# Galaxy Rotation Curve Fitting Using Machine Learning Tools

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## Abstract

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## 1. Introduction

## 2. Rotation Curve Data and Methodology

#### 2.1. Data Selection

#### 2.2. Mass Models of the MW

**(i)**The bulge is going to be modeled by two exponential spheroids according to [1,7], one to model the inner bulge and the other to model the main bulge. The matter density for such models, each one providing two free parameters, is:

**(ii)**The disk is going to be modeled by an exponential flat disk as studied in [7]. The surface mass density of such a disc provides for two free parameters (${\Sigma}_{d}$, ${a}_{d}$), and reads

**(iii)**Both the dark central compact object together with the DM halo will be modeled by the semi-analytical (extended) RAR model, which, as explained in the Introduction, is based on a self-gravitating system of fermions at finite temperature including for escape of particles and central fermion-degeneracy. This DM model was extensively studied in [2,6] and references therein for the Milky Way, and in [9] for other galaxy types. It has four free-parameters $(m,{\theta}_{0},{W}_{0},{\beta}_{0})$ with m the DM, particle mass, and (${\beta}_{0}$, ${\theta}_{0}$, ${W}_{0}$) the dimensionless parameters evaluated at the origin, reading for the temperature, degeneracy, and cut-off particle energy, respectively. The free-RAR model parameters enter in the underlying phase-space DF of the fermions at (quasi) equilibrium, whose formula is given in Equation (3) below. Interestingly, it can be demonstrated [12,13] that such a Fermi–Dirac-like DF is a quasi-stationary solution of a kinetic theory equation (of Fermionic–Landau form) via the application of a maximum entropy principle. Thus it is a most-probable coarse-grained DF at violent relaxation, extending the original results of Lynden–Bell on the subject.

#### 2.3. Gradient Descent Method: A Machine Learning Tool

**p**stands for the independent variable of the function F and $\gamma $ is a parameter called learning rate whose aim is to regulate the “length” of the steps. If this formula is implemented recursively, one can eventually go closer and closer to a minimum of F. An illustration of the procedure followed by the gradient descent method is shown in Figure 3. In that image can be seen a solid dark path, which is the result of evaluating F in the different points given by Equation (4). Since such a formula uses “minus” the gradient of the function, the path followed by the method is oriented to the direction of “maximum” decreasing, driving to the deepest point of F.

**p**is the vector of the (physical) free parameters that characterize the full mass model (baryons + DM). In this work we fit six free parameters (four in the RAR DM halo plus two of the Freeman disk), adopting the bulge free parameters as detailed in Section 2.2. The predicted circular velocities of the different mass models are denoted with $V({r}_{i},\mathbf{p})$ and ${v}_{i}$ are the observed ones, C is a normalization constant and N is the number of observations. The idea is to fit the free parameters of the model mentioned above to the overall rotation curve (milliparsec inner point + Grand RC).

## 3. Results

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | The compactness of the fermion-core is inversely proportional to m [6], and thus it is shown that for $m<48$ keV the core is too extended to fit within the S-2 star pericenter, while for $m>345$ keV the solutions are unstable since the critical value for collapse to a BH is reached at $m=345$ keV. |

2 | http://www.ioa.s.u-tokyo.ac.jp/~sofue/htdocs/2017paReview/ (accessed on 15 July 2022). |

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**Figure 1.**Overall Milky Way RC used to constrain the gravitational potential of the Galaxy. It is composed by the Grand Rotation Curve from [1] (light-blue and orange dots) and an inner (milliparsec scale) keplerian velocity (grey dot) of an S-cluster star as caused by the central object in SgrA*.

**Figure 2.**Density profile of the RAR model constrained in this work by the gradient descent method explained in the main text. It can be seen the constant-density core below the mili-pc scale (which is governed by quantum degeneracy pressure) and the transition to the plateau at ∼1 pc (where quantum effects are negligible), while far above ∼1 kpc, it follows a polytropic tail (see also [9]).

**Figure 3.**Illustration of the path followed by the gradient descent method to reach the minimum of a two-dimensional function. Taken from: https://easyai.tech/en/ai-definition/gradient-descent/ (accessed on 10 May 2023).

**Figure 4.**Loss function against the step or epochs of the method. It can be seen that it is a decreasing function, with little bumps at the tail on the right, indicating a clear minimization trend reaching the value of $0.000071979$ after 1500 epochs.

**Figure 5.**Best-fit circular velocity curve result of the implementation of the gradient descent method. It is remarkable the very good precision achieved in almost all the data-points in few hours CPU time, despite some minor deficiencies at the ∼10

^{−1}kpc scale.

**Table 1.**Best-fit parameters results after applying the gradient descent method to the set of Milky Way observables under the gravitational potential model described in the main text.

Parameter | Seed Value | Final Value |
---|---|---|

m [keV/c${}^{2}$] | $56.0$ | $54.809$ |

${\theta}_{0}$ | $37.766$ | $37.809$ |

${W}_{0}$ | $66.341$ | $66.449$ |

${\beta}_{0}$ | $1.1977\times {10}^{-5}$ | $1.1139\times {10}^{-5}$ |

${\Sigma}_{d}$ [${M}_{\odot}$/kpc${}^{2}$] | $5.9658\times {10}^{8}$ | $1.0882\times {10}^{9}$ |

${a}_{d}$ [kpc] | $4.9$ | $3.0039$ |

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**MDPI and ACS Style**

Argüelles, C.R.; Collazo, S.
Galaxy Rotation Curve Fitting Using Machine Learning Tools. *Universe* **2023**, *9*, 372.
https://doi.org/10.3390/universe9080372

**AMA Style**

Argüelles CR, Collazo S.
Galaxy Rotation Curve Fitting Using Machine Learning Tools. *Universe*. 2023; 9(8):372.
https://doi.org/10.3390/universe9080372

**Chicago/Turabian Style**

Argüelles, Carlos R., and Santiago Collazo.
2023. "Galaxy Rotation Curve Fitting Using Machine Learning Tools" *Universe* 9, no. 8: 372.
https://doi.org/10.3390/universe9080372