# Newtonian Fractional-Dimension Gravity and Galaxies without Dark Matter

## Abstract

**:**

## 1. Introduction

## 2. NFDG and the Ultra-Diffuse Galaxy AGC 114905

## 3. NFDG and the Dynamics of NGC 1052-DF2

## 4. NFDG and the Bullet Cluster

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. NFDG Virial Theorem and Spherical Structures

## Notes

1 | For a general overview of Newtonian fractional-dimension gravity, also see the NFDG website [4]. |

2 | The alternative radial profiles given by the direct interpolation of the mass densities (solid-orange and solid-blue curves in Figure 1) were also tested, but were found to yield results which were very similar to those based on the analytical profiles in Equation (4), so these alternative results will be omitted here. |

3 | For a more comprehensive analysis of NGC 1052-DF2 within the context of four popular alternative theories of gravity (MOND, CG, MOG, and Verlinde’s emergent gravity), see Ref. [55]. |

4 | This model by Giusti et al. [57,58,59] is based on a different way of extending the Poisson equation, $\Delta \mathrm{\Phi}\left(\mathbf{r}\right)=4\pi G\rho \left(\mathbf{r}\right)$ into a fractional Poisson equation, ${(-\Delta )}^{s}\mathrm{\Phi}\left(\mathbf{r}\right)=-4\pi G{l}_{0}^{2-2s}\rho \left(\mathbf{r}\right)$, where ${(-\Delta )}^{s}$ is the fractional Laplacian and $s\in [1,3/2]$ is the fractional index. The Newtonian case is recovered for $s=1$. This fractional Laplacian is defined in terms of fractional operators, while our NFDG is based on standard operators acting on a metric space of fractional dimension D [1,7]. |

5 | It should be noted that the positive/negative error values for ${\sigma}_{gc}$ correspond to the negative/positive error values for the dimension D, as shown in Equation (9). |

6 |

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**Figure 2.**NFDG results for AGC 114905. Top panel: NFDG variable dimension $D\left(R\right)$ based directly on the galaxy data (red-solid curve) and NFDG variable dimension ${D}_{m}\left(R\right)$ based on the mass dimension of a fractal system (red-dotted curve), are compared with fixed values $D=3.1$, $D=3.0$, and $D=2.9$ (dashed lines). Bottom panel: NFDG rotation curves (circular velocity vs. radial distance) compared to the original data [40,48] (black circles with error bars). The NFDG best fit for the variable dimension $D\left(R\right)$ is shown by the red-solid line; the NFDG fit for ${D}_{m}\left(R\right)$ is shown by the red-dotted line; while NFDG fits for fixed values $D=3.1$, $D=3.0$, and $D=2.9$ are shown, respectively, by the green-dashed, black-dashed, and blue-dashed lines. The original computation by Mancera Piña et al. [40,48] of the contributions expected from the stars (orange dot-dashed line), from gas (cyan dot-dashed line), and total baryonic mass (magenta, dot-dashed line) is also shown. The asymptotic flat velocity band (horizontal gray band) is instead based on the data/errors of the last experimental point.

**Figure 3.**NFDG velocity dispersion as a function of the variable dimension, following Equation (8) (red curve and related orange error band). The black-circle data point and related errors are assumed to correspond the stars’ velocity dispersion for $D=3$, while the red-circle data point and orange-circle error points are assumed to be related to the globular-cluster velocity dispersion with a variable dimension D.

R (kpc) | ${\mathit{v}}_{\mathit{circ}}$ (km s${}^{-1}$) | D (R) |
---|---|---|

$1.05$ | $8.1\pm 2.1$ | $2.69$ |

$3.18$ | $15.7\pm 4.3$ | $2.58$ |

$5.29$ | $22.3\pm 4.1$ | $2.16$ |

$7.41$ | $21.9\pm 4.7$ | $3.20$ |

$9.53$ | $22.5\pm 5.8$ | $2.92$ |

**Table 2.**NFDG fractional dimension D for the Bullet Cluster system for different choices of the reference mass ${M}_{ref}$.

${\mathit{M}}_{\mathbf{ref}}$ | D |
---|---|

${M}_{m}^{bar}+{M}_{b}^{bar}$ | $2.43$ |

${M}_{m}^{bar}$ | $2.45$ |

${M}_{b}^{bar}$ | $2.53$ |

$\frac{{M}_{m}^{bar}{M}_{b}^{bar}}{{M}_{m}^{bar}+{M}_{b}^{bar}}$ | $2.54$ |

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Varieschi, G.U.
Newtonian Fractional-Dimension Gravity and Galaxies without Dark Matter. *Universe* **2023**, *9*, 246.
https://doi.org/10.3390/universe9060246

**AMA Style**

Varieschi GU.
Newtonian Fractional-Dimension Gravity and Galaxies without Dark Matter. *Universe*. 2023; 9(6):246.
https://doi.org/10.3390/universe9060246

**Chicago/Turabian Style**

Varieschi, Gabriele U.
2023. "Newtonian Fractional-Dimension Gravity and Galaxies without Dark Matter" *Universe* 9, no. 6: 246.
https://doi.org/10.3390/universe9060246