The Science of Relativistic Celestial Mechanics.

The Science of Relativistic Celestial Mechanics.

Introduction for a layman.









Sergei Kopeikin









June 27-29, 2007 Astrocon 2007 1

The Founders



Albert Einstein



Hendric A. Lorentz Karl Schwarzschild





Willem de Sitter Vladimir A. Fock





Tullio Levi-Civita Leopold Infeld





Hans Thirring Arthur S. Eddington





Lev D. Landau

The Solar System: Hierarchy of Celestial Frames









Solar System

Barycentric Frame



Heliocentric Frame









Geocentric

Frame



Lunocentric Frame

Earth-Moon

June 27-29, 2007 Astrocon 2007 3

Barycentric Frame

Newtonian Gravity Field Equations

Gravitational potential Density of matter

(a scalar function) (a scalar function)





 

 (t , x )  4G (t , x ) Poisson equation



 (t , x )  0 Laplace equation





General solution of the gravity field equations

  

 (t , x )  int (t , x )  ext (t , x )

June 27-29, 2007 Astrocon 2007 4

Boundary Conditions and Reference Frames

 

Particular form of int (t , x ) and ext (t , x ) depends on the

boundary conditions imposed on the gravitational field .

They are defined by the choice of the referenceframe.





Body’s Frame



Barycentric Frame



r

M Field point



m

June 27-29, 2007 Astrocon 2007 5

Multipolar Fields in Body’s Frame





Solution of the Laplace equation in the body's frame (u , w) :

mass dipole intrinsic intrinsic

quadrupole octupole



  M (u ) I i (u ) wi I ij  (u ) wi w j I ijk  (u ) wi w j wk 

 (u, w)  G 

L

int     ...

 r r3 2r 5 6r 7 





where r  w ,





  1 1 

ext (u, w)  G Q(u )  Q i (u ) wi  Q ij  (u ) wi w j  Q ijk  (u ) wi w j wk  ...

L



 2 6 



monopole acceleration tidal tidal

quadrupole octupole

June 27-29, 2007 Astrocon 2007 6

Multipolar Fields in Global Frame

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

 (t , z )d 3 z

 (t , x )    int (t , x )  ext (t , x )

P P P



R3

xz





 (t , z )d 3 z   (t ) i (t )( xi  z i )  ij  (t )( xi  z i )( x j  z j ) 

 (t , x )  

P

G  1

 1 1

 ...

xz x  z1

int

x  z1 2 x  z1

3 5

V1 

 







 (t )    (t , z )d 3 z

V1



 i (t )    (t , z )( x i  z i )d 3 z

V1



 1 

 ij  (t )    (t , z ) ( x i  z i )( x j  z j )   ij ( x  z ) 2  d 3 z

V1  3 





 (t , z )d 3 z  1 

 (t , x )  

P

 G Q(t )  Qi (t )( x i  z1i )  Q ij  (t )( x i  z1i )( x j  z1j )  ...

xz

ext

V2  2 





Q(t )  ext (t , z1 ) ;

P

Qi (t )  iext (t , z1 ) ;

P

Qij  (t )  i  jext (t , z1 )

P





June 27-29, 2007 Astrocon 2007 7

The Frame Matching Technique

Matching Coordinate Transformations:

u t

  

w  x  z1





Matching Equations :

 

int (u , w)  int (t , x )

L P



 

ext (u , w)  ext (t , x )

L P









Matching Results :

M (u )  M(t ) Q(u )  Q(t )

I i (u )  I i (t ) Q i (u )  Q i (t )

I ij  (u )  I ij  (t ) Q ij  (u )  Q ij  (t )



June 27-29, 2007 Astrocon 2007 8

Microscopic Equations of Motion

Microscopic equations of motion in the barycentric frame:

 

 

 i  vi  0

t x

dvi  p

   i  i

dt x x



Microscopic equations of motion in the body's frame:

 

 

 i  i  0

u w

d i    p

     i  Qi   i

du  x  w

dz1i

 v 

i i



dt



Translational 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 9

Equations of Translational Motion in the Local Frame

 3  

 u w

 

d w   i  i d 3 w   d 3 w    i d 2 Ai 

u

d



du V1

d 3 w 

dM

du

0

V1 V1 V1 V1







M is constant,if there is no flux of matter through the boundary of the volumeV1





d i 3    p

 du

V1

 d w      i  Q i d 3 w   i d 3 w

V1 

w  V1

w







d int 3

P

1

  d w     d w   pd 2 Ai  Q ijk   w j wk  d 3 w  ...

i 3



du V1 w i

6

 

   V1 

V1



  V1



  

 Pi 0 0  I  jk









dP i 1 1

 Mai   Q ijk I ij   Q ijkp I  jkp  ...

du 6 24





June 27-29, 2007 Astrocon 2007 10

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



Picture The center of mass moves with

acceleration a with respect to

the world line of a spherically-symmetric

body.

S

For the Earth this acceleration

Moon amounts 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





The world-line of Earth’s

center of mass



World-line of a sperically-symmetric body

June 27-29, 2007 Astrocon 2007 11

Equations of Rotational Motion



Define the angular momentum (spin) of the body :



 i

S i   ijk  w j k d 3 w    w    d 3 w

V1 V1



d k 3    p 3

  w

ijk j

d w   ijk  w j  k  Q k d 3 w   ijk  w j d w

V1

du V1  w  V1

wk









j int 3

P

d ijk 1

  w  d w    w

j k 3 ijk

d w   ijk  pw j d 2 Ak   ijk Q kp   w j w p  d 3 w  ...

du V1 V1

wi V

2 V1



   

  1 

   

  

S i 0 0  I  jp









dS i 1

   ijk Q kp I  jp   ...

du 2



June 27-29, 2007 Astrocon 2007 12

Equations of Orbitalal Motion in the Barycentric Frame

dvi 3  3 p 3

  dt d w  V  x i d x  V x i d x

V1 1 1









d int 3

P

ext 3

P





du V1

v i d 3 w    

V1

w i

d w   pd 2 Ai   

V1 V1

w i

d w

 

        

0 0 P 

 Mv1

i

M iext ( z1 )  Mai









dv1i P 

 a i  iext ( z1 )

du

June 27-29, 2007 Astrocon 2007 13

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 14

Gravitational Field is not a Scalar!









June 27-29, 2007 Astrocon 2007 15

Building Blocks of General Relativity

 g 00 g 01 g 02 g 03 

 

g g11 g12 g13 

Scalar Field   Metric Tensor g   10

g g 21 g 22 g 23 

 20 

g g 33 

 30 g 31 g 32 





g g ,   g ,  g , 

 1 

Gravitational Force i  Affine Connection  

2



     

Tidal Force i  j  Curvature Tensor R    ,  -  ,      





1

Laplace's Operator   Einstein Tensor R  R  g R

2



Density of Matter   Stress- Energy Tensor T  u u    





June 27-29, 2007 Astrocon 2007 16

Field Equations and Gauge Freedom

1  8G

R  g R  4 T 

2 c



1 

( R  g R );  0  T  ;  0

2



Four identities point out to the gauge freedom of the metric tensor.

Four of ten componentsof the metric tensor are arbitrary.





( - g g  );  0





   - g g    





 1 2  16G

 

  2 2       4 T   

 c t 

  c



June 27-29, 2007 Astrocon 2007 17

Solving Einstein’s Equations



Small parameters:

  (size of the body)/(distance between the bodies)  L / R

  (speed of matter)/(speed of gravity)  v / c

  (gravitational radius of the body)/(size of the body)  GM / c 2 L



Post - Newtonian Approximations (non - analytic expansion, elliptic equations) :

  

    1   2 2   3 3  ...  8 ln   8  ...



Post - Minkowskian Approximations (analytic expansion, hyperbolic equations) :

 

    1   2 2   3 3  ...









June 27-29, 2007 Astrocon 2007 18

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 w

g ( x)  G ( w)   G ( w)    ,   ,   O( 2 )

x x 



Specific choice of coordinates is determined by the boundary

conditions imposed on the metric tensor components.



June 27-29, 2007 Astrocon 2007 19

Form-invariance of the Metric Tensor

2 2 2 1 2 2 2 1

g 00  1  2  4  O 5  G00  1  2  4  O 5 

c c c  c c c 

4 i 1 4 i 1

g 0 i  2  O 5  G0i  2  O 5 

c c  c c 

 2  1  2  1

g ij   ij 1  2   O 4  Gij   ij 1  2   O 4 

 c  c   c  c 



Field Equations in global coordinates : Field Equations in local coordinates :





 1 2   1 2 

- 2

 c t 2 

 2

 

  x   4G 1  2 v 2    2U  3 p  - 2

 c u 2 

 1

 

  w   4G 1  2 2 2    2W  3 p 

   c     c 

 1 2   1 2 

- 2

 c   x  i  4Gv i

 - 2

 c   w  i  4G i



 t 2   u 2 









June 27-29, 2007 Astrocon 2007 20

Reference Frames and Boundary Conditions

Global Coordinates Local Coordinates

 (t , x )  int (t , x )  ext (t , x )  (u , w)   int (u , w)   ext (u , w)

0 0



 i (t , x )  int (t , x )  ext (t , x )

i i

 i (u , w)   int (u , w)   ext (u , w)

i



0 0







lim  (t , x )  0 lim  int (u, w)  0 lim  ext (u, w)  0

x  w w 0



lim  i (t , x )  0 lim  iint (u, w)  0 lim  iext (u, w)  0

x  w w 0



   r  1   r      r  1   r  

lim   0 lim   0

r 

t  r / c  const. 

r c t  R 

u  R / c  const. 

R c u 



lim 

 

  r i



1    0

 r i

 lim 

 

  ri



1 

 ri   0



t  r / c  const. 

r  r c t  u  R / c  const. 

R  R c u 

   

June 27-29, 2007 Astrocon 2007 21

Global and Local Frames









R

L  rg









June 27-29, 2007 Astrocon 2007 22

Mathematical Techniques for Deriving Equations

of Motion



• Einstein-Infeld-Hoffmann

• Fock-Papapetrou

• Dixon-Synge

• Asymptotic Matching (D’Eath)









June 27-29, 2007 Astrocon 2007 23

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

Lagrangian-based theory of gravity





Field equations: tensor, vector, scalar



Boundary and initial conditions: Boundary and initial conditions:

External problem - global frame Internal problem - local frame(s)



External solution of the field equations: Internal solution of the field equations:

metric tensor + other fields in entire space metric tensor + other fields in a local domain;

external and internal multipole moments

Matching of external and internal solutions



Coordinate transformations External multipole moments in terms of

between the global and local frames external gravitational potentials



Laws of motion: external Laws of transformation of the Laws of motion: internal;

internal and external moments Fixing the origin of the local frame



Equations of motion: external Equations of motion: internal



of spherical

Effacing principle: equations of motionAstrocon 2007 and non-rotating bodies depend only on

June 27-29, 2007 24

their relativistic masses

Equations of Motion of Spherically-Symmetric Bodies



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)









4









1 3







2









June 27-29, 2007 Astrocon 2007 25

Geocentric coordinates (u,w) cover Barycentric coordinates (t,x ) cover

interior of the world tube bounded by the entire space-time. Metric tensor

radius of the lunar orbit. Metric tensor g αβ (t, x) . The two coordinate systems

Matching Global and Local Coordinates

Gαβ ( u, w ) overlaps admitting the matching

transformation:

1 1

u t  A(t, x)  B(t, x)  ...

c2 c4

1 1

wi  xi  xi (t )  Ai (t, x)  Bi (t, x) ...

E

c2 c4



u u u wi

g

 ( x) G00 (w)   G0i (w)  

Moon x x x x

wi u wi w j

G (w)  G (w)

i0 x

x  x x 

ij







Sun







Earth

June 27-29, 2007 Astrocon 2007 26

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 27

JPL Development Ephemeris (DE)

E. M. Standish, X.X. Newhall, J.G. Williams







• 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).







June 27-29, 2007 Astrocon 2007 28

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 29

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 30

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 31

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 32

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.









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June 27-29, 2007 Astrocon 2007 34