9

Northwest Analysis ---- Bozeman, Montana

knordtvedt@bresnan.net

LATOR's main science product is a part in 10 determination of PPN whose deviations

Kenneth Nordtvedt

LATOR: Its Science and Fly-by Orbits

9

from 1 signal modifications of gravity theory. Sun's gravitational binding energy

modifies the measurement to include new post-Newtonian features of theory.

In order to reach 10 precision in meas

urement of PPN , LATOR's light triangle

must be transversely located relative to the Sun with half-meter precision, or scientific

data must be desensitized to transverse position uncertainty. Spacecr

aft transits of Sun

line of sight can be designed to achieve this, eliminating need for drag-free spacecraft systems.

00 03

30 33

1 0 0 01 0 0

0 1 0 0"Rigid" 0 1 0 ( , ) "Flexible"

0 0 1 00 0 1

0 0 0 1

g g

g g r t

g g

1

Quarks Leptons( ?)g

,0A W

(8)G

Electro-Weak

Gluons

Standard Model plus Gravity

Arena field(s)Gravitational field(s)

e

e

v

u s t

d c b(3) x

Higgs scalarmass mechanism

Cosmologically Evolving Scalar Field 'Turns Itself Off'

( )f

o

t2

As

1 0

1 0, etc.

o

df

d

( )t

Damour and Nordtvedt 1994

1 2 12

C

22 2 2 4 2 212 1 2 12 1 2

1 2

2

Measure Light Triangle's , , with Laser Ranging

Measure using Interferometer, with loose Navigation

2sin

2

( ) 1 11 * .

sin / 2 sin / 2

EUC

SEUC

C CE

T T T

T T T T T T

T T

GM G

c R

..

S.C. 1

S.C. 2

12TERInterferometer

1T

2T

C

K Nordtvedt, 1 May 2006

2

Light Coordinate Speed Function: 0

ˆ( , , ) equally sensitive to spatial and temporal metric

field components, yields both "time delay" and deflection.

( ) (1 2 ... 1oo ab ab

g dx dx

c r t c

GM G GMg g

c R

2

2

)...

( ) ( )( ) 1 ...

c R

G M G Mc R

c R

Earth’s Mass-Energy consists of nuclear chromodynamic, electromagnetic, weak, kinetic, and gravity contributions.

All of Physics as we know it!

Gravitational Binding Water Molecule

Oxygen Nucleus

Proton

u u

d?

2

( ) 1 11 ...

sin

( )

( / 2 ) sin( /) 2 )( C CE

GM G

c R

M

M G

C

C C

By measuring with interferometer for fly-by configuration with

0, the sensitivity of measurement to uncertainty is minimizedEuc

Science signal doubled

2 2 2

2 2 4 4

*2 * 2

"Inner and Outer" Rescalings of Metric Tensor Gravity Potentials

1 2 2 4 2 2 1

* ( )1 2 ... ( ) ( ) 4 3

2

* 1

i ji i i ioo

i i i j ii i i ij i

i jioo

i ii ij

G m mGm Gm Gm vg

c r c r c r r c r

Gm mG M Gg M G M I

c r c r

t t U

2 2

* 1

* 1 4 3

S S

S

i iS

i ui i

r r U

G G U

Gm GmU

c r c r

2 22

2 2 4 4 3

4

* *2

31 2

2

1 ...

* ( )1 2

"Inner and Outer" Rescaling of Spatial Metric Tensor Potential

i ji i iab ab i i

i i i j ii i i ij i

ii ia b

i i

iab ab

i

G m mGm Gm Gmg v r

c r c r c r r c r

Gmv v

c r

G Mg

c r

* 2,

2*

3 2....

3 2

3 21 * / *

2

i j

i i j ij

i S S

Gm mM M I

c r

r r U G G U

2 2

2 2

2

( )( ) ( ) 1

( ) 2 ( ) 2( ) ( ) 1

1 ( ) ( ) ( ) ( ) 1 ( )

( )1 ...

( )

( )....Sun

Su

S iSi i

S S

S iS iSi i

S S

iS iS i S

n

GM GM I M I

c R c R

G M G MM I M I

c R c

M I

M G

R

M I M I M G M M I

GM G

c D

If Local Lorentz Invariance of Gravity is preserved under rescalingFrom matter in the external cosmos, some constraints exist

Sun Earth1.5 YearsLater

2 Space CraftVenus Mars

1.5 Year Orbit

.75 Year Orbit

1.5 Year Mission: 3.4 / sec

2.5 Year Mission: 2.0 / sec

km

km

To achieve spacecraft lines of sight passing oppositely by the Sun

LATOR Transverse Navigational Requirements

A. With Equal and Opposite Spacecraft Passages

B. Without Equal and Opposite Passages (Drag-Free System Reqd.)

30 kilometers

10 centimeters/secondC

C

X

V

4

13

70 centimeters

2 10 centimeters/second

10 drag-free performance

C

C

C

X

V

a g

Preferred Opposite Polar Passages

1.5 Y mission: 1/4 solar radius/day ---- 2.5 Y mission: 1/7 solar radius/day

2

2 2 2 2

2

2 2 3 3

Scientific Signals in Angle Deflection

1 1 1 11

1 11 ( ) ....

with modifications for spacecraft at solar latitude

S S

A B A B

S S

A B

GM GM

D Dc c D D

GM aJ F L

c D D

F L L

1 2 3 4

3

1

n

2 3

1 1 1

n n n

2

1/ 2 5 / 222 2

1 1 1 2

11 1

x

xx x

, , and J2 Deflection Signals

x

n

1 2 3 4

Z/R(sun) dgamma dJ2 dbeta dgamma* dJ2*

0 23 55 73 3.7 6.9

.25 28 82 95 3.8 7.9

.50 15 131 87 4.8 13.7

.75 9 25 10 3.6 22.1

1 11 10 18 1.9 4.4

-8

-92

-3

Multiply "dgamma" by 10 / ( )

Multiply "dJ " by 6 10 / ( )

Multiply "dbeta" by 3 10 / ( )

N eff

N eff

N eff

2

1 ( ) Modified Newtonian

(2 1) Non-linearity

(2 2) Gravitomagnetism

( 1/ 2) 2 Geodetic Precess

i o i

i

k kij

j i k i k jik jk

i j ijj i

i ij ij ij i

G

I

a t t g

gr r

v v g

v

mdG

Gdt

v g

m

g v

2 2

2

ion

ˆ( 1) 3( ) / 2 (2 1) Source Motion

1Inertial

2 2

ˆ ˆ(2 1) Misc. Inertial2

with ,

j i

j j ij ij j j ijj i

jj i i

j i ij

j ji j j ij ij i i i

j i j iij ij

i i ij

v v r g v v g

a v ar

a a a r r a v vr r

Gm g

3 , and , in 1 1 GRj

ji i ijj iij

Gmr g g

r

2

1

c

Post-Newtonian Equation of Motion for N Bodies

42

2 2

22

2 2,

2

The Post-Newtonian Lagrangian for N bodies

ˆ ˆ

8 2 2

2

4 2

in General Relativ t1

2

1

1

i y

i j i j i ij ij jii

i i j ij

i j i j ki j

i j i j kij ij ik

im m u u u r r uu G

L m cc r c

m m m m mG Gu u

c r c r

u

r

222 2

2

1 10

2 2

for each body 0

i i i i ii i i ii

j CECE i i i CE

i j i ij

i

i

LP p E p u L J r p

u

m d RG c PR m r c u V

E r E dt

d p L dP dE dJi

dt r dt dt dt

Orbit PolarizedIn fixed cosmic

direction

Cosmic or Sidereal EP Violation TestBecause lunar orbit’s perigee precession period is 8.9 years,

such an EP-violating perturbation is very near resonant.

14 2cosmic| | 10 /e ma a cm s

2

1 12 1 + .....

( ) ( )

Doubles Experimental Sensitivity

abA B

GM

c D t D t

22 2 2 2 2 42sin

2

a b a b

aba b

r R R R R rEuclid says

R R

aR

bR

rab

2 2

32 2

2 2

2 22

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

sin / 2 sin / 2

83.6 10

sin / 24

sin / 2 cos / 2

a b aba b e c c

e a b a b

c

cec c

e c c

GM GMT T t

c D D c R

GMR D D D D

c D D D

GM R x

c R x

is Sun's distance out

of light triangle's plane

x

aT

bT

abteR

( ) ( )

2

2

2 2 2

What about the Common Mode Angle ( )?

( ) ( ) ( ) ( ) ' ''

Either measure ( ) or fit-for ( ) and ( )

1 12

1 1

o

c

Dy Drc c o c o o c c

t

c c o c o

a b

a b

t

t t t t t dt dt

t t t

GM

c D D

GM

c D D

2 2 2

2 3 32

2 2 2

1 2cos 1 2cos

1 14

( )

S a b

a b

E

c a b

occ

GMR L L

J c D D

GMR

c D D

t t

Gamma Signal as Function of Time:Polar Passage vs. Equatorial Passage

t

Boost ofBoth SCsT = 0Boost of SC “B”

T = 1.5 Years

1

2

A

B

1.5 Year PeriodOrbit for SC “A+B”

Figure 4.

-9 -9

2 2

2

6

2

Measuring PPN to 10 is really Measuring ( ) / ( ) to 10

( ) ( )1 2 .... 1 2 ....

( ) ( ) 4 3 ( ) ( )

/ 4 10

1 2

i ioo ab ab

i ii i

PNi i i i i i

Sun Sun

iab ab

i i

M M G

GM G GMg g

c R c R

M G M I U M M I U

U E

Gmg

c r

2 2

2 4

22

4

2

....

.... So

But in presence of distant Spectator Matter is renormalized.

1 2 with

i ji

i iji i ij

PNi i

i i

sspec spec spec spec

s

G m mGm

c r c r r

Gm vOrder

c r

GmU U U U

c R

distant bodies s

0 0

22 ( ) ( )

'( ( )

0() ( )

0 0)

2

Model Fitting Issues

1 12 1 and signals

/ 2 / 2

is well-measured by light triangle

/ 2 ' ''

1 12

( )

)( ) (

SCM CM

A B

t tCM CM

ACC M

A B

MB

t t

S

GMJ

c d D d D

d D D

D D D t t dt a dtD

GM

t

c D

D t

t D

0 0

2

2 2

02 2 2 ( )

'( )

(

(

(

)

))

1 1....

( )

1 12 1

' ''

S

A B

SCM

A B

t tCM

CMt t

CM CM

GM

t c D D

GMt t

c D D D

dt a d

D

D

D

t

Figure 5.