Alagar Ramanujam.G1, Prakash.R2, Sivanesh
Pranav. U.V3
Abstract
The Concept of MOND (Modifying Newtonian Dynamics) was proposed by Mordehai Milgrom as a possible way to reconcile the difference between the experimentally observed high values and the calculated values (using Newton’s Law of Gravity) for the dynamical parameters of orbiting stars in a galaxy, without having to introduce the concept of dark matter. Milgrom’s MOND concept challenges the need for dark matter to account for the above difference.
The experimentally observed velocity rotation curves of stars in a galaxy show that for small values of r (distance of the star from the centre of the galaxy), the velocity observed (VO) for the orbiting star fairly agrees with values (VN) calculated using Newton’s law of gravity. But as r increases, the difference between VO and VN gradually increases. For very large values of r, VO increases with a constant slope. Finally, VO becomes fairly constant with distance. The above features of VO cannot be explained by Newton’s law of gravity. Milgrom successfully showed that the above features can be explained by modifying Newton’s law of gravity as
where μ is a function just added by Milgrom without a supporting theory behind and is assumed to have certain special properties to suit the purpose.
In this paper, we show that if we take into consideration the attenuation of the dark energy emitted by the objects due to absorption by the Space medium surrounding the objects, we are led to a modified Newton’s Law of Gravity, with a correction term in it. This correction term surprisingly gives rise to the required properties of the function μ added by Milgrom to the existing conventional law of gravity. The work presented here therefore can be considered as a theoretical support for the successful phenomenological scheme proposed
by Milgrom.
Key words: Dark Matter, Dark Energy, MOND, Alagar – Uma Mass formula, Axioms for Space.
1. Corresponding author, Former Professor of Physics, NGM College, Pollachi, India – 642002,
E-mail: gravity2003@gmail.com, Orchid-id: https://orchid.org/0000-0002-8024-0738
2. Engineer, M.E, M.A, Tamilnadu, India – 641004, E-mail: srfmetals@gmail.com
3. UG Scholar, Tamilnadu, India – 629171, E-mail: dr.sivaneshpranav@gmail.com
Introduction
The abnormally high experimental values (Krumm et al 1977, Salpeter et al 1978,
Bosma 1978,1981a,1981b, Rubin et al 1978,1980,1982) observed for the dynamical
properties associated with the motion of stars in a galaxy which are unexplainable by Newton’s Law of Gravity pose (Faber et al 1979, Rood 1981) a great challenge to the astrophysicists and induce them to consider the following three possibilities.
1. There exists a large quantity of unseen matter (Dark matter) in a galaxy, which boosts the values for the above-mentioned dynamical parameters beyond what would be expected on the basis of the visible mass alone.
2. Instead of requiring the existence of dark matter, the second possibility is that
Newton’s law of gravity needs a suitable modification and by this modification, the experimentally observed values for the dynamical parameters of stars in a galaxy can be accounted for.
3. There are scholars who need a modification of Newton’s law of gravity and also require the presence of dark matter to explain the above experimentally observed dynamical parameters of the motion of stars in a galaxy (Oliver Pignard 2022).
The first possibility leads to dark matter hypothesis and the second possibility leads to MOND proposed by Mordehai Milgrom (Milgrom 1983a, 1983b, 1983c and Bekenstein et al 1984).
As for the first possibility, the extra mass required to explain the difference is termed as dark matter (Missing mass or Hidden mass). For the past four decades various theories have been proposed regarding the nature of dark matter. As of today, there is no experimental support for the presence of any kind of dark matter. As an alternative approach, Mordehai Milgrom’s MOND scheme modifies Newton’s law of gravity to fit the experimental data without invoking dark matter. As a mere assumption, Milgrom replaces Newton’s law
and the function μ is tailored to suit the experimentally observed values.
Milgrom assumes that the acceleration a of a test particle in a gravitating system is given by
where gN is the conventional gravitational acceleration and a0 is a characterised constant for a given galaxy with the dimension of acceleration. The function
is so constructed that it has the following requirements.
Milgrom’s work, spanning over a few decades has met with a fair success and gained a good reputation among the current astrophysicists. However, his assumptions regarding the function μ and its properties do not stem from a theory
To quote Milgrom: “Successful as it may be, MOND remains at present as a limited phenomenological theory. By phenomenological, I mean that it has not been motivated by, and is not constructed on fundamental principles”. This observation of Milgrom provides a strong motivation for our work presented in this paper
2. Dark Energy and its Attenuation
Ever since Newton deduced his law of gravity from Kepler’s laws of planetary motion, the problem of deriving the gravity formula from suitable axioms is there posing a great challenge to the physics community. As Richard Feynman remarked: “No machinery has ever been invented that “explains” gravity without also predicting some other phenomenon that does not exist.” (Richard Feynman 1963) We have also the famous statement of Newton: “I do not pretend to know the cause of gravity……” (Newton’s Letter to Richard Bentley, 1692). About two decades ago, one of us (Alagar Ramanujam 2009), working on the premise that
Gravity is not attraction (Higgs 1964, Peebles et al 2003, Andrew Janiak 2004, Padmanabhan 2010, Eric Verlinde 2010, Davis et al 2022, Bernal Thalman 2023) but a phenomenon arising out of the interplay between the compressive space pressure on every object and the dark energy emitted by the object trying to mitigate the compressive pressure; obtained a formula for the mass (M) of a macro object as
M = 𝜷 𝑨 (𝑪 − 𝑹) ……………(1)
where A is the area of the object, C is the space pressure on the object, R is the dark energy repulsive flux emitted by the object and β is a universal constant
The above Alagar-Uma Mass Formula, was obtained from a set of axioms framed for the primordial space in which the universe is floating. Using the above mass formula, they (Alagar Ramanujam et al 2017, 2019, 2020a, 2020b and Vijay 2019) succeeded in obtaining the first ever derivation of Newton’s law of gravity.
In this paper their derivation is revisited taking into account the attenuation of the dark energy emitted by the objects due its absorption by the space surrounding the objects.
Let M be the mass of the supermassive object at the centre of a galaxy with surface area A and m be the mass of a star orbiting around it at a distance r. Let C be the compressive pressure on M due to space. The compressive thrust AC acting on the mass M from all directions is the same as the compressive thrust acting on the imaginary sphere of radius r. It is the compressive thrust that permeates through the imaginary sphere of radius r, finally arrives on the object M (Fig.1). So, the compressive pressure (Cr) due to space on the unit
area of the imaginary sphere will be
The intensity of the total repulsive flux AR emanating from the object M will go on
decreasing due to the attenuation by the space medium.
Let f(r) be the amount of repulsive flux absorbed by the space of volume
where x is the radius of the central massive object. Then (𝑨𝑹 − 𝒇(𝒓)) will be the repulsive total flux that will pass through the imaginary sphere. So, the repulsive pressure at the distance r will be given by
In view of this, the net inward pressure at the distance r will be
As r increases, f(r) will increase reaching a maximum at r = r0 when the total repulsive dark energy flux AR is completely absorbed. Hence, we write
where t is the flux absorbed by unit volume of the space.
The force 𝑭𝒎 on the orbiting star of mass m will be
𝑭𝒎 = 𝑲 𝒎 𝒑𝒓…………….(6)
where 𝑲 is a proportionality constant whose value depends on the nature of the
material of the mass m. The dimension of 𝑲 will be [L2 M-1]. Combining Eqns.(3) and (6), for the force acting on m we have
is a characteristic factor of the galaxy under consideration. Its dimension is [L-3]. Dimension of G is [L2 M-1] [LT-2] = [L3 M-1 T-2]. Eqn.(7) shows that the modified gravity force law between any two masses, here M and m, has a correction term and hence differs from the usual Newton’s law of gravity. With this correction term taken into account we may call Eqn.(7) as dark energy attenuation based Modified Newtonian Law of Gravity (MNLG) or Vethathirian Law of Gravity (VLG). In what follows we discuss the consequences of this extra correction term.
If we don’t consider the absorption of the repulsive dark energy flux by the space medium between the super massive object M and the star m, f(r) will be zero and in that case the Eqn. (7) reduces to the usual Newton’s law of gravity. Thus, the attenuation of the dark energy flux by the space gives rise to a modification to the existing Newton’s law of gravity.
3. Theoretical support to Milgrom’s MOND concepts
A significant aspect of the work presented here is our concept that every object emits dark energy waves in space and once emitted their intensity gradually decreases due to the gradual absorption of the dark energy waves by the space medium. Due to the attenuation of the dark energy waves by the space medium, the derived expression given in Eqn. (7) becomes different from the traditional Newton’s law of gravity. As a first application of the modified Newtonian law of gravity, we show below that the correction term(𝟏 + 𝛚 (𝒓𝟑 − 𝒙𝟑)) in Eqn.
(7) meets the requirements of the function 𝝁 of Milgrom as expressed in the Introduction. This comes as a total surprise!
From Eqn. (7) we can write for the modified acceleration of an orbiting star in the
galaxy as
𝒂𝒎 = 𝒈𝒏(𝟏 + 𝝎(𝒓𝟑 − 𝒙𝟑)) …………………..(8)
is the Newtonian acceleration of the star at a distance r from the centre of the galaxy and am is the acceleration of the star according to the modified Newtonian law of gravity. Since am will be always greater than gn, Eqn. (8) has the potential to explain the abnormally high experimental values for the accelerations of the stars. To speak in the language of Milgrom we have
As r decreases, Eqn. (9) shows that
tends to become a very large quantity and 𝒂𝒎 approaches 𝒈𝒏 since 𝝎(𝒓𝟑 − 𝒙𝟑) decreases. This property of Eqn. (9) supports Milgrom’s requirement that for large values of
tends to 1, so that 𝒂 = 𝒈𝒏. For large values of r beyond the value of r0, since the correction term 𝝎(𝒓𝟑 − 𝒙𝟑) becomes maximum and remains the same, we have from Eqn. (9)
and becomes a very small quantity since r ≫ r0 Let us consider a distance r in the periphery area of the galaxy such that
For this distance r, which is far greater than r0,
will be extremely small. Substituting for r2 in Eqn. (8) we get
Thus Eqn. (10) supports Milgrom’s another requirement expressed by him as follows. For small values of
Since the requirements of Milgrom’s function
are shown to flow from our correction term, the work presented here may be considered as a theoretical support to Milgrom’s work. It is our view that all his results based on his function
are physically possible and worth considering.
4. Conclusion
Derivation of Newton’s law of gravity presented here is a first of its kind. Our stand is, as already mentioned, gravity is not attraction but a phenomenon arising out of an interplay between space pressure on the object and the dark energy repulsive pressure emanating from the object. The attenuation of dark energy by the space medium gives rise to a correction term to the existing Newton’s law of gravity. It is this correction term that lends a theoretical
support to the work of Milgrom, spread over four decades, to explain the abnormally high experimental values for the dynamical parameters of the orbiting stars in a galaxy, without any necessity to introduce dark matter
Acknowledgement
It’s a pleasure to thank Mr. Umamaheswaran K and Mrs Vakeeswari S R for very many useful discussions
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