Newtonian fractional-dimension gravity and disk galaxies

Abstract

This paper continues previous work on a novel alternative model of gravity, based on the theory of fractional-dimension spaces applied to Newton’s law of gravitation. In particular, our Newtonian fractional-dimension gravity is now applied to axially symmetric structures, such as thin/thick disk galaxies described by exponential, Kuzmin, or other similar mass distributions. As in the case of spherically symmetric structures, which was studied in previous work on the subject, we examine a possible connection between our model and modified Newtonian dynamics, a leading alternative gravity model, which accounts for the observed properties of galaxies and other astrophysical structures without requiring the dark matter hypothesis. By relating the MOND acceleration constant \(a_{0} \simeq 1.2 \times 10^{-10}\,\text{ m } \text{ s}^{ -2}\) to a natural scale length \(l_{0}\) of our model, namely \(a_{0} \approx \mathrm{GM}/l_{0}^{2}\) for a galaxy of mass M, and by using the empirical Radial Acceleration Relation, we are able to explain the connection between the observed radial acceleration \(g_\mathrm{obs}\) and the baryonic radial acceleration \(g_\mathrm{bar}\) in terms of a variable local dimension D. As an example of this methodology, we provide a detailed rotation curve fitting for the case of the field dwarf spiral galaxy NGC 6503.

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Notes

  1. 1.

    Since \( \nabla _{D}\) is defined in terms of dimensionless coordinates, the physical dimensions for the gravitational potential \(\phi \) in Eq. (3) are the same as those for the gravitational field \({\mathbf {g}}\), i.e., both quantities will be measured in \(\text{ m } \text{ s}^{ -2}\). Therefore, the Newtonian potential is obtained as \(\phi _\mathrm{Newt} =l_{0}\phi _{D =3}\left( {\mathbf {w}}\right) \), from the NFDG potential in the first line of Eq. (3) for fixed \(D =3\).

  2. 2.

    Actually, MOND predictions were recovered for any positive value for the constant C in equation (9), showing that M can be considered as an arbitrary reference mass of the galactic structure being studied in our model.

  3. 3.

    For other possible expansions of the NFDG potential \(\phi \sim 1/\left| {\mathbf {w}} -{\mathbf {w}}^{ \prime }\right| ^{D -2}\), see also [54,55,56]. In particular, Ref. [55] illustrates additional expansions of the Euler kernel \(\left( z -x\right) ^{ -\nu }\), in terms of Jacobi, Gegenbauer, and Chebyshev polynomials. However, these alternative expressions have proven difficult to be used in the current work. Therefore, we opted to base our analysis on the expansion in Eq. (18).

  4. 4.

    We have also considered other possible choices for the \(\alpha _{R}\), \(\alpha _{z}\) values, such as having fractional dimension only in the radial direction (\(\alpha _{R} =D -2\), \(\alpha _{\varphi } =1\)), or only in the angular direction (\(\alpha _{R} =1\), \(\alpha _{\varphi } =D -2\)), for the thin-disk case (\(\alpha _{z} =1\)). The results do not differ much from those obtained with our preferred choice (\(\alpha _{R} =\alpha _{\varphi } =\frac{D -1}{2}\)), so we will not report them in this work. Also, results obtained with our preferred choice are somewhat in between those obtained with the other two extreme choices; thus, our choice for the \(\alpha _{i}\) parameters can be considered a good average between all possible alternatives.

  5. 5.

    The Pochhammer’s symbol \(\left( a\right) _{l}\) is defined as \(\left( a\right) _{0} =1\), \(\left( a\right) _{l} =a\left( a +1\right) \left( a +2\right) \ldots \left( a +l -1\right) =\Gamma \left( a +l\right) /\Gamma \left( a\right) \).

  6. 6.

    All the numerical computations (and some of the analytical ones) in this work were performed with Mathematica, Version 12.1.1.0, Wolfram Research Inc.

  7. 7.

    In this figure, as well as in the other similar figures of this section, we assumed \(D =3\) at the origin \(w_{R} =0\).

  8. 8.

    Using MOND functions \(\nu _{n}\), instead of \({\widehat{\nu }}_{n}\), yields very similar results for all values of n. Therefore, we have considered only the \({\widehat{\nu }}_{n}\) family of MOND interpolation functions as was also done in our previous paper I.

  9. 9.

    In the top-left panel of Fig. 3, we extrapolate the dimension functions below \(w_{R} \approx 0.45\) by assuming \(D =3\) at the origin.

  10. 10.

    In this figure, we limited the range of \(w_{r}\) between 0 and 10. Plotting the top-left panel for \(w_{r} \gg 10\) would show that \(D \rightarrow 2\) for large values of \(w_{r}\).

  11. 11.

    In this case, we set \(\genfrac(){}{}{g_\mathrm{obs}}{g_\mathrm{bar}} \rightarrow 1\) for \(w_{r} \rightarrow 0\) (Newtonian behavior at the origin) as was done in paper I. Thus, the ratio \(\genfrac(){}{}{g_\mathrm{obs}}{g_\mathrm{bar}}\) does not diverge for \(w_{r} \rightarrow 0\), as in the previous cases analyzed in this paper.

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Acknowledgements

This work was supported by a Faculty Sabbatical Leave granted by Loyola Marymount University, Los Angeles. The author also wishes to acknowledge Dr. Howard Cohl for his advice regarding computations with special functions, Dr. Federico Lelli for sharing NGC 6503 galactic data files and other useful information, and the anonymous reviewer for useful comments and suggestions.

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Varieschi, G.U. Newtonian fractional-dimension gravity and disk galaxies. Eur. Phys. J. Plus 136, 183 (2021). https://doi.org/10.1140/epjp/s13360-021-01165-w

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