In one dimension and in the presence of a drift velocity, v, along the x-direction, a diffusion is governed by Fick’s second equation for the concentration, C ( x ) , see Equation (4.5) in [6],

∂ C ∂ t = D ∂ 2 C ∂ x 2 − v ∂ C ∂ x , (1)

where v can take positive or negative values and D is the diffusion coefficient. The number density rises from 0 at x = a to a maximum value C_m at x = b and then falls again to 0 at r = c. The general solution to Equation (1) for a steady state is

C ( x ) = A + B e v x D . (2)

We now assume that v is negative: the solution is

C ( x ) = A + B e − v D x . (3)

The boundary conditions give

C a , b ( x ) = C m ( e − v x D − e − v a D ) − e − v a D + e − v b D   a ≤ x ≤ b     downstreamside , (4)

and

C b , c ( x ) = C m ( e − v c D − e − v x D ) − e − v b D + e − v c D . (5)

Given a 1D segment of length side we can implement a random walk with

step-length λ by introducing the numerical parameter NDIM = side λ . We now report the adopted rules when the injection is in the middle of the grid:

1) The first of NPART particles is chosen.

2) The random walk of the particle starts in the middle of the grid. The probability of taking one step in the negative direction (downstream), is p 1 = 1 2 − μ × 1 2 , and in the positive direction (upstream), p 2 = 1 2 + μ × 1 2 , where μ is a parameter that characterizes the asymmetry ( 0 ≤ μ ≤ 1 ).

3) When the particle reaches one of the two absorbing points, the motion starts again from (2) with a different diffusion pattern.

4) The number of visits is recorded on M , a 1D grid.

5) The random walk terminates when all the NPART particles have been processed.

6) For the sake of normalization, the one-dimensional visitation or number density grid M is divided by NPART.

There is a systematic change of the average particle position along the x-direction:

〈 d x 〉 = μ λ , (6)

for each time step. If the time step is d t = λ v t r where v t r is the transport velocity, the asymmetry, μ , that characterizes the random walk is

μ = v v t r . (7)

The PN A39 is extremely round and therefore can be considered an example of spherical symmetry, see for example Figure 1 in [7] or Figure 4.

The radius of the shell in A39, R s h e l l , is

R s h e l l = 2.42 × 10 18 Θ 77 D 21   cm = 0.78   pc , (19)

where Θ 77 is the angular radius in units of 77" and D 21 the distance in units of 2.1 kpc, see [7].

The expansion velocity has a range [ 32 ↔ 37   km s ] according to [8] and the age of the free expansion is 23,000 yr, see [7]. In our astrophysical units we will use a v = 3.5215 × 10 − 5 pc yr which corresponds to the average value of the above range in velocity. The angular thickness of the shell is

δ   r s h e l l = 3.17 × 10 17 Θ 10 D 21   cm = 0.103   pc , (20)

where Θ 10 is the thickness in units of 10.1" and the height above the galactic plane is 1.42 kpc, see [7]. For the sake of comparison with the observations, the length of some geometrical models will be expressed in arcsec. An occasional reader may ask whether A39 can be approximated by a sphere or not? One way to answer this question is to introduce the observational percentage of sphericity, ε obs , over the whole range of the polar angle θ ,

ε obs = 100 ( 1 − ∑ j | r obs − r num | j ∑ j r obs , j ) , (21)

where r num is the theoretical averaged radius of the nebula, r obs is the observed radius of the nebula, and the index j varies from 1 to the number of available observations. In the case of Figure 4, we have ε obs = 97 % over 67 directions when the central star is assumed to be the source of the expansion.

Figure 5 presents M ( x ) , the number of visits generated by the Monte Carlo simulation as well as the steady state mathematical solution with drift given by Formulas (4) and (5).

In the case of a transient diffusion, we present, in Figure 6, the parabolic solution that will be used in the theory of formation of the image.

The transfer equation in the presence of emission only, see for example [9] or [10], is

d I ν d s = − k ν ζ I ν + j ν ζ , (22)

where I ν is the specific intensity, 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 frequency of emission under consideration. The solution to Equation (22) is

I ν ( τ ν ) = j ν k ν ( 1 − e − τ ν ( s ) ) , (23)

where τ ν is the optical depth at frequency ν

d τ ν = k ν ζ d s . (24)

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 substituted with C ( s ) , our number density of particles. We now assume that the emissivity is proportional to the number density:

j ν ζ = K C ( s ) , (25)

where K is a constant function. This can be the case for synchrotron radiation in the presence of an isotropic distribution of electrons with a power law distribution in energy, N ( E ) ,

N ( E ) d E = K s E − γ f , (26)

where K s is a constant. In this case the emissivity is

j ν ρ ≈ 0.933 × 10 − 23 α ( γ f ) K s H ⊥ γ f + 1 2 ( 6.26 × 10 18 ν ) γ f − 1 2 erg s ⋅ cm 3 ⋅ Hz ⋅ rad 2 , (27)

where ν is the frequency and α ( γ f ) is a slowly varying function of γ f which is of the order of unity and is given by

α ( γ f ) = 2 ( γ f − 3 ) / 2 γ f + 7 / 3 γ f + 1 Γ ( 3 γ f − 1 12 ) Γ ( 3 γ f + 7 12 ) , (28)

for γ f ≥ 1 2 , see Formula (1.175) in [11]. Synchrotron emission is widely used to explain the radiation observed in SNR, see [12] - [17]. This non-thermal radiation continuum emission was also detected in a PN associated with a very long-period OH/IR variable star (V1018 Sco) [18], in the young PN IRAS 15103-5754 [19] and in IRAS 18062 + 2410 [20]. A discussion of thermal/non- thermal emission from PN’s can be found in [21] [22]. The intensity in the optically thin layer is therefore

I ν ( s ) = K ∫ s 0 s     C ( s ' ) d s '     optically   thin   layer     linearcase . (29)

In Monte Carlo experiments, the number density is stored on the grid M and the intensity is

I ( i , j ) = ∑ k     Δ s × M ( i , j , k )     optically   thin   layer         linearcase . (30)

where Δs is the spatial interval between the various values and the sum is performed over the interval of existence of the index k. The theoretical intensity is then obtained by integrating the intensity at a given frequency over the solid angle of the source.

The simplest model for the image is characterized by a radiative process with constant number density of emitting particles in a thin layer around the advancing sphere. We therefore assume that the number density C is constant in the spherical thin layer, and in particular rises from 0 at r = a to a maximum value C_m, remains constant up to r = b and then falls again to 0. 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 the Cartesian x-y plane and terminates at the external circle of radius b, see Figure 7. 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 . (31)

The number density C_m is constant between two spheres of radii a and b and therefore the intensity of radiation is

I 0 a = C m × 2 × ( b 2 − y 2 − a 2 − y 2 ) ;     0 ≤ y < a (32)

I a b = C m × 2 × b 2 − y 2 ;     a ≤ y < b . (33)

The ratio between the theoretical intensity at the maximum ( y = a ) and at the minimum ( y = 0 ) is given by

I 0 a = ( y = a ) I 0 a = ( y = 0 ) = b 2 − a 2 b − a . (34)

The quality of the fit is measured by χ 2 :

χ 2 = ∑ i = 1 n ( T i − O i ) 2 σ i 2 , (35)

where n is the number of points, T i is the theoretical value, O i is the observed value and σ i is the error with respect to the observed value.

A reduced merit function χ r e d 2 is evaluated by

χ r e d 2 = χ 2 / N F , (36)

where N F = n − k is the number of degrees of freedom, n is the number of elements and k is the number of parameters. A comparison of the observed data for A39 and the theoretical intensity of the simplest model are presented in Figure 8.

Figure 9 shows a spherical shell source of radius b between a spherical absorber of radius a and a spherical absorber of radius c. The number density rises from 0 at r = a to a maximum value C_m at r = b and then falls again to 0 at r = c. In order to continue, the 3D spherical diffusion is approximated by the 1D steady state diffusion with drift and the numbers density to be used are Formulas (4) and (5) once r = x 2 + y 2 is imposed. In this case, the operation of integration of the number density as given by Equation (29) can be performed only numerically, see Figure 10.

We assume that the number density rises from 0 at r = a to a maximum value C_m at r = b and then falls again to 0 at r = c following a parabolic behavior with vertex at r = b. The parabola that satisfies the above requirements is

C ( r ; a , b , c ) = − C m ( c − r ) ( a − r ) ( b − c ) ( a − b ) , (37)

and we continue inserting r = x 2 + y 2 . The integration which leads to the image, see Equation (29) and Figure 9, can be done and is

I 0 a = C m ( 3 y 2 arcsinh ( c 2 − y 2 y ) a + 3 y 2 arcsinh ( c 2 − y 2 y ) c               − 3 y 2 arcsinh ( a 2 − y 2 y ) a − 3 y 2 arcsinh ( a 2 − y 2 y ) c               − 3 c 2 − y 2   c a + c 2 − y 2   c 2 − 4 c 2 − y 2   y 2 − a 2 − y 2   a 2                   + 3 a 2 − y 2   a c + 4 a 2 − y 2   y 2 ) ;     0 ≤ y < a (38)

I a c = 1 3 ( b − c ) ( a − b ) × C m ( 3 y 2 arcsinh ( c 2 − y 2 y ) a               + 3 y 2 arcsinh ( c 2 − y 2 y ) c − 3 y 2 arcsinh ( a 2 − y 2 y ) a               − 3 y 2 arcsinh ( a 2 − y 2 y ) c − 3 c 2 − y 2   c a + c 2 − y 2   c 2               − 4 c 2 − y 2   y 2 − a 2 − y 2   a 2 + 3 a 2 − y 2   a c                   + 4 a 2 − y 2   y 2 ) ;     a ≤ y < c . (39)

In the case of the geometrical model the theoretical intensity of A39 is presented as a cut, see Figure 11 and as a surface, see Figure 12.

Figure 9 can model the formation of the image in the case of a transient state developed in Section 3 once a is replaced by zero, b is replaced by L/2 and c is replaced by L. The solution considered is Equation (9) for a parabolic density profile, see Equation (15). In this case the operation of integration which leads to the image is done numerically and is presented as a cut, see Figure 13 and as an image, see Figure 14.

The effect of the insertion of a threshold intensity, I t r , given by the observational techniques, will now be analysed. The threshold intensity can be parametrized by I max , the maximum value of intensity characterizing the ring: a typical image with a hole is visible in Figure 15 when I t r = I max / 2 .