4.2 Time

4.2.1 Basic Considerations

• There are 3 groups of time scales of importance:
  1. The time-dependent orientation of the Earth with respect to the inertial space (CTS vs CIS), due to the diurnal rotation of the Earth (Sidereal Time and Universal Time.
  2. For the description of the satellite motion: Dynamical Time given by the Mechanics of the orbital motion of the Earth around the Sun.
  3. For the precise measurements of signal travel times, it is necessary a uniform, precise and easily accessible time scale: Atomic Time related with Nuclear Physics phenomena.
• How timing errors in satellite geodesy are related to a position error of 1 cm:

  Earth rotation:

  

  Orbital motion:

  

  Signal travel time:

  

4.2.2 Sidereal Time and Universal Time

• The Local Apparent (i.e. true) Sidereal Time (LAST) is the local hour angle of the true vernal equinox (Aries point).
• For Greenwich: the Greenwich Apparent Sidereal Time (GAST).
• We have respectively the mean corresponding definitions, LMST and GMST.
• If is the nutation in longitude, the obliquity of the ecliptic, and is the longitude of the local meridian:

  

• The apparent (mean) sidereal time is used for astronomical observation and the mean sidereal time is used to construct a time scale: the Mean Sidereal Day (a difference of 0.0084 seconds with the Earth revolution period due to the precesion).
• For practical purposes a time scale tied to the Sun has been defined: the Mean Solar Time is measured by the hour angle of the mean Sun: the Universal Time (UT) is the Greenwich hour angle of the mean Sun (GHAMS), starting at midnight:

  

• Both kinds of time, based on the rotation of the Earth, differs basically due to the different reference: mean vernal equinox (Aries point) for sidereal time and mean Sun for universal time, i.e. due to the Earth orbit:
  1 mean sidereal day = 1 mean solar day - 3 55. 909
  
• But the raw universal time obtained from observations in a given station B is still afected by the location-dependent influences of the actual true pole position. Reducing to the conventional terrestrial pole (CTP), we have a change in longitude , giving UT1:

  

• UT1 is the fundamental time scale in geodetic astronomy and satellite geodesy, and also in navigation. However, as it contains all variations of the earth's rotation, is not an uniform time scale.
• The fundamental relation between GMST and UT1 is given by the International Astronomical Union (IAU) since 1984 as:

  

  being the time since J2000 (January 1 2000, 12h UT1) in Julian centuries of 365.25 days.

4.2.3 Dynamical Time

The Dynamical Time corresponds to the concept of the inertial time, i.e. the time scale that fulfills exactly the equation of motion of celestial bodies:

• Barycentric Dynamical Time (TDB) is derived from orbital motions referred to the barycenter of the solar system (Coordinate time in the terminology of General Relativity).
• Terrestrial Dynamical Time (TDT) is referred to the geocenter (Proper Time with periodic variations up to 1.6 milliseconds, but common between satellite and Earth).

4.2.4 Atomic Time

The International Atomic Time (TAI) is defined in such a way that its second equals to the TDT second (its unit was derived from the mean duration of the solar day between 1756 and 1895 and it has been adopted as the International System of units (SI) second):

The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the Cesium 133 atom

TAI is defined by a large number of Cesium clocks in various laboratories and agreed with the epoch of UT1 on January 1, 1958, and due to the Earth deceleration:

TAI-UT1 = = = = =

And:

For many applications, as navigation, an uniform scale but adapted to UT1: the Coordinated Universal Time (UTC): it is atomic but the difference with the UT1 should not exceed 0.7 seconds (leapseconds):

The GPS time (GPST):

• It has identical epoch than UTC on January 6, 1980 (starting epoch for GPST).
• It is an atomic scale and it is not incremented by the leap seconds.
• The unit of GPS is the SI (TAI) second.
• GPST is only derived from clocks of the Control segment, so it differ a bit referring to the TAI.

ranging between -1376 ns in Jan 1, 1989 to 232 ns in Jan 1, 1992.