 2000, 12:00:00 DD. The contribution of the remaining 126 series members is less than 0.08 ms in the years 1900 to 2100 (0.18 ms for the years 1000 to 3000).

## Other time systems

### International atomic time

The international atomic time TAI is based like the terrestrial time TT on the SI second with the same geopotential U geo  = U geo, 0 . Nevertheless, the rates of the two times can differ, since TAI, in contrast to TT, is based on an actual time measurement, in which random and systematic errors can occur. It is created by iterative averaging and scaling of the readings from more than 600 atomic clocks that are distributed all over the world. The BIPM publishes the results monthly in the Circular T. TAI is therefore not available in real time, but only afterwards.

The international atomic time is the basis for a frequently used realization of TT ,

${\ displaystyle {\ mathit {TT}} (\ mathrm {TAI}) = {\ mathit {TAI}} + 32 {,} 184 \, \ mathrm {ms}}$.

Since the TAI is not available in real time, the UTC (k) time signal of an individual institute must first be used for a time measurement, for example that of the PTB . Only after the publication of the new Circular T can this reading be converted into UTC and then into TAI .

For very demanding applications such as the observation of millisecond pulsars , the BIPM publishes a more precise implementation every year; the latest (as of February 2020) is

${\ displaystyle {\ mathit {TT}} (\ mathrm {BIPM19}) = {\ mathit {TAI}} + 32 {,} 184 \, \ mathrm {ms} + \ delta}$,

where δ is listed in tabular form every 10 days, beginning on June 26, 1975. The table is supplemented by an extrapolation formula . Currently (February 2020) that is

${\ displaystyle \ delta = 27 {,} 6800 \, \ mathrm {\ mu s} -0 {,} 02 \, \ mathrm {ns} \ cdot ({\ mathit {MJD}} - 58839)}$.

Here, MJD  - 58839 is the number of days that have passed since December 22, 2019, 00:00 UTC (MJD: Modified Julian Date ).

### Universal Time and Coordinated Universal Time

The Universal Time UT1 is not a strictly uniform time, as it is based on the Earth's rotation and this slows down and is also irregular; UT1 is following TT . The difference TT  - UT1 is referred to as Δ T ( Delta T ). It increased from −2.7 s at the beginning of 1900 (with the TT extrapolated into the past ) over 63.8 s at the beginning of 2000 to the current (beginning of April 2019) 69.3 s.

The Coordinated Universal Time UTC Although used since 1972 as opposed to universal time as the Terrestrial time the SI second , but it is by inserting leap seconds adjusted from time to time to the Universal Time. This ensures that (a) the deviation | UTC  - UT1 | Remains <0.9 s and (b) UTC differs from TAI by an integer number of SI seconds.

## example

The table below gives an impression of the size of the differences between the time measures discussed here. All times are for the event January 15, 2006, 21: 24: 37,500000 UTC on the beach in Hawaii ( 19 ° 28 ′ 52.5 ″  N , 155 ° 55 ′ 59.6 ″  W ); the location is important for converting TCG into TCB and TDB .

scale time
UTC Jan 15, 2006, 21:24: 37,500000
UT1 Jan 15, 2006, 21:24: 37.834055
TAI Jan 15, 2006, 21:25: 10.500000
TT Jan 15, 2006, 21:25: 42.684000
TCG Jan 15, 2006, 21:25: 43.322690
TDB Jan 15, 2006, 21:25: 42.684373
TCB Jan 15, 2006, 21:25: 56.893952

## Individual evidence

1. a b c d Dennis D. McCarthy, P. Kenneth Seidelmann: Time - From Earth Rotation to Atomic Physics . 2nd Edition. Cambridge University Press, 2018, ISBN 978-1-107-19728-2 , ch. 9 “Dynamical and Coordinate Timescales” (English, limited preview in Google Book Search).
2. Time / Dynamical Time. In: Encyclopædia Britannica . Accessed May 10, 2019 .
3. a b SOFA Time Scale and Calendar Tools. (PDF) Standards Of Fundamental Astronomy, accessed on May 10, 2019 .
4. Dynamical Time. In: Encyclopædia Britannica . Accessed May 10, 2019 .
5. Resolution A4: Recommendations from the Working Group on Reference Systems. (PDF) In: XXIst General Assembly, Buenos Aires, 1991. IAU, pp. 12–22 , accessed on May 10, 2019 (English, French, including Recommendations I (ds²), III (TCB, TCG), IV ( TT)).
6. a b Resolution 5 of Commissions 4, 19 and 31 on the designation of dynamical times. (PDF) In: XVIIth General Assembly, Montreal, 1979. IAU, p. 16 , accessed on May 10, 2019 .
7. Resolution B1.9: Re-definition of terrestrial time TT. (PDF) In: XXIVth General Assembly, Manchester, 2000. IAU, pp. 25–26 , accessed on January 10, 2020 (English, French).
8. P. Kenneth Seidelmann (ed.): Explanatory Supplement to the Astronomical Almanac . University Science Books, 2006, ISBN 1-891389-45-9 , pp. 691 f . (English, limited preview in the Google Book Search): “[The Supplement to the Astronomical Almanac 1984] gives the various resolutions… and the resulting equations that introduce the IAU (1976) System of Constants, the FK5 reference frame on J2000.0 , and the TDT and TDB time systems. "
9. Resolution 3: Re-definition of Barycentric Dynamical Time, TDB. (PDF) In: XXVIth General Assembly, Prague, 2006. IAU, pp. 5–6 , accessed on April 11, 2019 .
10. B. Guinot, P. K. Seidelmann: Time scales: their history, definition and interpretation . In: Astronomy & Astrophysics . tape 194 , 1988, pp. 304–308 , bibcode : 1988A & A ... 194..304G (English).
11. a b Gérard Petit, Brian Luzum (Ed.): IERS Conventions (2010) (=  IERS Technical Note . No. 36 ). Publisher of the Federal Agency for Cartography and Geodesy, 2010, ch. 10 (“General relativistic models for space-time coordinates and equations of motion”) - (English, full text (total and by chapters) ).
12. International Celestial Reference System (ICRS). USNO , 2017, accessed on April 10, 2019 (section “Standard Algorithms”).
13. a b Resolution B1.3: Definition of barycentric celestial reference system and geocentric celestial reference system. (PDF) In: XXIVth General Assembly, Manchester, 2000. IAU, pp. 5–11 , accessed on April 10, 2019 (English, French).
14. a b Resolution 2.2: Default orientation of the Barycentric Celestial Reference System (BCRS) and Geocentric Celestial Reference System (GCRS). (PDF) In: XXVIth General Assembly, Prague, 2006. IAU, p. 4 , accessed April 10, 2019 .
15. Martin Vermeer: Physical geodesy . School of Engineering, Aalto University, 2020, ISBN 978-952-60-8872-3 , pp. 10 (English, full text [PDF]): "In physical geodesy - unlike in physics - the potential is reckoned to be always positive ..."
16. a b Michael Soffel: Astronomical-geodetic reference systems. (PDF) 2016, pp. 38–41 , accessed on April 12, 2019 .
17. Wolfgang Torge : Geodesy . 2nd Edition. de Gruyter, 2003, ISBN 3-11-017545-2 , p. 76 f., 332–334 ( limited preview in Google Book search).
18. A conventional value for the geoid reference potential W 0 . (PDF) In: Unified Analysis Workshop 2017. German Geodetic Research Institute , pp. 5–7 , accessed on January 10, 2020 (English).
19. L. Fairhead, P. Bretagnon: An analytical formula for the time transformation TB – TT . In: Astronomy and Astrophysics . tape 229 , 1990, pp. 240–247 , bibcode : 1990A & A ... 229..240F (English, the 'TB' in the title stands for TDB).
20. These values ​​result from the summation of the amplitudes.
21. ^ Resolutions adopted at the 26th CGPM. BIPM, November 2018, accessed February 1, 2020 (English, French, Resolution 2: On the definition of time scales ).
22. The time scales TAI and EAL. PTB , accessed on May 17, 2019 .
23. a b BIPM (Ed.): BIPM Annual Report on Time Activities 2017 . 2018, ISBN 978-92-822-2268-3 , pp. 10–14 (English, full text [PDF]).
24. Atomic clocks participating in TAI statistics. In: BIPM Time Department Data Base. BIPM, accessed May 17, 2019 .
25. ^ Circular T. BIPM, accessed May 17, 2019 .
26. TT (BIPM19). BIPM, accessed February 4, 2020 .
27. ^ Historic Delta T and LOD. USNO, accessed May 14, 2019 .
28. Monthly determinations of Delta T. USNO, accessed May 14, 2019 .
29. TAI − UTC (Jan. 1, 1972 - Jun. 28, 2021). IERS , accessed July 24, 2020 .
30. Time scales. IERS , accessed on April 19, 2019 (English, query from UT1 − UTC).

## literature

• Dennis D. McCarthy, P. Kenneth Seidelmann: Time - From Earth Rotation to Atomic Physics . 2nd Edition. Cambridge University Press, 2018, ISBN 978-1-107-19728-2 , ch. 9 “Dynamical and Coordinate Timescales” (English, limited preview in Google Book Search).