We assume that the medium around the SN scales with the piece-wise dependence

ρ ( r ; r c , b ) = { ρ 0 r < r 0 ρ c ( b − r + r 0 ) b r 0 < r , (3)

where ρ c is the density at r = 0 , r₀ is the radius after which the density starts to decrease and b is an adjustable parameter. The mass swept, M₀, in the interval [0, r₀] is

M 0 ( ρ c , r 0 ) = 4 3 ρ c π r 0 3 .

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

M ( r ; r 0 , ρ c , b ) = 4 ρ c π r 0 3 3 + ρ c π r 3 ( − 3 r + 4 b + 4 r 0 ) 3 b − ρ c π r 0 3 ( r 0 + 4 b ) 3 b . (4)

The application of the conservation of energy gives the velocity as a function of the radius:

v ( r ; r 0 , b , v 0 ) = 2 b   r 0 3 2 v 0 4 b   r 3 − 3 r 4 + 4 r 3 r 0 − r 0 4 , (5)

or

d d t r ( t ) = v ( r ; r 0 , b , v 0 ) . (6)

The above ordinary differential equation (ODE) does not have an analytical solution. An approximation for the trajectory is obtained by a series solution of the above ODE to fourth order,

r ( t ; r 0 , v 0 , t 0 , b ) ≈ r 0 + ( r 0 b   v 0 r 0 b − v 0 ( r 0 b − r 0 b ) r 0 b ) ( t − t 0 )   + ( 3 v 0 2 ( r 0 b − 2 r 0 b ) 4 r 0 r 0 b − 3 v 0 2 ( r 0 b − r 0 b ) 4 r 0 b   r 0 ) ( t − t 0 ) 2   + v 0 3 ( 3 r 0   b 3 2 + 5 b r 0 b + 2 r 0 r 0 b ) ( t − t 0 ) 3 8 r 0 b   r 0 2 b . (7)

We assume that the medium around the SN scales with a trigonometric profile of the type

{ ρ c r < r 0 ρ c cos ( ( r − r 0 ) π 2 b ) r 0 < r , (8)

where ρ c is the density at r = 0 , r₀ is the radius after which the density starts to decrease and b is an adjustable parameter. The total mass swept, M ( r ; r 0 , ρ c , b ) , in the interval [0, r] is

M ( r ; r 0 , ρ c , b ) = 1 3 π 2 4 ρ c ( π 3 r 0 3 + 6 π 2 b   r 2 sin ( ( r − r 0 ) π 2 b )   + 24 π cos ( ( r − r 0 ) π 2 b ) b 2 r − 24 b 2 r 0 π − 48 b 3 sin ( ( r − r 0 ) π 2 b ) ) . (9)

The velocity as a function of the radius is

v ( r ; r 0 , b , v 0 ) = π 3 2 r 0 3 2 v 0 D , (10)

where

D = ( π 3 r 0 3 + 6 π 2 b   r 2 sin ( ( r − r 0 ) π 2 b ) + 24 π cos ( ( r − r 0 ) π 2 b ) b 2 r     − 24 b 2 r 0 π − 48 b 3 sin ( ( r − r 0 ) π 2 b ) ) 1 2 . (11)

The first order ODE to be solved is

d d t r ( t ) = π 3 2 r 0 3 2 v 0 D . (12)

A series solution for the above ODE to fourth order is

r ( t ; r 0 , v 0 , t 0 , b ) ≈ r 0 + ( r 0 π   v 0 π r 0 − v 0 ( r 0 π − π r 0 ) π r 0 ) ( t − t 0 )   + ( 3 v 0 2 ( r 0 π − 2 π r 0 ) 4 r 0 π r 0 − 3 v 0 2 ( r 0 π − π r 0 ) 4 π r 0   r 0 ) ( t − t 0 ) 2   + v 0 3 ( 3 r 0 π + 5 π r 0 ) ( t − t 0 ) 3 8 r 0 2 π r 0 . (13)

The numerical results for a medium with a linear decrease in density are presented in Table 2. The results for a medium with a cosine profile of density are presented in Table 3. Figure 1 presents the Taylor approximation of the trajectory for the linear profile as given by Equation (7) in the restricted range of time [27.75 yr, 55 yr].

Figure 2 presents the Taylor approximation of the trajectory in the trigonometric profile as given by Equation (13) in the restricted range of time [27.75 yr, 40 yr].

In the case of an inverse square profile for the density, we present a numerical solution as given by the numerical integration of the differential Equation (18). Figure 3 displays the theoretical model for SN 1993J 1993j as well as the observed trajectory [25] .

In ΛCDM cosmology the Hubble distance D H is defined as

D H ≡ c H 0 , (26)

where c is the speed of light and H₀ is the Hubble constant. We then introduce a parameter Ω M

Ω M = 8 π G ρ 0 3 H 0 2 , (27)

where G is the Newtonian gravitational constant and ρ 0 is the mass density at the present time. Another is Ω Λ

Ω Λ ≡ Λ c 2 3 H 0 2 , (28)

where Λ is the cosmological constant, see [29] . Once Ω Λ and H₀ are found, the

numerical value of the cosmological constant is derived, Λ ≈ 1.2 1 m 2 .

The two previous parameters are connected with the curvature Ω K by

Ω M + Ω Λ + Ω K = 1   . (29)

The comoving distance, D C , is

D C = D H ∫ 0 z d z ′ E ( z ′ ) , (30)

where E ( z ) is the “Hubble function”

E ( z ) = Ω M ( 1 + z ) 3 + Ω K ( 1 + z ) 2 + Ω Λ . (31)

The above integral cannot be done in analytical terms, except for the case of Ω Λ = 0 , but the Padé approximant, see Appendix A in [28] , allows us to derive an approximation for the indefinite integral. In the relativistic models, the angular diameter distance, D A [30] , is

D A = D L ( 1 + z ) 2 . (32)

We now introduce the minimax approximation. Let f ( x ) be a real function defined in the interval [ a , b ] . The best rational approximation of degree ( k , l ) evaluates the coefficients of the ratio of two polynomials of degree k and l, respectively, which minimizes the maximum difference of

max | f ( x ) − p 0 + p 1 x + ⋯ + p k x k q 0 + q 1 x + ⋯ + q l x l | , (33)

on the interval [ a , b ] . The quality of the fit is given by the maximum error over the considered range. The coefficients are evaluated through the Remez algorithm, see [31] [32] . The minimax approximation for the angular distance in the interval 0 < z < 8 with data as in Table 4 for the ΛCDM cosmology when k = 2 and p = 2 is

D A ,2,2 = − 0.08214096 + ( 297.0835166 + 2.721712660 z ) z 0.0672226102 + ( 0.0810609648 + 0.02497466229 z ) z Mpc (34)

maximumerror = 0.6925458 Mpc   .

A field of investigation in applied cosmology is the maximum of the angular distance as a function of the redshift [33] [34] , z max , which is finite in relativistic cosmologies and infinite in Euclidean cosmology. The numerical values of z max are presented in Table 5, as a reference z max = 1.594 for the flat Planck-ΛCDM cosmology [35] .

The starting point is Equation (1) for the luminosity distance in [36]

D L ( z ; c , H 0 , Ω M ) = c ( 1 + z ) H 0 ∫ 0 z 1 Ω M ( 1 + t ) 3 + 1 − Ω M d t , (35)

where the variable of integration, t, denotes the redshift.

By first changing the parameter Ω M by introducing

s = 1 − Ω M Ω M 3 (36)

the luminosity distance becomes

D L ( z ; c , H 0 , s ) = 1 H 0 c ( 1 + z ) ∫ 0 z 1 ( 1 + t ) 3 s 3 + 1 + 1 − ( s 3 + 1 ) − 1 d t . (37)

The following change of variable, t = s − u u , is performed for the luminosity distance, which becomes

D L ( z ; c , H 0 , s ) = − c H 0 s 2 ( 1 + z ) ( s 3 + 1 ) ∫ s s 1 + z u u 3 + 1 s 3 ( u 3 + 1 ) u 3 ( s 3 + 1 )   d u . (38)

The integral for the luminosity distance is

D L ( z ; c , H 0 , s ) = − 1 / 3 c ( 1 + z ) 3 3 / 4 s 3 + 1 s H 0   × ( F ( 2 s ( s + 1 + z ) 3 4 s 3 + s + z + 1 , 1 / 4 2 3 + 1 / 4 2 )   − F ( 2 3 4 s ( s + 1 ) s + 1 + s 3 , 1 / 4 2 3 + 1 / 4 2 ) ) , (39)

where s is given by Equation (36) and F ( ϕ , k ) is Legendre’s incomplete elliptic integral of the first kind,

F ( ϕ , k ) = ∫ 0 sin ϕ d t 1 − t 2 1 − k 2 t 2 , (40)

see [37] . The minimax approximation for the angular distance in the interval 0 < z < 8 with data as in Table 4 for the flat cosmology when k = 2 and p = 2 is

D A ,2,2 = − 0.03784870 + ( 274.0072985 + 2.188226163 z ) z 0.0640891726 + ( 0.0767235768 + 0.02582989582 z ) z Mpc (41)

maximumerror = 0.629028 Mpc   .

In the dynamical dark energy cosmology (wCDM), first introduced by [38] , the Hubble distance is

D H ( z ; Ω M , w , Ω D E ) = 1 ( 1 + z ) 3 Ω M + Ω D E ( 1 + z ) 3 + 3   w , (42)

where w is the equation of state, here considered a constant, see Equation (3.4) in [39] or Equation (18) in [40] for the luminosity distance. Here we assumed w to be constant but the case of w as function of z can also be considered, see Equation (19) in [40] . In the above cosmology the cosmological constant is absent. In flat cosmology

Ω M + Ω D E = 1, (43)

and the Hubble distance becomes

D H ( z ; Ω M , w ) = 1 ( 1 + z ) 3 Ω M + ( 1 − Ω M ) ( 1 + z ) 3 + 3   w . (44)

The indefinite integral in the variable z of the above Hubble distance, I z ≡ D C D H , is

I z ( z ; Ω M , w ) = ∫ D H ( z ; Ω M , w ) d z , (45)

where the new symbol I z underlines the mathematical operation of integration. In order to find the indefinite integral, we perform a change of variable 1 + z = t 1 / 3

I z ( t ; Ω M , w ) = 1 3 ∫ 1 − t ( ( − 1 + Ω M ) t w − Ω M ) t 2 / 3 d t . (46)

The indefinite integral is

I z ( t ; Ω M , w ) = − 2 F 2 1 ( 1 2 , − 1 6 w − 1 ; 1 − 1 6 w − 1 ; − t w − ( 1 − Ω M ) Ω M ) Ω M t 6 , (47)

where F 2 1 ( a , b ; c ; z ) is the regularized hypergeometric function, see [37] [41] [42] [43] [44] . We now return to the variable z, the redshift. The indefinite integral becomes

I z ( z ; Ω M , w ) = − 2 F 2 1 ( 1 2 , − 1 6 w − 1 ; 1 − 1 6 w − 1 ; − ( − z 3 + 3 z 2 + 3 z + 1 ) w ( 1 − Ω M ) − Ω M ) Ω M z 3 + 3   z 2 + 3   z + 1 6 . (48)

We denote by F ( z ; Ω M , w ) the definite integral

F ( z ; Ω M , w ) = I z ( z = z ; Ω M , w ) − I z ( z = 0 ; Ω M , w ) . (49)

The luminosity distance, D L , for the wCDM cosmology in the case of the analytical solution is

D L ( z ; c , H 0 , Ω M , w ) = c H 0 ( 1 + z ) F ( z ; Ω M , w ) , (50)

where F ( z ; Ω M , w ) is given by Equation (49) and the distance modulus is

m − M = 25 + 5 log 10 ( D L ( z ; c , H 0 , Ω M , w ) ) . (51)

More details can be found in [45] . The minimax approximation for the angular distance in the interval 0 < z < 8 with data as in Table 4 for the wCDM cosmology when k = 3 and p = 2 is

D A ,3,2 = 0.039674109 + ( 261.8382247 + ( 2.155808744 − 0.04094111183 z ) z ) z 0.0597028175 + ( 0.0839556948 + 0.02480097583 z ) z Mpc (52)

maximumerror = 0.0622614 Mpc   .

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

θ E = 4 G M ( θ E ) c 2 D d s D d D s , (56)

where M ( θ E ) is the mass enclosed inside the radius of the ER, D d , s , d s are the lens, source and lens-source distances, respectively, G is the Newtonian gravitational constant, and c is the speed of light, see Equation (20) in [47] and Equation (1) in [48] . The mass of the ER can be expressed in units of solar mass, M ⊙ :

M ( θ E , arcsec ) = 1.228 × 10 8 Θ E , arcsec 2 D d s , Mpc D s , Mpc D d , Mpc M ⊙ , (57)

where θ E , arcsec is the radius of the ER in arcsec and the three distances are expressed in Mpc.

The ring associated with the galaxy SDP.81, see [49] , is generally explained by a GL. In this framework we have a foreground galaxy at z = 0.2999 and a background galaxy at z = 3.042 [50] . This ring has been studied with the Atacama Large Millimeter/sub-millimeter Array (ALMA) by [50] - [55] . The system SDP.81 as analysed by ALMA presents 14 molecular clumps along the two main lensed arcs. We can therefore speak of the ring’s appearance as a “grand design” and we now test the hypothesis of circular symmetry. In order to test the departure from a circle, an observational percentage of reliability is introduced that uses both the size and the shape,

ε obs = 100 ( 1 − ∑ j | R obs − R ave | j ∑ j R obs j ) , (58)

where R obs is the observed radius in arcsec and R ave is the average radius in arcsec, which is R ave = 1.54   arcsec . Figure 4 presents the astronomical data of SDP.81; the percentage of reliability is ε obs = 92.78 % . According to Equation (55) the Euclidean average radius of the SDP.81 ring is 46.66 kpc when c = 299792.458   km ⋅ s − 1 , H 0 = 67   km ⋅ s − 1 ⋅ Mpc − 1 and α arcsec = 1.54 , see Table 6.

The object IAC J010127-334319 has been detected in the optical region with the Gran Telescopio CANARIAS; the background starburst galaxy is at z = 1.165 and the radius of the ER is θ E = 2.16   arcsec according to [48] and θ E = 2.25   arcsec according to [56] . As an example, inserting the above radius, D s , Mpc = 1192   Mpc , D l s , Mpc = 498   Mpc and D l , Mpc = 951   Mpc in Equation (57), we obtain a mass for the foreground galaxy of M ( θ E , arcsec ) = 1.3 × 10 12 M ⊙ . According to Equation (55) the Euclidean average radius of the Canarias ring is 36.19 kpc when c = 299792.458   km ⋅ s − 1 , H 0 = 67   km ⋅ s − 1 ⋅ Mpc − 1 and α arcsec = 2.16 , see Table 6.

We now simulate the ring connected with SDP.81 and its Euclidean radius of 46.66 kpc. The free parameters of the model with a linear profile are presented in Table 7, Figure 5 presents the law of motion and Figure 6 the behavior of the velocity as a function of time.

The free parameters of the model with a trigonometric profile are presented in Table 8, Figure 7 presents the numerical solution of the law of motion.

The Canarias ring is now simulated in a relativistic framework, see Section 3. The free parameters of the model with an inverse square profile of density are presented in Table 9 and Figure 8 presents the law of motion.

The transfer equation in the presence of emission and absorption, see for example Equation (1.23) in [57] or Equation (9.4) in [58] or Equation (2.27) in [59] , is

d I ν d s = − k ν ζ I ν + j ν ζ , (59)

where I ν is the specific intensity or spectral brightness, s is the line of sight, j ν the emission coefficient, k ν a mass absorption coefficient, ζ the mass density at position s, and the index ν denotes the involved frequency of emission. The solution to Equation (59) is

I ν ( τ ν ) = j ν k ν ( 1 − e − τ ν ( s ) ) , (60)

where τ ν is the optical depth at frequency ν

d τ ν = k ν ζ d s . (61)

We now continue analysing the case of an optically thin layer in which τ ν is very small (or k ν very small) and the density ζ is replaced by the number density of particles, n ( s ) . In the following, the emissivity is taken to be proportional to the number density

j ν ζ = K n ( s ) , (62)

where K is a constant. The intensity is therefore

I ν ( s ) = I 0 + K ∫ s 0 s n ( s ' ) d s ' , (63)

where I₀ is the intensity at the point s₀. The MKS units of the intensity are W∙m⁻²∙Hz⁻¹∙sr⁻¹. The increase in brightness is proportional to the number density integrated along the line of sight: in the case of constant number density, it is proportional only to the distance along the line of sight.

As an example, synchrotron emission has an intensity proportional to l, the dimension of the radiating region, in the case of a constant number density of the radiating particles, see formula (1.175) of [60] .

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

L m = 1 2 ρ A V 3 , (64)

where A is the considered area, V is the velocity of a spherical SB and ρ is the density in the advancing layer of the spherical SB. In the case of the spherical expansion of an SB, A = 4 π r 2 , where r is the instantaneous radius of the SB, which means

L m = 1 2 ρ 4 π r 2 V 3 . (65)

The units of the luminosity are W in MKS and erg∙s⁻¹ in CGS. The astrophysical version of the rate of kinetic energy, L m a , is

L m a = 1.39 × 10 29 n 1 r 1 2 v 1 3 ergs s , (66)

where n₁ is the number density expressed in units of 1 particle/cm³, r₁ is the radius in parsecs, and v₁ is the velocity in km/s. As an example, inserting r 1 = 46.66 × 10 3 , n 1 = 0.1 and v 1 = 26.08 in the above formula, the maximum

available mechanical luminosity is L m a = 5.37 × 10 41 ergs s . The spectral luminosity, L ν , at a given frequency ν is

L ν = 4 π D L 2 S ν , (67)

where S ν is the observed flux density at the frequency ν with MKS units W∙m⁻²∙Hz⁻¹. The observed luminosity at a given frequency ν can be expressed as

L ν = ε L m a , (68)

where ε is a conversion constant from the mechanical luminosity to the observed luminosity. More details on the synchrotron luminosity and the connected astrophysical units can be found in [59] .

We assume that the number density of the emitting matter n is variable, and in particular rises from 0 at r = a to a maximum value n_m, remains constant up to r = b , and then falls again to 0. This geometrical description is shown in Figure 9.

The length of the line 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 y in a Cartesian x-y plane and terminates at the external circle of radius b. The length of this locus is

l 0 a = 2 × ( b 2 − y 2 − a 2 − y 2 ) ;     0 ≤ y < a l a b = 2 × b 2 − y 2 ;     a ≤ y < b . (69)

When the number density of the emitting matter n_m is constant between two spheres of radii a and b, the intensity of radiation is

I 0 a = K I × n m × 2 × ( b 2 − y 2 − a 2 − y 2 ) ;     0 ≤ y < a I a b = K I × n m × 2 × b 2 − y 2 ;     a ≤ y < b , (70)

where K I is a constant. The ratio between the theoretical intensity at the maximum ( y = a ) and at the minimum ( y = 0 ) is given by

I ( y = a ) I ( y = 0 ) = b 2 − a 2 b − a . (71)

The parameter b is identified with the external radius, which means the advancing radius of an SB. The parameter a can be found from the following formula:

a = b ( ( I ( y = a ) I ( y = 0 ) ) obs 2 − 1 ) ( I ( y = a ) I ( y = 0 ) ) obs 2 + 1 , (72)

where ( I ( y = a ) I ( y = 0 ) ) obs is the observed ratio between the maximum intensity at

the rim and the intensity at the centre. The distance Δ y after which the intensity is decreased by a factor f in the region a ≤ y < b is

Δ y = 2 b 2 f 2 + a 2 − b 2 − a 2 f 4 − b 2 f 4 + 2 a 2 f 2 + 2 b 2 f 2 + a 2 − b 2 2 f . (73)

We can now evaluate the half-width half-maximum by analogy with the Gaussian profile H W H M U , which is obtained by the previous formula upon inserting f = 2 :

H W H M U = 1 2 a 2 + 3 b 2 − 1 4 25 a 2 − 9 b 2 . (74)

In the above model, b is associated with the radius of the outer region of the observed ring, a conversely can be deduced from the observed H W H M U :

a = 1 21 441 b 2 + 464 H W H M U 2 − 32 441 b 2 H W H M U 2 + 100 H W H M U 4 . (75)

As an example, inserting in the above formula b = 1.54 arcsec and H W H M U = 0.1 arcsec, we obtain a = 1.46 arcsec. A cut in the theoretical intensity of SDP.81, see Section 7.2, is presented in Figure 10 and a theoretical image in Figure 11.

The effect of the insertion of a threshold intensity, I t r , which is connected with the observational techniques, is now analysed. The threshold intensity can be parametrized to I max , the maximum value of intensity characterizing the ring: a typical image with a hole is visible in Figure 12 when I t r = I max / f a c , where f a c is a parameter which allows matching theory with observations.

For a comparison the observed image of the Einstein ring connected with SDP.81 is presented in Figure 13.