ASTROD and ASTROD I: Progress Report A. Pulido PatnPurple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008GWADW 2006, Isola dElba, May 27-June 2nd

The road towards ASTRODLaser Astrodynamics is proposed to study relativistic gravity and to explore the solar system, 2nd William Fairbank conference (Hong Kong), and International workshop on Gravitation and Fifth Force (Seoul) 1993.A multi-purpose astrodynamical mission is reached (1994) in 7th Marcel Grossmann, July, 1994, Stanford (California).ASTROD (Astrodynamical Space Test of Relativity using Optical Devices) presented at 31st COSPAR Scientific Assembly July 1996.ASTROD and its sensitivity to measurements presented in the Pacific conference on Gravitation and Cosmology. Seoul (Korea) February 1996.ASTROD and its related gravitational wave sensitivity presented at TAMA Gravitational Wave Workshop in Tokyo (Japan) 1997.The possibility of solar g-mode detection was presented in 3rd Edoardo Amaldi and 1st ASTROD Symposium (2001).

Current Collaborators Wei-Tou Ni1,2,3, Henrique Arajo4, Gang Bao1, Hansjrg Dittus5, Tianyi Huang6, Sergei Klioner7, Sergei Kopeikin8, George Krasinsky9, Claus Lmmerzahl5, Guangyu Li1,2, Hongying Li1, Lei Liu1, Yu-Xin Nie10, Antonio Pulido Patn1, Achim Peters11, Elena Pitjeva9, Albrecht Rdiger12, tienne Samain13, Diana Shaul4, Stephan Schiller14, Sachie Shiomi3,M. H. Soffel7, Timothy Sumner4, Stephan Theil5, Pierre Touboul15, Patrick Vrancken13, Feng Wang1, Haitao Wang16, Zhiyi Wei10, Andreas Wicht14, Xue-Jun Wu1,17, Yan Xia1, Yaoheng Xiong18, Chongming Xu1,17, Dong Peng6, Xie Yi6, Jun Yan1,2, Hsien-Chi Yeh19, Yuan-Zhong Zhang20, Cheng Zhao1, and Ze-Bing Zhou21

1 Center for Gravitation and Cosmology, Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing, 210008 China 2 National Astronomical Observatories, CAS, Beijing, 100012 China 3 Center for Gravitation and Cosmology, Department of Physics, Tsing Hua University, Hsinchu, Taiwan 30013 4 Department of Physics, Imperial College of Science, Technology and Medicine, London, SW7 2BW, UK 5 ZARM, University of Bremen, 28359 Bremen, Germany6 Department of Astronomy, Nanjing University, Nanjing, 210093 China7 Lohrmann-Observatorium, Institut fr Planetare Geodsie, Technische Universitt Dresden, 01062 Dresden, Germany8 Department of Physics and Astronomy, University of Missouri-Columbia, Columbia, Missouri 65221, USA 9 Institute of Applied Astronomy, Russian Academy of Sciences, St.-Petersburg, 191187 Russia 10 Institute of Physics, Chinese Academy of Sciences, Beijing, 100080 China 11 Department of Physics, Humboldt-University Berlin, 10117 Berlin, Germany 12 Max-Planck-Institut fr Gravitationsphysik, 85748 Grching, Germany13Observatoire de la Cte DAzur, 06460 Caussols, France14 Institute for Experimental Physics, University of Dsseldorf, 40225 Dsseldorf, Germany15 Office National Dtudes et de Recherches Aerospatiales, Chatillon Cedex, France16 College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016 China17 Department of Physics, Nanjing Normal University, Nanjing, 210097 China18 Yunnan Observatory, NAOC, Chinese Academy of Sciences, Kunming, 650011 China19 Division of Manufacturing Engineering, Nanyang Technological University, Singapore 63979820 Institute of Theoretical Physics, CAS, Beijing, 100080 China21 Department of Physics, Hua Zhong University of Science and Technology,Wuhan, 430074 China

International collaboration period2000: ASTROD proposal submitted to ESA F2/F3 call (2000)2001: 1st International ASTROD School and Symposium held in Beijing; Mini-ASTROD study began2002: Mini-ASTROD (ASTROD I) workshop, Nanjing2004: German proposal for a German-China ASTROD collaboration approved2005: 2nd International ASTROD Symposium (June 2-3, Bremen, Germany)2004-2005: ESA-China Space Workshops (1st &2nd, Noordwijk & Shanghai), potential collaboration discussed2006(7): Joint ASTROD (ASTROD I) proposal to be submitted to ESA call for proposals2006: 3rd ASTROD Symposium (July 14-16, Beijing) before COSPAR (July 16-23) in Beijing

ASTROD mission conceptASTROD mission concept is to use three drag-free spacecraft. Two of the spacecraft are to be in an inner (outer) solar orbit employing laser interferometric ranging techniques with the spacecraft near the Earth-Sun L1 Lagrange point. Spacecraft payload: a proof mass, two telescopes, two 1 W lasers, a clock and a drag-free system.

ASTROD scientific objectivesTest Relativistic gravity with 3-5 orders of magnitude improvement in sensitivity. That includes the measurement of relativistic parameters , , measurement of dG/dt, and the anomalous constant acceleration towards the Sun (Pioneer anomaly).Improvement by 3-4 orders of magnitude in the measurements of solar, planetary and asteroids parameters. That also includes a measurement of solar angular momentum via Lense-Thirring effect and the detection of solar g-modes by their changing gravity field.Detection of low frequency gravitational waves (5 Hz-5 mHz) from massive black hole and galactic binary stars. Background gravitational waves will also be explored.

ASTROD technological requirementsWeak-light phase locking to 100 fW.Heterodyne interferometry and data analysis for unequal-arm interferometry.Coronagraph design and development: sunlight in the photodetectors should be less than 1 % of the laser light.High precision space clock and/or absolute stabilized laser to 10-17.Drag-free system. Accelerometer noise requirement: (0.3-1)10-15[1+10(3mHz)2] ms-2Hz-1/2 at 0.1 mHz < < 100 mHz.Laser metrology to monitor position and distortion of spacecraft components for gravitational modeling.

ASTROD I ASTROD I is a simple version of ASTROD mission in which a single spacecraft in solar orbit and a ground laser station perform two-way interferometric and pulse laser ranging.

ASTROD orbit design featuresThe distance to the Sun of the inner spacecraft varies from 0.77 AU to 1 AU and for the outer spacecraft varies from 1 AU to 1.32 AU.The two spacecraft should go to the other side of the sun simultaneously to perform Shapiro time delay.To obtain better accuracy in the measurements of and asteroid parameters estimation, one spacecraft should be in inner orbit and the other in outer orbit. The two spacecraft at the other side of the sun should be near to each other for ranging in order to perform measurements of Lense-Thirring effect (measurement of solar angular momentum).

Relativistic parameter determination for ASTRODThe uncertainty of relativistic parameters (, and J2) assuming 1 ps accuracy and ASTROD acceleration noise ~ 310-18 m s-2 (0.1 mHz) are, 1200 days after launch:

1.0510-9; 1.3810-9 and J23.810-11

Relativistic parameter uncertainty evolution

Solar angular momentumLense-Thirring effect can be measured by taking the time difference between the light round trips SC1 - SC2 - Earth system basis and SC2 - SC1 - Earth System basis.The Newtonian time difference t1-t2 for 800-1034 days after launching gives about 10 ms. The Lense-Thirring effect has a totally different signature and for this period of time is about 100 ps.Assuming a laser stability of 10-15-10-13 one could achieve 10-5-10-7 level of uncertainty.Lense-Thirring effect is proportional to the solar angular momentum.

Evolution of G and Pioneer anomaly uncertainties

measurement and anomalous acceleration towards the SunG2.8210-15 yr-1 and the anomalous acceleration towards the sun Aa3.0210-17 m s-2 (again assuming 1 ps and 310-18 m s-2).By using an independent measurement of ASTROD would be able to monitor the solar mass loss rate. The expected solar mass loss rate: a) electromagnetic radiation ~ 710-14 Msunyr, b) solar wind ~10-14 Msunyr, c) solar neutrino ~ 210-15 Msunyr and d) solar axions ~ 10-15 Msunyr.

Gravitational wave detectionl1l212SC 2SC 1ERSGravitational detection topology. Path 1: ERS-SC1-ERS-SC2-ERS. Path 2: ERS-SC2-ERS-SC1-ERS. To minimize the arm-length difference.

For example if a monochromatic gravitational wave with + polarization arrives orthogonal to the plane formed by SC1, 2 and ERS, then the optical path difference for laser light traveling through path 1 and 2 and returning simultaneously at the same time t, is given by

With 1=2l1c and 2=2l2c.

Solar g-modesWhen the spacecraft moves in solar orbit the amplitude and direction of the solar oscillation signals are deeply modulated in addition to the modulation due to spacecraft maneuvering.Time constants for solar oscillations are about 106 yr for low-l g-modes and over 2-3 months for low-l p-modes. Close white dwarf binaries (CWDB) time constant are longer than 106 yr. Hence confusion background is steady in inertial space, only modulated by spacecraft maneuvering and not by spacecraft orbit motion.With this extra modulation due to orbit motion the solar oscillation signals can reach 5 orders lower than the binary confusion limit.

Technological requirements: ASTROD (I) drag-freeASTROD I acceleration noise of free fall test massesLISA acceleration noise of free fall test massesASTROD aims to improve LISA acceleration noise at 0.1 mHz by a factor 3-10, i.e., approx. 0.3-110-15 ms-2 Hz-1/2.ASTROD bandwidth 5 Hz5mHz

ASTROD Gravitational Reference Sensor (GRS) preliminary conceptMove towards true drag-free conditions and improving LISA drag-freeperformance by a factor 3-10.

GRS provide reference positioning only. Laser beam does not illuminate directly the proof mass (avoiding cross coupling effects and pointing ahead problem) surface but the GRS housing surface.Only one reference proof mass. GRS measures the center of mass position of the proof mass.Optical sensing could replace capacitive sensing. Capacitance could still be used for control purposes.Absolute laser metrology to measure structural changes due to thermal effects and slow relaxations.

LISA has adopted condition 1. Both conditions 1 and 2 avoid cross couplingdue to control forces aimed to keep the right orientation of the proof massmirror.

ASTROD will employ separate interferometry to measure the GW signal andthe proof mass-spacecraft relative displacement independently.

ASTROD GRSSchematic of possible GRS designs for ASTROD: a) a cubical proof mass free floating inside a housing anchored to the spacecraft, b) a spherical (cylindrical) proof mass is also considered. a)b)

Drag-free control conceptThrustersThrustersPM acceleration disturbance:p -KXnr + np + (ns + TNt)Ku-1-2

Direct proof mass acceleration disturbancesMagnetic interactions due to susceptibility () and permanent moment of the proof mass (Mr)Lorentz forces due to proof mass charging (Q).Thermal disturbances (radiometer, out gassing, thermal radiation pressure and gravity gradients).Impacts due to cosmic rays and residual gas.

Capacitive sensingCapacitive sensing needs very close metallic surfaces to achieve good readout sensitivity.Displacement readout sensitivity is proportional to d-1 or d-2 for different readout configurations.By decreasing the gap, readout sensitivity increases but also back action disturbances and stiffness terms increase.

Optical sensingA drawback of capacitive sensing is the need for close gaps between metallic surfaces to increase sensitivity. The sensitivity is proportional to the difference between capacitance (C1-C2), therefore proportional to d-2.Optical sensing allows us using larger gaps between the PM and surrounding metallic surfaces.Optical sensing provides a way of sensing essentially free of stiffness.Optical sensing sensitivity is limited by shot noise. Picometer sensitivity can be achieved with W of lasing power and 1.5 m wavelength.

Back action force, 2P/c, can be made negligible, with 1% compensation ~10-17 m s-2 Hz-1/2.

LISA and ASTROD accelerationnoise comparison

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ASTROD Capacitive sensing

LISA

ASTROD Optical Sensor

ASTROD cap low f

ASTROD opt low f

Frequency (Hz)

Acceleration noise (ms^{-2}Hz^{-1/2})

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ASTROD (capacitance)

LISA

ASTROD (Optical)

ASTROD low-f (capacitance)

ASTROD low-f (optical)

10^6 (6 months)

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10^5 (6 months)

10^6 (10 years)

10^7 (10 years)

10^5 (10 years)

Frequency (Hz)

Gravitational Wave Strain

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LISA

LISA Bender

ASTROD

Frequency (Hz)

Acceleration noise units ms^(-2)^Hz^(-1/2)

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LISA Bender extension

LISA, 1 yr int. time S/N=5

ASTROD, 1 yr int. time, S/N=5

Frequency (Hz)

Gravitational Wave Strain

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PressureSusceptibilityPermanent momentMagnetic shielding

0288100150

Total chargeEffective Charging RateElectrostatic shieldingThermal shielding

NumbersFrequencyCosmic RaysResidual GasMagnetic Susceptibility IPermanent Magnetic MomentMagnetic Susceptibility IILorentz ILorentz IIRadiometer effect (thermal)OutgassingThermal Radiation PressureQuantizationDielectric lossesVoltage 1Charging-voltage2Gravity Gradients.Total direct acc. Noise using capacitive sensingLISA Acc NoiseTotal acceleration noise using optical sensingU0 LISAU0 ASTRODINVSINCU0 LISAINVSINCU0 ASTRODShot noise LISAShot noise ASTRODSTRAIN due to Acc. Noise (LISA)TOTAL STRAIN (LISA)STRAIN ACCNOISE ASTROD (capacitive)TOTAL STRAIN ASTROD (capacitance)STRAIN Acc noise ASTROD (Optical)Total Strain ASTROD (Optical)RedshiftLuminosity integralFrequency 10^6-10^9 binary and 10^5-10^4 (6 months)GW Strain (6 months)Frequency (10 years)GW Strain (10 years)Frequency (100 years)GW Strain (100 years)FrequencyLISA Acceleration NoiseASTROD Acceleration noiseFrequencyASTROD opticalSTRAIN due to acceleration noise (LISA)FREQUENCY ASTRODSTRAIN AN ASTRODShot noise LISAShot noise ASTRODTOTAL LISA STRAINTOTAL STRAIN ASTRODAntenna transfer LISAAntenna transfer ASTRODFrequencyGW LISAGW ASTRODFrequencyASTROD

10.000001000000000000.00000100.00010471980.00314159271.00000000181.00000164490000000010.785660.000038183800.000012416300.000002700.00000100.0001001.80E-215.69E-2201.00000000180.00000101.00E-043.15E-16

20.000002000000000000.00000200.00020943950.00628318531.00000000731.00000657980000000051.8790.000019216700.000006248700.000001358800.00000200.000053.20E-1601.80E-215.69E-2201.00000000730.00000203.00E-063.30E-14

30.000003000000000000.00000300.00031415930.0094247781.00000001641.000014804600000000102.28881950.000013156900.000004278200.000000930300.00000300.00003001.80E-215.69E-2201.00000001640.0000030

40.000004000000000000.00000400.0004188790.01256637061.00000002921.000026319400000000202.6092654890.000008782800.000002855900.00000062100.00000400.00001001.80E-215.69E-2201.00000002920.0000040

50.000005000000000000.00000500.00052359880.01570796331.00000004571.000041124500000000402.8480570.000005781400.0000018800.000000408800.00000500.000003001.80E-215.69E-2201.00000004570.0000050

60.000006000000000000.00000600.00062831850.01884955591.00000006581.00005922010000000010.785660.000009054800.000002944400.000000640300.000006001.80E-215.69E-2201.00000006580.0000060

70.000007000000000000.00000700.00073303830.02199114861.00000008961.00008060630000000051.8790.00000455700.000001481800.000000322200.000007001.80E-215.69E-2201.00000008960.0000070

80.000008000000000000.00000800.0008377580.02513274121.0000001171.000105283500000000102.28881950.0000031200.000001014500.000000220600.000008001.80E-215.69E-2201.0000001170.0000080

90.000009000000000000.00000900.00094247780.02827433391.0000001481.000133252100000000202.6092654890.000002082700.000000677300.000000147300.000009001.80E-215.69E-2201.0000001480.0000090

100.00001000000000000.0000100.00104719760.03141592651.00000018281.000164512300000000402.8480570.00000137100.000000445800.000000096900.00001001.80E-215.69E-2201.00000018280.000010

200.00002000000000000.0000200.00209439510.06283185311.00000073111.00065827680000000010.785660.000002147200.000000698200.000000151800.00002001.80E-215.69E-2201.00000073110.000020

300.00003000000000000.0000300.00314159270.09424777961.00000164491.00148197630000000051.8790.000001080600.000000351400.000000076400.00003001.80E-215.69E-2201.00000164490.000030

400.00004000000000000.0000400.00418879020.12566370611.00000292431.002636751400000000102.28881950.000000739900.000000240600.000000052300.00004001.80E-215.69E-2201.00000292430.000040

500.00005000000000000.0000500.00523598780.15707963271.00000456931.00412420400000000202.6092654890.000000493900.000000160600.000000034900.00005001.80E-215.69E-2201.00000456930.000050

600.00006000000000000.0000600.00628318530.18849555921.00000657981.00594640200000000402.8480570.000000325100.000000105700.00000002300.00006000.00E+001.80E-215.69E-22001.00000657981.00E+000.000060

700.00007000000000000.0000700.00733038290.21991148581.00000895581.00810588650000000010.785660.000000509200.000000165600.00000003600.00007000.00E+001.80E-215.69E-22001.000008955810.000070

800.00008000000000000.0000800.00837758040.25132741231.00001169741.0106056790000000051.8790.000000256300.000000083300.000000018100.00008000.00E+001.80E-215.69E-22001.000011697410.000080

900.00009000000000000.0000900.0094247780.28274333881.00001480461.013449291800000000102.28881950.000000175400.000000057100.000000012400.00009003.00E-061.20E-171.80E-215.69E-22001.000014804610.0000902.85E-18

1000.00010000000000000000.00010000.01047197550.31415926541.00001827731.016640738500000000202.6092654890.000000117100.000000038100.000000008300.00010000.000101.80E-215.69E-22001.00001827731.01664073850.000102.38E-23

2000.00020000000000000000.00020000.0209439510.62831853071.00007311191.068959332100000000402.8480570.000000077100.000000025100.000000005500.00020000.000201.80E-215.69E-22001.00007311191.06895933210.000206.41E-24

3000.00030000000000000000.00030000.03141592650.94247779611.00016451231.16496662320000000010.785660.000161019600.000052359100.000011385800.00030000.000301.80E-215.69E-22001.00016451231.16496662320.000303.41E-24

4000.00040000000000000000.00040000.0418879021.25663706141.00029249261.32130639970000000051.8790.00008103600.000026350700.000005730100.00040000.000401.80E-215.69E-22001.00029249261.32130639970.000402.61E-24

5000.00050000000000000000.00050000.05235987761.57079632681.00045707231.570796326800000000102.28881950.00005548200.000018041200.000003923200.00050000.000501.80E-215.69E-22001.00045707231.57079632680.000502.55E-24

6000.00060000000000000000.00060000.06283185311.88495559221.00065827681.981959599500000000202.6092654890.00003703700.000012043400.000002618900.00060000.000601.80E-215.69E-22001.00065827681.98195959950.000602.92E-24

7000.00070000000000000000.00070000.07330382862.19911485751.0008961372.718255454200000000402.8480570.000024379900.000007927700.000001723900.00070000.000701.80E-215.69E-22001.0008961372.19911485750.000703.08E-24

8000.00080000000000000000.00080000.08377580412.51327412291.00117068944.27583732850000000010.785660.000679014100.000220796700.000048013500.00080000.000801.80E-215.69E-22001.00117068942.51327412290.000803.43E-24

9000.00090000000000000000.00090000.09424777962.82743338821.00148197639.14976664620000000051.8790.00034172600.000111119900.000024163700.00090000.000901.80E-215.69E-22001.00148197632.82743338820.000903.81E-24

10000.0010000000000000000.0010000.10471975513.14159265361.0018300456-976941262046814000000-0.0000000010-0.000000001102.28881950.000233965600.000076079200.000016543900.0010000.00101.80E-215.69E-22001.00183004563.14159265360.00104.19E-24

20000.0020000000000000000.0020000.20943951026.28318530721.00734840559769412620468140000000.00000000100.000000001202.6092654890.000156183600.000050786600.000011043800.0020000.00201.80E-215.69E-22001.00734840556.28318530720.00208.24E-24

30000.0030000000000000000.0030000.31415926549.42477796081.016640738525642546754255300000000.000000002600.0000000026402.8480570.000102809300.000033430800.000007269700.0030000.00301.80E-215.69E-22001.01664073859.42477796080.00301.24E-23

40000.0040000000000000000.0040000.418879020512.56637061441.02985316829769412620468140000000.00000000100.0000000010.0040000.00401.80E-215.69E-22001.029853168212.56637061440.00401.65E-23

50000.0050000000000000000.0050000.523598775615.70796326791.0471975512-1349733455948780000000-0.00000000140-0.00000000140.0050000.00501.80E-215.69E-22001.047197551215.70796326790.00502.06E-23

60000.0060000000000000000.0060000.628318530718.84955592151.0689593321-2564254675425530000000-0.00000000260-0.00000000260.0060000.00601.80E-215.69E-22001.068959332118.84955592150.00602.47E-23

70000.0070000000000000000.0070000.733038285821.99114857511.09550852834986295905063160000000.000000000500.00000000050.0070000.00701.80E-215.69E-22001.095508528321.99114857510.00702.88E-23

80000.0080000000000000000.0080000.83775804125.13274122871.12731463949769412620468140000000.00000000100.0000000010.0080000.00801.80E-215.69E-22001.127314639425.13274122870.00803.29E-23

90000.0090000000000000000.0090000.942477796128.27433388231.16496662326073517641501150000000.000000000600.00000000060.0090000.00901.80E-215.69E-22001.164966623228.27433388230.00903.70E-23

100000.010000000000000000.010001.047197551231.41592653591.209199576213497334559487800000000.000000001400.00000000140.010000.0101.80E-215.69E-22001.209199576231.41592653590.0104.12E-23

200000.020000000000000000.020002.094395102462.83185307182.418399152313497334559487800000000.000000001400.00000000140.020000.0201.80E-215.69E-22002.418399152362.83185307180.0208.23E-23

300000.030000000000000000.030003.141592653694.247779607725642546754255300274776310206491000000.000000106200.000000002800.00000000280.030000.0301.80E-215.69E-22003.141592653694.24777960770.0301.23E-22

400000.040000000000000000.040004.1887902048125.6637061436-4.836798304613497334559487800000-000.000000001400.00000000140.040000.0401.80E-215.69E-22004.1887902048125.66370614360.0401.65E-22

500000.050000000000000000.050005.235987756157.0796326795-6.0459978808-5724054125572580000-00-0.00000000060-0.00000000060.050000.0501.80E-215.69E-22005.235987756157.07963267950.0502.06E-22

600000.060000000000000000.060006.2831853072188.4955592154-2564254675425530027477631020649100000-0.000000106200.000000002800.00000000280.060000.0601.80E-215.69E-22006.2831853072188.49555921540.0602.47E-22

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7000000.70000000000000000.700073.30382858382199.1148575129-84.643970330911935763461837500000-000.000000001200.0000000012

8000000.80000000000000000.800083.77580409572513.274122871896.7359660925-572405412557258000000-0.00000000060-0.0000000006

9000000.90000000000000000.900094.24777960772827.4333882308-3771159070267650-4653113923953820000-0.00000001560-0.00000000050-0.0000000005

100000010000000000000001000104.71975511973141.5926535898-120.9199576156-9772681163864820000-00-0.0000000010-0.000000001

AT LOW FREQUENCIES

FrequencyNon thermalThermalCapacitive sensorTotal noise capacitiveTotal noise opticalShot noise ASTRODU0 ASTRODInvSIncU0STRAIN ACC NOISE (CAP)STRAIN ACC (OPT)GRAV WAVE (CAP,ASTROD)GRAV WAVE (OPT,ASTROD)

0.00011.26E-169.68E-1700.31415926541.01664073854.25E-213.27E-217.76E-256.01E-25

0.000038.40E-178.00E-172.10E-162.40E-16000.09424777961.00148197639.00E-204.35E-201.61E-237.76E-24

0.000011.00E-163.10E-155.80E-163.16E-15000.03141592651.00016451231.07E-171.05E-171.90E-211.87E-21

0.0000031.60E-163.30E-141.90E-153.31E-14000.0094247781.00001480461.24E-151.24E-152.21E-192.21E-19

Sheet2

Sheet3

Detecting Gravitational WavesTo estimate gravitational wave strength sensitivity

Lasing power (Pt): LISA 1W, ASTROD 10 WShot noise level (ASTROD) 1.210-21Acceleration noise (A0) is the dominant source of noise at low frequencies.

LISA and ASTROD GW strain sensitivity (S/N5, int. time 1yr)

Chart2

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ASTROD Capacitive sensing

LISA

ASTROD Optical Sensor

ASTROD cap low f

ASTROD opt low f

Frequency (Hz)

Acceleration noise (ms^{-2}Hz^{-1/2})

Chart1

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ASTROD (capacitance)

LISA

ASTROD (Optical)

ASTROD low-f (capacitance)

ASTROD low-f (optical)

10^6 (6 months)

10^7 (6 months)

10^5 (6 months)

10^6 (10 years)

10^7 (10 years)

10^5 (10 years)

Frequency (Hz)

Gravitational Wave Strain

Chart3

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LISA

LISA Bender

ASTROD

Frequency (Hz)

Acceleration noise units ms^(-2)^Hz^(-1/2)

Chart4

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LISA Bender extension

LISA, 1 yr int. time S/N=5

ASTROD, 1 yr int. time, S/N=5

Frequency (Hz)

Gravitational Wave Strain

Sheet1

0.00000010.0000030.00000001110

PressureSusceptibilityPermanent momentMagnetic shielding

0288100150

Total chargeEffective Charging RateElectrostatic shieldingThermal shielding

NumbersFrequencyCosmic RaysResidual GasMagnetic Susceptibility IPermanent Magnetic MomentMagnetic Susceptibility IILorentz ILorentz IIRadiometer effect (thermal)OutgassingThermal Radiation PressureQuantizationDielectric lossesVoltage 1Charging-voltage2Gravity Gradients.Total direct acc. Noise using capacitive sensingLISA Acc NoiseTotal acceleration noise using optical sensingU0 LISAU0 ASTRODINVSINCU0 LISAINVSINCU0 ASTRODShot noise LISAShot noise ASTRODSTRAIN due to Acc. Noise (LISA)TOTAL STRAIN (LISA)STRAIN ACCNOISE ASTROD (capacitive)TOTAL STRAIN ASTROD (capacitance)STRAIN Acc noise ASTROD (Optical)Total Strain ASTROD (Optical)RedshiftLuminosity integralFrequency 10^6-10^9 binary and 10^5-10^4 (6 months)GW Strain (6 months)Frequency (10 years)GW Strain (10 years)Frequency (100 years)GW Strain (100 years)FrequencyLISA Acceleration NoiseASTROD Acceleration noiseFrequencyASTROD opticalSTRAIN due to acceleration noise (LISA)FREQUENCY ASTRODSTRAIN AN ASTRODShot noise LISAShot noise ASTRODTOTAL LISA STRAINTOTAL STRAIN ASTRODAntenna transfer LISAAntenna transfer ASTRODFrequencyGW LISAGW ASTRODFrequencyASTROD

10.000001000000000000.00000100.00010471980.00314159271.00000000181.00000164490000000010.785660.000038183800.000012416300.000002700.00000100.0001001.80E-215.69E-2201.00000000180.00000101.00E-043.15E-16

20.000002000000000000.00000200.00020943950.00628318531.00000000731.00000657980000000051.8790.000019216700.000006248700.000001358800.00000200.000053.20E-1601.80E-215.69E-2201.00000000730.00000203.00E-063.30E-14

30.000003000000000000.00000300.00031415930.0094247781.00000001641.000014804600000000102.28881950.000013156900.000004278200.000000930300.00000300.00003001.80E-215.69E-2201.00000001640.0000030

40.000004000000000000.00000400.0004188790.01256637061.00000002921.000026319400000000202.6092654890.000008782800.000002855900.00000062100.00000400.00001001.80E-215.69E-2201.00000002920.0000040

50.000005000000000000.00000500.00052359880.01570796331.00000004571.000041124500000000402.8480570.000005781400.0000018800.000000408800.00000500.000003001.80E-215.69E-2201.00000004570.0000050

60.000006000000000000.00000600.00062831850.01884955591.00000006581.00005922010000000010.785660.000009054800.000002944400.000000640300.000006001.80E-215.69E-2201.00000006580.0000060

70.000007000000000000.00000700.00073303830.02199114861.00000008961.00008060630000000051.8790.00000455700.000001481800.000000322200.000007001.80E-215.69E-2201.00000008960.0000070

80.000008000000000000.00000800.0008377580.02513274121.0000001171.000105283500000000102.28881950.0000031200.000001014500.000000220600.000008001.80E-215.69E-2201.0000001170.0000080

90.000009000000000000.00000900.00094247780.02827433391.0000001481.000133252100000000202.6092654890.000002082700.000000677300.000000147300.000009001.80E-215.69E-2201.0000001480.0000090

100.00001000000000000.0000100.00104719760.03141592651.00000018281.000164512300000000402.8480570.00000137100.000000445800.000000096900.00001001.80E-215.69E-2201.00000018280.000010

200.00002000000000000.0000200.00209439510.06283185311.00000073111.00065827680000000010.785660.000002147200.000000698200.000000151800.00002001.80E-215.69E-2201.00000073110.000020

300.00003000000000000.0000300.00314159270.09424777961.00000164491.00148197630000000051.8790.000001080600.000000351400.000000076400.00003001.80E-215.69E-2201.00000164490.000030

400.00004000000000000.0000400.00418879020.12566370611.00000292431.002636751400000000102.28881950.000000739900.000000240600.000000052300.00004001.80E-215.69E-2201.00000292430.000040

500.00005000000000000.0000500.00523598780.15707963271.00000456931.00412420400000000202.6092654890.000000493900.000000160600.000000034900.00005001.80E-215.69E-2201.00000456930.000050

600.00006000000000000.0000600.00628318530.18849555921.00000657981.00594640200000000402.8480570.000000325100.000000105700.00000002300.00006000.00E+001.80E-215.69E-22001.00000657981.00E+000.000060

700.00007000000000000.0000700.00733038290.21991148581.00000895581.00810588650000000010.785660.000000509200.000000165600.00000003600.00007000.00E+001.80E-215.69E-22001.000008955810.000070

800.00008000000000000.0000800.00837758040.25132741231.00001169741.0106056790000000051.8790.000000256300.000000083300.000000018100.00008000.00E+001.80E-215.69E-22001.000011697410.000080

900.00009000000000000.0000900.0094247780.28274333881.00001480461.013449291800000000102.28881950.000000175400.000000057100.000000012400.00009003.00E-061.20E-171.80E-215.69E-22001.000014804610.0000902.85E-18

1000.00010000000000000000.00010000.01047197550.31415926541.00001827731.016640738500000000202.6092654890.000000117100.000000038100.000000008300.00010000.000101.80E-215.69E-22001.00001827731.01664073850.000102.38E-23

2000.00020000000000000000.00020000.0209439510.62831853071.00007311191.068959332100000000402.8480570.000000077100.000000025100.000000005500.00020000.000201.80E-215.69E-22001.00007311191.06895933210.000206.41E-24

3000.00030000000000000000.00030000.03141592650.94247779611.00016451231.16496662320000000010.785660.000161019600.000052359100.000011385800.00030000.000301.80E-215.69E-22001.00016451231.16496662320.000303.41E-24

4000.00040000000000000000.00040000.0418879021.25663706141.00029249261.32130639970000000051.8790.00008103600.000026350700.000005730100.00040000.000401.80E-215.69E-22001.00029249261.32130639970.000402.61E-24

5000.00050000000000000000.00050000.05235987761.57079632681.00045707231.570796326800000000102.28881950.00005548200.000018041200.000003923200.00050000.000501.80E-215.69E-22001.00045707231.57079632680.000502.55E-24

6000.00060000000000000000.00060000.06283185311.88495559221.00065827681.981959599500000000202.6092654890.00003703700.000012043400.000002618900.00060000.000601.80E-215.69E-22001.00065827681.98195959950.000602.92E-24

7000.00070000000000000000.00070000.07330382862.19911485751.0008961372.718255454200000000402.8480570.000024379900.000007927700.000001723900.00070000.000701.80E-215.69E-22001.0008961372.19911485750.000703.08E-24

8000.00080000000000000000.00080000.08377580412.51327412291.00117068944.27583732850000000010.785660.000679014100.000220796700.000048013500.00080000.000801.80E-215.69E-22001.00117068942.51327412290.000803.43E-24

9000.00090000000000000000.00090000.09424777962.82743338821.00148197639.14976664620000000051.8790.00034172600.000111119900.000024163700.00090000.000901.80E-215.69E-22001.00148197632.82743338820.000903.81E-24

10000.0010000000000000000.0010000.10471975513.14159265361.0018300456-976941262046814000000-0.0000000010-0.000000001102.28881950.000233965600.000076079200.000016543900.0010000.00101.80E-215.69E-22001.00183004563.14159265360.00104.19E-24

20000.0020000000000000000.0020000.20943951026.28318530721.00734840559769412620468140000000.00000000100.000000001202.6092654890.000156183600.000050786600.000011043800.0020000.00201.80E-215.69E-22001.00734840556.28318530720.00208.24E-24

30000.0030000000000000000.0030000.31415926549.42477796081.016640738525642546754255300000000.000000002600.0000000026402.8480570.000102809300.000033430800.000007269700.0030000.00301.80E-215.69E-22001.01664073859.42477796080.00301.24E-23

40000.0040000000000000000.0040000.418879020512.56637061441.02985316829769412620468140000000.00000000100.0000000010.0040000.00401.80E-215.69E-22001.029853168212.56637061440.00401.65E-23

50000.0050000000000000000.0050000.523598775615.70796326791.0471975512-1349733455948780000000-0.00000000140-0.00000000140.0050000.00501.80E-215.69E-22001.047197551215.70796326790.00502.06E-23

60000.0060000000000000000.0060000.628318530718.84955592151.0689593321-2564254675425530000000-0.00000000260-0.00000000260.0060000.00601.80E-215.69E-22001.068959332118.84955592150.00602.47E-23

70000.0070000000000000000.0070000.733038285821.99114857511.09550852834986295905063160000000.000000000500.00000000050.0070000.00701.80E-215.69E-22001.095508528321.99114857510.00702.88E-23

80000.0080000000000000000.0080000.83775804125.13274122871.12731463949769412620468140000000.00000000100.0000000010.0080000.00801.80E-215.69E-22001.127314639425.13274122870.00803.29E-23

90000.0090000000000000000.0090000.942477796128.27433388231.16496662326073517641501150000000.000000000600.00000000060.0090000.00901.80E-215.69E-22001.164966623228.27433388230.00903.70E-23

100000.010000000000000000.010001.047197551231.41592653591.209199576213497334559487800000000.000000001400.00000000140.010000.0101.80E-215.69E-22001.209199576231.41592653590.0104.12E-23

200000.020000000000000000.020002.094395102462.83185307182.418399152313497334559487800000000.000000001400.00000000140.020000.0201.80E-215.69E-22002.418399152362.83185307180.0208.23E-23

300000.030000000000000000.030003.141592653694.247779607725642546754255300274776310206491000000.000000106200.000000002800.00000000280.030000.0301.80E-215.69E-22003.141592653694.24777960770.0301.23E-22

400000.040000000000000000.040004.1887902048125.6637061436-4.836798304613497334559487800000-000.000000001400.00000000140.040000.0401.80E-215.69E-22004.1887902048125.66370614360.0401.65E-22

500000.050000000000000000.050005.235987756157.0796326795-6.0459978808-5724054125572580000-00-0.00000000060-0.00000000060.050000.0501.80E-215.69E-22005.235987756157.07963267950.0502.06E-22

600000.060000000000000000.060006.2831853072188.4955592154-2564254675425530027477631020649100000-0.000000106200.000000002800.00000000280.060000.0601.80E-215.69E-22006.2831853072188.49555921540.0602.47E-22

700000.070000000000000000.070007.3303828584219.91148575138.4643970331-498629590506316000000-0.00000000050-0.00000000050.070000.0701.80E-215.69E-22007.3303828584219.91148575130.0702.88E-22

800000.080000000000000000.080008.3775804096251.32741228729.673596609213497334559487800000000.000000001400.00000000140.080000.0801.80E-215.69E-22008.3775804096251.32741228720.0803.29E-22

900000.090000000000000000.090009.4247779608282.743338823125642546754255300-721116857899503000000.00000010620-0.00000000730-0.00000000730.090000.0901.80E-215.69E-22009.4247779608282.74333882310.0903.70E-22

1000000.10000000000000000.100010.471975512314.159265359-12.0919957616-5724054125572580000-00-0.00000000060-0.00000000060.10000.101.80E-215.69E-220010.471975512314.1592653590.104.11E-22

2000000.20000000000000000.200020.9439510239628.31853071824.1839915231-572405412557258000000-0.00000000060-0.0000000006

3000000.30000000000000000.300031.4159265359942.4777960769-25642546754255300-18492535930011900000-0.00000010620-0.00000000190-0.0000000019

4000000.40000000000000000.400041.88790204791256.6370614359-48.3679830462-5724054125572580000-00-0.00000000060-0.0000000006

5000000.50000000000000000.500052.35987755981570.796326794960.4599788078-977268116386482000000-0.0000000010-0.000000001

6000000.60000000000000000.600062.83185307181884.9555921539-25642546754255300-18492535930011900000-0.00000010620-0.00000000190-0.0000000019

7000000.70000000000000000.700073.30382858382199.1148575129-84.643970330911935763461837500000-000.000000001200.0000000012

8000000.80000000000000000.800083.77580409572513.274122871896.7359660925-572405412557258000000-0.00000000060-0.0000000006

9000000.90000000000000000.900094.24777960772827.4333882308-3771159070267650-4653113923953820000-0.00000001560-0.00000000050-0.0000000005

100000010000000000000001000104.71975511973141.5926535898-120.9199576156-9772681163864820000-00-0.0000000010-0.000000001

AT LOW FREQUENCIES

FrequencyNon thermalThermalCapacitive sensorTotal noise capacitiveTotal noise opticalShot noise ASTRODU0 ASTRODInvSIncU0STRAIN ACC NOISE (CAP)STRAIN ACC (OPT)GRAV WAVE (CAP,ASTROD)GRAV WAVE (OPT,ASTROD)

0.00011.26E-169.68E-1700.31415926541.01664073854.25E-213.27E-217.76E-256.01E-25

0.000038.40E-178.00E-172.10E-162.40E-16000.09424777961.00148197639.00E-204.35E-201.61E-237.76E-24

0.000011.00E-163.10E-155.80E-163.16E-15000.03141592651.00016451231.07E-171.05E-171.90E-211.87E-21

0.0000031.60E-163.30E-141.90E-153.31E-14000.0094247781.00001480461.24E-151.24E-152.21E-192.21E-19

Sheet2

Sheet3

Picowatt and femtowatt weak light phase locking LISA needs to achieve weak phase locking of the order of 85 pW. Because of longer armlengths ASTROD I and ASTROD have to probe weak phase locking of the order of 100 femtowatts (assuming 1 W lasing power from far spacecraft).u (t)

ud (t)

(2)

Output signal I2 (t)

Input signal I1 (t) (1)

PD

LF

Far-end Laser ~100 fwVCO

Laboratory research on weak light phase locking for ASTROD14

Low Power Beam Intensity (measured using oscilloscope)

20 nW

2 nW

200 pW

20 pW

2 pW

High Power Beam IntensitymW

2

2

0.2

0.2

0.2

Low Power Intensity Measured by Lock-in Amplifier

20.9 nW

2.15 nW

153 ~247 pW

N/A

N/A

r.m.s. Error signal VrmsmV

2.01

2.06

2.29

2.03

2.70

r.m.s Phase errorrad

0.0286

0.057

0.2

0.16

0.29

Phase-locking time

Longer than observation duration

Longer than observation duration

> 2 hours

> 2 hours

1.5 mins

Results on weak light phase locking for ASTROD (20 pW)

ConclusionsASTROD is a multi-purpose space mission employing pulse and interferometric ranging to measure relativistic and solar system parameters, and low-frequency gravitational waves. A ten-fold improvement in acceleration noise would allow us to reach relativistic parameter uncertainties at the ppb level.Close collaboration among the international scientific community is needed to achieve the scientific objectives and technological challenges required by ASTROD!

References[1] A. Bec-Borsenberger, J. Christensen-Dalsgaard, M. Cruise, A. Di Virgilio, D. Gough, M. Keiser, A. Kosovichev, C. Lmmerzahl, J. Luo, W.-T. Ni, A. Peters, E. Samain, P. H. Scherrer, J.-T. Shy, P. Touboul, K. Tsubono, A.-M. Wu and H.-C. Yeh, Astrodynamical Space Test of Relativity using Optical Devices ASTROD --- A Proposal Submitted to ESA in Response to Call for Mission Proposals for Two Flexi-Missions F2/F3, January 31, 2000; and references therein.[2] W.-T. Ni ASTRODan overview., Int J. Mod. Phys D. vol. 11 No 7 (2002) 947-962; and references therein. [3] W.-T. Ni, ASTROD and ASTROD I: an overview, General Relativity and Gravitation, Vol. 37, submitted, 2006.[4] Wei-Tou Ni, Henrique Arajo, Gang Bao, Hansjrg Dittus, Tianyi Huang, Sergei Klioner, Sergei Kopeikin, George Krasinsky, Claus Lmmerzah, Guangyu Li, Hongying Li, Lei Liu, Yu-Xin Nie, Antonio Pulido Patn, Achim Peters, Elena Pitjeva, Albrecht Rdiger, tienne Samain, Diana Shaul, Stephan Schiller, Sachie Shiomi,M. H. Soffel, Timothy Sumner, Stephan Theil, Pierre Touboul, Patrick Vrancken, Feng Wang, Haitao Wang, Zhiyi Wei, Andreas Wicht, Xue-Jun Wu, Yan Xia, Yaoheng Xiong, Chongming Xu, Jun Yan, Hsien-Chi Yeh, Yuan-Zhong Zhang, Cheng Zhao, and Ze-Bing Zhou ASTROD and ASTROD I: Progress Report Journal of Physics: Conference Series 32 (2006) 154-160. Sixth Edoardo Amaldi Conference on Gravitational Waves [5] Ni W-T, Bao Y, Dittus H, Huang T, Lmmerzahl C, Li G, Luo J, Ma Z, Mangin J, Nie Y, Peters A, Rdiger A, Samain , Schiller S, Shiomi S, Sumner T, Tang C-J, Tao J, Touboul P, Wang H, Wicht A, Wu X, Xiong Y, Xu C, Yan J, Yao D, Yeh H-C, Zhang S, Zhang Y and Zhou Z 2003 ASTROD I: Mission Concept and Venus Flybys Proc. 5th IAA Int. Conf. On Low-Cost Planetary Missions, ESTEC, Noordwijk, The Netherlands, 24-26 September 2003, ESA SP-542 79-86; ibid 2006 Acta Astronautica 58 in press [6]. S. Shiomi, and W.-T. Ni. Acceleration disturbances and requirements for ASTROD I. Submitted to Class. Quantum Grav, in press.[7] Pulido Patn A and Ni W-T 2006 The low-frequency sensitivity to gravitational waves for ASTROD General Relativity and Gravitation 37 in press; and references therein[8]. Ke-Xun Sun et al. Advanced gravitacional referente sensor for high precision space interferometer. Class. Quantum Grav. 22 (2005) S287-S296.[9]. C.C. Speake, S.M. Aston, Class. Quanutum Grav. 22 (2005) S269-S277)[10] F. Acernese et al. Class. Quantum Grav. 22 (2005) S279-S285.[11] S. Shiomi 2005. Acceleration disturbances due to local gravity gradients in ASTROD I, Journal of Physics: Conference Series 32 (2006) 186-191. Sixth Edoardo Amaldi Conference on Gravitational Waves[12] D. Shaul , T. Sumner, G. Rochester, Coherent Fourier components in the LISA measurement bandwidth from test mass charging: Estimates and suppression, International Journal of Modern Physics D, 14, pp51-71.[13] Ke-Xun Sun, Brett Allard, Saps Buchman, Scott Williams and Robert L Byer, LED deep UV source for charge management of gravitational reference sensors, Class.Quantum Grav. 23(2006) S141-S150.[14] An-Chi Liao, Wei-Tou Ni, Jow-Tsong Shy, Int. J. Mod. Phys. D 11, 1075 (2002)

Purple Mountain Observatory (Nanjing, China)

Back-up slides

Acceleration disturbancesPosition dependent (stiffness terms): a) sensor readout noise and b) external environmental disturbances affecting the spacecraft including thruster noise.Direct acceleration disturbances: a) environmental disturbances and b) sensor back action disturbances. For sensing proof mass-spacecraft relative displacement and control actuation both capacitive andor optical sensing will be considered.

Efforts towards optical sensing I Optical lever. A test mass displacement induces a transversal beam displacement which is detected by the position sensor. Relevant noise sources: a) shot noise and b) amplifier current noise (-1/2). Back action force disturbance depends on power fluctuations. In Acernese et al. [10] they achieve displacement readouts of the order of 10-9 m Hz-1/2 down to 1 mHz.

Efforts towards optical sensing IIFigure on the right side shows a GRS where the test masses are merged into a spherical proof mass [8]. They consider all-reflective grating beam splitters, minimizing optical path errors due to temperature dependence refractive index. They demonstrate an optical sensing of 30 pmHz-12.

Efforts towards optical sensing IIIIn Speake et al. [9] a prototype bench top polarization-based homodyne interferometer based on wavelength modulation technique achieve a shot limited displacement sensitivity of 3pmHz-1/2 above 60Hz (using 850 nm VCSEL with 60 nW optical power).

Experimental set-up

ASTROD I spacecraft: general featuresCylindrical spacecraft with diameter 2.5 m, 2 m height, and surface covered with solar panels. In orbit, the cylindrical axis is perpendicular to the orbit plane with the telescope pointing toward the ground laser station. The effective area to receive sunlight is about 5 m2 and can generate over 500 W of power.The total mass of spacecraft is 300-350 kg. That of payload is 100-120 kg.Science rate is 500 bps. The telemetry rate is 5kbps for about 9 hours in two days.

ASTROD I spacecraft schematic design

ASTROD I science objectivesTesting relativistic gravity and the fundamental laws of spacetime with three-order-of-magnitude improvement in sensitivity;Improving astrodynamics with laser ranging in the solar system, increasing the sensitivity of solar, planetary and asteroid parameter determination by 1-3 orders of magnitude;Improving the sensitivity in the 5Hz 5mHz low frequency gravitational-wave detection by several times (Auxiliary goal).

Proof mass acceleration disturbances

Sources of disturbances Expressions Frequency dependence Noise (units 10-16 ms-2Hz-1/2)

Cosmic rays

Residual gas

Magnetic susceptibility I (().

Magnetic susceptibility II (()

Permanent magnetic moment

Lorentz I.

Lorentz II.

Radiometer effect

Out gassing effect

Thermal radiation pressure

Gravity Gradient

Total proof mass acceleration noise at 0.1 mHz (ms-2Hz-1/2)

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Back action disturbances (Capacitive sensing)

Source of disturbances Expressions Frequency dependence Noise in units (10-16 ms-2 Hz-1/2)

Quantization

Dielectric losses

Voltage

Charging-Voltage 1

Charging-Voltage 2

Charging

Total sensor back action disturbance (capacitance)

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Coherent Fourier components

Arise due to steady build up of charge on the test mass Coulomb:Lorentz:

Coherent Charging Signals (CHS) CHS due to Coulomb forces are due to geometric (machining accuracy) and voltage offsets (non-uniformity in the sensor surfaces, to minimise work function differences, patch effects, etc) in the capacitive sensor.These signal increases at low frequencies.The magnitude of these signals have been shown to compromise the target acceleration noise sensitivity of LISA (see D. N. A. Shaul [9])Ways of dealing and/or suppressing these signals are also discussed in [9].

Discharging schemesAccumulation of charge in the test mass induces acceleration disturbances through Lorentz and Coulomb interactions (if employing capacitive sensing).Position dependent Coulomb forces also contributes to the coupling between the proof mass and the spacecraft.Discharging periodically the proof mass to maintain this signals under the allowable limits, introduce coherent Fourier components as mentioned before.These CHS can spoil the sensitivity for a mission as LISA and therefore for ASTROD. Therefore there is a need for looking into continuous discharging schemes to suppress CHS.Recently, a deep UV LED as the promising light source for charge management was identified by Ke-Xun Sun et al. (Hansen Experimental Physical Laboratory, Stanford University, CA 94305-4085, USA). This system could have advantages over the more traditional mercury lamp-based system in three key areas: power efficiency, lower weight and flexible functionalities including AC operation out of the science measurement band

Stiffness termsIn the case of employing optical sensing the stiffness will be limited by gravity gradients. In a preliminary study for ASTROD I a stiffness value associated to gravity gradients when considering a cylindrical spacecraft and a parallelepiped proof mass is about 3.510-10 s-2 and 5.710-8 s-2 when the proof mass is enclosed in a box (as in the case of capacitive sensing).

PM-spacecraft stiffness Expressions Parameter dependence Units (s-2) 0.1mHz

Image charges

Applied voltage

Patch fields

Gravity Gradient

Induced magnetic moment

Magnetic remnant moment

Total stiffness (Capacitance)

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ASTROD GRS parameters

Paraneter values used in the acceleration noise estimates.

Proof Mass

Mass (kg) 1.75

Density (kgm-3)

Cross Section (m2)

Temperature (K)

Magnetic Susceptibility: (

Permanent Magnetic Moment: Mr (Am2Kg-1)

Maximum charge build-up: UV light continuous discharging

Velocity [ms-1]

Electrostatic shielding factor (e

Magnetic shielding factor (m

Optical bench thermal shielding factor (TS

Residual gas pressure

Magnetic fields.

Local Magnetic field

Local Magnetic field gradient

Fluctuation in local magnetic field

Interplanetary magnetic field

Interplanetary magnetic field gradient

Gradient of time-varying nagnetic field

Capacitive sensing

Capacitance Cx [pF] 6

Capacitance to ground Cg [pF] 6

Total capacitance C [pF] 36

Gap d [mm] 4

Average voltage across opposite surfaces Vx0 [V] 0.5

Proof mass bias voltage VM0 [V] 0.6

Voltage difference to ground Vx0-Vg=V0g [V] 0.01

Voltage difference between opposite faces Vd [V] 10-4

Fluctuation voltage difference (Vd [V Hz-1/2]

Residual dc bias voltage on electrodes V0 [V] 10-2

Loss angle (

Gap asymmetry (d [(m] 10

Quantization

Net force on the proof mass: Fx0 [N]

Binary digit: N [bits] 16

Sampling frequency: (s [Hz} 100

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ASTROD I accelerometer parameters (1)

ASTROD I accelerometer parameters (2)

ASTROD I proof mass acceleration noise and back action

ASTROD I stiffness terms

ASTROD I requirements compared to LISA

Relativistic parameter uncertainties for ASTROD IAssuming 10 ps timing accuracy and 10-13 ms-2Hz-1/2 ( = 0.1 mHz), a simulation for 400 days (350-750 days after launch) we obtain uncertainties for, and J2 about 10-7, 10-7 and 3.810-9.