On the Trigonometric Correction of One Powerful Formula

An attempt is presented for the description of the magnitude of Newton’s gravitational force in the experiments with a horizontal torsion balance. There were developed many experimental arrangements in order to find experimentally the value of big G – the Newtonian gravitational constant – after the Michell-Cavendish experiment in 1798. The geometrical configurations of test and source masses play a very important role in these experiments. The old trigonometric function “sagitta” used by Johannes Kepler and Isaac Newton was newly employed as the trigonometric tool for the determination of the magnitude of Newton’s gravitational force between the source mass and the test mass. Based on the known experimental configurations with the horizontal torsion balance we have found that the “true” Michell-Cavendish configuration is not dependent on the space orientation. This “sagitta” function can be experimentally tested in the Karagioz-Izmailov configuration and the Karagioz-Izmailov-Gillies-Gershteyn configuration with the technology available at begin of our century. A proposal for the decomposition of the big G was presented. This concept could not be experimentally tested before the discoveries of the dipole in the cosmic microwave background radiation and the Pioneer anomaly.


Introduction
Gravitational interaction, described by the Newton's law (1687) of gravitation F = Gm 1 m 2 /d 2 where F represents force of attraction between two particles having masses m 1 and m 2 and d is the distance between these particles, continues to attract experimental and theoretical researchers.While the absolute values of other fundamental constants are known with high accuracy, the accuracy of experimental determination of G is below that of other fundamental constants.There are two main actual topics in this feld of the experimental physics.From one side the development of the experimental techniques steadily continues to design more precise instruments in order to come closer to the absolute value of G. From the other side the experimental research is focused on the elimination and description of numerous factors bringing their influence on the data distribution of obtained G values in different top laboratories.A determination of G is conceptually easy: measure the force two known masses arranged in a known geometry and compare the result with the Newton's law of gravitation.But the experimental realization of these measurements requires the greatest attention to all parameters: many of them are already known but some of them are still hidden.Big G measurements are full of elegance, complexity, subtlety, and beauty.Experimentalists try to answer the permanent question: can we find some hidden parameters behind these experiments and to find mathematical tools for their quantitative description and prediction?
The actual state of the art of the experimental determination of G was recently summarized in the special issue of the Philosophical Transactions A (2014): "Theo Murphy Meeting Issue 'The Newtonian constant of gravitation, a constant too difficult to measure?" T.J. Quinn and C.C. Speake concluded that main focus should be given to the explanation of the uncertainty of the value G. Contributions presented at this Meeting surveyed in details the actual experimental situation of the determination of G and proposed new guidelines for the near future research.B. M. Wood discussed the present situation of gravitational constant experimental results.See all 13 contributions focused on the big G in this special issue.
One aim of this contribution is to present a proposal for a geometric arrangement of the experiments with a horizontal torsion balance to precisely control the directions of the gravitational forces F 1 and F 2 -the directions of attracting forces between two source masses and two test masses.The second aim of this contribution is to present a possible trigonometric correction for the magnitudes of the gravitational forces │F 1 │ and │F 2 │ -the magnitudes of attracting forces of two source masses on two test masses.Finally, a working formula for the possible decomposition of big G is presented.

Directions of the Attracting Forces F 1 and F 2
The geometrical arrangement of these experiments plays a very important role.Two test masses are placed on ends of a horizontal torsion balance.This idea was proposed by John Michell and firstly experimentally demonstrated by Henry Cavendish.The positions of two source masses in the space around the test masses have to be precisely defined.During the past two hundred years numerous experimental arrangements were realized in order to study the influence of the spacial positions of the source masses around the test masses while their center of gravity should be identified with the center of the horizontal torsion balance.During last thirty years several new experimental configurations were realized.The most stable values of G were found for configurations with four source masses and four test masses suspended by a horizontal torsion balance.
There are two methods of measuring the gravitational constant: static and dynamic.The static method was realized by Michell-Cavendish.The dynamic method was developed in several stages by Ferdinand Reich (1852), Karl Braun (1897) and Rolánd Eötvös (1922).
There were proposed several alternatives to the horizontal torsion balance during the 19 th century.E.g., Phillip J.G. von Jolly (1881) performed these experiments with a common balance.Johannes Wilsing (1887) used a brass vertical pendulum.
In 1964 Jesse Wakefield Beams proposed a very important innovation for the determination of G.It is a classical Michell-Cavendish horizontal torsion balance but the whole system is placed on virtually frictionless rotary table.In this arrangement we can investigate possible space orientation influences of the attracting forces between the test masses and source masses.The first measured results for this arrangement published Rose et al. (1969).The rotary table experiments were further improved and developed by Gundlach and Merkowitz (2000) and Chao Xue et al. (2014).

Gravity Centers of Individual Test Masses and Source Masses Lay on the Same Horizontal Circle
This is a classical geometrical arrangement proposed by Michell-Cavendish.Two test masses have been placed on a horizontal torsion balance in such a way that their common center of gravity is in the middle of the torsion balance.Two source masses are so positioned that their individual centers of gravity lay on the same horizontal circle and their common center of gravity is identified with the center of the torsion balance.The whole system can be placed on the rotary table and we get the Michell-Cavendish-Beams arrangement.

Gravity Centers of Two (or Four) Test Masses Lay on the Inner Horizontal Circle, Gravity Centers of Two (or Four) Source Masses Lay on the Outer Circle and a Combined Configuration
Two test masses have been placed on a horizontal torsion balance in such a way that their common center of gravity is in the middle of the torsion balance.Horizontal rotation of these test masses forms an inner circle.Two source masses are so positioned that their individual centers of gravity lay on the outer horizontal circle in the same height as the inner circle.The common center of gravity of these source masses is identified with the center of the torsion balance.The whole system can be placed on the rotary table and we get the Heyl-Beams-Rose arrangement.Paul R. Heyl (1930) made very important contributions to this kind of experimental arrangement.This horizontal torsion balance system was further developed by Luther and Towler (1982), Quinn and Speake with their coworkers (2001, 2013) (Quinn-Speake configuration), Armstrong and Fitzgerald (2003), Hu and Luo (2004) (Hu-Luo configuration), Luo et al. (2009). www.ccsen

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A working Formula for the Newton's Gravitational Law
There is a tendency to decompose the Newton's gravitational constant into its parts in order to get some additional information about the gravitational field around the source mass.
Our working formula for the decomposition of the Newton's gravitational constant is: (6) This formula is composed from several parts: -the first part describes the reciprocal value of the density of gravitons in the spherical shell with radius R and with the layer thickness λ -the wavelength of the gravitational wave, -the second part is the frequency ν of the gravitational wave, -the third part is the specific mass evaporation constant H 0 describing the evaporation of gravitons with mass m graviton from the source mass M S per unit of time [kg kg -1 s -1 ].The numerical value is expected to be identical with the Hubble constant known from the astrophysical experiments, -the fourth part is the product of the source mass M S and the test mass M T per spherical surface.
This concept can be experimentally tested: If the source mass M S = 1 kg and the radius equals R = 1 m we can get from CODATA 2010 big G value = (6.67384± 0.00080)*10 -11 m 3 kg -1 s -2 the product c*H 0 = 8,386…*10 -10 ms -2 and H 0 = 86.26…kms - Mpc -1 .These values depend on the experimental value of the big G.In this case the letter c describes the longitudinal speed (there is the very well known constant |c| -the modulus of the light speed -used in the Maxwell equations).
There are three experimental fields that can be used for the test of this concept: 1. Modified Newtonian dynamics MOND -Mordeham Milgrom (1999) introduced a new approach to this field and with many followers continue to develop the MOND concept.
2. Hubble constant experimental search -this is a very active field of research since the time of Edwin Powell Hubble -see the web site of John P. Huchra (2008).The distance of objects for the determination of the Hubble constant plays a very important role.We should try to find this value for objects with very well defined short distances.It will be interesting if we will find the experimental value close to H 0 = 86.26…kms - Mpc -1 calculated from the big G.See numerous contributions of astrophysicists to this topic.
-Pioneer anomaly: John D. Anderson (1998) discovered the Pioneer anomaly and with co-workers found the anomalous acceleration a pio = (8.74± 1.33)*10 -10 ms -2 .Though, their interpretation goes into other direction, we should continue in this research and further experimentally tune this acceleration value.It will be interesting to investigate how the values of the big G and the Hubble constant will behave in further progress with more sophisticated instruments.

Conclusions
The geometrical configuration of source and test masses for the experimental determination of the Newtonian gravitational constant G plays a very critical role.The old trigonometric function "sagitta" was newly used and a possible trigonometric correction of big G was proposed.Three geometrical configurations were quantitatively analyzed with predictions: 1) "true" Michell-Cavendish configuration is not dependent on the space orientation -a null experiment.2) Karagioz-Izmailov configuration with two source masses is expected to be dependent on the space orientation towards the constellation Crater.3) Karagioz-Izmailov-Gillies-Gershteyn configuration with one source mass is expected to be dependent on the space orientation towards the constellation Crater.One working formula for the decomposition of big G was presented with proposals for further experimental research.This concept could not be experimentally tested before the discoveries of the dipole in the cosmic microwave background radiation and the Pioneer anomaly.