International Journal of Modern Physics: Conference Series

Vol. 1, No. 1 (2010) 1–5

World Scientific Publishing Company

DOI: 10.1142/insert DOI here

1

IMPACT OF HIGH PRECISION GRAVIMETRY IN

THE CONTEXT OF A FUTURE NEW SI

HENRI BAUMANN*

Mass and related Quantities, Federal Institute of Metrology METAS, Lindenweg 50

Bern-Wabern, Bern 3003, Switzerland henri.baumann@metas.ch

ALI L. EICHENBERGER

Electricity, Federal Institute of Metrology METAS, Lindenweg 50

Bern-Wabern, Bern 3003, Switzerland

ali.eichenberger@metas.ch

Received 18 December 2013

Revised 21 January 2014

In the early eighties, the development of ballistic absolute gravimeters based on laser

interferometer opened the doors to new research areas in various scientific domains such as

geodesy, geophysics or metrology. After a brief overview of the most used technique for gravity

measurements, the implication of gravity in the context of an improved SI, especially for a new

definition of the mass unit kg, will be presented.

Keywords: Gravity, Watt balance, Planck constant, International System of Units, SI

1. Introduction

The gravity force is a crucial physical quantity in many scientific areas. Since the first

experimental measurement of Galileo Galilei in 1604, the measurement techniques have

been continually improved. In the early eighties a technological breakthrough has been

achieved with the development of ballistic absolute gravimeters using laser

interferometer for the measurement of the position of a free falling body in the gravity

field g1. With these new types of absolute gravimeters, it was then possible to measure

gravity at the level of some µGal (Gal = 1 cm/s2) that opened the doors to new research

topics in various scientific domains such as earth science, geodesy, geophysics and

metrology.

* Presented by H. Baumann

2 H. Baumann and A. L. Eichenberger

1t 2t 3t

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Reflected beam

Falling prism

Reference

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SplitterPhoto-

Detector

1t 2t 3t

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Position

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1t 2t 3t

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reference beam

Reflected beam

Falling prism

Reference

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reference beam

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2. Absolute Gravimeters

Till the middle of the 20th

century, the technics used for the measurement of gravity were

based on simple or reversible pendulums. Nowadays, even if some new types of absolute

gravimeters are under development in different laboratories2, almost all absolute

gravimeters used in earth science or in metrology are ballistic instruments. These

instruments measure the position of a free falling body as a function of time to determine

gravity. The working principle of this type of gravimeters is illustrated in Figs. 1 and 2.

Fig. 1: Overview of the working principle of modern ballistic absolute gravimeters. The position of a free

falling optical prism is measured with a laser interferometer.

The position of a free falling body in the local gravity field is measured with an

interferometer. The light produced by a helium-neon laser is injected into the

interferometer using an optical fiber. The incoming beam is split into two beams, the

measuring beam and the reference beam. The measuring beam is reflected by a free

falling optical corner cube, while the reference beam is reflected by the fixed reference

corner cube. Both beams are then recombined to generate interferences. The optical

signal is then directed to a photodiode to be converted to an electrical signal. By timing

and counting the occurrence of the interference fringes (Fig. 2), the position of the falling

prism can be determined as a function of time. By knowing the vertical gradient, the

equation of motion, Eq. (1), is used to evaluate the gravity value go by a least square fit.

𝑥𝑖 = 𝑥𝑜 + 𝑣𝑜 +1

2𝑔𝑜 �̃�𝑖

2 +1

6𝛾𝑣𝑜 �̃�𝑖

3 +1

24𝛾𝑔𝑜�̃�𝑖

4 (1)

High Precision Gravimetry and Future New SI

where

�̃�𝑖 = 𝑡𝑖 −𝑥𝑖−𝑥𝑜

𝑐 (2)

and where go, vo and xo are the initial acceleration, velocity and position, respectively.

The factor γ is the vertical gravity gradient and c is the speed of light.

Fig. 2: Interference pattern measured by the photodiode. By timing and counting the fringes, the position of the

free falling body as a function of time can be determined.

Almost all ballistic gravimeters are free-fall gravimeters that measure only the falling

trajectory of a dropped corner cube3. Nevertheless, there are also other approaches that

are adopting a symmetric method4. Here, the corner cube is thrown and the complete rise

and fall of the trajectory is measured. However, both types of instruments are composed

of three main parts, the dropping chamber, the interferometer and the vibration isolation

system. Fig. 3 shows the main components of a free-fall absolute gravimeter5. The upper

part of the instrument (the dropping chamber) is a vacuum chamber that contains the

mobile arm of the interferometer. In this vacuum chamber, a cart (the drag free chamber)

lifts the optical corner cube to the top. After a stabilization time, the cart is accelerated to

put the corner cube in free-fall conditions. To minimize the drag forces from the residual

gas molecules, the drag free chamber follows the prism during its drop and gently catches

it at its end.

A major issue of gravimetry is coming from environmental perturbations. Different

techniques have been developed to overcome this problem. One possibility is to measure

the environmental noise and to introduce it as observable in the system of equation6.

Another approach would be to place the reference corner cube on a kind of mechanical

4 H. Baumann and A. L. Eichenberger

filter. This method is illustrated in Fig. 3 by the superspring7. This system mimics the

behavior of a spring of about 1 km length to reach a free period of approximately 60 s.

Fig. 3: General overview of a free-fall gravimeter

3. Implication of Gravity in the New Definition of the Mass Unit, the Kilogram

Today, the kilogram is the last unit of the International System of Units (SI) still based on

an artifact, the international prototype of the kilogram (IPK), kept at the Bureau

International des Poids et Mesures (BIPM). The IPK prototype, machined in 1878, is a

cylinder of a platinum-iridium alloy (Pt 90% - Ir 10% in mass) the height of which (39

mm) is equal to its diameter. Each country signatory of the meter convention received a

copy of the IPK to materialize its national mass unit. To survey the evolution of the

different copies relative to IPK, three comparisons have been made since 1880. The

results of these comparisons have shown a relative drift of about 0.5 µg/yr between the

copies and the IPK. Because the instability of the mass unit affects also other units of the

SI (such as the ampere or the mole), there is now a general consensus that the time for a

redefinition of the kilogram has come. Up to now, one of the most successful approaches

has been the watt balance proposed by B. Kibble8 in 1975. The principle of this

experiment is to link the mass unit to the Planck constant h by a comparison between the

mechanical power and the electrical power.

In 1997, the development of a watt balance started at the Federal Institute of

Metrology (METAS). After more than ten years of continuous improvements, systematic

characterization and thorough investigations, a final result for the Planck constant has

High Precision Gravimetry and Future New SI

been published9. With this result, the apparatus had reached its limits. Additional

improvements needed to be implemented to reduce the uncertainty significantly, became

incompatible with the conception of the experimental setup. For this reason, it has been

decided to start the development of a new watt balance at METAS, in strong

collaboration with industrial partners (METTLER TOLEDO, Maxon), universities

(Laboratoire de Systèmes Robotiques, LSRO, from the Ecole Polytechnique de

Lausanne, EPFL) and research institutes (Centre Européen de Recherche Nucléaire,

CERN)10

.

4. The Watt Balance Principle

The concept of the watt balance has been intensely discussed in other places11

. The

experiment is performed in two steps with the same experimental setup: the static or

weighing mode, and the dynamic or induction mode (Fig. 4).

Fig. 4: Principle of the watt balance experiment. Static mode (left): The electromagnetic force acting on the

current carrying coil is balanced against the weight of the test mass. Dynamic mode (right): The coil is moved

in the vertical direction through the magnetic field and the induced voltage is measured.

In the static mode the force generated by a mass m, placed in the local gravity field g, is

balanced by the vertical component of the electromagnetic force produced by a current I

flowing through a coil immersed in a magnetic field B. The electromagnetic force can be

expressed as

�⃗� = 𝐼 ∙ ∮ 𝑑𝑙⃗⃗⃗⃗ × �⃗⃗� , (3)

where l is the conductor length of the coil. In the dynamic mode, the coil is moved

vertically at a velocity v through the magnetic field B. This motion induces a voltage U

across the coil that can be expressed as

𝑈 = − ∮(𝑑𝑙⃗⃗⃗⃗ × �⃗⃗�) ∙ �⃗�. (4)

6 H. Baumann and A. L. Eichenberger

If the mechanical dimensions of the coil and the magnetic field are strictly identical in

both modes, and under the hypothesis that the coil passes through its weighing position

during the velocity mode, the combination of both modes leads to the expression

𝑈 ∙ 𝐼 = 𝑚 ∙ 𝑔 ∙ 𝑣. (5)

The experiment thus allows a comparison between electrical and mechanical power.

Using the expressions of the Josephson, Eq. (6), and quantum Hall effects, Equ. (7),

𝑈 = 𝐶𝐽 ∙ 𝑈𝐽 = 𝐶𝐽 ∙ 𝑛𝐽 ∙ℎ

2𝑒∙ 𝑓𝐽, (6)

𝑅 = 𝐶𝐻 ∙ 𝑅𝐻 = 𝐶𝐻 ∙ℎ

𝑛𝐻∙𝑒2 , (7)

Eq. (5) can be rewritten as

𝑚 = 𝐶 ∙𝑓𝑗∙𝑓𝑗

′

𝑔∙𝑣∙ ℎ , (8)

where CJ, CH and C are calibration constants, fj and fj’ are the Josephson frequencies used

during the static and the dynamic phases, nJ and nH are the step number, e the elementary

charge and h Planck's constant. The watt balance experiment allows therefore relating

the unit of mass to the meter, the second and the Planck constant.

From all the quantities that have to be measured in equation (5), only the local value of

the Earth gravity field g cannot be accessed in a direct way. For a new definition of the

kilogram, the CCM (Consultative Committee for Mass and Related Quantities) requests

that the watt balances should be in agreement with a relative standard uncertainty of 2 ×

10−8

. This implies a contribution associated with g of the order of some μGal. That is why

an appropriate method for the evaluation of g at the position of the test mass has been

developed. The method, described in Ref. 12 monitors the time dependent variations of

the Earth gravity field by using an absolute gravimeter and by establishing a three-

dimensional model of the gravity field of the watt balance laboratory. The 3D model

served to determine the difference in g between the absolute instrument and the point of

measurement of the watt balance.

High Precision Gravimetry and Future New SI

5. Conclusion

The knowledge of gravity at a high level of precision plays an important role in various

scientific domains. Especially in metrology, the development of instruments able to

determine g at the level of 10−9

opened the door to a possible new definition of the mass

unit by relaying it to the Planck constant.

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