The giant arcs in the cluster of galaxies start to be observed as narrow-like shape by    and the first theoretical explanation was the gravitational lensing, see     . The determination of the statistical parameters of the giant arcs has been analyzed in order to derive the cosmological parameters, see  , in connection with the ΛCDM cosmology, see  , in the framework of the triaxiality and substructure of CDM halo, see  , in connection with the Wilkinson Microwave Anisotropy Probe (WMAP) data, see  , including the effects of baryon cooling in dark matter N-body simulations, see  and in order to derive the photometric properties of 105 giant arcs in the Second Red-Sequence Cluster Survey (RCS-2), see  . The gravitational lensing is the most common theoretical explanation; we select some approaches among others:  evaluated the mass distribution inside distant clusters;  studied the statistics of giant arcs in flat cosmologies with and without a cosmological constant;  analyzed how the gravitational lensing influences the surface brightness of giant luminous arcs and  used the warm dark matter (WDM) cosmologies to explain the lensing in galaxy clusters. Another theoretical line of research explains the giant arcs as shells originated by the galaxies in the cluster:   analyzed the limb-brightened shell model, the gravitational lens model and the echo model;  suggested that the Gamma-ray burst (GRB) explosions are the sources of the shells with sizes of many kpc, and  suggested a connection between the Einstein ring associated to SDP.81 and the evolution of a superbubble (SB) in the intracluster medium.
This paper analyzes in Section 2 three cosmologies in order to calibrate the transversal distance which allows to convert the arcsec in pc. Section 3 is devoted to the evolution of a SB in the intracluster medium. Section 4 reports the observations of the giant arcs and the first phase of a SB. Section 5 reports the various steps which allow to reproduce the shape of the giant arc A2267 and the multiple arcs visible in the cluster of galaxies. Section 6 is dedicated to theory of the image: analytical formulae explain the hole in the central part of the SBs and numerical results reproduce the details of a giant arc.
2. Adopted Cosmologies
In the following we review three cosmological theories.
2.1. ΛCDM Cosmology
The basic parameters of ΛCDM cosmology are: the Hubble constant, , expressed in km∙s−1∙Mpc−1, the velocity of light, c, expressed in kms−1, and the three numbers , , and , see  for more details. In the case of the Union 2.1 compilation, see  , the parameters are , and . To have the luminosity distance, , as a function of the redshift only, we apply the minimax rational approximation, which is characterized by two parameters, p and q. The luminosity distance, , when and
The transversal distance in ΛCDM cosmology, , which corresponds to the angle expressed in arcsec is
2.2. Flat Cosmology
The two parameters of the flat cosmology are , the Hubble constant expressed in km∙s−1∙Mpc−1, and which is
where G is the Newtonian gravitational constant and is the mass density at the present time. In the case of m = 2 and n = 2 the minimax rational expression for the luminosity distance, , when and , is
The transversal distance in flat cosmology, , which corresponds to the angle expressed in arcsec is
2.3. Modified Tired Light
In an Euclidean static framework the modified tired light (MTL) has been introduced in Section 2.2 in  . The distance in MTL is
The distance modulus in the modified tired light (MTL) is
Here is a parameter comprised between 1 and 3 which allows to match theory with observations. The number of free parameters in MTL is two: and . The fit of the distance modulus with the data of Union 2.1 compilation gives , , see  , which means the following distance
The transversal distance in MTL, , which corresponds to the angle expressed in arcsec is
We report the angular distance for a fixed delta as function of redshift for the three cosmologies, see Figure 1. The angular distance in flat and ΛCDM cosmology does not increase with z, see  , in contrast with the modified tired light.
3. The Motion of a SB
We now summarize the adopted profile of density and the equation of motion for a SB.
3.1. The Profile
The density is assumed to have the following hyperbolic dependence on Z which is the third Cartesian coordinate,
Figure 1. Angular distances in kpc for the three cosmologies here considered when .
where the parameter fixes the scale and is the density at . In spherical coordinates the dependence on the polar angle is
Given a solid angle the mass swept in the interval is
The total mass swept, , in the interval is
and its approximate value at high values of r is
The density can be obtained by introducing the number density, , expressed in particles cm−3, the mass of hydrogen, , and a multiplicative factor f, which is chosen to be 1.4, see  ,
The astrophysical version of the total approximate swept mass as given by Equation (14), expressed in solar mass units, , is
where , and are , and r expressed in pc.
3.2. The Equation of Motion
The conservation of the classical momentum in spherical coordinates along the solid angle in the framework of the thin layer approximation states that
where and are the swept masses at and r, and and v are the velocities of the thin layer at and r. This conservation law can be expressed as a differential equation of the first order by inserting :
The velocity as a function of the radius r is
The differential equation which models the momentum conservation in the case of a hyperbolic profile is
where the initial conditions are and when .
The variables can be separated and the radius as a function of the time is
As a consequence the velocity as function of the time is
More details as well the exploration of other profiles of density can be found in  . We now continue evaluating the flux of kinetic energy, , in the thin emitting layer which is supposed to have density
The volume of the thin emitting layer, , is approximated by
where is thickness of the layer; as an example  quotes . The two approximations for mass, Equation (14), and volume, Equation (28), allows to derive an approximate value for the density in the thin layer
Inserted in Equation (27) the radius, velocity and density as given by Equation (21), Equation (26) and Equation (29), we obtain
We now assumes that the amount of luminosity, , reversed in the shocked emission is proportional to the flux of kinetic energy as given by Equation (30)
The theoretical luminosity is not equal along all the SB but is function of the polar angle . In this framework is useful to introduce the ratio, , between theoretical luminosity at and that one at ,
The above model for the theoretical luminosity is independent from the image theory, see Section 6, and does not explains the hole of luminosity visible in the shells.
4. Astrophysical Environment
We now analyze the catalogue for the giant arcs, the two giant arcs SDP.81 and A2267 and the initial astrophysical conditions for the SBs.
4.1. The Catalogue
Some parameters of the giant arcs as detected as images by cluster lensing and the supernova survey with Hubble (CLASH) which is available as a catalogue at http://vizier.u-strasbg.fr/viz-bin/VizieR, see  . We are interested in the arc length which is given in arcsec, the arc length to width ratio, the photometric redshift, and in the radial distance from the arc center to the cluster center in arcsec. Table 1 reports the statistical parameters of the radial distance from the arc center to the cluster center in kpc and Figures 2-4 the histogram of the frequencies in the framework of flat, ΛCDM and MTL cosmology respectively.
4.2. Single Giant Arcs
The ring associated with the galaxy SDP.81, see  , is characterized by a foreground galaxy at and a background galaxy at . This ring has been studied with the Atacama Large Millimeter/sub-millimeter Array (ALMA) by  -  and has the observed parameters as in Table 2.
Another giant arc is that in A2667 which is made by three pieces: A, B and C, see Figure 1 in  . The radius can be found from the equation of the circle
Table 1. Statistical parameters of the radial distance from the arc center to the cluster center in kpc.
Table 2. Observed parameters of the giants arcs.
Figure 2. Histogram of the radial distance from the arc center to the cluster center in Mpc in flat cosmology with conversion formula (5).
Figure 3. Histogram of the radial distance from the arc center to the cluster center in Mpc in ΛCDM cosmology with conversion formula (2).
Figure 4. Histogram of the radial distance from the arc center to the cluster center in Mpc in MTL cosmology with conversion formula (9).
4.3. The Initial Conditions
We review the starting equations for the evolution of the SB    which can be derived from the momentum conservation applied to a pyramidal section. The parameters of the thermal model are , the number of SN explosions in yr, , the distance of the OB associations from the galactic plane, , the energy in 1051 erg usually chosen equal to one, , the initial velocity which is fixed by the bursting phase, , the initial time in yr which is equal to the bursting time, and t the proper time of the SB. With the above definitions the radius of the SB is
and its velocity
In the following, we will assume that the bursting phase ends at (the bursting time is expressed in units of 107 yr) when SN are exploded
The two following inverted formula allows to derive the parameters of the initial conditions for the SB with ours expressed in pc and expressed in km∙s−1 are
Table 3. Radius of the giant arcs in kpc.
5. Astrophysical Simulation
We simulate a single giant arc, A2267, and then we simulate the statistics of many giant arcs.
5.1. Simulation of A2667
Figure 7 reports the 2D section of the SB as well the three pieces of the giant arc connected with A2267. The similarity between the observed radius of curvature of the giant arc as well the theoretical one is reported in a zoom, see Figure 8.
We can understand the reason for which the giant arc A2267 has a limited angular extension of ≈31˚ by plotting the ratio , equation (37), between the theoretical luminosity as function of and the theoretical luminosity at with parameters as in Table 4, see Figure 9. As a practical example at ≈31˚/2, where the factor two arises from the symmetry of the framework, the theoretical luminosity is decreased of a factor in respect to the value at . We now introduce the threshold luminosity, , which is an observational parameter. The theoretical luminosity will scale as function of the polar angle as where is the theoretical luminosity at and has been defined in Equation (39). When the inequality is verified the giant arc is impossible to detect and only the zone characterized by low values of the polar angle will be detected.
5.2. Simulation of Many Giants Arcs
The presence of multiple giants arcs in the CLASH cluster, see as an example Figure 11 in  , can be simulated adopting the following steps:
Table 4. Theoretical parameters of the SB connected with A2267.
Figure 5. 3D surface of the SB connected with A2267, parameters as in Table 4 and axes in pc.
• A given number of SBs, as an example 15, are generated with variable lifetime, t, see Figure 11.
• For each SB we select a section around polar angle equal to zero characterized by a fixed angle of ≈31˚ and we randomly rotate it around the origin, see Figure 12.
Figure 6. 2D section in the z = 0 plane of the SB connected with A2267, parameters as in Table 4 and axes in kpc.
Figure 7. 2D section in the z = 0 plane of the SB connected with A2267, parameters as in Table 4 (full green points) and the three pieces of the giant arc in A2667 (empty red stars); axes in arcsec.
Figure 8. Enlarged view of the three pieces of the giant arc in A2667 (empty red stars) and the theoretical radius (full green points); axes in arcsec.
Figure 9. Normalized luminosity as function of the polar angle in deg, parameters as in Table 4.
Figure 10. Velocity in km/s as function of the polar angle in deg.
Figure 11. Multiple sections of the SB with time, t comprised in [106 yr, 108 yr] and other parameters as in Table 4.
Figure 12. Multiple sections of the SB as in Figure 11 with angular extension of the polar angle, θ, of ≈31˚ and progressive rotation of the selected piece of section.
• The centers of the SBs are randomly placed in a squared box with side of 300 kpc, see Figure 13.
Table 5 reports the theoretical statistical parameters of the above simulation for the radial distance from the arc center to the cluster center in kpc. A comparison should be done with the astronomical parameters for the CLASH clusters of Table 1.
6. Theory of the Image
We now review the theory of the image for the case of optically thin medium both from an analytical and an analytical point of view.
6.1. The Elliptical Shell
A real ellipsoid represents a first approximation of the asymmetric giants arcs and has equation
in which the polar axis is the z-axis.
We are interested in the section of the ellipsoid which is defined by the following external ellipse
We assume that the emission takes place in a thin layer comprised between the external ellipse and the internal ellipse defined by
Table 5. Statistical parameters of the radial distance from the theoretical arc center to the cluster center in kpc.
Figure 13. Multiple sections of SB as in Figure 12 with random shift of the origin of the selected SB (green empty stars). The random shift denotes the galaxies (red crosses).
see Figure 14. We therefore assume that the number density C is constant and in particular rises from 0 at (0, a) to a maximum value , remains constant up to (0, a-c) and then falls again to 0. The length of sight, when the observer is situated at the infinity of the x-axis, is the locus parallel to the x-axis which crosses the position z in a Cartesian x-z plane and terminates at the external ellipse. The locus length is
In the case of optically thin medium, the intensity is split in two cases
Figure 14. Internal and external ellipses when , and .
where is a constant which allows to compare the theoretical intensity with the observed one. A typical profile in intensity along the z-axis is reported in Figure 15. The ratio, , between the theoretical intensity at the maximum, ( ), and at the minimum, ( ), is given by
As an example the values , , gives . The knowledge of the above ratio from the observations allows to deduce c once a and b are given by the observed morphology
The above analytical model explains the hole in luminosity visible in the astrophysical shells such as supernovae and SBs. More details can be found in  .
Figure 15. The intensity profile along the z-axis when , and .
6.2. The Numerical Shell
The source of luminosity is assumed here to be the flux of kinetic energy, ,
where A is the considered area, V is the velocity and is the density. In our case , where is the considered solid angle and the temporary radius along the chosen direction. The observed luminosity along a given direction can be expressed as
where is a constant of conversion from the mechanical luminosity to the observed luminosity.
We review the algorithm that allows to build the image, see  :
• An empty memory grid which contains NDIM3 pixels is considered.
• We first generate an internal 3D surface of revolution by rotating the ideal image of 360˚ around the polar direction and a second external surface of revolution at a fixed distance from the first surface. As an example, we fixed , where R is the momentary radius of expansion. The points on the memory grid which lie between the internal and external surfaces are memorized on by a variable integer number according to formula (52) and density proportional to the swept mass.
• Each point of has spatial coordinates which can be represented by the following matrix, A,
The orientation of the object is characterized by the Euler angles and therefore by a total rotation matrix, E. The matrix point is represented by the following matrix, B,
• The intensity map is obtained by summing the points of the rotated images along a particular direction.
The image of A2267 built with the above algorithm is shown in Figure 16.
The threshold intensity, , is
where , is the maximum value of intensity characterizing the ring and is a parameter which allows matching theory with observations and was previously defined in Equation (39). A typical image with a hole is visible in Figure 17. The opening angle of the visible arc can be parametrized as function of the ratio , see Figure 18. An opening of ≈31˚ is reached at .
7.1. The Equation of Motion
The giants arcs are connected with the visible part of the SBs which advance in the intracluster medium surrounding the host galaxies. The chosen profile of density is hyperbolic, see Equation (10), and the momentum conservation along
Figure 16. Contour map of I for A2267, the x and y axes are in arcsec. The three Euler angles characterizing the orientation are , and , and NDIM = 400.
Figure 17. The same as Figure 16 parameters as in Figure 16 and .
Figure 18. The opening angle of the visible simulated arc as function of ; parameters as in Figure 16.
a given direction allows to derive the equation of motion as function of the polar angle, see Equation (21).
7.2. The Image
According to the theory here presented the giants arcs are the visible part of an advancing SB. An analytical explanation for the limited angular extent of the giant arcs is represented by the theoretical luminosity as function of the polar angle, see Equation (39). An increase in the polar angle produces a decrease of the theoretical luminosity and the arc becomes invisible. Selecting a given numbers of SBs with variable lifetime and randomly inserting them in a cubic box of side ≈ 600 kpc is possible to simulate the giants arcs visible in the clusters of galaxies, see Figure 13 and relative statistical parameters in Table 5.
This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France.