PHYSICAL REVIEW LETTERS week ending PRL98,150801(2007) 13 APRIL 2007 Laboratory Test of Newton's Second Law for Small Accelerations J.H.Gundlach,S.Schlamminger,C.D.Spitzer,and K.-Y.Choi Center for Experimental Nuclear Physics and Astrophysics,University of Washington,Seattle,Washington 98195,USA B.A.Woodahl Physics Department,Indiana University-Purdue University,Indianapolis,Indiana 46202,USA J.J.Coy Earth and Space Science Department,Saint Joseph's College,Rensselaer.Indiana 47978,USA E.Fischbach Physics Department,Purdue University,West Lafayette,Indiana 47907,USA (Received 12 February 2007;published 13 April 2007) We have tested the proportionality of force and acceleration in Newton's second law,F=ma,in the limit of small forces and accelerations.Our tests reach well below the acceleration scales relevant to understanding several current astrophysical puzzles such as the flatness of galactic rotation curves,the Pioneer anomaly,and the Hubble acceleration.We find good agreement with Newton's second law at accelerations as small as 5 x 10-14 m/s2. DOI:10.1103/PhysRevLett.98.150801 PACS numbers:06.30.Gv.04.80.Cc Newton's second law is the equation of motion defining standard Newtonian dynamics.The functional form of the the field of dynamics.In its nonrelativistic form,F=ma transition between the two regimes is not specified.A is perhaps the most famous and most often used equation of smooth transition can be obtained by multiplying the physics.Together with its relativistic and quantum me- right side of F=ma by u(a/ao)=a/ao(1+a2/ap)-12 chanical variants,this law is implicitly tested in many so that for a>do the function u1 and standard applications and experiments,and its validity is simply Newtonian mechanics is recovered.The characteristic ac- assumed at all acceleration scales.Any deviation from celeration ao was determined from fits [4]to galactic F=ma would have profound consequences as it would rotation curves to be ao≈l.2×l0-iom/s2 imply a violation of crucial conservation laws such as Further testing of Newton's second law is motivated by energy and momentum in their conventional definition. the Pioneer anomaly.Doppler-tracking data of the Pioneer At very small accelerations a deviation from Newton's 10 and 11 spacecraft shows an unmodeled acceleration at second law could remain hidden in most laboratory scale distances >15 AU of ao 9x 10-10 m/s2 roughly point- experiments,but might appear in astrophysical and cos- ing towards the Sun [5].Exhaustive efforts have been mological observations. undertaken to find a conventional explanation for this One observed fact is the flatness of galactic rotation effect,so far without success.It is also interesting to note curves.The tangential velocity of stars measured as a that the Hubble accelerationacH7X 10-10 m/s2 function of distance from the galactic center rises first provides a natural acceleration scale,with H being the and flattens for larger distances.Newton's second law Hubble constant.Furthermore,it is also conceivable that together with the gravitational effect of known matter a violation of Newton's second law could play a role in predicts a decrease in the velocities for larger distances, explaining the acceleration of the Universe at large dis- and dark matter has been introduced to resolve this dis- tances,which is usually attributed to dark energy. crepancy [1].Alternatively,Milgrom discovered that Here we report the results of a laboratory experiment to Newton's second law can be modified with a single addi- test Newton's second law using a torsion pendulum.Our tional parameter ao to describe the measured galactic system is significantly different from that used in an earlier rotation curves extremely well without invoking dark mat- experiment by Abramovici and Vager [6],who interfero- ter [2,3].While Milgrom's full formalism MOND (modi- metrically measured the acceleration of a pendulum mass fied Newtonian dynamics)is untestable in the laboratory, in response to an applied electric field.Abramovici and since it requires the absence of accelerations in all direc- Vager found agreement with Newton's second law at ac- tions,a modification of Newtonian dynamics provides a celerations as small as 3 x 10-11 m/s2.In our experiment simple explanation of the galactic rotation curves.Milgrom we utilize the fact that as a torsion pendulum passes suggested that Newton's second law would smoothly tran- through equilibrium its acceleration (relative to the labo- sition from F x a to Fx a2/do at a ao.Hence,for a< ratory)is zero.If the torsional amplitude is small,the time ao a force would yield a larger acceleration as compared to spent experiencing small accelerations and small forces 0031-9007/07/98(15)/150801(3) 150801-1 2007 The American Physical SocietyLaboratory Test of Newton’s Second Law for Small Accelerations J. H. Gundlach, S. Schlamminger, C. D. Spitzer, and K.-Y. Choi Center for Experimental Nuclear Physics and Astrophysics, University of Washington, Seattle, Washington 98195, USA B. A. Woodahl Physics Department, Indiana University-Purdue University, Indianapolis, Indiana 46202, USA J. J. Coy Earth and Space Science Department, Saint Joseph’s College, Rensselaer, Indiana 47978, USA E. Fischbach Physics Department, Purdue University, West Lafayette, Indiana 47907, USA (Received 12 February 2007; published 13 April 2007) We have tested the proportionality of force and acceleration in Newton’s second law, F
ma, in the limit of small forces and accelerations. Our tests reach well below the acceleration scales relevant to understanding several current astrophysical puzzles such as the flatness of galactic rotation curves, the Pioneer anomaly, and the Hubble acceleration. We find good agreement with Newton’s second law at accelerations as small as 5 1014 m=s2. DOI: 10.1103/PhysRevLett.98.150801 PACS numbers: 06.30.Gv, 04.80.Cc Newton’s second law is the equation of motion defining the field of dynamics. In its nonrelativistic form, F~
ma~ is perhaps the most famous and most often used equation of physics. Together with its relativistic and quantum me￾chanical variants, this law is implicitly tested in many applications and experiments, and its validity is simply assumed at all acceleration scales. Any deviation from F~
ma~ would have profound consequences as it would imply a violation of crucial conservation laws such as energy and momentum in their conventional definition. At very small accelerations a deviation from Newton’s second law could remain hidden in most laboratory scale experiments, but might appear in astrophysical and cos￾mological observations. One observed fact is the flatness of galactic rotation curves. The tangential velocity of stars measured as a function of distance from the galactic center rises first and flattens for larger distances. Newton’s second law together with the gravitational effect of known matter predicts a decrease in the velocities for larger distances, and dark matter has been introduced to resolve this dis￾crepancy [1]. Alternatively, Milgrom discovered that Newton’s second law can be modified with a single addi￾tional parameter a0 to describe the measured galactic rotation curves extremely well without invoking dark mat￾ter [2,3]. While Milgrom’s full formalism MOND (modi- fied Newtonian dynamics) is untestable in the laboratory, since it requires the absence of accelerations in all direc￾tions, a modification of Newtonian dynamics provides a simple explanation of the galactic rotation curves. Milgrom suggested that Newton’s second law would smoothly tran￾sition from F / a to F / a2=a0 at a  a0. Hence, for a  a0 a force would yield a larger acceleration as compared to standard Newtonian dynamics. The functional form of the transition between the two regimes is not specified. A smooth transition can be obtained by multiplying the right side of F~
ma~ by
a=a0
a=a01a2=a2 0 1=2, so that for a a0 the function
’ 1 and standard Newtonian mechanics is recovered. The characteristic ac￾celeration a0 was determined from fits [4] to galactic rotation curves to be a0  1:2 1010 m=s2. Further testing of Newton’s second law is motivated by the Pioneer anomaly. Doppler-tracking data of the Pioneer 10 and 11 spacecraft shows an unmodeled acceleration at distances >15 AU of a0  9 1010 m=s2 roughly point￾ing towards the Sun [5]. Exhaustive efforts have been undertaken to find a conventional explanation for this effect, so far without success. It is also interesting to note that the Hubble acceleration aH
cH  7 1010 m=s2 provides a natural acceleration scale, with H being the Hubble constant. Furthermore, it is also conceivable that a violation of Newton’s second law could play a role in explaining the acceleration of the Universe at large dis￾tances, which is usually attributed to dark energy. Here we report the results of a laboratory experiment to test Newton’s second law using a torsion pendulum. Our system is significantly different from that used in an earlier experiment by Abramovici and Vager [6], who interfero￾metrically measured the acceleration of a pendulum mass in response to an applied electric field. Abramovici and Vager found agreement with Newton’s second law at ac￾celerations as small as 3 1011 m=s2. In our experiment we utilize the fact that as a torsion pendulum passes through equilibrium its acceleration (relative to the labo￾ratory) is zero. If the torsional amplitude is small, the time spent experiencing small accelerations and small forces PRL 98, 150801 (2007) PHYSICAL REVIEW LETTERS week ending 13 APRIL 2007 0031-9007=07=98(15)=150801(3) 150801-1 © 2007 The American Physical Society