relativistic scaling of astronomical quantities and the system of

1

Relativistic scaling of

astronomical quantities and

the system of astronomical units _

Sergei A.Klioner

Lohrmann Observatory, Dresden Technical University

Lohrmann Observatory, 4 May 2007

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Content

• Relativistic time scales and reasons for them:

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

• Scaled-BCRS

• Scaled-GCRS

• Astronomical units in Newtonian and relativistic frameworks

• Do we need astronomical units?

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

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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:

T = t1

c2

A(t) + vE

ir

E

i( ) +1

c4

B(t) + Bi (t)r

E

i + Bij (t)r

E

ir

E

j +C t, x( )( ) +O c5( )

TCG = TCG(TCB,xi )

xi= x

obs

i (t)

( r

E

i= x

ix

E

i (t) )

TCG = TCG(TCB,xobs

i (TCB)) = TCG(TCB)

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Relativistic Time Scales: TCB and TCG

• Important special case gives the TCG-TCB relation

at the geocenter: x

i= x

E

i (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:

xobs

i (t) and/or Xobs

a (T )

d

dt= g

00t,x

obs(t)( )

2

cg

0it,x

obs(t)( ) x

obs

i (t)1

c2g

ijt,x

obs(t)( ) x

obs

i (t) xobs

j (t)

1/ 2

d

dT= G

00T ,X

obs(T )( )

2

cG

0aT ,X

obs(T )( ) X

obs

a (T )1

c2G

abT ,X

obs(T )( ) X

obs

a (T ) Xobs

b (T )

1/ 2

6

Relativistic Time Scales: proper time scales

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

d

dT= 1

1

c2

1

2X

obs

2 (T ) +WE

T ,Xobs

( ) + "tidal terms" +O c4( )

1017

is the height above the geoid

is the velocity relative to the rotating geoid

• Idea: let us define a time scale linearly related to T=TCG, but which

is numerically close to the proper time of an observer on the geoid:

TT = (1 L

G) TCG, L

G6.969290134 10-10

d

d TT= 1

1

c2

"terms h,i"+ "tidal terms"+ ...( ) + ...

can be neglected

in many cases

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Relativistic Time Scales: TT

h is the height above the geoid

iis the velocity relative to the rotating geoid

• Idea: let us define a time scale linearly related to T=TCG, but which

is numerically close to the proper time of an observer on the geoid:

TT = (1 L

G) TCG, L

G6.969290134 10-10

d

d TT= 1

1

c2

"terms h,i"+ "tidal terms"+ ...( ) + ...

can be neglected

in many cases

• 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:

L

G6.969290134 10

-10

• 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:

T

eph= R TCB +T

eph0

• The coefficients are different for different ephemerides.

• The user has NO information on those coefficients from the ephemeris.

• The coefficients could only be restored by some additional numerical

procedure (Fukushima’s “Time ephemeris”)

• Teph is de facto defined by a fixed relation to TT:

by the Fairhead-Bretagnon formula based on VSOP-87

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Relativistic Time Scales: TDB-2

The 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,

• TDB0 6.55 10 5 s.

TDB = TCB LB

JDTCB

T0

( ) 86400 +TDB0

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Linear drifts between time scales

Pair Drift per year

(seconds)

Difference at J2007

(seconds)

TT-TCG 0.021993 0.65979

TDB-TCB 0.489307 14.67921

TCB-TCG 0.467313 14.01939

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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 (μ= GM) of each body

WHY THREE SCALINGS?

t*= F TCB + t

0

*

x*= F x

μ*= F μ

F = 1 L

B

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• These three scalings together leave the dynamical equations unchanged:

• for the motion of the solar system bodies:

(first published in 1917!)

• 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

a*= F a

P*= F P

n*= F

1n

a3n

2= μ a

*3n

*2= μ

*

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Scaled GCRS

• If one uses TT being a scaled version TCG one effectively uses three scaling:

• time


• masses (μ= GM) of each body

• International Terrestrial Reference Frame (ITRF) uses such scaled GCRS coordinates and quantities

T**= TT = L TCG

X**= L X

μ**= L μ

L = 1 L

G

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Quantities:

numerical values and units of measurements

A = A{ }

XX

AXX

• Arbitrary quantity can be expressed by a numerical value

in some given units of measurements :

A{ }XX

A

XX

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

• Notations taken from ISO 31-0

“Quantities and units – Part 0: General principles”, 1992

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Quantities:

numerical values and units of measurements

B = F A, F = const

• 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 BXX

= AXX

B{ }XX

= F A{ }XX

A = A{ }

XX

AXX

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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

• Official definitions from the SI document, published by BIPM:

Units of measurements: SI

tSI

second s

xSI

meter m

MSI

kilogram kg

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• System of astronomical units

• time

• length

• mass

• Day is by definition 86400 SI seconds

• The mass of the Sun expressed in astronomical units of mass is by

definition 1

Any possible variability of the solar mass is ignored!

Astronomical units in the Newtonian framework

tA

day

xA

AU

MA

Solar mass 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 (IAU 1938, VIth General Assembly in Stockholm)

• 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 Newtonian Keplerian motion around the Sun


tA

= d tSI

, d = 86400

xA

= xSI

MA

= MSI

G{ }A

k2

0.017202098952

2 / k 365.256898326328…days

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• Geometrized units are defined implicitly by setting three fundamental constant to unity:

• Gravitational constant

• speed of light

• Planck constant

• Why possible?

Astronomical units vs. geometrized units

G{ }geo= 1

c{ }geo= 1

h{ }geo= 1

GSI= m3kg 1s 2

cSI= m s 1

hSI= m2kg s 1

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• Both system of units – geometrized and astronomical - can be used

without any relation to the “directly measurable” SI units

• The relations to the SI units of length, time and mass can only be obtained

from experimental data.

Astronomical units vs. geometrized units

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• Values in SI and astronomical units:

• distance (e.g. semi-major axis)

• time (e.g. period)

• GM

• Mass parameter of a body in SI can be directly expressed as


x{ }A= 1

x{ }SI

t{ }A= d

1t{ }

SI

μ{ }A= d

2 3 μ{ }SI

μ{ }SI= k

2d

2 3M{ }

A

MSun

{ }A

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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

• Four constants define the system of units:

tA*

day* = d*

tSI

xA*

AU* = *

xSI

MA*

SM* = *M

SI

G{ }A*

k*( )

2

d*,

*,

*, k

*

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• Considering that

• the only constraint on the constants reads

k*( )

2

MSun

*{ }A*

k2

MSun{ }

A

*3

d*

d

2

= F = 1 LB

μ*{ }A*= k

*( )2

MSun

*{ }A*

,

μ{ }A= k( )

2

MSun{ }

A

,

μ*{ }A*= F

d*

d

2*

3

μ{ }A

28


• Possibility I: Standish, 1995

• This leads to

d = d* = 86400

k = k*

MSun

*{ }A*

= MSun{ }

A

= 1

* = F1/3

x*{ }

A*

= F2/3

x{ }A

t*{ }

A*

= F t{ }A

μ*{ }A*

= μ{ }A

strange scaling…

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

• This leads to

d = d* = 86400

k*( )

2

MSun

*{ }A*

= F k2

MSun{ }

A

* =

x*{ }

A*

= F x{ }A

t*{ }

A*

= F t{ }A

μ*{ }A*

= F μ{ }A

The same scaling as with SI:

x*{ }

SI

= F x{ }SI

t*{ }

SI

= F t{ }SI

μ*{ }SI

= F μ{ }SI

Either k is different

or the mass of the Sun

is not 1 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

μ

Sun

*{ }SI

= μSun

*{ }A*

*( )3

86400-2

=1.32712440018 1020

*= 1.49597870691 10

11

μSun

*{ }A*

k2 = 2.959122082855911025 10

-4

μSun{ }

SI

=1

1 LB

μSun

*{ }SI

=1.32712442076 1020

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Scaled GCRS

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

• time


• masses (μ=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

T**= L TT

X**= L X

μ**= L μ

L = 1 LG

μEarth

**{ }SI

= 398600441.5± 0.4( ) 106

μEarth

{ }SI

=1

1 LG

μEarth

**{ }SI

= 398600441.8 ± 0.4( ) 106

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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

μEarth

**{ }SI

= 398600441.5± 0.4( ) 106

μEarth

{ }SI

=1

1 LG

μEarth

**{ }SI

= 398600441.8 ± 0.4( ) 106

μEarth

*{ }SI

= 1 LB

( ) μEarth

{ }SI

= 398600435.6 ± 0.4( ) 106

μEarth

*{ }SI

= 398600435.6( ) 106

μEarth

*{ }SI

= 398600432.9( ) 106

Should the SLR mass be

used for ephemerides?

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• The reason to introduce astronomical units was that the angular

measurements were many orders 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 in their current form?

NO!

Do we need astronomical units?

MSun

/ MSun

10 13 yr 1

relativistic scaling of astronomical quantities and the system of

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