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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