By Kathleen A. Rosser
December 15, 2020

Abstract

It is shown that the mathematical expression for galactic orbital acceleration a_g, as proposed by the modified theory of gravity known as Modified Newtonian Dynamics (MOND), takes the same form as that for the total surface area S of a cone, including the area of the base, such that S=ηa_g where η is a conversion factor of dimension [rt²]. The physical meaning of this coincidence, if any, is unknown. For a galaxy of mass M, there exists an infinite set of corresponding cones. The radius R of the base of the cone depends on the selection of η, while these quantities in turn fix the height H. Two models are presented here. The first employs a cone of height H=r, for r the distance from the galactic center to the orbiting star. The second model employs a cone with R=r.

I. INTRODUCTION

The equation for the total surface area S of a cone of height H and radius of base R is

S = πR² + πR²√(1 + H²/R²)                       (1)

where the first term is the area of the base, and the second term is the surface area of the body of the cone. Interestingly, this equation takes the same form as the equation for the observed radial acceleration a_g of a star orbiting the center of a galaxy, as given by the formalism known as Modified Newtonian Dynamics (MOND) under the simple interpolation function discussed by H.S. Zhao, B. Famaey and J. Binney [1,2], to be defined below.

The MOND formalism was first proposed in 1983 by Mordehai Milgrom [3], and provides a phenomenological description of the observed discrepancy in the rotation velocity v(r) of galaxies, where r is the radial distance from the galactic center. This discrepancy, apparent in differential Doppler shift observations, arises from the fact that the outer stars and hydrogen clouds of galaxies orbit too fast to be accounted for by the Newtonian gravitational attraction of ordinary baryonic matter alone. Indeed, outside the bright galactic disk, v(r) does not fall off as r^─1/2, as would be expected from Newtonian gravity [4], but approaches a constant as r increases. This anomaly, referred to as the flattening of the galactic rotation curve, extends outward to the limits of observation, often many times the visible radius of the galaxy.

Astronomers generally attribute the excess velocity to the presence of an invisible dark matter halo, which if it existed, would consist of mysterious particles as yet unidentified possessing properties incompatible with the standard model of particle physics.  Hence, the rotation anomaly remains one of the most baffling problems in astrophysics.  MOND, in contrast, depends on ordinary baryonic matter alone without invoking dark matter, and has proven highly accurate in describing the orbital motion of galaxies [5].

 

II. THE MOND FORMALISM

The MOND formalism is based on the empirical relation

μ(a_g/a₀) a_g = a_N                                   (2)

in which observed radial acceleration a_g is expressed in terms of expected Newtonian gravitational acceleration a_N = ─ GM/r². This relation, by means of the interpolating function μ(a_g/a₀), describes the smooth transition from the inner galaxy, where acceleration falls off as 1/r², to the outlying region beyond the visible galactic disk, called the deep MOND region, where acceleration falls off as 1/r.

The quantity a₀ is assumed to be a universal constant with dimensions of acceleration. Its value is given by

a₀ = 1.2×10^─8cm/sec² ≅ H₀/2π = c²/R_U.

Here H₀ is the Hubble parameter [6], and R_U is roughly 2π times the radius of the visible universe, or the de Sitter radius.

Using the so-called simple interpolating function

μ(a_g/a₀) = (a_g/a₀) / (1 + a_g/a₀)

investigated by Zhao, Famaey and Binney, the MOND relation of Eq. (2) can be expressed, after some algebra, as a quadratic equation with solution

a_g = ─ GM/2r² ─ GM/2r² √(1+4r²a₀/GM)

=  a_N /2  +  (a_N /2)√(1─ 4a₀/a_N)                   (3)

for a_N the Newtonian acceleration. Equivalently

a_g = a_N /2 + (a_N /2)√(1+4r²/R₀²).

where R₀, to be called the MOND radius, lies inside the edge of the bright galactic disk and is defined here as

R₀ := √(GMR_U/c²) = √(GM/a₀).

 

III. THE CORRESPONDENCE BETWEEN CONIC AREA AND MOND ACCELERATION

Comparing Eqs. (1) and (3), it is clear that the expression for observed acceleration a_g takes the same form as that for the total surface area S of a cone

S = πR² + πR²√(1 + H²/R²)

with a_N /2 corresponding to the area πR² of the base, and 4r²a₀/GM corresponding to H²/R². To calculate this correspondence, we relate the first terms in Eqs. (1) and (3) by means of a conversion factor η with dimension [rt²] that transforms acceleration into area, such that

πR² =  ─ η GM/2r² = η a_N / 2.

Hence the conversion factor η is given by

η = 2πR² / a_N.                                  (4)

Furthermore, the height H and radius R of the cone obey the relation

H²/R² = ─ 4a₀/a_N.

Hence height H may be expressed as

H = 2R√( ─ a₀ / a_N ).                            (5)

The radius R of the base is so far undetermined, and may correspond to any physical length. This degree of freedom allows a variety of models in which orbital acceleration corresponds to cones of different sizes and orientations. Two simple models will be considered below.

IV. TWO MODELS FOR CORRESPONDING CONES

The two simplest models for cones of area S=ηa_g are characterized by the correspondences: 1) H=r, and 2) R=r..

Model 1) is characterized by the correspondence H=r. We thus have from Eq. 5:

r = 2R√( ─ a₀ / a_N ).

Solving for R, we obtain

R = (r/2) √( ─ a₀ / a_N )

which can be expressed in terms of the MOND radius R₀ of the galaxy as

R=R₀/2.

The conversion factor  can be found from Eq. (4) as

η = ─ πr² / 2a₀ = ─ A_r / 2a_0,

where A_r=πr² is the area enclosed by the orbit of the rotating star. Thus, model 1) employs a cone whose height H is the radial distance to the star, and whose base has a radius R equal to half the MOND radius of the galaxy.

Model 2) is characterized by the correspondence R=r. From Eq. (5) we obtain

H = 2r√( ─ a₀ / a_N ) = 2r² / R₀

with conversion factor

η = ─ 2A_r / 2a_N.

Thus, for model 2), the area of the base of the cone corresponds to the area A_r of the star’s orbit, while the height of the cone is proportional to A_r and inversely proportional to the MOND radius of the galaxy.

V. DISCUSSION

The two models presented here are the simplest known examples of the correspondence between total conic surface area and the MOND acceleration. These models, however, offer little insight into why galaxies exhibit MOND behavior. Other more complex models have been studied, possibly motivating further speculation.

The galactic rotation curve has eluded explanation for many decades, and poses a dilemma of such magnitude it may be necessary to explore unusual ideas that appear to lack physical basis. Indeed, MOND itself is purely phenomenological, with no known basis in physical principle.

References:

[1] Zhao, H.S. and Famaey, B., Refining MOND interpolating function and TeVeS Lagrangian, arXiv:astro-ph/0512425v3 (2006)

[2] Benoit Famaey and James Binney, Modified Newtonian Dynamics in the Milky Way, arXiv:astro-ph/0506723v2 (2005)

[3] Milgrom M., ApJ 270, 365 (1983)

[4] Jacob D. Bekenstein, Relativistic gravitation theory for the modified Newtonian dynamics paradigm, Phys Rev D 70, 083509 (2004)

[5] Robert H. Sanders and Stacy S. McGaugh, Modified Newtonian Dynamics as a alternative to dark matter, arXiv:astro-ph/0204521v1 2002

[6] Lasha Berezhiani and Justin Khoury, Theory of dark matter superfluidity, Phys Rev D 92, 103510 (2015)