# Gravitational Lensing of Rays through the Levitating Atmospheres of Compact Objects

## Abstract

**:**

## 1. Introduction

## 2. Theory

## 3. Numerical Results: A Variety of Novel Orbits

#### 3.1. A Stable Circular Orbit for Electromagnetic Rays

#### 3.2. Periodic Orbits

#### 3.3. Frequency Windows

#### 3.3.1. Ray Trapping

#### 3.3.2. Ray Escape

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**

**Top**panel: Λ as a function of angular momentum L is plotted for $E={\omega}_{\infty}=1$ as the thick black line. On this panel we plot the value of $\Lambda =1$ as a dashed line, $1/2$, $1/3$, $1/4$, $1/5$, $1/6$ as thin solid lines and $\Lambda =1/7$ as a dash-dotted line. The intersections of the $\Lambda =1$ and $\Lambda =1/7$ lines with $\Lambda \left(L\right)$ are marked as white discs. The effective potential corresponding to the angular momentum specified by these discs are plotted in the

**lower**panel. The shaded region represents the interior of the CO and the heavy black lines are the effective potentials experienced by the ray. The $\Lambda =1$ and $\Lambda =1/7$ cases are represented by dashed and dash-dotted lines respectively.

**Figure 2.**Periodic orbits with angular momentum L found from $\Lambda =1$, $1/2$, $1/3$, $1/4$, $1/5$, $1/6$, $1/7$ and the stable circular orbit. The CO is the shaded disc, and the unstable circular orbit is marked with a dashed line.

**Figure 3.**A family of periodic orbits with L found from $\Lambda =1/5$, $2/5$, $3/5$ and $4/5$. The orbits reflect off of the plasma shell 5 times, and the orbital path intersects itself 0, 5, 10 and 15 times, respectively. The CO is the shaded disc, and the unstable circular orbit is marked with a dashed line.

**Figure 4.**An example of precession as a function of angular momentum for $\Lambda =1$ (

**middle**panel). For $\Lambda >1$ we find precession in the counter-clockwise direction (

**top**panel) and for $\Lambda <1$ the orbit precesses in the opposite sense (

**bottom**panel). Initial positions are plotted as black dots and final positions are squares.

**Figure 5.**Trapped and scattered rays in the APW. The

**upper**panel plots the paths of 10 rays emitted from the CO surface, R, at $\varphi =\pi /2$. These rays are evenly spaced between $L=0$ (dashed), and ${L}_{\mathrm{max}}$ (dashed-dotted). Also plotted are the paths of 10 rays that approach the ECS externally from the $+x$ direction, and are reflected by the potential boundary. These external rays are plotted as the solid black curves. The ECS radius is the dotted circle. The

**lower**panel shows the corresponding effective potential for the bound rays $L=0$ (dashed) and $0.99{L}_{\mathrm{max}}$ (solid). The square of the asymptotic frequency ${\omega}_{\infty}^{2}$ is the horizontal line. The shaded region is the CO interior, and the vertical dotted line is the ECS radius. Both interior and exterior radially directed rays ($L=0$) have frequencies ${\omega}_{\infty}={\omega}_{\infty p}$. These rays, and all rays with lower frequency, cannot propagate through the plasma.

**Figure 6.**Trajectories of rays in the EW launched from the surface of the CO with frequency ratios ${\omega}_{\infty}/{\omega}_{\infty +}=0.90$ (

**top**panel); $0.93$ (

**second**panel) and $0.97$ (

**third**panel); For rays higher than the EW, we also show the case for ${\omega}_{\infty}/{\omega}_{\infty +}=1.05$ (

**fourth**panel) and $1.50$ (

**bottom**panel).

**Figure 7.**The general Woods-Saxon function. The dashed curve ($a=1$, $C=8$) shows a density $N\left(r\right)$ with a discrete maximum at ${r}_{\mathrm{ECS}}$. This function does not vanish at the CO surface at R, shown by the shaded portion on the left of the figure. The solid curve ($a=1$, $C=0$) represents the extreme case in which the density does not monotonically decrease toward the star. Both of these functions produce regions of constant density above the stellar surface. The vertical dotted line denotes the position of ${r}_{\mathrm{ECS}}$.

© 2017 by the author. 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 ( http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Rogers, A.
Gravitational Lensing of Rays through the Levitating Atmospheres of Compact Objects. *Universe* **2017**, *3*, 3.
https://doi.org/10.3390/universe3010003

**AMA Style**

Rogers A.
Gravitational Lensing of Rays through the Levitating Atmospheres of Compact Objects. *Universe*. 2017; 3(1):3.
https://doi.org/10.3390/universe3010003

**Chicago/Turabian Style**

Rogers, Adam.
2017. "Gravitational Lensing of Rays through the Levitating Atmospheres of Compact Objects" *Universe* 3, no. 1: 3.
https://doi.org/10.3390/universe3010003