_{1}

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A superbubble which advances in a symmetric Navarro-Frenk-White density profile or in an auto-gravitating density profile generates a thick shell with a radius that can reach 10 kpc. The application of the symmetric and asymmetric image theory to this thick 3D shell produces a ring in the 2D map of intensity and a characteristic “U” shape in the case of 1D cut of the intensity. A comparison of such a ring originating from a superbubble is made with the Einstein’s ring. A Taylor approximation of order 10 for the angular diameter distance is derived in order to deal with high values of the redshift.

A first theoretical prediction of the existence of gravitational lenses (GL) is due to Einstein [

At the moment of writing, the GL is used routinely as an explanation for lensed objects, see the following reviews [

The strong lensing is verified when the light from a distant background source, such as a galaxy or quasar, is deflected into multiple paths by an intervening galaxy or a cluster of galaxies producing multiple images of the background source: examples are the ER and the multiple arcs in cluster of galaxies, see https://apod.nasa.gov/apod/ap160828.html.

In the case of weak lensing the lens is not strong enough to form multiple images or arcs, but the source can be distorted: both stretched (shear) and magnified (convergence), see [

We now introduce supershells, which were unknown when the GL was postulated. Supershells started to be observed firstly in our galaxy by [

This paper derives, in Section 2, an approximate solution for the angular diameter distance in flat cosmology. Section 3 briefly reviews the existing knowledge of ERs. Section 4 derives an equation of motion for an SB in a Navarro-Frenk-White (NFW) density profile. Section 5 adopts a recursive equation in order to model an asymmetric motion for an SB in an auto-gravitating density profile.

Section 6.3 applies the symmetrical and the asymmetrical image theory to the advancing shell of an SB.

Following Equation (2.1) of [

where

where

More details on the analytical solution for the luminosity distance in the case of flat cosmology can be found in [

The goodness of the Taylor approximation is evaluated through the percentage error,

As an example,

Taylor order | |
---|---|

2 | 22.6% |

8 | 2.2% |

10 | 0.61% |

Another useful distance is the angular diameter distance,

see [

As a practical example of the above equation, the angular scale of 1 arcsec is 7.73 kpc at

This section reviews the simplest version of the ER and reports the observations of two recent ERs.

In the case of a circularly symmetric lens and when the source and the length are on the same line of sight, the ER radius in radiant is

where

where

The ring associated with the galaxy SDP.81, see [

where

The object IAC J010127-334319 has been detected in the optical region with the Gran Telescopio CANARIAS; the radius of the ER is

The density is assumed to have a Navarro-Frenk-White (NFW) dependence on

where

The total mass swept,

The conservation of momentum in spherical coordinates in the framework of the thin layer approximation states that

where

where

and

The integration of the above differential equation of the first order gives the following non-linear equation:

The above non-linear equation does not have an analytical solution for the radius,

where ^{7} yr, ^{51} erg, ^{−}^{3} (density

The initial condition for

Once we have fixed the standard radius of SDP.81 at

The pair of values of

Name | b(pc) | ||||
---|---|---|---|---|---|

SDP.81 | 10,000 | 1 | 1 | 1000 |

versely reports the actual velocity of the SB associated with SDP.81 as function of

The swept mass can be expressed in the number of solar masses,

In order to simulate an asymmetric SB we briefly review a numerical algorithm developed in [

where

We now analyze the case of an expansion that starts from a given galactic height

The following two recursive equations are found when momentum conservation is applied:

where

where

The physical units for the asymmetrical SB have not yet been specified: parsecs for length and 10^{7} yr for time are perhaps an acceptable astrophysical choice. With these units, the initial velocity ^{7} yr) and should be converted into km/s; this means that

We are now ready to present the numerical evolution of the SB associated with SDP.81 when

The degree of asymmetry can be evaluated introducing the radius along the polar direction up,

We can evaluate the radius and the velocity as function of the direction plotting the radius and the direction in section in the

Direction | |
---|---|

Equatorial | 10,906 |

Polar up | 11,077 |

Polar down | 11,068 |

We now briefly review the basic equations of the radiative transfer equation, the conversion of the flux of energy into luminosity, the symmetric and the asymmetric theory of the image.

The transfer equation in the presence of emission and absorption, see for example Equation (1.23) in [

where

where

We now continue analysing the case of an optically thin layer in which

where

where

As an example, synchrotron emission has an intensity proportional to

The ultimate source of the observed luminosity is assumed to be the rate of kinetic energy,

where

The units of the luminosity are W in MKS and erg s^{−}^{1} in CGS. The astrophysical version of the rate of kinetic energy,

where

where

where

We assume that the number density of the emitting matter

When the number density of the emitting matter

where

The parameter

where

the rim and the intensity at the center. The distance

We can now evaluate the half-width half-maximum by analogy with the Gaussian profile

In the above model,

As an example, inserting in the above formula

The effect of the insertion of a threshold intensity,

with the observational techniques, is now analysed. The threshold intensity can be parametrized to

The theoretical flux profiles for IAC J010127-334319, see Section 3.3, is reported in

The linear relation between the angular distance, in pc, and the projected dis-

tance on the sky in arcsec allows to state the following

Theorem 1. The “U” profile of cut in theoretical flux for a symmetric ER is independent of the exact value of the angular distance.

We now explain a numerical algorithm which allows us to build the complex image of an asymmetrical SB.

・ An empty (value = 0) memory grid

・ We first generate an internal 3D surface by rotating the section of

・ Each point of

The orientation of the object is characterized by the Euler angles

・ The intensity map is obtained by summing the points of the rotated images along a particular direction.

・ The effect of the insertion of a threshold intensity,

An ideal image of the intensity of the Canarias ring is shown in

The theoretical flux which is here assumed to scale as the flux of kinetic energy as represented by Equation (28), is reported in

Flat cosmology: In order to have a reliable evaluation of the radius of SDP.81 we have provided a Taylor approximation of order 10 for the luminosity distance in the framework of the flat cosmology. The percentage error between analytical solution and approximated solution when

Symmetric evolution of an SB: The motion of a SB advancing in a medium with decreasing density in spherical symmetry is analyzed. The type of density profile here adopted is a NFW profile which has three free parameters,

and an approximation of the above relationship is

Symmetric Image theory: The transfer equation for the luminous intensity in the case of optically thin layer is reduced in the case of spherical symmetry to the evaluation of a length between lower and upper radius along the line of sight, see Equation (32). The cut in intensity has a characteristic “U” shape, see Equation (33), which also characterizes the image of ER associated with the galaxy SDP.81.

Asymmetric Image theory:

The layer between a complex 3D advancing surface with radius,

The real data of

Zaninetti, L. (2017) The Ring Produced by an Extra-Galactic Superbubble in Flat Cosmology. Journal of High Energy Physics, Gravitation and Cosmology, 3, 339-359. https://doi.org/10.4236/jhepgc.2017.32029