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### Earth Orientation Parameters (EOP): Definitions and theiruse in transformations

Time ScalesTime ScalesTime Scales

Outline

- Reference Systems and Reference Frames
- Earth Orientation Parameters
- Time
- Software

What is a Reference System?

z

y

- A 4-D coordinate system
- 3 spatial coordinates, 1 time coordinate
- A mathematical abstraction that allows us to combine, compare, or otherwise relate positional measurements taken at different times and/or from different places
- Defined by specification of
- Origin of coordinates
- Direction of axes
- Standardized algorithms (software) that allow raw measurements to be transformed into the system
- Some reference systems are defined by their relationship to others

x

z

z

z

y

x

x

Primary Reference Systems

Two primary reference systems are of interest:

- Barycentric Celestial Reference System (BCRS)

Also referred to as the International Celestial Reference System (ICRS)

= The system of star catalog data

replacing the “J2000.0 system” or the “FK5 system”

- International Terrestrial Reference System (ITRS)

= The world geodetic system = WGS-84 = GPS

= Earth-Centered-Earth-Fixed (ECEF)

Today’s Problem: BCRS ITRS

Celestial coordinates expressed in BCRS (ICRS)

At any given time, what is the relationship?

Terrestrial coordinates expressed in ITRS

(ECEF)

How to transform vectors between the BCRS (ICRS) and the ITRS (ECEF)

Vectors are sets of 4 numbers (3 spatial coordinates and 1 time coordinate) representing location (or its time derivatives) or direction (or its time derivatives)

There are some intermediate reference systems that are often used as waypoints:

BCRS GCRS CIRS TIRS ITRS

Rephrased, Today’s Problem is . . .For Later!

relativistic rotations

transformation (a function of EOPs)

Real-World Applications

- Looking up: In what direction is a ground-based sensor pointed?
- Looking down: What spot on Earth will be “seen” by a space-based sensor?
- Navigation: What is the position, velocity, acceleration, and/or rotation of a vehicle in inertial space? With respect to the ground?

A Few Ideas About the Primary Reference Systems

- The orientation of the coordinate axes must be specified, in practice, by the position vectors of real objects

“control points” or “anchors”

- The ensemble of these defining objects, and their coordinates, are referred to as a “reference frame”

The objects themselves are provided by nature; their assigned coordinates define the directions of the coordinate axes

The Terrestrial Reference System

- ITRS specifications:
- Origin at geocenter (center of mass of Earth + oceans + atmosphere)
- Relativistic “local Earth frame” with unit of time given by TCG second and unit of length given by SI meter
- Direction of axes
- Initially given by BIH system orientation at 1984.0
- Axes rotate with the crust — time evolution of orientation will create no residual global rotation with respect to the crust
- Realizations — reference frames:
- ITRFyyyy: 3-dimensional site coordinates and velocities

(yyyy indicates year of solution; e.g., ITRF2005 or ITRF2008)

- Site selection criteria: 3 years’ observations available, on rigid plates away from crustal deformations, velocities known to better than 3 mm/year

The Celestial Reference System

- BCRS (ICRS) specifications:
- Origin at solar system barycenter
- Relativistic metric tensor specified in IAU resolution B1.3 of 2000
- Defining objects are unresolved, stationary, and stable extragalactic radio sources
- Direction of axes
- Fixed in space — do not rotate with respect to defining objects
- “As near as possible” to system defined by the Earth mean equator and equinox of J2000.0 (i.e., ICRS “J2000.0 system”)
- Independent of time and specific realizations (if the objects that define their direction eventually change)
- Realizations two reference frames:
- ICRF2: Catalog of 3414 extragalactic radio sources (295 defining sources)
- HCRF: Catalog of ~100,000 stars from the Hipparcos catalog

(85% of entire catalog; astrometrically well-behaved stars)

Densification of Reference Frames

Addition of new control points, with coordinates consistent with those of defining points

- Allows for higher spatial density
- Allows for more even coverage of the Earth or celestial sphere
- On the celestial sphere, allows for reference frame to be represented at wavelengths other than radio (=5 cm) or optical (=0.5 m)

For example, IR, mm, or UV

Continued…

Densification of Reference Frames (cont.)

- On the Earth, any civilian GPS receiver effectively densifies the ITRF at 10-meter-level accuracy (a few meters or better for DGPS)
- Celestial extensions of the ICRF and HCRF:
- VLBA Calibrator Survey (VCS6)
- Tycho-2 Catalog
- UCAC2 (to V=16) and UCAC3 (UCAC4 in process)
- USNO B1.0
- 2MASS (IR)
- Solar system ephemerides (e.g., JPL DE405) aligned to ICRF

Accuracy of representation varies considerably — Let the buyer beware!

Complications in Establishing Reference Frames

- Problem is over-determined: only need two objects to define orientation of axes
- Therefore, if N objects in a catalog, there are ~N2/2 independent reference frame definitions — which will not, in general, be consistent due to errors in coordinate values
- Not too bad a problem as long as errors are random
- If errors are a function of position, the reference frame is warped (systematic distortions)
- Also problematic if errors are a function of other characteristics:
- For stars, brightness or color
- For Earth stations, type of instrument / technique

Using a warped reference frame — errors in coordinates a function of position

Using a warped reference frame — errors in coordinates a function of position

Each measured position of the target is slightly wrong, because the star positions on which it is based are slightly wrong

What our tracking measurements actually mean

Observed trajectory

What our tracking measurements seem to mean

Observed trajectory

Why Star Positions, and the Reference

Frames They Define, Degrade with Time

1950

1976

1990

Measured Path

2005

Actual Path

Even Many of the ICRF Radio Sources Seem to Move

From USNO Radio Reference Frame Image Database (RRFID), A. Fey et al.

BCRS GCRS CIRS TIRS ITRSFinally: The Barycentric to Geocentric Transformation

relativistic rotations

transformation (a function of EOPs)

How do we go from the Barycentric Celestial Reference System (BCRS) to the Geocentric Celestial Reference System (GCRS)?

Note: GCRS = Earth-Centered Inertial (ECI)

That is, how do we transform star coordinates from the solar system barycentric system of the star catalogs to the geocentric system in which we actually observe the stars on a given date?

BCRS to GCRS Transformation

- Positions in star catalogs represent where the stars would be seen from an idealized barycentric system at a certain epoch. To get observed geocentric coordinates of stars for a given date and time, they need to be adjusted for:
- Orbital motion, if part of a binary or multiple system
- Proper motion (3-D space velocity)
- Parallax
- Gravitational light deflection
- Aberration of light
- There are comparable adjustments for planetary or spacecraft ephemerides expressed in the BCRS

Relativistic terms in the aberration formula effectively implement the BCRS to GCRS transformation

Why All This is Important

- Any list of real objects and their coordinates defines a reference frame
- It can be good, bad, or something in between
- It may not be consistent with the ICRF or HCRF (for celestial coordinates) or the latest ITRF (for terrestrial coordinates)
- If it is based on old data, its accuracy has probably degraded significantly
- It is a bad idea to keep using some list of coordinates handed down from generation to generation
- The FK5 is obsolete!
- Please consult USNO!

Definition of Earth Orientation Parameters

- Earth orientation parameters (EOPs) describe the direction of axes fixed to the Earth in space
- used to transform between ground based coordinates (terrestrial reference frame) and “inertial” coordinates (celestial reference frame)
- 5 EOPs
- 2 polar motion (PM-x and PM-y)
- 1 Earth rotation angle (UT1-UTC)
- 2 celestial pole ( and or X and Y)

Three Aspects of Earth Rotation

1

1 Path of rotation axis in space (wrt GCRS)

precession / nutation

2

Path of rotation axis across Earth’s crust (wrt ITRS)

polar motion

- Rotation angle

UT1

All are functions of time

3

Two schemes for transforming between the ITRS and GCRS

Polar Motion

Frame Bias

“OLD”:

Nutation

Sidereal Time

Precession

“NEW”:

Precession + Nutation + Frame Bias

Polar Motion

Earth Rotation

Two schemes for transforming between the ITRS and GCRS

Polar Motion

Frame Bias

“OLD”:

Nutation

Sidereal Time

Precession

“NEW”:

Precession + Nutation + Frame Bias

Polar Motion

Earth Rotation

Polar Motion

- Left-handed coordinate system
- 2 dominant motions
- Chandler wobble (430 d)
- Annual wobble (365 d)
- Other smaller amplitude stochastic motions
- Causes of polar motion
- Oceans (pressure>current)
- Atmosphere (press.>wind)
- Hydrology?

Spectrum

POLAR MOTION

Prograde Annual Wobble

Power

NUTATION

ChandlerWobble

Prograde Semi-annual Wobble

Precession

Nutations

Free Core Nutation

Atmospheric Tides

-2

-1

1

2

0

Frequency in the Terrestrial Reference Frame (cycles per day)

-1

0

1

2

3

Frequency in the Celestial Reference Frame (cycles per day)

Two schemes for transforming between the ITRS and GCRS

Polar Motion

Frame Bias

“OLD”:

Nutation

Sidereal Time

Precession

“NEW”:

Precession + Nutation + Frame Bias

Polar Motion

Earth Rotation

UT1-UTC

- Dominant motions
- Trend
- Decadal
- Annual/semiannual
- Tidal
- Other smaller amplitude motions
- Causes of UT1-UTC
- Tidal deceleration
- Internal changes in inertia tensor
- Atmosphere (winds)
- Solid Earth tides

UT1-UTC Spectrum

annual

quasi-biennial oscillation

atmospheric modes

southern oscillation

solid Earth and ocean tides

semi -annual

40-50 -day oscillations

Power

monthly

fortnightly

decade fluctuations

(from core?)

atmospheric tides

0.1 year-1

0.2 year-1

1 year-1

0.1 month-1

Frequency

Leap Seconds

- Leap seconds are a conventional attempt to align Earth rotation time (angle) with clock time (UTC)
- |UT1-UTC| < 0.9 s
- Leap seconds occur at end of June or December but the time between leap seconds is unpredictable
- Expected to occur with increasing frequency
- Problematic for operational systems
- International Telecommunications Union (ITU) is considering a redefinition of UTC to eliminate leap seconds
- UT1 and UTC would diverge

No impact if system

doesn’t assume that UT1≈UTC, and

makes no assumption regarding size of UT1-UTC

Impact if system

assumes that UT1≈UTC, or

assumes a limit on the size of UT1-UTC

Potential Impact of Leap Second EliminationSuggestions Regarding Leap Second

If you are using UT1≈UTC, need to determine impact of changing to input UT1-UTC into operational procedures

If you assume numerical restrictions on UT1-UTC, need to determine impact on eliminating restrictions

Any significant problem with operational procedures should be reported to USNO

Two schemes for transforming between the ITRS and GCRS

Polar Motion

Frame Bias

“OLD”:

Nutation

Sidereal Time

Precession

“NEW”:

Precession + Nutation + Frame Bias

Polar Motion

Earth Rotation

Precession and Nutation

Precession / nutation: The changing direction of the Earth’s axis in space (now specifically with respect to the GCRS) due to torques exerted by the Moon, Sun, and planets

- Precession: the long-term, smooth, secular motion

23.5° circle around the ecliptic pole in 25,800 years

- Nutation: the smaller, short-term periodic motions

Largest component: 18x12 arcsecond ellipse traced out in 18.6 years

(Diagram from G. Beutler)

Celestial Pole

- Dominant motions at frequencies
- 18.6-year
- Semiannual
- Fortnightly
- Causes of precession-nutation are gravitational pull of celestial bodies on the equatorial bulge of Earth
- Very well determined
- Small component not caused by gravity

Celestial Pole Models

- Since direction of celestial pole in space is affected mainly by gravitational attraction of Sun, Moon, and planets, it is very predictable
- Current models IAU 2000 nutation and IAU 2006 precession
- Residuals between models and observations (d and d or dX and dY) due to
- Free Core Nutation (FCN)
- Other variable amplitude terms (e.g. annual)
- Errors in models
- Estimated error in models ~0.5 mas

Precession / Nutation ObservationsLatest Results — wrt P03 / IAU2000A

Model errors

~0.5 mas

(Figure from Capitaine et al. 2008)

Celestial Pole Models

- Older models IAU 1980 nutation and IAU 1976 precession
- Residuals between models and observations (d and d) predominantly due to
- Errors in models
- Estimated error in models ~10 mas
- Error is systematic, not random

Precession / Nutation ObservationsLate 90s Results — wrt Lieske / Wahr

Model errors

~10 mas

(Figure from Ma et al. 1998)

Basics

- Concepts
- Frequency
- Repeatable phenomenon
- Length of the time unit
- Epoch
- Naming convention

- Sources of Time
- Astronomy
- Based on Earth’s rotation
- Atomic Clocks
- Based on frequency of an atomic transition

Time Scales

Earth Rotation

- UT1 (Universal Time)
- Measure of Earth’s rotation angle wrt Sun
- Determined by conventional expression
- Earth Rotation Angle
- Greenwich Mean Sidereal Time
- UT0 = UT1 including polar motion
- UT2 = UT1 with conventional Seasonal Correction

- Sidereal Time
- Measure of Earth’s rotation angle wrt Celestial Reference Frame
- Determined by conventional expression

Earth Revolution

Atomic Time

Ephemeris Time

Echelle Atomique Libre (EAL)

Dynamical Time

Terrestrial Dynamic Time (TDT)

Barycentric Dynamic Time (TDB)

International Atomic Time (TAI)

Space-Time Coordinates

Coordinated Universal Time (UTC)

* TAI corrected by leap seconds

* Basis for civil time

Terrestrial Time (TT)

Geocentric Coordinate Time (TCG)

Barycentric Coordinate Time (TCB)

Time Scales

Earth Rotation

- UT1 (Universal Time)
- Measure of Earth’s rotation angle wrt Sun
- Determined by conventional expression
- Earth Rotation Angle
- Greenwich Mean Sidereal Time
- UT0 = UT1 including polar motion
- UT2 = UT1 with conventional Seasonal Correction

- Sidereal Time
- Measure of Earth’s rotation angle wrt Celestial Reference Frame
- Determined by conventional expression

Earth Revolution

Atomic Time

Ephemeris Time

Echelle Atomique Libre (EAL)

Dynamical Time

Terrestrial Dynamic Time (TDT)

Barycentric Dynamic Time (TDB)

International Atomic Time (TAI)

Space-Time Coordinates

Coordinated Universal Time (UTC)

* TAI corrected by leap seconds

* Basis for civil time

Terrestrial Time (TT)

Geocentric Coordinate Time (TCG)

Barycentric Coordinate Time (TCB)

Solar Time

Time measured by the Earth’s rotation with respect to the Sun

Elementary conceptual definition based on the diurnal motion of the Sun

Mean solar time reckoned from midnight on the Greenwich meridian

Traditional definition of the second used in astronomy

Mean solar second = 1/86 400 mean solar day

Three Forms

UT1 is measure of Earth’s rotation angle

Defined

By observed sidereal time using conventional expression

GMST= f1(UT1)

by Earth Rotation Angle

q = f2(UT1)

UT0 is UT1 plus effects of polar motion

UT2 is UT1 corrected by conventional expression for annual variation in Earth’s rotational speed

Universal Time (UT)inclination effect

eccentricity effect

Mean Time vs. Apparent Time

- Time interval between successive passages of the Sun over a meridian is not constant
- Inclination of the Earth’s axis
- Eccentricity of the Earth’s orbit
- Difference between mean time and apparent time is called the “equation of time”
- Mean noon precedes apparent noon by 14.5 minutes on February 12
- Apparent noon precedes mean noon by 16.5 minutes on November 3

Time Scales

Earth Rotation

- UT1 (Universal Time)
- Measure of Earth’s rotation angle wrt Sun
- Determined by conventional expression
- Earth Rotation Angle
- Greenwich Mean Sidereal Time
- UT0 = UT1 including polar motion
- UT2 = UT1 with conventional Seasonal Correction

- Sidereal Time
- Measure of Earth’s rotation angle wrt Celestial Reference Frame
- Determined by conventional expression

Earth Revolution

Atomic Time

Ephemeris Time

Echelle Atomique Libre (EAL)

Dynamical Time

Terrestrial Dynamic Time (TDT)

Barycentric Dynamic Time (TDB)

International Atomic Time (TAI)

Space-Time Coordinates

Coordinated Universal Time (UTC)

* TAI corrected by leap seconds

* Basis for civil time

Terrestrial Time (TT)

Geocentric Coordinate Time (TCG)

Barycentric Coordinate Time (TCB)

- First cesium-133 atomic clock established at National Physical Laboratory (NPL) by Essen and Parry in 1955
- Frequency of cesium transition measured in 1955 in terms of the second of UT2
- 9 192 631 830 10 Hz
- Frequency of cesium transition measured by Markowitz, Hall, Essen, and Parry during 1955 – 1958 in terms of the second of ET
- 9 192 631 770 20 Hz
- Definition of the SI second adopted by the 13th CGPM in 1967
- Second of atomic time = second of Ephemeris Time (ET)

Second = duration of 9 192 631 770 periods of radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom

International Atomic Time (TAI)

- Coordinate time scale in a geocentric reference frame with the SI second realized on the rotating geoid as the scale unit
- Continuous atomic time scale determined by Bureau International de l’Heure (BIH) since 1958, now maintained by Bureau International des Poids et Mesures (BIPM)
- TAI = UT2 on January 1, 1958 0 h
- Follow-on from
- A.1 (maintained at USNO with input from 9 other laboratories originally. - now only USNO)
- AM (at BIH with input from many laboratories)
- A3 at BIH with input from 3 best laboratories
- Others
- Became AT (or TA) in 1969, TAI in 1971

Earth Rotation

- UT1 (Universal Time)
- Measure of Earth’s rotation angle wrt Sun
- Determined by conventional expression
- Earth Rotation Angle
- Greenwich Mean Sidereal Time
- UT0 = UT1 including polar motion
- UT2 = UT1 with conventional Seasonal Correction

- Sidereal Time
- Measure of Earth’s rotation angle wrt Celestial Reference Frame
- Determined by conventional expression

Earth Revolution

Atomic Time

Ephemeris Time

Echelle Atomique Libre (EAL)

Dynamical Time

Terrestrial Dynamic Time (TDT)

Barycentric Dynamic Time (TDB)

International Atomic Time (TAI)

Space-Time Coordinates

Coordinated Universal Time (UTC)

* TAI corrected by leap seconds

* Basis for civil time

Terrestrial Time (TT)

Geocentric Coordinate Time (TCG)

Barycentric Coordinate Time (TCB)

Coordinated Universal Time (UTC)

- Name “Coordinated Universal Time” (UTC) adopted by IAU in 1967
- From 1961 to 1972 UTC contained both frequency offsets and fractional (less than 1 s) steps to maintain agreement with UT2 within about 0.1 s
- In 1970 formalized by International Radio Consultative Committee (CCIR) of International Telecommunication Union (ITU) in 1962 ITU-R TF.460-6 STANDARD-FREQUENCY AND TIME-SIGNAL EMISSIONS(1970-1974-1978-1982-1986-1997-2002)
- Incorporated by reference into the Radio Regulations

Leap Seconds may be introduced as the last second of any UTC month.

December and June preferred, March and September second choice.

Adjustments

TAI

In 1972 present UTC system was implemented, with 1 s (leap second) steps but no frequency offsets to maintain agreement with UT1 within 0.9 s

Definition of UTC is a compromise to provide both the SI second and an approximation to UT1 for celestial navigation in same radio emission

Coordinated Universal Time (UTC)Standard Time

U. S. Code specifies UTC offset by integral hours as standard time

Federal (DoD and DoT) Radionavigation Systems document specifies USNO as time/frequency standard for all U. S. radionavigation services

DoD Master Position Navigation and Timing Plan specifies USNO as standard time/frequency for DoD

SEC regulations specify NIST as standard for timing sales of securities

Federal Standard 1002A - 1990 (Time and Frequency Information in Federal Telecommunications Systems) specifies NIST as standard

USNO and NIST have Memorandum of Agreement

UTC(USNO) and UTC(NIST) do not differ by more than 100 nanoseconds.

Typically they do not differ by more than a few tens of nanoseconds

Time scales of both institutions are traceable to each other at the nanosecond level

- Each satellite carries a suite of cesium or rubidium atomic clocks
- Satellite and global tracking network atomic clocks are used to form a common statistical time scale known as GPS Time
- No leap seconds
- Origin is midnight of January 5/6, 1980 UTC
- Steered to within 1 s of UTC(USNO), except no leap seconds are inserted
- Relationships with TAI and UTC (within statistical error)
- GPS Time = TAI – 19 s = constant

- Re-computed every 15 minutes based on satellite ranging measurements made by GPS monitor stations
- OCS software estimates clock differences of GPS satellite and monitor station clocks
- Satellite clock differences uploaded to each satellite approximately once a day
- The additional correction contained in the GPS broadcast message allows a GPS timing user to produce TAI or UTC time.

USNO continuously monitors GPS satellites

Provides GPS Master Control Station differences between GPS Time and UTC(USNO)

Master Control Station Kalman Filter (MCSKF) generates clock solutions to minimize UTC(USNO)-GPS

Corrections to create both GPS Time and GPS’s delivered prediction of UTC(USNO) are applied in GPS receiver by applying correction contained in theGPS data message

Implementing GPS TimeTelling GPS Time

GPS Time counts in weeks and seconds of a week from midnight of January 5/6, 1980 UTC

Weeks begin at the Saturday/Sunday transition

Days of the week are numbered, with Sunday being 0; Saturday is day 6

GPS week 0 began at the beginning of the GPS Time Scale. ( 60 x 60 x 24 x 7)

- Within each week the time denoted as the second of the week
- Number between 0 and 604,800 (60 x 60 x 24 x 7)

Time scales from GPS

GPS

USNO

Monitor Station Clocks

Cesium

Monitoring Data

GPS System Time

SV Clocks

Cesium

Rubidium

GPS Time – SV Time

UTC(USNO) - GPS Time

SV Nav Message

SV Time

GPS Time

UTC(USNO)

USERS

Earth Rotation

- UT1 (Universal Time)
- Measure of Earth’s rotation angle wrt Sun
- Determined by conventional expression
- Earth Rotation Angle
- Greenwich Mean Sidereal Time
- UT0 = UT1 including polar motion
- UT2 = UT1 with conventional Seasonal Correction

- Sidereal Time
- Measure of Earth’s rotation angle wrt Celestial Reference Frame
- Determined by conventional expression

Earth Revolution

Atomic Time

Ephemeris Time

Echelle Atomique Libre (EAL)

Dynamical Time

Terrestrial Dynamic Time (TDT)

Barycentric Dynamic Time (TDB)

International Atomic Time (TAI)

Space-Time Coordinates

Coordinated Universal Time (UTC)

* TAI corrected by leap seconds

* Basis for civil time

Terrestrial Time (TT)

Geocentric Coordinate Time (TCG)

Barycentric Coordinate Time (TCB)

- Ephemeris Time (ET) was based on the Newtonian theory of gravitation and made no distinction between proper time and coordinate time
- Proper time: Reading of an ideal clock in its own rest frame
- Coordinate time: Time coordinate in given space-time coordinate system
- Between 1976 and 2000, the IAU adopted new relativistic time scales consistent with the general theory of relativity whose unit is the SI second
- Note that relativistic times are theoretical
- Time not kept on a clock

Earth Rotation

- UT1 (Universal Time)
- Measure of Earth’s rotation angle wrt Sun
- Determined by conventional expression
- Earth Rotation Angle
- Greenwich Mean Sidereal Time
- UT0 = UT1 including polar motion
- UT2 = UT1 with conventional Seasonal Correction

- Sidereal Time
- Measure of Earth’s rotation angle wrt Celestial Reference Frame
- Determined by conventional expression

Earth Revolution

Atomic Time

Ephemeris Time

Echelle Atomique Libre (EAL)

Dynamical Time

Terrestrial Dynamic Time (TDT)

Barycentric Dynamic Time (TDB)

International Atomic Time (TAI)

Space-Time Coordinates

Coordinated Universal Time (UTC)

* TAI corrected by leap seconds

* Basis for civil time

Terrestrial Time (TT)

Geocentric Coordinate Time (TCG)

Barycentric Coordinate Time (TCB)

- In 1991 IAU renamed TDT as Terrestrial Time (TT)
- Unit is the SI second on the geoid and is defined by atomic clocks on the surface of the Earth
- Origin of January 1, 1977 0 h
- TT = TAI + 32.184 s
- Maintains continuity with Ephemeris Time (ET)
- Theoretical equivalence of time measured by quantum mechanical atomic interaction and time measured by gravitational planetary interaction
- To be used as the time reference for apparent geocentric ephemerides
- Any difference between TAI and TT is a consequence of the physical defects of atomic time standards, and has probably remained within the approximate limits of ± 10µs. It may increase slowly in the future as time standards improve. In most cases, and particularly for the publication of ephemerides, this deviation is negligible

TT

ET

1977.0

After F.R. Stephenson and L.V. Morrison, Phil. Trans. R. Soc. LondonA351, 165 – 202 (1995)

Julian Date

- Used for continuous counts of days
- Joseph Justus Scaliger in 1583 used three cycles :
- The Julian calendar in use until the Gregorian calendar reform of 1582
- The “Golden Number” cycle associated with Meton in the fifth century B.C., equated 19 years with 235 months.
- A tax cycle or “indiction” of 15 years used in legal and financial organization of the Roman Empire
- Year could be characterized by its position (S) within a 28-year solar cycle, its position (G) within the 19-year cycle of Golden Numbers, and its position (I) within the Roman tax cycle

- Because 15, 19 and 28 have no common factors S, G, and I repeat after 28×19×15 = 7980 years.
- Scaliger called this a Julian period, not in honor of his father as is frequently asserted, but because of the involvement of the Julian Calendar.
- Knowing that the year 1 B.C. was characterized by S=1, G=1, I=3, Scaliger computed that S=1, G=1, I=1 occurred for 4713 B.C. This provided him with the starting year
- Modified Julian Date (MJD) is defined as JD−2400000.5
- Timescale should be specified when precision matters, e.g., MJD 47479.25 TAI.

Traceability

International Organization for Standardization (ISO) International Vocabulary of Basic and General Terms in Metrology, traceability is defined as:

The property of a result of a measurement or the value of a standard whereby it can be related to stated references, usually national or international standards, through an unbroken chain of comparisons all having stated uncertainties.

GPS Time can be made directly traceable to UTC

Measured difference between UTC and GPS Time is reported in the BIPM Circular T, with an accuracy of 10 ns.

GPS Time directly traceable to UTC by being traceable to UTC(USNO), which is also traceable to UTC via the Circular T.

Alternate means of direct traceability may also be established via any other national laboratory, such as NIST.

Philosophy is that only a user should decide traceability issues because only that user knows the specifics of their system

National Standard

Time / Frequency Transfer

International Standard (BIPM)

User

Master Clock

Alternate Master Clock

Voice

Computer

Modem

Internet

(± 1ms)

GPS

(± 8 ns)

Two-way Satellite

Time Transfer

(± 1ns)

- GPS
- DSCS
- Satellite Tracking Ranges
- Satellite Ground Control Stations

- NGA
- DSN
- Loran-C
- NAVSECGRU

- Ships, Submarines, Aircraft
- Ground Forces
- Shore Activities
- Space Commands
- NGA
- GPS

- CommunicationsCenters
- Operation Centers

- GPS
- DSCS
- Classified Programs

USERS

Scope

- Authoritative software
- Tied to international standards
- Produced or vetted by DoD SMEs
- “Free”
- Source-code products
- Web-based calculator

- Naval Observatory Vector Astrometry Software (NOVAS)
- Standards of Fundamental Astronomy (SOFA)

Complete software package for positional astronomy

Instantaneous coordinates and radial velocity of any star or planet

e.g., apparent, topocentric, astrometric places

Observer at geocenter, on or near surface, on near-Earth spacecraft

Terrestrial → celestial transformation

“Building blocks” and support functions for these calculations

e.g., precession, nutation, frame bias, star-catalog transformations, etc.

Produced and supported by USNO

Used for US sections of The Astronomical Almanac

Status

Version 3.1 released in March 2011

Python edition nearly completed

NOVAS

Naval Observatory Vector Astronomy Subroutines

Collection of routines which implement official algorithms for fundamental-astronomy computations

Primarily supports celestial ↔ terrestrial transformation

“astronomy” routines

e.g., calendars, time scales, precession, nutation, star space motion, and star catalog transformations

“vector/matrix” routines

Service of IAU Division I

Managed by international review board

Basis for IERS Conventions Chapter 5 software

Used for UK sections of The Astronomical Almanac

Status

Latest release 2010-12-01

SOFA

Standards of Fundamental Astronomy

Similarities

- Both come from authoritative sources
- Both support latest IAU resolutions
- Both are consistent with IERS Conventions for practical levels of accuracy
- Both have Fortran and C versions
- Both contain “testbed” and demonstration programs

Differences

- Developed independently
- Except NOVAS uses SOFA code for IAU 2000A nutation
- NOVAS currently has broader scope
- i.e., instantaneous coordinates of celestial bodies (Sun, Moon, planets, stars, etc.)
- NOVAS contains its own improved “low-precision” nutation model
- 2000K: 0.1 mas level, 1700 to 2300 (truncated IAU 2000A)
- Documented in USNO Circular 181
- NOVAS uses an alternative algorithm for location of CIO

Differences (continued)

- SOFA includes support for older models
- e.g., IAU 1976 precession, IAU 1980 nutation
- SOFA includes small corrections to IAU 2000A nutation
- SOFA has finer structure

NOVAS/SOFA Numerical Comparison

- Test: Compare ITRF → GCRS transformation using SOFA and NOVAS
- February 2009: latest Fortran SOFA and NOVAS F3.0g (beta)
- May 2010: latest C editions of SOFA and NOVAS
- Results:
- Differences ~ 1.8 μas
- Attributable to very small terms from IAU 2006 precession model that were included in SOFA implementation of IAU 2000A nutation model
- Differences drop to ~ 0.2 μas when these terms are removed
- Full discussion in NOVAS User’s Guides

Which Product Should I Use?

- Use NOVAS When…
- You need coordinates of celestial bodies (Sun, Moon, planets, stars, etc.)
- Your project requires use of DoD or GOTS code
- Use SOFA When…
- Your project requires use of official IAU code
- You need to compute the actual transformation matrices
- You also need code for now-obsolete models

Both products are authoritative, well-documented, relatively easy to use, and comply with IERS Conventions and IAU recommendations.

USNO Equinox-Based ITRF-to-GCRS CalculatorBackground

- Upgraded equinox-based transformation matrix calculator is being tested at the USNO EO Department server at http://maia.usno.navy.mil/t2c36e/t2c36e.html
- The IERS Conventions (2010) TN36-based validation code for computing the equinox-based ITRF to GCRS transformation
- Written in FORTRAN
- Relying heavily on code from http://tai.bipm.org/iers/conv2010/conv2010.htmland SOFA.
- Observable quantities are obtained from a version of finals2000A.daily
- Later observables can be obtained from evaluation of the NGA 5-liner
- Nutation and tidal model changes
- Major changes are updates to the nutation and long-period tidal models
- Interpolation of observable quantities, intermediate quantities, and other output quantities are similar to the TN32-based calculator

Equinox-Based ITRF-to-GCRS Calculator User Interface (continued)

- User chooses dates and time intervals
- Code produces a file containing the ITRF-to-GCRS transformation and desired intermediate quantities
- Standard output is the transformation matrix from terrestrial to celestial frames
- Optional quaternion/euler-parameter output
- Options for intermediate quantities include the polar motion, GMST, Equation of the Equinoxes, Precession, Nutation, and combined bias-precession-nutation matrices or quaternions
- Future developments include
- CIO/CIP compatible (as opposed to equinox based) using 2006 IAU Precession Model

Calculator Output

Terrestrial-to-celestial (T2C)

rotation matrix

Intermediate rotation matrices

are below the T2C matrix

Resources — A Sample

- IERS Conventions (2010): http://maia.usno.navy.mil/conv2010/conventions.html
- Conventions users’ survey available at http://maia.usno.navy.mil/conv2010/Conventions_Survey.html
- USNO Circular 179 (PDF version): http://aa.usno.navy.mil/publications/docs/Circular_179.php
- IERS FAQs: http://www.iers.org/IERS/EN/Service/FAQs/faq__cont.html
- IERS Earth rotation data (Bulletins A & B): http://www.iers.org/IERS/EN/DataProducts/EarthOrientationData/eop.html?__nnn=true
- The Astronomical Almanac: http://aa.usno.navy.mil/publications/docs/asa.php

NOVAS

http://www.usno.navy.mil/USNO/astronomical-applications/software-products/novas/

SOFA

http://www.iausofa.org/

USNO Earth Orientation Matrix Calculator

http://maia.usno.navy.mil/t2crequest/t2crequest.html

Software Resources

Polar Motion

“OLD” Transformation:

- “NEW” Transformation:

- xP and yP are the pole coordinates
- s’ is new and is given by

- Terms with periods <2 days and amplitudes <10mas
- Available in IERS Conventions
- Usually neglected

- SOFA:
- iau_SP00

- NOVAS:
- Contained in WOBBLE

GPS Time Nomenclature

ICD-GPS-200C

“GPS time is established by the Control Segment and is referenced to a UTC (as maintained by the U.S. Naval Observatory) zero time-point defined as midnight on the night of January 5, 1980/morning of January 6, 1980. The largest unit used in stating GPS time is one week defined as 604,800 seconds. GPS time may differ from UTC because GPS time shall be a continuous time scale, while UTC is corrected periodically with an integer number of leap seconds. There also is an inherent but bounded drift rate between the UTC and GPS time scales. The OCS shall control the GPS time scale to be within one microsecond of UTC (Modulo one second).”

2003 CJCS Master Positioning, Navigation and Timing Plan (CJCSI 6130.01C )

“The accuracy of the GPS transfer of UTC is measured at the Master Clock of the United States, which is located at the USNO. The USNO calculates the accuracy and delivers it to the Operational Control Segment (OCS). The OCS maintains accurate time transfer by calculating GPS time, calculating individual space vehicle (SV) corrections to GPS time, and uploading corrections into the SV databases for subsequent re-broadcast to users. The OCS maintains a steady state time transfer accuracy of the GPS signal to an error of less than or equal to 20 nanoseconds (nsec) (95 percent) relative to UTC (USNO) (threshold) and 10 nsec (95 percent) relative to UTC (USNO) (objective).”

“GPS Time” is used to indicate the internal GPS System Time as defined in section 3.3.4 of ICD-GPS-200C (2003).

"UTC(USNO) via GPS" is used to indicate GPS’s delivered prediction of UTC (USNO).

In mathematical expressions where "UTC(USNO) via GPS" might be considered to be awkward, “tUTC” may be substituted.

Only in text files that do not permit the display of subscripts should “t_UTC” be used as a substitute for “tUTC”

This nomenclature may be extended, if required, to specific applications of GPS for other means of time dissemination by adding the technique employed, e.g. “UTC(USNO) via GPS common view“

Dynamical Time

- In 1976 IAU defined dynamical time scales consistent with general relativity to distinguish between time scales with origins at the geocenter and the barycenter.of the solar system
- Named Terrestrial Dynamical Time (TDT) and Barycentric Dynamical Time (TDB) in 1979
- At the instant 1977 January 01 d 00h 00m 00s TAI, the value of the new time scale for apparent geocentric ephemerides is 1977 January 1d 00h 00m 32.184 exactly.
- The unit is a day of 86400 SI seconds at mean sea level.
- The timescales for equations of motion referred to the barycenter of the solar system is such that there will be only periodic variations between these timescales and those of the apparent geocentric ephemerides.
- TDT maintains continuity with ET
- By choosing an appropriate scaling factor TDB determined from TDT by a conventional mathematical expression

Barycentric Ephemeris Time (Teph)

- Coordinate time related to TCB by an offset and a scale factor
- Ephemerides (e.g. DEnnn) based upon the coordinate time Teph are automatically adjusted in the creation process so that the rate of Teph has no overall difference from the rate of Terrestrial Time (TT)
- Therefore also no overall difference from the rate of Barycentric Dynamical Time (TDB).
- Space coordinates obtained from the ephemerides are consistent with TDB.

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