The basic parameters of ΛCDM cosmology are: the Hubble constant, H 0 , expressed in km∙s⁻¹∙Mpc⁻¹, the velocity of light, c, expressed in kms⁻¹, and the three numbers Ω M , Ω K , and Ω Λ , see [22] for more details. In the case of the Union 2.1 compilation, see [23] , the parameters are H 0 = 69.81   km   ⋅ s − 1 ⋅ Mpc − 1 , Ω M = 0.239 and Ω Λ = 0.651 . To have the luminosity distance, D L ( z ; H 0 , c , Ω M , Ω Λ ) , 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, D L ,3,2 , when p = 3 and q = 2

D L , 3 , 2 = − 7.761 − 1788.53 z − 3203.06 z 2 − 65.8463 z 3 − 0.438025 − 0.334872 z + 0.0203996 z 2 Mpc   for     0.001 < z < 4. (1)

The transversal distance in ΛCDM cosmology, D T ,3,2 , which corresponds to the angle δ expressed in arcsec is

D T ,3,2 = 4.84813 δ ( 2.328 + 502.067 z + 113.03 z 2 ) 0.124085 + 0.149501 z + 0.0932928 z 2 pc (2)

The two parameters of the flat cosmology are H 0 , the Hubble constant expressed in km∙s⁻¹∙Mpc⁻¹, and Ω M which is

Ω M = 8 π   G ρ 0 3 H 0 2 , (3)

where G is the Newtonian gravitational constant and ρ 0 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, d L , m ,2,2 , when H 0 = 70   km ⋅ s − 1 ⋅ Mpc − 1 and Ω M = 0.277 , is

d L , m ,2,2 = 0.0889 + 748.555 z + 5.58311 z 2 0.175804 + 0.206041 z + 0.068685 z 2 Mpc (4)

The transversal distance in flat cosmology, D T f ,3,2 , which corresponds to the angle δ expressed in arcsec is

D T f ,3,2 = 4.84813 δ ( 0.0889 + 748.555 z + 5.58311 z 2 ) 0.175804 + 0.206041 z + 0.0686854 z 2 pc (5)

In an Euclidean static framework the modified tired light (MTL) has been introduced in Section 2.2 in [24] . The distance in MTL is

d = c H 0 ln ( 1 + z ) . (6)

The distance modulus in the modified tired light (MTL) is

m − M = 5 2   β ln ( z + 1 ) ln ( 10 ) + 5 1 ln ( 10 ) ln ( ln ( z + 1 ) c H 0 ) + 25 (7)

Here β is a parameter comprised between 1 and 3 which allows to match theory with observations. The number of free parameters in MTL is two: H 0 and β . The fit of the distance modulus with the data of Union 2.1 compilation gives β = 2.37 , H 0 = 69.32 ± 0.34 , see [22] , which means the following distance

d = 4324.761 ln ( 1 + z ) ( 1 + z ) 1.185 Mpc . (8)

The transversal distance in MTL, d T , which corresponds to the angle δ expressed in arcsec is

d T = 20967 δ ln ( 1 + z ) ( 1 + z ) 1.185 pc (9)

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 [25] , in contrast with the modified tired light.

The density is assumed to have the following hyperbolic dependence on Z which is the third Cartesian coordinate,

ρ ( Z ; Z 0 , ρ 0 ) = { ρ 0 if   z ≤ Z 0   ρ 0 Z 0 z if   z > Z 0   (10)

where the parameter Z 0 fixes the scale and ρ 0 is the density at Z = Z 0 . In spherical coordinates the dependence on the polar angle is

ρ ( r ; θ , Z 0 , ρ 0 ) = { ρ 0 if   cos ( θ ) ≤ Z 0 ρ 0 Z 0 r cos ( θ ) if   r   cos ( θ ) > Z 0 (11)

Given a solid angle Δ Ω the mass M 0 swept in the interval [ 0, r 0 ] is

M 0 = 1 3   ρ 0   r 0 3 Δ Ω   . (12)

The total mass swept, M ( r ; r 0 , Z 0 , α , θ , ρ 0 ) , in the interval [ 0, r ] is

M ( r ; r 0 , Z 0 , α , θ , ρ 0 ) = ( 1 3 ρ 0 r 0 3 + 1 2 ρ 0 Z 0 ( r 2 − r 0 2 ) cos ( θ ) ) Δ Ω   . (13)

and its approximate value at high values of r is

M ( r ; Z 0 , α , θ , ρ 0 ) ≈ 1 2 r 2 ρ 0 Z 0 cos ( θ ) Δ Ω   . (14)

The density ρ 0 can be obtained by introducing the number density, n 0 , expressed in particles cm⁻³, the mass of hydrogen, m H , and a multiplicative factor f, which is chosen to be 1.4, see [26] ,

ρ 0 = f m H n 0 . (15)

The astrophysical version of the total approximate swept mass as given by Equation (14), expressed in solar mass units, M ⊙ , is

M ( r p c ; Z 0, p c , n 0 , θ ) ≈ 0.0172   n 0 z 0, p c r p c 2 cos ( θ )   M ⊙ Δ Ω , (16)

where Z 0, p c , and r 0, p c are Z 0 , and r expressed in pc.

The conservation of the classical momentum in spherical coordinates along the solid angle Δ Ω in the framework of the thin layer approximation states that

M 0 ( r 0 ) v 0 = M ( r ) v , (17)

where M 0 ( r 0 ) and M ( r ) are the swept masses at r 0 and r, and v 0 and v are the velocities of the thin layer at r 0 and r. This conservation law can be expressed as a differential equation of the first order by inserting v = d r d t :

M ( r ) d r d t − M 0   v 0 = 0. (18)

The velocity as a function of the radius r is

v ( r ; r 0 , Z 0 , v 0 , θ ) = 2 r 0 3 v 0 cos ( θ ) 2 r 0 3 cos ( θ ) − 3 r 0 2 Z 0 + 3 r 2 Z 0 . (19)

The differential equation which models the momentum conservation in the case of a hyperbolic profile is

( 1 3 r 0 3 + 1 2 Z 0 ( − r 0 2 + ( r ( t ) ) 2 ) cos ( θ ) ) d d t r ( t ) − 1 3 r 0 3 v 0 = 0, (20)

where the initial conditions are r = r 0 and v = v 0 when t = t 0 .

The variables can be separated and the radius as a function of the time is

r ( t ; t 0 , r 0 , Z 0 , v 0 , θ ) = H N H D , (21)

where

H N = − 3 3 ( 2 cos ( θ ) 3 3 r 0 − 3 3 3 Z 0 − ( − 9 Z 0 3 / 2 + ( ( 9 t − 9 t 0 ) v 0 + 9 r 0 ) cos ( θ ) Z 0                   + 3 27 A H N ) 2 / 3 ) r 0 , (22)

with

A H N = ( 8 ( cos ( θ ) ) 2 r 0 3 27 + Z 0 ( ( t − t 0 ) 2 v 0 2 + 2 r 0 ( t − t 0 ) v 0 − 1 3 r 0 2 ) cos ( θ )                     − 2 v 0 Z 0 2 ( t − t 0 ) ) cos ( θ ) (23)

and

H D = 3 Z 0 × − 9 Z 0 3 / 2 + ( ( 9 t − 9 t 0 ) v 0 + 9   r 0 ) cos ( θ ) Z 0 + 9 B H D 3 , (24)

with

B H D = ( 8 ( cos ( θ ) ) 2 r 0 3 27 + Z 0 ( ( t − t 0 ) 2 v 0 2 + 2 r 0 ( t − t 0 ) v 0 − 1 3 r 0 2 ) cos ( θ )                     − 2   v 0 Z 0 2 ( t − t 0 ) ) cos ( θ ) . (25)

As a consequence the velocity as function of the time is

v ( t ; t 0 , r 0 , Z 0 , v 0 , θ ) = d r ( t ; t 0 , r 0 , Z 0 , v 0 , θ ) d t . (26)

More details as well the exploration of other profiles of density can be found in [27] . We now continue evaluating the flux of kinetic energy, F e k , in the thin emitting layer which is supposed to have density ρ l

F e k ( t ; t 0 , r 0 , Z 0 , v 0 , θ ) = 1 2 ρ l 4 π r ( t ) 2 v ( t ) 3 . (27)

The volume of the thin emitting layer, V l , is approximated by

V l = 4 Δ π r 2 , (28)

where Δ is thickness of the layer; as an example [28] quotes Δ = r 12 . The two approximations for mass, Equation (14), and volume, Equation (28), allows to derive an approximate value for the density in the thin layer

ρ l = 1 8 ρ 0 Z 0 f cos ( θ ) r π . (29)

Inserted in Equation (27) the radius, velocity and density as given by Equation (21), Equation (26) and Equation (29), we obtain

F e k ( t ; t 0 , r 0 , Z 0 , v 0 , θ ) = F N F D , (30)

where

F N = − 27 ( − 3 3 Z 0 3 / 2 + 27 F 1 cos ( θ ) + 3 F 5 ) 3 f   × ( 2 3 3 F 3 + ( − 9 Z 0 3 / 2 + F 2 + 27 F 1 cos ( θ ) 3 ) 2 / 3 ) 3 r 0 4   × cos ( θ ) v 0 3 Z 0 ( 2 3 3 F 3 − ( − 9 Z 0 3 / 2 + F 2 + 27 F 1 cos ( θ ) 3 ) 2 / 3 ) 3 3 ρ 0 (31)

and

F D = 108   F 1 cos ( θ ) × ( − 9 Z 0 3 / 2 + F 2 + 27 F 1 cos ( θ ) 3 ) 13 / 3 F 4 (32)

being

F 1 = 8 ( cos ( θ ) ) 2 r 0 3 27 + ( ( t − t 0 ) 2 v 0 2 + 2 r 0 ( t − t 0 ) v 0 − 1 3 r 0 2 ) Z 0 cos ( θ )             − 2 v 0 Z 0 2 ( t − t 0 ) , (33)

F 2 = ( ( 9 t − 9 t 0 ) v 0 + 9 r 0 ) cos ( θ ) Z 0 , (34)

F 3 = cos ( θ ) r 0 − 3 / 2 Z 0 (35)

F 4 = 8 ( cos ( θ ) ) 2 r 0 3 + 27 ( ( t − t 0 ) 2 v 0 2 + 2   r 0 ( t − t 0 ) v 0 − 1 3 r 0 2 ) Z 0 cos ( θ )               − 54   v 0 Z 0 2 ( t − t 0 ) , (36)

F 5 = ( v 0 ( t − t 0 ) + r 0 ) cos ( θ ) 3 Z 0 . (37)

We now assumes that the amount of luminosity, L t h e o , reversed in the shocked emission is proportional to the flux of kinetic energy as given by Equation (30)

L t h e o ( t ; t 0 , r 0 , Z 0 , v 0 , θ ) ∝ F e k ( t ; t 0 , r 0 , Z 0 , v 0 , θ ) . (38)

The theoretical luminosity is not equal along all the SB but is function of the polar angle θ . In this framework is useful to introduce the ratio, κ , between theoretical luminosity at θ and that one at θ = 0 ,

κ = L t h e o ( t ; t 0 , r 0 , Z 0 , v 0 , θ ) L t h e o ( t ; t 0 , r 0 , Z 0 , v 0 , θ = 0 ) . (39)

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.

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 [29] . 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.

The ring associated with the galaxy SDP.81, see [30] , is characterized by a foreground galaxy at z = 0.2999 and a background galaxy at z = 0.3042 . This ring has been studied with the Atacama Large Millimeter/sub-millimeter Array (ALMA) by [31] - [36] 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 [37] . The radius can be found from the equation of the circle

given the three points A, B and C, see Table 3. The three pieces can be digitalized for a further comparison with a simulation, see empty red stars in Figure 8.

We review the starting equations for the evolution of the SB [38] [39] [40] which can be derived from the momentum conservation applied to a pyramidal section. The parameters of the thermal model are N * , the number of SN explosions in 5.0 × 10 7 yr, Z OB , the distance of the OB associations from the galactic plane, E 51 , the energy in 10⁵¹ erg usually chosen equal to one, v 0 , the initial velocity which is fixed by the bursting phase, t 0 , 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

R = 111.56 ( E 51 t 7 3 N * n 0 ) 1 5 pc , (40)

and its velocity

V = 6.567 1 t 7 2 / 5 E 51 N * n 0 5   km ⋅ s − 1 . (41)

In the following, we will assume that the bursting phase ends at t = t 7 , 0 (the bursting time is expressed in units of 10⁷ yr) when N S N SN are exploded

N S N = N * t 7 , 0 × 10 7 5 × 10 7 . (42)

The two following inverted formula allows to derive the parameters of the initial conditions for the SB with ours r 0 expressed in pc and v 0 expressed in km∙s⁻¹ are

t 7,0 = 0.05878095238 r 0 v 0 , (43)

and

N * = 2.8289 × 10 − 7 r 0 2 n 0 v 0 3 E 51 . (44)

The final stage of the SB connected with A2267 is simulated with the parameters reported in Table 4; in particular Figure 5 displays the 3D shape and Figure 6 reports the 2D section.

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 κ , equation (37), between the theoretical luminosity as function of θ and the theoretical luminosity at θ = 0 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 κ = 0.987 in respect to the value at θ = 0 . We now introduce the threshold luminosity, L t r , which is an observational parameter. The theoretical luminosity will scale as function of the polar angle as L t h e o ( θ ) ∝ L 0 ∗ r where L 0 is the theoretical luminosity at θ = 0 and κ has been defined in Equation (39). When the inequality L t h e o < L t r is verified the giant arc is impossible to detect and only the zone characterized by low values of the polar angle will be detected.

In our model the velocity with parameters as in Table 4 is function of the polar angle, see Figure 10, and has range 37   km / s < v ( θ ) < 142   km / s . As a comparison a velocity 50   km / s < v < 75   km / s is measured in A2267, see Figure 5 in [37] .

The presence of multiple giants arcs in the CLASH cluster, see as an example Figure 11 in [29] , can be simulated adopting the following steps:

• 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.

• 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.

A real ellipsoid represents a first approximation of the asymmetric giants arcs and has equation

z 2 a 2 + x 2 b 2 + y 2 d 2 = 1 , (45)

in which the polar axis is the z-axis.

We are interested in the section of the ellipsoid y = 0 which is defined by the following external ellipse

z 2 a 2 + x 2 b 2 = 1. (46)

We assume that the emission takes place in a thin layer comprised between the external ellipse and the internal ellipse defined by

z 2 ( a − c ) 2 + x 2 ( b − c ) 2 = 1 , (47)

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 C m , 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

l I = 2 a 2 − z 2 b a (48)

when ( a − c ) ≤ z < a

l I I = 2 a 2 − z 2 b a − 2 a 2 − 2 a c + c 2 − z 2 ( b − c ) a − c (49)

when 0 ≤ z < ( a − c )

In the case of optically thin medium, the intensity is split in two cases

I I ( z ; a , b ) = I m × 2 a 2 − z 2 b a (50)

when ( a − c ) ≤ z < a

I I I ( z ; a , c ) = I m × ( 2 a 2 − z 2 b a − 2 a 2 − 2 a c + c 2 − z 2 ( b − c ) a − c ) (51)

when 0 ≤ z < ( a − c )

where I m 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, κ , between the theoretical intensity at the maximum, ( z = a − c ), and at the minimum, ( z = 0 ), is given by

I I ( z = a − c ) I I I ( z = 0 ) = κ = 2   a − c b c a . (52)

As an example the values a = 6   kpc , b = 4   kpc , c = a 12   kpc gives κ = 3.19 . The knowledge of the above ratio from the observations allows to deduce c once a and b are given by the observed morphology

c = 2 a b 2 a 2 r 2 + b 2 . (53)

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 [41] .

The source of luminosity is assumed here to be the flux of kinetic energy, L m ,

L m = 1 2 ρ A V 3 , (54)

where A is the considered area, V is the velocity and ρ is the density. In our case A = r 2 Δ Ω , where Δ Ω is the considered solid angle and r ( θ ) the temporary radius along the chosen direction. The observed luminosity along a given direction can be expressed as

L = ϵ L m , (55)

where ϵ is a constant of conversion from the mechanical luminosity to the observed luminosity.

We review the algorithm that allows to build the image, see [42] :

• An empty memory grid M ( i , j , k ) 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 Δ R from the first surface. As an example, we fixed Δ R = R / 12 , 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 M ( i , j , k ) by a variable integer number according to formula (52) and density ρ proportional to the swept mass.

• Each point of M ( i , j , k ) has spatial coordinates x , y , z which can be represented by the following 1 × 3 matrix, A,

A = [ x y z ] . (56)

The orientation of the object is characterized by the Euler angles ( Φ , Θ , Ψ ) and therefore by a total 3 × 3 rotation matrix, E. The matrix point is represented by the following 1 × 3 matrix, B,

B = E ⋅ A     . (57)

• 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, I t r , is

I max   κ = I max , (58)

where I max , is the maximum value of intensity characterizing the ring and κ 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 κ , see Figure 18. An opening of ≈31˚ is reached at κ ≈ 0.95 .

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

a given direction allows to derive the equation of motion as function of the polar angle, see Equation (21).

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.