# PossibleExistence of Dark-Matter-Admixed Pulsar in the Disk Region of the Milky Way Galaxy

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Interior Spacetime

**effective**energy density and ${p}_{eff}$ is the

**effective**isotropic pressure.

## 3. Study of Physical Properties

#### 3.1. Energy Density and Pressure

**be**obeyed, including the pressure, which goes to zero at the boundary.

#### 3.2. Energy Conditions

- (i)
- NEC: $\rho \ge 0$
- (ii)
- WEC: $\rho +p\ge 0$, $p\ge 0$
- (iii)
- SEC: $\rho +p\ge 0$, $\rho +3p\ge 0$
- (iv)
- DEC: $\rho >\left|p\right|$

#### 3.3. Matching Conditions

#### 3.4. Mass-Radius Relation and Surface Red-Shift

#### 3.5. TOV Equation

#### 3.6. Speed of Sound and Adiabatic Index

## 4. Discussion and Concluding Remarks

**it**based on

- (i)
- the singular isothermal sphere (SIS) profile for pulsars in the galactic halo region of different galaxies [73].
- (ii)
- the universal rotational curve (URC) profile for pulsars in the galactic halo region of Milky Way galaxy [75].

**Heintzmann**metric. Density and pressure at the interior of the pulsar are well behaved (Figure 1 and Figure 2). Here, we assume the value of the constants ($a=0.00138889$ km${}^{-2}$, $C=1.34164$, $m=0.025$, ${r}_{s}=20$ kpc, and ${\rho}_{s}=0.26$ GeV/c.c.) in such a way that all of the physical required conditions must satisfy. We also note that the value of ${r}_{s}$ and ${\rho}_{s}$ taken here are applicable for the Milky Way galaxy, and we have checked our model with the pulsars, namely, PSR J0045-7319 and PSR J0537-6910, located in the disk region of Milky Way galaxy.

## Author Contributions

## Funding

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Energy density ($\rho $) variation with radial distance (r) for $a=0.00138889$ km${}^{-2}$, $C=1.34164$, $m=0.025$, ${r}_{s}=20$ kpc, and ${\rho}_{s}=0.26$ GeV/c.c.

**Figure 2.**Pressure (p) variation with radial distance (r) for $a=0.00138889$ km${}^{-2}$, $C=1.34164$, $m=0.025$, ${r}_{s}=20$ kpc, and ${\rho}_{s}=0.26$ GeV/c.c.

**Figure 3.**Energy condition variation with radial distance at the pulsar interior for $a=0.00138889$ km${}^{-2}$, $C=1.34164$, $m=0.025$, ${r}_{s}=20$ kpc, and ${\rho}_{s}=0.26$ GeV/c.c.

**Figure 4.**Radial dependence mass function, $M\left(r\right)$ for $a=0.00138889$ km${}^{-2}$, $C=1.34164$, $m=0.025$, ${r}_{s}=20$ kpc, and ${\rho}_{s}=0.26$ GeV/c.c.

**Figure 5.**Radial dependence compactness ($u\left(r\right)$) for $a=0.00138889$ km${}^{-2}$, $C=1.34164$, $m=0.025$, ${r}_{s}=20$ kpc, and ${\rho}_{s}=0.26$ GeV/c.c.

**Figure 6.**Radial dependence red-shift (${Z}_{s}$) for $a=0.00138889$ km${}^{-2}$, $C=1.34164$, $m=0.025$, ${r}_{s}=20$ kpc, and ${\rho}_{s}=0.26$ GeV/c.c.

**Figure 7.**Radial dependence of gravitational force $\left({F}_{g}\right)$ and hydrostatic force $\left({F}_{h}\right)$ for $a=0.00138889$ km${}^{-2}$, $C=1.34164$, $m=0.025$, ${r}_{s}=20$ kpc, and ${\rho}_{s}=0.26$ GeV/c.c.

**Figure 8.**Radial dependence of sound velocity (${V}_{r}^{2}$) for $a=0.00138889$ km${}^{-2}$, $C=1.34164$, $m=0.025$, ${r}_{s}=20$ kpc, and ${\rho}_{s}=0.26$ GeV/c.c.

**Figure 9.**Radial dependence of Adiabatic Index ($\gamma $) for $a=0.00138889$ km${}^{-2}$, $C=1.34164$, $m=0.025$, ${r}_{s}=20$ kpc, and ${\rho}_{s}=0.26$ GeV/c.c.

PSR | Distanc (kpc) | Observed Mass [103] | Radius from Model (km) | Compactness from Model | Red-Shift from Model |
---|---|---|---|---|---|

J 0045-7319 | 57 | ${1.58}_{-0.34}^{+0.34}$ | ${10.8436}_{-0.98}^{+0.88}$ | ${0.215152}_{-0.029}^{+0.027}$ | ${0.324885}_{-0.064}^{+0.067}$ |

PSR | Distance (kpc) | Equation of State | Observed Mass [103] | Radius from Model (km) | Compactness from Model | Red-Shift from Model |
---|---|---|---|---|---|---|

J 0537-6910 | 52.122 | BSk 20 | ${1.83}_{-0.04}^{+0.04}$ | ${11.5007}_{-0.10}^{+0.10}$ | ${0.234957}_{-0.003}^{+0.003}$ | ${0.373495}_{-0.0079}^{+0.0079}$ |

J 0537-6910 | 52.122 | BSk 21 | ${2.11}_{-0.05}^{+0.04}$ | ${12.1875}_{-0.12}^{+0.094}$ | ${0.25564}_{-0.003}^{+0.003}$ | ${0.430441}_{-0.01}^{+0.0083}$ |

J 0537-6910 | 52.122 | APR | ${2.05}_{-0.03}^{+0.04}$ | ${12.0441}_{-0.072}^{+0.096}$ | ${0.251328}_{-0.002}^{+0.003}$ | ${0.417985}_{-0.006}^{+0.0083}$ |

**Table 3.**Comparison between the parameters evaluated for different dark matter profiles (pulsar PSR J 0045-7319).

Dark Matter Profile | Radius from Model (km) | Compactness from Model | Red-Shift from Model |
---|---|---|---|

NFW | ${10.8436}_{-0.98}^{+0.88}$ | ${0.215152}_{-0.029}^{+0.027}$ | ${0.324885}_{-0.064}^{+0.067}$ |

URC | ${9.45519}_{-0.90}^{+0.81}$ | ${0.246746}_{-0.033}^{+0.029}$ | ${0.405098}_{-0.083}^{+0.089}$ |

SIS | ${10.0366}_{-0.93}^{+0.84}$ | ${0.232452}_{-0.031}^{+0.028}$ | ${0.367048}_{-0.074}^{+0.078}$ |

**Table 4.**Comparison between the parameters evaluated for different dark matter profiles (pulsar PSR J 0537-6910).

Equation of State | Dark Matter Profile | Radius from Model (km) | Compactness from Model | RedShift from Model |
---|---|---|---|---|

BSk 20 | NFW | ${11.5007}_{-0.10}^{+0.10}$ | ${0.234957}_{-0.003}^{+0.003}$ | ${0.373495}_{-0.0079}^{+0.0079}$ |

BSk 20 | URC | ${10.0604}_{-0.09}^{+0.09}$ | ${0.268595}_{-0.003}^{+0.003}$ | ${0.469936}_{-0.0106}^{+0.0107}$ |

BSk 20 | SIS | ${10.6601}_{-0.097}^{+0.096}$ | ${0.253485}_{-0.003}^{+0.003}$ | ${0.424174}_{-0.009}^{0.009}$ |

BSk 21 | NFW | ${12.1875}_{-0.12}^{+0.094}$ | ${0.25564}_{-0.003}^{+0.003}$ | ${0.430441}_{-0.01}^{+0.0083}$ |

BSk 21 | URC | ${10.698}_{-0.11}^{+0.088}$ | ${0.291234}_{-0.004}^{0.003}$ | ${0.547587}_{-0.014}^{+0.012}$ |

BSk 21 | SIS | ${11.3142}_{-0.11}^{+0.090}$ | ${0.275374}_{-0.004}^{+0.003}$ | ${0.491953}_{-0.0124}^{+0.0101}$ |

APR | NFW | ${12.0441}_{-0.072}^{+0.096}$ | ${0.251328}_{-0.002}^{+0.003}$ | ${0.417985}_{-0.006}^{+0.0083}$ |

APR | URC | ${10.5645}_{-0.067}^{+0.089}$ | ${0.286529}_{-0.002}^{+0.003}$ | ${0.530439}_{-0.008}^{+0.0114}$ |

APR | SIS | ${11.1774}_{-0.069}^{+0.091}$ | ${0.270817}_{-0.002}^{+0.003}$ | ${0.477046}_{-0.007}^{+0.0099}$ |

**Table 5.**The values of the metric parameters and dark matter parameters used in different star modellings.

Dark Matter Profile | a (in km${}^{-2}$) | C | m | K | ${\mathit{r}}_{\mathit{s}}$ (in kpc) | ${\mathit{\rho}}_{\mathit{s}}$ (in $\frac{\mathrm{GeV}}{c.c.})$ | ${\mathit{r}}_{0}$ (in kpc) | ${\mathit{\rho}}_{0}$ (in $\frac{\mathit{g}\mathit{m}}{\mathit{c}.\mathit{c}.}$) |
---|---|---|---|---|---|---|---|---|

NFW | $0.00138889$ | $1.34164$ | $0.025$ | * | 20 | $0.26$ | * | * |

URC | $0.003$ | $0.8$ | $0.1$ | * | * | * | $9.11$ | $5\times {10}^{-24}$ |

SIS | $0.002$ | $1.14$ | $0.01$ | ${10}^{-7}$ | * | * | * | * |

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Rahman, N.; Murshid, M.; Molla, S.; Kalam, M.
PossibleExistence of Dark-Matter-Admixed Pulsar in the Disk Region of the Milky Way Galaxy. *Universe* **2022**, *8*, 652.
https://doi.org/10.3390/universe8120652

**AMA Style**

Rahman N, Murshid M, Molla S, Kalam M.
PossibleExistence of Dark-Matter-Admixed Pulsar in the Disk Region of the Milky Way Galaxy. *Universe*. 2022; 8(12):652.
https://doi.org/10.3390/universe8120652

**Chicago/Turabian Style**

Rahman, Nilofar, Masum Murshid, Sajahan Molla, and Mehedi Kalam.
2022. "PossibleExistence of Dark-Matter-Admixed Pulsar in the Disk Region of the Milky Way Galaxy" *Universe* 8, no. 12: 652.
https://doi.org/10.3390/universe8120652