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Physics 798G Spring 2007

Lecture 7Gauss’s Law Test of the 1/r2 Law

Using an SGG

Ho Jung Paik

University of Maryland

February 15, 2007

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Newtonian null sources

• Uniform spherical shell: The gravity field vanishes inside the shell.

• Infinitely long cylindrical shell: 2-D spherical shell.

• Infinite plane slab: 1-D spherical shell. The field is constant on either side of the plane.

Newton Gauss’s law

Problem: The spherical shell must be made in two halves with holes to support a detector inside.⇒ Source metrology error

Problem: The cylindrical shell must be truncated.⇒ Newtonian error from the missing mass

Problem: The plane slab must be truncated.⇒ Newtonian error from the missing mass

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Newtonian null detector?

• Gauss’s law:

• Gravitational field (vector):

⇒ Cannot be distinguished from platform acceleration.

• Gravity gradient (tensor):

⇒ In vacuum, (traceless)

• Gravity gradiometer:

ρG

mGdaS

4π

π4total

−=⋅∇⇒

−=⋅≡Φ ∫g

ng

Total flux of field lines ∝ Total mass enclosed

jiij xx ∂∂−∂≡ /Γ 2φ

0π4Γ 2 =−=−∇=∑ ρφ Gi

ii

The output sum of a 3-axis diagonal-component gravity gradiometer must remain constant as a source mass is moved outside.⇒ “Source-independent” null test! Paik, PRD 19, 2320 (1979)

jji

jj

ijiji x

xxx

xgxgxg δφδ

∂∂∂

−=∂∂

≈−2

1,2, )()(

ii xg ∂−∂= /φ

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Gauss’s law test of the 1/r2 law

• Source mass: 1.5-ton Pb pendulum (driven pnematically)

• Detector: 3-axis Superconducting Gravity Gradiometer (SGG)Finite baseline (δx ~ 20 cm) ⇒ Only a near-null detectorFinite baseline error: O(δx/r)2 ⇒ Negligible for geophysical scale exp.

First result:

Chan, Moody, & Paik, PRL 49, 1745 (1982).

True null detector

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Superconducting linear accelerometer

pLL

mdAB

0

0

22m

20

,1

2

≡

++=

γ

µγωω

• Large energy coupling:

• Low noise:

m = 1.2 kg, l = 0.19 m, ω0/2π ≈ 10 Hz, Q ≈ 106, T = 4.2 K, β ≈ η ≈ 0.5, EA(f) = (1 + 0.1 Hz/f) 5 × 10−31 J Hz−1 ⇒ S

optimal :21

121 2

00

22

0

m ≈+

=

−=

dmABωµγω

ωβ

+= )(

24)(

200

B fEQ

Tkm

fS Aa βηωω

Hz 1.0 ,Hz s m 102)( 2/12132/1 >×≈ −− ff -a

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Design of component accelerometers

Angular accelerometer (αi)Linear accelerometer (ai)

I1 − I2

L3

TEST MASS

L1 L2

SQUID

R

I1 I2

Cantilever spring and niobium pancake coil

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Superconducting Gravity Gradiometer

• Test masses are coupled by persistent currents in a differencing circuit to a dc SQUID. ⇒ Γii

• The ratio IS2/IS1 is adjusted to balance out the common mode.

• Test masses are coupled by persistent currents in a summing circuit to another SQUID. ⇒ ai

• Angular acceleration is separately measured. ⇒ αI

• Intrinsic gradient noise:

⇒ Highly sensitive and stable.

TEST MASS 2

TEST MASS 1

SQUID

Lt1

LS3

RSP

LL1

LL2

LS1

LS2

RSS

RL

RBS

LB2

LB1

IL IS2

IS1

IS2 - IS1

IB2

IB1

RBP

Lt2

E) 103(

s 10 E 1 ,Hz E102)(

)(2

8)(

3Earth,zz

29-2/132/1

200

2

×≈Γ

≡×≈

+=

−−−

Γ

ΓfS

fEQ

Tkm

fS AB βηωω

l

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Gravity source, detector, and cryostat

LASER

TURNTABLE

MICROMETER

RUBBER TUBE

VOICE COILACTUATOR

LIQUIDHELIUM

SQUIDHOLDER

SGG

SAA

PHOTODIODE

MUMETAL(2-WALL)

MIRROR(HIDDEN)

1.5-ton Pb pendulumAngular position read by optical shaft encoder

3-axis SGG in umbrella orientationMoody et al, RSI 73, 3957(2002)

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Major errors in SGG

• Linear acceleration sensitivity due to axis misalignment:

• Angular acceleration sensitivity due to misconcentricity:

Residual balance: sensitivity to ai and αi is reduced to <10-7 by error compensation.

• Orientation sensitivity:

• Centrifugal acceleration sensitivity:

• Temperature sensitivity due to variation of penetration depth with temperature:

410ˆ ,ˆ1 −≈⋅=Γ nana δδδ r

l

( ) .10ˆ ,ˆˆ 4−≈⋅×=Γ lr

l δ n αδδ α nˆ,l l

1m

2m

1n

2n

( ) )(,' 2,

2 Ω+ΩΩ−ΩΩ−=ΓΩ−ΩΩ−Γ=Γ Ω δδδδδ Okkjjiiijjiijij

)(),()( 1 δθδδθδθ Γ≈Γ= − OiiΓRRΓ'

⇒ Critical error due to nonlinearity

⇒ Low-pass filtered by soft mounting

2/14c ])/(1/[)0()( −−= TTT λλ

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Performance of SGG

⇒ 10-15 m Hz-1/2

Gravity gradient noise PSD

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Additional errors in Gauss’s law test

• 3-axis SGG measures

• Scale factor error can be removed by rotating 1-axis SGG about vertical.

• Upon rotation and summation (Parke 1990),

Non-orthogonality Angular acceleration Linear acceleration

• Orthogonality error can be nulled by choosingsource-detector orientation such that

• Source-driven accelerations are averaged out except for αz .⇒ Had to detect αz with fiber optics gyro

and remove the error.

• Finite baseline error is computed and removed.

)baseline finite()ityorthogonal()factor scale('3'3'2'2'1'1

δδδδσδσδσ

++≈++ ggg

( )( ) ( ) ( )[ ] ( ) ( )3/ˆˆˆ3ˆˆˆˆ3ˆˆˆˆ33'

''c

jjj hznzgrzxzzxzz δδδαδ +⋅⋅+−⋅×⋅+⋅⋅Γ⋅=Γ∑

r&&r

ll

rt

.0≈Γzz

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Experimental data

• Inverse-square law data • Data was taken over 33 nights, with SGG rotated automatically by 120° at a time.

The data for Gradiometer 2 showed a sign of a current leakage and was discarded.

Source-driven acceleration errors are mostly in even harmonics and filtered out.

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1/r2 law result

• Resolution of the Gauss’s law test • Double null experimentSource: Cylindrical shell Detector: 3-axis SGG

m 5.1at 10)6.49.0(

)2( 10)10.358.0(4

4

=×±=

×±=Γ−

−∑λα

σii

Moody & Paik, PRL 70, 1195 (1993)