Inverse Square Law Of Gravity Research Articles

A Celebration of Kepler’s Third Law: Harmony, Power Laws, and Keplerian Orbits

Kepler’s greatest achievements are contained in his three laws, which today reappear constantly at the cutting edge of modern astrophysics. The modified third law has been the major workhorse of astronomy and astrophysics in determining the mass of objects throughout the universe. Kepler never revealed why he chose the integers, 2 and 3, to test using Tycho’s data. Because of his attachment to Pythagorean ideas and his knowledge of musical theory, I suggest he was guided by the musical interval known as the perfect fifth, for which the ratio of frequencies is 3/2. The perfect fifth is the most consonant of all intervals except the octave, and, as such, is the basis of all the tuning of stringed instruments. If Kepler had recognized the significance of the perfect fifth in analyzing Tycho’s data, it suggests a very pleasing historical parallelism between music and astronomy. In discovering the third law, Kepler also chanced upon the world's first known power law, which is now found in many forms throughout the earth and heavens. In discovering that the Galilean moons of Jupiter also obeyed the third law, Kepler encountered the phenomenon of scale independence, which is responsible for the ubiquity of power laws across the universe. The third law also played a crucial role in Newton’s discovery of the inverse square law of gravity in 1666. Not only did it provide Newton with a crucial mathematical step, but the third law also had the authority of Tyco’s observations. Christopher Wren and Edmund Halley relied similarly upon the third law for their apparent independent discoveries of the inverse square law.

Gravitational lensing and missing mass

The mass of an astrophysical object can be estimated by the amount of gravitational lensing of another object that it causes. To arrive at the estimation however, one assumes the validity of the inverse square law of gravity, or equivalently an attractive 1/r potential. We show that the above, augmented by a logarithmic potential at galactic length scales, proposed earlier to explain the flat galaxy rotation curves, predicts a larger deflection angle for a given mass. In other words, the true mass of the object is less than its estimated value. This may diminish the importance and role of dark matter in explaining various observations.

Lorentz violation and Sagnac gyroscopes

Sagnac gyroscopes with increased sensitivity are being developed and operated with a variety of goals including the measurement of General-Relativistic effects. We show that such systems can be used to search for Lorentz violation within the field-theoretic framework of the Standard-Model Extension, and that competitive sensitivities can be achieved. Special deviations from the inverse square law of gravity are among the phenomena that can be effectively sought with these systems. We present the necessary equations to obtain sensitivities to Lorentz violation in relevant experiments.

A search for deviations from the inverse square law of gravity at nm range using a pulsed neutron beam

Recently published results and ongoing experimental efforts to search for deviations from the inverse square law of gravity at the nanometer length scale using slow neutron scattering from the noble gases are discussed. Using the pulsed slow neutron beamline BL05 at the Materials and Life Sciences Facility at J-PARC, we measured the neutron momentum transfer (q) dependence of the differential scattering cross section for the noble gases He, Ne, Ar, Kr, and Xe. By comparing to the distributions obtained using pseudo-experimental Monte Carlo simulations and forming ratios between Xe and He, we placed an upper bound on the strength of a new interaction as a function of interaction length λ which improved upon previous results in the region λ < 0.1 nm, and remains competitive in the larger λ region. Additionally we describe how we are using our technique to extract relative values of the total neutron scattering cross sections of the noble gases, as well as how we plan to measure the neutron-electron scattering length using the NOVA instrument on BL21 at J-PARC.

Search for deviations from the inverse square law of gravity at nm range using a pulsed neutron beam

We describe an experimental search for deviations from the inverse square law of gravity at the nanometer length scale using neutron scattering from noble gases on a pulsed slow neutron beamline. By measuring the neutron momentum transfer ($q$) dependence of the differential cross section for xenon and helium and comparing to their well-known analytical forms, we place an upper bound on the strength of a new interaction as a function of interaction length $\lambda$ which improves upon previous results in the region $\lambda < 0.1\,$nm, and remains competitive in the larger $\lambda$ region. A pseudoexperimental simulation developed for this experiment and its role in the data analysis described. We conclude with plans for improving sensitivity in the larger $\lambda$ region.

Ronald S. Calinger. Leonhard Euler: Mathematical Genius in the Enlightenment. xvii + 669 pp., illus., apps., bibl., index. Princeton, N.J./Oxford: Princeton University Press, 2016. $55, £40.95 (cloth). Leonhard Euler. Correspondence. Edited by Franz Lemmermeyer and Martin Mattmüller. 2 parts. (Opera Omnia, 4A: Commercium Epistolicum, 4.) xiii + 1,248 pp., illus., figs., bibl., index. Basel: Springer, 2015. $458, £148.50 (cloth).

Ronald S. Calinger. <i>Leonhard Euler: Mathematical Genius in the Enlightenment</i>. xvii + 669 pp., illus., apps., bibl., index. Princeton, N.J./Oxford: Princeton University Press, 2016. $55, £40.95 (cloth). Leonhard Euler. <i>Correspondence</i>. Edited by Franz Lemmermeyer and Martin Mattmüller. 2 parts. (Opera Omnia, 4A: Commercium Epistolicum, 4.) xiii + 1,248 pp., illus., figs., bibl., index. Basel: Springer, 2015. $458, £148.50 (cloth).

Quantised inertia from relativity and the uncertainty principle

It is shown here that if we assume that what is conserved in nature is not simply mass-energy, but rather mass-energy plus the energy uncertainty of the uncertainty principle, and if we also assume that position uncertainty is reduced by the formation of relativistic horizons, then the resulting increase of energy uncertainty is close to that needed for a new model for inertial mass (MiHsC, quantised inertia) which has been shown to predict galaxy rotation without dark matter and cosmic acceleration without dark energy. The same principle can also be used to model the inverse square law of gravity, and predicts the mass of the electron.

22pAX-7 マイクロメートルスケールにおける重力の逆二乗則検証のためのNewton-Vの開発

22pAX-7 マイクロメートルスケールにおける重力の逆二乗則検証のためのNewton-Vの開発

An experimental 'Life' for an experimental life: Richard Waller's biography of Robert Hooke (1705).

Richard Waller's 'Life of Dr Robert Hooke', prefixed to his edition of Hooke's Posthumous Works (1705), is an important source for the life of one of the most eminent members of the early Royal Society. It also has the distinction of being one of the earliest biographies of a man of science to be published in English. I argue that it is in fact the first biography to embrace the subject's natural-philosophical work as the centre of his life, and I investigate Waller's reasons for adopting this strategy and his struggle with the problem of how to represent an early experimental philosopher in print. I suggest that Waller eschews the 'Christian philosopher' tradition of contemporary biography - partly because of the unusually diverse and fragmentary nature of Hooke's intellectual output - and draws instead upon the structure of the Royal Society's archive as a means of organizing and understanding Hooke's life. The most quoted phrase from Waller's biography is that Hooke became 'to a crime close and reserved' in later life; this essay argues that Waller's biographical sketch was fashioned in order to undo the effects of that reserve. In modelling his approach very closely on the structure of the society's records he was principally concerned with making Hooke's work and biography accessible, intelligible and useful to the fellowship in a context familiar to them, a context which had provided the institutional framework for most of Hooke's adult life. I argue that Waller's 'Life' was also intended to make the largest claims for Hooke's intellectual standing that the author dared in the context of the enmity between Hooke and Isaac Newton once the latter became president of the Royal Society. However, I also adduce fresh manuscript evidence that Waller actually compiled, but did not publish, a defence of Hooke's claim to have discovered the inverse square law of gravity, allowing us to glimpse a much more assertive biography of Hooke than the published version.

Testing long-distance modifications of gravity to 100 astronomical units

There are very few direct experimental tests of the inverse square law of gravity at distances comparable to the scale of the Solar System and beyond. Here we describe a possible space mission optimized to test the inverse square law at a scale of up to 100 AU. For example, sensitivity to a Yukawa correction with a strength of $10^{-7}$ times gravity and length scale of 100 AU is within reach, improving the current state of the art by over two orders of magnitude. This experiment would extend our understanding of gravity to the largest scale that can be reached with a direct probe using known technology. This would provide a powerful test of long-distance modifications of gravity including many theories motivated by dark matter or dark energy.

Preliminary limits on deviation from the inverse-square law of gravity in the solar system: a power-law parameterization

New physics beyond the standard model of particles might cause a deviation from the inverse-square law of gravity. In some theories, it is parameterized by a power-law correction to the Newtonian gravitational force, which might originate from the simultaneous exchange of particles or modified and extended theories of gravity. Using the supplementary advances of the perihelia provided by INPOP10a (IMCCE, France) and EPM2011 (IAA RAS, Russia) ephemerides, we obtain preliminary limits on this correction. In our estimation, we take the Lense-Thirring effect due to the Sun's angular momentum into account. The parameters of the power-law correction and the uncertainty of the Sun's quadrupole moment are simultaneously estimated with the method of minimizing χ2. From INPOP10a, we find N = 0.605 for the exponent of the power-law correction. However, from EPM2011, we find that, although it yields N = 3.001, the estimated uncertainty in the Sun's quadrupole moment is much larger than the value given by current observations. This might be caused by the intrinsic nonlinearity in the power-law correction, which makes the estimation very sensitive to the supplementary advances of the perihelia.

Gravity resonance spectroscopy constrains dark energy and dark matter scenarios.

We report on precision resonance spectroscopy measurements of quantum states of ultracold neutrons confined above the surface of a horizontal mirror by the gravity potential of Earth. Resonant transitions between several of the lowest quantum states are observed for the first time. These measurements demonstrate that Newton's inverse square law of gravity is understood at micron distances on an energy scale of 10-14 eV. At this level of precision, we are able to provide constraints on any possible gravitylike interaction. In particular, a dark energy chameleon field is excluded for values of the coupling constant β>5.8×108 at 95% confidence level (C.L.), and an attractive (repulsive) dark matter axionlike spin-mass coupling is excluded for the coupling strength gsgp>3.7×10-16 (5.3×10-16) at a Yukawa length of λ=20 μm (95%C.L.).

New upper limits on deviation from the inverse-square law of gravity in the solar system: a Yukawa parameterization

New physics beyond the standard model of particles might cause deviation from the inverse-square law of gravity. In many theoretical models of modified gravity, it is parameterized by the Yukawa correction to the Newtonian gravitational force in terms of two parameters α and λ. Here α is a dimensionless strength parameter and λ is a length scale. Using the supplementary advances in perihelia provided by INPOP10a and EPM2011 ephemerides, we obtain new upper limits on the deviation from the inverse-square law when the uncertainty of the Sun's quadrupole moment is taken into account. We find that INPOP10a yields the upper limits as α = 3.1 × 10−11 and λ = 0.15 au, and EPM2011 gives α = 5.2 × 10−11 and λ = 0.21 au. In both of them, α is at least 10 times less than the previous results.

On the (im)possibility of testing new physics in exoplanets using transit timing variations: deviation from inverse-square law of gravity

Ground-based and space-borne observatories studying exoplanetary transits now and in the future will considerably increase the number of known exoplanets and the precision of the measured times of transit minima. Variations in the transit times can not only be used to infer the presence of additional planets, but might also provide opportunities for testing new physics in the places beyond the Solar system. In this work, we take deviation from the inverse-square law of gravity as an example, focus on the fifth-force-like Yukawa-type correction to the Newtonian gravitational force which parameterizes this deviation, investigate its effects on the secular transit timing variations and analyze their observability in exoplanetary systems. It is found that the most optimistic values of Yukawa-type secular transit timing variations are at the level of $\sim 0.1$ seconds per year. Those values unfortunately appear only in rarely unique cases and, most importantly, they are still at least two orders of magnitude below the current capabilities of observations. Such a deviation from the inverse-square law of gravity is likely too small to detect for the foreseeable future. Meanwhile, systematic uncertainties, such as the presence of additional and unknown planets, will likely be exceptionally difficult to remove from a signal that should be seen.

Was Newton right? A search for non-Newtonian behavior of weak-field gravity

Empirical tests of Einstein's metric theory of gravitation, even in the non- relativistic, weak-field limit, could play an important role in judging theory-driven extensions of the current Standard Model of fundamental interactions. Guided by Galileo's work and his own experiments, Newton formulated a theory of gravity in which the force of attraction between two bodies is independent of composition and proportional to the inertia of each, thereby transparently satisfying Galileo's empirically informed conjecture regarding the Universality of Free Fall. Similarly, Einstein honored the manifest success of Newton's theory by assuring that the linearized equations of GTR matched the Newtonian formalism under classical conditions. Each of these steps, however, was explicitly an approximation raised to the status of principle. Perhaps, at some level, Newtonian gravity does not accurately describe the physical interaction between uncharged, unmagnetized, macroscopic bits of ordinary matter. What if Newton were wrong? Detecting any significant deviation from Newtonian behavior, no matter how small, could provide new insights and possibly reveal new physics. In the context of physics as an empirical science, for us this yet unanswered question constitutes sufficient motivation to attempt precision measurements of the kind described here. In this paper we report the current status of a project to search for violation of the Newtonian inverse square law of gravity.

Yukawa effects on the clock onboard a drag-free satellite

The Yukawa correction to the Newtonian gravitational force is accepted as a parameterization of deviations from the inverse-square law of gravity which might be caused by new physics beyond the standard model of particles and the general theory of relativity. We investigate these effects on the clock onboard a drag-free satellite: dynamics of the satellite and influence on the time transfer link. We find the Yukawa signal in the time transfer is much harder to detect with current state of clocks than those effects on the dynamics, especially the secular change of periastron, by laser ranging in the case of an artificial Earth satellite carrying a frequency standard with an orbit of $a=10^7$\,m and $e=0.01$.

Noniterative Instantaneous Impact Point Prediction Algorithm for Launch Operations

T HE instantaneous impact point (IIP) in launch operations is defined as the touchdown point of a rocket with an assumption of an immediate end of the propelled flight [1–3]. The IIP is very important information to judge whether the flight is normal or not, and it should be continuously monitored during the whole flight so that proper actions be taken in an emergency to protect human life and property. Therefore, the real-time prediction and monitoring of the IIP is one of the key safety requirements. The IIP prediction problem seeks for the intersection point between the Earth’s surface and the free-flight trajectory of a rocket subject to gravity and other disturbing forces. The current position and velocity vectors are given as inputs to the problem. A numerical integration of the dynamicmodel including all force elementswith an initial position and velocity could yield a very accurate answer to the problem with relative ease. But when the real-time calculation is considered as a constraint, the numerical integration that requires a lot of computational resources is not a feasible option in the majority of launch operations. There has been some research on developing the IIP prediction algorithm that provides the real-time solution within acceptable accuracy with limited computational resources. To calculate the IIP for shortto midrange rockets, simplified models in the localvertical–local-horizontal frame are frequently used [3–6]. These assume that the trajectory of a rocket is parabolic subject to the constant vertical downward gravitational acceleration and often add correction terms to compensate for the modeling errors. Montenbruck and Markgraf [6] developed a linearized IIP calculation algorithm and implemented it for the launch operation of the Maxus-5 sounding rocket. Their algorithm was based on a simple parabolic flight model, and its prediction accuracy was improved by considering the effects of the Earth curvature, the Coriolis force, and the gravity field variation. The Keplerian algorithm predicts the IIP using the equations describing the motion of a rocket under the inverse-square law of gravity. Regan and Anandakrishnan [4] and Zarchan [7] developed a series of formulations that can be used as the base of theKeplerian IIP prediction, such as the expressions for the flight angle and the flight time. Currently, an iterativeKeplerian algorithm is frequently used to predict the IIP for long-range space vehicles [8,9]. The iterative Keplerian algorithm can calculate the true IIP solution for the Keplerian motion of a rocket using an iterative procedure (e.g., successive substitution) to find parameters such as Kepler’s f and g series expansions, the eccentric anomaly of the impact point, and the surface radius at impact. Considering that the IIP prediction for an actual launch operation should be carried out in real time, the iterative procedure that has the risk of computational overload is not very desirable to be included in the prediction algorithm. Especially when the algorithm is implemented in an onboard computer that has limited computing resources but is responsible for multiple mission-critical tasks such as event sequencing, navigation, and attitude control, the potential damage associated with the risk is significantly increased. This engineering Note presents a Keplerian IIP prediction algorithm that does not require any iterative procedure and is thus remarkably faster than the iterative Keplerian algorithm. First, an exact and closed solution for the Keplerian IIP with the spherical Earth model is developed. Then, a noniterative procedure to compensate for the effect of the oblateness of the Earth is added. Finally, the validity of the proposed algorithm is proved through the performance comparison with the iterative algorithm in terms of computational efficiency and the prediction accuracy.

QBounce, the Quantum Bouncing Ball Experiment

We give a short description of the qBounce-experiment. The dynamics of the wavefunction of a neutron in the gravity potential has been observed and transitions between discrete energy level have been realized. The experiment allows to test Newton's inverse-square law of gravity with quantum objects by means of quantum interference at short distances.

From Moon-fall to motions under inverse square laws

The motion of two bodies, along a straight line, under the inverse square law of gravity is considered in detail, progressing from simpler cases to more complex ones: (a) one body fixed and one free, (b) both bodies free and identical mass, (c) both bodies free and different masses and (d) the inclusion of electrostatic forces for both bodies free and different masses. The equations of motion (EOM) are derived starting from Newton's second law or from conservation of energy. They are then reduced to dimensionless EOM using appropriate scales for time and distance. Solutions of the dimensionless EOM as well as the original EOM are given. The time interval for the bodies to fall is expressed as a function of the distance fallen. Formulae for the inverse were obtained. The coalescence times for the different cases are (a) where L is the initial separation of the two bodies and m1 is the mass of the fixed body, (b) and (c) where mT is the total mass of the two bodies and (d) where and is a measure of the ratio of the electrostatic force to gravity. The last formula may also be used when with the interpretation that there is no collision if t is infinity or imaginary. We also discuss this motion along the straight line as a special case of the general elliptic motion of two bodies. I believe that this paper will be useful to university tutors as well as undergraduate and even graduate students who prefer to consider the special case before the general case, and their relationship.

Development of a second generation torsion balance based on a spherical superconducting suspension

This paper describes the development of a second generation superconducting torsion balance to be used for a precision measurement of the Casimir force and a short range test of the inverse square law of gravity at 4.2 K. The instrument utilizes niobium (Nb) as the superconducting element and employs passive damping of the parasitic modes of oscillation. Any contact potential difference between the torsion balance and its surroundings is nulled to within approximately 50 mV by applying known DC biases and fitting the resulting parabolic relationship between the measured torque and the applied voltage. A digital proportional-integral-derivative servo system has been developed and characterized in order to control the azimuthal position of the instrument. The angular acceleration and displacement noise are currently limited by the capacitive sensor at the level 3x10(-8) rad s(-2)/ squarerootHz and 30 nm/ squarerootHz at 100 mHz. The possibility of lossy dielectric coatings on the surface of the torsion balance test masses is also investigated. Our measurements show that the loss angles delta are (1.5+/-2.3)x10(-4) and (2.0+/-2.2)x10(-4) at frequencies of 5 and 10 mHz, respectively. These values of loss are not significant sources of error for measurements of the Casimir force using this experimental setup.