# Orbits of Particles and Photons around Regular Rotating Black Holes and Solitons

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Internal Structure of de Sitter–Kerr Compact Objects

## 3. Circular Equatorial Orbits and Light Rings

#### 3.1. Particle Orbits

**The particle orbits**are described by the equation — for time-like geodesics [83]:

**Marginally bound orbits**are the limiting orbits defined by ${E}^{2}=1$ [83]. The innermost stable and direct marginally bound orbits exist in the minima of the potentials (${V}_{p}^{\prime}=0$), described by [80]

**Marginally stable orbits**satisfy the equation ${V}^{\prime}={V}^{\u2033}=0$ and form the next group of limiting orbits. Analysis of their behavior has been studied by applying the orbit equation in terms of the metric function $g\left(r\right)=1-2G\mathcal{M}\left(r\right)/r$, which reads as follows [80]:

#### 3.2. Light Rings

**Closed photon orbits**are described by the null geodesics: [80]

#### 3.3. All Limiting Orbits

## 4. Observational Signatures

^{3}detector, as in the IceCUBE, one can expect up to 300 events per year [85].

#### Shadows of de Sitter–Kerr Black Holes

**Basic equations defining the contour of a shadow:**The basic parameters characterizing photon orbits are [83]

**Identification of a de Sitter–Kerr black hole by its shadow:**The celestial coordinates $(x,y)$ are introduced as the impact parameters, shown in Figure 6 (left) [108]. These are related to the orbit parameters (Equation (21)) for the innermost photon orbits, which form the boundary of the gravitational capture cross section, since photons falling in with such impact parameters are captured on the closed innermost orbits [83,87]. The celestial coordinates are defined as follows [108]:

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The first type of interior, horizons, and ergosphere (

**left**) and second type of interior (

**right**).

**Figure 2.**$\mathcal{S}$ surface, horizons, and ergospheres for $a=0.314$ (

**left**) and $a=0.4$ (

**right**).

**Figure 3.**Dependence of radii on a for the marginally bound orbits (

**left**) and for the marginally stable orbits (

**right**), where ${x}_{g}={r}_{g}/{r}_{0}$.

**Figure 4.**Photon orbits, horizons, and ergospheres depending on a (

**left**) and their enlarged image near the double horizon (

**right**), where ${x}_{g}={r}_{g}/{r}_{0}$.

**Figure 5.**Limiting orbits in the de Sitter–Kerr geometry depending on a (

**left**) and an enlarged image near the double horizon (

**right**), where ${x}_{g}={r}_{g}/{r}_{0}$.

**Figure 6.**Impact parameters as celestial coordinates (

**left**) and the asymmetry parameter D in the shadow contour (

**right**).

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Dymnikova, I.; Dobosz, A.
Orbits of Particles and Photons around Regular Rotating Black Holes and Solitons. *Symmetry* **2023**, *15*, 273.
https://doi.org/10.3390/sym15020273

**AMA Style**

Dymnikova I, Dobosz A.
Orbits of Particles and Photons around Regular Rotating Black Holes and Solitons. *Symmetry*. 2023; 15(2):273.
https://doi.org/10.3390/sym15020273

**Chicago/Turabian Style**

Dymnikova, Irina, and Anna Dobosz.
2023. "Orbits of Particles and Photons around Regular Rotating Black Holes and Solitons" *Symmetry* 15, no. 2: 273.
https://doi.org/10.3390/sym15020273