Technology Readiness Level:

Transcription

1 Technology Readiness Level: We plan to raise the TRL of the model with an acceleration noise performance requirement of < m sec -2 Hz -1/2 at frequencies between 1 mhz and 1 Hz, from TRL 3 to TRL 4, i.e. a lab model ready for environmental testing. At the proposed requirement of m sec -2 Hz -1/2 for the sensor, maps of the Earth s gravitational potential could be made with higher resolution than GRACE, enabling improved studies of large scale geopotential variations and better large scale baselines for more local interferometry radar studies of subsidence and other phenomena. Since drag-free sensors using spheres have already been successfully flown on the TRIAD and GPB missions, Technology Readiness at Level 3 is well established. However, none have flown with the wide gaps needed to reach high performance, or with low force optical readout. Analytical and experimental critical function has been reported in the literature [6,7,9,19,20]. For our design the TM position readout is considered a critical function to be demonstrated in the laboratory and initial successful results have been documented [8, 12]. Information on the status of the readout and other components is given below. 1. Test Mass (TM) Mass Unbalance and Non-Sphericity The proposed TM is the same as used in MGRS, a 1.25 cm radius sphere of beryllium copper with a mass of 0.07 kg. Table II shows the magnetic properties of magnetically clean BeCu acquirable from assorted vendors. The specifications are appropriate for performances better than two to four orders of magnitude from the present proposal and thus will significantly reduce the requirements on the magnetic properties of the satellite. A thin layer of magnetic shielding material forms the outer layer of the DFPA unit. The acceleration goal a DFPA = ms -2 and the SRP acceleration a SRP 10-7 ms -2. The requirements for the TM mass unbalance and surface uniformity are given by: μ Max = ν Max = 1 2 d 4 a DFPA a SRP μ Max = ν Max 1.5μm (1) This is an easily achievable uniformity and readily available from commercial manufacturers for specs a factor of 5 better. Note that the above maximum estimate for and assumes an unrealistically worst case, with the TM immobile in the satellite frame while at the same time centered around the center of mass. The TM and the inner surface of its housing will be coated with TiC, a standard technology successfully used for the MGRS. The residual gas in the accelerometer will spin down the TM from any collisional spin-up by the un-caging system with a time constant given by: τ = 3 m TM 10 P r2 k BT TM P 10 4 Pa T 300K m 2π m g kg τ 5.5 yr (2) g P, T, and m g are the pressure, temperature and molecular mass of the residual gas and k B the Boltzmann constant. With a spin-down time constant of years, the TM will continue to tumble through the life of any mission. A spin-up effect will be caused by the torque due to the solar radiation pressure preload on the TM mass unbalance: ω TM 1 2 μ m TM a SPR I TM = 5 μ a SPR 4 r2 ω TM sec 2 2 Hz = TM yr This effect will modify only slightly the initial tumbling of the TM, as the torque is not fixed with respect to the TM and the angular velocity increase per year is very small. Note that the TM tumbling provides a pseudo-random averaging process for all disturbance mechanisms caused by TM and TM-housing inhomogeneity, imperfections, asphericity, charge patches, etc. This (3)

2 averaging is expected to reduce the calculated torques by at least an order of magnitude. To conclude, the BAA driven requirements <3x10-5 emu and < 5x10-6 emu/g are at least two orders of magnitude less demanding than the state of the art and make the TM procurable from commercial vendors. This reduces the risk in one of the key DFPA components and enables a lower cost, and small volume design leveraging commercial products. Table II. Magnetic Properties of Commercial BeCu Remnant Susceptibility Material Supplier moment (emu) (emu/g) BeCu 125 Materion 1.7x x10-7 binary BeCu NGK Berylco 1.9x x10-8 Ames Research binary BeCu Iowa State 9.7x x10-8 BeCu 3HP Materion 6.5xl x Electrostatic Suspension For the maximum nominal acceleration of 10-7 ms -2 Hz -½ the bandwidth of the suspension system will have to be above 0.5 Hz. We plan to use a 0.5 Hz system, similar to Gravity Probe (GP-B). Six suspension electrodes with a r e = 1.25 cm radius each, in three orthogonal pairs on the faces of the cubic housing, will have a d 0 1 cm spacing and a minimum capacitance of about 1 pf to the TM. For the m = Kg mass of the beryllium copper r = 1.25 cm radius TM the acceleration provided by one electrode a e is given by: F P a P m = C 0 d 0 V P 2, a P 10 6 m s 2, V P = a P m d 0 C 0 27V (4) Thus V P, the preload voltage of the suspension system in operational mode, will be 27 V, with a secondary 100 V preload for TM levitation, capture and initialization. The suspension voltage change V e corresponding to the ms -2 detection requirement is given by (for two electrodes on one axis): δv e = V PL 4 a DFPA a P 1 mv (5) an easily achievable measurement level. The GP-B Gyro Suspension System (GSS) is our heritage for the DFPA electro-static suspension. The GSS was fully demonstrated in flight. Figure Error! No text of specified style in document.-1 shows one of the four actual flight GSS units. Unlike the DFPA, the GSS was required to operate both in flight and in ground test covering an operational range of over 9 orders of magnitude. Table III compares a summary of the GP-B GSS functions with the much simpler DFPA requirements. Note that all difficult requirements and much of the functionality (including the high voltage ground suspension) are not needed, thus requiring only a very simple version of the GSS for DFPA. This enables size, weight, and power savings to be realized on DFPA. Table III. Electrostatic suspension system requirements: GP-B flight versus DFPA Requirement / System GP-B ( flight) DFPA Resolution as accelerometer < g < g Suspension system Hybrid digital/analog Digital only Operational range 10-9 g to 1 g (0.05 V -1,000 V) (9 orders of magnitude) 10-8 g to 10-7 g (24 V 100 V) (2 orders of magnitude) Adaptive control algorithm Adaptive LQE digital control Not required; PDI only

3 Backup TM suspension system 3 backup, analog PD controllers Not required TM position sensing Capacitive to 0.15 nm/ Hz Not required (see DOSS) TM spin alignment Preload modulation at SC roll Not required TM charge measurement 3 axes force modulation Not required Figure Error! No text of specified style in document.-1. GP-B flight Gyro Suspension System Figure Error! No text of specified style in document.-2. Two layer MGRS/DFPA prototype and CAD. 4. Thermal Control The largest contribution to the disturbance budget is the radiometer effect (pink line A8 in Figure Error! No text of specified style in document.-3, left). Residual gas pressure in the DFPA is about 10-4 Pa and we require that temperature fluctuations and gradients be less than 50 mk. Temperature control is achieved with a double enclosure system, which is identical to the MGRS design. Note that the MGRS requires active temperature control for noise performance at the level, while the DFPA does not as passive thermal control is more than sufficient for the 10-9 level. This allows for SWaP savings for DFPA while still exceeding BAA performance requirements. Figure Error! No text of specified style in document.-2 shows the two-layer MGRS prototype and the CAD drawing, identical to the DFPA design (caging not shown). 5. Size Weight and Power DFPA size, weight, and power are summarized in Table V. Accelerometer size and weight easily meet goal values. Accelerometer power will depend on the level of thermal control required for the sensor which is a function of the satellite environment and thermal interfaces. We allocate 5W for sensor thermal control, the DOSS and charge control system, and the electro-static suspension. This brings the total accelerometer power to 30 W which meets the goal value. Table V. DFPA Size, Weight and Power Parameter Capability Threshold Goal Size Sensor 1,500 cm 3 Electronics cm 3 Total 2,400 cm 3 10,000 cm 3 8,000 cm 3 Weight Sensor 2 kg Electronics + 3 kg Total 5 kg 10 kg 8 kg Power

4 Sensor (allocation) 5 W Electronics + 25 W Total 30 W 50 W 30 W Expected performance of the DFPA The DFPA system is based on the MGRS design, whose detailed error budget is given in Figure Error! No text of specified style in document.-3. The DFPA requirement is defined as ms -2 up to 1 Hz and ms -2 Hz -1/2 above 1 Hz, a factor of 100 less demanding than the MGRS design (and hence not shown in the figure). Dominant contributions to the total MGRS error are due to two terms: a. The radiometer effect caused by the unbalanced residual gas pressure under thermal inhomogeneity around the TM (pink line A8 in Figure Error! No text of specified style in document.-3 left). Thermal and residual pressure requirements were discussed in Section 2.6. b. The stiffness term, equivalent to a spring between TM and housing, caused by motion of the TM in the gravitational and electromagnetic fields of its surroundings (black line A0 in Fig. 2-6, left and black line total in Fig. 2-6, right). For MGRS, the stiffness term is mainly due to the interaction of the residual magnetization of the TM with the spacecraft magnetic field (blue line A0mag in Figure Error! No text of specified style in document.-3, right). For the DFPA the applied voltage term (yellow line A0s3 in Figure Error! No text of specified style in document.-3, right) will be enhanced by less than 10 3, resulting in a disturbance of <10-11 ms -2. Comparing to the MGRS error budget, we expect the DFPA to exceed the derived 2x10-8 ms - 2 Hz -1/2 requirement and significantly exceed the 5 ng and 20 ng noise floor performance goals. We analyze the expected performance of the accelerometer in the frequency band from below 10-2 Hz to about 10-5 Hz. The three contributions to the specific force on the accelerometer TM are the acceleration by the electrostatic suspension system, the specific forces acting between the TM and the satellite, and the noise in the optical read-out that senses the position of the TM relative to its housing. The requirement of 20 m position accuracy integrated over 12 hours, results in a requirement for a residual acceleration of less than m s -2. For the GPS constellation the dominant acceleration is caused by the solar radiation pressure at m s -2, with the nominal values of the accelerations due to the non-gravitational forces estimated at: Force Acceleration (m s -2 ) Solar Radiation Pressure Thermal re-radiation y-bias Earth Radiation Pressure Along track Antenna recoil While 20% accuracy in measuring the residual acceleration of a GPS spacecraft should suffice to meet the 20 m in 12 hours requirement, we propose to design the accelerometer to a m s -2 requirement, m s -2 goal, thus allowing a margin of 20 by requirement and 200 by goal. The sum of all spacecraft to TM forces will be maintained below m s -2 by design, modeling, and testing, and the 3 axes accelerometer forces will be measured and calibrated to 1% or better. Continuous measurements of the 3 rotational coordinates of the satellite with accuracies of less or equal to one degree will be required.

5 MGRS Design Radiometer effect dominates MGRS Design Magnetic term dominated by S/C fields Figure Error! No text of specified style in document.-3. MGRS acceleration noise complete error tree (left) and the detailed stiffness term (right). The effect of the forces due to the TM on the satellite may be ignored since they are considerably smaller than the other forces acting on the satellite. Two significant effects are due to any displacement d of the TM center from the satellite center of mass. These effects are measurable and are going to be instrumental in the on-orbit calibration of the accelerometer. The first effect is the gravity gradient force F gg m a gg, caused by the TM (of mass m) following a different orbit than the satellite. The direction and time dependence of this force is determined by the orbit and the attitude and roll control of the satellite, its instantaneous magnitude is proportional to the d projection on the gravity gradient direction and for d 1m, a gg m s 2. The second effect, F cf m a cf, is due to the centrifugal force on the TM caused by rotation of the satellite around its center of mass. For a 12 hour GPS orbit with an Earth pointing satellite, a cf m s 2. As orbit radius and attitude and rotation angles will be known to better than 10-3, a gg and a cf will provide two accelerometer calibration methods to at least a factor of ten better than the m s -2 design requirement. With this assumption, the equations of motion for the geometric center of the accelerometer housing ( housing that is rigidly connected to spacecraft), c, and the centerof-mass of the accelerometer test mass (TM), r, are: M d2 c = F dt 2 ext, m d2 r dx dt2 + β + kx = f + f dt f (6) F ext represents the external forces acting on the satellite. M and m are the masses of the satellite and the TM, respectively. The vector x is the displacement of the center-of-mass of the TM from the geometric center of the housing, r = c + x. Here the geometric center of housing is assumed to coincide with the pick-off null for the position readout system. Any very low frequency offsets do not change the analysis that follows, and high frequency offsets are included as noise in the position readout system. The coefficients and k are the damping coefficient and the effective spring constant for any velocity dependent or position dependent forces acting on the TM due to the housing. For the proposed ESS system we can

6 set β = 0. All disturbance forces acting on the TM are represented by f, with the exception of the feedback control force applied by the electrostatic suspension system (ESS), which is denoted f f, where the goal is to minimize f and accurately measure f f such that: t+12h t+12h ( F ext f f t M m dt2 ) ( f f t m dt2 ) 10 2 (7) Solutions of these differential equations are discussed below, where the force due to the ESS, f f, is determined by the ESS control system. From equation (6) the equation of motion for the TM is: m d2 x + kx = f + f dt 2 f m d2 c (8) dt 2 The dynamic solution for the equations of motion along any axis may be found by taking the Laplace transform of the equations of motion. The Laplace transform of the ESS feedback force is given by: f f (s) = H(s) [x(s) n(s)] (9) H(s) is the Laplace transform of the compensation network and n(s) is the Laplace transform of the optical read-out position noise. These equations may be solved to find the relative position of the housing and the TM, the feedback force supplied by the ESS control system, and the acceleration of the housing and the TM. Below the bandwidth of the ESS control system, the accelerations of the center of the housing is given by: s 2 c(s) = F ext(s) + f(s) G(s)H(s)M m (ms2 + k) n(s) (10) m where m G(s) = (11) M(ms 2 +k) The closed loop expression for the feedback force may be found by substituting expression (10) into equation (8). Then, below the bandwidth of the servo system, the feedback force on the supported accelerometer becomes: f f (s) = ms 2 c(s) f(s) (ms 2 + k)n(s) (12) This result shows that below the bandwidth of the servo system, the control effort signal (the accelerometer read-out) is dominated by the acceleration of the housing/satellite and has contributions from the residual forces acting on the accelerometer and the noise in the position sensing bridge. The capacitance C 0 and spacing d 0 of the TM to each of six electrodes (in three orthogonal pairs) is about C 0 1pF, d 0 1cm. The mass m of the BeCu spherical TM of r = 1.25cm radius of is m = 0.070kg. With a margin factor of ten the acceleration preload of the ESS is a P 10 6 m s 2, corresponding to a maximum electrode voltage V P of: F P a P m = C 0 d 0 V P 2, V P = a P m d 0 C 0 27V (4 ) Adding an additional margin of four to account for the TM being off housing center we establish the maximum required ESS voltage per electrode of V Pmax 100V. We next estimate the two principal sources of ESS noise caused by a) the ESS spring constant k and b) the DAC generating the suspension voltage V P. a) The spring constant k is given by: k = 2 C 0 d 0 2 V P 2 = kg s 2 (13)

7 The noise of the TM position read-out, the Differential Optical Shadow Sensor (DOSS) is n DOSS 10nm/ Hz. The DOSS noise and the spring constant combine to noise acceleration: a k = k n DOSS = m s 2 / Hz m (14) b) The noise acceleration on the TM due to the DAC is given by: a DAC = a P δv q /V P, δv q q/ 12 ν c (15) where δv q is the quantization noise, q is the DAC quantization step and ν c is the DAC conversion rate. For a 16 bit DAC converter operating over a range of V Pmax = 100V with a conversion rate of 50 Hz, the quantization noise is δv q = 60μV/ Hz, resulting in: a DAC = m s 2 / Hz (16) The acceleration noise due to the 1μW/ Hz variation in optical power of the DOSS is, as expected very small: a DOSS m s 2 / Hz (17) Charge on the TM will be controlled to Q max 1pC. As charge management is passive there is no charge measurement noise. An asymmetry ΔC 3pF in the ground plane (housing) to TM will result in in a spring constant: k GP = ΔC d2 ( Q max 0 C 0 (18) and therefore in an acceleration noise: a GP = k GP n DOSS m ) 2 = kg s 2 = m s 2 / Hz (19) The patch effects are negligible at the 1 cm TM to housing spacing. By design or after calibration, the acceleration disturbances are well below the requirement and the goal of 10 9 m s 2 and m s 2 respectively. The two large contributions from the gravity gradient and the centrifugal forces are in effect the principal absolute calibration methods for the accelerometer. 8. Self Gravity Self-gravity from the surrounding spacecraft generates constant and variable biases for the accelerometer making it desirable to place the TM towards the center of the spacecraft. Ideally we would require that (point or distributed masses): GM i (r i r TM ) Gρ (r r i r TM ) i i dv < m a r i r TM 3 r i r TM 3 DFPA (20) In practice we require good knowledge of the placement of all masses in the spacecraft to the level: M(kg) < α DFPA (kg m 2 )r 2 (m 2 ) α DFPA 0.15kg m 2 (21) where DFPA is consistent with the BAA requirements (with a factor of 10 margin). Inevitable mass distribution unbalances will be compensated with weights placed on the outer thermal enclosure of the accelerometer at about 5 cm from the center of TM. For example, an

8 extraneous mass of 1 kg at 1 m would require a compensating mass of only 2.5x10-3 kg at 5 cm.