The density is assumed to have the following exponential dependence on z in Cartesian coordinates:

ρ ( z ; b , ρ 0 ) = ρ 0 exp ( − z / b ) , (1)

where b represents the scale. In spherical coordinates, the density has the following piecewise dependence

ρ ( r ; r 0 , b , ρ 0 ) = ( ρ 0 if   r ≤ r 0 ρ 0 exp ( − r cos ( θ ) b ) if   r > r 0 (2)

which has a jump discontinuity at r = r 0 when θ > 0 . Given a solid angle Δ Ω , the total mass swept, M ( r ; r 0 , b , θ , ρ 0 , Δ Ω ) , in the interval [ 0, r ] is

M ( r ; r 0 , b , θ , ρ 0 , Δ Ω ) = ( 1 3 ρ 0 r 0 3 − b ( r 2 ( cos ( θ ) ) 2 + 2 r b cos ( θ ) + 2 b 2 ) ρ 0 ( cos ( θ ) ) 3 e − r cos ( θ ) b         + b ( r 0 2 ( cos ( θ ) ) 2 + 2 r 0 b cos ( θ ) + 2 b 2 ) ρ 0 ( cos ( θ ) ) 3 e − r 0 cos ( θ ) b ) Δ Ω (3)

The density is assumed to have the following Gaussian dependence on z in Cartesian coordinates:

ρ ( z ; b , ρ 0 ) = ρ 0 e − 1 2 z 2 b 2 , (4)

where b represents a parameter. In spherical coordinates, the density is

ρ ( r ; r 0 , b , ρ 0 ) = ( ρ 0 if   r ≤ r 0 ρ 0 e − 1 2 z 2 b 2 if   r > r 0 (5)

and presents a jump discontinuity at r = r 0 when θ > 0 . The total mass swept, M ( r ; r 0 , b , θ , ρ 0 ) , in the interval [ 0, r ] is

M ( r ; r 0 , b , θ , ρ 0 ) = ( 1 3 ρ 0 r 0 3 + ρ 0 ( − r b 2 ( cos ( θ ) ) 2 e − 1 2 r 2 ( cos ( θ ) ) 2 b 2 + 1 2 b 3 π 2 ( cos ( θ ) ) 3 erf ( 1 2 2 cos ( θ ) r b ) )         − ρ 0 ( − r 0 b 2 ( cos ( θ ) ) 2 e − 1 2 r 0 2 ( cos ( θ ) ) 2 b 2 + 1 2 b 3 π 2 ( cos ( θ ) ) 3 erf ( 1 2 2 cos ( θ ) r 0 b ) ) ) Δ Ω (6)

where erf ( x ) is the error function, defined by

erf ( x ) = 2 π ∫ 0 x e − t 2 d t . (7)

The density is assumed to have the following dependence on z in Cartesian coordinates,

ρ ( z ; z 0 , ρ 0 ) = ρ 0 ( 1 + z z 0 ) − 2 . (8)

In this paper, we will adopt the following density profile in spherical coordinates

ρ ( r ; r 0 , b , ρ 0 ) = ( ρ 0 if   r ≤ r 0 ρ 0 ( 1 + r cos ( θ ) z 0 ) − 2 if   r > r 0 (9)

where the parameter z 0 fixes the scale and ρ 0 is the density at z = z 0 . The above density presents a jump discontinuity at r = r 0 when θ > 0 . The mass M 0 swept in the interval [ 0, r 0 ] is

M 0 = 1 3   ρ 0   r 0 3 Δ Ω (10)

The total mass swept, M ( r ; r 0 , z 0 , θ , ρ 0 , Δ Ω ) , in the interval [ 0, r ] is

M ( r ; r 0 , z 0 , θ , ρ 0 , Δ Ω )

= ( 1 3   ρ 0 r 0 3 + ρ 0 b 2 r ( cos ( θ ) ) 2 − 2 ρ 0 b 3 ln ( r cos ( θ ) + b ) ( cos ( θ ) ) 3 − ρ 0 b 4 ( cos ( θ ) ) 3 ( r cos ( θ ) + b )         − ρ 0 b 2 r 0 ( cos ( θ ) ) 2 + 2 ρ 0 b 3 ln ( r 0 cos ( θ ) + b ) ( cos ( θ ) ) 3 + ρ 0 b 4 ( cos ( θ ) ) 3 ( r 0 cos ( θ ) + b ) ) Δ Ω (11)

The usual Navarro-Frenk-White (NFW) distribution has a dependence on r in spherical coordinates of the type

ρ ( r ; r 0 , b , ρ 0 ) = ρ 0 r 0 ( b + r 0 ) 2 r ( b + r ) 2 , (12)

where b represents the scale, see [14] for more details. The NFW profile along the axis z can be obtained by substituting r with r   cos ( θ ) = z

ρ ( r ; r 0 , b , ρ 0 , θ ) = ρ 0 r 0 ( b + r 0 ) 2 r cos ( θ ) ( b + r cos ( θ ) ) 2 , (13)

The piece-wise density is

ρ ( r ; r 0 , b , ρ 0 θ ) = { ρ 0   if   r ≤ r 0 ρ 0 r 0 ( b + r 0 ) 2 r cos ( θ ) ( b + r cos ( θ ) ) 2   if   r > r 0 (14)

and has a jump discontinuity at r = r 0 when θ > 0 . The total mass swept, M ( r ; r 0 , b , ρ 0 θ ) , in the interval [ 0, r ] is

M ( r ; r 0 , b , θ , ρ 0 , Δ Ω ) = ( 1 3 ρ 0 r 0 3 + ρ 0 ( ( b + r cos ( θ ) ) ln ( b + r cos ( θ ) ) + b ) ( b + r 0 ) 2 r 0 ( cos ( θ ) ) 3 ( b + r cos ( θ ) ) =         − ρ 0 ( ( b + r 0 cos ( θ ) ) ln ( b + r 0 cos ( θ ) ) + b ) ( b + r 0 ) 2 r 0 ( cos ( θ ) ) 3 ( b + r 0 cos ( θ ) ) ) Δ Ω (15)

In the case of constant density of the ISM, the analytical solution for the trajectory is

r ( t ; t 0 , r 0 , v 0 ) = 4 r 0 3 v 0 ( t − t 0 ) + r 0 4 4 , (18)

and the velocity is

v ( t ; t 0 , r 0 , v 0 ) = r 0 3 v 0 ( 4   r 0 3 v 0 ( t − t 0 ) + r 0 4 ) 3 / 4 , (19)

where r 0 and v 0 are the position and the velocity when t = t 0 , see [15] [16].

In the case of an exponential density profile for the ISM, as given by Equation (2), the differential equation that models momentum conservation is

( 1 3 r 0 3 − b ( ( r ( t ) ) 2 ( cos ( θ ) ) 2 + 2 r ( t ) b cos ( θ ) + 2 b 2 ) ( cos ( θ ) ) 3 e − cos ( θ ) r ( t ) b + b ( r 0 2 ( cos ( θ ) ) 2 + 2 r 0 b cos ( θ ) + 2 b 2 ) ( cos ( θ ) ) 3 e − r 0 cos ( θ ) b ) d d t r ( t ) = 1 3 r 0 3 v 0 . (20)

There is no analytical solution.

In the case of a Gaussian density profile for the ISM, as given by Equation (5), the differential equation that models momentum conservation is

( 1 3 ρ 0 r 0 3 + ρ 0 ( − r ( t ) b 2 ( cos ( θ ) ) 2 e − 1 2 ( r ( t ) ) 2 ( cos ( θ ) ) 2 b 2 + 1 2 b 3 π 2 ( cos ( θ ) ) 3 erf ( 1 2 2 cos ( θ ) r ( t ) b ) ) − ρ 0 ( − r 0 b 2 ( cos ( θ ) ) 2 e − 1 2 r 0 2 ( cos ( θ ) ) 2 b 2 + 1 2 b 3 π 2 ( cos ( θ ) ) 3 erf ( 1 2 2 cos ( θ ) r 0 b ) ) ) d d t r ( t ) = 1 3 ρ 0 r 0 3 v 0 . (21)

In the case of an inverse square density profile for the ISM, as given by Equation (9), the differential equation that models the momentum conservation is

( 1 3 ρ 0 r 0 3 + ρ 0 z 0 2 r ( t ) ( cos ( θ ) ) 2 − 2 ρ 0 z 0 3 ln ( r ( t ) cos ( θ ) + z 0 ) ( cos ( θ ) ) 3 − ρ 0 z 0 4 ( cos ( θ ) ) 3 ( r ( t ) cos ( θ ) + z 0 ) − ρ 0 z 0 2 r 0 ( cos ( θ ) ) 2 + 2 ρ 0 z 0 3 ln ( r 0 cos ( θ ) + z 0 ) ( cos ( θ ) ) 3 + ρ 0 z 0 4 ( cos ( θ ) ) 3 ( r 0 cos ( θ ) + z 0 ) ) d d t r ( t ) − 1 3 ρ 0 r 0 3 v 0 = 0. (22)

There is not an analytical solution for this differential equation.

In the case of a Navarro-Frenk-White density profile for the ISM, as given by Equation (13), the differential equation that models momentum conservation is

( 1 3 ρ 0 r 0 3 + r 0 ρ 0 ( ( b + r ( t ) cos ( θ ) ) ln ( b + r ( t ) cos ( θ ) ) + b ) ( b + r 0 ) 2 ( cos ( θ ) ) 3 ( b + r ( t ) cos ( θ ) ) − r 0 ρ 0 ( ( b + r 0 cos ( θ ) ) ln ( b + r 0 cos ( θ ) ) + b ) ( b + r 0 ) 2 ( cos ( θ ) ) 3 ( b + r 0 cos ( θ ) ) ) d d t r ( t ) = 1 3 ρ 0 r 0 3 v 0 . (23)

A first approximated solution of this differential equation can be given as a series of order 4

r ( t ; t 0 , r 0 , v 0 , b ) = r 0 + v 0 ( t − t 0 ) − 3 2 ( b + r 0 ) 2 v 0 2 ( t − t 0 ) 2 r 0 cos ( θ ) ( b + r 0 cos ( θ ) ) 2       + 1 2 ( b + r 0 ) 2 v 0 3 ( ( cos ( θ ) ) 3 r 0 2 − cos ( θ ) b 2 + 9 r 0 2 + 18 b r 0 + 9 b 2 ) ( t − t 0 ) 3 r 0 2 ( cos ( θ ) ) 2 ( b + r 0 cos ( θ ) ) 4 (24)

Figure 1 reports a comparison between numerical and series solution.

To find a second approximate solution for this differential equation of the first order in r, we separate the variables and we integrate. The following non-linear equation is obtained

N D = t − t 0 , (25)

where

N = − 6 ( b + r 0 ) 2 ( b + r 0 cos ( θ ) ) ( 1 / 2 r cos ( θ ) + b ) ln ( b + r 0 cos ( θ ) )     + 6 ( b + r 0 ) 2 ( b + r 0 cos ( θ ) ) ( 1 / 2 r cos ( θ ) + b ) ln ( b + r cos ( θ ) )     − 6 cos ( θ ) ( r − r 0 ) ( − 1 / 6 r 0 3 ( cos ( θ ) ) 4 − 1 / 6 b r 0 2 ( cos ( θ ) ) 3     + 1 / 2 r 0 ( b + r 0 ) 2 cos ( θ ) + b ( b + r 0 ) 2 ) , (26)

and

D = v 0 r 0 2 ( cos ( θ ) ) 4 ( b + r 0 cos ( θ ) ) . (27)

In this case, it is not possible to find an analytical solution for the radius, r, as a function of time. Therefore, we apply the Padé rational polynomial, see [17] [18] [19] [20]. We choose an approximation of degree 2 in the numerator and degree 1 in the denominator about the point r = r 0 to the left-hand side of Equation (25). The resulting equation of second degree is

N N D D = t − t 0 , (28)

where

N N = − ( r 0 − r ) ( 4 ( cos ( θ ) ) 3 r 0 3 + 2 ( cos ( θ ) ) 3 r 0 2 r + 12 ( cos ( θ ) ) 2 r 0 2 b     + 8 r 0 cos ( θ ) b 2 − 2 cos ( θ ) b 2 r − 9 r 0 3 − 18 b r 0 2 + 9 r 0 2 r         − 9 r 0 b 2 + 18 r 0 b r + 9 b 2 r ) , (29)

and

D D = 2 cos ( θ ) v 0 ( b + r 0 cos ( θ ) ) ( 2 cos ( θ ) r 0 2 + r 0 r cos ( θ ) + 4 b r 0 − b r ) . (30)

The resulting Padé approximant for the trajectory, the radius r 2,1 , is the second approximated solution

r 2,1 ( t ; t 0 , r 0 , v 0 , b ) = N N N D D D . (31)

where

N N N = ( cos ( θ ) ) 3 t r 0 2 v 0 − ( cos ( θ ) ) 3 r 0 2 t 0 v 0 − ( cos ( θ ) ) 3 r 0 3     − 6 ( cos ( θ ) ) 2 r 0 2 b − cos ( θ ) b 2 t v 0 + cos ( θ ) b 2 t 0 v 0     − 5 r 0 cos ( θ ) b 2 + 9 r 0 b 2 + 18 r 0 2 b + 9 r 0 3 + A , (32)

and

D D D = 2 ( cos ( θ ) ) 3 r 0 2 − 2 b 2 cos ( θ ) + 9 b 2 + 18 b r 0 + 9 r 0 2 , (33)

and

A = ( b + r 0 cos ( θ ) ) 2 cos ( θ ) ( ( 3 r 0 + v 0 ( t − t 0 ) ) 2 r 0 2 ( cos ( θ ) ) 3     − 2 ( 3 r 0 + v 0 ( t − t 0 ) ) ( − 3 r 0 + v 0 ( t − t 0 ) ) r 0 b ( cos ( θ ) ) 2     + ( − 3 r 0 + v 0 ( t − t 0 ) ) 2 b 2 cos ( θ ) + 54   r 0 v 0 ( b + r 0 ) 2 ( t − t 0 ) ) . (34)

Figure 2 reports a comparison between the numerical and the series solution.

The two approximations that we have used here cover the range in time after which the percent error is ≈10%: 1360 yr for the Taylor series and 194,285 yr for the Padé approximant. We conclude that in our case the Padé approximant has a wider radius of convergence in respect to the Taylor series.

The starting equations for the evolution of the SB [15] [21] are defined by the following parameters: N * , which is the number of SN explosions in 5.0 × 10⁷ yr; Z OB , which is the distance of the OB associations from the galactic plane; E 51 ,

which is the energy in 10⁵¹ erg and is usually chosen equal to one; v 0 , which is 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, which is the proper time of the SB. The radius of the SB is

R = 111.56 ( E 51 t 7 3 N * n 0 ) 1 5 pc , (35)

and its velocity

V = 6.567 1 t 7 2 / 5 E 51 N * n 0 5 km / s . (36)

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 SNs are exploded

N S N = N * t 7,0 × 10 7 5 × 10 7 . (37)

The two following inverted formula allow us to derive the parameters of the initial conditions for the SB in terms of r 0 expressed in pc and v 0 expressed in km∙s⁻¹

t 7,0 = 0.05878 r 0 v 0 , (38)

and

N * = 2.8289 × 10 − 7 r 0 2 n 0 v 0 3 E 51 . (39)

The LB has been recently observed in the X-ray in the 0.1 - 1.2 keV region by [3], whose Figure 7 reports six configurations of the LB along great-circle cuts through the Galactic pole and the Galactic plane. As a target of the simulation, we have chosen the cut characterized by galactic longitude, l, between 120˚ and 300˚. An observational percentage reliability, ϵ obs , is introduced over the whole range of the polar angle θ ,

ϵ obs = 100 ( 1 − ∑ j | r obs − r num | j ∑ j r obs , j ) , (40)

where r num is the theoretical radius of the local LB, r obs is the observed radius of the local LB, and the index j varies from 1 to the number of available observations. The observational percentage of reliability allows us to fix the theoretical parameters.

The numerical solution is reported as a cut in the x-z plane: see Figure 3 for an exponential density profile as given by Equation (2); see Figure 4 for a Gaussian

density profile as given by Equation (5); see Figure 5 for an inverse square density profile as given by Equation (9); and see Figure 6 for a NFW density profile as given by Equation (13).

The theory of the asymmetrical expansion already developed is independent of the azimuthal angle φ and therefore the 3D advancing surface of a LB can be obtained by rotating a cut in x-z plane, see Figure 7.

Some radioisotopes on Earth, such as ⁶⁰Fe (half life of 1.5 × 10⁶ yr [22]), were measured in a deep-sea ferromanganese crust: the concentration of ⁶⁰Fe increased 2.8 Myr ago, see [6]. These measurements have triggered some simulations that can explain the LB in the framework of SN explosions [12] [23] [24]. The encounter between the advancing shell of the LB and the Sun is here simulated in 2D assuming a constant density, see Equation (18). The following distances are involved:

1) r 0 the initial radius of the LB,

2) r e the radius of the LB when encounters the LB,

3) r a the actual radius of the LB,

4) D the distance between the sun and the LB, D = r a − r e , and they are reported in Figure 8. The times of the 2D simulation are

1) t 0 the time at which the radius of the LB is r 0 ,

2) t F 60 e the time at which ⁶⁰Fe was deposited on the Earth,

3) t a the actual time of the LB,

4) t e the time of the encounter between LB and Sun, t e = t a − t F 60 e .

The distance LB-Sun, D, is reported in Figure 9 as function of the elapsed time.

The LB is a part of other bubbles which show a Swiss-cheese structure, see Figure 10. We simulate this network with the multiple explosion of N bubbles in 2D assuming a constant density, see Equation (18). We choose N = 7 and the time is allowed to vary in a random way in the interval ( t 0 , t max ) , the position of the explosion on the two Cartesian axis is randomly generated in the interval ( 0, s i d e ) , see Figure 11.