# New Strong Constraints on the Central Behaviour of Spherical Galactic Models

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Statement

**STATEMENT:**

## 3. A Five-Dimensional Class of Spherically Symmetric Galactic Models

## 4. Mathematical Discussion

- 1.
**NFW model**—It corresponds to $\alpha =1$, $\beta =3$, $\gamma =1$. The integrated mass profile is$$M\left(r\right)=4\pi {\rho}_{0}\phantom{\rule{0.166667em}{0ex}}{a}^{3}\left[\mathrm{ln}\left(1+\frac{r}{a}\right)-\frac{r}{r+a}\right]\phantom{\rule{3.33333pt}{0ex}},$$- 2.
**Dehnen models**—They correspond to $\alpha =1$, $\beta =4$, $\gamma <3$. The integrated mass profile is$$M\left(r\right)=\frac{4\pi {\rho}_{0}{a}^{3}}{3-\gamma}{\left(\frac{r}{r+a}\right)}^{3-\gamma}\phantom{\rule{3.33333pt}{0ex}}.$$- 3.
**Hernquist model**—It is the particular case of the Dehnen models with $\gamma =1$. Hence, conditions (2) and (3) are met but condition (4) is not, and we have $g\left(r\right)\to -2\pi G{\rho}_{0}\phantom{\rule{0.166667em}{0ex}}a$ as $r\to 0$. As a consequence, the Hernquist model is physically incorrect in the neighbourhood of the centre.- 4.
**Jaffe model**—It is a particular case of the Dehnen models with $\gamma =2$. Thus, only condition (2) is obeyed but conditions (3) and (4) are not, and we have ${v}_{c}^{2}\left(r\right)\to 4\pi G{\rho}_{0}\phantom{\rule{0.166667em}{0ex}}{a}^{2}$ as $r\to 0$ and $g\left(r\right)\to -\phantom{\rule{0.166667em}{0ex}}\infty $ as $r\to 0$. Therefore, the Jaffe model is physically incorrect towards the centre.- 5.
**Pseudo-isothermal sphere**—It corresponds to $\alpha =2$, $\beta =2$, $\gamma =0$. The integrated mass profile is$$M\left(r\right)=4\pi {\rho}_{0}\phantom{\rule{0.166667em}{0ex}}{a}^{2}\left[r-\phantom{\rule{0.166667em}{0ex}}a\phantom{\rule{0.166667em}{0ex}}{\mathrm{tan}}^{-1}\left(\frac{r}{a}\right)\right]\phantom{\rule{3.33333pt}{0ex}}.$$- 6.
**Modified Hubble profile**—It corresponds to $\alpha =2$, $\beta =3$, $\gamma =0$. The integrated mass profile is$$M\left(r\right)=\frac{4\pi {\rho}_{0}\phantom{\rule{0.166667em}{0ex}}{a}^{3}}{{r}^{2}+{a}^{2}}-\left[({r}^{2}+{a}^{2})\phantom{\rule{0.166667em}{0ex}}{\mathrm{sinh}}^{-1}\left(\frac{r}{a}\right)-a\phantom{\rule{0.166667em}{0ex}}r{\left(1+\frac{{r}^{2}}{{a}^{2}}\right)}^{1/2}\right]\phantom{\rule{3.33333pt}{0ex}}.$$- 7.
**Perfect sphere model**—It corresponds to $\alpha =2$, $\beta =4$, $\gamma =0$. The integrated mass profile is$$M\left(r\right)=\frac{2\pi {\rho}_{0}\phantom{\rule{0.166667em}{0ex}}{a}^{3}}{{r}^{2}+{a}^{2}}\phantom{\rule{0.166667em}{0ex}}\left[({r}^{2}+{a}^{2})\phantom{\rule{0.166667em}{0ex}}{\mathrm{tan}}^{-1}\left(\frac{r}{a}\right)-a\phantom{\rule{0.166667em}{0ex}}r\right]\phantom{\rule{3.33333pt}{0ex}}.$$- 8.
**Plummer sphere model**—It corresponds to $\alpha =2$, $\beta =5$, $\gamma =0$. The integrated mass profile is$$M\left(r\right)=\frac{4}{3}\pi {\rho}_{0}{r}^{3}{\left(1+\frac{{r}^{2}}{{a}^{2}}\right)}^{-3/2}\phantom{\rule{3.33333pt}{0ex}}.$$

## 5. Real Spheroidal Ellipticals and Bulges

- Gebhardt et al. (2000) find $A=1.2\pm 0.2$ and $\mathrm{\Gamma}=3.75\pm 0.3$ [16]. These authors choose the definition of $\sigma \left(0\right)$ within the slit aperture of length $2{R}_{e}$, where ${R}_{e}$ is the bulge effective radius.
- Merritt and Ferrarese (2001) obtain $A=1.30\pm 0.36$ and $\mathrm{\Gamma}=4.72\pm 0.36$ [17]. They use the standard definition to evaluate $\sigma \left(0\right)$ inside ${R}_{e}/8$.
- Tremaine et al. (2002) obtain $A=1.36\pm 0.19$ and $\mathrm{\Gamma}=4.02\pm 0.32$ [15]. They estimate $\sigma \left(0\right)$ with a variety of techniques.

- (1)
- Radial distance: $r/{R}_{e}$.
- (2)
- Mass density: $\rho /{\rho}_{0}$.
- (3)
- Integrated mass profile: $M\left(r\right)/\left(4\pi \phantom{\rule{0.166667em}{0ex}}{\rho}_{0}\phantom{\rule{0.166667em}{0ex}}{R}_{e}^{3}\right)$.
- (4)
- Square circular velocity: ${v}_{c}^{2}\left(r\right)/\left(4\pi G\phantom{\rule{0.166667em}{0ex}}{\rho}_{0}\phantom{\rule{0.166667em}{0ex}}{R}_{e}^{2}\right)$.
- (5)
- Gravitational field: $g\left(r\right)/\left(2\pi G\phantom{\rule{0.166667em}{0ex}}{\rho}_{0}\phantom{\rule{0.166667em}{0ex}}{R}_{e}\right)$.

## 6. NFW Model

#### 6.1. NFW Model and Regular Galaxy Clusters

#### 6.2. NFW Model and Dark Matter Halos

## 7. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Plummer, H.C. On the Problem of Distribution in Globular Star Clusters. Mon. Not. R. Astron. Soc.
**1911**, 71, 460. [Google Scholar] [CrossRef] - Jaffe, W.A. Simple model for the distribution of light in spherical galaxies. Mon. Not. R. Astron. Soc.
**1983**, 202, 995. [Google Scholar] [CrossRef] - Hernquist, L. An Analytical Model for Spherical Galaxies and Bulges. Astrophys. J.
**1990**, 356, 359. [Google Scholar] [CrossRef] - Kent, S.M. Dark matter in spiral galaxies. I. Galaxies with optical rotation curves. Astron. J.
**1986**, 91, 1301. [Google Scholar] [CrossRef] - Navarro, J.F.; Frenk, C.S.; White, S.D.M. A Universal Density Profile from Hierarchical Clustering. Astrophys. J.
**1997**, 490, 493. [Google Scholar] [CrossRef] - Dehnen, W. A Family of Potential-Density Pairs for Spherical Galaxies and Bulges. Mon. Not. R. Astron. Soc.
**1993**, 265, 250. [Google Scholar] [CrossRef] - Dehnen, W.; Gerhard, O.E. Two-integral models for oblate elliptical galaxies with cusps. Mon. Not. R. Astron. Soc.
**1994**, 268, 1019. [Google Scholar] [CrossRef] - Tremaine, S.; Richstone, D.O.; Byun, Y.-I.; Dressler, A.; Faber, S.M.; Grillmair, C.; Kormendy, J.; Lauer, T.R. A family of models for spherical stellar systems. Astron. J.
**1994**, 107, 634. [Google Scholar] [CrossRef] - Binney, J.; Tremaine, S. Galactic Dynamics; Princeton University Press: Princeton, NJ, USA, 2008. [Google Scholar]
- de Zeeuw, P.T. Elliptical galaxies with separable potentials. Mon. Not. R. Astron. Soc.
**1985**, 216, 273. [Google Scholar] [CrossRef] - Mo, H.; van den Bosch, F.; White, S. Galaxy Formation and Evolution; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Zhao, H.S. Analytical models for galactic nuclei. Mon. Not. R. Astron. Soc.
**1996**, 278, 488. [Google Scholar] [CrossRef] - Kormendy, J.; Richstone, D. Inward Bound—The Search For Supermassive Black Holes In Galactic Nuclei. Ann. Rev. Astron. Astrophys.
**1995**, 33, 581. [Google Scholar] [CrossRef] - Magorrian, J.; Tremaine, S.; Richstone, D.; Bender, R.; Bower, G.; Dressler, A.; Faber, S.M.; Gebhardt, K.; Green, R.; Grillmair, C.; et al. The Demography of Massive Dark Objects in Galaxy Centers. Astron. J.
**1998**, 115, 2285. [Google Scholar] [CrossRef] - Tremaine, S.; Gebhardt, K.; Bender, R.; Bower, G.; Dressler, A.; Faber, S.M.; Filippenko, A.V.; Green, R.; Grillmair, C.; Ho, L.C.; et al. The Slope of the Black Hole Mass versus Velocity Dispersion Correlation. Astrophys. J.
**2002**, 574, 740. [Google Scholar] [CrossRef] - Gebhardt, K.; Bender, R.; Bower, G.; Dressler, A.; Faber, S.M.; Filippenko, A.V.; Green, R.; Grillmair, C.; Ho, L.C. A Relationship between Nuclear Black Hole Mass and Galaxy Velocity Dispersion. Astrophys. J.
**2000**, 539, L13. [Google Scholar] [CrossRef] - Merritt, D.; Ferrarese, L. The M
_{•}-σ Relation for Supermassive Black Holes. Astrophys. J.**2001**, 547, 140. [Google Scholar] [CrossRef] - Rix, H.-W.; de Zeeuw, P.T.; Cretton, N.; van der Marel, R.P.; Carollo, C.M. Dynamical Modeling of Velocity Profiles: The Dark Halo around the Elliptical Galaxy NGC 2434. Astrophys. J.
**1997**, 488, 702. [Google Scholar] [CrossRef] - Hernandez, X.; Gilmore, G. Inferring Dark Halo Structure from Observed Scaling Law of Late-Type Galaxies and LSBs. Mon. Not. R. Astron. Soc.
**1998**, 294, 595. [Google Scholar] [CrossRef] - Cappellari, M.; Bacon, R.; Bureau, M.; Damen, M.C.; Davies, R.L.; De Zeeuw, P.T.; Emsellem, E.; Falcón-Barroso, J.; Krajnovic, D.; Kuntschner, H.; et al. The SAURON project - IV. The mass-to-light ratio, the virial mass estimator and the Fundamental Plane of elliptical and lenticular galaxies. Mon. Not. R. Astron. Soc.
**2006**, 366, 1126. [Google Scholar] - Jiang, G.; Kochanek, C.S. The Baryon Fractions and Mass-to-Light Ratios of Early-Type Galaxies. Astrophys. J.
**2007**, 671, 1568. [Google Scholar] [CrossRef] - Ciotti, L.; Morganti, L. Two-Component Galaxy Models: The Effect of Density Profile at Large Radii on the Phase-Space Consistency. Mon. Not. R. Astron. Soc.
**2009**, 303, 179. [Google Scholar] - Samurovic, S. Dynamical Constant Mass-to-Light Ratio Models of NGC 5128. Astron. Astrophys.
**2010**, 514, A95. [Google Scholar] [CrossRef] - Ragone-Figueroa, C.; Granato, G.L. Puffing up Early-Type Galaxies by Baryonic Mass Loss: Numerical Experiments. Mon. Not. R. Astron. Soc.
**2011**, 414, 3690. [Google Scholar] [CrossRef] - Tortora, C.; Napolitano, N.R.; Romanowsky, A.J.; Jetzer, P. Central Dark Matter Trend in Early-Type Galaxies. Mem. Della Soc. Astron. Ital. Suppl.
**2012**, 9, 302. [Google Scholar] - Tortora, C.; Napolitano, N.R.; Romanowsky, A.J.; Jetzer, P. Stellar Mass-to-Light Ratio Gradients in Galaxies: Correlations with Mass. Mon. Not. R. Astron. Soc.
**2011**, 418, 1557–1564. [Google Scholar] [CrossRef] - Lingam, M.; Nguyen, P.H. The Double-Power Approach to Spherically Symmetric Astrophysical Systems. Mon. Not. R. Astron. Soc.
**2014**, 440, 2636. [Google Scholar] [CrossRef] - Eadie, G.M.; Eadie, G.M.; Harris, W.E.; Widrow, L.M. Estimating the Galactic Mass Profile in the Presence of Incomplete Data. Astrophys. J.
**2015**, 806, 54. [Google Scholar] [CrossRef] - Ebrova, I.; Lokas, E.L. Galaxies with Prolate Rotation in Illustris. Astrophys. J.
**2017**, 850, 144. [Google Scholar] [CrossRef] - Zoldan, A.; Lucia, G.D.; Xie, L.; Fontanot, F.; Hirschmann, M. Structural and Dynamical Properties of Galaxies in a Hierarchical Universe: Sizes and Specific Angular Momenta. Mon. Not. R. Astron. Soc.
**2018**, 481, 1376. [Google Scholar] [CrossRef] - de Nicola, S.; Saglia, R.P.; Thomas, J.; Dehnen, W.; Bender, R. Non-parametric Triaxial Deprojection of Elliptical Galaxies. Mon. Not. R. Astron. Soc.
**2020**, 496, 3076. [Google Scholar] [CrossRef] - Caravita, C.; Ciotti, L.; Pellegrini, S. Jeans Modeling of Axisymmetric Galaxies with Multiple Stellar Populations. Mon. Not. R. Astron. Soc.
**2021**, 506, 1480. [Google Scholar] [CrossRef] - Carlberg, R.G.; Yee, H.K.C.; Ellingson, E.; Morris, S.L.; Abraham, R.; Gravel, P.; Pritchet, C.J.; Smecker-Hane, T.; Hartwick, F.D.A.; Hesser, J.E.; et al. The Average Mass Profile of Galaxy Clusters. Astrophys. J.
**1997**, 485, L13. [Google Scholar] [CrossRef] - van der Marel, R.P.; Magorrian, J.; Carlberg, R.G.; Yee, H.K.C.; Ellingson, E. The Velocity and Mass Distribution of Clusters of Galaxies from the CNOC1 Cluster Redshift Survey. Astron. J.
**2000**, 119, 2038. [Google Scholar] [CrossRef] - Adami, C.; Mazure, A.; Ulmer, M.P.; Savine, C. Central matter distributions in rich clusters of galaxies from z∼0 to z∼0.5. Astron. Astrophys.
**2001**, 371, 11. [Google Scholar] [CrossRef] - Lin, Y.T.; Mohr, J.J.; Stanford, S.A. K-Band Properties of Galaxy Clusters and Groups: Luminosity Function, Radial Distribution, and Halo Occupation Number. Astrophys. J.
**2004**, 610, 745. [Google Scholar] [CrossRef] - Adams, J.J.; Simon, J.D.; Fabricius, M.H.; van den Bosch, R.C.E.; Barentine, J.C.; Bender, R.; Gebhardt, K.; Hill, G.J.; Murphy, J.D.; Swaters, R.A.; et al. Dwarf galaxy dark matter density profiles inferred from stellar and gas kinematics. Astrophys. J.
**2014**, 789, 63. [Google Scholar] [CrossRef] - Oh, S.-H.; Hunter, D.A.; Brinks, E.; Elmegreen, B.G.; Schruba, A.; Walter, F.; Rupen, M.P.; Young, L.M.; Simpson, C.E.; Johnson, M.C.; et al. High-resolution mass models of dwarf galaxies from little things. Astron J.
**2015**, 149, 180. [Google Scholar] [CrossRef] - Blumenthal, G.R.; Faber, S.M.; Flores, R.; Primack, J.R. Contraction of Dark Matter Galactic Halos due to Baryonic Infall. Astrophys. J.
**1986**, 301, 27. [Google Scholar] [CrossRef] - Tollet, E.; Macció, A.V.; Dutton, A.A.; Stinson, G.S.; Wang, L.; Penzo, C.; Gutcke, T.A.; Buck, T.; Kang, X.; Brook, C.; et al. NIHAO—IV: Core Creation and Destruction in Dark Matter Density Profiles across Cosmic Time. Mon. Not. R. Astron. Soc.
**2016**, 456, 3542. [Google Scholar] [CrossRef] - Macció, A.V.; Crespi, S.; Blank, M.; Kang, X. NIHAO— XXIII. Dark Matter Density Shaped by Black Hole Feedback. Mon. Not. R. Astron. Soc.
**2020**, 495, L46. [Google Scholar] [CrossRef] - Pontzen, A.; Governato, F. Cold Dark Matter Heats Up. Nature
**2014**, 506, 171. [Google Scholar] [CrossRef] - Dekel, A.; Ishai, G.; Dutton, A.A.; Macciò, A.V. Dark-Matter Halo Profiles of a General Cusp/Core with Analytic Velocity and Potential. Mon. Not. R. Astron. Soc.
**2017**, 468, 1005. [Google Scholar] [CrossRef] - Freundlich, J.; Jiang, F.; Dekel, A.; Cornuault, N.; Ginzburg, O.; Koskas, R.; Lapiner, S.; Dutton, A.; Macciò, A.V. The Dekel-Zhao profile: A mass-dependent dark-matter density profile with flexible inner slope and analytic potential, velocity dispersion, and lensing properties. Mon. Not. R. Astron. Soc.
**2020**, 499, 2912. [Google Scholar] [CrossRef]

**Figure 1.**We report $\rho /{\rho}_{0}$ on the vertical axis and $r/{R}_{e}$ on the horizontal axis, both in logarithmic scale.

**Figure 2.**We show $M\left(r\right)/\left(4\pi \phantom{\rule{0.166667em}{0ex}}{\rho}_{0}\phantom{\rule{0.166667em}{0ex}}{R}_{e}^{3}\right)$ on the vertical axis and $r/{R}_{e}$ on the horizontal axis, both in logarithmic scale.

**Figure 3.**We exhibit ${v}_{c}^{2}\left(r\right)/\left(4\pi G\phantom{\rule{0.166667em}{0ex}}{\rho}_{0}\phantom{\rule{0.166667em}{0ex}}{R}_{e}^{2}\right)$ on the vertical axis and $r/{R}_{e}$ on the horizontal axis, both in logarithmic scale.

**Figure 4.**We report $g\left(r\right)/\left(2\pi G\phantom{\rule{0.166667em}{0ex}}{\rho}_{0}\phantom{\rule{0.166667em}{0ex}}{R}_{e}\right)$ on the vertical axis and $r/{R}_{e}$ on the horizontal axis, both in logarithmic scale.

**Figure 5.**We show the projected Jaffe and Hernquist models as well as the De Vaucouleurs law versus $r/{R}_{e}$, both in logarithmic scale. In all cases, $\mathrm{I}/{\mathrm{I}}_{\mathrm{ref}}$ is the dimensionless surface brightness.

**Figure 6.**We exhibit $\rho \left(r\right)/{\rho}_{0}$ on the vertical axis and $r/{a}_{\mathrm{NFW}}$ on the horizontal axis, both in logarithmic scale.

**Figure 7.**We report ${v}_{c}^{2}\left(r\right)/\left(4\pi G\phantom{\rule{0.166667em}{0ex}}{\rho}_{0}\phantom{\rule{0.166667em}{0ex}}{a}_{\mathrm{NFW}}^{2}\right)$ on the vertical axis and $r/{a}_{\mathrm{NFW}}$ on the horizontal axis, both in logarithmic scale.

**Figure 8.**We show $g\left(r\right)/\left(2\pi G\phantom{\rule{0.166667em}{0ex}}{\rho}_{0}\phantom{\rule{0.166667em}{0ex}}{a}_{\mathrm{NFW}}\right)$ on the vertical axis and $r/{a}_{\mathrm{NFW}}$ on the horizontal axis, both in logarithmic scale.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Roncadelli, M.; Galanti, G.
New Strong Constraints on the Central Behaviour of Spherical Galactic Models. *Astronomy* **2023**, *2*, 193-205.
https://doi.org/10.3390/astronomy2030014

**AMA Style**

Roncadelli M, Galanti G.
New Strong Constraints on the Central Behaviour of Spherical Galactic Models. *Astronomy*. 2023; 2(3):193-205.
https://doi.org/10.3390/astronomy2030014

**Chicago/Turabian Style**

Roncadelli, Marco, and Giorgio Galanti.
2023. "New Strong Constraints on the Central Behaviour of Spherical Galactic Models" *Astronomy* 2, no. 3: 193-205.
https://doi.org/10.3390/astronomy2030014