Why is it so hard to measure the strength of gravity?Newton’s Law of GravityWhat is the quantity of matter?What is the distance dependence?What is the strength of gravity?Why is it so hard to measure the strength of gravity?Henry Cavendish (1798)Cavendish ExperimentG MeasurementsMeasurement Error1998 Relative SituationThe Big G ControversyNBS G MeasurementLimitations of Time of Swing MethodUniversity of Washington G MeasurementMethodGravitational acceleration of a torsion balanceThin Plate PendulumMinimize other gravity momentsApparatusBig G ApparatusPendulum positionPendulumTurntable ReadoutPendulum-Turntable FeedbackSignal SpectrumBad DataFitting for GAttractor Mass MetrologyAttention to DetailsOther TestsG Data SetsValue for G2000 StatusPTB ResolutionUniversity of Zurich G Measurement2002 Value AdjustmentConclusionsWhy is it so hard to measure the Why is it so hard to measure the strength of gravity?strength of gravity?Stephen M. MerkowitzNASA/GSFCMarch 8, 20072G UMD - 3/8/07Newton’s Law of GravityNewton’s Law of Gravity• What is the quantity of matter?• What is the distance dependence?• What is the strength of gravity?FG= GMmr2“there is a power of gravity pertaining to all bodies, proportional to the several quantities of matter which they obtain... the force of gravity towards the several particles of any body is inversely as the square of the distances of places from the particles” - Isaac Newton3G UMD - 3/8/07What is the quantity of matter?What is the quantity of matter?Inertial mass = Gravitational mass?Equivalence Principle TestsLunar Laser RangingTorsion Pendulum Tests4G UMD - 3/8/07What is the distance dependence?What is the distance dependence?Inverse Square Law Tests5G UMD - 3/8/07What is the strength of gravity?What is the strength of gravity?• G is a fundamental physical constant• Planetary motion - used to weight the planets and the sun• Hydrostatic equilibrium for stars• Planck scale• May play a central role in unification at high energies (short distances)• Fundamental constant in metrology?6G UMD - 3/8/07Why is it so hard to measure the strength of gravity?Why is it so hard to measure the strength of gravity?• Absolute measurement dependent on mass, length, & time.• Gravitational interaction ~1000 times smaller than electromagnetic interaction.• Gravitational force cannot be shielded.• G has no known dependence on another fundamental constants.• Measuring instrument of choice, torsion pendulum, is subject to a variety of parasitic couplings and systematic effects.7G UMD - 3/8/07Henry Cavendish (1798)Henry Cavendish (1798)Density of Earth = 5.448 x Density of WaterCornu & Baille (1873) used G as scale factor“Experiments to determine the density of the Earth”G = (6.754 ± 0.05) ×10−11Nm2/kg2I. Falconer, Measur. Sci. Tech. 10, 470 (1999).G =3g4πR⊕ρ⊕8G UMD - 3/8/07Cavendish ExperimentCavendish Experiment“As the force with which the balls are attracted by these weights is excessively minute, not more than 1/50 000 000 of their weight, it is plain, that a very minute disturbing force will be sufficient to destroy the success of the experiment” - Henry Cavendish Apparatus by John Michell9G UMD - 3/8/07G MeasurementsG MeasurementsG ·1011(m3/kg/s2)10G UMD - 3/8/07Measurement ErrorMeasurement ErrorUncertainty of G has increased only by about an order of magnitude per century!11G UMD - 3/8/071998 Relative Situation1998 Relative Situation12G UMD - 3/8/07The Big G ControversyThe Big G Controversy13G UMD - 3/8/07NBS G MeasurementNBS G MeasurementSource PPMPosition of large masses 10Mass of large masses 1Length of small mass 22Thickness of small mass 36Density of small mass 6Moment of inertia of mirror 23Frequency difference 40Total 64G. G. Luther and W. R. Towler, Phys. Rev. Lett. 48, 121 (1982).G = (6.6726 ± 0.0005) ×10−11Nm2/kg2Time of swing methodω2= Kf+ Kg()IKg=d2Udϕ2G =∆ω2()Ikg, kg= KgG14G UMD - 3/8/07Limitations of Time ofLimitations of Time ofSwing MethodSwing Method• Changes in environment• Changing environmental mass distribution• Uncertainty of pendulum mass distribution• 1/f noise• Anelastic fiber properties (Kuroda effect)K. Kuroda, Phys. Rev. Lett., 75, 2796 (1995).S. Matsumura et al., Phys. Lett. A 244, 4 (1998).15G UMD - 3/8/07University of Washington G MeasurementUniversity of Washington G MeasurementJ. H. Gundlach and S. M. Merkowitz, Phys. Rev. Lett. 85, 2869 (2000)G = (6.674255 ± 0.000092) × 10−11Nm2/kg2Source PPMPendulum Width 0.4Pendulum Thickness 4.0Attractor Diagonal 7.1Ball-bar Calibration 1.4Attractor Vertical 5.2Sphere Diameter 2.6Attractor Temperature 6.9Sphere Mass 0.4Air Humidity 0.5Remaining Twist 0.3Magnetic Field 0.6Rot. Temperature Grad. 0.4Time Base: 0.1Data Reduction 2.0Statistical Error 5.8Total 13.716G UMD - 3/8/07MethodMethod• Start pendulum turntable at constant rate• Turn on acceleration feedback so torsion fiber does not twist– No Kuroda effect– Angular acceleration transferred to turntable– Angle encoding absolutely calibrated• Attractor mass on different turntable rotating at constant difference speedpendulum, thin vertical platevacuum chamber on a turntableAttractor mass17G UMD - 3/8/07Signal periodic at twice per revolutionϕπαϕαimmmmmmeQqmIGllllllll∑∑∑+−=∞=+−==1,1214)(rdrYrrqmm3*pend)ˆ()(lll∫=ρrdrYrrQmm3)1(source)ˆ()(lll+−∫=ρϕπα2sin516222222QIqG−=Biggest component is22αGravitational acceleration of a torsion balanceGravitational acceleration of a torsion balance18G UMD - 3/8/07Thin Plate PendulumThin Plate Pendulumα22=−24π5GQ22sin2ϕα22=−16π5Gq22IQ22sin2ϕ q22I=ρr r p()Y22θp,φp()rp2d3rp∫ρr r p()sin2θprp2d3rp∫2D →  1532πq22I=w2−t2w2+ t21532πFinite thickness plate:19G UMD - 3/8/07Minimize other Minimize other gravity momentsgravity momentsαl2l=odd vanish by symmetrical design of pendulum & sourceα42q42vanishes for rectangular plate with()222103twh +=Q42 vanishes for vertical sphere separation ofρ32=zα62, α82Small and calculableHigher terms small, fall of as2−lRrr=pendulum radiusR=attractor mass radiuszρ20G UMD - 3/8/07ApparatusApparatus21G UMD - 3/8/07Big G ApparatusBig G Apparatus22G UMD - 3/8/07Pendulum positionPendulum position• 4 bounces off pendulum plate gives angle amplification of 8• Insensitive to other rotations23G UMD - 3/8/07PendulumPendulum24G UMD - 3/8/07Turntable ReadoutTurntable Readout25G UMD -