June 27-29, 2007 Astrocon 2007 1

The Science of Relativistic Celestial Mechanics.Introduction for a layman.

Sergei KopeikinSergei Kopeikin

The Founders

Albert Einstein

Hendric A. Lorentz

Willem de Sitter

Leopold InfeldTullio Levi-Civita

Hans Thirring

Vladimir A. Fock

Karl Schwarzschild

Arthur S. Eddington

Lev D. Landau

June 27-29, 2007 Astrocon 2007 3

The Solar System: Hierarchy of Celestial Frames

Solar SystemBarycentric Frame

Heliocentric Frame

Geocentric Frame

Lunocentric FrameEarth-MoonBarycentric Frame

June 27-29, 2007 Astrocon 2007 4

Newtonian Gravity Field Equations

equation Laplace 0),(

equationPoisson ),(4),(

xt

xtGxt

Gravitational potential (a scalar function)

Density of matter(a scalar function)

),(),(),(

equations fieldgravity theofsolution General

extint xtxtxt

June 27-29, 2007 Astrocon 2007 5

Boundary Conditions and Reference Frames

frame. reference theof choice by the defined areThey

. field nalgravitatio on the imposed conditionsboundary

on the depends ),( and ),( of form Particular extint xtxt

M

m

Barycentric Frame

Body’s Frame

rField point

June 27-29, 2007 Astrocon 2007 6

Multipolar Fields in Body’s Frame

...)(6

1)(

2

1)()(),(

, where

...6

)(

2

)()()(),(

:),( frame sbody' in theequation Laplace theofSolution

ext

753int

kjiijkjiijiiL

kjiijkjiijiiL

wwwuQwwuQwuQuQGwu

wr

r

wwwuI

r

wwuI

r

wuI

r

uMGwu

wu

mass dipole intrinsic quadrupole

intrinsic octupole

monopole acceleration tidal quadrupole

tidal octupole

June 27-29, 2007 Astrocon 2007 7

Multipolar Fields in Global Frame

3

1

3

int ext

31 1

int 31 1

Solution of the Poisson equation in the barycentric frame ( , ) :

( , )( , ) ( , ) ( , )

( )( ) ( )( )(( , ) ( )( , )

P P P

R

i i i ij i i jP

V

t x

t z d zt x t x t x

x z

t x z t x z xt z d z tt x G

x z x z x z

1

1

1

2

15

1

3

3

2 3

3

ext 1 1 1

)...

2

( ) ( , )

( ) ( , )( )

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

3

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

2

j

V

i i i

V

ij i i j j ij

V

P i i i ij i i j j

V

z

x z

t t z d z

t t z x z d z

t t z x z x z x z d z

t z d zt x G t t x z t x z x z

x z

ext 1 ext 1 ext 1

) ...

Q( ) ( , ) ; Q ( ) ( , ) ; Q ( ) ( , )P i P ij Pi i jt t z t t z t t z

June 27-29, 2007 Astrocon 2007 8

The Frame Matching Technique

)(Q)( )(I)(

)(Q)( )(I)(

)Q()( )M()(

:Results Matching

),(),(

),(),(

:Equations Matching

extext

intint

1

tuQtuI

tuQtuI

tuQtuM

xtwu

xtwu

zxw

tu

ijijijij

iiii

PL

PL

Matching Coordinate Transformations:

June 27-29, 2007 Astrocon 2007 9

Microscopic Equations of Motion

1

Microscopic equations of motion in the barycentric frame:

0

Microscopic equations of motion in the body's frame:

0

Tr

ii

i

i i

ii

ii

i i

ii i

vt x

dv p

dt x x

u w

d pQ

du x w

dzv

dt

anslational equations of motion of the bodies are derived by means

of the volume integration of the microscopic equations of motion both

in the body's frame and in the barycentric frame.

Rotational equations of motion of the bodies are derived by means

of the volume integration of the microscopic equations of motion in

the body's frame.

June 27-29, 2007 Astrocon 2007 10

Equations of Translational Motion in the Local Frame

...24

1

6

1

...6

1

volume theofboundary ough thematter thr offlux no is thereif constant, is

0

3

0

2

0

3int3

333

1

32333

1111

111

11111

jkpijkpijijkii

I

kj

V

ijki

VVi

P

P

V

i

Vi

V

ii

V

i

V

i

V

i

VV

ii

V

IQIQMadu

dP

wdwwQAdpwdw

wddu

d

wdw

pwdQ

wwd

du

d

VM

du

dMwd

du

dAdwd

uwd

wwd

u

jki

June 27-29, 2007 Astrocon 2007 11

Picture

S

The world-line of Earth’s center of mass

World-line of a sperically-symmetric body

The center of mass of a massive body having non-zero intrinsic multipoles moves with acceleration with respect to a spherically-symmetric body because of the coupling of the intrinsic and external multipoles

The center of mass moves withacceleration a with respect to the world line of a spherically-symmetricbody.

For the Earth this accelerationamounts to 3.10-11 m/s2. As its orbital acceleration around the Sun is about 6.10-3 m/s2 , the relative effect is of order 5.10-9 . (taken into account in JPL ephemerides)

Earth

Moon

June 27-29, 2007 Astrocon 2007 12

Equations of Rotational Motion

...2

1

...2

1

:body theof (spin) momentumangular theDefine

3

0

2

0

3int3

333

33

1111

111

11

jpkpijki

I

pj

V

kpijkk

V

jijk

Vi

Pjijk

S

V

kjijk

Vk

jijk

V

kk

jijk

V

kjijk

V

i

V

kjijki

IQdu

dS

wdwwQAdpwwdw

wwdwdu

d

wdw

pwwdQ

wwwd

du

dw

wdwwdwS

jpi

June 27-29, 2007 Astrocon 2007 13

Equations of Orbitalal Motion in the Barycentric Frame

)( 1ext1

)(

3ext

0

2

0

3int3

333

1ext

111

1

1

111

zadu

dv

wdw

Adpwdw

wdvdu

d

xdx

pxd

xwd

dt

dv

Pi

ii

MazM

Vi

Pi

VVi

P

Mv

V

i

Vi

Vi

V

i

iPi

i

June 27-29, 2007 Astrocon 2007 14

Einstein’s Definition of Relativity

"Put your hand on a hot stove for a minute, and it seems like an hour. Sit with a pretty girl for an hour, and it seems like a minute. THAT's relativity."

A. Einstein.

June 27-29, 2007 Astrocon 2007 15

Gravitational Field is not a Scalar!

June 27-29, 2007 Astrocon 2007 16

Building Blocks of General Relativity

uuT

RgRR

R

gggg

gggg

gggg

gggg

gggg

g

ji

i

Tensor Energy -Stress Matter ofDensity

2

1Tensor Einstein Operator sLaplace'

- Tensor Curvature Force Tidal

2

1 Connection Affine Force nalGravitatio

Tensor Metric FieldScalar

,,

,,,

33323130

23222120

13121110

03020100

June 27-29, 2007 Astrocon 2007 17

Field Equations and Gauge Freedom

Tc

G

tc

g

g

TRgR

Tc

GRgR

42

2

2

;

;;

4

161

g-

0)g-(

arbitrary. are tensor metric theof components ten ofFour

tensor.metric theof freedom gauge theout topoint identitiesFour

0 0)2

1(

8

2

1

June 27-29, 2007 Astrocon 2007 18

Solving Einstein’s Equations

...

:equations) hyperbolic expansion, (analytic ionsApproximatn Minkowskia-Post

... ln...

:equations) elliptic expansion, analytic-(non ionsApproximatNewtonian -Post

/body) theof ebody)/(siz theof radius onal(gravitati

/gravity) of peedmatter)/(s of (speed

/bodies) ebetween th tancebody)/(dis theof (size

:parameters Small

33

22

1

88

33

22

1

2

LcGM

cv

RL

June 27-29, 2007 Astrocon 2007 19

Residual Gauge Freedom and Coordinates

The gauge conditions simplify Einstein's equations

but the residual gauge freedom remains. It allows us

to perform the post-Newtonian coordinate transformations:

( )

( ) ( )

w x x

w wg x G w G

x x

2

, ,( ) ( )

Specific choice of coordinates is determined by the boundary

conditions imposed on the metric tensor components.

w O

June 27-29, 2007 Astrocon 2007 20

Form-invariance of the Metric Tensor

iix

x

ijij

i

i

vGt

pUvc

Gt

cO

cg

cO

cg

cO

ccg

4c

1-

322

14c

1-

:scoordinate globalin Equations Field

121

14

1221

2

2

2

222

2

2

42

520

54

2

200

iiw

w

ijij

i

i

Gu

pWc

Gu

cO

cG

cO

cG

cO

ccG

4c

1-

3221

14c

1-

:scoordinate localin Equations Field

121

14

1221

2

2

2

222

2

2

42

520

54

2

200

June 27-29, 2007 Astrocon 2007 21

Reference Frames and Boundary Conditions

int ext

0

int ext

0

/ const.

/ const.

( , ) ( , ) ( , )

( , ) ( , ) ( , )

lim ( , ) 0

lim ( , ) 0

1lim 0

1lim 0

i i i

x

i

x

rt r c

i i

rt r c

t x t x t x

t x t x t x

t x

t x

r r

r c t

r r

r c t

int ext

0

int ext

0

int ext0

int ext0

/ const.

( , ) ( , ) ( , )

( , ) ( , ) ( , )

lim ( , ) 0 lim ( , ) 0

lim ( , ) 0 lim ( , ) 0

1lim

i i

w w

i i

w w

Ru R c

u w u w u w

u w u w u w

u w u w

u w u w

r r

R c u

/ const.

0

1lim 0

i i

Ru R c

r r

R c u

Global Coordinates Local Coordinates

June 27-29, 2007 Astrocon 2007 22

LR

gr

Global and Local Frames

June 27-29, 2007 Astrocon 2007 23

Mathematical Techniques for Deriving Equations of Motion

• Einstein-Infeld-Hoffmann

• Fock-Papapetrou

• Dixon-Synge

• Asymptotic Matching (D’Eath)

June 27-29, 2007 Astrocon 2007 24

Derivation of equations of motion. The internal-structure effacing principle

Lagrangian-based theory of gravity

Laws of transformation of theinternal and external moments

Boundary and initial conditions:External problem - global frame

Field equations: tensor, vector, scalar

Laws of motion: external

External multipole moments in terms of external gravitational potentials

Matching of external and internal solutions

Boundary and initial conditions:Internal problem - local frame(s)

External solution of the field equations:metric tensor + other fields in entire space

Internal solution of the field equations:metric tensor + other fields in a local domain;external and internal multipole moments

Coordinate transformations between the global and local frames

Laws of motion: internal;Fixing the origin of the local frame

Equations of motion: external Equations of motion: internal

Effacing principle: equations of motion of spherical and non-rotating bodies depend only on their relativistic masses

June 27-29, 2007 Astrocon 2007 25

1

2

3

4

Spherical symmetry of a moving body is ill-defined in the global frame because of the Lorentz (special-relativistic) and Einstein (general-relativistic) contractions. Spherical symmetry can be physically defined only in the body’s local frame (tides are neglected)

Equations of Motion of Spherically-Symmetric Bodies

June 27-29, 2007 Astrocon 2007 26

Geocentric coordinates (u,w) cover interior of the world tube bounded byradius of the lunar orbit. Metric tensor

Barycentric coordinates (t,x ) cover the entire space-time. Metric tensor

. The two coordinate systemsoverlaps admitting the matching transformation:

Sun

1 1( , ) ( , ) ...2 4

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

2 4

( ) ( ) ( )00 0

( ) ( )0

u t A t x B t xc c

i i i i iw x x t A t x B t xE c c

iu u u wg x G w G wix xx x

ji iw u w wG w G wi ijx xx x

Earth

Moon

u, wGαβ ( ) Matching Global and Local Coordinatesαβg (t, x)

June 27-29, 2007 Astrocon 2007 27

Einstein-Infeld-Hoffmann Force in the Global Reference Frame

The JPL Solar System Ephemeris specifies the past and future positions of the Sun, Moon, and nine planets in three-dimensional space. Many versions of this ephemeris have been produced to include improved measurements of the positions of the Moon

and planets and to conform to new and improved coordinate system definitions.

June 27-29, 2007 Astrocon 2007 28

• The DE100-series ephemeris is in the B1950 coordinate system

• The DE200 series is in the J2000 system• The DE400 series is in the reference frame

defined by the International Earth Rotation Service (IERS).

JPL Development Ephemeris (DE)E. M. Standish, X.X. Newhall, J.G. Williams

June 27-29, 2007 Astrocon 2007 29

Planetary positions are generated by a computer integration fit to the best available observations of the positions of the Sun, Moon, planets, and five largest asteroids. The computer integration involves stepwise computation of the position of each planet as determined by the gravitation of all of the other objects in the solar system.

June 27-29, 2007 Astrocon 2007 30

The observation are mainly from:

• transit circles since 1911, • planetary radar ranging since 1964, • lunar laser ranging since 1969, • distances to the Viking lander on Mars since 1976, • Very Long Baseline Interferometry since 1987.

The computer calculations have been extended as far as 3000 BC to 3000 AD, but positions for the 1850-2050 range are the most accurate.

June 27-29, 2007 Astrocon 2007 31

Subtle differences exist between:• the best ephemeris model coordinates and the standard definitions of B1950

and J2000, • the coordinate systems defined by star positions and the B1950 and J2000

standards,

• the coordinate systems defined by stars and radio sources.

These differences, which start at the level of a couple of milliseconds and a few tenths of an arcsecond, are very important to pulsar timing and radio interferometry. With care and consistency, all-sky accuracies of a few hundred nanoseconds and a few milliarcseconds are currently being achieved

June 27-29, 2007 Astrocon 2007 32

A Sketchy History of DE Versions• DE118

This was the best available planetary ephemeris as of 1983, spanning the 1850-2050 time range, based on transit circle measurements since 1911, planetary radar since 1964, lunar laser ranging since 1969, and Viking spacecraft ranging on Mars since 1974. Its larger time span companion was DE102, which covered 1411 BC to 3002 AD. The major ephemerides leading to DE118 were DE96, DE102, DE108, and DE111. All of these ephemerides, including DE118 are in the B1950 coordinate system (FK4 catalogue)

• DE200 : (includes nutations but not librations) This is DE118 rotated into the J2000 coordinate system. DE200 has been the basis for the calculation of Astronomical Almanac planetary tables since 1984.

• DE125 Created in July 1985 for the Voyager encounter with Uranus.

• DE130 Created in October 1987 for the Voyager encounter with Neptune.

• DE202 This is DE130 rotated into the J2000 coordinate system. DE202 is more accurate for the outer planets than is DE200.

• DE403 : (includes both nutations and librations) A new ephemeris aligned with the (J2000) reference frame of the Radio Source Catalog of the International Earth Rotation Service (IERS). It it based on planetary and reference frame data available in 1995.

• DE405 : (includes both nutations and librations) It is based upon the International Celestial Reference Frame (ICRF). (DE200 is within 0.01 arcseconds of the frame of the ICRF). DE405 was created in May-June 1997.

• DE406 : the New "JPL Long Ephemeris" (includes neither nutations nor librations) This is the same ephemeris as DE405, though the accuracy of the interpolating polynomials has been lessened. For DE406, the interpolating accuracy is no worse than 25 meters for any planet and no worse than 1 meter for the moon.

June 27-29, 2007 Astrocon 2007 33

Other Ephemeris Programs

• Planetary Ephemeris Program (PEP)This is the MIT Harvard Smithsonian Astrophysics Center ephemeris. Originally generated by I. Shapiro, M. Ash, R. King in 1967. Significantly improved in the spring of 1975 by Bob Goldstein. John Chandler has maintained PEP since the middle of 80th. PEP has the same accuracy as DE.

• Ephemerides of Planets and the Moon (EPM) This is the Institute of Applied Astronomy, St. Petersburg ephemeris code. Created by Geogre Krasinsky in 1974. Major contributions and improvements by Elena Pitjeva, Michael Sveshnikov. Previous versions: EPM87, EPM98, EPM2000. Current version EPM2006 has the same acuracy as DE405/414, and it is maintained by G. Krasinsky and E. Pitjeva. There are ephemeris programs in the Institute of Applied Mathematics and the Space Flight Control Center.

• Variations Seculaires des Orbites Planetaires (VSOP) Institute de Mechanique Celeste et de Calcul des Ephémérides (IMCCE). Created by P.

Bretagnon and G. Francou in 1988. Recent developments by A. Fienga and J.-L. Simon (VSOP2002) which includes the Moon, 300 asteroids, solar oblateness, and relativity. Diverges from DE405 up to 100 meters over 30 years.

June 27-29, 2007 Astrocon 2007 34