Lecture 10 Satellite Tests of General Relativity: GP-B, STEP, and TRIOFrame draggingGravitomagnetic fieldPrecession of magnetic momentLAGEOS 1 and 2Lens-Thirring orbit precessionGravity Probe BGP-B gyros and readoutGP-B telescope and spacecraftGP-B missionGP-B follow-up with SGG?Why test the EP in Earth orbit?Coupling to gravity gradientsSTEP test masses STEP accelerometerSTEP missionTRIO (Test of Relativity In Orbit)EP test on TRIOISL Test on TRIO 1ISL Test on TRIO 2LLI test on TRIO 1LLI test on TRIO 2Paik-1Lecture 10Satellite Tests of General Relativity:GP-B, STEP, and TRIOHo Jung PaikUniversity of MarylandFebruary 27, 2007 Physics 798G Spring 2007Paik-2Frame dragging• According to GR, spacetime is curved around any mass (or energy). • In 1919, Lense and Thirring predicted that a mass could deform spacetime in a second way − through frame-dragging. • In 1960, Schiff proposed a relativistic gyroscope experiment: If the local spacetime was curved or was twisting, the gyroscope's position and spin axis would change to follow this curve or twist.Paik-3Gravitomagnetic field• Field equations: EM: GR: (φ≈ 0, v « c)GE field ⇒ curvature GM field ⇒ twist(). ,//1 where,cπ41 ,0,01 ,π4ABAEJEBBBEE×∇=∂∂−∇==∂∂−×∇=⋅∇=∂∂+×∇=⋅∇tc -tctcφρ().1 , ,1, ,/ where,16/ ,0,0/ ,π400021==≈+−≈×∇=∂∂−∇−=−≈∂∂−×∇≈⋅∇≈∂∂+×∇≈⋅∇GcgAgt ttiigggggggφφπρρABAEvEBBBEEggEgBPaik-4Precession of magnetic momentSμmcq2=Sμcg1−=• Precession of magnetic moment:EM: GR: : spin angular momentum• Precession rate of an orbiting gyro: ()[]for Earth 108 and 107 whereˆˆˆˆ3ˆˆ231202310200−−×=≡×=≡−⋅+×=ΩωμεμωεωcrGJrcGMJrJrvrrGeodetic precessionFrame dragging precessionPaik-5LAGEOS 1 and 2Launch S.M. axis Inclination PeriodLAGEOS 1 1976 12,270 km 109.84 deg 225 minLAGEOS 2 1992 12,210 km 52.64 deg 223 min• Laser-ranged satelliteswith 426 corner cubes. (~400 kg, 60 cm dia.)• An Earth-orbiting satellite is a gyroscope, and therefore its orbit will experience a frame-dragging.Paik-6Lens-Thirring orbit precession• GR predicts a LT effect of 31.0 mas/yr on LAGEOS 1 node, 31.5 mas/yr on LAGEOS 2 node.• With the aid of the recent Earth gravity model, the only relevant uncertainty in the orbit of the LAGEOS satellites is δJ2~ 10-7J2, in the Earth's quadrupole moment. • Ciufolini and Pavlis, Nature 431, 958 (2004): Analysis of nearly 11 years of laser-ranging data, from January 1993 to December 2003, led to a detection of the LT effect with 10% uncertainty. Raw nodal residualAfter removing six-periodic signalGR predictionPaik-7Gravity Probe BSix prerequisites to a successful relativity mission with gyroscopes: 1. Drift-free gyroscope: < 10-11degrees/hour 2. Sensitive gyro readout: To determine changes in spin angle to 0.1 milliarc-second without disturbing the gyroscope (width of human hair at 100 miles)3. Stable reference: Telescope and mechanical structure of referring the gyro readout to the guide star 4. Trustworthy guide star: A bright, properly located star whose motion with respect to inertial space is known 5. Technique for separating relativity effects: An orbit and a data processing method that together allow the frame-dragging and geodetic effects to be separated 6. Credible calibration scheme: In-flight calibration tests to ensure that the gyroscopes -- and the entire instrument -- are free from errors that might masquerade as relativity signalsPaik-8Paik-9GP-B gyros and readoutSuperconducting gyros• Four superconducting gyros in a polar orbit• Material: fused quartz spheres, coated with Nb • Sphericity: < 8 × 10−9m• Homogeneity: < 2 ppmLondon moment readout• A spinning superconductor generates a magnetic field.• London moment ∝ spin speed &exactly aligned with the spin axis.• The precession of the London moment is detected by a SQUID.Paik-10GP-B telescope and spacecraft•A quartz telescope for accurate pointing• Cooled to 2 K by superfluid helium• Spacecraft under drag-free control locked to a guide starIM PegasiThe aberration of starlight is used for absolute calibration of the gyro sensitivity.Paik-11GP-B mission• After over 40 years of development (and over $600M), GP-B was finally launched on April 20, 2004!P.I.: Francis Everitt at Stanford• Liquid helium lasted for 17 months.• All four gyros worked well with a spin-down time of 10,000 years.Paik-12GP-B follow-up with SGG?• The Riemann (gravity gradient) tensor due to the gravitomagnetic field in a space-fixed frame in the polar orbit at altitude h:where a = RE+ h and ψ= ω0t is the phase of the orbit.•A two-axis in-line SGG with axes at 45° from the orbit plane can measure ΓGMdirectly (Mashoon, Paik, and Will, PRD 39, 2825, 1989).• To resolve ΓGMwith S/N = 100 in a year (as GP-B), an SGG sensitivity of 3 × 10−6E Hz−1/2is required at f0= 1.7 × 10−4Hz.⇒ An SGG with levitated test masses will meet the requirement.• Pointing requirement for the spacecraft: 10−3arcsec Hz−1/2at f0⇒ May require a quartz telescope or superconducting gyros.⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡+−+−−−=02cos02cos02sin02sin0621213GMψψψψμaGMΓPaik-13Why test the EP in Earth orbit?• Test masses can fall a long time.• Nearly the full gravitational acceleration of the Earth can be used.⇒ Signal 103times larger than the torsion balance experiments• A very quiet environment can be created by a drag-free spacecraft.⇒ More than 103times quieter than any place on Earth• Satellite Test of the Equivalence Principle (STEP) aims at η= 10−18.Paik-14Coupling to gravity gradients• Force on test mass A by source with the Newtonian potential USand an EP violation force Φ:• Differential acceleration between test masses A and B:• Near masses couple to test masses through higher multipole moments.⇒ Helium confinement and test mass metrology requirement.AiSijkAjkSijkAjkSijASiAAiUmUmUmUmFjΦ+⋅⋅⋅+∂+∂+∂+∂=ll!31!21Total mass Dipole Quadrupole Octupole ()()⎟⎟⎠⎞⎜⎜⎝⎛Φ−Φ+⋅⋅⋅+∂⎟⎟⎠⎞⎜⎜⎝⎛−+∂⎟⎟⎠⎞⎜⎜⎝⎛−+∂−+∂−=Δ−BBiAAiSijkBBjkAAjkSijkBBjkAAjkSijjBCMjACMSiBAimmUmmmmUmmmmUxxUalll!31!2111,,Monopole coupling (~GM/r2) Dipole coupling (~GM/r3)Vanishes identically Drops out by CM matchingQuadrupole coupling (~GM/r4) Octupole coupling (~GM/r5) Violation