1 ephemerides in the relativistic framework: _ time scales, spatial coordinates, astronomical...

1

Ephemerides in the relativistic framework:_

time scales, spatial coordinates, astronomical constants and units

Sergei A.Klioner

Lohrmann Observatory, Dresden Technical University

Paris Observatory, 6 April 2006

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Contents

• Newtonian astrometry and Newtonian equations of motion

• Why relativity?

• Coordinates, observables and the principles of relativistic modelling

• IAU 2000: BCRS and GCRS, metric tensors, transformations, frames

• Relativistic time scales and reasons for them:

TCB, TCG, proper times, TT, TDB, Teph

• Scaled-BCRS

• Astronomical units in Newtonian and relativistic frameworks

• Do we need astronomical units?

• Scaled-GCRS

• TCB/TCG-, TT- and TDB-compatible planetary masses

• TCB-based or TDB-based ephemeris: notes, rules and recipes

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Accuracy of astrometric observations

1 mas

1 µas10 µas

100 µas

10 mas

100 mas

1“

10”

100”

1000”

1 µas10 µas

100 µas

1 mas

10 mas

100 mas

1”

10”

100”

1000”

1400 1500 1700 1900 2000 21000 1600 1800

Ulugh Beg

Wilhelm IVTycho Brahe

HeveliusFlamsteed

Bradley-Bessel

FK5

Hipparcos

Gaia

SIM

ICRF

GC

naked eye telescopes space

1400 1500 1700 1900 2000 21000 1600 1800

Hipparchus

4.5 orders of magnitude in 2000 years

further 4.5 orders in 20 years

1 as is the thickness of a sheet of paper seen from the other side of the Earth

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Modelling of astronomical observations in Newtonian physics

M. C. Escher

Cubic space division, 1952

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Astronomical observation

physically preferred global inertialcoordinates

observables are directly related to the inertial coordinates

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Modelling of positional observations in Newtonian physics

• Scheme:• aberration• parallax• proper motion

• All parameters of the model are defined in the preferred global coordinates:

• Newtonian equations of motion:

( , ), ( , ), ,

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Why general relativity?

• Newtonian models cannot describe high-accuracy observations:

• many relativistic effects are many orders of magnitude larger than the observational accuracy

space astrometry missions or VLBI would not work without relativistic modelling

• The simplest theory which successfully describes all available observational data:

APPLIED RELATIVITYAPPLIED RELATIVITY

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physically preferred global inertialcoordinates

observables are directly related to the inertial coordinates

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no physicallypreferred coordinates

observables have to be computed ascoordinate independent quantities

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General relativity for astrometry

Coordinate-dependentparameters

Equations ofsignal

propagation

Astronomicalreference

frames

Observational data

Relativisticequationsof motion

Definition ofobservables

Relativisticmodels

of observables

A relativistic reference system

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General relativity for astrometry

Coordinate-dependentparameters

Relativistic reference systems

Equations ofsignal

propagation

Astronomicalreference

frames

Observational data

Relativisticequationsof motion

Definition ofobservables

Relativisticmodels

of observables

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The IAU 2000 framework in Manchester…

13

The IAU 2000 framework

• Three standard astronomical reference systems were defined

• BCRS (Barycentric Celestial Reference System)

• GCRS (Geocentric Celestial Reference System)

• Local reference system of an observer

• All these reference systems are defined by

the form of the corresponding metric tensors.

Technical details: Brumberg, Kopeikin, 1988-1992 Damour, Soffel, Xu, 1991-1994 Klioner, Voinov, 1993 Klioner,Soffel, 2000 Soffel, Klioner,Petit et al., 2003

BCRS

GCRS

Local RSof an observer

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Relativistic Astronomical Reference Systems

particular reference systems in the curved space-time of the Solar system

• One can use any

• but one should fix one

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The Barycentric Celestial Reference System

• The BCRS is suitable to model processes in the whole solar system

200 2 4

0 3

2

2 21 ( , ) ( , ) ,

4( , ) ,

21 ( , ) .

ii

ij ij

g w t w tc c

g w tc

g w tc

x x

x

x

23 3 3

2 2

00 2 0

( , ) 1 ( , )( , ) ( , ) | | , ( , ) ,

| | 2 | |

/ , / , is the BCRS energy-momentum tensor

ii

kk i i

t tw t G d x G d x t w t G d x

c t

T T c T c T

x x

x x x x xx x x x

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N-body problem in the BCRS

Equations of motion

in the PPN-BCRS:

Einstein-Infeld-Hoffman (EIH) equations:

General relativity:

Lorentz, Droste, 1916

EIH, 1936

Damour, Soffel, Xu, 1992

PPN formalism:

Will, 1973; Haugan, 1979;

Klioner, Soffel, 2000

used in the JPL ephemerissoftware (usually ==1)

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Geocentric Celestial Reference System

The GCRS is adopted by the International Astronomical Union (2000) to model physical processes in the vicinity of the Earth…

Why not BCRS?

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Geocentric Celestial Reference System• Imagine a sphere (in inertial coordinates of special relativity), which is then forced to move in a circular orbit around some point…

• What will be the form of the sphere for an observer at rest relative to that point?

Lorentz contraction deforms the shape…

Direction ofthe velocity

Additional effect due to acceleration (not a pure boost) and gravitation (general relativity, not special one)

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The GCRS is adopted by the International Astronomical Union (2000) to model physical processes in the vicinity of the Earth:

A: The gravitational field of external bodies is represented only in the form of a relativistic tidal potential.B: The internal gravitational field of the Earth coincides with the gravitational field of a corresponding isolated Earth.

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The GCRS is adopted by the International Astronomical Union (2000) to model physical processes in the vicinity of the Earth:


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The GCRS is adopted by the International Astronomical Union (2000)to model physical processes in the vicinity of the Earth:


200 2 4

0 3

2

2 21 ( , ) ( , ) ,

4( , ) ,

21 ( , ) .

aa

ab ab

G W T W Tc c

G W Tc

G W Tc

X X

X

X

1( , ) ( , ) ( ) ( , ), ( , ) ( , ) ( ) ( , ).

2a a a c a

E a T E abc b TW T W T Q T X W T W T W T C T X W T X X X X X X

internal + inertial + tidal external potentials

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BCRS-GCRS transformation

52 4

42

1 1( ) ( ) ( ) ( ) , ,

1 1( ) ( ) ( )

2

i i i i ij i jE E E E E

a a i i j j ij j ijk j ki E E E E E E E

T t A t v r B t B t r B t r r C t O cc c

X R t r v v r D t r D t r r O cc

x

• The coordinate transformations:

with 3, , ( )X r i i iE E EC T O r x x t

where ( )iEx t are the BCRS position and velocity of the Earth,( )i

Ev tand

( )a ai iR T

( ), ( ), ( ), ( ), ( , ), ( ), ( )i ij ij ijkA t B t B t B t C t D t D tx

and the orientation is CHOSEN to be kinematically non-rotating:

are explicit functions,

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Local reference system of an observer

The version of the GCRS for a massless observer:

A: The gravitational field of external bodies is represented only in the form of a relativistic tidal potential.

• Modelling of any local phenomena: observation, attitude, local physics (if necessary)

, :aW W internal + inertial + tidal external potentials

observer

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Celestial Reference Frame

• All astrometric parameters of sources obtained from astrometric observations are defined in BCRS coordinates:

• positions• proper motions• parallaxes• radial velocities• orbits of (minor) planets, etc.• orbits of binaries, etc.

• These parameters represent a realization (materialization) of the BCRS

• This materialization is „the goal of astrometry“ and is called

Celestial Reference Frame

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Relativistic Time Scales: TCB and TCG

• t = TCB Barycentric Coordinate Time = coordinate time of the BCRS

• T = TCG Geocentric Coordinate Time = coordinate time of the GCRS

These are part of 4-dimensional coordinate systems so that

the TCB-TCG transformations are 4-dimensional:

• Therefore:

• Only if space-time position is fixed in the BCRS TCG becomes a function of TCB.

52 4

1 1( ) ( ) ( ) ( ) ,i i i i ij i j

E E E E ET t A t v r B t B t r B t r r C t O cc c

x

( , )iTCG TCG TCB x

( )i iobsx x t

( ( ) ) i i iE Er x x t

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Relativistic Time Scales: TCB and TCG

• Important special case gives the TCG-TCB relation at the geocenter:

( )i iEx x t

linear drift removed:

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Relativistic Time Scales: proper time scales

• proper time of each observer: what an ideal clock moving

with the observer measures…

• Proper time can be related to either TCB or TCG (or both) provided that the trajectory of the observer is given:

The formulas are provided by the relativity theory:

and/or( ) ( )i aobs obsx t X T

1/ 2

00 0 2

2 1, ( ) , ( ) ( ) , ( ) ( ) ( )i i j

obs i obs obs ij obs obs obs

dg t t g t t x t g t t x t x t

dt c c

x x x

1/ 2

200 0 2

2 1, ( ) , ( ) ( ) , ( ) ( ) ( ) /a a b

obs a obs obs ab obs obs obs

dG T T G T T X T G T T X T X T c

dT c c

X X X

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Relativistic Time Scales: proper time scales

• Specially interesting case: an observer close to the Earth surface:

2 42

1 11 ( ) , "tidal terms"

2

Xobs E obs

dX T W T O c

dT c

1710

• But is the definition of the geoid! 21( ) , const

2 X

obs E obs GX T W T U

• Therefore

2

11 " terms ( ), ( )" ...

iG

dU h T T

dT c

h is the height above the geoid

iis the velocity relative to the rotating geoid

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Relativistic Time Scales: TT

• Idea: let us define a time scale linearly related to T=TCG, but which numerically coincides with the proper time of an observer on the geoid:

with

(1 ) GTT L TCG

• Then 2

11 " terms , " ...

idh

d TT c

2

1G GL U

c

• To avoid errors and changes in TT implied by changes/improvements in the geoid, the IAU (2000) has made LG to be a defined constant:

-106.969290134 10 GL

• TAI is a practical realization of TT (up to a constant shift of 32.184 s)

• Older name TDT (introduced by IAU 1976): fully equivalent to TT

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Relativistic Time Scales: TDB-1

• Idea: to scale TCB in such a way that the scaled TCB remains close to TT

• IAU 1976: TDB is a time scale for the use for dynamical modelling of the Solar system motion which differs from TT only by periodic terms.

• This definition taken literally is flawed: such a TDB cannot be a linear function of TCB!

But the relativistic dynamical model (EIH equations) used by e.g. JPL is valid only with TCB and linear functions of TCB…

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Relativistic Time Scales: Teph

• Since the original TDB definition has been recognized to be flawed

Myles Standish (1998) introduced one more time scale Teph differing

from TCB only by a constant offset and a constant rate:

0 eph ephT K TCB T

• The coefficients are different for different ephemerides.

• The user has NO information on those coefficients!

• The coefficients could only be restored by some numerical procedure (Fukushima’s “Time ephemeris”)

•For JPL only the transformation from TT to Teph which matters…

• VSOP-based analytical formulas (Fairhead-Bretagnon) are used for this

transformations

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Relativistic Time Scales: TDB-2The IAU Working Group on Nomenclature in Fundamental Astronomy suggested to re-define TDB to be a fixed linear function of TCB:

• TDB to be defined through a conventional relationship with TCB:

• T0 = 2443144.5003725 exactly,

• JDTCB = T0 for the event 1977 Jan 1.0 TAI at the geocenter and increases by 1.0 for each 86400s of TCB,

• LB = 1.550519768×10−8 exactly,

• TDB0 = −6.55 ×10−5 s exactly.

Using this “new TDB”, it is trivial to convert from TDB to TCB and back.

0 086400 B TCBTDB TCB L JD T TDB

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Gaia needs: TCB

• With all this involved situation with TDB/Teph the only unambiguous way

is to use TCB for all aspects of data processing:

• solar system ephemeris

• Gaia orbital data

• time parametrization of proper motions

• time parametrization of orbital solutions (asteroids and stars)

• …

• TCB was officially agreed to be the fundamental time scale for Gaia

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Scaled BCRS: not only time is scaled

• If one uses scaled version TCB – Teph or TDB – one effectively uses three scaling:

• time

• spatial coordinates

• masses (= GM) of each body

(from now on “*” refer to quantities defined in the scaled BCRS; these quantities are called TDB-compatible ones)

WHY THREE SCALINGS?

* *0

*

*

x x

t K TCB t

K

K

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• These three scalings together leave the dynamical equations unchanged:

• for the motion of the solar system bodies:

• for light propagation:

Scaled BCRS

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• These three scalings lead to the following:

semi-major axes

period

mean motion

the 3rd Kepler’s law

Scaled BCRS

*

*

* 1

3 2 *3 *2 *

a K a

P K P

n K n

a n a n

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Quantities: numerical values and units of measurements

XX XXA A A

• Arbitrary quantity can be expressed by a numerical value

in some given units of measurements :

A XXA

XX

A

• XX denote a name of unit or of a system of units, like SI

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Quantities: numerical values and units of measurements

, const B K A K

• Consider two quantities A and B, and a relation between them:

• No units are involved in this formula!

• The formula should be used on both sides before

numerical values can be discussed.

• In particular,

is valid if and only if

XX XXA A A

XX XX

B K A

XX XX

B A

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• For the scaled BCRS this gives:

Scaled BCRS

• Numerical values are scaled in the same way as quantities if and only if the same units of measurements are used.

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• SI units

• time

• length

• mass

• Astronomical units

• time

• length

• mass

Astronomical units in the Newtonian framework

second s

meter m

kilogram kg

SI

SI

SI

t

x

M

day

Solar mass

A

A

A

t

x AU

M SM

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• Astronomical units vs SI ones:

• time

• length

• mass

• AU is the unit of length with which the gravitational constant G takes

the value

• AU is the semi-major axis of the [hypothetic] orbit of a massless particle

which has exactly a period of

in the framework of unperturbed Keplerian motion around the Sun


, 86400

A SI

A SI

A SI

t d t d

x x

M M

2 20.01720209895 A

G k

2 / 365.256898326328 days k

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• Values in SI and astronomical units:

• distance (e.g. semi-major axis)

• time (e.g. period)

• GM


1

1

2 3

A SI

A SI

A SI

x x

t d t

d

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Be ready for a mess!

Astronomical units in the relativistic framework

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• Let us interpret all formulas above as TCB-compatible astronomical units

• Now let us define a different TDB-compatible astronomical units

• The only constraint on the constants:

* *

*

* *

*

* *

*

2* *

*

day

SM

A SI

A SI

A SI

A

t d t

x AU x

M M

G k

2 3 2* * * *Sun *

2Sun

1

AB

A

k M dK L

dk M

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• Possibility I: Standish, 1995

• This leads to

*

*

*Sun Sun*

* 1/ 3

86400

1

A A

d d

k k

M M

K

* 2 / 3

*

*

*

*

AA

AA

AA

x K x

t K t

strange scaling…

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• Possibility II: Brumberg & Simon, 2004; Standish, 2005

• This leads to

*

2* * 2Sun Sun*

*

86400

A A

d d

k M K k M

*

*

*

*

*

AA

AA

AA

x K x

t K t

K

The same scaling as with SI:

*

*

*

SISI

SISI

SISI

x K x

t K t

K

Either k is differentor the mass of the Sunis not one or both!

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• From the DE405 header one gets:

TDB-compatible AU:

• Using that (also can be found in the DE405 header!)

one gets the TDB-compatible GM of the Sun expressed in SI units

• The TCB-compatible GM reads (this value can be found in IERS Conventions 2003)

How to extract planetary masses from the DEs

3* * * -2 20Sun Sun *

86400 =1.32712440018 10 SI A

* 111.49597870691 10

* 2 -4Sun *

2.959122082855911025 10 Ak

* 20Sun Sun

1=1.32712442076 10

1 SI SI

BL

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• The reason to introduce astronomical units was that the angular

measurements were many order of magnitude more accurate than

distance measurements.

• Arguments against astronomical units

• The situation has changed crucially since that time!

• Solar mass is time-dependent just below current accuracy of ephemerides

• Complicated situations with astronomical units in relativistic framework

• Why not to define AU conventionally as fixed number of meters?

• Do you see any good reasons for astronomical units?

Do we need astronomical units?

13 1/ 10 yr Sun SunM M

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Scaled GCRS

• Again three scalings (“**” denote quantities defined in the scaled GCRS;these TT-compatible quantities):

• time

• spatial coordinates

• masses (=GM) of each body

• the scaling is fixed

• Note that the masses are the same in non-scaled BCRS and GCRS…

• Example: GM of the Earth from SLR (Ries et al.,1992; Ries, 2005)

• TT-compatible

• TCG-compatible

**

**

**

-101 1 6.969290134 10

X X

G

T L TT

L

L

L L

** 6

Earth

** 6

Earth Earth

398600441.5 0.4 10

1398600441.8 0.4 10

1

SI

SI SIGL

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• GM of the Earth from SLR:

• TT-compatible

• TCG/B-compatible

• TDB-compatible

• GM of the Earth from DE:

• DE403

• DE405

TCG/TCB-, TT- and TDB-compatible planetary masses

** 6

Earth

** 6

Earth Earth

* 6

Earth Earth

398600441.5 0.4 10

1398600441.8 0.4 10

1

1 398600435.6 0.4 10

SI

SI SIG

BSI SI

L

L

* 6

Earth

* 6

Earth

398600435.6 10

398600432.9 10

SI

SI

Should the SLR mass beused for ephemerides?

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• Note 1: Teph defined by a fixed relation to TT may be a source of inconsistency since newer ephemerides are not fully compatible with the Teph –TT relation used for their development (derived on the basis of VSOP87/DE200)

• Note 2: No good reasons to develop more accurate analytical formulas: just like with the ephemeredes too many terms…

• Note 3: With fixed scaling constant K=1-LB (that is, with the re-defined TDB) it is impossible to have different post-fit residuals when using TDB or TCB.

The fits must be absolutely equivalent!

• Note 4: Once a TDB ephemeris is constructed, it is trivial to convert it to TCB and vice verse: just use the three scalings given above!

TCB-based or TDB-based ephemeris?

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Iterative procedure to construct ephemeris with TCB or TDB in a fully consistent way

1. Use some apriori T(C/D)B–TT relation (based on some older ephemeris) to convert the observational data from TT to T(C/D)B

2. (Re-) Construct the new ephemeris

3. Update the T(C/D)B–TT relation by numerical integrationusing the new ephemeris

4. Convert the observational data from TT to T(C/B)D using the updated T(C/D)B–TT relation

• This scheme works even if the change of the ephemeris is (very) large

• The iterations are expected to converge very rapidly (after just 1 iteration)

1 ephemerides in the relativistic framework: _ time scales, spatial coordinates, astronomical...

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