Torsion Pendulums, Fluid Flows, and the Coriolis Farce

Paul Boynton* and Philip Peters

Department of Physics

University of Washington

Seattle, WA 98195

ABSTRACT

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We review the current status of the Index experiments with particular attention to

the significant Coriolis effect that has been identified in the data from the second series

of experiments in that program, Index II. This extraneous effect is neither experted

nor observed in the Index I experiments reported in PRL 59, 1385 (1987 ) . nor have we

yet discovered a mechanism to explain the small signal suggesting or mimicking a fifth

forre that is marginally detected in that work.

* Also Department of Astronomy

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INTRODUCTION Early in 1986, shortly after Ephraim Fischbach presented a stimulating seminar

at the University of Washington regarding a possible reinterpretation of the classical

Eiitvos experiment, Paul Boynton, David Crosby, Phil Ekstrom, and graduate student

Anthony Szumilo decided jointly that the prospect of carrying out a table-top

experiment to detect a possible new fundamental force was too appealing to let pass

by .

Our basic design principle was to detect the differential acceleration of a

composition dipole by suspending it horizontally from a torsion fiber and by measuring

the change in oscillation frequency of this torsion pendulum as a function of the

or.tentation angle {) between the dipole moment and an adjacent source mass. The

signature of a composition dependent force whose range is not much greater than

the characteristic dimension of the source is a simple cos {) variation in oscillation

frequency, with the dipole moment parallel to the source direction for {) = 0 or 7r .

Our rationale was to trade the somewhat thorny problem of measuring sub-arcsecond

angular deflections of the Eotvos experiment for the relative simplicity of measuring a

frequency with the modest resolution of a part per ten million. Around that concept,

we designed an exceedingly simple, portable, battery-powered instrument that could

operate at remote sites. Sites may then be chosen without regard for the availability of

"laboratory amenities" , but specifically for their topographical appeal. We soon settled

on the "Lower Town Wall'' near Index, Washington, a 300 meter granite intrusion

with a nearly vertical face for the first 130 meters, and a favorite of rock climbers.

Subsequently we have referred to our project as the "Index collaborat:on" and were

recently joined in this effort by Phil Peters.

RECENT WORK

In 1987 we reported the outcome of the first series of experiments at the Index

site 1 . The result was. in a sense, both too small and too large. If treated as an upper

limit on the strength of a putative composition-dependent interaction, our value was

about 50 times too small to be consistent (in the context of a simple model) with the

ear lier reported detection by Peter Thieberger2. On the other hand, the rms precision

of our strength measurement was even smaller than the small value observed, and this

3.5a discrepancy was too large to be easily dismissed as a statistical fluke.

Since that time our effort has been devoted to developing new strategies for

improving instrument performance in order to get a clearer view of the effect observed

in the first Index experiments . If found to be old physics in a new guise, we would

attempt to eliminate or at least suppress that mechanism in order to look with still higher sensitivity for possible new physics.

Through a combination of modifications to the instrument, statistical measurement

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noise has been reduced by more than an order of magnitude below the level reported

in 1987 (see Appendix) . Improving the mechanical rigidity and thermal stability of

the apparatus , taking advantage of a favorable scaling of noise power with pendulum

mass, and developing statistically more efficient analysis techniques have all contributed

to this increment in performance. At the present time we can measure differential

acceleration with an rms statistical uncertainty of about 10- 13g in three hours.

This improved precision has enabled us to make substantial gains in diagnosing

and suppressing or at least precisely measuring various systematic effects, the real

limitations to this kind of work. On the other hand. we have yet to provide either an

answer to the obvious question regarding the reproducibility of our earlier result, or

further insight to the possible nature of that marginally significant effect detected in

the Index I experiments.

A new sequence of experiments, Index II. was begun in the summer of 1987,

and ended with the discovery of a previously unrecognized systematic effect resulting

from a design change following Index I. The diagnosis and remedy of this effect is

discussed later in this paper. The next effort began in the early spring of 1988.

The experimental sequence involves a long list of diagnostic runs to evaluate gravity

gradients, temperature effects. magnetic and electrostatic effects, instrument tilt. and

more. A particular configuration of the instrument must, of course, be subject to the

entire gamut of tests. Unfortunately, the failure of the torsion fiber late in the 1988

experimental sequence necessitated starting over. The most recent series, Index III,

should be completed within the next few months .

DISCUSSION

New strategies don't always work. The copper-polyethylene composition dipole

constructed for the Index II experiments was based on the same "ring" configuration

of the Be-Al dipole used in Index I. To avoid the obvious electrostatic implications

of an exposed dielectric surface, the polyethylene semi-ring had to be enclosed in a

conducting shell. In order to achieve reasonable hydrodynamic symmetry, that square

cross section, thin-wall, annular shell of electroformed copper was extended around

the remaining half of the dipole to enclose an equal mass of copper, which meant the

interior of that half was mostly empty because of the rather small [CH2]n /Cu density

ratio. The fabrication technique involved a polyethylene semi-ring with an chemically

activated surface attached to an aluminurn s."mi-ring of identical dimensions but with

six equispaced cylindrical copper rods filling six holes drilled through the fiat faces at the mean radius. This uniform cross section ring assembly was then plated with 0.1

mm of copper, and the aluminum core was etched out with NaOH through two 3mm

holes drilled into the bottom face between the first and second and the fifth and sixth

copper rods. These two holes were the only breach in the copper shell.

The data obtained with this composition dipole ring were qualitatively similar to

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those of the Index I experiment, with one exception. After accounting for the effects of

gravity gradients. the residual cos () variation of the pendulum oscillation frequency was

found to be significantly larger and phase shifted by roughly 90c . That is, the measured

effect was relatively small when the composition dipole axis pointed perpendicularly to

the cliff face, and had extreme values when roughly parallel. The azimuthal symmetry

of the signal was better described in terms of the compass points - not oriented by

magnetic north, but true north. In such a coordinate systems , the extreme variation in

pendulum oscillation frequency occurred with the composition dipole moment oriented

along the east-west line.

This discussion of the configuration of the Cu-CH2 detector ring and the character of

the data obtained with it is prefatory to the following development of the hydrodynamic

consequences of the two, innocently drilled holes venting the cavity on the copper side

of 1;he ring.

The fact that the experiment was carried out in the rotating frame of the earth

implies that in addition to forces arising from gravity gradients, centrifugal and Coriolis

forces are present also. These forces act on the rigid ring as well as on the air

surrounding the ring. The centrifugal force, since it can be expressed as a gradient of a

centrifugal potential, acts in a way similar to a gravitational force. But the centrifugal

force acting on the ring is found not to generate a cos () signal, the kind that would

be confused with the signature of a "fifth force" . Furthermore, the centrifugal force

acting on the air causes an alteration of the pressure in the air, but that change does

not produce any torque about the pendulum rotation axis, and thus no variation in

oscillation frequency.

The Coriolis force. however, does give rise to two interesting, observable effects.

TlH' first of these generates a cos 0 variation in the damping constant, which then

affects the measured frequency. since the measured frequency depends on both the

resonant frequency of the torsion pendulum and the damping constant. Because both

the frequency and damping constant are measured in the experiment , this effect is

easily removed , since one is interested only in changes in the resonant frequency. The

other effect directly generates a cos 0 variation in the resonant frequency and thus

mimics a fifth force signal. Sue h an effect must be understood in order to render it

negligible by appropriate modification of the apparatus. This understanding and its

successful application is the tale told here.

The presence of an east-west antisymmetry m the oscillation frequency of the

pendulum suggests that the Coriolis force is acting, because the local angular velocity

of the rotating frame, We. has a component to the north as well as one in the vertical

direction. But an examination of the direction of the Coriolis force,

Fe = -2mwe x v, (1)

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shows that there is no effect of that kind on the oscillating pendulum itself. The velocity

of any mass element of the pendulum is in the azimuthal direction, which mearis that

the force on that element is perpendicular to the local azimuthal direction. Therefore

the force does not produce a torque about the vertical rotation axis, only a horizontal

torque which causes a small rocking of the pendulum out of the horizontal plane. A

more subtle mechanism is required to explain the observed behavior.

The Coriolis force acts also on the air surrounding the system, although one might

think that because the density of the air is so much smaller than the density of the ring

that these forces would be negligible. Moreover, one can argue on dimensional grounds

that the non-linear term in the hydrodynamic equations is small, and consequently the

dominant motion of the air is in the azimuthal direction. This argument is supported

by an approximate calculation showing that the observed magnitude of the damping

of the pendulum is due primarily to the viscous torque arising from the gradient in the

azimuthal flow at the surface of the ring. But again, if the velocity of the air is in the

azimuthal direction at all points, the Coriolis force should not give rise to any torques

about the rotation axis. Or so it would seem.

But in order to understand clearly what effect a given force has in a hydrodynamic

system, the force must first be expressed as the sum of two terms: one for which the

curl vanishes and the other for which the divergence vanishes. These two terms produce

different effects in the hydrodynamic equations. To simplify the discussion, the air may

be treated as incompressible so that the Na vier-Stokes equation applies,

av- - - �P 2 -- - + v . \l v = f - - + v v v at Po (2)

where v is the velocity field. p is the pressure, p0 is the density, f is the force per unit

mass, and v is the k inematic viscosity. As stated earlier, estimates of the magnitude of

the nonlinear term show that it is small, as is the inertial term av/ at, implying that

the angular velocity of the pendulum changes at a rate which is sufficiently small that

a steady flow can be assumed as a first approximation. The force with zero curl can be expressed as the gradient of a scalar potential.

In the steady flow approximation, such a term will cause an adjustment in pressure

throughout the system, but will not cause a non-steady flow to occur. However, for the

force term that has zero divergence, no such adjustment is possible, and a non-steady

flow results.

To see how this works in practice, consider the Coriolis force acting on a fluid in

uniform rotation, in which case v = w x r. This is, of course, a poor approximation

outside the ring, but a good approximation for the air inside the cavity in the copper

half of the ring. The Coriolis force per unit mass is then

f, = - 2we x (w x i') = -2[w(we . r) - r(w . we) ] . (3)

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By adding and subtracting the term w.(w · r) we can write fc = f1 + £; . where

and

(5)

For this decomposition of fc. fi has zero curl and f2 has zero divergence. Note that the

term that was added and subtracted is in the direction of we, a term that has a nonzero

horizontal component and consequently a non-zero azimuthal component for a typical

mass element in the flow.

In the steady flow approximation, the force fi produces a pressure readjustment

within the cavity,

(6)

No pressure readjustment can cancel the force fz , and so it results in a circulation of the

air within the cavity. but this flow can be shown not to give rise to a torque about the

pendulum rotation axis. The pressure force, of course, acts normal to the walls of the

cavity. For the top, bottom, and sides of the cavity, this force is in the radial or vertical

direction, so it generates no torque either. However, the force acts also at the ends of

the semi-circular cavity, and because of the existence of the azimuthal pressure gradient

within the cavity, the resulting forces and torques do not cancel. In fact, analysis of

the forces shows that they result in a cos () modulation of the damping constant for the

srstem, which could be seen in the observed pendulum frequency before correcting to

determine the resonant frequency. Therefore the torque about the rotation axis is due

to a term not explicitly present in the standard expression for the Coriolis force, but

appears when the force is expressed as the sum of gradient and curl terms. For this

example, the most important term in the hydrodynamic equation was the one that was

not there.

A modulation of the damping constant is suggested by the data, but it is far

too small to explain the substantial (L:..w/w � 10-5) cos () variation in the observed

frequency under the assumption of a constant resonant frequency. Essentially all the

observed effect must then be explained as a direct modulation of the resonant frequency

itself. For that we look further.

The treatment of the Coriolis force on the air outside the ring is more complicated

because the angular velocity of rotation of the air is not uniform. If the nonlinear

term in the Navier-Stokes equation is again ignored, a trial solution for the velocity

representing differential rotation in the azimuthal flow can be posed. Letting v =

w(p , z, t) x f. where w is in the z direction and p is the cylindrical radial coordinate, the

boundary conditions on the fluid flow around this damped oscillator can be satisfied

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for all times if w (p , z) satisfies the differential equation

o2w 3 OW o2w . . ZJ [- + - - + -] = 1 (w1 - n)w op2 p op oz2 (7)

where w = Wr on the boundary of the ring, and Wr = w1ei(w1 -i1)t. This, of course,

represents the asymptotic behavior of the fluid. There will also be transients that

depend on the initial air circulation; but those damp out since their boundary condition

at the ring is zero velocity. We do not need to solve equation (7) for w (p , z) , since just

the form is adequate for estimating Coriolis effects.

The breakup of the Coriolis force into irrotational and rotational terms is more

involved when there is differential rotation. Using standard techniques of vector

analysis, the Coriolis force per unit mass can be expressed as

where

and

and finally

fc = - Vm + '\7 , c

<P = c; • . ( � x A )

c = (w • . V)A ,

A = � j v(r')d3r•

27r l r- r l

(8)

(9)

(10)

( 1 1)

The scalar potential causes a pressure change, op = p0We · B , where we define B =

'\7 x A. But, unlike the case of the cavity, the forces on the outside of the ring act in

the p or z directions only and generate no azimuthal torques.

The equation for the vector potential bears a close resemblance to that of the

magnetic vector potential for an azimuthal current density. Since B = '\7 x A, B is

analogous to the magnetic field of such a current density, in this case a field with only

p and z components. The rotational part of the Coriolis force can therefore be written

as

frot = (we · "7)B . ( 12)

Because B has the form B = Bpp + B, k, the derivative with respect to x (the direction

of the horizontal component of w.) creates an azimuthal force component and thus

torques on the fluid about the rotation axis.

Of course. the existence of torques on fluid elements does not mean there is an

observable effect on the resonant frequency. Carrying out the analysis of these torques

for a fluid satisfying equation (7) shows that the torque per unit mass about the rotation

508

axis is actually proportional to the local value of aw/ az. This effect was not present

in the cavity calculation, where the angular velocity was uniform. However, if the

exterior of the ring were completely uniform, by symmetry this differential fluid flow

term would not generate a cos () signal. Unfortunately, the method of creating a cavity

in the copper left two holes in the bottom of the ring. These holes serve to transmit

the azimuthal pressure gradient in the cavity, which is present as a result of the air

being accelerated, to the air exterior to the side of the ring containing the cavity.

thereby creating an azimuthal and z-axis asymmetry in the air flow and consequently

in aw I az. This asymmetry is then reflected in an asymmetry in the Coriolis forces

on the air, resulting in a net torque on the torsion pendulum. This effect produces

a variation in the resonance frequency proportional to cos () with just the east-west

antisymmetry observed. Although a detailed numerical computation was not carried

out to evaluate the magnitude of the variation in aw/az, a 10% asymmetry in the

angular velocity profile would fully account for the observed effect.

This possible explanation of the puzzling outcome of the second series of Index

experiments has had two principal benefits. First, the pendulum ring was redesigned

as a sealed unit whose external appearance gives no hint of the orientation of the

composition dipole.3 Second, from equation 14, the effect is absent for symmetry in

the flow about the horizontal midplane of the ring. Thus, the Al-Be ring of the Index

I experiment� with 22 holes drilled vertically through the aluminum half of the ring

preserves symmetry about the midplane and thus would not be expected to show this

effect. Moreover, the "signal" observed in Index I is nearly orthogonal to that generated

by the Coriolis effect.

Because of this experience we have examined with some care a number of other

possible fluid effects on this type of experiment and none have been large enough to

observe even with our current sensitivity. It is interesting to note that the Coriolis

effect reviewed here does not scale with fluid density and is therefore undiminished

until the ambient fluid pressure is lowered to the point that the mean free collision

path is comparable to the characteristic linear scale of the apparatus.

CONCLUSIONS A list will suffice:

• We can understand the puzzling outcome of the Index II experiments as a rather

subtle but sizeable Coriolis effect on the air viscously entrained by an oscillating ring

with azimuthal and z-axis asymmetry. The outcome of the Index III experiments

will indicate whether this mechanism was fully responsible for the observed effect . • Because of design differences this effect would not be expected to arise in the

Index I experiments, and in fact it was not observed. Moreover, we have as

yet no understanding of the effect that was marginally observed in that series of

experiments.

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• The new CH2-Cu ring currently employed in the Index III experiments was designed

to avoid this Coriolis effect.

• We are continuing our program of measurements at the Index site, which is

specifically well-suited to a sensitive search for a force whose range is in the interval

from hundreds of meters to a few kilometers. Our rms measurement noise in unit

observing t ime is now more than an order of magnitude below the level of Index I

with comparable limits on systematic effects.

REFERENCES

1 . P. E. Boynton, D . Crosby, P. Ekstrom, and A . Szumilo. Phys. Rev. Lett . , 59, 1385

( 1987) .

2. P. Thieberger, Phys. Rev. Lett., 56, 2347 ( 1986 ) .

3. A picture of the Index III Cu-CH2 composition-dipole torsion pendulum is shown

in Physics Today, July 1988, page 21 .

4 . P. E. Boynton. Proceedings of the XXIII Recontres de Moriond, Ed . by 0. Fackler

and J. Tran Thanh Van, Editions Frontieres ( 1988) .

5. P. G. Roll, R. Krotkov, and R. H . Dicke. :inn. Phys. (NY). 26, ( 1964) .

Appendix: Instrument Performance The Index III experiment currently operates within a factor of 2 of the thermal

noise limit discussed in reference 4. As a noise power density (fractional rms frequency

fluctuation per root Hertz) this is roughly a factor of 20 improvement relative to the

Index I experiment, and is a little smaller than that reported by Roll, Krotkov, and

Dicke.5 A combination of modifications to the earlier instrument contributed to this

gain:

• Suppression of possible air convection through improved temperature stratification.

• Improved oscillation frequency stability resulting from active thermal control.

• Reduction in thermal noise from appropriate scaling of pendulum parameters.

Of these, the scaling advantage requires further clarification. Mechanical thermal

noise, of course, represents a fundamental limitation to the precision of oscillator

frequency measurement. Following reference 4, eq. 7, we write the mean square noise

in unit observing time as

2 b N thermal ex: �()2 K 0

where b is the damping constant and K the torsion spring constant of the suspension

fiber. The oscillation amplitude Oo is taken to be fixed and therefore suppressed in the

following expressions and discussion.

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Being near the thermal noise limit does not mean the system i s necessarily quiet,

but only that Nthermal dominates other noise sources . In our case this is because b is

relatively large due to air friction. How can we make the best of this situation? Our

motivation is to increase the S/N ratio by an appropriate choice of design parameters

K, f., and m, where € and m are the pendulum radius and mass. The viscous damping

constant b for the ring geometry of our pendulum scales as R.3 , and for a composition­

dependent force coupling approximately to mass, S ex mf./ K. Consequently,

m (S/N)thermal ex f.l/2 ·

Thus we are urged to increase m (to increase the signal) and to decrease R. (since the

noise is thereby reduced more strongly than the signal) . The scaling law in the vacuum

case would be quite different.

There is no explicit dependence of the S/N ratio on "· but m and K are related.

Making m large implies a necessary increase in fiber cross-section, but the torsional

rigidity of the fiber is approximately proportional to the cross-section squared, thus

K ex m2 . This relation can be interpreted as a sea.ling of the suspended mass while

maintaining fiber tension as a fixed fraction of fiber strength.

The S /N ratio depends only weakly on R. relative to m, but not so for w . Because

w depends more strongly on R. than m, constraints arise from the relation between the

magnitude of w and measurement error that led us to fix R. and increase m by a factor

of 6 to increase correspondingly the S /N ratio. A larger increase is certainly feasible,

but there are limitations. Many systematic effects sea.le approximately as S; but there

is a. notable exception, the temperature dependence of the torsion oscillator frequency.

Hence, the necessity for active control of the system temperature mentioned earlier.

An attempt to increase S/N by a further increment in m would depress N (note that

N ex m-2 ) and the noise budget would become dominated by temperature variations.

Clearly an additional large reduction in noise could result from operation in a

sufficiently high vacuum to turn off both gas viscosity and thermal conductance. This

feature is included in an instrument now under construction.