A first theoretical prediction of the existence of gravitational lenses (GL) is due to Einstein [1] where the formulae for the optical properties of a gravitational lens for star A and B were derived. A first sketch which dates back to 1912 is reported at p. 585 in [2] . The historical context of the GL is outlined in [3] and ONLINE information can be found at http://www.einstein-online.info/. After 43 years a first GL was observed in the form of a close pair of blue stellar objects of magnitude 17 with a separation of 5.7 arc sec at redshift 1.405, 0957 + 561 A, B, see [4] . This double system is also known as the “Twin Quasar” and a Figure which reports a 2014 image of the Hubble Space Telescope (HST) for objects A and B is available at https://www.nasa.gov/content/goddard/hubble-hubble-seeing-double/.

At the moment of writing, the GL is used routinely as an explanation for lensed objects, see the following reviews [5] [6] [7] [8] . As an example of current observations, 28 gravitationally lensed quasars have been observed by the Subaru Telescope, see [9] , where for each system a mass model was derived. Another example is given by the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS), see [10] , where 13 two-image quasar lenses have been observed and the relative Einstein’s radius was reported in arcsec. A first classification separates the strong lensing, such as an Einstein ring (ER) and the arcs from the weak lensing, such as the shape deformation of background galaxies.

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 [11] and [12] . The first cluster of galaxies observed with the weak lensing effect is reported by [13] .

We now introduce supershells, which were unknown when the GL was postulated. Supershells started to be observed firstly in our galaxy by [14] , where 17 expanding H I shells were classified, and secondly in external galaxies, see as an example [15] , where many supershells were observed in NGC 1569. In order to model such complex objects, the term super bubble (SB) has been introduced but unfortunately the astronomers often associate the SBs with sizes of 10 - 100 pc and the supershells with ring-like structures with sizes of 1 kpc. At the same time an application of the theory of image explains the limb-brighten- ing visible on the maps of intensity of SBs and allows associating the observed filaments to undetectable SBs, see [16] .

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 [17] , the luminosity distance in flat cosmology, , is

where is the Hubble constant expressed in, is the velocity of light expressed in, is the redshift, is the scale factor, and is

where is the Newtonian gravitational constant and is the mass density at the present time. An analytical solution for the luminosity distance exists in the complex plane, see [18] . Here we deal with an approximate solution for the luminosity distance in the framework of a flat universe adopting the same cosmological parameters of [19] which are, and. An approximate solution for the luminosity distance, , is given by a Taylor expansion of order 10 about for the argument of the integral (1)

More details on the analytical solution for the luminosity distance in the case of flat cosmology can be found in [20] and Figure 1 reports the comparison between the above analytical solution, and Taylor expansion of order 10, 8 and 2.

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

As an example, Table 1 reports the percentage error at z = 4 for three order of expansion; is clear the progressive decrease of the percentage error with the increase in the order of expansion.

Another useful distance is the angular diameter distance, , which is

see [21] and the Taylor approximation for the angular diameter distance,

As a practical example of the above equation, the angular scale of 1 arcsec is 7.73 kpc at when [19] quotes 7.78 kpc: this means a percentage error of 0.63% between the two values. Another check can be done with the Ned Wright’s Cosmology Calculator [22] available at http://www.astro.ucla.edu/wright/CosmoCalc.html: it quotes a scale of 7.775 kpc/arcsec which means a percentage error of 0.57% with respect to our value. In this section we have derived the cosmological scaling that allows to fix the dimension of the ER.

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

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

where represents the scale, see [31] for more details. The piece-wise density is

The total mass swept, , in the interval is

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

where and are the swept masses at and, and and are the velocities of the thin layer at and. The velocity is, therefore,

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, , as a function of time. The astrophysical units are pc for length and yr for time. With these units, the initial velocity is . The energy conserving phase of an SB in the presence of constant density allows setting up the initial conditions, and the radius is

where is the time expressed in units of 10⁷ yr, is the energy expressed in units of 10⁵¹ erg, is the number density expressed in particles cm⁻³ (density, where) and is the number of SN explosions in and therefore is a rate, see Equation (10.38) in [32] . The velocity of an SB in such a phase is

The initial condition for and are now fixed by the energy conserving phase for an SB evolving in a medium at constant density. The free parameters of the model are reported in Table 2; Figure 3 reports the law of motion and Figure 4 the behaviour of the velocity as a function of time.

Once we have fixed the standard radius of SDP.81 at, we evaluate the pair of values for (the scale) and for (the time) that allows such a value of the radius, see Figure 5.

The pair of values of (initial number density) and (the time) which produces the standard value of the radius is reported in Figure 6; Figure 7 con-

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, , and, with parameters as in Table 2, is

In order to simulate an asymmetric SB we briefly review a numerical algorithm developed in [16] . We assume a number density distribution as

where is the density at, is a scaling parameter, and sech is the hyperbolic secant ( [33] [34] [35] [36] ).

We now analyze the case of an expansion that starts from a given galactic height, denoted by, which represents the OB associations. It is not possible to find analytically and a numerical method should be implemented.

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

where, , are the temporary radius, the velocity, and the total mass, respectively, is the time step, and is the index. The advancing expansion is computed in a 3D Cartesian coordinate system () with the center of the explosion at (0, 0, 0). The explosion is better visualized in a 3D Cartesian coordinate system () in which the galactic plane is given by. The following translation, , relates the two Cartesian coordinate systems.

where is the distance in parsec of the OB associations from the galactic plane.

The physical units for the asymmetrical SB have not yet been specified: parsecs for length and 10⁷ yr for time are perhaps an acceptable astrophysical choice. With these units, the initial velocity is expressed in units of pc/(10⁷ yr) and should be converted into km/s; this means that where is the initial velocity expressed in km/s.

We are now ready to present the numerical evolution of the SB associated with SDP.81 when, see Figure 8.

The degree of asymmetry can be evaluated introducing the radius along the polar direction up, , the polar direction down, and the equatorial direction,. In our model all the already defined three radii are different, see Table 3.

We can evaluate the radius and the velocity as function of the direction plotting the radius and the direction in section in the plane, see Figure 9 and Figure 10.

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.

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 (the redshift of SDP.81) is.

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. The available astronomical data do not allow to close the equations at kpc (the radius of SDP.81). A numerical relationship which connects the number density with the lifetime of a SB is reported in Figure 6

and an approximation of the above relationship is when and.

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, , function of two angles in polar coordinates (external surface) and (internal surface) is filled with N random points. After a rotation characterized by three Euler angles which align the 3D layer with the observer, the image is obtained by summing a 3D visitation grid over one index, see image 4. The variations for the Canarias ER of the flux counts in ADU as function of the angle can be modeled because radius, velocity and therefore flux of kinetic energy are different for each chosen direction, see Figure 17.

The real data of Figure 17 were kindly provided by Margherita Bettinelli. The real data of Figure 2 were digitized using WebPlotDigitizer, a Web based tool to extract data from plots, available at http://arohatgi.info/WebPlotDigitizer/.

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