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1
Millimeter Laser Ranging to the Moon:a comprehensive theoretical model for advanced
data analysis
Dr. Sergei Kopeikin
Dr. Erricos Pavlis (Univ. of Maryland)
Despina Pavlis (SGT, Inc.)
2
Next 25 slides:
• The basics of LLR• Historical background• The Newtonian Motion• General Relativity at a glimpse• PPN equations of motion• Motivations behind PPN • Gauge freedom in the lunar motion• IAU 2000 theory of reference frames• Lunar theory in a local-inertial reference frame• Magnitude of synodic relativistic terms
3
No celestial body has required as much labor for the study of its motion as the Moon!
True longitude of the Moon = the mean longitude (20905 km) + 377' sin M ECCENTRIC-1 (period 27.3 days)
+ 13' sin 2M ECCENTRIC-2 (period 13.7 days)(3699 km) + 76' sin (2D - M) EVECTION (period 31.8 days)(2956 km) + 39' sin 2D VARIATION (period 14.7 days)(833 km) – 11' sin M' ANNUAL INEQUALITY (period 365.25 days)(110 km) – 2' sin D + ... PARALLACTIC INEQUALITY(period 29.5 days)
M – the mean anomaly of the MoonM' – the mean anomaly of the SunD = M - M'
Earth
Moon
Sun
M
M'
The ascending node
4
Historical Background (before Einstein)• Newton – the first theoretical explanation of the main
lunar inequalities (1687)• Clairaut – lunar theory with the precision of 1.5 arc-
minute (1752)• Laplace – the lunar theory with the precision of 0.5 arc-
minute; secular acceleration; speed of gravity (1802)• Hansen – the lunar theory and tables with the precision
of 1 arc-second (1857)• Delaune – an elliptic unperturbed orbit; 230 terms in the
perturbing function; perturbation of the canonical set of elements; precision 1 arc-second (1860)
• Hill – rotating coordinates; Hill’s equation; Hill’s intermediate orbit; precision 0.1 arc-second (1878)
• Brown – extension of Hill’s theory; Brown’s tables; precision 0.01 arc-second (1919)
5
Historical Background (after Einstein)• De Sitter – relativistic equations of the Moon; geodetic precession (1919)• Einstein-Infeld-Hoffmann – relativistic equations of N-body problem; massive
bodies as singularities of space-time (1938)• Fock-Petrova – relativistic equations of N-body problem; massive bodies as
extended fluid balls (1940)• Brumberg – relativistic Hill-Brown theory of the Moon based on the EIH
equations; eccentricity in relativistic term e = 0 (1958)• Baierlein – extension of Brumberg’s theory for e ≠ 0 (1967) • Apollo 11 - LLR technique gets operational; ranging precision = a few meters
(1969)• Nordtvedt – testing the strong principle of equivalence with LLR (1972)• Standish – JPL numerical ephemeris of the Moon and planets (DE/LE)• Brumberg-Kopeikin – relativistic theory of reference frames in N-body
problem; matching technique (1989)• Damour-Soffel-Xu - relativistic theory of reference frames in N-body problem;
relativistic multipole moments (1991)• IAU 2000 – relativistic resolutions on time scales and reference frames based on
the BK-DSX papers• APOLLO – new LLR technology at the Apache Point Observatory (2005);
ranging precision 1 millimeter
6
Newtonian Equations of the Lunar Motion
2
2
Figure's effectsEarth's gravity Sun and planets
2
2
Figure's effectsMoon's gravity Sun and p
cm
lanets
( ) ( )
( )
;
( )
LE L L L
EL
L E L
E E E
d xU x Q U x
dt
M M M Mx
d xU x Q U
t
M
xd
2cm
cm2
Earth-Moon gravity force Tidal gravity force from
2
2
; ;
( ) ( ) ( ) (tidal terms)
( ) ( ) ( )
(
)
L EL E
E L L E L E
L E E L E
d x M MU x U
x M x r x
x U xdt M M
d rU x U x U x U x
t
x
d
the Sun and planets.Gradient of the perturbing potential.
From Minkowski to Riemann geometry
null
cone
null
cone null cone
jiij
ii dxdxgcdtdxgdtcgds 0
2200
2 2222222 dzdydxdtcds
<>
9
• The metric tensor ten gravitational potentials• The affine connection the force of gravity
• The Riemann tensor the relative (tidal) force of gravity• The Principle of Equivalence the covariant derivative • The Gravity Field Equations
General Theory of Relativity at a glimpse
1
2
g g gg
x x x
4
1 8
2
GR R T
c
Matter tells space-timehow to curve: field eqs.
Space-time tells matter how to move: eqs. of motion
00
TG
10
PPN metric tensor for a spherical body
Conventional tests of the metric tensor
| - 1|| - 1|
PPN parameters
?
5-10% (expected)
11
the Newtonian gravity force
non-linearity of the gravity fi "gravitomagnetic-like" forcd e
2 2
l
2
e
1ˆ3 4 6
2
4
3
i
i j ijj i
i
k kij
j i k i k jik jk
i j i ij ij
a g
gr r
v v v r g
v
g
v g
special-relativistic corrections to the gravity force
an inductive acceleration-dependent gravity force
17
2
ij j j i ij i ij i
jj j ij ij
j i ij
v v v g v v
a a r rr
2the post-Newtonian modification of E m
2
c
13
2j
i i i i i ij i ij
v a a v v ar
EIH equations of motion
12
PPN equations of motion of extended bodies
the Newtonian gravity forceviolation of SEPtime-dependent G
non-linearity of the gravity field
01 ( ) 1
2 1 2 2
i i i
k kij
j i k i k jik jkG
Ga t t g
G
gr r
Lorentz-invariance of the gr
"gravitomagnet
22
avity fo
ic-like"
r
2
f
ce ( effe
orce
cts)
1ˆ2 1 2 2 3 4 2
2preferre
i j i ij ij ij j j i ij i ij
d fr
i j ijj i
am
i
e
v v
v v g
v r g g v v v g v v
an inductive acceleration-dependent gravity force the post-Newtonian modi
21 14 3 2 1
2 2j j
j j ij ij i i i i i ij i j iij ij
a a r r v a a v v ar r
2fication of E mc
a “gravitomagnetic-field” parameter introduced by Soffel et al. (PRD 2008)
Solution of these equations must be substituted to the solution of equation of a laser pulse propagation (time-delay equation). The PPN time-delay equation has many terms being identical to those in the PPN equations of motion of extended bodies.
13
‘Conventional’ PPN ranging model• Any coordinate reference system can be used in relativity to
interpret the data.True, but making use of inappropriate coordinates
easily leads to misinterpretation of gravitational physics.• Modern computer technology is highly advanced. Data processing
can be done in any coordinates irrespectively of the complexity of the equations of motion.
True, but making use of inappropriate coordinates mixes up the spurious, gauge-dependent effects with real
physical effects and makes them entangled. There is no unambiguous way to clearly separate gravitational physics from coordinate effects.
• Any post-Newtonian term in the PPN equations of motion has physical meaning and, in principle, can be measured.
Not true. The PPN equations of motion of the Moon have an enormous number of spurious, gauge-dependent terms that have no physical meaning.
14
The Gauge Freedom The gauge condition is imposed on the metric tensor. It simplifies
the gravity field equations making their solution mathematically simpler.
However, the residual gauge freedom remains. It is defined
'new' coordinates 'old' coordinates
by the gauge
functions , which obey certain equations and introduce a number of
spurious (unphysical) terms to the metric tensor (= gravity field potentials)
w x
the gauge functions
2, ,
The spurious terms enter relativistic equations of motion of both the bodies
and photons. They must be carefully disentangled from
( )
( ) ( ) ( ) ( )
x
w wg x g w g w O
x x
the real physical
effects exisiting in the motion of the celestial bodies. The Moon-Earth-Sun
system admits a large number of the gauge degrees of freedom, which can be
eliminated after transformation to the local inertail frame of the EM barycenter.
15
Lorentz and Einstein contractions as the gauge modesMagnitude of the contractions is about 1 meter! Ellipticity of the Earth’s orbit leads to their annual oscillationof about 2 millimeters. Are they observable by means of LLR?
Earth
The Lorentzcontraction
The Einsteincontraction
1
2
3
4
Shape of a moving body can be defined in the global frame but it faces major difficulties because of the Lorentz contraction and other (non-linear) frame-dependent coordinate effects. One needs a local frame to work out a such definition.
To maintain the shape of the celestial body in the global frame, one has to introduce a spurious stress and strain inside the body to compensate the Lorentz contraction (physics does not work in this way)
Shape of a moving body in the global frame
17
Ranging model of a gauge-invariant theory of gravity
2 1
1
2
( ) ( ) Newtonian orbit Gauge-dependent terms Physical PN perturbations
( ) Newtonian ERP Gauge-dependent terms Physical PN perturbations
( ) Newtonian LRP Gauge-dependent terms Physical
L Ex t x t
r t
t
contaGauge-independentobservable time del
ins the gauge-dependent terms contains the gauge-dependent termay
2 1 1
s
2 1 2
PN perturbations
( ) ( ) ( ) ( ) ( ) PN time delay (Sun)L Ec x t x t t r t
all together these terms are gauge-independent
PN time delay (Earth)
1( )Ex t
2( )Lx t
1( )r t
2( )t
Solar system
barycenter
2 1( )ck
Earth Moon
Sun
More details in: Brumberg & Kopeikin, Nuovo Cimento B, 103, 63 (1989)
18
What is happening in the ‘conventional’ PPN ranging model?
2 1
1
2
( ) ( ) Newtonian orbit Gauge-dependent terms Physical PN perturbations
( ) Newtonian ERP Gauge-dependent terms Physical PN perturbations
( ) Newtonian LRP Gauge-dependent terms Physic
L E Gx t x t
r t
t
cGauge-independentobservable ti
ontains the gauge-dependent terms contains the gauge-dependent me de
termla
2 2 1 2 1
y
1
al PN perturbations
( ) ( ) ( ) ( ) ( ) PN time delay (Sun)L Ec x t x t t r t
all together these terms are NOT gauge-independent but proportional to ( 1)
s
PN time delay (Earth)
G
1( )Ex t
2( )Lx t
1( )r t
2( )t
Solar system
barycenter
2 1( )ck
Earth Moon
Sun
19
Correcting the PPN ranging model
2 1
1
2
( ) ( ) Newtonian orbit Gauge-dependent terms Physical PN perturbations
( ) Newtonian ERP Gauge-dependent terms Physical PN perturbations
( ) Newtonian LRP Gauge-dependent terms Ph
G
G
G
L Ex t x t
r t
t
Gauge-independentobservable time delay
contains the gauge-dependent terms contains the gauge-depende
2 1 2 1 2 1
n
ysical PN perturbations
( ) ( ) ( ) ( ) ( ) PN time delay (Sun)GL Ec x t x t t r t
all together these terms are gauge-independent that is does NOT depend on the paramet
t term
er
s
PN time delay (Earth)
G
1( )Ex t
2( )Lx t
1( )r t
2( )t
Solar system
barycenter
2 1( )ck
Earth Moon
Sun
Some details in: Kopeikin & Vlasov, Physics Reports, 2004
Magnitude of the synodic relativistic terms in the radial coordinate of the Moon
2
24
22
Schwarschild 1 cm
Lense-Thirring 0.3 mm
PN Quadrupole
from a few meters Gauge-dependent terms ...
2 10 mm
GM
c
R vR
c c
GM RJ
vvr
c
c
c
r
2
2 2
2
2
PN Gravitomagnetic a few mm
PN Gravitoelectric a fe
w cm
Non-linear
ity of gravit
down to a f w
y
e mm
n vvr
n c c
n vr
n c
n GM
n c
0.1 mm
Gauge-invariant theory of reference frames – IAU 2000(Brumberg & Kopeikin 1988; Damour, Soffel & Xu 1989)
Field equations for the metric tensor
PN approximation
Gauge and boundary conditions
Global frame (BCRF) (t, x)Resolution B1.3
Local frame (GCRF) (u, w)Resolution B1.3
Coordinate transformation (t, x) (u, w)Resolutions B1.3 and B1.5
Matching metric tensor in two frames. Residual gauge freedom
Laws of conservation Translational and rotational equations of motion
Multipole moments Resolution B1.4
23
Global RF (t,xi)
Local RF (u,wi)
Earth
Moon
Sun
Jupiter
Geodesic world-line
World-line of the geocenter
CCR RF ˆˆ( , )i
),(
),(
wu
wuii
),(
),(
xtww
xtuuii
Observer RF ),( i
Reference Frames
24
Lunar theory in the local-inertial frame. • Earth-Moon system being considered locally, is a binary system on a curved space-time
background (Sun, planets). • Equations of motion of the Earth-Moon system are those of the deviation of geodesics
perturbed by the mutual gravitational interaction between Earth and Moon.• There is a considerable similarity between this problem and that of the evolution of the
cosmological perturbations in expanding universe.• Earth-Moon equations of motion have enormous gauge freedom leading to spurious
gauge-dependent modes in motion of the celestial bodies participating in three-body problem.
• The main goal of the advanced lunar theory is – to remove all gauge modes,– to construct and to match reference frames in the Earth-Moon system with a sub-millimeter
tolerance,– to ensure that ‘observed’ geophysical parameters and processes are real.
• This is not trivial mathematical problem that requires a peer attention of experts in relativity!
25
Relativistic mass, center-of-mass and the Earth/Moon figure
• Definition of mass, center of mass and other multipoles must include the post-Newtonian corrections
• Definition of the body’s local reference frame• Definition of figure in terms of distribution of
intrinsic quantities: density, energy, stresses• Relativistic definition of the equipotential surface
– geoid/celenoid (Kopeikin S., 1991, Manuscripta Geodetica, 16, 301)
26
Rotation of the Earth/Moon in the Local Frame(Kopeikin & Vlasov, Physics Reports, 2004)
• Define the intrinsic angular momentum S = I · of the rotating body in the locally-inertial frame of the body
• Derive equations of the rotational motion in the locally-inertial frame of the body
2 4
d=(body's quadrupole)×(tidal octupole of the Sun, Earth and planets)+...
dτ1 1
+ post-Newtonian relativistic torque (neglectibly small)c c
S