Improving the angular detection sensitivity of

a torsion pendulum by an electrostatic spring

Y Z Bai, H Yin, L Liu, D Y Tan, Z B Zhou ([email protected])

Center for Gravitational Experiments, School of Physics, Huazhong Univ. of Science & Technology, Wuhan, China

An electrostatic torsion pendulum aiming at improving the angular detection sensitivity without increasing torque noise floor is presented. Theoretical analysis shows that it

could be used to release requirement of angular measurement, and is useful for gravitational experiments with much higher precision requirement. In this poster, the

principle of the electrostatic pendulum system is introduced, and its sensitivity and noise analysis are presented.

The power spectrum of the thermal fluctuation of the electrostatic pendulum, th,e2

i. Introduction

The torsion pendulum plays a paramount role in the field of precision measurement and

gravitational experiments due to its high-precision sensitivity. Such as investigating the performance

of a gravitational sensor and charge management for LISA.

k

I keff (1 i m ) th,e e,n

keff

keff km ke

2

V

k 0 f a ye (a 3 12a l 2 )

xe

xe e

3

e

6

d

e

th,e

2

4kBT

k m

(keff I 2 ) 2 km 2

Resolution of electrostatic torsion pendulum can be given by

H s,e ( )

1

k

I ikm

2

eff

2

min,e 2

eq

4kBTkm

2

e,n

2

H s,e ( )

Compare the electrostatic torsion pendulum with the typical balance as follows

Ground testing for LISA

(University of Trento)

Charge management

(University of Washington)

ii. A Typical Torsion Balance and Its Potential Sensitivity

A typical torsion balance is very sensitive to probe

force or torque which induced by weak signals, and its

resolution is limited by the thermal noise of the

pendulum and the readout noise, namely, the angular

detection level.

at low frequency: <<0
H ( ) 2 H ( ) 2
s
s,e
2
2
min,e min
at high frequency: >>0

H ( ) 2 H ( ) 2

s

s,e

2

2

min,e

min

iv. Example

The torsion

pendulum presented by

Washington University to study the charge

effects for gravitational-wave is used as an

example to be discussed, whose main

parameters are listed in Table I, and a

couple electrodes are added as add

additional electrostatic spring, whose

parameters are listed in Table II.

The torsion balance is set in a high vacuum chamber,

where viscous damping can be ignored. The motion

equation of the system in this case can be written as

Figure 1

I km (1 i ) th

where is the structure loss angle, th is a random force with a white spectral density,

and is presented as follows

th 2 4kBTR

R Re[ Z ( )]

Z ( ) iI k i k

m

m

th 2

4kBT

k m

( k m I 2 ) 2 k m 2

The power spectrum of the minimum detectable torque can be obtained as follows

n 2 th 2 eq 2

The schematic of the electrostatic pendulum is shown in

Figure 2, where one pair of electrodes are added in sided

of the test mass, where de is the distance between the test

mass each electrode, le is the vertical distance from the

centre of the electrode to the fiber, Vf is the voltage

applied on the electrodes to adjust the performance the

system, and Se=axe aye is the area of each electrode.

Assuming the angle measurement

noise is 510-8 (1+10-2/f)1/2 rad/Hz1/2.

Resolutions of the torsion pendulum

system with or without an

electrostatic spring are shown in

Figure 3. The electrostatic noise of

electrostatic pendulum induced by

fluctuation of voltage 10 V/Hz1/2 is

within 10-15 Nm/Hz1/2, which can be

neglected. If the requirement of

torque detection is 510-15 Nm/Hz1/2

above 0.1mHz, the requirement of

the angle measurement device is

shown in Figure 4.

The motion equation of the electrostatic torsion

pendulum is given by

Conclusions:

H s ( )

1

k

m

min 2

I 2 ikm

eq 2

2

4kBTkm

2

2

H s ( )

H s ( )

n

iii. Electrostatic Torsion Pendulum

I km (1 i ) th,e e e,n

The electrostatic spring can release the

requirement of the angular detection of a

torsion pendulum, which is very important

for the much higher precise torsion

pendulum experiments, such as to

investigate the effects of a test mass for

LISA and advanced LISA projects.

Figure 2(a)

where e is the electrostatic torque, and e,n is the

random torque induced by applied voltage

fluctuations.

e

0Vf2 a ye

2

1

le axe

2

1

le axe

2

x

x

dx

2

2

d x d x

e

e

e,n 2 0 axe a ye leVf Vf ,n / de2

Figure 3

References:

Figure 2(b) Top view

Figure 4

1.M Hueller, A Cavalleri et al, Torsion pendulum facility for ground testing of gravitational sensors for LISA,

Class. Quantum Grav. 19 (2002)1757-1765.

2.S.E.Pollack, M.D.Turner, S.Schlamminger, et al, Charge Management for gravitational-wave observatories using

UV LEDs, Phys. Rev. D. 81, 021001(R)(2010).

3.Q. L. Wang, H. C. Yeh, Z. B. Zhou et al, Improving the sensitivity of a torsion pendulum by using an optical

spring method, Phy. Rev. A, 80, 043811(2009)

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