Paper published on New Astronomy, 10 March 1998; 3(3) 175-218 (PDF file)

Proposed noncryogenic, nondrag-free test of the equivalence principle in space

[a] A.M. Nobili *

[a] D. Bramanti *

[a] G. Catastini *

[b] E. Polacco *

[c] G. Genta *

[c] E. Brusa *

[d] V.P. Mitrofanov *

[e] A. Bernard

[e] P. Touboul *

[f] A.J. Cook

[g] J. Hough *

[h] I.W. Roxburgh *

[h] A. Polnarev *

[i] W. Flury *

[j] F. Barlier *

[e] C. Marchal

Received 3 April 1997; accepted 9 September 1997
Communicated by Francesco Melchiorri 

Abstract

Ever since Galileo scientists have known that all bodies fall with the same acceleration regardless of their mass and composition. Known as the Universality of Free Fall, this is the most direct experimental evidence of the Weak Equivalence Principle, a founding pillar of General Relativity according to which the gravitational (passive) mass  and the inertial mass  are always in the same positive ratio in all test bodies. A space experiment offers two main advantages: a signal about a factor of a thousand bigger than on Earth and the absence of weight. A new space mission named GALILEO GALILEI (GG) has been proposed (Nobili et al., 1995; GALILEO GALILEI, 1996) aimed at testing the weak Equivalence Principle (EP) to 1 part in  in a rapidly spinning () drag-free spacecraft at room temperature, the most recent ground experiments having reached the level of  (Adelberger et al., 1990; Su et al., 1994). Here we present a nondrag-free version of GG which could reach a sensitivity of 1 part in . The main feature of GG is that, similarly to the most recent ground experiments, the expected (low frequency) signal is modulated at higher frequency by spinning the system, in this case by rotating the test bodies (in the shape of hollow cylinders) around their symmetry axes, the signal being in the perpendicular plane. They are mechanically suspended inside the spacecraft and have very low frequencies of natural oscillation (due to the weakness of the springs that can be used because of weightlessness) so as to allow self-centering of the axes; vibrational noise around the spin/signal frequency is attenuated by means of mechanical suspensions. The signal of an EP violation would appear at the spin frequency as a relative (differential) displacement of the test masses perpendicularly to the spin axis, and be detected by capacitance sensors; thermal stability across the test masses and for the required integration time is obtained passively thanks to both the fast spin and the cylindrical symmetry. In the nondrag-free version the entire effect of atmospheric drag is retained, but a very accurate balancing of the test bodies must be ensured (through a coupled suspension) so as to reach a high level of Common Mode Rejection and reduce the differential effects of drag below the target sensitivity. In so doing the complexities of a drag-free spacecraft are avoided by putting more stringent requirements on the experiment. The spacecraft must have a high area-to-mass ratio in order to reduce the effects of nongravitational forces; it is therefore a natural choice to have three pairs of test masses (in three experimental chambers) rather than one as by Nobili et al. (1995) and the (mission called GALILEO GALILEI, 1996). The GG setup is specifically designed for space; however, a significant EP test on the ground is possible - because the signal is in the transverse plane - by exploiting the horizontal component of the gravitational and the centrifugal field of the Earth. This ground test is underway.

Printable version available (1368251 bytes) 

Table of contents

1. Introduction
2. Experiment setup and orbit choice
3. Self-centering in supercritical rotation
4. Axial centering and Earth tides
5. Inertial forces
6. Room temperature effects
7. The capacitance read out system
8. Coupling to higher mass moments of the test bodies
9. Electrostatic and magnetic effects
10. Initial unlocking in supercritical rotation
11. Conclusions

• Acknowledgement
• References

Body of article in one file

Background data sets

table1.tbl
Inertial properties of the rigid bodies constituting the FEM model 

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• (32786 bytes)
  Fig. 1. Schematic section through the spin axis of the spacecraft showing the spacecraft, the suspended laboratory and the three experimental chambers containing two test masses each. The spacecraft is a cylinder of  height and  base diameter. The figure is to scale and shows that the experiment can be performed in a very small and compact satellite (the area-to-mass ratio is . The central chamber contains two masses made of the same (dense) material for a null test. Each of the other two chambers contains two test masses of different materials, including low density ones. The numbers 1-8 are referred to in Section 3 . (to main text)
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  Fig. 2. Section through the spin axis of the spacecraft showing (not to scale) the spacecraft, the PGB laboratory and (for simplicity) only one experimental chamber. The PGB laboratory and the test masses are suspended with springs and their equilibrium positions can be stabilized by means of electrostatic active dampers (see Fig. 19 for a top view). The suspensions of the test masses also employ "elastic" gimbals (i.e. gimbals pivoted with torsion wires) on two movable rods for the balancing of inertial forces discussed in Section 5.2 . The axial position of each half of these rods can be finely adjusted by means of piezoelectric actuators (see also Fig. 13 ). The capacitive plates of the read out system, between the test masses, are attached to inch-worms for adjusting their distance from the surfaces of the test masses. (to main text)
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  Fig. 3. Noise reduction factor (i.e. amplitude of disturbing vibration at the suspended mass over amplitude of vibration at the suspension point) as function of frequency for a suspended laboratory of  and mechanical suspensions with stiffness  and quality factor of order 1.  and all the rest of the system (spacecraft plus test masses). (to main text)
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  Fig. 4. Section across the spin axes of two test bodies with their centres of mass displaced by a distance  due to an equivalence principle violation  in the field of the Earth. For  (and with mechanical properties as given in Section 3.2 ) we get . The centres of mass of the two bodies rotate independently around  and  respectively. The direction of the displacement  changes with respect to inertial space as the satellite orbits around the Earth in . (to main text)
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  Fig. 5. A mathematical model of two axially symmetric bodies of masses  and  coupled by springs of stiffness k.  and  are the inertial and the rotating plane respectively. z and  are the complex variables in the two planes. (to main text)
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  Fig. 6. Sketch of the FEM (Finite Element Method) model of GG. The figure shows the final FEM model used to analyse the rotordynamics of the active controlled system with DYNROT. The beam elements have been drawn on the left side of the picture: the white parts correspond to zero mass beams with structural stiffness. The nodes are shown on the right side of the sketch, each node corresponding to two translational and two rotational degrees of freedom which describe the lateral dynamics. In order to provide an understandable overview each node has been located on the corresponding beam element, instead of on the rotation axis of the satellite, as it actually is. Since each active damper has been connected to the central rod there are shorter beam elements near the gimbals and two nodes very close to each other. (to main text)
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  Fig. 7. Schematic representation of the various components of one experimental chamber placed next to one another in order to show the different kind of connections between them (springs, gimbals and electrostatic dampers). The outer and the inner test mass are respectively connected by spring elements (continuous line) to the pair of movable supports. Gimbals join the movable supports to the central rod, while the electrostatic dampers act between the test masses and the central rod (dotted line). (to main text)
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  Fig. 8. Mode shapes of GG satellite. The mode shapes found by the DYNROT FEM code can be divided into three types: the first includes 7 cylindrical modes forward and backward, the second 7 backward mainly conical modes and the third 8 conical forward modes. A sample of each set including the  and  forward modes and the  backward mode is shown by giving the position of the axis of rotation (continuous line) and the location of the nodes (the "" symbols). The x- is a coordinate along the spacecraft axis (in cm); the y- is an adimensional normalized coordinate (the mode automatically scales all the modes to the maximum value). (to main text)
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  Fig. 9. Electrostatic damping of whirling motions. The circular instability motions of the rotation axis of the inner shaft, which have the natural (low) frequency of the suspended masses , can be actively damped by means of the electrostatic force obtained by applying a voltage pulse V of short duration (for example for about one fourth of the spin period, i.e. about ) to each plate rotating at the (rapid) spin frequency  when it is passing through the position  before the point of its nearest approach to the inner shaft. This happens, for each plate, at a frequency which is equal to the spin frequency  minus the natural frequency . (to main text)
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  Fig. 10. Simple scheme of Earth tidal forces on two test bodies which rotate around the same axis but are displaced along it. The figure shows how the component of the tidal force towards the Earth changes phase by  every half orbital period of the satellite around the Earth. Only this component does produce a differential displacement of the centres of mass which can be recorded by the spinning capacitors. It is apparent that a differential force due to a violation of the equivalence principle would not change sign every  orbit and would not go to zero with the separation distance . (to main text)
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  Fig. 11. Qualitative representation, in the orbital plane and for one pair of test masses, of the differential displacements obtained from the synchronous demodulation of the 2-phase  signal. The x-axis is in the Earth-to-satellite direction and the vector  is the differential displacement, directed along the x-axis and constant in amplitude, of the two masses due to an EP violation. The perturbation  due to the initially unbalanced atmospheric drag will be found in the area between the two dotted lines crossing in P: the angle between them is about , and is due to the fact that the drag has a variable component in the radial direction because of the solar radiation pressure (of amplitude about  times the atmospheric drag and in the Sun-satellite direction). Smaller contributions to the  vector come from the Earth albedo, the Earth infrared radiation and, by a smaller amount, from a possible small eccentricity of the orbit. By finely adjusting the lengths of the suspension arms the point D is displaced up or down inside this area, and this balancing of the drag should be continued until D is as close as possible to P. In doing so, also the radial component is automatically balanced. The resonant variations of the drag (not shown) will oscillate inside the same area. The vector  is the instability due to the internal dissipation of the springs (Section 3.2 ), slowly rotating and increasing: it must be actively damped until Q is as close as possible to D (and P). The circle around point Q represents the error in the measurement due to the thermal noise of the mechanical oscillations in a few days of integration time. The actual values of all these quantities are discussed in the text. (to main text)
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  Fig. 12. Testing of suspension springs. By means of a torsion pendulum one can make ground tests of the longitudinal (a) and of the transversal (b) characteristics of the very thin and weak springs that will be used in orbit at zero g. One can test one spring at a time or, as shown in this figure, measure the differences between two of them. (to main text)
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  Fig. 13. The system of piezoelectric actuators placed in the two balancing rods. The  and  signs represent the intrinsic polarization of the actuators, i.e. how each one of them must be oriented when mounted. Control voltages are applied to the actuators (when they are applied with the opposite polarity they should not exceed a certain value, which however is relatively high, so as not to risk to depolarize the piezoelectrics): the sum  determines the relative axial position of the barycentres of the test masses and is used for axial centering (Section 4 ). The voltage difference  can be used to change the lengths of the four halves of the rods so as to balance out the effect of transverse inertial forces (in particular the along-track component of the air drag). (to main text)
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  Fig. 14. Outline of the read-out circuit. The two variable capacitors  and  and the two halves of the inductor L form an LC bridge whose output is proportional to the difference between the two capacitances. (to main text)
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  Fig. 15. Each capacitor of the read-out system (see also Fig. 16 ) is formed by two surfaces, one for each of the two grounded masses, and one plate, to which a sinusoidal voltage is applied. Any differential displacement of the test masses with respect to the plates causes a loss of balance of the system and therefore an output signal. (to main text)
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  Fig. 16. The surfaces of the capacitors before and after: a) a common mode displacement and b) a differential mode displacement. (to main text)
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  Fig. 17. Scheme of the inch-worm. Lateral piezoelectric actuators alternately fasten and release the extremities of the inch-worm to the sides of its container while the inner part is made to expand and contract by means of the other piezoelectrics. In this way the inch-worm can move on a relatively long path in successive very small steps. (to main text)
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  Fig. 18. In an equatorial orbit the direction of the Earth's magnetic field is not more than about  away from the perpendicular to the orbit. Therefore its effects in the plane of the orbit will be reduced to about  times those of the full field. These effects are distinguishable from a violation of the equivalence principle because they change sign every half orbit. (to main text)
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  Fig. 19. Top view of a set of four inch-worms actuators for locking and unlocking the suspended masses. Each mass needs two such sets placed at its two axial ends (see also Fig. 2 ). Between the inch-worms are the electrostatic plates used for active damping. The rod, hence the suspended masses, is locked during launch and until the spacecraft has reached the final spin angular velocity . Then the inch-worms equipped with pressure sensors sensitive to  are used for initial centering until the centrifugal forces detected by the pressure sensors become smaller than the forces that can be generated by the electrostatic plates. At this point the inch-worm will be retracted and the electrostatic system will complete the centering and will keep it stable. (to main text)

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Last  edited   May 14, 1998