Human's understanding of gravity began with the observation of planetary motion.Johannes Kepler, a German astronomer, summarized a large number of detailed observation data from Tycho Brahe, a Danish astronomerThree laws of planetary motionIt perfectly describes the motion law of the planets around the sun, but does not point out the reason why the planets move in an elliptical orbit.About half a century later, Isaac Newton proposedLaw of universal gravitation。This law not onlyKepler's three lawsIt provides a dynamic explanation, and points out that the force that keeps the planet moving in an elliptical orbit is essentially the same as the force that makes the apple fall on the earth.This force is everywhere, from basic particles to cosmic objects, and is called "universal gravitation".
In 1687, Newton《Mathematical Principles of Natural Philosophy》The book "Mathematical Principles of Natural Philosophy" systematically introduces the law of universal gravitation. Its contents are as follows: any two particles in the universe have mutual attraction. Its size is proportional to the product of the mass m1 and m2 of the two particles, and inversely proportional to the square of the distance r between them. It is mathematically expressed as
Where the scale coefficient G is calledUniversal gravitational constant(Universal Gravitational Constant)。It is a universal constant and is not affected by the size, shape, composition and other factors of objects.The gravitational constant G is a basic physical constant closely related to theoretical physics, astrophysics and geophysics.It is closely related to celestial motion, celestial evolution and structural model [2].G plays a very important role in the modern theoretical research of particle and field theory, cosmology and gravitational physics.For example, the Planck length, time and mass describing the basic constant system of nature are given by different combinations of the three basic physical quantities Planck constant ħ, the universal gravitational constant G, and the speed of light c.
Since Newton's《Mathematical Principles of Natural Philosophy》Since its publication more than 300 years ago, the theoretical and experimental research of gravity has been one of the hot spots in the scientific community.Theoretical workers are committed to studying the nature and origin of gravity and its role in physics, trying to unify the four basic interactions.Experimenters put forward a series of questions about the law, such as: What is the exact value of G?Is it a constant or will it change with time and place?Is gravity strictly inversely proportional to the square of distance?Is it related to the composition of two objects?Is gravity related to the state of motion of objects?wait.Although the results of introducing G into the increasing discussion of physics and astrophysics will be different, in-depth research on G will help toGravitational interactionKnowledge of nature.Professor Cook, who was once the director of Cavendish Laboratory at Cambridge University, even raised the question whether G could be regarded as the basic constant in the measurement system like the speed of light in the international unit.He also proposed that if it is possible to define it as a basic constant, then there must be higher requirements for the measurement accuracy of G[1]。
Development history
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The early experiments on gravity research all used geophysical methods, whose purpose was to measure the average density of the earth, but the inherent errors related to geological characteristics of this method made it impossible to give a high precision.In 1798, Cavendish published a famous article entitled "Experiments to Determine the Density of the Earth" in the then Proceedings of the Royal Society (Philosophy).In his article, he introduced how to use the Torsion Balance made by Michell to study the universal gravitation between two objects in the laboratory, and for the first time, he accurately concluded that the density of the earth is 5.48 times that of water, from which later generations gave the corresponding G value of the experiment as
。Therefore, we often say that Cavendish is the first person in history to "weigh" the earth's mass, and therefore he became the first person in history to measureUniversal gravitational constantG scientists.
After Cavendish, the representative work of measuring G belongs to the experiment of Boys in 1895, whose principle is completely the same as Cavendish's torsion balance method.It was not until 1942 that their results were replaced by the results measured by Heyl using the torsion balance periodic method [25, 26].The biggest advantage of the torsion balance periodic method proposed by Heyl to measure the gravitational constant G is that the measurement of weak force is transformed into the measurement of time.Since the accurate measurement of time is relatively easy to achieve, the G value given by Heyl has a high precision. It can be said that the advent of his measurement results marks the beginning of the accurate measurement of the absolute value of G.There are still several groups in the world using this method to measure G.
In the second half of the 20th century, compared with other periods in the history of science, human beings carried out more research on measuring the gravitational constant.Scientists not only pay attention to the absolute value of the gravitational constant, but also to the changes of the gravitational constant with time and space and some anomalies related to gravity.Many scholars have made useful summaries of these experimental research works before the 1980s [27~31].
Difficulty in measurement
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In the past 200 years, people haveUniversal gravitational constantGreat efforts have been made in the measurement of G, but the improvement of the measurement accuracy of the gravitational constant G is very slow, which increases by an order of magnitude almost every century.The reason why the research progress in this field is so slow is well known.First, gravity is the weakest of the four basic interactions in nature.For example, the electromagnetic interaction between an electron and a proton is about 1039 times the gravitational interaction between them.Weak gravitational signals are easily annihilated by other interference signals, so the interference of electromagnetic force, ground vibration, temperature change and other factors must be overcome in the experiment, and the measurement must be carried out in some laboratories with special measures.Secondly, the universal gravitation cannot be shielded, so the inspection quality will inevitably be interfered by the gravitation of other objects except the gravitation mass specially set for the experiment, such as experimental instruments, experimental background quality, and experimental personnel.In addition, moving mass, such as vehicles passing near the laboratory and pedestrians, will bring gravitational disturbance to the experiment.Even in a very remote and quiet laboratory, changes in cloud pressure, rain, snow and other weather will interfere with the measurement results.Third, up to now, no definite relationship has been found between G and any other basic constant, so it is impossible to use other basic constants to indirectly determine the value of G, but only according toNewton's law of universal gravitation。Fourthly, the accuracy of the experiment is limited by the accuracy of the measuring instrument.At present, the measurement accuracy of G basically represents the level of existing machining and measurement.Finally, tools used to detect weak gravity, such as various forms of torsion scales and balances, have various parasitic coupling effects and system errors, which ultimately limit the improvement of measurement accuracy.
measuring method
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Space survey
The measurement of universal gravitational constant G can be roughly divided intogeophysicsMethodological measurement, spatial measurement and laboratory measurement.The geophysical method for measuring G uses large natural objects (such as mountains, mines and lakes with regular shapes) as the attractive mass [12, 13].The main advantage of this method is that the natural object as the attractive mass is very large, and the gravitational effect is obvious.However, because the scale, density and distribution of the attractive mass cannot be measured accurately, the accuracy of the experiment is relatively low.With the development of space technology, people expect to carry out G measurement experiments in space [49~53].The space measurement method can avoid two major problems encountered in the ground laboratory: one is the effect of the additional background gravitational field in the ground experimental environment, and the other is the interference of the ground vibration noise. However, as far as the current situation is concerned, the space measurement method G faces many new technical problems and is still under exploration.The common tools for measuring the gravitational constant G in the laboratory are precision torsion scales and balances.Compared with the geophysical method, the biggest advantage of the precision torsion balance is that the gravitational interaction between the test mass to be measured and the attraction mass is placed in the water plane orthogonal to the direction of the earth's gravity field, which greatly reduces the influence of gravity and its waves in the experimental design [54].The balance can tilt up and down in the vertical plane around the knife edge to detect the gravitational action in the vertical direction [55~57].Common measurement methods include direct tilt method, compensation method, resonance method, periodic method and free fall method[1]。
Torsion balance period method
The motion equation of a freely suspended torsion balance can be written in the form of a harmonic oscillator:
(2)
Where I is the moment of inertia of the torsion balance, γ is the damping coefficient, and k is the torsional elastic coefficient of the torsion wire.The eigenfrequency of the torsion scale is
(3)
Figure 1 Schematic diagram of torsion balance cycle method
If a large attraction mass is placed near the torsion scale, as shown in Figure 1 (a), when the connection of the attraction mass Ma, Mb is parallel to the balance position of the torsion scale (short range configuration), the attraction of the attraction mass to the inspection mass provides the torsion scale system with additional positive recovery torque, so that the total recovery torque increases, and its vibration frequency becomes
(4)
Where subscript n refers to the short-range configuration, Kn refers to the elastic coefficient of the suspension wire under this configuration, Cgn refers to the gravitational coupling constant determined by the mass distribution of the attraction mass and the inspection mass, and I refers to the moment of inertia of the torsion balance.As shown in Figure 1 (b), when the connection of the attraction mass Ma and Mb is perpendicular to the balance position of the torsion scale (remote configuration), the attraction of the attraction mass to the inspection mass provides an additional negative restoring moment for the torsion scale system, reducing the total restoring moment and changing its vibration frequency to
(5)
The subscript f indicates remote configuration.The G value is determined by measuring the period of the torsion scale in two configurations:
(6)
among
，
Is the anelastic effect of twisted wire [58].
The torsion balance periodic method for measuring G was proposed by Heyl in 1931 [25] and developed with his colleagues.In 1942, the G value measured by Heyl et al. [26] using the torsion balance period method wasInternational Committee on Scientific and Technological DataAs the first recommended value of G value in CODATA-73.In the subsequent adjustment of CODATA-86, Cohen and Taylor [33] further improved the measurement accuracy by using the torsion balance cycle method, so they were recommended as the new round of G value.The LANL-97, TR&D-98 (TR&D-96) and HUST-99 (HUST-05) collected by CODATA for the last three times all adopt the torsion balance cycle method.
The torsion balance period method is a dynamic measurement method, which takes advantage of the high sensitivity of the torsion balance, and converts some physical quantities that are difficult to measure (such as geometric angular displacement θ) into other physical quantities that are relatively easy to measure (such as time), which can achieve high measurement accuracy.The measurement of geometric parameters in the torsion balance period method, such as the moment of inertia I and the gravitational coupling constant Cg, does not involve the movement of the torsion balance, so it is relatively easy and convenient to measure.Compared with other experimental methods using torsion balance (such as direct tilting method), the periodic method is not very sensitive to the drift of the balance position of torsion balance.The torsion balance period method actually uses the principle of "difference", and all the same parameters (or the same change rule) in the near and remote configuration have the same effect on the experimental results.
Of course, the torsion balance periodic method also has some difficulties.The period of high sensitivity torsion balance is generally longer, so the measurement time is also longer, which puts forward higher requirements for the stability of the external environment.It takes a long time for the torsion balance to reach a relatively stable state, and it is necessary to measure the movement of the torsion balance in a long time interval to give an accurate frequency difference.However, in the long-term measurement process, the change of the background environment (such as temperature) may be relatively large, which inevitably affects the experiment.In addition, the torsion balance period method is extremely dependent on the torsion wire.The in-depth study of various characteristics of the selected suspension wire (metal wire or quartz wire) plays an important role in the torsion balance periodic method experiment.In our latest experiment [46, 47], we used the torsion balance period method to measure G, and conducted in-depth analysis and research on the thermal elasticity [59, 60], nonlinear [61, 62], anelasticity [58], aging and other effects of the torsion wire, as well as the high-precision extraction method of the torsion balance period [63~66].Especially in the experiment, we used a quartz wire with a Q value of 3.6 × 10 ^ 5 to directly measure the anelasticity effect for the first time [58], and the measurement results are consistent with the anelasticity hypothesis given by Kuroda in 1995 (the correction of anelasticity effect to G value is 1/π Q) within the error range [67].The experimental results HUST-99 (HUST-05) measured by our experimental group using the torsion balance periodic method were collected by CODATA for nearly three times.The G value given by our latest G measurement experiment [46, 47] is
，Relative uncertaintyUp to 26ppm.This result is one of the six experimental results with relative accuracy better than 50ppm in the world.
Angular acceleration compensation method of torsion balance
The basic principle of measuring universal gravitation constant G by angular acceleration compensation method of torsion balance.Place the torsion balance as the inspection mass in a multipolar gravitational field with a spherical attraction mass, and its angular acceleration can be expressed as
(7)
Where lmq is the multipole moment of the torsion balance, lmQ is the multipole moment of the attraction mass ball, I is the rotational inertia of the torsion balance, and Φ is the phase difference of the relative position between the attraction mass and the torsion balance.In Formula (7), the term with the greatest angular acceleration effect is
(8)
The relationship between the multipole moment and the moment of inertia of the torsion balance is as follows:
(9)
Where w, t are the width and thickness of the torsion balance.When the torsion balance is an ideal thin two-dimensional flat plate with upper and lower symmetry, equation (9) can be approximately expressed as
(10)
Therefore:
(11)
If the torsion balance and the attracted mass are respectively placed on two independent turntables and rotated, let the torsion balance obtain an inertia moment I β (β is the acceleration of the torsion balance turntable rotation) and gravitational moment τ=GCg, then the motion equation of the torsion balance becomes
(12)
Where ω s and ω p are the angular velocities of the mass attracting rotary table and the torsion balance rotary table respectively.By adjusting the rotation speed of the two turntables, the inertia torque of the torsion balance compensates the gravitational torque of the attraction mass to the torsion balance, so that the torsion balance is always in its balance position. At this time, I β=GCg.The G value can be given by recording the rotating acceleration of the torsion balance, calculating the moment of inertia I of the torsion balance and the gravitational coupling coefficient Cg with the attractive mass.
In 1969, Rose et al. [68] proposed the experimental method of measuring G by angular acceleration compensation of torsion balance, and carried out preliminary experiments with this method.After considering various error factors, they get the following results:
。They also pointed out a series of improvement measures and claimed that this method could further improve the experimental accuracy by two orders of magnitude.Because of the complexity of the system in this method, no one in the world continued to use this method to test G for a long time thereafter.Until the end of the 20th century,University of WashingtonGundlach et al. [69, 70] improved the angular acceleration compensation method of the torsion balance, proposed to place the attractive mass on another turntable and rotate it, and realized high-precision tracking of the two turntables. The advantage of this is that it can effectively reduce the influence of the background gravitational field of the experimental environment.Through further optimization design of the experimental system, the experimental results given by Gundlach and Merkowitz in 2000 are [42] (Uwash-00)
，Relative uncertaintyIt is better than 14ppm, and this result is the most accurate G measurement result reported internationally at present.In the angular acceleration compensation method of torsion balance, a series of optimized configurations (mainly symmetry) can greatly reduce the measurement requirements for the geometric size and density uniformity of torsion balance.Compared with the torsion balance periodic method, the suspension wire in this method is not twisted relative to the inspection quality, so the dependence of the experimental results on the suspension wire is reduced, especially the inevitable anelasticity effect in the periodic method is greatly suppressed [58].The measurement of the experiment is the angular acceleration α 22, and the constant angular displacement or angular velocity caused by other reasons will not affect the results, especially the creep of the suspension wire, whose influence can be ignored.Since the gravitational moment is very small, the resulting angular acceleration is also very small, which means that the angular velocity changes very slowly, so the measurement accuracy can be very high.The experiment does not need a long time of measurement to obtain a good measurement result, and the measurement time is much shorter than the periodic method.In the experiment, because the attractive mass is also rotating, the background gravitational field is naturally averaged out, which has little impact on the experimental results.
The disadvantage of the angular acceleration compensation method of torsion balance is that the experimental system is very complex and needs to be configured with a high-precision turntableclosed-loop control Realize that the torsion balance remains relatively static relative to the turntable.In the experiment, the rotation speed stability of the turntable needs to reach the order of 10 ^ - 7rad/s, and the angular acceleration difference stability of the two turntables needs to reach the order of 10 ^ - 12rad/s2, which puts forward high requirements for the turntable itself.At the same time, the interference of external environment, such as temperature, pressure, magnetic field and vibration, will still affect the experimental results.
Direct tilting/compensation method of torsion balance
Fig. 2 General device for G measurement experiment by torsion balance tilting method
The general device of the torsion balance tilt method G measurement experiment is shown in Figure 2 (a).The inspection mass of m is suspended at both ends of the torsion arm of the torsion scale, and the torsion arm is suspended with suspension wire.Place two large attractive masses M at the horizontal position away from the inspection quality r.Due to the gravitational effect of the attracted mass, the balance position of the torsion balance will produce a deflection angle of θ. At this time, the restoring torque of the torsion wire is balanced with the gravitational moment. The gravitational constant G can be calculated by measuring the θ angle and the relevant parameters of the torsion balance system:
(13)
Where I is the moment of inertia of the torsion scale, T is the swing period of the torsion scale, and b is the half arm length of the torsion scale.In the direct tilt method experiment, absolute measurement of θ angle is required, and the measurement error of G caused by the error of θ angle is
(14)
The measurement error of G value caused by the distance r between the attraction mass and the inspection mass center is
(15)
The value of r is obtained through the measurement of θ angle, so the error of θ angle is transferred to r, and it is also magnified twice.θ angle itself is very small (mrad level), and it is impossible to measure accurately, so θ will be larger, so the error is large.If you want to increase θ angle, you can only increase the attraction mass, but using a large attraction mass will affect the stability of the whole device, such as tilting the device, which will bring other errors.In addition, the gravitational moment is measured by measuring the torsional moment of the balanced torsion wire, so the stability of the torsional moment is very important.Torsion torque is very sensitive to temperature changes, and the defects of the twisted wire material will also make the twisted wire move irregularly, as well as the nonlinear, thermoelastic and aging effects of the twisted wire itself will also produce errors.In addition to the influence of the twisting wire itself, the changes of the surrounding objects will cause the drift of its balance position.
The principle of the electrostatic compensation method is similar to that of the direct tilt method, but the difference is that the additional electrostatic torque is used to compensate the gravitational torque, so that the twisted wire maintains its original static state.If the direct measurement of angular displacement in the direct tilt method is converted into the measurement of electrical signal, the measurement is relatively easy.Since the twisted wire only plays the role of suspending the inspection quality and does not participate in the measurement, the nonlinear factors of the twisted wire will not affect the experimental results.Because the electrostatic compensation method is still a static method, even if the gravitational moment is balanced by the electrostatic moment, the torsion balance cannot be completely static due to external interference.In addition, the calculation of the ideal parallel plate capacitance is very simple, while the calculation of the actual capacitance is quite complex, especially the influence of the boundary effect.
After Cavendish, many experimental physicists used the torsion balance tilt method to measure G.Fitzgerald and Armstrong [71] of New Zealand National Standards Laboratory have used electrostatic compensation method to measure G since 1995. The experimental principle is shown in Figure 2 (b).The gravitational torque generated by attracting mass and the torque generated by servo feedback voltage are balanced. Their experimental results published in 1995 are
。In 1999, they published the improvement experiment result as [72] (MSL-99)
。Subsequent further improvement experiments are given [45] (MSL-03)
，Relative uncertaintyUp to 40ppm.DeBoer [9] tried to use the buoyancy of mercury to support the quality inspection in 1987, and used electrostatic compensation to balance the leading moment, and the G value obtained is
。This result is much larger than other subsequent experimental results, and in their later research, they also found that there was an undetermined systematic error in this experiment [74], so in CODATA-02Basic physical constantsWhen adjusting, the result was not adopted.
