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New force or thermal convection in the differential-accelerometer experiment?
Volume 192, number 1,2
PHYSICS LETTERS B
25 June 1987
N E W FORCE OR T H E R M A L C O N V E C T I O N IN T H E D I F F E R E N T I A L - A C C E L E R O M E T E R E X P E R I M E N T ? ~ Y.E. KIM Department of Physics, Purdue University, West Lafayette, IN 47906, USA Received 5 March 1987
Recently Thieberger reported that the result of his differential-accelerometer experiment is consistent with the existence of a substance-dependent medium-range repulsive force. It is shown that the effect due to thermal convection is not negligible and can be large enough to account for his data. Methods of measuring and minimizing the thermal convection effect, as well as several ways of improving the differential-accelerometer experiment, are proposed.
Recently Thieberger [ 1 ] reported on the results of his differential-accelerometer measurements and suggested that the observed effect may be due to the existence of a hypothetical substance-dependent medium-range repulsive force. Based on the requirement of the local baryon gauge invariance, such a substance-dependent repulsive (long-range) force was originally suggested by Lee and Yang in 1955 [2]. The existence of such a hypothetical (intermediate-range) force has been suggested recently from the reanalysis [ 3] of the experiment of E~Stv6s, Pek~ir, and Fekete [4] but has been shown to be inconclusive [5-9]. In this paper, an alternative explanation, that thermal convection may be responsible for the effect observed by Thieberger [ 1 ], is proposed. The magnitude of the thermal convection effect is estimated using the steady convection model of Turcotte [ 10,11 ]. The estimated thermal convection effect is found to be large and may account for the effect observed by Thieberger [ 1 ]. Since the observed motion in Thieberger's measurements is directed mostly in one direction (nearly toward east), the experimental set-up used by Thieberger can be approximated as a two-dimensional hydrodynamic problem with boundaries as shown in fig. 1. Because of a large heat capacity of the ground in comparison with that of the air, it is reasonable to assume that there exists a very small temperature Work supported in part by the National Science Foundation.
236
x=O z=OI
T=To
L_/ -',, 7,,,'(inlond)
ili
r\
ix
IA i~"~"~
//,,.i \
x=X/2
", EAST I ~ [ V° (cliff)
*'i
z= d l \ \ \ \ \ \ \ \ \ \ ~ /-o-":ff~q, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \ \ \ \ I T= Td Fig. 1. Velocity profiles for two different models of thermal convection for Thieberger's differential-accelerometer measurements. The dashed arrows (one clockwise and another counter clockwise) are for the case of two-cell convection. The straight arrows labeled by the horizontal velocity, Uo, and vertical velocity, Vo, are for the steady convection model or Turcotte [ 10,11 ].
gradient of ATao=Td-To (with 0 < Td(west) -Td(east)
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 192, number 1,2
Rac= (n/8)4(1 +62) 3 ,
PHYSICS LETTERS B
(1)
where ~ is the aspect ratio defined as = (2/2)/d = 2/2d.
(2)
For Thieberger's experiment [ 1 ], 2/2 ~ 67 cm (or 67/2=33.5 cm for the case of two-cell convection shown as dashed curves in fig. 1) d_=43 cm, and - ~~ 0.6418 (or 1.2836), which yield a critical value of Ra~=665 (or 1097).
(3)
The minimum value of Rac=657 occurs at 2/2 = x/~d or n ~ - ~= n / x / 2 ~, 2.2. Ra~ = 665 occurs at rt~- t = 2.016, and Ra~ = 1097 occurs at rcO- l = 4.032. The Rayleigh number is defined as Ra = gpt~ ( Td -- To)d3/xrl,
(4)
where g is the gravitational acceleration ( ~ 9 7 8 cm s - l ) , p is the density, a is the volumetric coefficient of thermal expansion, and t/is the viscosity of the fluid, x is the thermal diffusivity defined as x = k/pc, where k is the thermal conductivity and c is the specific heat at a constant pressure. For Ra < Ra~, disturbances will decay with time and hence the system becomes stable since the buoyancy force due to the density variation caused by the thermal gradient is weaker than the viscous force. For R a > Rao disturbances will grow exponentially with time until a steady convection is established since the buoyancy force is now stronger than the viscous force of the fluid. To calculate the maximum temperature gradient ATdo required to maintain the stability of the water for the case of Thieberger's experiment [ 1 ], we use the following properties [ 13] of the water at 4°C and 1 atm: p=0.999975 g m c m -3, t/=1.567×10 -2 gm cm -I s - j , k = 5 . 6 8 × 1 0 3 W c m -~ K ~,c=4.2028 J gm -t K -j and x = 1 . 3 5 0 9 × 1 0 -3 s -l. Assuming that the water density is a maximum, p = 1 gm cm -3, at T = 3 . 9 8 ° C = 2 7 7 . 1 3 K [13], a is parameterized as
o t = 2 a , ( T - To) ,
(5)
with To=3.98°C=277.13 K and o q = 6 . 7 5 X 1 0 -6 K - 2. The above parameterization of a (eq. ( 5 )) fits both a = 0 at T = 3 . 9 8 ° C and a = +0.27X 10 - 6 K-1 at T = 4 ° C [ 13 ]. With the above values for the prop-
25 June 1987
erties of the water, R a = R a c = 6 6 5 , g = 9 7 8 cm s -2 and d = 4 3 cm in eq. (4), we obtain the following value of ATdo for the Thieberger experiment [ 1 ]: ATdo = 0 . 0 0 3 6 6 ° C ,
(6)
which is at least two orders of magnitude smaller than the uncertainties of +0.2°C of the water temperature at (4 + 0.2) ° C in the Thieberger experiment [ 1 ]. Therefore, an instability and hence thermal convection are expected to be present in his experiment. In the following, an estimate for the magnitude of the thermal convection is made using the steady convection model originally developed by Turcotte and Oxburgh [ 10,11 ]. To Obtain an accurate estimate of the amplitude of the thermal convection, it is necessary to consider a set of nonlinear (Navier-Stokes) partial differential equations with appropriate boundary conditions since the linear stability analysis cannot predict the magnitude of finite-amplitude convection currents. However, the order of magnitude estimate can be made using the steady convection model of Turcotte, originally developed for studying the continental drift [ 10,11 ]. In Turcotte's model [ 11 ], the horizontal velocity, Uo, of the steady convection current shown in fig. 1 is given by X
t~7/3
(Ra'~
u0= 2 n (1 +~,)2/3 \~----~]
2/3
•
(7)
With 2/2~67 cm and ATdo~0.00366 K (eq. (6)) corresponding to R a = R a c = 6 6 5 , the maximum velocity, Uo, is calculated from eq. (7) to be u0 = 28.8 m m / h .
(8)
The average horizontal velocity in Turcotte's model [ 12] is Uo/2 and hence the calculated average horizontal velocity, u, of the center of mass of the copper sphere [ 1 ] is expected to be u ~ ½(Uo/2) ~ 7.2 mm/h, which is the same order of magnitude as the observed velocity of ~ 4.5 mm/h. Eq. (8) shows that the effect of the thermal convection is large and may account for the effect observed by Thieberger. The value of U=Uo/4_~7.2 mm/h from eq. (8) is an upper limit and may be reduced to be comparable to or smaller than the experimental value of ~4.5 mm/h by adjusting and using more realistic boundary conditions. The fact that l u l ~ 1 mm/h was observed in the absence of a cliff but under otherwise similar 237
Volume 192, number 1,2
PHYSICS LETTERS B
conditions is not inconsistent with the thermal convection effect since it is very sensitive to the temperature gradients, ( T ~ - T o ) , at different locations. Although the t e m p e r a t u r e o f the external west wall o f the box containing the water (see fig. 1 ) was raised by an average o f 6 ° C a b o v e the east wall temperature for 14 h during the 5-day p e r i o d o f the Thieberger e x p e r i m e n t [ 1 ] (which m a y have changed the water t e m p e r a t u r e inside slightly), the thermal convection effect would not change drastically if the t e m p e r a t u r e gradient, ( T d - T o ) , a n d the thermal diffusivity, x, d i d not change significantly. There are several m e t h o d s to check if the effect observed by Thieberger is due to the thermal convection. In the following, some m e t h o d s are proposed. (1) To d e t e r m i n e the m a g n i t u d e o f the thermal convection, one can measure up(z) as a function o f z using either the same copper sphere or two similar spheres simultaneously, one near the top ( u o ~ Utop) and one near the b o t t o m (Up ~ Ubottom) o f the water. The thermal convection effect can be then e s t i m a t e d using Up(Z). The difference, (Utop--Ubot~om), can be attributed or related to the new or other effects. If the substance-dependent effect is present but is substantially smaller after a sizeable correction is m a d e for thermal convection, it m a y turn out that the effect is consistent with the substance-dependent long-range force originally p r o p o s e d by Lee and Y a n g [ 2 ] and recently by D e Rtijula [8]. (2) A n o t h e r test is to repeat the differential-accele r o m e t e r m e a s u r e m e n t using the spheres o f the same material but o f different radii to check whether the observed velocities at the same depth are the same or not. F o r a given acceleration due to the hypothetical s u b s t a n c e - d e p e n d e n t force, the observed velocities will be different and will be p r o p o r t i o n a l to the inverse square o f the radius o f the sphere, an observation that is inconsistent with thermal convection. (3) The thermal convection effect can be minim i z e d if the Rayleigh n u m b e r (eq. ( 4 ) ) can be m a d e
238
25 June 1987
to be smaller or c o m p a r a b l e to the critical value (eq. ( 3 ) ) . The t e m p e r a t u r e uncertainties o f the fluid can be m a d e much smaller than + 0 . 2 ° C , but it m a y be difficult to keep the t e m p e r a t u r e o f the fluid within ( T d - - T o ) <<0.0037°C for a long d u r a t i o n to maintain R a < R a c ~ 665. (4) Ultimately, it is essential to establish a stand a r d null result by eliminating thermal convection and other effects using i m p r o v e d differential accelerometers with an o p t i m a l choice o f geometry and viscosity at a location with fiat terrain and uniform ground mass density. After establishing the null result, a heavy load o f materials can be brought to one side o f the experimental set-up to test whether the spheres (one or two) m o v e away from the heavy load. The a u t h o r wishes to t h a n k T.R. Palfrey for helpful suggestions and discussions on the subject.
References [ 1] P. Thieberger, Phys. Rev. Lett. 58 (1987) 1066. [2] T.D. Lee and C.N. Yang, Phys. Rev. 98 (1955) 1501. [3] E. Fischbach et al., Phys. Rev. Lett. 56 (1986) 3. [4] R.V. EStv6s, D. Pekar and E. Feteke, Ann. Phys. (Leipzig) 68 (1922) 11. [5] Y.E. Kim, Phys. Lett. B 177 (1986) 255, and references therein. [6] Y.E. Kim, and D.J. Klepacki, Theoretial analysis for geophysical determination of the gravitational constant, Purdue Nuclear Theory Group preprint PNTG-86-11 (July 1986). [7] Y.E. Kim, D.J. Klepacki and W.J. Hinze, Mass-density profile and the geophysical determination of the gravitational constant, Purdue Nuclear Theory Group preprint PNTG86-12 (July 1986), Nature, to be published. [8] A. De Rtijula, Phys. Lett. B 180 (1986) 213. [9] S.Y. Chu and R.H. Dicke, Phys. Rev. Lett. 57 (1986) 1823. [ 10] D.L. Turcotte and E.R. Oxburgh, J. Fluid Mech. 28 (1967) 29. [ 11 ] D.L. Turcotte and G. Schubert, Geodynamics (Wiley, New York, 1982). [12] Lord Rayleigh, Phil. Mag. 32 (1916) 529. [ 13 ] R.C. Weast, Handbook of chemistry and physics, 57th ed. (CRC Press, Cleveland, OH, 1976).
PHYSICS LETTERS B
25 June 1987
N E W FORCE OR T H E R M A L C O N V E C T I O N IN T H E D I F F E R E N T I A L - A C C E L E R O M E T E R E X P E R I M E N T ? ~ Y.E. KIM Department of Physics, Purdue University, West Lafayette, IN 47906, USA Received 5 March 1987
Recently Thieberger reported that the result of his differential-accelerometer experiment is consistent with the existence of a substance-dependent medium-range repulsive force. It is shown that the effect due to thermal convection is not negligible and can be large enough to account for his data. Methods of measuring and minimizing the thermal convection effect, as well as several ways of improving the differential-accelerometer experiment, are proposed.
Recently Thieberger [ 1 ] reported on the results of his differential-accelerometer measurements and suggested that the observed effect may be due to the existence of a hypothetical substance-dependent medium-range repulsive force. Based on the requirement of the local baryon gauge invariance, such a substance-dependent repulsive (long-range) force was originally suggested by Lee and Yang in 1955 [2]. The existence of such a hypothetical (intermediate-range) force has been suggested recently from the reanalysis [ 3] of the experiment of E~Stv6s, Pek~ir, and Fekete [4] but has been shown to be inconclusive [5-9]. In this paper, an alternative explanation, that thermal convection may be responsible for the effect observed by Thieberger [ 1 ], is proposed. The magnitude of the thermal convection effect is estimated using the steady convection model of Turcotte [ 10,11 ]. The estimated thermal convection effect is found to be large and may account for the effect observed by Thieberger [ 1 ]. Since the observed motion in Thieberger's measurements is directed mostly in one direction (nearly toward east), the experimental set-up used by Thieberger can be approximated as a two-dimensional hydrodynamic problem with boundaries as shown in fig. 1. Because of a large heat capacity of the ground in comparison with that of the air, it is reasonable to assume that there exists a very small temperature Work supported in part by the National Science Foundation.
236
x=O z=OI
T=To
L_/ -',, 7,,,'(inlond)
ili
r\
ix
IA i~"~"~
//,,.i \
x=X/2
", EAST I ~ [ V° (cliff)
*'i
z= d l \ \ \ \ \ \ \ \ \ \ ~ /-o-":ff~q, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \ \ \ \ I T= Td Fig. 1. Velocity profiles for two different models of thermal convection for Thieberger's differential-accelerometer measurements. The dashed arrows (one clockwise and another counter clockwise) are for the case of two-cell convection. The straight arrows labeled by the horizontal velocity, Uo, and vertical velocity, Vo, are for the steady convection model or Turcotte [ 10,11 ].
gradient of ATao=Td-To (with 0 < Td(west) -Td(east)
0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 192, number 1,2
Rac= (n/8)4(1 +62) 3 ,
PHYSICS LETTERS B
(1)
where ~ is the aspect ratio defined as = (2/2)/d = 2/2d.
(2)
For Thieberger's experiment [ 1 ], 2/2 ~ 67 cm (or 67/2=33.5 cm for the case of two-cell convection shown as dashed curves in fig. 1) d_=43 cm, and - ~~ 0.6418 (or 1.2836), which yield a critical value of Ra~=665 (or 1097).
(3)
The minimum value of Rac=657 occurs at 2/2 = x/~d or n ~ - ~= n / x / 2 ~, 2.2. Ra~ = 665 occurs at rt~- t = 2.016, and Ra~ = 1097 occurs at rcO- l = 4.032. The Rayleigh number is defined as Ra = gpt~ ( Td -- To)d3/xrl,
(4)
where g is the gravitational acceleration ( ~ 9 7 8 cm s - l ) , p is the density, a is the volumetric coefficient of thermal expansion, and t/is the viscosity of the fluid, x is the thermal diffusivity defined as x = k/pc, where k is the thermal conductivity and c is the specific heat at a constant pressure. For Ra < Ra~, disturbances will decay with time and hence the system becomes stable since the buoyancy force due to the density variation caused by the thermal gradient is weaker than the viscous force. For R a > Rao disturbances will grow exponentially with time until a steady convection is established since the buoyancy force is now stronger than the viscous force of the fluid. To calculate the maximum temperature gradient ATdo required to maintain the stability of the water for the case of Thieberger's experiment [ 1 ], we use the following properties [ 13] of the water at 4°C and 1 atm: p=0.999975 g m c m -3, t/=1.567×10 -2 gm cm -I s - j , k = 5 . 6 8 × 1 0 3 W c m -~ K ~,c=4.2028 J gm -t K -j and x = 1 . 3 5 0 9 × 1 0 -3 s -l. Assuming that the water density is a maximum, p = 1 gm cm -3, at T = 3 . 9 8 ° C = 2 7 7 . 1 3 K [13], a is parameterized as
o t = 2 a , ( T - To) ,
(5)
with To=3.98°C=277.13 K and o q = 6 . 7 5 X 1 0 -6 K - 2. The above parameterization of a (eq. ( 5 )) fits both a = 0 at T = 3 . 9 8 ° C and a = +0.27X 10 - 6 K-1 at T = 4 ° C [ 13 ]. With the above values for the prop-
25 June 1987
erties of the water, R a = R a c = 6 6 5 , g = 9 7 8 cm s -2 and d = 4 3 cm in eq. (4), we obtain the following value of ATdo for the Thieberger experiment [ 1 ]: ATdo = 0 . 0 0 3 6 6 ° C ,
(6)
which is at least two orders of magnitude smaller than the uncertainties of +0.2°C of the water temperature at (4 + 0.2) ° C in the Thieberger experiment [ 1 ]. Therefore, an instability and hence thermal convection are expected to be present in his experiment. In the following, an estimate for the magnitude of the thermal convection is made using the steady convection model originally developed by Turcotte and Oxburgh [ 10,11 ]. To Obtain an accurate estimate of the amplitude of the thermal convection, it is necessary to consider a set of nonlinear (Navier-Stokes) partial differential equations with appropriate boundary conditions since the linear stability analysis cannot predict the magnitude of finite-amplitude convection currents. However, the order of magnitude estimate can be made using the steady convection model of Turcotte, originally developed for studying the continental drift [ 10,11 ]. In Turcotte's model [ 11 ], the horizontal velocity, Uo, of the steady convection current shown in fig. 1 is given by X
t~7/3
(Ra'~
u0= 2 n (1 +~,)2/3 \~----~]
2/3
•
(7)
With 2/2~67 cm and ATdo~0.00366 K (eq. (6)) corresponding to R a = R a c = 6 6 5 , the maximum velocity, Uo, is calculated from eq. (7) to be u0 = 28.8 m m / h .
(8)
The average horizontal velocity in Turcotte's model [ 12] is Uo/2 and hence the calculated average horizontal velocity, u, of the center of mass of the copper sphere [ 1 ] is expected to be u ~ ½(Uo/2) ~ 7.2 mm/h, which is the same order of magnitude as the observed velocity of ~ 4.5 mm/h. Eq. (8) shows that the effect of the thermal convection is large and may account for the effect observed by Thieberger. The value of U=Uo/4_~7.2 mm/h from eq. (8) is an upper limit and may be reduced to be comparable to or smaller than the experimental value of ~4.5 mm/h by adjusting and using more realistic boundary conditions. The fact that l u l ~ 1 mm/h was observed in the absence of a cliff but under otherwise similar 237
Volume 192, number 1,2
PHYSICS LETTERS B
conditions is not inconsistent with the thermal convection effect since it is very sensitive to the temperature gradients, ( T ~ - T o ) , at different locations. Although the t e m p e r a t u r e o f the external west wall o f the box containing the water (see fig. 1 ) was raised by an average o f 6 ° C a b o v e the east wall temperature for 14 h during the 5-day p e r i o d o f the Thieberger e x p e r i m e n t [ 1 ] (which m a y have changed the water t e m p e r a t u r e inside slightly), the thermal convection effect would not change drastically if the t e m p e r a t u r e gradient, ( T d - T o ) , a n d the thermal diffusivity, x, d i d not change significantly. There are several m e t h o d s to check if the effect observed by Thieberger is due to the thermal convection. In the following, some m e t h o d s are proposed. (1) To d e t e r m i n e the m a g n i t u d e o f the thermal convection, one can measure up(z) as a function o f z using either the same copper sphere or two similar spheres simultaneously, one near the top ( u o ~ Utop) and one near the b o t t o m (Up ~ Ubottom) o f the water. The thermal convection effect can be then e s t i m a t e d using Up(Z). The difference, (Utop--Ubot~om), can be attributed or related to the new or other effects. If the substance-dependent effect is present but is substantially smaller after a sizeable correction is m a d e for thermal convection, it m a y turn out that the effect is consistent with the substance-dependent long-range force originally p r o p o s e d by Lee and Y a n g [ 2 ] and recently by D e Rtijula [8]. (2) A n o t h e r test is to repeat the differential-accele r o m e t e r m e a s u r e m e n t using the spheres o f the same material but o f different radii to check whether the observed velocities at the same depth are the same or not. F o r a given acceleration due to the hypothetical s u b s t a n c e - d e p e n d e n t force, the observed velocities will be different and will be p r o p o r t i o n a l to the inverse square o f the radius o f the sphere, an observation that is inconsistent with thermal convection. (3) The thermal convection effect can be minim i z e d if the Rayleigh n u m b e r (eq. ( 4 ) ) can be m a d e
238
25 June 1987
to be smaller or c o m p a r a b l e to the critical value (eq. ( 3 ) ) . The t e m p e r a t u r e uncertainties o f the fluid can be m a d e much smaller than + 0 . 2 ° C , but it m a y be difficult to keep the t e m p e r a t u r e o f the fluid within ( T d - - T o ) <<0.0037°C for a long d u r a t i o n to maintain R a < R a c ~ 665. (4) Ultimately, it is essential to establish a stand a r d null result by eliminating thermal convection and other effects using i m p r o v e d differential accelerometers with an o p t i m a l choice o f geometry and viscosity at a location with fiat terrain and uniform ground mass density. After establishing the null result, a heavy load o f materials can be brought to one side o f the experimental set-up to test whether the spheres (one or two) m o v e away from the heavy load. The a u t h o r wishes to t h a n k T.R. Palfrey for helpful suggestions and discussions on the subject.
References [ 1] P. Thieberger, Phys. Rev. Lett. 58 (1987) 1066. [2] T.D. Lee and C.N. Yang, Phys. Rev. 98 (1955) 1501. [3] E. Fischbach et al., Phys. Rev. Lett. 56 (1986) 3. [4] R.V. EStv6s, D. Pekar and E. Feteke, Ann. Phys. (Leipzig) 68 (1922) 11. [5] Y.E. Kim, Phys. Lett. B 177 (1986) 255, and references therein. [6] Y.E. Kim, and D.J. Klepacki, Theoretial analysis for geophysical determination of the gravitational constant, Purdue Nuclear Theory Group preprint PNTG-86-11 (July 1986). [7] Y.E. Kim, D.J. Klepacki and W.J. Hinze, Mass-density profile and the geophysical determination of the gravitational constant, Purdue Nuclear Theory Group preprint PNTG86-12 (July 1986), Nature, to be published. [8] A. De Rtijula, Phys. Lett. B 180 (1986) 213. [9] S.Y. Chu and R.H. Dicke, Phys. Rev. Lett. 57 (1986) 1823. [ 10] D.L. Turcotte and E.R. Oxburgh, J. Fluid Mech. 28 (1967) 29. [ 11 ] D.L. Turcotte and G. Schubert, Geodynamics (Wiley, New York, 1982). [12] Lord Rayleigh, Phil. Mag. 32 (1916) 529. [ 13 ] R.C. Weast, Handbook of chemistry and physics, 57th ed. (CRC Press, Cleveland, OH, 1976).